Imperfect Inventory Systems: Inventory and Production Management 303056973X, 9783030569730

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Ata Allah Taleizadeh

Imperfect Inventory Systems

Inventory and Production Management

Imperfect Inventory Systems

Ata Allah Taleizadeh

Imperfect Inventory Systems Inventory and Production Management

Ata Allah Taleizadeh School of Industrial Engineering University of Tehran Tehran, Iran

ISBN 978-3-030-56973-0 ISBN 978-3-030-56974-7 https://doi.org/10.1007/978-3-030-56974-7

(eBook)

© Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Imperfect Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Multi-product Single Machine . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Quality Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

1 2 2 3 3 4 4 4

2

Imperfect EOQ System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Deterioration, Perishability, and Lifetime Constraints . . . . . 8 2.2.2 Imperfect-Quality Items . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 EOQ Model with No Shortage . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 Imperfect Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Maintenance Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.3 Screening Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.4 Learning Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.5 EOQ Models with Imperfect-Quality Items and Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.6 Buy and Repair Options . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.7 Entropy EOQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.4 EOQ Model with Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.4.1 Imperfect Quality and Inspection . . . . . . . . . . . . . . . . . . . 70 2.4.2 Multiple Quality Characteristic Screening . . . . . . . . . . . . . 74 2.4.3 Rejection of Defective Supply Batches . . . . . . . . . . . . . . . 82 2.4.4 Rework and Backordered Demand . . . . . . . . . . . . . . . . . . 87 2.4.5 Learning in Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.4.6 EOQ Model for Imperfect-Quality Items . . . . . . . . . . . . . . 109 v

vi

Contents

2.5

EOQ Model with Partial Backordering . . . . . . . . . . . . . . . . . . . . . 2.5.1 EOQ Model of Imperfect-Quality Items . . . . . . . . . . . . . . . 2.5.2 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Reparation of Imperfect Products . . . . . . . . . . . . . . . . . . . 2.5.4 Replacement of Imperfect Products . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112 112 118 126 142 147

3

Scrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 No Shortage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Continuous Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Discrete Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fully Backordered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Continuous Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Partial Backordered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Continuous Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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153 153 155 155 166 186 186 223 223 232 233

4

Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 No Shortage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Imperfect Item Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Rework Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Imperfect Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Quality Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Simple Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Defective Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Random Defective Rate: Same Production and Rework Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Rework Process and Scraps . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Rework and Preventive Maintenance . . . . . . . . . . . . . . . . 4.4.6 Random Defective Rate: Different Production and Rework Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.7 Imperfect Rework Process . . . . . . . . . . . . . . . . . . . . . . . 4.5 Partial Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Immediate Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Repair Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Multi-delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Multi-delivery Policy and Quality Assurance . . . . . . . . . . 4.6.2 Multi-delivery and Partial Rework . . . . . . . . . . . . . . . . . 4.6.3 Multi-delivery Single Machine . . . . . . . . . . . . . . . . . . . . 4.6.4 Multi-product Two Machines . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

235 235 236 239 239 244 249 254 261 261 268

. 277 . 293 . 300 . . . . . . . . . .

307 312 318 318 324 331 331 337 340 345

Contents

vii

4.6.5 Shipment Decisions for a Multi-product . . . . . . . . . . . . . . 351 4.6.6 Pricing with Rework and Multiple Shipments . . . . . . . . . . 357 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 5

Multi-product Single Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 No Shortage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Discrete Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Scrapped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Defective Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Multidefective Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Interruption in Manufacturing Process . . . . . . . . . . . . . . . 5.4.4 Immediate Rework Process . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Repair Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Partial Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Repair Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Scrapped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Immediate Rework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

367 367 368 371 371 375 380 385 391 401 401 409 419 433 443 449 449 460 461 470 479 494 494

6

Quality Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 EOQ Model with No Return . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 No Return Without Shortage . . . . . . . . . . . . . . . . . . . . . 6.3.2 Two Quality Levels with Backordering . . . . . . . . . . . . . . 6.3.3 Learning in Inspection with Backordering . . . . . . . . . . . . 6.3.4 Partial Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 EOQ Model with Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Inspection and Sampling . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Inspection Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Different Defective Quality Levels and Partial Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 EPQ Model Without Return . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Quality Assurance Without Shortage . . . . . . . . . . . . . . . . 6.5.2 Quality Screening and Rework Without Shortage . . . . . . .

. . . . . . . . . . .

497 497 500 501 502 502 506 506 515 515 522

. . . .

525 534 535 536

viii

Contents

6.6

7

EPQ Model with Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Continuous Quality Characteristic Without Shortage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Inspections Errors Without Shortage . . . . . . . . . . . . . . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 537 . . . .

537 542 546 547

Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 No Shortage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Imperfect Preventive Maintenance . . . . . . . . . . . . . . . . . 7.2.3 Imperfect Maintenance and Imperfect Process . . . . . . . . . 7.2.4 Aggregate Production and Maintenance Planning . . . . . . . 7.3 Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Partial Backordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

549 549 550 550 561 569 574 578 578 582 582 582 582

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

Chapter 1

Introduction

In today’s manufacturing environment, managing inventories is one of the basic concerns of enterprises dealing with materials according to their activities, because material as the principal inventories of enterprises specially production ones composes the large portion of their assets. As a result, managing inventories influences directly financial, production, and marketing segments of enterprises so that efficient management of inventories leads to improving their profits. In addition, the effect of managing inventories on the selling prices of finished products is undeniable because more than half of production systems’ revenues are spent to buy materials or production components. On the other hand, customers expect to receive their orders at a lower price apace. So, an efficient managing inventories and production planning are key managerial and operational tools to achieve the main goals, which are satisfying the customers’ demand and becoming lower-cost producer, in order to increase market share. Economic production quantity (EPQ) model is a well-known economic lot size model used in production enterprises that internally produce products. However, traditional EPQ model is utilized for perfect production process to determine the optimal production lot size so that overall production/inventory costs are minimized. In reality, a perfect production run rarely exists. Breakdown is an inevitable issue in production processes. Indeed, after a production period, a production process often shifts to out-control state owing to machine wear or corrosion which leads to generating defective items with loss cost. In order to reimburse these costs, some production strategies including reworking and repairing defective items, quality control, and maintenance planning to reduce the defective or scrape item costs are employed. So, the main prophecy of this book is to introduce all mentioned production strategies which can lessen unexpected imperfect item costs. The main focus of this book is to introduce mathematical models of imperfect inventory control systems in which at least one of imperfect items, scraped item, rework

© Springer Nature Switzerland AG 2021 A. A. Taleizadeh, Imperfect Inventory Systems, https://doi.org/10.1007/978-3-030-56974-7_1

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1 Introduction

policy, quality control or maintenance planning may be used. In the following a brief introduction about each chapter is presented.

1.1

Imperfect Items

Since the introduction of the economic order quantity (EOQ) model by Harris (1913), frequent contributions have been made in the literature toward the development of alternative models that overcome the unrealistic assumptions embedded in the EOQ formulation. For example, the assumption related to the perfect quality items is technologically unattainable in most supply chain applications (Cheng 1991). In contrast, products can be categorized as “good quality,” “good quality after reworking,” “imperfect quality,” and “scrap” (Chan et al. 2003). In practice, the presence of defective items in raw material or finished products inventories may deeply affect supply chain coordination, and, consequently, the product flows among supply chain levels may become unreliable (Roy et al. 2011). In response to this concern, the enhancement of currently available production and inventory order quantity models, which accounts for imperfect items in their mathematical formulation, has become an operational priority in supply chain management (Khan et al. 2011). This enhancement may also include the knowledge transfer between supply chain entities in order to reduce the percentage of defective items. In the second chapter of this book, the main focus is on introducing several mathematical models of EOQ inventory systems with imperfect items considering different kinds of shortages under different assumptions.

1.2

Scrap

The economic order quantity (EOQ) model was first introduced in 1913. Seeking to minimize the total cost, the model generated a balance between holding and ordering costs and determined the optimal order size. Later, the EPQ model considered items produced by machines inside a manufacturing system with a limited production rate, rather than items purchased from outside the factory. Despite their age, both models are still widely used in major industries. Their conditions and assumptions, however, rarely pertain to current real-world environments. To make the models more applicable, different assumptions have been proposed in recent years, including random machine breakdowns, generation of imperfect and scrap items, and discrete shipment orders. The assumption of discrete shipments using multiple batches can make the EPQ model more applicable to real-world problems. The EPQ inventory models assume that all the items are manufactured with high quality and defective items are not produced. However, in fact, defective items appear in the most of manufacturing systems; in this sense, researchers have been developing EPQ inventory models for

1.4 Multi-product Single Machine

3

defective production systems. In these production systems, defective items are of two types: scrapped items and reworkable items. Usually non-conforming products are scrapped and are removed from the systems’ inventories. This strategy is employed for production enterprises in which either imperfect items cannot be repaired or both repair/reworking cost is more than their selling revenues. In turn, enterprises prefer to reject imperfect items instead of performing reworking/repair procedure. In the third chapter of this book, the EPQ model with scraped items under different kinds of shortages and both continuous and discrete delivery are introduced.

1.3

Rework

Rework is one of the key drivers of production designs applied in imperfect production systems in which their production lines face defectives. It helps producers reproduce the non-conforming items, which are detected within/after inspecting process, and sell them as healthy ones. Although a reworking process makes an additive cost for production companies, it causes the producers to profit from buying the reworked items more than their reworking costs, so they prefer to rework the imperfect items in order to reduce their unexpected expenses. In the fourth chapter of this book, rework process in imperfect EPQ model under different assumptions is introduced. Indeed, several mathematical models of EPQ problem with defective and rework process are presented.

1.4

Multi-product Single Machine

The economic production quantity (EPQ) is a commonly used production model that has been studied extensively in the past few decades. One of the considered constraints in the EPQ inventory models is producing all items by a single machine. Since all of the products are manufactured on a single machine with a limited capacity, a unique cycle length for all items is considered. It is assumed there is a real constant production capacity limitation on the single machine on which all products are produced. If the rework is placed, both the production and rework processes are accomplished using the same resource, the same cost, and the same speed. The first economic production quantity inventory model for a single-product single-stage manufacturing system was proposed by Taft (1918). Perhaps Eilon (1985) and Rogers (1958) were the first researchers that studied the multi-products single manufacturing system. Eilon (1985) proposed a multi-product lot-sizing problem classification for a system producing several items in a multi-product single-machine manufacturing system. In the fifth chapter of this book, multi-

4

1 Introduction

product single-machine EPQ model with defective and scraped items and also rework process under different assumptions are presented.

1.5

Quality Considerations

Traditional economic order quantity (EOQ) models offer a mathematical approach to determine the optimal number of items a buyer should order to a supplier each time. One major implicit assumption of these models is that all the items are of perfect quality (Rezaei and Salimi 2012). However, presence of defective products in manufacturing processes is inevitable. There is no production process which can guarantee that all its products would be perfect and free from defect. Hence, there is a yield for any production process. Basic and classical inventory control models usually ignore this fact. They assume all output products are perfect and with equal quality; however, due to the limitation of quality control procedures, among other factors, items of imperfect quality are often present. So it has given researchers the opportunity to relax this assumption and apply a yield to investigate and study its impact on several variables of inventory models such as order quantity and cycle time. In the sixth chapter of this book, several inventory control models under quality considerations such as sampling, inspections, return, etc. with different assumptions of inventory systems are presented.

1.6

Maintenance

The role of the equipment condition in controlling quality and quantity is wellknown (Ben-Daya and Duffuaa 1995). Equipment must be maintained in top operating conditions through adequate maintenance programs. Despite the strong link between maintenance production and quality, these main aspects of any manufacturing system are traditionally modeled as separate problems. In the last chapter of this book, maintenance and inventory systems are considered together, and several mathematical models are presented.

References Ben-Daya, M., & Duffuaa, S. O. (1995). Maintenance and quality: The missing link. Journal of Quality in Maintenance Engineering, 1(1), 20–26. Chan, W. M., Ibrahim, R. N., & Lochert, P. B. (2003). A new EPQ model: Integrating lower pricing, rework and reject situations. Production Planning & Control, 14(7), 588–595. Cheng, T. C. E. (1991). EPQ with process capability and quality assurance considerations. Journal of the Operational Research Society, 42(8), 713–720.

References

5

Eilon, S. (1985). Multi-product batch production on a single machine—A problem revisited. Omega, 13, 453–468. Harris, F. W. (1913). What quantity to make at once. In The library of factory management. Operation and costs (The factory management series) (Vol. 5, pp. 47–52). Chicago, IL: A.W. Shaw Co.. Khan, M., Jaber, M. Y., & Bonney, M. (2011). An economic order quantity (EOQ) for items with imperfect quality and inspection errors. International Journal of Production Economics, 133(1), 113–118. Rezaei, J., & Salimi, N. (2012). Economic order quantity and purchasing price for items with imperfect quality when inspection shifts from buyer to supplier. International Journal of Production Economics, 137(1), 11–18. Rogers, J. (1958). A computational approach to the economic lot scheduling problem. Management Science, 4, 264–291. Roy, M. D., Sana, S. S., & Chaudhuri, K. (2011). An economic order quantity model of imperfect quality items with partial backlogging. International Journal of Systems Science, 42(8), 1409–1419. Taft, E. W. (1918). The most economical production lot. Iron Age, 101(18), 1410–1412.

Chapter 2

Imperfect EOQ System

2.1

Introduction

Since the introduction of the economic order quantity (EOQ) model by Harris (1913), frequent contributions have been made in the literature toward the development of alternative models that overcome the unrealistic assumptions embedded in the EOQ formulation. For example, the assumption related to the perfect-quality items is technologically unattainable in most supply chain applications. In contrast, products can be categorized as “good quality,” “good quality after reworking,” “imperfect quality,” and “scrap” (Chan et al. 2003; Pal et al. 2013). In practice, the presence of defective items in raw material or finished product inventories may deeply affect supply chain coordination, and, consequently, the product flows among supply chain levels may become unreliable (Roy et al. 2015). In response to this concern, the enhancement of currently available production and inventory order quantity models, which accounts for imperfect items in their mathematical formulation, has become an operational priority in supply chain management (Khan et al. 2011). This enhancement may also include the knowledge transfer between supply chain entities in order to reduce the percentage of defective items (Adel et al. 2016). Also some related works can be found in Hasanpour et al. (2019), Keshavarz et al. (2019), Taleizadeh et al. (2015, 2016a, 2018a, b), Taleizadeh and Zamani-Dehkordi (2017a, b), Salameh and Jaber (2000), Maddah and Jaber (2008), and Papachristos and Konstantaras (2006). The EOQ models with imperfect-quality items in three categories are categorized and their subcategories are shown in Fig. 2.1. The common notations of imperfect EOQ models are shown in Table 2.1.

© Springer Nature Switzerland AG 2021 A. A. Taleizadeh, Imperfect Inventory Systems, https://doi.org/10.1007/978-3-030-56974-7_2

7

8

2

Learning effects

No shortage

Maintenance actiuons

Imperfect quality Fully backordered

Batch rejection

Learning effects

Partial backordered

Rejection

Imperfect EOQ system

No shortage

Imperfect EOQ System

Reparation

Inspections and screening

Entropy EOQ

Fully backordered

Imperfect quality

Inspections and screening

Partial backordered

Imperfect quality

Screening

Fig. 2.1 Categories of EOQ model of imperfect-quality items

2.2

Literature Review

The academic literature related to inventory control for imperfect-quality items is multidisciplinary in nature and, for reviewing/presentation purposes in this chapter, is thematically organized around two main streams: (1) deterioration, perishability, and shelf lifetime constraints and (2) model formulations and related solution techniques that consider imperfect-quality items (Adel et al. 2016).

2.2.1

Deterioration, Perishability, and Lifetime Constraints

The terms “deterioration,” “perishability,” and “obsolescence” are used interchangeably in the literature and may often be perceived as ambiguous because they are linked to particular underlying assumptions regarding the physical state/fitness and behavior of items over time. Usually, deterioration refers to the process of decay, damage, or spoilage of a product, i.e., the product loses its value of characteristics and can no longer be sold/used for its original purpose (Wee 1993). In contrast, an item with a fixed lifetime perishes once it exceeds its maximum shelf lifetime and then must be discarded (Ferguson and Ketzenberg 2005). Obsolescence incurs a partial or a total loss of value of the on-hand inventory in such a way that the value for a product continuously decreases with its perceived utility (Song and Zipkin 1996; Also some related works can be found in works of Nobil, et al. (2019), Lashgary et al. (2016, 2018), Kalantary and Taleizadeh (2018), Diabat et al.

2.2 Literature Review

9

Table 2.1 Notations P D R s v C CR CJ Cb CT Cd g b π K KS h h1 hR γ CI x p E[p] f( p) T t f(γ) y B β E[.]

Production rate (units per unit time) Demand rate (units per unit time) Repair rate (units per unit time) Selling price for good-quality items ($/unit) Selling price for imperfect or salvage value per items ($/unit) Production/purchasing cost ($/unit) Rework cost per unit ($/unit) Reject cost per unit (including transportation, handling, and damage cost) ($/unit) Backordering cost ($/unit/unit time) Transportation cost per unit ($/unit) Disposal cost per unit ($/unit) Goodwill cost per unit ($/unit) Lost sale cost per unit ($/unit) Fixed setup/ordering cost ($/lot) Fixed transportation or shipment cost ($/lot) Holding cost per unit per unit time ($/unit/unit time) The holding cost for defective items per unit per unit time ($/unit/unit time) The holding cost for reworked items per unit per unit time ($/unit/unit time) Fraction of imperfect items (percent) The unit screening or inspection cost ($/unit) Inspection rate (units per unit time) Imperfect rate (units per unit time) Expected imperfect rate Probability density function of p Ordering cycle duration (time) Screening time (time) Probability density function of imperfect products (γ) Production/ordering quantity (unit) Backordered level (unit) Partial backordering rate (%) 0 < β
1 Expected value of a random variable

(2017), Mohammadi et al. (2015), Tat et al. (2015), Hasanpour et al. (2019), Taleizadeh (2014), Taleizadeh and Rasouli-Baghban, (2015, 2018), Taleizadeh et al. (2013a, b, 2015, 2016, 2019), Taleizadeh and Nemattolahi (2014) and Tavakkoli and Taleizadeh (2017), Bakker et al. (2012).

2.2.2

Imperfect-Quality Items

The classical EOQ has been a widely accepted model for inventory control purposes due to its simple and intuitively appealing mathematical formulation. However, it is true to say that the operation of the model is based on a number of explicitly or

10

2

Imperfect EOQ System

implicitly made unrealistic mathematical assumptions that are never actually met in practice (Jaber et al. 2004). Salameh and Jaber (2000) developed a mathematical model that permits some of the items to drop below the quality requirements, i.e., a random proportion of defective items are assumed for each lot size shipment, with a known probability distribution. The researchers assumed that each lot is subject to a 100% screening, where defective items are kept in the same warehouse until the end of the screening process and then can be sold at a price lower than that of perfectquality items. Huang (2004) developed a model to determine an optimal integrated vendor–buyer inventory policy for flawed items in a just-in-time (JIT) manufacturing environment. Maddah and Jaber (2008) developed a new model that rectifies a flaw in the one presented by Salameh and Jaber (2000) using renewal theory. Jaber et al. (2008) extended it by assuming that the percentage defective per lot reduces according to a learning curve. Jaggi and Mittal (2011) investigated the effect of deterioration on a retailer’s EOQ when the items are of imperfect quality. In their research, defective items were assumed to be kept in the same warehouse until the end of the screening process. Jaggi et al. (2011) and Sana (2012) presented inventory models, which account for imperfect-quality items under the condition of permissible delay in payments. Moussawi-Haidar et al. (2014) extended the work of Jaggi and Mittal (2011) to allow for shortages. In a real manufacturing environment, the defective items are not usually stored in the same warehouse where the good items are stored. As a result, the holding cost must be different for the good items and the defective ones (e.g., Paknejad et al. 2005). With this consideration in mind, Wahab and Jaber (2010) presented the case where different holding costs for the good and defective items are assumed. They showed that if the system is subject to learning, then the lot size with the same assumed holding costs for the good and defective items is less than the one with differing holding costs. When there is no learning in the system, the lot size with differing holding costs increases with the percentage of defective items. For more details about the extensions of a modified EOQ model for imperfect-quality items, see Khan et al. (2011). Here are some main models in literature with their mathematical model, solution procedure, and numerical examples. In the next sections, these models starting from the basic to complicated ones are presented. First, EOQ models with imperfect quality items are studied considering no shortage, back-ordering shortage, and partial back-ordering.

2.3 2.3.1

EOQ Model with No Shortage Imperfect Quality

In this section, two imperfect EOQ models developed by Salameh and Jaber (2000) and Maddah and Jaber (2008) are presented. Consider the EOQ model with a

2.3 EOQ Model with No Shortage Fig. 2.2 Inventory level (Salameh and Jaber 2000; Maddah and Jaber 2008)

11

Inventory

y

Py

Time

t T

demand rate of D units per unit time. An order of size y is placed every time the inventory level reaches zero and is assumed to be delivered instantaneously. The fixed ordering cost is K, the fixed shipping of imperfect-quality items is KS, the unit purchasing cost is C, and the inventory holding cost is h per unit per unit time. Each order contains a fraction P of defective items, a random variable with support in [0, 1]. Each order is subjected to a 100% inspection process at a rate of x units per unit time, xD. The screening cost is d per unit. Upon completion of the screening process, items of imperfect quality are sold as a single batch at a reduced price of v per unit. The price of a perfect-quality item is s per unit, s < v. The behavior of the inventory level in an ordering cycle is shown in Fig. 2.2, where T is the ordering cycle duration (T ¼ (1  p)y/D, and t ¼ y/x). Salameh and Jaber (2000) assumed that (1  p) y  Dt, or, equivalently, p
1  D/x, in order to avoid shortages. Under the above assumptions, the expected profit is presented as: Revenue

cost zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ Fixed z}|{ TPðyÞ ¼syð1  pÞ þ υyp  K 

Purchasing cost

z}|{ Cy

Inspection cost



z}|{ CI y



Holding cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   ½yð1  pÞ2 py2 h þ 2D x

ð2:1Þ

Then the expected profit per unit time is derived as:   TPðyÞ E½TPUðyÞ ¼ E T After some simplifications,

ð2:2Þ

12

2

Imperfect EOQ System

      y hy K 1 E E ½TPUðyÞ ¼ D s  υ þ h þ D υ   C  C I  x x y 1p 

hyð1  E ½pÞ 2

ð2:3Þ

And the optimal order quantity is derived as: ySJ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KDE ½1=ð1  pÞ ¼ h½1  E½p  2Dð1  E ½1=ð1  pÞÞ=x

ð2:4Þ

Then, Maddah and Jaber (2008) corrected Eq. (2.2) as: E½TPUðyÞ ¼

E ½TPðyÞ E ½T 

ð2:5Þ

and derived a new expected profit function as: E½TPðyÞ ¼ syð1  E ½pÞ þ υyE ½p  K  Cy  C I y i 0 h 1 y 2 E ð 1  pÞ 2 2 E ½ p y A þ  h@ 2D x Since E[T] ¼ (1  E[p])y/D, then Eq. (2.5) is rearranged as:

E ½TPUðyÞ ¼

h i ½sð1  E ½pÞ þ υE½p  C  CI D  KD=y  hyðE ð1  pÞ2 =2 þ E ½pD=xÞ 1  E ½p

ð2:6Þ After proofing the concavity of Eq. (2.6) with respect to y, the optimum order size is derived as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD i  y ¼u t  h h E ð1  pÞ2 þ 2E½pD=x 

ð2:7Þ

The expected profit in Eq. (2.6) has several terms independent of y. In subsequent analysis, these are dropped, and the objective function is redefined in terms of minimizing the expected “relevant” cost per unit time as (Maddah and Jaber 2008):

2.3 EOQ Model with No Shortage

ECðyÞ ¼

h  h i i 1 KD=y þ hy E ð1  pÞ2 =2 þ E ½pD=x 1  E ½p

13

ð2:8Þ

Maddah and Jaber (2008) showed that for a large inspection rate, the optimum order size in Eq. (2.7) converges to: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD i y ¼u t  h h E ð1  pÞ2 

ð2:9Þ

In real cases, it is not optimal to ship imperfect-quality items as a single batch in each ordering cycle (Maddah and Jaber 2008). So Maddah and Jaber assumed that shipping any number of imperfect-quality batches has a fixed cost of KS and developed their previous model under multiple batches. Now the decision variables are order size ( y) and ordering cycle number (n), and they derived the expected cost per unit time of perfect-quality items similar to Eq. (2.8) as below: h  h i i E ½CPðyÞ ¼ KD=y þ hy E ð1  pÞ2 =2 =ð1  E½pÞ Figure 2.3 corresponds to the case when n ¼ 3, where the imperfect-quality inventory is held for two ordering cycles and then shipped upon completing the screening of the last order. According to Maddah and Jaber (2008), let Ti be the duration of ordering period i of a shipping cycle, i ¼ 1, . . ., n. Note that Ti ¼ (1  Pi) y/D, where Pi is the fraction of imperfect-quality items in order i of a shipping cycle. Then, the expected holding cost per shipping cycle is: 3 Imperfect quality inventory Imperfect inventory cost accumulated 7 6 7 6 cost from an order in thenth period,which is carried 7 6 7 6 7 6 carried over the ordering 7 6 for aduration of y=x 7 6 7 6 period of the order itself 7 6 before being shipped 7 6 " zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflffl ffl }|fflfflffl ffl { 7 6 n1 n2 n1 n X X X X P y2 7 6 i 7 ECIh ðy,nÞ ¼ hE 6 P ð Þy=D þ P y ð 1P Þy=D y 1P þ i i i i 7 6 x 7 6 i¼1 i¼1 j¼iþ1 i¼1 7 6 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl } 7 6 7 6 Imperfect inventory cost from an order 7 6 7 6 7 6 carried through subsequent ordering 7 6 7 6 7 6 periods during a shipping cycle 5 4 excluding thenth ordering period 2

ð2:10Þ Assuming that P1, . . ., Pn are independent and identically distributed, the expression for ECIh(y, n) in Eq. (2.10) after some simplifications changes to:

14

2

Imperfect EOQ System

 i hy2 nðn  1Þ y E ½pð1  E½pÞ þ nE½pD  ðn  1Þvar½p ECIh ðy, nÞ ¼ 2 x D  Knowing E

n P

 ð1  Pi Þy=D ¼ nð1  E ½pÞy=D, the expected imperfect-quality

i¼1

item cost per unit time is: E ½CIðy, nÞ ¼

n 1 KS D 1  E ½p n y  

ð n  1Þ D n1 þ hy E½pð1  E ½pÞ þ E ½p  var½p 2 x n

And total cost will be: E ½TCðy, nÞ ¼

 h i 2ð n  1Þ K S D hy var½p þ ½E ð1  pÞ2  n 2 n y o D þðn  1ÞE½pð1  E½pÞ þ 2E ½p  x ð2:11Þ

1 1  E ½p

n

Kþ

Because of convexity of Eq. (2.11), one can easily derive that: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2ðK þ ðK S =nÞÞD u i i y ð nÞ ¼ u h h uh E ð1  pÞ2  ð2ðn 1Þ=nÞvar½pþ ðn  1ÞE½pð1 E ½pÞþ 2E ½pðD=xÞ u t|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 

γ ðnÞ

ð2:12Þ Maddah and Jaber (2008) showed that optimal values of n can be found by optimizing the expected total cost presented in Eq. (2.13) which is presented in Eq. (2.14): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ECT1 ðnÞ ¼ ECTðn, y ðnÞÞ ¼ 2κ ðnÞγ ðnÞD ð2:13Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi h i u uK 1 E ð1  pÞ2  2ð1  K=K S Þvar½p  E ½pð1  E ½pÞ þ 2E½pðD=xÞ t e n¼ KE ½pð1  E½pÞ ð2:14Þ Then, the optimal value of n is one of the two integers which is closest to e n, whichever leads to lower value of ECT1(n). That is, n ¼ argmin(ECT1(n)) where [x] is the largest integer
x and [x] is the smallest integer x. Finally, the optimal order quantity is found from Eq. (2.12) as y* ¼ y*(n*) (Maddah and Jaber 2008).

2.3 EOQ Model with No Shortage Fig. 2.3 Perfect and imperfect inventory levels when shipments are consolidated, n ¼ 3 (Maddah and Jaber 2008)

15

Perfect Quality Inventory

(1 -P 1 )y

Time

t Imperfect Quality Inventory

P 1y T1

T2

T3

Time

Example 2.1 Maddah and Jaber (2008) developed numerical results similar to those in Salameh and Jaber (2000). This illustrates the application of their model and allows comparing their results with those of Salameh and Jaber. Consider a situation with the following parameters: demand rate, D ¼ 50,000 units/year; ordering cost, K ¼ $100/cycle; holding cost, h ¼ $5/unit/year; screening rate, x ¼ 175,200 units/ year; screening cost, CI ¼ $0.5/unit; purchasing cost, C ¼ $25/unit; selling price of good-quality items, s ¼ $50/unit; selling price of imperfect-quality items, v ¼ $20/ unit; and the fraction of imperfect-quality item, p, uniformly distributed on (a, b), 0 < a < b < 1, i.e., P ~ (a, b). With p ~ U(a, b), E[p] ¼ (a + b)/2, Var[p] ¼ (b  a)/ 12 and: h i E ð 1  pÞ 2 ¼

1 ba

Z

b a

ð1  pÞ2 dp ¼

a2 þ ab þ b2 þ1ab 3

ð2:15Þ

Assuming a ¼ 0 and b ¼ 0.04, then the optimal order quantity using Eq. (2.7) becomes y* ¼ 1434 units, and the related cost from Eq. (2.6) is E[TPU (y*)] ¼ $1,212,274. Assuming shipping of imperfect-quality items has a fixed cost of KS ¼ $50 with same values for other parameters used in the previous example, in the following, the continuous value of n that minimizes ECT1(n) is e n ¼ 4.93. So, n* is either 4 or 5 where ECT1(4) ¼ 7614 > ECT1 (5) ¼ 7600. So, n* ¼ 5. The optimal order quantity is then given from Eq. (2.11) as y*(5) ¼ 1447.

16

2

2.3.2

Imperfect EOQ System

Maintenance Actions

The objective of the analysis in this section is to determine the optimal lot size y such that the expected total cost is minimized when maintenance and reworking actions are taken into account. For describing this section, some new notations are used as presented in Table 2.2 (Porteus 1986). Before presenting the model, first we should take notice of the remark presented by Hou et al. (2015) in Eq. (2.16). He derived the expected number of unhealthy item as below: E ðN Þ ¼ θ y 

y X

! q

j

ð2:16Þ

j¼1

and  PrfX ¼ jg ¼

qj q, 0
j
y qy , j ¼ y

ð2:17Þ

Then, they showed that: E ðX Þ ¼ q

y1 X

jqj þ yqy ¼

j¼1

y X

qj

ð2:18Þ

j¼1

Finally, the number of defective items in y is N ¼ θ(y  X) and the E(N ) is what presented in Eq. (2.16). Now based on Eq. (2.18), the expected cyclic cost of rework process will be: C R E ðN Þ ¼ C R θ y 

y X

! q

j

ð2:19Þ

j¼1

Since the related cost to maintenance should be considered when the manufacturing process is out-of-control at the end of a production uptime for a lot of size y, the expected cyclic-related cost is:

Table 2.2 Notations of a given problem Q q

The probability that the system from in-control state shifts to out-of-control state The probability that the system stays in-control state during the production of an item and q ¼1q

θ X Cm

The percentage of defective items when the process is in the out-of-control state Random variable representing number of items produced in the in-control state Maintenance cost per unit ($/unit)

2.3 EOQ Model with No Shortage

17

C m ð1  qy Þ

ð2:20Þ

So the cyclic total cost is (Hou et al. 2015): Fixed cost

z}|{ TCðyÞ ¼ K þ

! Maintenance cost y zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ X hy2 y j þ C m ð1  q Þ þCR θ y  q 2D |{z} j¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Holding cost

ð2:21Þ

Rework cost

And the expected cost per unit of time becomes: f ðyÞ ¼ TCðyÞ=T

" # y X DK h D y j ¼ þ y þ CR Dθ þ C m ð1  q Þ  C R θ q y 2 y j¼1

ð2:22Þ

It should be noticed that for q ¼ 0, the production system is always in the in-control state and the produced items are healthy, and Eq. (2.22) reverses to the traditional EOQ model with healthy item. But Cm ¼ 0 means all produced items are defective, and Eq. (2.22) will reduce to the approximated model (using Taylor series expansion) in Porteus (1986) as presented in Eq. (2.23): f p ð yÞ ¼

DK y þ ðh þ CR DqÞ y 2

ð2:23Þ

and derive an approximately optimal lot size as follows (Hou et al. 2015): yp

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK ¼ h þ C R Dq

ð2:24Þ

Since Eq. (2.23) was not a good approximation, Hou et al. (2015) presented a comprehensive method to derive the optimal values. They provided the bounds for searching the optimal lot size y that minimizes f( y) of Eq. (2.22) as β ¼ C m  CRqθq and using necessary condition for optimal points ( f0(y) ¼ 0) derived optimal values. They prove that y exists and is unique when q equals 0 or 1 such that f 0(y) ¼ 0 satisfies. But for 0 < q < 1, let: h gðyÞ ¼ y2 f 0 ðy Þ ¼ DK þ y2  Dβð1  qy þ yqy ln q Þ 2

ð2:25Þ

since g( y) is a continuous function with limþ gðyÞ ¼ D and lim gðyÞ ¼ 1 > 0. y!0

y!1

Furthermore, the first derivative of g( y) is given by Hou et al. (2015):

18

2

h i g0 ðyÞ ¼ y h  Dβð ln q Þ2 qy

Imperfect EOQ System

ð2:26Þ

After some algebra, Hou et al. (2015) proposed the optimal lot size y when 0 < q < 1: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðK þ Cm Þ y1 ¼ h

rffiffiffiffiffiffiffiffiffiffi 2DK y2 ¼ h

ð2:27Þ

And proved that: if if

β
0 ! 0 < y
y2
y1

β > 0 ! 0 < y2 < y < y1

ð2:28Þ

Using Eqs. (2.27) and (2.28), the following algorithm is proposed to find the optimal values. Algorithm 2.1 Step 1: Let ε > 0, and compute β, y2, and y1. Step 2: If β
0, set yL ¼ 0, yU ¼ y2; otherwise, set yL ¼ y2, yU ¼ y1. U Step 3: Set yopt ¼ yL þy 2 . Step 4: If |g(yopt)| < ε, go to Step 6; otherwise, go to Step 5. Step 5: If |g(yopt)| < 0, set yL ¼ yopt; however, if |g(yopt)| > 0, yU ¼ yopt. Then, go to Step 3. Step 6: Set y ¼ yopt and compute f(y). Example 2.2 Consider K ¼ $600/cycle, h ¼ $8/unit/year, D ¼ 1000 units/year, and CR ¼ $5/unit, Cm ¼ $200/cycle, θ ¼ 0.75, and θ ¼ 0.1. Then, it can be verified that β ¼ 166.25. Using Algorithm 2.1, y ¼ 437.68 units and f(y) ¼ $7251.43 (Hou et al. 2015).

2.3.3

Screening Process

In this section, now consider a general EOQ model for items with imperfect quality under varying demand, defective items, screening process, and deterioration rates for an infinite planning horizon presented by Alamri et al. (2016). Assume that each lot is subject to a 100% screening where items that are not conforming to certain quality standards are stored in a different warehouse. Therefore, different holding costs for the good and defective items are considered. Items deteriorate while they are in storage, with demand, screening, and deterioration rates being arbitrary functions of time. The percentage of defective items per lot reduces according to a learning curve. After a 100% screening, imperfect-quality items may be sold at a discounted price as a single batch at the end of the screening process or incur a disposal penalty charge. Moreover, a general step-by-step solution procedure is provided for continuous intra-cycle periodic review applications.

2.3 EOQ Model with No Shortage

19

Table 2.3 Notations of a given problem D(t) x(t) δ(t) pj j Qj

Demand rate (units per unit time) Screening rate (units per unit time) Deterioration rate (units per unit time) The percentage defective per lot reduces according to a learning curve Cycle index ( j ¼ 1, 2,. . .) Lot of size delivered at the beginning of each cycle j (unit)

Qj

Inventory level

Qj (1 – pj ) – yj

pj Qj 0

T1j

T2j

Time

Cycle length Fig. 2.4 Inventory variation of an economic order quantity (EOQ) model for one cycle (Alamri et al. 2016)

Some related notation for this problem is presented in Table 2.3. Alamri et al. (2016) assumed that a single item held in stock lead time is zero and no restrictions exist. Moreover, any order arrives before the end of that same cycle. In order to avoid the shortage, Alamri et al. (2016) assumed (1  pj)x(t)  D(t), 8t  0. Lot size covers both deterioration and demand during both the first phase (screening) and the second phase (non-screening). Each lot is subjected to a 100% screening process that starts at the beginning of the cycle and ceases by time T1j, by which point in time Qj units have been screened and yj units have been depleted, which is the summation of demand and deterioration. During this phase, items not conforming to certain quality standards are stored in a different warehouse. The variation in the inventory level during the first and second phase (please refer to Fig. 2.4) and the variation in the inventory level for the defective items (Alamri et al. 2016) are presented in Eq. (2.29):

20

2

dI gj ðt Þ ¼ Dðt Þ  p j xðt Þ  δðt ÞI gj ðt Þ, dt

Imperfect EOQ System

0
t
T 1j

ð2:29Þ

Using boundary condition Igj(0) ¼ Qj: Z

T 1j

Qj ¼

xðuÞdu,

ð2:30Þ

0

dI gj ðt Þ ¼ Dðt Þ  δðt ÞI gj ðt Þ, dt

ð2:31Þ

And with the boundary condition Igj(T2j) ¼ 0: dI dj ðt Þ ¼ p j xðt Þ, dt

0
t
T 1j

ð2:32Þ

knowing Idj(0) ¼ 0. After some complicated algebra, the solutions of the above differential equations are (Alamri et al. 2016):

I gj ðt Þ ¼ e

ðgðt Þgð0ÞÞ

ZT 1j xðuÞdu  e

gðt Þ

0

Zt



DðuÞ þ p j xðuÞ egðuÞ du,

0
t

0


T 1j

ð2:33Þ I gj ðt Þ ¼ e

gðt Þ

ZT 2j

DðuÞegðuÞ du,

0
t
T 1j

ð2:34Þ

t

Zt I gj ðt Þ ¼

p j xðuÞdu,

0
t
T 1j

ð2:35Þ

0

gðt Þ ¼ ιζδðt Þdt

ð2:36Þ

The per cycle cost components for the given R T inventory system are as follows. The total purchasing cost during the cycle ¼ C 0 1j xðuÞdu. Note that this cost includes the defective and deterioration costs. Holding cost ¼ h[Igj(0, T1j) + Igj(T1j, T2j)] + h1 Idj(0, T1j). Thus, the total cost per unit time of the underlying inventory system during the cycle [0, T2j], as a function of T1j and T2j, say Z(T1j, T2j) is given by: Gðt Þ ¼ ζegðtÞ dt

ð2:37Þ

Our objective is to find T1j and T2j that minimize Z(T1j, T2j). However, the variables T1j and T2j are related to each other as follows:

2.3 EOQ Model with No Shortage

egð0Þ

Z

T 1j

21

0 < T 1j < T 2j

Z

T 2j

xðuÞdu ¼

0

DðuÞegðuÞ du þ

ð2:38Þ Z

0

T 1j

p j xðuÞegðuÞ du

ð2:39Þ

0

Thus, their goal is to solve the following optimization problem, which they shall call problem (m). (m) ¼ {minimize Z(T1j, T2j) given by Eq. (2.37) subject to Eq. (2.39) and hj ¼ 0: h j ¼ egð0Þ

Z

T 1j

Z

T 1j

xðuÞdu 

0

p j xðuÞegðuÞ du 

0

Z

T 2j

DðuÞegðuÞ du

ð2:40Þ

0

It can be noted from Eq. (2.40) that T1j ¼ 0 ) T2j ¼ 0 and T1j > 0 ) T1j < T2j. Thus Eq. (2.40) implies constraint (Eq. 2.38). Consequently, if they temporarily ignore the monotony constraint (Eq. 2.38) and call the resulting problem as (m1), then it does satisfy any solution of (m1). Hence, (m) and (m1) are equivalent. Moreover, T1j > 0 ) RHS of Eq. (2.33) > 0, i.e., Eq. (2.39) guarantees that the number of good items is at least equal to the demand during the first phase. First, Alamri et al. (2016) noted from Eq. (2.30) that T1j can be determined as a function of Qj, say:   T 1j ¼ f 1j Q j

ð2:41Þ

Taking also into account Eq. (2.40), they found that T2j can be determined as a function of T1j, and thus of Qj, say:   T 2j ¼ f 2j Q j

ð2:42Þ

Thus, if they substitute Eqs. (2.40)–(2.42) in Eq. (2.36), then problem (m) will be converted to the following unconstrained problem with the variable Qj (which they shall call problem (m2)):   Z f 1j Z f 1j   1 W Qj ¼ ðC þ C I Þ xðuÞdu þ h Gð0Þegð0Þ xðuÞdu f 2j 0 0  Z f 1j 

Z f 2j Z f 1j

p j xðuÞGðuÞegðuÞ du þ DðuÞGðuÞegðuÞ du þh1 f 1j  u p j xðuÞdu þ K þ 0

0

0

ð2:43Þ Now, the necessary condition for having a minimum for problem (m2) is:

22

2

Letting W ¼

w f 2j ,

Imperfect EOQ System

dW ¼0 dQ j

ð2:44Þ

w0Q j f 2j  f 02j,Q j w dW ¼ dQ j f 22j

ð2:45Þ

then:

where w0Q j and f 02j,Q j are the derivatives of w and f2j w.r.t. Qj, respectively. Hence, Eq. (2.45) is equivalent to (Alamri et al. 2016): w0Q j f 2j ¼ f 02j,Q j w

ð2:46Þ

Also, taking the first derivative of both sides of Eq. (2.40) w.r.t. Qj, one obtains: egð0Þ  p j egð

f 1j Þ

  ¼ f 02j,Q j D f 2j egð

f 2j Þ

ð2:47Þ

From which and Eqs. (2.38)–(2.40) it can be obtained:        w0Q j ¼ ðC þ C I Þ þ h G f 2j  Gð0Þ egð0Þ þ G f 1j Z f 1j   gð f Þ i h1 1j   p j xðuÞdu:  G f 2j p j e þ x f 1j 0 w0Q j w ¼ 0 W¼ f 2j f 2j,Q j

ð2:48Þ ð2:49Þ

where W is given by Eq. (2.40) and w0Q j is given by Eq. (2.48). Equation (2.49) can be used to determine the optimal value of Qj and its corresponding total minimum cost and then the optimal values of T1j and T2j (Alamri et al. 2016). Example 2.3 Alamri et al. (2016) presented an example to illustrate the efficiency of their mathematical model and solution procedures. They considered x(t) ¼ at + b, D τ l (t) ¼ at + r, p j ¼ Cb þe γj, and δðt Þ ¼ zβtwhere b, d, l, τ, Cb, z > 0; a, r, γ, β, t  0; and βt < z. Alamri et al. (2016) adopted the values considered in the study by Wahab and Jaber (2010), as presented in Table 2.4. The optimal values of Qj*, T1j*, T2j*, and ωj*, the corresponding total minimum cost for ten successive cycles, are obtained, and the results are shown in Table 2.5.

2.3 EOQ Model with No Shortage

23

Table 2.4 Input parameters (Alamri et al. 2016) Parameter C CI h h1 K a b γ

Value 100 ($/unit) 0.5 ($/unit) 20 ($/unit/year) 5 ($/unit/year) 3000 ($/cycle) 1000 (unit/year) 100,200 (unit/year) 0.7932 (unit/year)

Parameter α r l z β τ Cb

Value 500 (unit/year) 50,000 (unit/year) 1 (unit/year) 20 (unit/year) 25 (unit/year) 70.067 (unit/year) 819.76 (unit/year)

Table 2.5 Optimal results for varying demand, screening, and deterioration rates with pj (Alamri et al. 2016) j 1 2 3 4 5 6 7 8 9 10

pj 0.08524 0.08497 0.08436 0.08305 0.08030 0.07482 0.06502 0.05042 0.03369 0.01944

T1j 0.035424 0.035419 0.035407 0.035380 0.035324 0.035212 0.035013 0.034715 0.034376 0.034088

T2j 0.06482 0.06483 0.06485 0.06489 0.06498 0.06516 0.06548 0.06594 0.06644 0.06686

2.3.4

Learning Effects

2.3.4.1

Different Holding Costs

Qj* 3550 3550 3548 3546 3540 3529 3509 3479 3445 3416

pjQj* 303 302 299 294 284 264 228 175 116 66

ωj* 5.4 5.4 5.4 5.4 5.4 5.5 5.5 5.6 5.7 5.8

Wj * 5,585,464 5,583,830 5,580,142 5,572,240 5,555,724 5,523,107 5,465,734 5,382,467 5,290,159 5,214,030

wj* 362,030 361,980 361,850 361,580 361,020 359,900 357,890 354,900 351,490 348,600

Salameh and Jaber (2000) developed a model to determine the economic lot size by maximizing the expected total profit per unit time. Each delivered lot has defective items with a known probability function and is screened completely. Then the defective items are sold as a single batch at a discounted price at the end of the screening period (Wahab and Jaber 2010). In Salameh and Jaber’s model, it is observed that they use the same holding cost for both good items and defective items. However, in the real manufacturing environment, the good items and the defective items are treated in a different way. So, the holding cost, h ¼ iC, must be different for the good items and the defective items (e.g., Paknejad et al. 2005). With this consideration, they assigned holding costs h and h1 (where h > h1) for a unit of good item per period and a unit of defective item per period, respectively. In Fig. 2.5, inventory of defective items is depicted by the shaded area. In this section, the work of Wahab and Jaber (2010) based on Salameh and Jaber (2000), Maddah and Jaber (2008), and Jaber et al.

24

2

Fig. 2.5 The inventory level over time (Wahab and Jaber 2010)

Imperfect EOQ System

Inventory level

py

y t

T

Time

(2008) with different holding costs for the good items and defective items is presented. Figure 2.5 presents the inventory level of problem on hand. Let N(y, p) be the number of good items in each lot size y where p is a random variable and is given by: N ðy, pÞ ¼ y  py ¼ yð1  pÞ

ð2:50Þ

To avoid any shortage, good items’ quantity should not be less than the demand during the screening time, t, meaning: N ðy, pÞ  Dt

ð2:51Þ

Using Eqs. (2.50) and (2.51) and replacing t by y ¼ x: p
1

D ! E½p
1  D=x x

ð2:52Þ

Let TR( y) and TC( y) be the total revenue and the total cost per cycle, respectively. TR( y) consists of revenues from the good and defective items and is given by TR( y) ¼ sy(1  p) + vyp, and the total cost per cycle is (Wahab and Jaber 2010): Fixed and purchasing cost

TCðyÞ ¼

zfflfflfflffl}|fflfflfflffl{ K þ Cy

þ

þ

CI y |{z} Screening cost

The holding costs of good items per cycle

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{  2 ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl yð1  pÞT py h þ 2 2x

þ

 2 py h1 2x |fflfflfflfflffl{zfflfflfflfflffl}

Holding costs of defective items

ð2:53Þ

2.3 EOQ Model with No Shortage

25

The total profit per cycle, TP( y), is determined as the total revenue per cycle, TR ( y), minus the total cost per cycle, TC( y), and given as (Wahab and Jaber 2010): TPðyÞ ¼ syð1  pÞ þ vyp  2    2  yð1  pÞT py py  K þ Cy þ C I y þ h þ h1 þ 2 2x 2x

ð2:54Þ

The total profit per unit time, TPU( y), is determined by TP( y) ¼ T and expressed in Eq. (2.55). The expected value of the total profit per unit time is given in Eq. (2.56):     hy h y D TPUðyÞ ¼ D s  v þ þ 1 þ 2x 2x 1p   hyð1  pÞ K hy h1 y  v   C  CI   ð2:55Þ  2 y 2x 2x       hy h1 y K hy h1 y 1 E ½TPUðyÞ ¼ D s  v þ þ þ D v   C  CI   E 2x 2x y 2x 2x 1p hyð1  E ½pÞ  2 ð2:56Þ It can be easily shown that the E[TPU( y)] is concave in y, and thus by minimizing the expected total profit per unit time, the optimal lot size is determined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DKE ½1=ð1  pÞ y1 ¼ hð1  E½pÞ þ ðD=xÞðh þ h1 ÞE½1=ð1  pÞ  ðD=xÞðh þ h1 Þ

ð2:57Þ

Let the optimal lot size be y2 when the holding cost of the defective items and the good items is equal (i.e., h ¼ h1) and y2 is given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DKE ½1=ð1  pÞ y2 ¼ h½ð1  E ½pÞ  2ðD=xÞð1  E ½1=ð1  pÞÞ

ð2:58Þ

Regarding Eq. (2.58), the error appearing on Salameh and Jaber (2000) has been corrected on Cárdenas-Barrón (2001). A simple approach for determining the economic production quantity for an item with imperfect quality is also presented in Goyal and Cárdenas-Barrón (2002). Next, they have compared the optimal lot size derived in Maddah and Jaber (2008) with the one that they assigned different holding costs for the defective and good items. Since the cycle length T depends on the percentage rate of defective items, the cycle length is a random variable. Knowing Þ yð1E ½pÞ T ¼ yð1p (Wahab and Jaber 2010), the expected profit per cycle is: D and E ½T  ¼ D

26

2

Imperfect EOQ System

E½TPðyÞ ¼ syð1  E½pÞ þ vyE½p h i1 2 3 0   y 2 E ð 1  pÞ 2 2 2 E ½ p y E ½ p y 5 A þ h1 4K þ Cy þ C I y þ h@ þ 2D 2x 2x

ð2:59Þ

As a renewal process, the expected profit per unit time will be: Dfsð1  E ½pÞ þ vE½p  C  C I g KD  yð1  E ½pÞ ð 1  E ½ p Þ h i1 2 0 3 2   E ð 1  p Þ E ½ p  E ½ p  yD 4 h@ 5 A þ h1  þ D x x 2ð1  E ½pÞ

E½TPUðyÞ ¼

Because of the concavity of E[TPU( y)], the optimum lot size is: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2DK  i y3 ¼ u t h 2 hE ð1  pÞ þ ðD=xÞE ½pðh þ h1 Þ By considering h1 ¼ h in expression (2.61): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2DK  i i y4 ¼ u t h h h E ð1  pÞ2 þ 2E½pD=x

ð2:60Þ

ð2:61Þ

ð2:62Þ

which is the same as the one in Maddah and Jaber (2008). Jaber et al. (2008) considered learning effects in an EOQ model with imperfect items when the holding costs for the good and defective items are the same. Now different holding costs are assigned to the good and defective items. In this case, the total profit per unit time is the same as in Eq. (2.54). However, p is replaced with p(n), which is the percentage of defective per shipment n. For example, p(n) is expressed using a S-shaped logistic learning curve model as follows (Wahab and Jaber 2010): pð nÞ ¼

α γ þ eβn

ð2:63Þ

Including the learning effects, the total profit per unit time is given as (Wahab and Jaber 2010):      hy h y hy hy D K TPUðyn Þ ¼ D s  v þ n þ 1 n þ v   C  CI  n  1 n yn 2x 2x 2x 2x 1  pðnÞ hyn ð1  pðnÞÞ  2 ð2:64Þ In Eq. (2.64), n is not a decision variable. For a given n, p(n) is a constant. Hence, the total profit per unit time is concave in yn. By setting the first derivative equal to zero, the optimal yn can be obtained as (Wahab and Jaber 2010):

2.3 EOQ Model with No Shortage

27

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK ½1=ð1  pðnÞÞ yn ¼ hð1  pðnÞÞ þ ðD=xÞðh þ h1 Þ½1=ð1  pðnÞÞ  ðD=xÞðh þ h1 Þ

ð2:65Þ

Let the optimal lot size be zn when the holding costs of defective and good items are equal (i.e., h1 ¼ h) and n is given by (Wahab and Jaber 2010): zn

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK ¼ hð1  pðnÞÞ½ð1  pðnÞÞ þ ð2D=xÞ½pðnÞ=ð1  pðnÞÞ

ð2:66Þ

which is the same as the one in Jaber et al. (2008). Example 2.4 (Without Learning Effects) In order to illustrate the behavior of the optimal lot sizes y1, y2, y3, and y4, let us consider an example with D ¼ 50,000, C ¼ $100, K ¼ $3000, s ¼ $200, v ¼ $50, x ¼ 100, CI ¼ $0.5, h ¼ $20 unit/year, and h1 ¼ $5 unit/year. The percentage of defective item, p, is uniformly distributed, i.e., p ~ U[a, b], where E[p] ¼ (a + b)/2; E[1/(1  p)] ¼ [1/(a  b)]. They considered a ¼ 0 and b has been varied from 0.001 to 0.5 (Wahab and Jaber 2010). The optimal lot sizes for different values of b are given in Table 2.6 and y1, y2, y3, and y4 (Wahab and Jaber 2010). Optimal lot sizes y2 and y4 are computed by substituting h1 ¼ h ¼ $20 unit/year. Example 2.5 (With Learning Effects) In this example, they considered the same parameters that they used in Example 2.1 except p(n), as given in Eq. (2.63), and the values of α ¼ 70.067 and β ¼ 0.7932. Using Eqs. (2.65) and (2.66), yn* and zn* are computed and values yn* and zn* are presented in Table 2.7. As the learning takes Table 2.6 Comparison of order quantities (Wahab and Jaber 2010) b 0.001 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5

f( p) 1000.00 100.00 50.00 33.33 25.00 20.00 16.67 14.29 12.50 11.11 10.00 5.00 3.33 2.50 2.00

E[p] 0.0005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.1 0.15 0.2 0.25

E [1/(1  p)] 1.0005003 1.0050336 1.0101354 1.0153069 1.0205499 1.0258659 1.0312567 1.0367242 1.0422701 1.0478964 1.0536052 1.1157178 1.1889165 1.2770641 1.3862944

E [(1  p)2] 0.999000 0.990033 0.980133 0.970300 0.960533 0.950833 0.941200 0.931633 0.922133 0.912700 0.903333 0.813333 0.730000 0.653333 0.583333

y1* 3874.32 3886.34 3899.74 3913.19 3926.70 3940.26 3953.87 3967.52 3981.22 3994.98 4008.77 4149.06 4292.60 4437.47 4580.86

y2* 3873.95 3882.67 3892.34 3901.97 3911.58 3921.16 3930.70 3940.21 3949.68 3959.11 3968.50 4059.62 4143.91 4218.35 4279.33

y3* 3874.32 3886.31 3899.65 3912.99 3926.33 3939.68 3953.03 3966.37 3979.71 3993.04 4006.37 4138.72 4267.73 4390.69 4504.47

y4* 3873.95 3882.66 3892.27 3901.83 3911.32 3920.75 3930.11 3939.40 3948.63 3957.78 3966.85 4053.02 4129.32 4193.61 4243.91

28

2

Table 2.7 Comparison of order quantities with learning effects (Wahab and Jaber 2010)

n 1 2 3 4 5 6 7 8 9 10

P(n) 0.08524 0.08497 0.08436 0.08305 0.08030 0.07482 0.06502 0.05042 0.03369 0.01944

Imperfect EOQ System

zn 4064.53 4063.82 4062.26 4058.92 4051.91 4038.16 4014.13 3979.72 3942.29 3911.99

yn 4105.47 4104.70 4103.02 4099.38 4091.74 4076.55 4049.45 4009.30 3963.66 3925.09

place in the system, the number of defective items is going to decrease. Therefore, both yn* and zn* decrease as the number of shipment increases.

2.3.4.2

Transfer of Learning

As presented before, Salameh and Jaber (2000) extended the traditional EOQ model by accounting for imperfect-quality items under 100% screening and poor-quality items. The behavior of inventory was presented in Fig. 2.2. In this section, the work of Salameh and Jaber (2000) under learning in inspection which is developed by Khan et al. (2010) will be introduced. They considered a learning curve fits well to the power form of learning suggested by Wright (1936) as below: π n ¼ π 1 nb

ð2:67Þ

where p1 is the time to perform the first repetition, b is the learning exponent 0 < b < 1, and n is the cumulative number of repetitions. Also according to the work of Khan et al. (2010), the forgetting curve takes the form: πm ¼ π1 m f where π 1 ¼ π 1 mð f þbÞ and π m ¼ π m , where m is the equivalent number of items that could have been produced. The forgetting exponent in cycle i is determined by Jaber and Bonney (1996) as (Khan et al. 2010): fi ¼

bð1  bÞ log ðui þ yi Þ log ð1 þ V=λi Þ

ð2:68Þ

where V is the time for total forgetting to occur and is assumed to be an input parameter, ui is the experience remembered in cycle i, and li is the time to inspect (ui + yi) items without interruption. So (Khan et al. 2010):

2.3 EOQ Model with No Shortage

29

Table 2.8 Notations of a given problem x1 b fi V yi ui Zi Bi tsi tBi

Initial inspection rate (units/unit time) Learning exponent Forgetting exponent in the ith cycle Time for total forgetting to occur (time) Batch size in the ith cycle (unit) Experience of screening remembered from the previous i cycles Lost sales quantity in the ith cycle (unit) Backorder quantity in the ith cycle (unit) Time when the screening rate is equal to the demand in the ith cycle (time) Time to inspect Bi and DtBi units in the ith cycle (time)

λi ¼

ðui þ yi Þ1b x 1 ð 1 þ bÞ

ui þ 1 ¼ ðui þ yi Þð

f i þbÞ=b  f i =b Ri

ð2:69Þ

with: h i1=ð1bÞ Ri ¼ x1 ð1  bÞðT  τi Þ þ ðui þ yi Þð1bÞ

ð2:70Þ

where Ri is the equivalent number of items that could have been screened if no interruption occurs of length T  ti, and ti is the screening P time in cycle i. In case of breaks in screening, one should note that 0 < ui < i1 j¼1 y j when T  τi < V, for P partial forgetting; ui ¼ 0 when T  τi  V, for total forgetting; and ui ¼ i1 j¼1 y j , when V becomes infinite, for total transfer of learning will be used to determine the equivalent experience remembered at the start of each cycle, in case of partial and total transfer of learning (Khan et al. 2010). In order to model the presented problem, some new notations which are specifically used are shown in Table 2.8. Inspection is usually a manual task where an inspector tests incoming units for specific quality characteristics to determine if the units conform to the quality requirements. Time to inspect (or screen) each unit reduces as the number of inspected units increase as given in Eq. (2.67) and is represented as xn ¼ x1nb, where x1 ¼ 1/π 1 and xn ¼ 1/π n, whereas Salameh and Jaber (2000) assumed x1 ¼ xn > D. Here, it is assumed that x1 < D for yS < y. If xn < D, then the demand met is xn and the rate at which units lost or backordered is D  xn and D otherwise. The length of the stock-out period or the period over which backorders are accumulated is given as (Khan et al. 2010):

30

2 uZi þyi

t si ¼ ui

1 b n dn  x1

uZi þyi

ysi þui

Imperfect EOQ System

ðy þ ui Þ1b  u1b 1 b i n dn ¼ si x1 ð1  bÞx1

ð2:71Þ



1=ð1bÞ where ysi ¼ ð1  bÞx1 t s þ u1b  ui and ysi ¼ (D/x1)1/b  ui. Now the time i to screen yi items in a cycle (Khan et al. 2010) is: n τi ¼

ðyi þ ui Þ1b  u1b i

o ð2:72Þ

ð1  bÞx1

where ui is computed from Eq. (2.69). In case of no transfer of learning, that is, a worker does not retain any knowledge from earlier cycles (ui ¼ 0), it will be taken as (Khan et al. 2010): ys ¼ ysi ¼ ðD=x1 Þ1=b t s ¼ t si ¼ τ ¼ τi ¼

ð2:73Þ

D1b=b

ð2:74Þ

1=b

ð1  bÞx1

y1b ð1  bÞx1

ð2:75Þ

Two cases (lost sales and backorders) will be considered now to deal with the shortages, in each of the three scenarios for the transfer of learning from one cycle to another. These models are a direct extension to the work of Salameh and Jaber (2000) extended by Khan et al. (2010). In the case of lot sales, the demand that cannot be fulfilled due to slow screening will be taken as lost sale. The inventory level figure of this case is presented in Fig. 2.6 and the inventory level is shown in Eq. (2.76): 8 1=ð1bÞ > < yi  ½ð1  bÞx1 t  I I ðt Þ ¼ yi  ysi  Dðt  t si Þ > : ð1  pÞyi  ysi  Dðt  t si Þ

0
t < t si t si
t < τi

ð2:76Þ

τi < t
T i

At time t ¼ Ti, the inventory level is zero, i.e., (1  p)yi  ysi  D(Ti  tsi) ¼ 0, and the cycle time is given as (Khan et al. 2010): Ti ¼

ð1  pÞyi ysi  þ t si D D

ð2:77Þ

The holding costs for the three different behaviors of inventory shown in Fig. 2.9. are determined, respectively, from Eq. (2.76) as (Khan et al. 2010):

2.3 EOQ Model with No Shortage

31

Fig. 2.6 Learning in inspection with lost sales (Khan et al. 2010)

Ztsi h i HC1I ¼ h yi  ½ð1  bÞx1 t 1=ð1bÞ dt 0



¼ hyi t si  h

 1b 2b=1b ½ð1  bÞx1 1=ð1bÞ t si 2b

ð2:78Þ

Zτi HC2I ¼ h

½yi  ysi  DðT i  t si Þdt ts

¼ hðyi  ysi þ Dt si Þðτi  t si Þ 

 hD  2 τi  t 2si 2

ð2:79Þ

ZT i HC3I ¼ h

½ð1  pÞyi  ysi  DðT i  t si Þdt τi

ð2:80Þ

 hD  2 ¼ hðyi  ysi þ Dt si ÞðT i  τi Þ  hpyi ðT i  τi Þ  T i  τ2i 2 Adding the costs for three different behaviors, they can get the total holding cost for the lost sales case as (Khan et al. 2010):

32

2

Imperfect EOQ System

HCL ¼ hð1  pÞyi T i þ hpyi τi  hT i ðysi  Dt si Þ þ hysi t si    hD  2 1b 2b=1b T i  t 2si  h  ½ð1  bÞx1 1=ð1bÞ t si 2 2b

ð2:81Þ

After some simplifications and substitutions,

HCL ¼

h

i

h 2 h y ð1  pÞ2 þ 2yi Z i ð1  pÞ þ Z 2i  t2si þ hysi tsi þ 2D i 2D   2b 1b ½ð1  bÞx1 1=ð1bÞ t1b h si 2b

n o hpyi ðyi þ ui Þ1b  u1b i x1 ð1  bÞ

ð2:82Þ Cost of the lost sales ¼ b π ðDt si  ysi Þ ¼ b πZ n o C I ðyi þ ui Þ1b  u1b i Cost of inspection ¼ CI τi ¼ ð1  bÞx1

ð2:83Þ ð2:84Þ

So the total profit per cycle is (Khan et al. 2010): h i h 2 yi ð1  pÞ2 þ 2yi Z i ð1  pÞ þ Z 2i TPL ¼ ½sð1  pÞ þ vp  Cyi  K  b πZ i  2D n o 1b   1b ð hpy þ C Þ ð y þ u Þ  u I i i i i hD 1b 2b=1b þ t2si  ½ð1  bÞx1 1=ð1bÞ tsi  hysi tsi þ h 2 2b x1 ð 1  b Þ

ð2:85Þ And its expected value becomes: E ½TPiL  ¼ fsð1  E ½pÞ þ vE ½p  Cgyi  K  b πZ i n h i o h hD y2i E ð1  pÞ2 þ 2yi Z i ð1  E ½pÞ þ Z i 2 þ t 2si  2D 2 n o 1b   1b ðhyi E½p þ CI Þ ðyi þ ui Þ  ui 1b 2b=1b  ½ð1  bÞx1 1=ð1bÞ t si  hysi t si þ h 2b x 1 ð 1  bÞ ð2:86Þ The expected cycle time E[TiL] can be written as (Khan et al. 2010): E ½T iL  ¼ Using renewal reward theorem:

f1  E½pgyi ysi  þ t si D D

ð2:87Þ

2.3 EOQ Model with No Shortage

33

E½TPUiL  ¼

E ½TPiL  E ½T iL 

ð2:88Þ

According to the Khan et al. (2010), the experience gained in each cycle i will be taken as: ui ¼

i1 X

yj

ð2:89Þ

j¼1

This implies that the worker does not lose any knowledge in his break while he is not screening. In case of total forgetting, screening in each cycle starts with no prior knowledge, which means that the worker loses all the experience gained in the earlier cycles. In this case, the previous equations change to: τ¼ HCL ¼

y1b ð1  bÞx1

h i h 2 hD 2 hpy2b y ð1  pÞ2 þ 2yZ ð1  pÞ þ Z 2  ts þ 2D 2 x 1 ð 1  bÞ   1b þhys t s  h ½ð1  bÞx1 1=ð1bÞ t 2b=1b s 2b

ð2:90Þ

ð2:91Þ

In the expected profit per cycle and the expected annual profit of this case, Eqs. (2.90) and (2.91) should be replaced (Khan et al. 2010). But for the backorder case, the screening rate becomes equal to the demand rate at tsi. The dotted line shows that the backorder that piles up till tsi is fulfilled at the time (tsi + tBi). Inventory level diagram is presented in Fig. 2.7. So the maximum backorder level is: Bi ¼ Dt si  ysi

ð2:92Þ

Now, following its definition in the notations, yBi can be written as: yBi ¼ Dt Bi þ Bi þ ysi ¼ Dðt Bi þ t si Þ The time tBi can be written as: ZyBi t Bi ¼ 0

Substituting yBi:

1 b n dn  x1

Zysi 0

y1b y1b 1 b Bi Bi n dn ¼  x1 x 1 ð 1  bÞ x 1 ð 1  bÞ

ð2:93Þ

34

2

Fig. 2.7 Learning in inspection with backorders (Khan et al. 2010)

Imperfect EOQ System

Inventory level

tsi

Bi

tBi

yi

τi

pyi

Bi

Ti

t Bi ¼

½Dðt Bi þ t si Þ1b D1b=b  t si or t Bi þ t si ¼ 1=b x 1 ð 1  bÞ x1 ð1  bÞ1=b

Time

ð2:94Þ

This time can be taken as (Khan et al. 2010): t x ¼ t Bi þ t si ¼

D1b=b 1=b

x1 ð1  bÞ1=b

ð2:95Þ

Again, with the help of learning, the screening rate will become equal to or more than the demand rate, and there will not be any backorders after some cycles. The three scenarios of learning discussed in the lost sales case will be considered here to develop the expected annual profit of a buyer (Khan et al. 2010). The inventory level in Fig. 2.7 can be represented as (Khan et al. 2010): 8 > yi  ½ð1  bÞx1 t 1=ð1bÞ > > > < yi  ysi  Dðt  t si Þ I B ðt Þ ¼ > yi  Dt > > > : ð1  pÞyi  Dt

0
t
t si t si
t
t si þ t Bi t si þ t Bi
t
τi τi < t < T i

ð2:96Þ

At time t ¼ Ti, the inventory level is zero, i.e., (1  p)yi  DTi ¼ 0. The cycle time Ti is (Khan et al. 2010):

2.3 EOQ Model with No Shortage

35

Ti ¼

ð1  pÞyi D

ð2:97Þ

The holding costs for the four different time intervals are determined respectively from Eq. (2.96) as: 

 1b 2b=1b ½ð1  bÞx1 1=ð1bÞ t si 2b h i hD HC2Bi ¼ hðyi  ysi þ Dt si Þt Bi  ðt si þ t Bi Þ2  t 2si 2 h i hD 2 τi  ðt si þ t Bi Þ2 HC3Bi ¼ h½τi  ðt si þ t Bi Þyi  2  hD  2 T i  τ2i HC4Bi ¼ hð1  pÞðT i  τi Þyi  2 HC1Bi ¼ hyt si  h

ð2:98Þ ð2:99Þ ð2:100Þ ð2:101Þ

Adding Eqs. (2.98)–(2.101), the holding cost for the backorder case is:   h i 1b hD 2b=1b HCBi ¼ hyi t si  h ½ð1  bÞx1 1=ð1bÞ t si þ hðyi  ysi þ Dtsi Þt Bi  ðt si þ tBi Þ2  t 2si 2b 2 h i hD 2 hD  2 2  2 τ  ðtsi þ t Bi Þ þ hð1  pÞðT i  τi Þyi  T i  τi þh½τi  ðtsi þ t Bi Þyi  2 i 2

Using Eq. (2.97) and simplifying the above expression results in:  hD  2 HCBi ¼ hð1  pÞyi T i þ hpyi τi þ ht x B þ hysi t si  T i  t 2si 2   1b 1=ð1bÞ 2b=1b h ½ð1  bÞx1  t si 2b

ð2:102Þ

Substituting ti and Ti in terms of yi from Eqs. (2.110) and (2.96) respectively, the above expression can be simplified as (Khan et al. 2010):

HCBi ¼

h

i

n o hpyi ðyi þ ui Þ1b  u1b i

h 2 hD 2 y ð 1  pÞ 2  t þ 2D i 2 si x 1 ð 1  bÞ   1b 2b=1b þhysi t si þ ht x B  h ½ð1  bÞx1 1=ð1bÞ t si 2b

ð2:103Þ

It should be noted that the above holding cost reduces to the one in Salameh and Jaber (2000) once b, tx, tsi, ysi, and ui become zero (Khan et al. 2010).   Now the backorder cost in a cycle ¼ Cb2ts Bi þ C b ðt x  t si ÞBi ¼ C b t x  t2si Bi

36

2





Imperfect EOQ System

n o ðhpyi þ C I Þ ðyi þ ui Þ1b  u1b i

t si B 2 i x 1 ð 1  bÞ   h i h 2 hD 1b 2b=1b  ½ð1  bÞx1 1=ð1bÞ t si y ð1  pÞ2 þ t 2si  hysi t si  ht x B þ h 2D i 2 2b

TPiB ¼ ½sð1  pÞ þ υp  C yi  K  C b t x 

ð2:104Þ And the expected total profit per cycle is (Khan et al. 2010):   t E ½TPiB  ¼ fsð1  E ½pÞ þ υE ½p  C gyi  K  Cb t x  si Bi 2 n o 1b 1b n h io ðhyi E½p þ CI Þ ðyi þ ui Þ  ui h hD 2 y2i E ð1  pÞ2 þ t  hysi t si   2D 2 si x 1 ð 1  bÞ   1b 2b=1b ht x B þ h ½ð1  bÞx1 1=ð1bÞ t si 2b ð2:105Þ The expected cycle time E[TiB] can be written using Eq. (2.96) as: E ½T iB  ¼

ð1  E ½pÞyi D

So, the expected annual profit can be written as: E ½TPUiB  ¼

E ½TPiB  E ½T iB 

ð2:106Þ

In the case of backorders for total transfer of learning, the worker will retain all the experience gained in the earlier cycles. This experience will be calculated using Eq. (2.92). The holding cost, the expected profit per cycle, and the expected annual profit in Eqs. (2.103), (2.105), and (2.106), respectively, will be determined using this experience in each cycle (Khan et al. 2010). In the case of backorders for total forgetting, the experience ui in cycle i becomes zero, and the inspection time will be determined by Eq. (2.93). The holding cost in Eq. (2.103) will be written as (Khan et al. 2010): HCB ¼

h i h 2 hD 2 hpy2b y i ð 1  pÞ 2 þ t si þ þ hys t s þ ht x B 2D 2 x 1 ð 1  bÞ   1b 2b=1b h ½ð1  bÞx1 1=ð1bÞ t si 2b

ð2:107Þ

2.3 EOQ Model with No Shortage

37

The expected profit per cycle and the expected annual profit in Eqs. (2.105) and (2.106), respectively, will be determined using Eqs. (2.93) and (2.107) (Khan et al. 2010).

2.3.5

EOQ Models with Imperfect-Quality Items and Sampling

2.3.5.1

Inspection Shifts from Buyer to Supplier

In contrast to traditional EOQ models, which implicitly assume that all items are completely perfect, Salameh and Jaber (2000) have formulated the problem in situations where not all items are perfect. The imperfect items are separated from perfect ones by a full inspection and are used in another inventory situation. The implicit assumption of Salameh and Jaber’s (2000) model, however, is that the supplier does not perform a full inspection; otherwise the received batches are expected to be completely perfect. In fact, the very presence of imperfect items in a batch depends on whether or not the supplier carries out a full inspection, which is why they outlined two different possible scenarios here (Rezaei and Salimi 2012): Scenario 1. The supplier does not perform a full inspection, and, as a result, the batches received by the buyer contain some imperfect items. This implies that the buyer should conduct a full inspection. Scenario 2. The supplier performs a full inspection, and, as a result, the batches received by the buyer contain no imperfect items. The first scenario was formulated and analyzed in Salameh and Jaber (2000), while the second scenario is the implicit assumption of traditional EOQ models. Based on these two scenarios, they examined the following research questions: Assuming there is no relationship between the buyer’s selling price, the purchasing price, and customer demand. According to Salameh and Jaber (2000) and Maddah and Jaber (2008), the buyer’s expected profit per ordering cycle is as follows (Rezaei and Salimi 2012): TPðyÞ ¼ Total sales of good‐quality items þ Total sales of imperfect‐quality items Ordering cost  Purchasing cost  Inspection cost  Holding costs or equivalently:

38

2 Income

zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ TPðyÞ ¼syð1  pÞ þ vyp  |{z} K 

Purchasing cost

z}|{ Cy

Fixed cost

Imperfect EOQ System



 ½yð1  pÞ2 py2 þ h ð2:108Þ 2D x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Holding cost

Then the buyer’s expected total profit per time unit is:  ETPUðyÞ ¼

 h i  ðsð1  E ½pÞ þ υE ½p  C  C I ÞD  KD=y  hy E ð1  pÞ2 =2 þ E ½pD=x 1  E ½p

ð2:109Þ Consequently, the optimal order quantity would be: 0

11=2

2KD i A ySMJ ¼ @  h h E ð1  pÞ2 þ 2E ½pD=x

ð2:110Þ

Given completely perfect batches ( p ¼ 0), the total profit per time unit is: TPUp ¼ 0ðyÞ ¼ Ds  Dc0 

KD hy  y 2

ð2:111Þ

And the optimal order quantity for this condition is the traditional EOQ: y

trade

rffiffiffiffiffiffiffiffiffiffi 2KD ¼ h

ð2:112Þ

Now the maximum purchasing price for batches without imperfect items should be determined. First, determine the difference between the total profit per time unit when there are no imperfect items and the expected profit per time unit when there are p% imperfect items on average in each batch. If one considers c0 as a variable here, then (Rezaei and Salimi 2012):     KD hytrade TPU ytrade , c0  ETPU ySMJ ¼ Dðs  c0 Þ  trade  2 y  SMJ   ETPU y

ð2:113Þ

The buyer accepts to pay more if and only if:     TPU ytrade , c0  ETPU ySMJ  0 which yields to:

ð2:114Þ

2.3 EOQ Model with No Shortage

c0
s 

39

K

y

 trade

hytrade þ 2ETPUðySMJ Þ 2D

ð2:115Þ

The right-hand side of this equation determines the maximum unit purchasing price (Mc) the buyer is willing to pay for batches without imperfect items. To determine the optimal buyer’s selling price and order quantity, they consider the expected total profit equation (Eq. 2.109) as an objective function, while both the buyer’s selling price s and the order quantity y are decision variables. The expected total profit of the inventory problem should be maximized subject to the price– demand relationship function:  max π ðy, sÞ ¼

 h i  ðsð1  E½pÞ þ υE½p  C  C I ÞD  KD=y  hy E ð1  pÞ2 =2 þ E ½pD=x 1  E ½ p

s:t: D ¼ f ðsÞ

ð2:116Þ To obtain the optimal values of y, s, and D, the following Lagrangian function is used (Rezaei and Salimi 2012): L ¼ π ðy, sÞ  λðD  f ðsÞÞ

ð2:117Þ

Then the partial derivation of the Lagrangian function should be set with respect to y, s, D, and l to zero:   ∂L 1 K hyE½p  λ ¼ 0 ð2:118Þ ¼ sð1  E ½pÞ þ υE ½p  C  CI   x y ∂D 1  E½p ∂L ¼ D þ λ f 0 ðsÞ ¼ 0 ∂s   h i  ∂L 1 KD 2 ¼  h E ð 1  p Þ =2 þ E ½ p D=x ¼0 ∂y 1  E ½p y2 ∂L ¼ D þ f ðsÞ ¼ 0 ∂λ

ð2:119Þ ð2:120Þ ð2:121Þ

Then solving simultaneously, the equation system yields to (Rezaei and Salimi 2012):

40

2

Imperfect EOQ System

0

s

11=2 2KD i A y ¼ @  h h E ð1  pÞ2 þ 2E ½pD=x   D K hyE ½p =ð1  E ½pÞ ¼  0  υE ½p  C  C I   x y f ðsÞ D ¼ f ðsÞ

ð2:122Þ

ð2:123Þ ð2:124Þ

In Eq. (2.123), there is no specific price–demand relationship, which means it is a general formula designed to obtain the optimal value of buyer’s selling price s. To determine the maximum purchasing price (Mc), Rezaei and Salimi (2012) first calculated the difference between the total profit when there are no imperfect items and the expected total profit when there are p% imperfect items on average in every batch:   k hy π ðs, y, c0 Þ  π ðy , s Þ ¼ D s  c0    π ðy , s Þ y 2

ð2:125Þ

Here, the buyer agrees to pay more for each item in batches without imperfect items if and only if: π ðs, y, c0 Þ  π ðy , s Þ  0

ð2:126Þ

Consequently, one obtains (Rezaei and Salimi 2012): c0
s 

K hy þ 2π ðy , s Þ  2D y

ð2:127Þ

The right-hand side of this equation is the maximum purchasing price. To determine the highest value of c0, one should find its maximum value (R). As R is a function of variables s and y, Rezaei and Salimi (2012) considered it a function that should be maximized subject to the price–demand relationship function: max Rðs, yÞ ¼ s  s:t: D ¼ f ðsÞ

K hy þ 2π ðy , s Þ  2D y

ð2:128Þ

To obtain the optimal values of s, y, and D, they first made the following Lagrangian function:

2.3 EOQ Model with No Shortage

L¼s

41

K hy þ 2π ðy , s Þ  λ ðD  f ðsÞÞ  2D y

ð2:129Þ

The partial derivation of the Lagrangian function with respect to y, s, D, and l are: ∂L hy þ 2π ðy , s Þ ¼ λ¼0 ∂D 2D2

ð2:130Þ

∂L ¼ 1 þ λ f 0 ðsÞ ¼ 0 ∂s

ð2:131Þ

∂L K h ¼ 2 ¼0 2D y ∂y

ð2:132Þ

∂L ¼ D þ f ðsÞ ¼ 0 ∂λ

ð2:133Þ

Solving simultaneously equation system yields to (Rezaei and Salimi 2012): sp ¼

ð f 0 ðsÞð2π ðy , s Þ þ hyÞ=2Þ f 0 ðsÞ   2KD 1=2 yp ¼ h

1=2

 f ð 0Þ

ð2:134Þ ð2:135Þ

This is a general model to determine the values of s and y, which leads to determine the maximum value of c0 , Mc. Specifying a suitable price–demand relationship function, they found the optimal values of s and y. Putting the value of sp and yp on Eq. (2.127), the maximum value of c0 can be determined. In order to make an optimal decision, Rezaei and Salimi (2012) proposed a decision rule as below which is presented in Fig. 2.8. Example 2.6 Determining Mc assuming there is no relationship between the buyer’s selling price, the purchasing price, and customer demand, Rezaei and Salimi (2012) adopted the same data as used in Salameh and Jaber (2000) and Maddah and Jaber (2008) as follows:  f ð pÞ ¼

25, 0
p
0:04 0, otherwise

h i ) E½p ¼ 0:02 and E ð1  pÞ2 ¼ 0:96

D ¼ 50,000 units/year, C ¼ $25/unit, K ¼ $100/cycle, h ¼ $5/unit/year, x ¼ 1 unit/min, CI ¼ $0.5/unit, s ¼ $50/unit, v ¼ 20/unit, and the inventory operation operates on an 8 h/day, for 365 days a year. Using Eqs. (2.109)–(2.112) and (2.115), y* ¼ 1434.48, ETPU* ¼ 1,212,274.30, ytrade* ¼ 1414.21, TPUtrade* ¼ 1,242,929, and Mc ¼ 25.61.

42

2

Imperfect EOQ System

Fig. 2.8 Decision rule Table 2.9 Average rate of imperfect items in each batch and the corresponding maximum purchasing price (Mc) (Rezaei and Salimi 2012)

E[p] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13

Mc 25.56 25.61 25.67 25.73 25.79 25.85 25.92 25.98 26.05 26.12 26.19 26.26 26.33

E[p] 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

Mc 26.40 26.48 26.56 26.64 26.72 26.80 26.89 26.98 27.07 27.16 27.25 27.35

Table 2.9 shows the corresponding maximum purchasing price (Mc) for different average rates of imperfect items. As becomes clear, the higher the average rate of imperfect items E[p], the higher the maximum purchasing price, Mc. Example 2.7 To determining Mc assuming there is a relationship between the buyer’s selling price, the purchasing price, and customer demand, Rezaei and Salimi (2012) considered the same data presented in Example 2.6. Commonly, two demand functions have been considered in literature: (1) the constant price–elasticity function and (2) the linear demand function (e.g., Rezaei and Davoodi in press). Here, they supposed a linear price–demand relationship function as D ¼ 100,000–1000s. Using Eqs. (2.116) and (2.122)–(2.124), s** ¼ 62.8477; y** ¼ 1238.39; D** ¼ 37,152.32; and y ¼ 1,377,260.25. Table 2.10 shows the optimal value of these variables (s**, y**, D**, y, sp, yp, and Mc) for different values of E[p].

2.3 EOQ Model with No Shortage

43

Table 2.10 The optimal value of s**, y**, D**, p*(s, y), sp, yp, and Mc for different values of E[p] (Rezaei and Salimi 2012) E[p] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

2.3.5.2

s** 62.819 62.848 62.877 62.907 62.937 62.968 63.000 63.033 63.066 63.100 63.135 63.170 63.207 63.244 63.282 63.321 63.361 63.402 63.444 63.487 63.531 63.576 63.623 63.670 63.719

y** 1229.16 1238.39 1247.66 1256.98 1266.33 1275.72 1285.14 1294.59 1304.06 1313.54 1323.04 1332.54 1342.05 1351.55 1361.03 1370.50 1379.94 1389.34 1398.71 1408.02 1417.28 1426.47 1435.58 1444.61 1453.55

D** 37,180.96 37,152.32 37,123.08 37,093.23 37,062.75 37,031.61 36,999.80 36,967.28 36,934.05 36,900.07 36,865.32 36,829.77 36,793.40 36,756.17 36,718.06 36,679.03 36,639.06 36,598.10 36,556.12 36,513.09 36,468.96 36,423.68 36,377.22 36,329.53 36,280.56

p*(s, y) 1,379,381.21 1,377,260.25 1,375,096.78 1,372,889.51 1,370,637.11 1,368,338.19 1,365,991.29 1,363,594.91 1,361,147.48 1,358,647.34 1,356,092.80 1,353,482.06 1,350,813.26 1,348,084.47 1,345,293.63 1,342,438.63 1,339,517.24 1,336,527.14 1,333,465.88 1,330,330.93 1,327,119.59 1,323,829.08 1,320,456.45 1,316,998.61 1,313,452.31

sp 62.819 62.848 62.877 62.906 62.937 62.968 63.000 63.032 63.065 63.099 63.134 63.169 63.205 63.242 63.280 63.319 63.359 63.400 63.442 63.485 63.529 63.574 63.620 63.668 63.717

yp 1219.53 1219.06 1218.58 1218.09 1217.59 1217.08 1216.56 1216.03 1215.48 1214.92 1214.35 1213.77 1213.17 1212.56 1211.93 1211.29 1210.63 1209.96 1209.27 1208.55 1207.83 1207.08 1206.31 1205.52 1204.71

Mc 25.56 25.61 25.67 25.73 25.79 25.85 25.92 25.98 26.05 26.12 26.19 26.26 26.33 26.40 26.48 26.56 26.64 26.72 26.80 26.89 26.98 27.07 27.16 27.25 27.35

DMc 0.556 0.057 0.058 0.059 0.061 0.062 0.063 0.065 0.066 0.068 0.069 0.071 0.072 0.074 0.076 0.078 0.080 0.082 0.084 0.086 0.088 0.090 0.093 0.095 0.098

Sampling Inspection Plans

Rezaei (2016) considers a situation where a buyer wants to decide on economic order quantity of a product, where the received order contains a p percentage of imperfects, with a known probability density function, f( p). In order to model the presented problem, some new notations which are specifically used are shown in Table 2.11. Upon receiving the lot, the buyer draws a sample of size n from the lot, and based on the findings from the sample, one of the three following decisions is made: • If p is above p1, the whole lot will be rejected, and the supplier should send a same lot without any defective. • If p is less than p1 and above p0, the lot is accepted. The buyer performs a full inspection. The imperfect items are separated from the perfect ones and used in another inventory situation. • If p is below p0, the lot is accepted and no inspection is conducted.

44

2

Table 2.11 Notations of a given problem

Imperfect EOQ System

Maximum level of imperfect item (%) Minimum level of imperfect item (%) Imperfect item quantity (unit) Minimum level of imperfect item quantity (unit) Maximum level of imperfect item quantity (unit)

p1 p0 θ α0 α1

Case I: p > p1 Considering the lot size y and selling price s, the total profit is: Fixed cost

Revenue

z}|{ TPðyÞ ¼ sy 

Cy |{z}



z}|{ K 

Purchasing cost

hy2 2D |{z}

ð2:136Þ

Holding cost

Dividing TP( y) by the inventory cycle length T ¼ Dy gives the buyer’s total profit per time unit, as follows: TPUðyjp  p1 Þ ¼ ðs  CÞD 

KD hy  y 2

ð2:137Þ

So the optimal order quantity is: yR

rffiffiffiffiffiffiffiffiffiffi 2DK ¼ h

ð2:138Þ

Case II: p0 < p < p1 This case has been proposed by Maddah and Jaber (2008). The total profit of this case is calculated as follows: Revenue of perfect items zfflfflfflfflffl}|fflfflfflfflffl{ TPðyÞ ¼ syð1  pÞ þ

Inspection cost

z}|{ CI y

Fixed cost

 υyp |{z} Revenue of imperfect items

  ½yð1  pÞ2 py2 þ h 2D x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

z}|{ K 

Cy |{z}



Purchasing cost

ð2:139Þ

Holding cost

Dividing TP( y) by the expected inventory cycle length E½T  ¼ ð1ED½pÞy , the buyer’s expected total profit per time unit is:

2.3 EOQ Model with No Shortage

45

E½TPUðyjp0
p
p1 Þ ¼

 h i  ððsð1  E½pÞ þ υE½p  C  C I ÞD  KD=y  hy E ð1  pÞ2 =2 þ E½pD=x 1  E ½ p ð2:140Þ So, the optimum order size is: 0

112

2KD i A yAl ¼ @  h h E ð1  pÞ2 þ 2E ½pD=x

ð2:141Þ

Case III: p < p0 The total profit function of this case contains elements that are similar to the previous case, except that in this case, the buyer spends no cost on inspection, but has to spend for return costs. When a customer returns an item, the customer receives a new one. However, as no inspection is conducted in this case, there is still a chance (though very slim) that the new item delivered to the customer will also be imperfect. That is to say, from the order quantity y, py units are imperfect. This means that the buyer will initially receive py returned items. These customers are given new items. Again, from this py items, p(py) are returned. If they considered the whole process including the next times, theoretically one obtains:   The total number of returned items ¼ y p þ p2 þ p3 þ ⋯

ð2:142Þ

Inside the bracket is the sum of a geometric series, for which one obtains:   p þ p2 þ p3 þ ⋯ ¼ lim p þ p2 þ p3 þ ⋯ þ pn ¼ n!1

p 1p

ð2:143Þ

This means that The total number of returned items ¼ y The total costs of return ¼ C J y And finally the total profit becomes:

p 1p

p 1p

ð2:144Þ ð2:145Þ

46

2

Imperfect EOQ System

Revenue of perfect and imperfect items

zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ syð1  pÞ þ υyp

TPðyÞ ¼

 |{z} K  Fixed cost

Purchasing cost

z}|{ Cy

Return cost

zfflfflfflfflffl}|fflfflfflfflffl{ y2 ð1  pÞ p h  CJ y 2D 1  p |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

ð2:146Þ

Holding cost

Dividing TP( y) by the expected inventory cycle length E[T], the buyer’s expected total profit per time unit is calculated as follows:

E ½TPUðyjp
p0 Þ ¼

½pÞ ðsð1  E ½pÞ þ υE ½pÞD  KD=y  CD  h yð1E  C J DE 2

h

i

p 1p

1  E ½ p ð2:147Þ

So the optimum order quantity for this case would be: yAnI

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK ¼ hð1  E ½pÞ

ð2:148Þ

In order to plan a sampling, Rezaei (2016) suggested the following steps: Step 1. Draw a random sample of n items from the lot, inspect the sample, and count θ. Step 2. Based on the observed number θ, follow one of the following strategies: (a) If θ > a1, reject the entire lot. (b) If a0 < θ < a1, accept the entire lot and conduct a 100% inspection. (c) If θ < a0, accept the entire lot, and there is no need for inspection. The expected total profit of the problem considering the sampling plan is: ETPUðyjp U ða, bÞÞ ¼Prðθ  α1 ÞTPUðyjp  p1 Þ þPrðα0
θ
α1 ÞETPUðyjp0
p
p1 Þ

ð2:149Þ

þPrðθ
α0 ÞETPUðyjp
p0 Þ Before continuing, the sampling plan should be incorporated into the total profit functions of all cases. Case IV: p > p1 with Sampling Plan In this case, a new cost element (sampling costs) which is nCI is added to the total profit function of the first case. In addition, if the buyer rejects a lot, the buyer receives a complete perfect lot, while, if the supplier later investigates and realizes that the lot has been wrongly rejected, or in fact p
p1,

2.3 EOQ Model with No Shortage

47

the buyer is penalized for CW. The chance of such an unjustified rejection by the buyer is (Rezaei 2016): ψ ¼ Prðp
p1 jθ  α1 Þ Finally, the total profit of the fourth case is: Revenue‐purchasing cost

TPUðyjp  P1 Þ ¼

zfflfflfflfflffl}|fflfflfflfflffl{ ðs  cÞD

Holding cost

z}|{ hy 2

Fixed cost

z}|{ KD   y

Inspection cost

zffl}|ffl{ C I nD y



Buyer’s penalty cost



zfflfflffl}|fflfflffl{ ψC W D y

ð2:150Þ

Case V: p0 < p < p1 with Sampling Plan In this case, since a 100% inspection is conducted and n is considered as part, the sampling cost is considered as part of the inspection cost. So the total profit of this case is similar to the second case. Case VI: p < p0 with Sampling Plan In this case, inspection cost should be added to the total profit of the third case: E ½TPUðyjp
p0 Þ ¼

½pÞ ðsð1  E ½pÞ þ υE ½pÞD  ðK þ C I nÞD=y  CD  h yð1E  CJ DE 2

h

p 1p

i

1  E ½ p ð2:151Þ

Because of type I error and type II error and different parts of the probability density function f( p) used for the three cases, the expected numbers of imperfect items for the three cases are slightly different. The expected numbers of imperfect items in a received lot, given the observed number of imperfect items in the sample is (1) greater than a1, E1[p], (2) is between a0 and a1, and (3) is less than a0, are calculated as follows: E1 ½p ¼ Prðθ  α1 \ p
p0 ÞE p
p0 ½p þ Prðθ  α1 \ p0
p
p1 ÞE p0
p
p1 ½p þPrðθ  α1 \ p  p1 ÞE pp1 ½p ð2:152Þ E2 ½p ¼ Prðα0
θ
α1 \ p
p0 ÞEp
p0 ½p þ Prðα0
θ
α1 \ p0
p
p1 ÞE p0
p
p1 ½p þPrðα0
θ
α1 \ p  p1 ÞEpp1 ½p ð2:153Þ

48

2

Imperfect EOQ System

E3 ½p ¼ Prðθ
α0 \ p
p0 ÞE p
p0 ½p þ Prðθ
α0 \ p0
p
p1 ÞE p0
p
p1 ½p þPrðθ
α0 \ p  p1 ÞE pp1 ½p ð2:154Þ E1[p], E2[p], and E3[p] are considered as the expected numbers of imperfect items in a lot when they respectively reject the lot, accept the lot and conduct the full inspection, and accept the lot and conduct no inspection. Considering Eqs. (2.140) and (2.150)–(2.154), Eq. (2.149) can now be rewritten as follows:   KD C I nD hy ψRD E ½TPUðyjp U ða, bÞÞ ¼ Prðθ  α1 Þ ðs  C ÞD     y y 2 y i  h     ð s 1  E 2 ½p þ υE2 ½p  C  C I D  KD=y  hy E 2 ð1  pÞ2 =2 þ E 2 ½pD=x þPrðα0
θ
α1 Þ 1  E 2 ½p        y 1  E 3 ½ p p  C J DE 3 s 1  E3 ½p þ υE3 ½p D  ðK þ C I nÞD=y  CD  h 2 1p þPrðθ
α0 Þ 1  E 3 ½p

ð2:155Þ Using Eq. (2.155), the optimum order quantity considering the sampling plan becomes (Rezaei 2016): ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u 2 Prðθ  α1 Þ þ Prðθ
α0 Þ= 1  E 3 ½p ðKD þ C I nDÞ u    u þ Prðα0
θ
α1 Þ= 1  E 2 ½p KD þ Prðθ  α1 ÞψRD u h     yIntg ¼ t h   h Prðθ  α1 Þ þ Prðα0
θ
α1 Þ= 1  E 2 ½p E 2 ð1  pÞ2 þ 2E 2 ½p=x þ Prðθ
α0 Þ

ð2:156Þ Example 2.8 To illustrate the proposed model, Rezaei (2016) considered D ¼ 50,000 units/year, C ¼ $25/unit, K ¼ $100/cycle, h ¼ $5/unit/year, x ¼ 1 unit/min, CI ¼ $0.5/unit (the inventory operation operates on an 8 h/day, for 365 days a year), s ¼ $50/unit, v ¼ $20/unit, CJ ¼ $15/unit returned, CW ¼ $70 and  f ð pÞ ¼

4, 0,

0
p
0:25 : otherwise

ð2:157Þ

It should note that for a uniform probability density function P ¼ U(a, b): E ½ p ¼

aþb 2

ð2:158Þ

2.3 EOQ Model with No Shortage

h i E ð 1  pÞ 2 ¼

1 ba

49

Zb ð1  pÞ2 dp ¼

a2 þ ab þ b2 abþ1 3

ð2:159Þ

a



 Zb  ln ð1  bÞ  b þ ln ð1  aÞ þ a p 1 p E dp ¼ ¼ ba 1p ba 1p

ð2:160Þ

a

The buyer should first define a sampling strategy, identifying n, a0, and a1. As mentioned before, the buyer and supplier should agree on the maximum limit, while the minimum limit is identified by the buyer. Using the algorithm presented in the last section, and considering the data, they found b ¼ 0.063707, which helps the buyer identify the minimum limit. They assumed that the buyer selects b ¼ 0.06 for the minimum limit and, together with the supplier, chooses b ¼ 0.15 for the maximum limit. Based on this information, the general probability density function can be further divided into three probability density functions, as follows:  f ðpÞreject ¼

10 0:15
p
0:25 0



f ðpÞaccept and inspection ¼ f ðpÞaccept and no‐inspection ¼

otherwise 11:1 0 

0:06
p
0:15 otherwise

16:6, 0

0
p
0:06 otherwise

ð2:161Þ ð2:162Þ ð2:163Þ

The expected values of these three probability density functions are 0.2, 0.105, and 0.03, respectively. A random sample of 20 items (n ¼ 20) is selected and inspected, based on the results of which the following decisions are made: • If θ > 4, the buyer rejects the entire lot and receives a replacement lot including no imperfect items. • If 1 < θ < 4 (two or three imperfect items), the buyer accepts the entire lot, and then conducts a 100% inspection, after which the perfect items are sold to the customers ($50/unit), while the imperfect items are sold as a single batch at the end of inspection process in another inventory situation ($20/unit). • If θ < 1, the entire lot is accepted and the buyer does not inspect the lot. The probability of observing θ imperfect items in a sample of size n can be calculated using a binomial distribution function as follows:   n θ f ðθ, n, pÞ ¼ p ð1  pÞnθ θ

ð2:164Þ

50

2

Imperfect EOQ System

Because the buyer makes decisions based on the sample, there is a chance of making wrong decision (the so-called type I and II errors), which is why the expected values, E[p], have to be adjusted. For instance, the first element of this matrix shows the probability of observing zero or one imperfect items and accepting the lot, meaning that the imperfect items of the sample follow the distribution function. In other words, it is likely (0.186) that they accepted the sample (h > 1), while the actual imperfect rate of the lot follows from the distribution function, and it is also likely (0.018) that the lot actually follows from the distribution function. The other elements of the matrix are interpreted in the same way. One has: E 1 ½p ¼ 0:196; E2 ½p ¼ 0:117; E 3 ½p ¼ 0:049

ð2:165Þ

The optimal order quantity of the entire problem is calculated using Eq. (2.156), and is y*Intg ¼ 1485, with an expected total profit of:   ~ ða, bÞ ¼ 1, 216, 570 E TPU yjpU

ð2:166Þ

The economic order quantity and expected total profit of these two models are shown below (Rezaei 2016): Case II. EOQ ¼ 1541.04, E[TPU] ¼ 1,178,298.12 (Maddah and Jaber 2008) Case III. EOQ ¼ 1511.86, E[TPU] ¼ 1,077,530.75

2.3.5.3

Instantaneous Replenishment Model with Sampling

Moussawi-Haidar et al. (2013) considered a periodic review EOQ-type inventory model with random supply and imperfect items. Upon receiving an order, an acceptance sampling plan is used to decide whether to accept the lot based on the number of defective items found in the sample or to reject the lot. It is assumed that a rejected lot is submitted to 100% screening. The acceptance sampling plan is characterized by two parameters, a sample of size n and an acceptance number an. The problem is to jointly determine the optimal lot size y and the optimal sampling plan, i.e., the sample size, n, and acceptance number an, that maximize the total expected profit subject to a constraint limiting the proportion of uninspected defective items passed to the customer. In order to model the presented problem, some new notations which are specifically used are shown in Table 2.12. In Moussawi-Haidar et al. (2013)’s model, the production of each lot varies randomly with probability density function f(P). Then, the distribution of the number of lot defectives, N, in a lot of size y is given by the integral:

2.3 EOQ Model with No Shortage Table 2.12 Notations of a given problem

51

an N X Pa ICU QCU E[TRU(y, n, c)]

Acceptance number/level Number of lot defective (unit) Number of sample defectives (unit) Probability of accepting the lot Average inventory cost per unit of time Quality-related cost per unit time Expected total revenue per unit time

Z1 gðN Þ ¼

bðN, y, PÞf ðPÞdP 0

They showed that when P follows a beta distribution with parameters a and b, g (N ) follows the beta-binomial distribution, with parameters y, a, and b. The number of sample defectives, X, in a sample of size n is given by the conditional distribution t (XjN), assumed to be a hypergeometric distribution. The joint distribution of N and X is obtained as: PrðN, X Þ ¼ t ðXjN ÞgðN Þ And the marginal distribution of X is beta binomial with parameters n, a, and b, which is given by: hðX Þ ¼

X PrðN, X Þ Y

Upon screening a sample of size n, if the number of sample defective items X is less than the acceptance level an, the lot is accepted. So Pa becomes (MoussawiHaidar et al. 2013): Pa ¼ PðX
an Þ ¼

an X

hð X Þ

X¼0

Inventory-related costs. The inventory cost consists of the fixed ordering cost and the inventory holding cost. To compute the expected inventory cost, the dynamics of the inventory level are needed to consider as depicted in Fig. 2.9, for each of the two cases: (I) when the lot is accepted based on the sample defectives and (II) when the lot is rejected and subject to 100% screening (i.e., XPc). The behavior of the inventory level for case I is illustrated by the solid line in Fig. 2.9, where T1 represents the cycle time. Screening the sample is completed at time t1 ¼ n/x, at which the inventory level drops by np, the average number of sample defectives withdrawn from the inventory at the end of the screening process. Thus, the inventory level at time t1 drops to

52

2

Imperfect EOQ System

Fig. 2.9 Behavior of the inventory level over time, under two scenarios: X < an and X > an (Moussawi-Haidar et al. 2013)

y  np. The cycle time becomes T1 ¼ (y  np)/D. Referring to Fig. 2.9, the average inventory per cycle for this case, denoted by AII, can be written as follows: 

2

ðy  npÞ n AII ¼ np þ x 2D

ð2:167Þ

Using Eq. (2.167) and the renewal–reward theorem, Theorem 3.6.1 of MoussawiHaidar et al. (2013), the average inventory per unit time for case I is: h AIUI ¼ AII =E ðT 1 Þ ¼

  i ðn=xÞnp þ ðy  npÞ2 =2D E ðT 1 Þ

ð2:168Þ

In Case II which occurs with probability 1  Pa, the entire lot is subject to 100% screening. The inventory behavior is shown by the dotted line in Fig. 2.9, with T2 being the cycle time and t2 the screening time. Screening the lot is completed at time t2 ¼ y/x, at which the inventory level drops by yp, the average number of lot defectives. The cycle time is T2 ¼ (y  yp)/D. Referring to Fig. 2.9, the average inventory per cycle for case II, denoted by AIII, can be written as:   ðy  ypÞ2 n AIII ¼ yp þ x 2D Using Eq. (2.169), in case II:

ð2:169Þ

2.3 EOQ Model with No Shortage

AIUII ¼ AIII =EðT 2 Þ ¼

53

h   i ðy=xÞyp þ ðy  ypÞ2 =2D E ðT 2 Þ

ð2:170Þ

And finally: E ½ICU ¼ Pa  AIUI þ ð1  Pa Þ  AIUII     3 2 3 ðn=xÞnp þ ðy  npÞ2 =2D ðy=xÞyp þ ðy  ypÞ2 =2D 5 þ ð1  Pa Þ4 5 ¼ Pa 4 E ðT 1 Þ E ðT 2 Þ 2

To compute the expected inventory ordering cost per unit time, they divided the ordering cost per cycle, K, by the expected cycle length, E[T], which is obtained as the expectation over E[T1] and E[T2], as follows: E½T  ¼ Pa  E ½T 1  þ ð1  Pa Þ  E½T 2  ¼ Pa ¼

ðy  npÞ y  yp þ ð1  Pa Þ D D

yð1  p þ pPa Þ  npPa D

ð2:171Þ

Finally, E[ICU], including the expected ordering, purchasing, and holding cost per unit time, is given as:   8 2 3 2 < ð n=x Þnp þ ð y  np Þ =2D K þ Cy 5 E½ICUðy, n, cÞ ¼ þ h Pa 4 E ðT Þ E ðT 1 Þ :   39 ðy=xÞyp þ ðy  ypÞ2 =2D = 5 þð1  Pa Þ4 E ðT 2 Þ ; 2

ð2:172Þ

The quality-related cost consists of the cost of accepting the lot caused by the non-inspected defective items that are encountered in an accepted lot and cost of inspection. If the lot is accepted, only the sample is inspected. If the lot is rejected, the entire lot is inspected. To compute the quality-related cost, QCU can be expressed as: E ½QCUðy, n, an Þ ¼ and

Cd ðy  nÞpPa þ C I nPa þ C I yð1  Pa Þ E ðT Þ

ð2:173Þ

54

2

Imperfect EOQ System

E ½TCUðy, n, an Þ ¼ E ½ICUðy, n, an Þ þ E ½QCUðy, n, an Þ   8 2 3 < ðn=xÞnp þ ðy  npÞ2 =2D K þ Cy 5 ¼ þ h Pa 4 E ðT Þ E ðT 1 Þ :   2 39 ðy=xÞyp þ ðy  ypÞ2 =2D = 5 þð1  Pa Þ4 E ðT 2 Þ ; þ

ð2:174Þ

Cd ðy  nÞpPa þ C I nPa þ C I yð1  Pa Þ E ðT Þ

Moussawi-Haidar et al. (2013) first expressed the expected revenue function. Then, they defined the integrated inventory-quality problem as an integer nonlinear program. So: ETRUðy, n, an Þ ¼

sðy  npÞPa þ sðy  ypÞð1  Pa Þ E ðT Þ

ð2:175Þ

and E ½TPUðy, n, an Þ ¼ E½TRUðy, n, an Þ  E½TCUðy, n, an Þ ½sðy  npÞPa þ sðy  ypÞð1  Pa ÞD ¼ yð1  p þ pPa Þ  npPa  2

ypDð1  Pa Þ ðy  npÞPa yð1  pÞð1  Pa Þ n pDPa h    2 2 xðy  npÞ x ð 1  pÞ yðC þ C I  C I Pa þ C d pPa ÞD þ ðK þ CI nPa  Cd npPa ÞD  yð1  p þ pPa Þ  npPa

ð2:176Þ

Proposition 2.1 For a given acceptance level an, there exist a lot size y and a sample size n for which the Hessian matrix of the expected profit function in Eq. (2.176) is not negatively semidefinite (Moussawi-Haidar et al. 2013). Proof See Moussawi-Haidar et al. (2013). Lemma 2.1 The profit function E[TRU(y, n, C)] given in Eq. (2.176) is concave in the order size y if the following condition holds: Cd < K/(npPa) + CI/p (MoussawiHaidar et al. 2013). Proof For proof and more details, see Moussawi-Haidar et al. (2013). Their objective was to maximize the expected profit per unit time in order to jointly determine the optimal y, n, and an using an integer nonlinear program with the following constraints:

2.3 EOQ Model with No Shortage

55

max E ½TPUðy, n, an Þ s.t. Pa

pð y  nÞ
AOQL y

quality of the outgoing material should be less than the AOQL an < n < y y, n, an  0,

Integers

ð2:177Þ

Moussawi-Haidar et al. (2013) developed the following numerical solution method: Step 1. Fix an, and solve the nonlinear programming problem using a nonlinear program search method such as gradient search. Then, determine the optimal sample size n(an) and lot size y(an) and the corresponding profit E[TPU(y(an), n (an))]. Step 2. Set an ¼ 1, 2, . . ., upper bound (e.g., upper bound ¼ 30). For each value of an, repeat Step 1. They noted that after some value of c sets the probability of acceptance to 1, the optimal solution will remain the same. Hence, they may not need to reach the upper bound. Step 3. Determine the optimal value of c by comparing the maximal expected profits E[TPU(y, n, an1 )], E[TPU(y, n, an2 )], . . ., E[TPU(y, n, anupper bound )]. The c value associated with the highest of this list of profits is the optimal value of the acceptance level, an, with associated optimal sample size n*, lot size y*, and optimal profit E[TPU(y*, n*, an)]. Example 2.9 Moussawi-Haidar et al. (2013) considered a situation with the following parameters: D ¼ 5000 units/year, K ¼ $100/cycle, h ¼ $5/unit/year, x ¼ 175,200 units/year, C ¼ $50/unit, CI ¼ $1/unit, Cd ¼ $25 per unit defective, s ¼ $70/unit, and p ¼ 0.04. The probability of acceptance, Pa ¼ P(X > n), is binomially distributed with parameters n and p and can be well approximated by a normal distribution when n is large or np > 5. In developing the numerical examples, they used the normal approximation of the binomial, with mean np and variance np (1  p), so PaN(np, np (1  p)). Table 2.13 presents numerical results illustrating the effect of varying CI, h, and Cd, one at a time, on the optimal solution, (y*, n*, an), and optimal profit, assuming that AOQL ¼ 2.5%.

56

2

Table 2.13 Effect of varying CI, h, and Cd, for AOQL ¼ 2.5% (MoussaviHaidar et al., 2013)

CI

h

Cd

2.3.6

0.5 1 1.5 2 2.5 5 10 15 20 25 5 10 25 40 60

an 16 15 13 12 11 12 9 8 8 8 9 10 12 13 15

n* 170.17 169.98 167.97 164.89 159.95 165.09 116.12 95.702 84.33 75.88 143.71 153.05 165.09 168.05 170.01

Imperfect EOQ System

y* 453.95 453.84 453.03 453.89 453.12 452.54 320.30 261.72 226.81 202.97 451.50 451.84 452.54 453.26 453.90

Optimal profit $89,832.2 $88,880.56 $87,928.5 $86,976.61 $86,024.56 $88,880.15 $87,953.60 $87,242.81 $86,643.76 $86,115.89 $91,417.40 $90,783.30 $88,880.15 $86,976.70 $84,438.77

Buy and Repair Options

In this section, a new model which is developed by Jaber et al. (2014) is presented. They developed an imperfect inventory system in which both repair and buy options for defective items are considered and compared. In order to model the presented problem, some new notations which are specifically used are shown in Table 2.14.

2.3.6.1

Model I: Repair Case of Jaber et al. (2014)

According to Fig. 2.10, the maximum level of inventory or lot size is y and period length is T. The lot is subjected to a 100% inspection at a rate X > D, where ρy units (ρ is a percentage) of imperfect quality are withdrawn from inventory at the end of the screening period, tI, and sent to a repair shop. Repaired items are returned after tR units of time, which includes transportation and repair times where tI + tR < T and T ¼ y/D. Note that this model does not consider the case when tI + tR > T as in our opinion it will logically be expensive favoring the buy option described in Case II as one has to account for backordering costs. They left this case as a mathematical exercise for interested readers. Further, assume that the repair process at the shop is always in control, which is not necessarily true. There are cases where the repair process may go out of control and restored through performing preventive maintenance (e.g., Jaber 2006; Liao and Sheu 2011, Liao 2012). Figure 2.10 illustrates the behavior of inventory for the first case of the repair option. So, to repair the ρy items, the repair shop encounters the following costs: total cost to the repair shop ¼ KR + 2Ks + ρy (C1 + 2CT + h2tR) where KR is the repair setup cost, Ks is the transportation fixed cost (it is assumed that the transportation of

2.3 EOQ Model with No Shortage

57

Table 2.14 Notations of a given problem E [.] m tR C1 KR h2 CE ρ tR hE CR

Expected value of a random variable Markup percentage by the repair store Transportation and repair time of imperfect products (time) Material and labor cost to repair an item ($/unit) Repay setup cost ($/setup) Holding cost at the repair facility ($/unit/unit time) Unit purchasing cost of an emergency order of a random variable ($/unit) Fraction of defective items ($/unit/unit time) Total transport time of defective units from the buyer to the repair shop and back to the buyer (time) Holding cost of emergency order ($/unit/unit time) Unit repair cost charged to the buyer ($/unit)

Fig. 2.10 Inventory level for the repair option for the defective items (Case I) (Jaber et al. 2014)

Inventory level y

py tI

py tR Time

T

repairable items is charged to the repair shop), C1 is the combined material and labor cost to repair an item, CT is the unit transportation cost (from the inventory system to the repair shop and back to the system), and h2 is the holding cost at the repair facility. Thus, each repaired item cost CR, which could be agreed upon as a unit cost times a markup percentage, is: 

K þ 2K S þ C 1 þ 2C T þ h2 t R C R ðyÞ ¼ ð1 þ mÞ R ρy

 ð2:177Þ

where m is the markup percentage by the repair shop, tR ¼ ρy/R + tT, tT is the total transport time of ρy units from the inventory system to the repair shop and back to the system, and R is the repair rate (where R > D), and these cost components are incurred by the repair shop. From this point onward, CR( y) will be used, which is paid by the firm for each repaired item. The holding cost per cycle is determined from Fig. 2.10 as:

58

2

 HC ¼ h

y 2 ð 1  ρÞ 2 y2 þρ 2D X



Imperfect EOQ System

 2    y y ρy y2 þ hR ρ  ρy þ þ t T  ρ2 ð2:178Þ X R D 2D

where hR is the holding cost of a repaired item. The cyclic total cost becomes: Fixed and purchasing cost

Inspection cost

Repair cost

zfflfflfflffl}|fflfflfflffl{ zfflfflfflffl}|fflfflfflffl{ z}|{ TCðyÞ ¼ K þ Cy þ CI y þ C R ðyÞρy  2   2    y ð 1  ρÞ 2 y2 y y ρy y2 þh þρ þ hR ρ  ρy þ þ t T  ρ2 X R 2D X D 2D |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð2:179Þ

Holding cost

Now, the total unit time profit is the total revenue per cycle less the total cost per cycle divided by the cycle time and is give as:   y ð 1  ρÞ 2 KD yD TPUðyÞ ¼ sD   CD  CI D  C R ðyÞρD  h þρ y X 2     y ρy y hR ρy  ρD þ þ t T  ρ2 X R 2 KD ¼ sD   CD y   ðK þ 2K S ÞD ρy  ð1 þ mÞ C 1 þ 2C T þ h2 þ h2 t T ρD CI D  ð1 þ mÞ R y R   2     y ð 1  ρÞ yD y ρy y  hR ρy  ρD þ þ t T  ρ2 þρ h X X R 2 2 ð2:180Þ whose solution is given by setting the first derivative of Eq. (2.180) to zero and solving for y to get: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ðK þ ð1 þ mÞðK R þ 2K S ÞÞD u   y ¼t    ρ2  2 yð1ρÞ2 yD h þ ρ X þ ð1 þ mÞh2 ρRD þ hR ρ  ρ DX þ ρD 2 R  2 

The expected value of Eq. (2.180) is given as:

ð2:181Þ

2.3 EOQ Model with No Shortage

KD þ ð1 þ mÞðK R þ 2K S ÞD  CD  CI D E½TPUðyÞ ¼ sD  y   E ½ρ2 y þ h2 t T E ½ρ D ð1 þ mÞ C1 E½ρ þ 2CT E ½ρ þ h2 R   2 yð1  2E½ρ þ E ½ρ Þ yD h þ E ½ ρ 2 X    y y y 2 hR E ½ρy  D E½ρ þ E ρ þ t T E ½ρ  E ρ2 X R 2

59

ð2:182Þ

whose solution is given in a similar manner to Eq. (2.181) as: y ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðK þ ð1 þ mÞðK R þ 2K S ÞÞD         2 h 1 þ E ½ρ2  þ E ½ρ DX  1 þ 2ð1 þ mÞh2 E½ρR D þ hR 2E½ρ 1  DX  E ½ρ2  2D R þ1

ð2:183Þ If the repaired items are received by time y(1  ρ) ¼ D as shown in Fig. 2.9, then Eq. (2.181) is rewritten as (Jaber et al. 2014):   yð1  ρÞ2 KD yD TPUðyÞ ¼ sD   CD  CI D  C R ðyÞρD  h þρ y X 2 y  hR ρ2 ð2:184Þ 2 The expected value of Eq. (2.184) is given as: ðK þ 2K S ÞD KD  CD  C I D  ð1 þ mÞ R y y   E ½ρ2 y D ð1 þ mÞ ðC 1 þ 2CT ÞE½ρ þ h2 R   y yð1  2E½ρ þ E½ρ2 Þ yD h þ E ½ ρ  hR E ρ2 2 X 2

E ½TPUðyÞ ¼ sD 

ð2:185Þ

whose solution is given in a similar manner to Eq. (2.181) as (Jaber et al. 2014): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2ðK þ ð1 þ mÞðK R þ 2K S ÞÞD u   y ¼t  ð2:186Þ   2 2 h ð1 þ E ½ρ Þ þ 2E ½ρ DX  1 þ 2ð1 þ mÞ h2 E½ρR D þ hR E ½ρ2  

60

2

2.3.6.2

Imperfect EOQ System

Model II: Buy Case of Jaber et al. (2014)

In the buy scenario, like in Salameh and Jaber (2000), the imperfect-quality items are withdrawn from inventory by the end of the inspection/screening period and are sold (salvaged) as a single batch for v. The imperfect-quality items are substituted with an emergency order at a cost of CE each, where CE < C < v. The behavior of inventory for this model is the same as shown in Fig. 2.11 (Jaber et al. 2014). The expected unit time profit is similar to Eq. (2.182), which is: KD  CD  C I D y   y yð1  2E ½ρ þ E ½ρ2 Þ yD þ E ½ ρ  hE E ρ2 CE E½ρD  h 2 X 2 E ½TPUðyÞ ¼ sD þ vE½ρD 

ð2:187Þ

Finally: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD   y ¼ h 1  2E½ρ þ E ½ρ2  þ 2E½ρ DX þ hE E½ρ2  

ð2:188Þ

The behaviors of Models I and II are illustrated numerically in the next section. Example 2.10 In this section, the input parameters of the numerical examples are adopted from Salameh and Jaber (2000) and Jaber et al. (2014) as provided below in Table 2.15. Using data:

Fig. 2.11 Inventory level for the repair option for the defective items (Case II) (Jaber et al. 2014)

Inventory level y

py tI

py tR T

Time

2.3 EOQ Model with No Shortage

61

Table 2.15 Data for the numerical analysis from Salameh and Jaber (2000) (Jaber et al. 2014) Symbol D X s K C v h CI ρ f(ρ) hE

Value 50,000 175,200 50 100 25 20 5 0.5 U(0, 0.04) 1/(0.04–0) 8

Units Units/year Units/year $/unit $ $/unit $/unit $/unit/year $/unit

Zb ρf ðρÞdρ ¼

Zb

ρ

Zb ρ f ðρÞdρ ¼

ρ2

2

a

Units $ $ $/unit $/unit $/unit/year Units/year Year $/unit/year $/unit

1 b þ a 0:04 þ 0 dρ ¼ ¼ ¼ 0:02 ba 2 2

a

b

E ρ2 ¼

Value 100 200 2 5 4 50,000 2/220 6 20% 40

$/unit/year

Zb E ½ρ ¼

Symbol KR KS CT C1 h2 R tT hR m CE

1 a2 þ ab þ b2 0 þ 0 þ ð0:04Þ dρ ¼ ¼ ba 3 3

2

a

¼ 5:33E  04: The optimal policy for Model I Case I occurs when y* ¼ 3732 and E[TPU (y*)] ¼ 1,195,455, where CR ¼ 18.89. Solving for Case II when repaired items are received by time y(1  p)/D, the optimal policy occurs when y ¼ 3792 and E[TPU ( y)] ¼ 1,195,743, where CR ¼ 18.72. It should be noted that CR can be computed from Eq. (2.177) by substituting ρ ¼ E[ρ] (i.e., E[CRρy]/E[ρy] ¼ yE[CRρ]/yE[ρ] ¼ E [CRρ]/[ρ]). The optimal policy for Model II occurs when y* ¼ 1434 and E[TPU (y*)] ¼ 1,198,026 (Jaber et al. 2014).

2.3.7

Entropy EOQ

2.3.7.1

Model 1: Entropy EOQ Without Screening

In this section, the work of Jaber et al. (2009) in which entropy cost for EOQ model is considered will be presented. Jaber et al. (2009) analogized a production system with physical thermodynamic system. They expressed that a production system resembles a physical system operating within surroundings, including the market and the supply system. Similarly, a physical thermodynamic system is defined by its temperature, volume, pressure, and chemical composition. A production system could be described analogously by its characteristics, for example, the price (s)

62

2

Imperfect EOQ System

Table 2.16 Notations of a given problem s so ρ θ h1 CE ye TCE( y)

The reduced selling price per unit ($/unit) The higher selling price per unit ($/unit) The Markov transition probability of production process (machine) shifts to an out-ofcontrol state and the production process begins to produce The expected number of defective in a lot (unit) The holding cost adjusted for the average additional cost of reworking defective items h1 > h ($/unit/unit time) Entropy cost per unit ($/unit) Optimal lost size under entropy cost (unit) The total cost of EOQ model under entropy cost ($)

that the system ascribes to the commodity (or collection of commodities) that it produces. Reducing the price of the commodity below the market price may increase demand and produce a commodity flow (sales) from the system to its surroundings. This is similar to the flow of heat from a high temperature (source) to a low temperature (sink) in a thermodynamic system, where part of this heat is converted into useful work and some of the heat is lost from the system and wasted. So in the work of Jaber et al. (2009), a similar behavior between price in supply system and temperature in thermodynamic system presented and used a demand function D ¼ W(s  so) to investigate the effects of price changes such as temperature changes. Noting that when s  so (analogues to difference in temperature in a thermal system), the direction of the commodity (heat) flow is from the system to its surrounding, i.e., from a high-temperature reservoir (low price) to a low-temperature reservoir (high price). The second law of thermodynamics applies to the spontaneous flow of heat from hot (low price) to cold (higher price). Heat flow ceases when the system and its surrounding have the same temperature (Jaber et al. 2009). The new notations used in this section are presented in Table 2.16. The EOQ model assumes constant commodity flow, an infinite planning horizon, no shortages (lost sales or backorders are not allowed), instantaneous replenishment of orders, zero lead time, no quantity discounts, no deterioration, and all units conform to quality and that the input cost parameters are constant over time. The EOQ model adopts the classical approach of minimizing the sum of the conflicting holding and procurement costs. The EOQ cost function is given as (Jaber et al. 2009): TCðyÞ ¼

KW ðs  s0 Þ KD y y  CW ðs  s0 Þ þ h þ CD þ h ¼  y y 2 2

ð2:189Þ

The optimal solution is obtained by setting the first derivative to zero to get (Jaber et al. 2009):

2.3 EOQ Model with No Shortage

63

rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KW ðs  s0 Þ 2KD y ¼ ¼ h h 

ð2:190Þ

where the optimal cycle time is T ¼ y/D ¼ y/W(s  s0). Excluding the material cost CW(s  s0), the optimal cost could be computed as (Jaber et al. 2009): TCðy Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hKW ðs  s0 Þ

ð2:191Þ

Porteus (1986) denoted θ as the expected number of defectives in a lot of size y, given that the process is in control before beginning the lot. Then (Jaber et al. 2009):

¼ ρy

n1 X i¼1

θ ¼ ρy þ βdQ1 , where ðβ ¼ 1  ρÞ

n1 X β 1 þ ðy  1Þβy  Qβy1 ð1  β y Þ i i ρ β ρ iβ ¼ ρy ρ ρ2 i¼1 ¼yβ

ð1  β y Þ ρ

where θ could also be computed as θ ¼

y P

ρðy  iÞð1  ρÞi1 . Porteus (1986)

i¼1

suggested that for very small ρ values, θ  ρy2/2. This approximation has been adopted by several researchers (e.g., Chand 1989; Jaber 2006). The cost function (2.189) is then modified as follows (Jaber et al. 2009): KD y y þ CD þ h1 þ C R ρD y 2 2 KW ðs  s0 Þ y y  CW ðs  s0 Þ þ h1  C R ρW ðs  s0 Þ ¼ y 2 2 TCðyÞ ¼

ð2:192Þ

The holding cost for a single unit h1 ¼ iC + iρCR/2 ¼ h + hρCR/2C, where i ¼ h/ C. Equation (2.192) can now be written as (Jaber et al. 2009):   KD C y y þ CD þ h 1 þ ρ R þ CR ρD y 2 2C 2   KW ðs  s0 Þ C y y  CW ðs  s0 Þ þ h 1 þ ρ R ¼  CR ρW ðs  s0 Þ y 2 2C 2 TCðyÞ ¼

ð2:193Þ

The optimal solution is given by setting the first derivative of Eq. (2.193) to zero

64

2

y



Imperfect EOQ System

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD ¼ hð1 þ ρðC R =2CÞÞ þ ρCR D sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KW ðs  s0 Þ ¼ hð1 þ ρðC R =2CÞÞ þ ρCR W ðs  s0 Þ

ð2:194Þ

When ρ ¼ 0 Eq. (2.194) reduces to Eq. (2.192). Substituting Eq. (2.194) in Eq. (2.193), and excluding the material cost, the optimal cost could be computed as: rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i   C TCðy Þ ¼ h 1 þ ρ R þ ρC R W ðs  s0 Þ  ½2KW ðs  s0 Þ 2C 

ð2:195Þ

Considering Entropy Cost Jaber et al. (2009) used the following equation as entropy cost per unit: CE ¼ 

s0 s ðs  s0 Þ

ð2:196Þ

Accounting for entropy cost is done by adding the entropy cost per unit time, CE/ T where T ¼ y/D, to the other cost terms and substituting D ¼ E(s  s0) in Eq. (2.193):   KD C y y s sW þ CD þ h 1 þ ρ R þ C R ρD þ 0 y 2 y 2C 2   KW ðs  s0 Þ CR y y s sW  CW ðs  s0 Þ þ h 1 þ ρ ¼  CR ρW ðs  s0 Þ þ 0 y 2 y 2C 2 ð2:197Þ TCE ðyÞ ¼

The optimal solution is given by setting the first derivative of Eq. (2.197) to zero to get (Jaber et al. 2009): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD þ 2s0 sW hð1 þ ρðC R =2C ÞÞ þ ρC R D sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KW ðs0  sÞ þ 2s0 sW ¼ hð1 þ ρðC R =2C ÞÞ þ ρC R W ðs  s0 Þ

ye  ¼

ð2:198Þ

Substituting Eq. (2.198) in Eq. (2.197), and excluding the material cost, the optimal cost could be computed as (Jaber et al. 2009):

2.3 EOQ Model with No Shortage

  TCE ye ¼

65

rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i   CR þ ρC R W ðs0  sÞ  ½2KW ðs  s0 Þ þ 2s0 sW  h 1þρ 2C

When ρ ¼ 0, Eq. (2.198) reduces to (Jaber et al. 2009): ye

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KW ðs0  sÞ þ 2s0 sW 2KD þ 2s0 sW ¼ ¼ h h

ð2:199Þ

which is the entropic version of Eq. (2.190) whose cost function reduces from Eq. (2.197) to (Jaber et al. 2009): b E ðyÞ ¼ KD þ CD þ h y þ s0 sW C y 2 y ¼

KW ðs0  sÞ y s sW  CW ðs0  sÞ þ h þ 0 y 2 y

ð2:200Þ

Similarly, substituting Eq. (2.200) in Eq. (2.199), and excluding the material cost, the optimal be computed as (Jaber ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi et al. 2009):   costpcould TCE ye ¼ h½2KW ðs0  sÞ þ 2s0 sW  Example 2.11 Jaber et al. (2009) considered an inventory situation where the product price s ¼ 100, the market equilibrium price s0 ¼ 105, and W ¼ 20 represents the change in the flux for a change in the price of a commodity (e.g., units/day/$), corresponding to a commodity flow (demand) rate D ¼ W(sos) ¼ 20(105100) ¼ 100. Also h ¼ 0.10, C ¼ 60, CR ¼ 30, K ¼ 100, and ρ ¼ 0.0001. In the numerical examples below, the term CD ¼ CW(ss0) is excluded from the cost functions because it actually dominates the cost function (CD ¼ 60(100) ¼ 6000) although it does not affect the optimization. In the case when there are no defects, y ¼ 447.21 and TC( y) ¼ 44.72. When the entropy cost is included in the cost function, the optimal  entropic order quantity using Eq. (2.199) ye ¼ 2097.62, which minimizes, TCE ye ¼ 209.76, is obtained from Eq. (2.200). For a more detailed information about the model, readers can see Jaber et al. (2009).

2.3.7.2

Model 2: Entropy EOQ with Screening

Jaber et al. (2013) revisited their previous work (Jaber et al. 2009) by adding a screening topic. In this section, the work of Jaber et al. (2013) is presented. The behavior of inventory in this problem is depicted in Fig. 2.12 (Jaber et al. 2013). Upon replenishing the inventory instantaneously, the received lot is subjected to a 100% screening process. Items that do not confirm to quality are withdrawn from the inventory and sold at a discounted price as a single batch.

66

2

Imperfect EOQ System

Fig. 2.12 Salameh and Jaber’s model (Jaber et al. 2009)

Inventory level

y

py

Time

T

Table 2.17 Notations of a given problem s0 g0 g D(s, g) W Cinv

Is the selling price of a good-quality item of a competitor brand in the market ($/unit), where s0 > s ($/unit) Is the quality index of an item of a competitor brand (1/unit), where 0 < g0 < 1 Is the quality index of an item (1/unit), where 0 < g < 1 Demand rate (unit/year) as a function of s and g, where s/g > s0/g0 Elasticity (unit/year/$) of the demand function D(s, g) Is the investment required to increase quality index g by 1% ($/1%/year)

In order to model the presented problem, some new notations which are specifically used are shown in Table 2.17. The total revenue per cycle is the sum of the revenue from selling good-quality items Py(1  ρ) and the revenue from salvaging imperfect-quality items, vyρ, and is given as (Jaber et al. 2013): TRðyÞ ¼ syð1  ρÞ þ vyρ The total cost per cycle is (Jaber et al. 2013):  TCðyÞ ¼ K þ Cy þ C I y þ h

y ð 1  ρÞ ρy2 Tþ 2 x

Therefore, the profit function per unit of time is:



2.3 EOQ Model with No Shortage

67

TRðyÞ  TCðyÞ T D ¼ y ð 1  ρÞ   

y ð 1  ρÞ ρy2 Tþ  syð1  ρÞ þ vyρ  K  Cy  CI y  h 2 x

TPUðyÞ ¼

Then: TPUðyÞ ¼ Dðs  v þ hy=xÞ þ Dðv  hy=x  C  C I  K=yÞ h

1 1ρ

y ð 1  ρÞ 2

ð2:201Þ

whose expected value is:  ETPUðyÞ ¼ Dðs  v þ hy=xÞ þ Dðv  hy=x  C  C I  K=yÞE h

y ð 1  E ½ ρ Þ 2

1 1ρ



ð2:202Þ

The optimal solution is given as (Jaber et al. 2013): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KDE ½1=ð1  ρÞ y ¼ hð1  E½ρ  2ðd=xÞÞð1  E ½1=ð1  ρÞÞ 

ð2:203Þ

But in order to develop the thermodynamic version of the proposed model, Jaber et al. (2013) applied the per unit time profit function and the optimal lot size of Maddah and Jaber as below: E ½TRðyÞ  E ½TCðyÞ E ½TPðyÞ ¼ E ½T  E ½T  h i 8 9 2 < = E ð 1  ρ Þ KD D þ hy E ½ρ þ hy Dfsð1  E ½ρÞ þ vE ½ρ  C  CI g  2 x : y ;

E½TPUðyÞ ¼

¼

1  E ½ ρ ð2:204Þ

68

2

Imperfect EOQ System

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD i y ¼u t h hE ð1  ρÞ2 þ 2hE½ρD=x 

ð2:205Þ

Jaber et al. (2013) applied the laws of thermodynamics to inventory systems, where they postulated that commodity flow (demand) is similar to energy flow in thermal systems as presented in the previous model. They proposed commodity flow to be of the form: Dðt Þ ¼ W ðsðt Þ  s0 ðt ÞÞ

ð2:206Þ

where s(t) and s0(t) are analogous, respectively, to the system temperature TH (high) and the surrounding temperature TL (low) and W (analogous to a thermal capacity) is the change in the flux for a change in the price (s(t)  s0(t)) of a commodity and is measured in additional units per year per change in unit price, e.g., dollar (units/year/ $). The difference between the temperature of a thermodynamic system and that of its surrounding (TH  TL > 0) creates a flow from the thermodynamic system to its surrounding (Jaber et al. 2013). Jaber et al. (2013) assumed that the flow of commodity is one dimensional (price). The quality of the product was implicitly assumed to be the same as that of the competitor’s brand in the market and was set at 1 (dimensionless measure between 0 and 1). They proposed a commodity flow function of the form: 

sðt Þ s0 ðt Þ Dðt Þ ¼ W ðV ðt Þ  V 0 ðt ÞÞ ¼ W  gð t Þ g0 ð t Þ

 ð2:207Þ

where 0 < g(t) < 1 and 0 < g0(t) < 1 are the goodness (quality) measures of the firm’s product and that of its competitor brand, respectively, where V(t) and V0(t) are the values as perceived by the customer. The smaller V(t) is with respect to V0(t), the better it is for the customer as it is getting a higher-quality product for less. So, substituting g(t) ¼ 1 and g0(t) ¼ 1 in Eq. (2.207) reduces it to Eq. (2.206), making it a special case of the latter. For commodity flow gsððttÞÞ < gs0 ððttÞÞ for every t > 0. The 0 entropy generated per cycle of length T is determined from Eq. (2.207) as (Jaber et al. 2013): ZT 

 V ðt Þ V 0 ðt Þ þ  2 dt V 0 ðt Þ V ðt Þ

σ ðT Þ ¼

ð2:208Þ

0

Since the demand is considered as the constant rate, so assuming Eqs. (2.206)– (2.207) for constant situation, V(t) ¼ V and V0(t) ¼ V0 for every t > 0, Eq. (2.206) reduces to: D(t) ¼ D(s, g) ¼  W(V  V0) ¼  W(s/g  s0/g0) (Jaber et al. 2013). The entropy cost per cycle is determined from Eqs. (2.207) and (2.208) as (Jaber et al. 2013):

2.3 EOQ Model with No Shortage

RT En ¼ EnðT Þ ¼

0

W

69

W ðs=g  s0 =g0 Þdt

RT  0

s=g s0 =g0

ss0  ¼  sg0  s0 g =g0 þ s0s=g  2 dt

ð2:209Þ

where g0s < s0g ) g0/g < s0/s, s < s0g/g0. Note that improving the quality of a product is usually associated with investment, and the function is cinv(g  g0)/g0, where cinv is the investment required to increase the quality by 1%. In this work, the salvage price is a decision variable as the imperfect-quality items may as well follow a commodity flow function like the one in Eq. (2.207) and is given as (Jaber et al. 2013): 

 v C yρ ¼ WT  gs g0   W v C  y¼ T ρ gs g0

ð2:210Þ ð2:211Þ

where yρ is the demand rate that occurs once in a cycle at y/x units of time and gs is the quality of a salvaged item. The entropy cost for the flow of imperfect items is computed in a similar manner to Eq. (2.209) and is given as (Jaber et al. 2013):  σ s ðT Þ ¼ WT

v=gs C=g0 þ 2 C=g0 v=gs

 ð2:212Þ

The entropy cost per cycle to manage the flow of imperfect-quality items is determined from Eqs. (2.210) and (2.212) in a similar manner to determining Eq. (2.209) as (Jaber et al. 2013): Ens ¼ Ens ðT Þ ¼

ρy vC where v < Cgs =g0 ¼ vg σ s ðT Þ 0  Cgs

ð2:213Þ

In this section, they modified the model in Eq. (2.204) by replacing the constant D with D(s,g) and subtracting the entropy cost per unit of time expressions and the investment cost from Eq. (2.204), which becomes (Jaber et al. 2013): i 0 h 1

E ð1  ρÞ2 E ½ ρ D ð s,g Þ K A þ Dðs,gÞ sð1  E½ρÞ þ vE ½ρ   C  C I  hy@ y 2 x 

E½TPUE ðyÞ ¼

1  E ½ρ    Dðs,gÞ c ðg  g0 Þ ss0 vC þ  inv þ g0 ð1  E½ρÞy ðsg0  s0 gÞ ðvg0  Cgs Þ

ð2:214Þ The optimal solution that maximizes is given as (Jaber et al. 2013):

70

2

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u2Dðs, gÞ K  ss0  vC ð sg s g Þ ð Cg Þ vg u 0 0 s h i0 yE ¼ t Dðs, gÞ 2 hE ð1  ρÞ þ 2h x E ½ρ

Imperfect EOQ System

ð2:215Þ

Equation (2.214) can be optimized for v and s solving a nonlinear programming model (NLPP) as (Jaber et al. 2013): Maximize E ½TPUE ðs, vÞ, s:t: s > C,

ð2:216Þ

sg0 < s0 g, 0 < v < Cgs =g0 The proposed NLPP model can be solved by any suitable optimization package which can be used.

2.4

EOQ Model with Backordering

In this section, the imperfect EOQ model with backordering will be presented.

2.4.1

Imperfect Quality and Inspection

In this section, an EOQ model with imperfect items with backordering according to the work of Eroglu and Ozdemir (2007) is presented. In order to model the presented problem, some new notations which are specifically used are shown in Table 2.18. Assume that y is replenished instantaneously at unit purchasing price of c and fixed cost of K per order similar to the previous models. The percentage of defectives in each lot is p, with a known probability density function, f( p). A 100% screening policy with screening rate per unit time of x is applied. (1  θ)% of the defective items is imperfect and θ% is scrap items. When screening process is finished, imperfect-quality items are sold as a single lot, and scrap items are subtracted from inventory with unit cost of Cd. The selling prices of good- and imperfect-quality items are s and v per unit, respectively, where s < v. Figure 2.13 presents the behavior of the inventory level. The rate of good-quality items which are screened during t2 is (1  p). A part of these good-quality items meet the demand with a rate of D, and the remaining is used to eliminate backorders with a rate of (1  p) x  D ¼ x(1  p  D/x) (Eroglu and Ozdemir 2007). The screening process finishes up at the end of time interval of t3, and defective items of py are subtracted from inventory. Since the demand has been met from

2.4 EOQ Model with Backordering Table 2.18 Notations of a given problem

Fig. 2.13 Behavior of the inventory level over time (Eroglu and Ozdemir 2007)

71 Time to build up a backorder level of “w” units (time) Time to eliminate the backorder level of w units (time) Time to screen y units ordered per cycle (time) The backorder level (unit)

t1 t2 t3 B

Inventory Level

)x -p (1

y

D

z z1+py

py

z1 xA

D

D

B

t1

t2

Time t3 t

perfect-quality items, the period length t is calculated by dividing the amount of perfect-quality items in a period to amount of demand in unit time: t¼

ð1  pÞy D

ð2:217Þ

Since the percentage of defective items, p, is a random variable, the expected value of period length is given by: E ½t  ¼

ð1  E½pÞy D

ð2:218Þ

Referring to Fig. 2.13, the findings are as follows. The time, t1, needed to build up a backorder level of “B” units is: t1 ¼

B D

ð2:219Þ

The time, t2, needed to eliminate the backorder level of “B” units is: t2 ¼

B x½1  p  D=x

ð2:220Þ

72

2

t2 ¼

Imperfect EOQ System

yz ð1  pÞx

ð2:221Þ

Using Eqs. (2.229) and (2.221): z¼y

ð1  pÞB ½1  p  D=x

ð2:222Þ

According to Fig. 2.13: t 3 ¼ y=x

ð2:223Þ

t 3  t 2 ¼ ðz  z1  pyÞ=D

ð2:224Þ

z1 ¼ ½1  p  D=xy  B

ð2:225Þ

And:

Finally using Eqs. (2.217)–(2.225), the total cost function will be (Eroglu and Ozdemir 2007): Shortage cost

zfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflffl{  zfflfflfflfflffl}|fflfflfflfflffl{ zffl}|ffl{ zfflfflfflffl}|fflfflfflffl{ C b ðt 1 þ t 2 ÞB TC ¼ ðCy þ K Þ þ ðC I yÞ þ ðC d θpyÞ þ 2   

t ðy þ zÞ ðt 3  t 2 Þðz þ z1 þ pyÞ ðt  t 1  t 3 Þz1 þ þ þ h 2 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Purchasing cost

Screening cost

Disposal cost

ð2:226Þ

Holding cost

  2  D=x ð1  p  D=xÞ2 2 h ¼ ðC þ C I þ C d θpÞy þ K þ  y þ 2 D x 

hð1  pÞBy ðh þ C b Þð1  pÞB2 þ D 2Dð1  p  D=xÞ

Moreover, the total revenue, TR, of both imperfect and good items is: TR ¼ sð1  pÞy þ υð1  θÞpy

ð2:227Þ

Since cycle length is a variable, using the renewal reward theorem, the expected total profit per unit time is given as:

2.4 EOQ Model with Backordering

73

E ðTRÞ  E ðTCÞ E ðt Þ υDð1  θÞEðpÞ DðC þ CI þ Cd θpÞ KD hE 4 y ðh þ C b ÞE2 B2    þ hB  ¼ sD þ E1 E1 yE 1 2E1 2yE 1 ð2:228Þ

E ½TPU ¼

where   Þ 1p E1 ¼ 1  E( p), E2 ¼ E 1pD=x þ E 3. , E3 ¼ E[(1  p  D/x)2], E 4 ¼ Dð2D=x x Since the expected total profit is strictly concave, setting its partial derivatives with respect to B and y gives: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD  y ¼u t  hE 2 h E 4  ðhþCb1ÞE2 

B ¼

hE 1 y ðh þ C b ÞE 2

ð2:229Þ

ð2:230Þ

But it should be noted that the derived optimal values are valid if both next conditions hold: 1. In order to avoid the backorders from the beginning of each cycle: xE ð1  p  D=xÞ > 0 or EðpÞ < 1  D=x

ð2:231Þ

x>D

ð2:232Þ

and

2. t3, must be at least equal or greater than E(t2). So: E ðt 2 Þ
t 3 or Eð1  p  D=xÞE 2 h
h þ Cb 1  E ð pÞ

ð2:233Þ

If shortage cost is infinite, and scrap rate and unit scrap cost are zero, then the model with no shortages is attained. Thus, the following reduced forms of Eqs. (2.228)–(2.230) are achieved (Eroglu and Ozdemir 2007):

74

2

E ½TPU ¼ sD þ

Imperfect EOQ System

υDE ðpÞ DðC þ CI Þ KD hE4 y    E1 E1 yE1 2E 1 rffiffiffiffiffiffiffiffiffiffi 2KD  y ¼ hE 4 B ¼ 0

ð2:234Þ ð2:235Þ ð2:236Þ

Suppose that defective’s fraction, p, is zero. This yields t3 and CI, are zero and x, is infinite. So, E1 ¼ E2 ¼ E3 ¼ E4 ¼ 1 and Eqs. (2.228)–(2.230) are reduced to the following equations which are the same equations as those given by classical EOQ model with shortages: ðh þ C b ÞB2 KD hy  þ hB  y 2 2y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KDðh þ Cb Þ y ¼ hC b

TPU ¼ ðs  C ÞD 

B ¼

hy ðh þ C b Þ

ð2:237Þ ð2:238Þ ð2:239Þ

Example 2.12 Eroglu and Ozdemir (2007) presented a firm that orders a product as lots to meet outside demand. The defective fraction in each lot has a uniform distribution with the following probability density function:  10, 0
p
0:1 f ð pÞ ¼ 0, otherwise And the other model parameters are given as follows D ¼ 15,000 units/year, K ¼ $400/cycle, h ¼ $4 unit/year, Cb ¼ 6, x ¼ 60,000 unit/year, CI ¼ $1/unit, C ¼ $35/unit, s ¼ $60/unit, v ¼ $25/unit, θ ¼ 0.2, and Cd ¼ $2/unit. By using the above parameters, E( p) ¼ 0.05, E1 ¼ 0.95, E2 ¼ 1.357752, E3 ¼ 0.490833, and E4 ¼ 0.928333, the optimum values of solution are calculated as y* ¼ 2128.06 units, B* ¼ 595.59 units, and E[TPU]* ¼ $341,116.89. In addition, Eqs. (2.231)–(2.233) are valid herein.

2.4.2

Multiple Quality Characteristic Screening

Assume an imperfect single-item inventory system with constant demand rate and permitted backordering in which the lead time is negligible. Suppose that after the replenishment of items arrives from the supplier, the items have to go through n screening processes on each quality characteristic before delivering to customers. Denote by Si the screening process for quality characteristic i and by xi the corresponding screening rate of Si, where i ¼ 1, . . ., n. Without loss of generality,

2.4 EOQ Model with Backordering

75

they supposed x1  x2  ⋯  xn. The processes are performed in parallel and are completed when the last process is finished. Screening rate is greater than the demand rate, i.e., xi > D (i ¼ 1, . . ., n), and defective items exist in lot size y and are returned to the supplier when replenishment items arrive (Tai 2015). In order to model the presented problem, some new notations which are specifically used are shown in Table 2.19. Let pi be the proportion of items in the batch that cannot pass Si. Here they assumed that all pi is independent of each other in the sense that whether an item can pass Si does not depend on the result of other screening processes. For any i < j, the screening process Si will finish before Sj. The first screening process S1 starts at time 0 and finishes at time y/x1. During this period, p1y items cannot pass S1, and they are removed from the inventory at time y/x1. To save the transportation cost and administrative work, the items which cannot pass the screening process are removed from the inventory as one lot at the end of each screening process. The next screening process S2 finishes at time y/x2  y/x1 since x2
x1. The number of items which cannot pass S2 is p2y. The items to be screened out are items which pass S1 but not S2. Hence, the number of items screened out by S2 is p2(1  p1) y. By similar arguments, the items to be screened out by Si are items which pass S1, . . ., Si  1 but not Si. Because all screening processes are independent, the proportion of items in y that is screened out after Si, which is denoted by ρi, is: 8 for i ¼ 1 < p1 i 1 ρi ¼ Q : ð1  pk Þpi for i ¼ 2, . . . , n k¼1

P P The total number and portion of defective are ni¼1 ρi y and ρ ¼ ni¼1 ρi , respectively (Tai 2015). The inventory level in a replenishment cycle is illustrated in Fig. 2.14. At the beginning of each cycle, all screening processes proceed simultaneously. The rate of items to complete all the screening processes depends only on the lowest screening Table 2.19 Notations of a given problem n Si ρi t1 t2 t3 t4 TR( y) TC (y, B)

Number of quality characteristics of the product to be inspected Screening process for quality characteristic i (i ¼ 1, . . ., n) Random variable representing the proportion of items in y that screen out after Si (i ¼ 1, . . ., n) The length of time to make up backorders in a replenishment cycle (time) The length of time of positive inventory level in a replenishment cycle (time) The length of time to backorder shortage in a replenishment cycle (time) The length of time of no backorder (time) Cyclic total revenue ($) Cyclic total cost ($)

76

2

Imperfect EOQ System

y

ρ1y ρ2y

y x2 y x1

ρn y

t3 time

B t1

t4 t2 T

Fig. 2.14 The inventory level in a replenishment cycle inventory level (Tai 2015)

process rate xn. The inventory level decreases at a rate of (1  ρ)xn, which is the rate of non-defective items that completed all the screening processes. After completing all the screening processes, the items are first shipped to satisfy the demands. The remaining items are then used to clear the outstanding backorders. Hence, the backordering quantity decreases at a rate of (1  ρ)xn  D (see the left colored region in Fig. 2.14). Here Tai (2015) assumed that: prob ½ð1  ρÞxn  D > 0 ¼ 1

ð2:240Þ

The backorders are made up in time period t1. Then, the inventory level decreases at a rate of D and reaches 0 at the end of time period t2. Shortage is backordered in time period t3 until the end of the replenishment cycle (see the right colored region in Fig. 2.14). The screening time for Si is y/xi. After the screening process Si, the defective items (ρiy units) are transferred to another inventory warehouse. They remarked that consolidating the defective items is a common policy in inventory management. The inventory level of defective items is illustrated in Fig. 2.15. The number of items delivered to customers in a replenishment cycle is (1  ρ) y. In which (1  ρ)y  B of them are used to satisfy the demands in the cycle. The remaining B items are used to make up the shortage in the last replenishment cycle. The same number of demands is backordered to the next cycle. So using Figs. 2.14 and 2.15 (Tai 2015):

defective items level

2.4 EOQ Model with Backordering

77

ρn y ρ2 y ρ1 y y y x1 x2

y xn

T

time

Fig. 2.15 The inventory level of defective items inventory level (Tai 2015)

ð1  ρÞy D B t1 ¼ ð1  ρÞxn  D T¼

ð1  ρÞy  B D B t3 ¼ D

t2 ¼ T  t1 ¼

t4 ¼ t2  t1 ¼

ð1  ρÞy  B B  D ð1  ρÞxn  D

TR( y) is the sum of the total sales of good-quality items and the amount received from the supplier for the return of the imperfect-quality items. One has (Tai 2015): TRðyÞ ¼ ð1  ρÞys þ ρyv TC(y, B) consists of purchasing, fixed, screening, holding, andPshortage costs. n The fixed, and screening costs are Cy, K, and i¼1 ðC Ii  yÞ ¼ Pn purchasing,  C respectively. The perfect items’ holding cost is: y I , i¼1 i

78

2

"

n t 2 D t 2 ð1  ρÞxn X y h 4 þ 1 ρi y þ t 1 t 4 D þ x 2 2 i i¼1

Imperfect EOQ System

#

And after some simplifications, the holding cost becomes: "

X ðð1  ρÞy  BÞ2 B2 y ρy þ h þ xi 2D 2ðð1  ρÞxn  DÞ i¼1 i n

#

The defective items’ holding cost is: " h1

n X i¼1

"  #  # n X ð 1  ρÞ 1 y ¼ h1 y2  ρi y T  ρi D xi x i i¼1 " !# n ρð 1  ρÞ X ρi  ¼ h1 y2 D x i i¼1

The penalty cost for shortage is:  Cb

   t1  B t3  B 1 1 þ ¼ Cb þ B2 2 2 2D 2ðð1  ρÞxn  DÞ   1ρ ¼ Cb B2 2Dðð1  ρÞ  D=xn Þ

For simplicity, let: h i P1 ¼ E ð1  ρÞ2 ,

P2 ¼ E½ρð1  ρÞ,

P3 ¼ 1E1 ½ρ ,

E ½ρ P4 ¼ 1E ½ρ

,

R¼

h 1E½ρ i. Let also. E

1ρ ð1ρÞD=xn

P P A1 ¼ ni¼1 Ex½ρi i  which depends on ρi, xi (i ¼ 1, . . ., n) and A2 ¼ ni¼1 C Ii which depends on CIi (i ¼ 1, . . ., n), since R > 0. Then the cyclic expected net profit is (Tai 2015): E ½TPðy, BÞ ¼ ð1  E ½ρÞys þ E½ρyυ  3 2 P1 2 1  E ½ρ B2 2 Cy þ A2 y þ h By þ y  þ A1 y 6 7 ð2:241Þ D 2D 2RP3 D 7 6 4 5 h i P2 Cb 2 2 þh1  A1 y þ B þK D 2RP3 D And the expected cycle length (Tai 2015) is:

2.4 EOQ Model with Backordering

79

E ½T  ¼

ð1  E ½ρÞy D

ð2:242Þ

Using renewal reward theorem:  E½TPðy, BÞ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ðh þ C b ÞB2 E ½TPUðy, BÞ ¼  hB ¼sD þ υDP4  ðC þ A2 ÞDP3  2Ry E ½T   

hP1 KDP3 þ þ hA1 D þ h1 ðP2  A1 DÞ P3 y þ 2 y Independent of B & y

ð2:243Þ The goal is to maximize the expected total profit using optimal decision variables y and B. According to Eq. (2.243), maximizing Eq. (2.243) can be reduced to minimizing f(y, B) as below (Tai 2015): f ðy, BÞ ¼

  ðh þ C b ÞB2 hP1  hB þ þ hA1 D þ h1 ðP2  A1 DÞ P3 y 2Ry 2 þ

KDP3 y

ð2:244Þ

Differentiating f(y, B) in Eq. (2.244) with respect to B and y, respectively, gives (Tai 2015): ðh þ C b ÞB ∂f h ¼ Ry ∂B

  ðh þ Cb ÞB2 ∂f hP1 KDP3 ¼ þ þ hA1 D þ h1 ðP2  A1 DÞ P3  2 2 2y2 ∂y 2Ry The second-order partial derivatives are (Tai 2015): 2

ðh þ C b Þ ∂ f ¼ Ry ∂B2 2

∂ f ðh þ C b ÞB2 2KDP3 ¼ þ y3 ∂y2 Ry3 2

ðh þ Cb ÞB ∂ f ¼ ∂B∂y Ry2  2 2 2 2 2ðh þ Cb ÞKDP3 ∂ f ∂ f ∂ f  ¼ ∂B∂y Ry4 ∂B2 ∂y2

ð2:245Þ ð2:246Þ

80

2 2

2

∂ f ∂ f Since R > 0, one has ∂B 2 > 0 and ∂B2

2

∂ f ∂y2





2

∂ f ∂y∂B

2

Imperfect EOQ System

> 0. This implies that f(y, B)

is strictly convex for positive B and y. Hence, the unique global minimum for ∂f ∂f positive B and y can be obtained by solving ∂B ¼ 0 and ∂y ¼ 0, which gives (Tai 2015): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD y ¼ h hP1 þ 2hA1 D þ 2h1 ðP2  A1 DÞ  ðhþCRb ÞP3 

ð2:247Þ

So: B ¼

hR  y h þ Cb

ð2:248Þ

When n ¼ 1, the proposed model reduces to the one presented in Hsu and Hsu (2012), with: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD i y ¼u t h hR h P1 þ 2A1 D  ðhþC b ÞP3 

ð2:249Þ

And: B ¼

hR  y h þ Cb

ð2:250Þ

They remarked that the above expressions are simpler than those obtained in Hsu and Hsu (2012). As stated in Hsu and Hsu (2012), if the defective percentage p1 follows uniform distribution with probability density function: 8 0. In order to model the expected cost, one needs to understand the system operation over time. Since defective batches are rejected, the length of inventory cycles is a random variable, depending on the respective history of consecutive previous defective deliveries and the probability of a defective batch p. Specifically, let X be a random variable representing the number of consecutive defective deliveries. Using this, the length of any inventory cycle can be directly expressed as T0 ¼ (X + 1) T. Since defective delivery occurrences are independent and X is a geometric random variable with parameter p and probability distribution function P(X ¼ x) ¼ px(1  p), x  0. Therefore, under imperfect quality, there can be an infinite number of cycles of length T0 (each occurring with a known probability). For any such cycle, they could directly determine the starting and ending inventory levels. Since the total quantity of successive defective deliveries becomes available with the first acceptable delivery (Assumption 7), the start-of-cycle inventory is y  B (for any cycle length). The end-of-cycle inventory (representing backorders), however, depends on the cycle length and is B + Xy (i.e., planned backorders the total quantity of the X successive defective deliveries, Xy) (Skouri et al. 2014). Figure 2.16 depicts a possible situation with two different cycles, the first of length T0 ¼ 2T (where a defective delivery at time T has been rejected) followed by a

y B E[TPU(y, B)] y1* B1* E[TPU(y1*, B1*)]

1624.85 384.34 1,217,432.76 1630.52 384.90 1,217,432.72

S1

1679.81 477.24 1,192,509.48 1682.95 477.72 1,192,509.47

S2 1638.40 379.32 1,213,159.67 1662.38 381.61 1,213,158.94

S3 1699.16 474.22 1,213,382.37 1712.65 476.31 1,213,382.17

S4

Table 2.21 Numerical results for the screening processes Si (i ¼ 1, . . ., 7) (Tai 2015) 1534.16 209.17 1,222,940.58 1573.85 208.45 1,222,937.97

S5

1664.90 368.63 1,204,203.81 1732.15 374.58 1,204,198.33

S6

1542.35 194.28 1,214,227.78 1651.62 191.32 1,214,208.42

S7

2.4 EOQ Model with Backordering 83

84

2

Imperfect EOQ System

Inventory level

a typical cycle

a typical cycle

Q T

T

Time

B

T

Fig. 2.16 Inventory realization with two consecutive cycles of length 2T and T (Skouri et al. 2014)

cycle of length T0 ¼ T. Observe that the batch quantity effectively rejected at delivery in time T has been made available at the delivery in time 2T (so both inventory cycles have the same start-of-cycle inventory). Also observe the end-of-cycle backorders for the first cycle which includes the quantity of the rejected delivery in time T (so backorders are B + y). According to the above description, the problem has two decision variables (y, B). So they need to express system cost in terms of these variables. The cyclic inventoryrelated cost is: hðy  BÞ2 C b B2 C X2T 2D þ þ Cb BXT þ b 2 2D 2D hðy  BÞ2 C b B2 Cb BXy C b X 2 y2 ¼ þ þ þ D 2D 2D 2D

ICðy, B, X Þ ¼

ð2:255Þ

And the total cost in a cycle consists of setup and inventory-related costs and is (Skouri et al. 2014): hðy  BÞ2 C b B2 C X2T 2D þ þ Cb BXT þ b 2 2D 2D 2 2 2 hðy  BÞ C B C BXy C b X y2 ¼ ðX þ 1ÞK þ þ þ b þ b D 2D 2D 2D

cðy, B, X Þ ¼ ðX þ 1ÞK þ

ð2:256Þ

Since now X is a geometric random variable with parameter p, its expected value and second moment are respectively given by (Lefebvre 2008):

2.4 EOQ Model with Backordering

85

E ðX Þ ¼   E X 2 ¼ VarðX Þ þ ½E ðX Þ2 ¼

p 1p

pð 1 þ pÞ p p2 þ ¼ 2 ð 1  pÞ ð 1  pÞ 2 ð 1  pÞ 2

Therefore, the expected value of Eq. (2.256) is given as: Cðy, BÞ ¼

hðy  BÞ2 C b B2 C y 2 pð 1 þ pÞ K C Byp þ þ b þ þ b 1p 2D 2D Dð1  pÞ 2Dð1  pÞ2

ð2:257Þ

Now, recall that the length of any inventory cycle T0 ¼ (X + 1)T, which is also a random variable. So, the expected length of each inventory cycle is: EðT 0 Þ ¼ ½EðX Þ þ 1T ¼ ½EðT Þ þ 1

y y ¼ D ð1  pÞD

ð2:258Þ

Therefore, the following standard practice (e.g., Maddah and Jaber 2008; Nasr et al. 2013), expected cost per unit time, is finally obtained by invoking the renewal– reward theorem (Ross 1996, Theorem 3.6.1): TCðy, BÞ ¼

C ðy, BÞ E ðT 0 Þ 2

¼K þ

C ð1  pÞB2 D hð 1  pÞ ð y  B Þ þ þ b þ C b pB y 2y 2y C b pð1 þ pÞy 2ð 1  pÞ

ð2:259Þ

In order to solve the model, the first-order derivatives of cost function with respect to y and B are: ∂TCðy, BÞ KD hð1  pÞ C b pð1 þ pÞ ðh þ C b Þð1  pÞB2 ¼ 2 þ þ  2 2ð 1  p Þ y 2y2 ∂y ∂TCðy, BÞ ð1  pÞðh þ Cb ÞB ¼ pð h þ C b Þ  h þ y ∂B

ð2:260Þ ð2:261Þ

By equating relations (2.260) and (2.261) to zero (Skouri et al. 2014): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KDð1  pÞðh þ Cb Þ y ¼ C b ½h þ pðh þ C b Þ 

ð2:262Þ

86

2

B ¼

Imperfect EOQ System

½h  pðh þ C b Þy 0 ð 1  pÞ ð h þ C b Þ

ð2:263Þ

Since B should be nonnegative, Eqs. (2.261) and (2.263) indicate that two cases should be examined: (A) h  p(h + Cb)  0 and (B) h  p(h + Cb) < 0 (denominator is always positive) (Skouri et al. 2014). h To determine the nature of optimal point, the second-order Case A: p
hþC b derivatives are derived (Skouri et al. 2014): 2

∂ TCðy, BÞ ð1  pÞðh þ C b Þ >0 ¼ y ∂B2

ð2:264Þ

∂ TCðy, BÞ 2KD ðh þ C b Þð1  pÞB2 ¼ 3 þ >0 y y3 ∂y2

ð2:265Þ

2

2

2

∂ TCðy, BÞ ∂ TCðy, BÞ ðh þ C b Þð1  pÞB ¼ ¼

0 , > 0 , and ¼  2 2 2 2 ∂y ∂y ∂y∂B ∂B ∂B 2KDðhþC b Þð1pÞ > 0 , so Eqs. (2.262) and (2.263) are optimal values and replacing y4 Since

them in Eq. (2.257) yields to (Skouri et al. 2014): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCðy , B Þ ¼ 2KDC b

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h þ pð h þ C b Þ ð1  pÞðh þ C b Þ

ð2:267Þ

ðy, BÞ h Case B: p > hþC From relation (2.240), ∂TC∂B is positive for all B > 0 and b therefore TC(y, B) is increasing in B. So the minimum is located on the boundary of the constraint set B ¼ 0. From the first-order condition for a minimum:

∂TCðy, B ¼ 0Þ KD hð1  pÞ C b pð1 þ pÞ ¼ 2 þ þ ¼0 2 2ð 1  pÞ y ∂y They evaluated the respective order quantity: rffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD 1p y ¼ h ð1  pÞ2 þ C pð1þpÞ 

ð2:268Þ

h

The point (y*, B* ¼ 0) is the unique global minimum and using Eq. (2.257), the respective expected cost per unit time is:

2.4 EOQ Model with Backordering

pffiffiffiffiffiffiffiffiffiffi TCðy , B ¼ 0Þ ¼ 2KD

87

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hð 1  pÞ 2 þ C b pð 1 þ pÞ 1p

ð2:269Þ

Example 2.14 Skouri et al. (2014) presented an example using data D ¼ 4000 per year, K ¼ $250 per delivery, h ¼ $2 per unit per year, Cb ¼ $8 per backordered per year, and p ¼ 0.10. According to the parameter value since p
h/(h + Cb), the solution is given by Case A. So, using Eqs. (2.262) and (2.263), y* ¼ 866, and B* ¼ 96, TC(y*, B*) ¼ 2309.4$.

2.4.4

Rework and Backordered Demand

Consider a production system allowing shortages which are backorders. A percentage of manufactured items are imperfect. The faction of imperfect items is a random variable and its distribution function is known. In order to have a quality control, during production, all items manufactured are inspected. Thus, during the production period, the imperfect items are identified and kept separately from perfect items. During production period, shortages do not occur. This means that production rate is greater than demand rate plus the product of percent of imperfect items by production rate; mathematically speaking this is P  D  γP > 0 (Taleizadeh et al. 2016c). Here, the imperfect items are reworkable. But repair of them in the production systems is not possible due to some restriction such as avoiding interruptions in the production program. On the other hand, the imperfect products have a significant value to the company; therefore, the rework of imperfect items is outsourced. It is assumed that after repair process, the products are as good as perfect ones. The repair cost and the holding cost of repaired products which is higher than the initial holding cost are taken into account. The total cost at repair shop is comprised of fixed and variable costs. The fixed cost is comprised of the repair setup cost and round trip fixed cost. The variable cost consists of the unit transportation cost, unit setup cost, and unit holding cost at the repair facility. Additionally, it is assumed that in the repair shop, the repair process is always in control and all imperfect products can be repaired. Also, repaired items are added to inventory in the same production cycle according to the following three cases (Taleizadeh et al. 2016c): Case I. The repaired products are received when the inventory level is positive. Case II. The repaired products are received when the inventory level is zero. Case III. The repaired products are received when shortage quantity is equal to imperfect products’ quantity. The main objective of the proposed inventory model is to determine the optimal value for production lot size and backorder level in order to maximize the total profit (Taleizadeh et al. 2016c). In order to identify imperfect products, during production period, all manufactured products are screened. The production rate (P) is greater than the

88

2

Imperfect EOQ System

demand rate (D) plus the product of percent of imperfect items (γ) by production rate (P). This means P  D  γP > 0 or ð1  γ Þ  DP > 0. All imperfect items are sent to the repair store. Repair duration is tR and it contains the repair time at rate R and total transportation time tT of imperfect products (tR ¼ γy/R + tT). The fixed cost at the repair shop is calculated with KR + 2Ks where KR is the repair setup fixed cost and Ks is the transportation fixed cost. The variable cost per imperfect product is calculated with C1 + 2CT + h2tR where C1 is the material and labor cost per unit, CT is the transportation cost per unit, and h2 is the unit holding cost in the repair store. So the cost at the repair shop per unit is given by CR ¼ (KR + 2Ks)/γy + (C1 + 2CT + h2tR). The repairer claims this cost with an m margin as repair charges. Thus, the total repair charge per unit is determined with: 

 K R þ 2K S ð1 þ mÞ þ C 1 þ 2C T þ h2 t R γy    K R þ 2K S γy þ tT þ C 1 þ 2CT þ h2 ð1 þ m Þ R γy

ð2:270Þ

Case I The behavior of inventory level is shown in Fig. 2.17. In this case, when the repaired items are returned to the company, the inventory level is positive. After receiving the repaired items, then the inventory level increases by repaired items’ quantity. The shortages occur immediately after the inventory level reaches zero. The inventory cycle is divided into four sections. The production time starts at the beginning of t1 (t 1 ¼ Pð1γB ÞD ). During t1 backorders and current demand are covered. At the end of the production time, the inventory level is Imax. Thus, the I max production period is equal to t1 + t2 where t 2 ¼ Pð1γ ÞD. During the production time, a lot size of y units is manufactured. Thus: t1 þ t2 ¼

y B þ I max ¼ P P ð1  γ Þ  D

Consequently, Imax is equal to: h i D B I max ¼ y ð1  γ Þ  P

ð2:271Þ

At the end of the production period, imperfect items are sent to the repair store. The time t3 represents the time in which the Imax is consumed, and it is given by t 3 ¼ I maxDþγy. Repaired items after tR period are returned to inventory as perfect items. In this case, tR is always shorter than t3. During t4 (t4 ¼ B/D), the shortages occur and these are accumulated until B units. Inventory cycle duration (T ) is equal to t1 + t2 + t3 + t4 and by substitution the ti then T is obtained as:

2.4 EOQ Model with Backordering

89

Inventory level

Imax I1 )-D

γ

(1

tR

P

t1

D

I2

t2

t3

t4

t

T

B

Fig. 2.17 Behavior of inventory level in Case I (Taleizadeh et al. 2016c)

y I max þ γy B þ þ P D D y T¼ D

T ¼ t1 þ t2 þ t3 þ t4 ¼

ð2:272Þ

Also, I1 and I2 are calculated as: I 1 ¼ ðt 3  t R ÞD ¼ I max þ γy  t R D     D γy B þ tT D I1 ¼ y 1  P R   y γy þ tT D I 2 ¼ I max  t R D ¼ ðPð1  γ Þ  DÞ  B  P R

ð2:273Þ ð2:274Þ

The total sales revenue per time unit is: sy ¼ sD T

ð2:275Þ

The total fixed setup cost per time unit is: K KD ¼ T y The total production and rework costs per time unit are, respectively:

ð2:276Þ

90

2

Imperfect EOQ System

Cy γQ ¼ CD and C R T T

ð2:277Þ

The total inspection cost per time unit is: CI y ¼ CI D T

ð2:278Þ

The total holding cost (HC) per time unit is:     h I max t 2 I max I max hR γ 2 y2 HC ¼ þ þ γyðt 3  t R Þ  T 2 2D T 2D i  hð 1  γ Þ  h h ð1  γ Þ D HC ¼ Q ð1  γ Þ   2B þ h R i B2 D 2 P 2y ð1  γ Þ  P      D γy γy B D þ tT þγhR y 1  P 2 R

ð2:279Þ

The total backordering cost (BC) per time unit is:   Cb t 1 B t4 B þ 2 T 2   2 2 C ð1  γ Þ C D B B

B2 BC ¼ b ¼ b þ y 2ðPð1  γ Þ  DÞ 2D 2y ð1  γ Þ  DP BC ¼

ð2:280Þ

Therefore, the total manufacturer’s profit is determined by the total sales revenues minus the total costs incurred expressed as:   KD γy TP ¼ sD  CD þ þ C I D þ C R þ HC þ BC y T

ð2:281Þ

Substituting and simplifying HC, BC, T, and CR into Eq. (2.366), the total profit becomes: TP ¼ sD 0

   1 KD K R þ 2K S γy þ C þ t D þ γD ð 1 þ m Þ þ 2C þ h CD þ þ C I 1 T 2 T C B y R γy C B C B  h i  hð1  γ Þ D C B y ð 1  γ Þ  þ  2B C B B 2 P C C B   h   i ðh þ Cb Þð1  γ Þ 2 C B D γy γy A @ þγhR y 1  B D þ tT þ h iB D P 2 R 2y ð1  γ Þ  P ð2:282Þ

2.4 EOQ Model with Backordering

91

Obviously, the total profit is maximized if and only if the mathematical expression in brackets is minimized. It is important to remark that the mathematical expression in brackets is the total cost. Also the terms which are independent of decision variables (y, B) can be eliminated from the total cost and is represented by N (y, B). Hence, the total cost N(y, B) is given by:   ðh þ C b Þð1  γ Þ 2 KD K þ 2K S γyh2 þ γDð1 þ mÞ R þ þ h iB D y γy R 2y ð1  γ Þ  P i   h   i hR ð 1  γ Þ  h D D γQ γy y ð1  γ Þ   2B þ γhR y 1  B D þ tT þ 2 P P 2 R ð2:283Þ

N ðy, BÞ ¼

In order to obtain the optimal value for the decision variables, it is necessary to prove that N(y, B) is convex. In other words, it is sufficient to show that the Hessian matrix of N(y, B) is positive definite. It is worth to mention that if N(y, B) is minimized, then the total profit (TP) is maximized. The detailed optimization procedure is given below (Taleizadeh et al. 2016c): ∂N ðy, BÞ DðK þ ð1 þ mÞðK R þ 2K S ÞÞ γ 2 Dh2 ð1 þ mÞ ¼ þ R y2 ∂y h i h   i h ð1  γ Þ ðh þ C b Þð1  γ Þ 2 D D γ γD ð1  γ Þ  þ R þ γhR 1     h iB D 2 P P 2 R 2y2 ð1  γ Þ  P ð2:284Þ ∂N ðy, BÞ ðh þ C b Þð1  γ Þ

B ¼ hð1  γ Þ  γhR þ ∂B y ð1  γ Þ  DP

ð2:285Þ

2

∂ N ðy, BÞ 2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ ðh þ C b Þð1  γ Þ 2

B ¼ þ 3 y3 ∂y2 y ð1  γ Þ  DP

ð2:286Þ

2

∂ N ðy, BÞ ðh þ C b Þð1  γ Þ

B ¼ 2 ∂B∂y y ð1  γ Þ  DP

ð2:287Þ

2

∂ N ðy, BÞ ðh þ C b Þð1  γ Þ

B ¼ 2 ∂y∂B y ð1  γ Þ  DP

ð2:288Þ

2

∂ N ðy, BÞ ðh þ C b Þð1  γ Þ

¼ ∂B2 y ð1  γ Þ  DP

ð2:289Þ

Note that ð1  γ Þ  DP > 0 and the second derivatives of N(y, B) with respect to y and B are both positive, and the Hessian matrix determinant is positive (Taleizadeh et al. 2016c):

92

2

Imperfect EOQ System

2Dðh þ C b Þð1  γ ÞðK þ ð1 þ mÞðK R þ 2K S ÞÞ

>0 y4 ð1  γ Þ  DP Then the Hessian matrix is positive definite. Therefore, N(y, B) is convex. In order to obtain the optimal value for the decision variables it is sufficient to set the first derivatives of N(y, B) with respect to y and B equal to zero. Thus, the optimal values of lot size ( y) and backorder level (B) are given by (Taleizadeh et al. 2016c): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u 2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ ð1  γ Þ  DP þ ðh þ C b Þð1  γ ÞB2 u y ¼ t



 

 2 ð1  γ Þ  DP 2Dh2 γRð1þmÞ þ hð1  γ Þ ð1  γ Þ  DP þ 2γhR 1  DP  2γ  γD R

B¼

D

yðhð1  γ Þ þ γhR Þ ð1  γ Þ  P ðh þ C b Þð1  γ Þ

ð2:290Þ ð2:291Þ

Substituting Eq. (2.291) into Eq. (2.290), the optimal value of y independent of B is obtained as shown below: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ u y ¼ t

 

 ðhð1γÞþγhR Þ2

 2Dh2 γ 2 ð1þmÞ  ðhþCb Þð1γÞ ð1  γ Þ  DP þ hð1  γ Þ ð1  γ Þ  DP þ 2γhR 1  DP  2γ  γD R R

ð2:292Þ Note that γ is a random variable, so the expected value of γ must be substituted in Eqs. (2.281), (2.291), and (2.292). Thus, the expected value of total profit, economic production quantity, and backorder level are (Taleizadeh et al. 2016c):    3 Eðγ Þy KD K R þ 2K S þ C þ C CD þ þ t D þ E ð γ ÞD ð 1 þ m Þ þ 2C þ h I 1 T 2 T 7 6 y γy R 7 6 7 6  h i  7 6 hð1  Eðγ ÞÞ ðh þ C b Þ ð1  E ð γ Þ Þ 2 D 7 6þ B  2B þ y ð 1  E ð γ Þ Þ  h i TP ¼ sD  6 7 D P 2 7 6 2y ð1  E ðγ ÞÞ  7 6 P     7 6  5 4 E ðγ Þy Eðγ Þy D B D þ tT þEðγ ÞhR y 1  P 2 R 2

ð2:293Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ  y ¼ u h i  E ðγ Þ E ðγ ÞD u 2E ðγ 2 ÞDhð1 þ mÞ D D u þ 2E ð γ Þh  1  þ h ð 1  E ð γ Þ Þ ð 1  E ð γ Þ Þ   R u P P R 2 R u 0 1 h i u D 2 u u B½ðhð1  E ðγ ÞÞ þ E ðγ Þh1 Þ ð1  E ðγ ÞÞ  P C u @ A t ðh þ C b Þð1  E ðγ ÞÞ

ð2:294Þ

2.4 EOQ Model with Backordering

93

Inventory level

Imax -D g)

D

(1

tR

P gQ

t1

B

t2

t3

t4 t5

t

T

Fig. 2.18 Behavior of inventory level in Case II (Taleizadeh et al. 2016c)

  yðhð1  E ðγ ÞÞ þ Eðγ ÞhR Þ ð1  E ðγ ÞÞ  DP B¼ ðh þ C b Þð1  E ðγ ÞÞ

ð2:295Þ

Case II Here, when the repaired items are returned to the company, the inventory level is zero. After receiving the repaired products, the inventory level increases as much as the repaired products’ quantity, and then the inventory level is positive. Figure 2.18 illustrates the behavior of the inventory level for Case II. In this case, the inventory cycle is divided into five sections. The times t1, t2, and the maximum inventory level (Imax) are the same as in Case I (Taleizadeh et al. 2016c). At the beginning of t1 (t 1 ¼ Pð1γB ÞD) period, backordered shortage has its maximum value, and the production period is started. At the end of this period, the inventory level is I max zero. The production period is equal to t1 + t2 where t 2 ¼ Pð1γ ÞD . Thus, Imax is derived as follows: t1 þ t2 ¼

h i y B þ I max D ¼ B ! I max ¼ y ð1  γ Þ  P Pð1  γ Þ  D P

ð2:296Þ

At the end of the production period, the imperfect items are sent to the repair store. The t3 (t 3 ¼ I max D ) is a fraction of cycle period in which production is not occurring but the inventory level is positive; and the inventory is being consumed by demand. Repaired items after tR period enter into inventory and t4 period is given by B t 4 ¼ γy D. Obviously, in this case, tR is equal to t3. Also t5 (t 5 ¼ D) is a fraction of cycle period in which shortage is occurring. Cycle time duration (T ) is equal to t1 + t2 + t3 + t4 + t5, and by substitution ti then T is obtained as T ¼ t 1 þ t 2 þ t 3 þ γy B t 4 þ t 5 ¼ Py þ I max D þ D þ D (Taleizadeh et al. 2016c). Hence:

94

2

T¼

Imperfect EOQ System

y D

ð2:297Þ

The total sales revenue per time unit is given by: sy ¼ sD T

ð2:298Þ

The fixed setup cost, the production cost, the inspection cost, and the outsourced rework cost are the same as Case I. The total holding cost (HC) per time unit is expressed as: HC ¼

    h t 2 I max t 3 I max h t I þ þ R 4 1 T 2 2 T 2

Here I1 is given by γQ: HC ¼

hð 1  γ Þ h γ2y ðy½ð1  γ Þ  D=P  2BÞ þ R 2 2 hð 1  γ Þ þ B2 2y½ð1  γ Þ  D=P

ð2:299Þ

The total backordered cost (BC) per time unit is derived as:   Cb t 1 B t5 B þ 2 T 2   2 2 C ð1  γ Þ C D B B

B2 BC ¼ b ¼ b þ y 2ðPð1  γ Þ  DÞ 2D 2y ð1  γ Þ  DP BC ¼

ð2:300Þ

So, the total manufacturer’s profit is determined by the total sales revenues minus the total costs incurred:   KD γy TP ¼ sD  CD þ þ C I D þ CR þ HC þ BC y T

ð2:301Þ

The mathematical expressions of HC, BC, T, and CR are substituted into Eq. (2.301) and the total profit (TP) becomes: TP ¼ sD 2

   3 KD K R þ 2K S γy þ CI D þ Dγ ð1 þ mÞ þ tT CD þ þ C 1 þ 2C T þ h2 7 6 y R γy 7 6  h i  7 6  6 hð1  γ Þ y ð1  γ Þ  D  2B 7 2 7 6 ð h þ C Þ ð 1  γ Þ h γ y P 5 4þ þ R þ h b i B2 D 2 2 2y ð1  γ Þ  P ð2:302Þ

2.4 EOQ Model with Backordering

95

It is easy to see that the total profit is maximized if and only if the mathematical expression in brackets is minimized. The constant terms can be eliminated from the total profit function; consequently N(y, B) is expressed as:  h i  D  2B h ð 1  γ Þ y ð 1  γ Þ  KD K þ 2K S γh2 y P N ðB, yÞ ¼ þ Dγ ð1 þ mÞ R þ þ y R γy 2 

þ



hR γ 2 y ð h þ C b Þ ð 1  γ Þ 2 þ h iB D 2 2y ð1  γ Þ  P ð2:303Þ

In order to obtain the optimal value for decision variables y and B, it is necessary to prove that N(y, B) is convex, and it is sufficient to show that the Hessian matrix of N(y, B) is positive definite. The detailed optimization procedure is given below: ∂N ðy, BÞ KD Dð1 þ mÞðK R þ 2K S Þ Dh2 γ 2 ð1 þ mÞ hð1  γ Þ½ð1  γ Þ  D=P hR γ 2 þ ¼ 2  þ þ R 2 2 y y2 ∂y ðh þ C b Þð1  γ Þ 2 B  2 2y ½ð1  γ Þ  D=P

ð2:304Þ ∂N ðy, BÞ ðh þ C b Þð1  γ Þ

B ¼ hð1  γ Þ þ ∂B y ð1  γ Þ  DP

ð2:305Þ

2

∂ N ðy, BÞ 2D½K þ ð1 þ mÞðK R þ 2K S Þ ðh þ C b Þð1  γ Þ 2

B ¼ þ 3 y3 ∂y2 y ð1  γ Þ  DP

ð2:306Þ

2

∂ N ðy, BÞ ðh þ C Þð1  γ Þ

B ¼  2 b ∂B∂y y ð1  γ Þ  DP

ð2:307Þ

2

ðh þ C Þð1  γ Þ ∂ N ðy, BÞ

B ¼  2 b ∂y∂B y ð1  γ Þ  DP

ð2:308Þ

2

∂ N ðy, BÞ ðh þ C b Þð1  γ Þ

¼ ∂B2 y ð1  γ Þ  DP

ð2:309Þ

Note that ð1  γ Þ  DP > 0 and the second derivatives of N(y, B) with respect to y and B are positive. Also the Hessian matrix determinant is positive: 2Dð1  γ Þðh þ Cb Þ½K þ ð1 þ mÞðK R þ 2K S Þ

>0 y4 ð1  γ Þ  DP

96

2

Imperfect EOQ System

Since the Hessian matrix is positive definite, therefore the N(y, B) is convex. Setting the first derivatives of N(y, B) with respect to y and B equal to zero, then the optimal values for the decision variables y and B are derived as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u u2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ ð1  γ Þ  DP þ ðh þ C b Þð1  γ ÞB2   y¼t ð2:310Þ



2 ð1  γ Þ  DP 2Dh2 γRð1þmÞ þ hð1  γ Þ ð1  γ Þ  DP þ h1 γ 2

hy ð1  γ Þ  DP B¼ ðh þ C b Þ

ð2:311Þ

And by substituting Eq. (2.310) into Eq. (2.311), the optimal value for y independent of B is obtained as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ u  y ¼ t

 hC  2Dh2 γ 2 ð1þmÞ b þ h1 γ 2 þ ð1  γ Þ ð1  γ Þ  DP ðhþC R bÞ

ð2:312Þ

It is important to remark that γ is a random variable, so in Eqs. (2.302), (2.310), and (2.311), the expected value of γ must be substituted. Therefore, the expected values of the total profit, economic production quantity, and optimal amount of backordered shortage are given by:    3 Eðγ Þy KD K R þ 2K S þ C CD þ þ t D þ DE ð γ Þ ð 1 þ m Þ þ 2C þ h þ C I 1 T 2 T 7 6 y R Eðγ Þy 7 6 7  h i  2 TP ¼ sD  6 h1 Eðγ Þy ðh þ Cb Þð1  E ðγ ÞÞ 2 7 6 hð1  Eðγ ÞÞ D 4þ y ð1  E ðγ ÞÞ   2B þ þ h iB 5 D P 2 2 2y ð1  Eðγ ÞÞ  P 2

ð2:313Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ u  y ¼ t

 hC  ð2:314Þ 2DhE ðγ 2 Þð1þmÞ b 2 Þ þ ð1  E ðγ ÞÞ ð1  E ðγ ÞÞ  D þ h E ð γ 1 P ðhþCb Þ R

hy ð1  E ðγ ÞÞ  DP B¼ ðh þ C b Þ

ð2:315Þ

Case III In this case, when the repaired items return to the system, the inventory level is negative; in other words, shortage exists. Notice that in this case, the shortage quantity is exactly the same as the imperfect products’ quantity. After receipt, the items that repaired the inventory levels are increased. The shortages are covered and the inventory level reaches zero and shortage time continues again. The cycle time is divided into five sections. At the beginning of t1 period (t 1 ¼ Pð1γB ÞD), backordered shortage has its maximum value and production period starts. At the end of t1, the

2.4 EOQ Model with Backordering

97

Inventory level

Imax

D

-D g) 1

P(

t1

tR t2

t3

t4

t5

t

gQ B

T

Fig. 2.19 Behavior of inventory level in Case III (Taleizadeh et al. 2016c) I max inventory level is zero. Production time is equal to t1 + t2 where t 2 ¼ Pð1γ ÞD. The maximum inventory level (Imax) is calculated as:

t1 þ t2 ¼

h i y B þ I max D ¼ B ! I max ¼ y ð1  γ Þ  P Pð1  γ Þ  D P

ð2:316Þ

At the end of production time, the imperfect items are sent to the repair store. Here, t3 (t 3 ¼ I max D ) is a fraction of cycle period in which the manufacturing system is not producing but the inventory level is positive. The repaired items after tR period are returned to inventory, this time occurring also at the end of t4 ¼ γy/D. In t4 period, the shortage occurs. In this case, tR is equal to t3 + t4. Also t5 (t 5 ¼ DB ) is a fraction of cycle time in which the manufacturing system does not produce items and shortage is occurring. Figure 2.19 shows the behavior of the inventory level in this case. Inventory cycle duration (T ) is equal to t1 + t2 + t3 + t4 + t5. The corresponding mathematical expression for each ti (i ¼ 1, 2, 3, 4, 5) is substituted. Thus, T is obtained as (Taleizadeh et al. 2016c): T ¼ t1 þ t2 þ t3 þ t4 þ t5 ¼

y I max γy B þ þ þ P D D D

Hence: T¼

y D

ð2:317Þ

The total sales revenue and the fixed setup cost, the production cost, the inspection cost, and the outsourced rework cost are the same as Case I and Case II.

98

2

Imperfect EOQ System

Thus, the holding cost (HC) per time unit is:   h t 2 I max t 3 I max þ T 2 2  h i 2 hð 1  γ Þ D HC ¼ h B i y ð1  γ Þ  D P 2y ð1  γ Þ  P i  hð 1  γ Þ  h hð 1  γ Þ D

B2 y ð1  γ Þ   2B þ HC ¼ 2 P 2y ð1  γ Þ  DP HC ¼

ð2:318Þ

The total backordered cost (BC) per time unit is:   C b t 1 B t 4 γy t 5 B þ þ 2 2 T 2  Cb D B2 γ 2 y2 B2 BC ¼ þ ¼ þ y 2ðPð1  γ Þ  DÞ 2D 2D BC ¼

C b ð1  γ Þ C γ2y h i B2 þ b D 2 2y ð1  γ Þ  P ð2:319Þ

Then the total manufacturer’s profit is determined by the total sales revenues minus the total costs incurred: 

KD γy þ C I D þ CR þ HC þ BC TP ¼ sD  CD þ y T

 ð2:320Þ

The mathematical expressions for HC, BC, T, and CR are substituted into Eq. (2.320) and the total profit is rewritten as: TP ¼ sD 2

   3 KD K R þ 2K S γy þ tT 7 þ C 1 þ 2C T þ h2 6 CD þ y þ C I D þ Dγ ð1 þ mÞ R γy 7 6 7  h i  ðh þ C Þð1  γ Þ 6 2 7 6 hð 1  γ Þ D C γ y b b 5 4þ y ð1  γ Þ   2B þ h i B2 þ D 2 P 2 2y ð1  γ Þ  P ð2:321Þ

Notice that the total profit in Case II and Case III (Eqs. 2.313 and 2.321) differs on the holding cost of reworked items and the backorder cost. Thus, the concavity of the total profit function in Case III is proved. Therefore, setting the first derivatives of the total profit with respect to y and B equal to zero, then the optimal value for the lot size ( y) and shortage level (B) are determined as (Taleizadeh et al. 2016a):

2.4 EOQ Model with Backordering

99

∂TP ¼0! ∂y 2

3 h i D hð1  γ Þ ð1  γ Þ  2 2 D ð 1 þ m Þ ð K þ 2K Þ Dh γ ð 1 þ m Þ ð h þ C Þ ð 1  γ Þ KD C γ 6 7 R S P  h b 4 2  þ 2 þ iB2 þ b 5 ¼ 0 D 2 2 R y2 y 2y2 ð1  γ Þ  P

ð2:322Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u u2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ ð1  γ Þ  DP þ ðh þ C b Þð1  γ ÞB2   ð2:323Þ y¼t



2 ð1  γ Þ  DP 2Dh2 γRð1þmÞ þ hð1  γ Þ ð1  γ Þ  DP þ Cb γ 2 ∂TP ¼0! ∂B

ðh þ Cb Þð1  γ Þ hð1  γ Þ þ h iB¼0 D y ð1  γ Þ  P

D hy ð1  γ Þ  P B¼ ðh þ C b Þ

ð2:324Þ

ð2:325Þ

By substituting Eq. (2.325) into Eq. ((2.323), the optimal value of y independent of B is calculated as (Taleizadeh et al. 2016c): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2DðK þ ð1 þ mÞðK R þ 2K s ÞÞ u  y ¼ t

 hC  2Dh2 γ 2 ð1þmÞ b þ C b γ 2 þ ð1  γ Þ ð1  γ Þ  DP ðhþC R bÞ

ð2:326Þ

Again note that γ is a random variable, so in Eqs. (2.321), (2.325), and (2.326), the expected value of γ must be substituted. Thus, the expected values for the total profit, economic production quantity, and backorder level are given by (Taleizadeh et al. 2016c):    3 E ðγ Þy KD K R þ 2K S þ C CD þ D þ DE ð γ Þ ð 1 þ m Þ þ 2C þ h þ t þ C I 1 T 2 T 7 6 y R Eðγ Þy 7 6 7  h i  2 TP ¼ sD  6 C b E ðγ Þy ðh þ C b Þð1  Eðγ ÞÞ 2 7 6 h ð1  E ðγ Þ Þ D 4þ y ð1  Eðγ ÞÞ   2B þ þ h iB 5 D P 2 2 2y ð1  E ðγ ÞÞ  P 2

ð2:327Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2DðK þ ð1 þ mÞðK R þ 2K S ÞÞ u  y ¼ t

 hC  2Dh2 E ðγ 2 Þð1þmÞ b 2 Þ þ ð1  E ðγ ÞÞ ð1  E ðγ ÞÞ  D þ C E ð γ b P ðhþC b Þ R

hy ð1  E ðγ ÞÞ  DP B¼ ðh þ C b Þ

ð2:328Þ ð2:329Þ

100

2

Imperfect EOQ System

Table 2.22 Data for the numerical example (Salameh and Jaber 2000; Taleizadeh et al. 2016c) Description Percentage of defectives Probability density function

Symbol γ f(γ)

Value U ~ (0, 0.04) 1/(0.04–0)

Table 2.23 Data for the numerical example (Jaber et al. 2014; Taleizadeh et al. 2016c) Description Repair setup cost Transportation fixed cost Unit transportation cost Unit material and labor cost Unit holding cost in repair shop Repair rate Total transport time Holding cost of repaired product Markup percentage Table 2.24 Data for the numerical example (Hsu and Hsu, 2014; Taleizadeh et al. 2016c)

Table 2.25 The optimal value for each case (Taleizadeh et al. 2016c)

Symbol KR KS CT C1 h2 R tT hR m

Description Selling price Demand rate Production cost Production rate Holding cost Backorder cost Inspection cost

Value 100 200 2 5 4 50,000 2/220 6 20%

Symbol s D C P h Cb CI

TP 197,178.22 197,225.99 197,224.96

Case I Case II Case III

Units $/setup $/trip $/unit $/unit $/unit/year Units/year Year $/unit/year

Value 300 1000 100 3000 5 10 0.5

y 949.65 972.26 971.77

Units $/unit Units/year $/unit Units/year $/unit/year $/unit/year $/unit

B 209.72 209.58 209.47

T 0.95 0.97 0.97

Example 2.15 This section presents a numerical example. The data for the numerical example is given in Tables 2.22, 2.23, and 2.24. Table 2.22 contains data from Salameh and Jaber (2000). Table 2.23 gives data from Jaber et al. (2014), and Table 2.24 presents data from Hsu and Hsu (2016). The imperfect percentage follows a uniform distribution (U ~ (0, 0.04)). Thus, the Rb Rb 1 expected value of the defective products (γ) is E ðγ Þ ¼ a γf ðγ Þdγ ¼ a γ ba dγ ¼ bþa 2

¼ 0:04þ0 ¼ 0:02 2

, and ðbaÞ2 12 þ

the E(γ 2) and (1  E(γ 2))2 are given by E ðγ 2 Þ ¼ 2

Þ 2 E 2 ðγ Þ ¼ ð0:04 varðγ Þ þ E2 ðγ Þ ¼ 12 þ ð0:02Þ ¼ 0:0005333333. Firstly, for each profit function, the following condition must be satisfied: ð1  γ Þ  DP ¼ 0:65 > 0. Since the condition is satisfied, then the problem can be optimized. Table 2.25 shows the optimal results for the expected total profit and the decision variables and cycle duration (Taleizadeh et al. 2016c).

2.4 EOQ Model with Backordering Fig. 2.20 Behavior of the inventory level over time (Konstantaras et al. 2012)

101

Inventory level

qi

p iq i

Bi

Time

Table 2.26 Notations of a given problem pi qi n TR( y)

The fraction of defective items in shipment of size qi (percent) ith shipment size (unit) Number of cycle Total revenue per cycle ($)

According to Table 2.25, it is easy to see that the highest profit for this example is obtained in Case II. It is important to mention that the total profit in Case II and Case III differs only in the holding cost of repaired items and backorder cost. Here, it is easy to show that if the holding cost of repaired items is lower than the backordering cost, then the total profit in Case II is higher than the total profit in Case III and vice versa.

2.4.5

Learning in Inspection

Konstantaras et al. (2012) developed an EOQ imperfect system with learning in inspection. They assumed that 100% inspection of items is performed for each shipment, and the screening rate is faster than the demand rate. The defective items are sold at a discounted price; the fraction of defective items follows a learning curve that is either of an S-shape or of a power form learning curve. To avoid shortages during the screening time, it is assumed that the number of good items in shipment i of order size qi is at least equal to the demand during the screening time, i.e., qi ð1  pi Þ  Dqx i ) pi
1  Dx , where pi is the fraction of defective items in shipment of size qi, D is the demand rate, and x is screening rate in units per unit of time. The behavior of the inventory level in a given cycle i is shown in Fig. 2.20, which is the same as that in Wee et al. (2007). In order to model the presented problem, some new notations which are specifically used are shown in Table 2.26. The total revenue per cycle, TR(qi), is the sum of revenues from selling good- and imperfect-quality items and is given as (Konstantaras et al. 2012):

102

2

Imperfect EOQ System

TRðqi Þ ¼ sqi ð1  pi Þ þ υqi pi

ð2:330Þ

where s is the unit price of a good-quality item and v is the unit price of an imperfectquality item. The total cost per cycle, TC(qi, Bi), is the sum of ordering cost in addition to penalty cost purchasing cost, screening cost, holding cost, and backordering cost and is given as (Konstantaras et al. 2012): Cq TCðqi , Bi Þ ¼ |{z} þ C I qi K þ |{z}i |{z} Fixed Purchasing Inspection cost cost cost 

Cb B2i ðqi  pi qi  Bi Þ2 pi qi 2 þh þ þ 2D x 2D |ffl{zffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Holding cost

ð2:331Þ

Shortage cost

where K is the fixed cost, C is the unit purchasing cost, CI is the unit screening cost, h is the holding cost per unit per unit of time, Cb is the backordering cost per unit per unit of time, and Bi is the maximum backordering level in cycle i. The profit per unit of time is obtained from Eqs. (2.330) and (2.331) as (Konstantaras et al. 2012): TRðqi ÞTCðqi ,Bi Þ Ti ðhþCb ÞB2i ðC þCI ÞD hDpi qi hð1pi Þqi p KD ¼ sDþυD i   þhBi    1pi 1pi 2 qi ð1pi Þ 2qi ð1pi Þ ð1pi Þx TPðqi ,Bi Þ ¼

ð2:332Þ where Ti is the length. Note that i in Eq. (2.332) is an input parameter, where pi is a constant. The objective here is to maximize the profit per unit time given in Eq. (2.332). By taking the partial derivatives of TP(qi, Bi) with respect to qi and Bi and by setting the results to zero, one has (Konstantaras et al. 2012): ∂TPðqi , Bi Þ ðh þ C b Þ hð1  pi Þqi ¼h B ¼ 0 where Bi ¼ h þ Cb qi ð 1  pi Þ i ∂Bi

ð2:333Þ

∂TPðqi , Bi Þ ðh þ C b Þ 2 hð 1  pi Þ hDpi KD ¼ 0 ð2:334Þ ¼ þ B   2 ð1  pi Þqi 2 2qi 2 ð1  pi Þ i ð1  pi Þx ∂qi After some simple algebra, the unique solution is determined by substituting Eq. (2.333) in Eq. (2.334) and solving for qi to get: qi

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KDxðh þ C b Þ u i ¼t h h xð1  pi Þ2 Cb þ 2Dpi ðh þ C b Þ

ð2:335Þ

2.4 EOQ Model with Backordering

103

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2KDhxð1  pi Þ2 h i Bi ¼ t ðh þ Cb Þ xC b ð1  pi Þ2 þ 2Dpi ðh þ Cb Þ

ð2:336Þ

The second partial derivatives of TP(qi, Bi) are: 2

∂ TPðqi , Bi Þ ðh þ C b Þ ¼ 0 q4i ð1  pi Þ2

is satisfied and: 2

∂ TPðqi , Bi Þ

> > > < s:t: ð pÞ Dðsi  si1 Þ > x ¼ ti þ > > i y s ð 1  pi Þ > : 0
si1
t i
si where

106

2

Imperfect EOQ System

FTPðn, si , t i Þ ¼ TPðn, si , t i Þ 2 3  X Zt i n  n X Dðsi  si1 Þ 4C b ¼ nK þ ðC þ C I Þ Dðt  si1 Þdt5 þ 1  pi i¼1 i¼1 þ

n X i¼1

2

4h

si1

Zxi 



p Dðsi  si1 Þ Dt þ Dsi þ i dt þ h 1  pi

ti



n  X i¼1

Zsi xi

p Dðsi  si1 Þ sDðsi  si1 Þ þ υ i 1  pi

3

Dðsi  t Þdt 5

ð2:351Þ



To develop a solution procedure for the problem (P), the initial value of n is fixed and the second constraint 0
si  1
ti
si is ignored. Take the first-order derivatives of FTP(n, si, ti) with respect to ti and si and equate them to zero yields to (Konstantaras et al. 2012):     ∂FTPðn, si , ti Þ pi Dðsi  si1 Þ ∂xi pi Dðsi  si1 Þ ¼ h Dxi þ Dsi þ  Dt i þ Dsi þ 1  pi 1  pi ∂t i ∂t i   ∂x þh Dðsi  xi Þ i þ Cb Dðt i  si1 Þ ¼ 0 ∂t i ∂FTPðn, si , ti Þ t  si1 h ¼ ðh þ C b Þðti  si1 Þ  hðsi  si1 Þ ¼ 0 ) i ¼ ) si  si1 h þ C b ∂t i

ð2:352Þ Next, Konstantaras et al. (2012) presented that for t1P0 the constraints in Eq. (2.350) satisfy the necessary conditions obtained from Eq. (2.353). Example 2.16 In order to illustrate the behavior of the optimal policy for the finite planning horizon model, which is more complex than the infinite one, an example with the following input parameters is considered: D ¼ 500 units/year, x ¼ 17,520 units/year, K ¼ $300, h ¼ $1/unit/year, Cb ¼ $5/unit/year, C ¼ $25/ unit, s ¼ $50/unit, CI ¼ $0.5/unit, v ¼ $20/unit, and H ¼ 6. In this example, the percentage of defectives per shipment n, pn, is expressed using an S-shaped logistic learning curve model as pn ¼ gþea nm where a ¼ 70.067, g ¼ 819.76, and m ¼ 0.7932 which are positive model parameters (Konstantaras et al. 2012). The results in Table 2.27 show that the optimal replenishment policy occurs when n0 ¼ 5 where the maximum profit is TP(n, si, ti) ¼ 68,985.0. The table also shows the corresponding values of si, ti, and qi, i ¼ 1, 2, . . ., 5, for the optimal solution. The optimal solution was determined by programming the solution described in solution procedure in Mathematica 6.0. The results in P Table 2.28 show that as the learning parameter m increases, the total order quantity ( ni¼1 qi ) decreases and the profits increase. Learning in quality suggests ordering in smaller lots as the fraction of defective items decrease with

n TP n¼5 ti si qi

0.1978 1.1868 645.21

1 64,110.1

1.3851 2.3760 646.52

2 67,589.0 2.5749 3.5698 649.02

3 68,552.7 3.7705 4.7740 654.67

4 68,891.0 4.9784 6 666.52

5 68,985.0

7 68,896.3 – – –

6 68,966.1 – – –

Table 2.27 Total profit for different values of n and the overall optimal replenishment policy (Konstantaras et al. 2012)

– – –

8 68,809.6

– – –

9 68,720.4

– – –

10 68,626.3

2.4 EOQ Model with Backordering 107

108

2

Imperfect EOQ System

Table 2.28 The optimal ti, si, qi, n, and TP(n, si, ti) values for increasing values of m under S-shaped learning curve (Konstantaras et al. 2012) m m ¼ 0.00

m ¼ 0.10

m ¼ 0.20

m ¼ 0.30

m ¼ 0.40

m ¼ 0.50

m ¼ 0.60

m ¼ 0.70

m ¼ 0.80

ti 0.20000 1.40000 2.60000 3.80000 5.00000 0.19996 1.39977 2.59967 3.79968 4.99980 0.19991 1.39939 2.59910 3.79910 4.99943 0.19982 1.39878 2.59816 3.79809 4.99877 0.19967 1.39782 2.59661 3.79639 4.99760 0.19945 1.39630 2.59412 3.79355 4.99560 0.19910 1.39393 2.59015 3.78891 4.99222 0.19856 1.39024 2.58387 3.78141 4.98664 0.19775 1.38459

si 1.20000 2.40000 3.60000 4.80000 6.00000 1.19979 2.39967 3.59966 4.79976 6.00000 1.19944 2.39911 3.59905 4.79932 6.0000 1.19890 2.39819 3.59800 4.79852 6.0000 1.19804 2.39669 3.59624 4.79713 6.00000 1.19670 2.39429 3.59332 4.79472 6.00000 1.19462 2.39047 3.58855 4.79067 6.00000 1.19140 2.38447 3.58090 4.78397 6.00000 1.18647 2.37519

qi 656.01 656.01 656.01 656.01 656.01 655.84 655.89 655.95 656.00 656.08 655.57 655.69 655.84 656.02 656.25 655.14 655.36 655.64 656.03 656.55 654.46 654.79 655.28 656.02 657.10 653.38 653.86 654.65 655.94 658.06 651.68 652.35 653.57 655.77 659.70 649.02 649.93 651.76 655.38 662.44 644.92 646.14

n 5

TP(n, si, ti) 68,949.5

5

68,950.2

5

68,951.3

5

68,952.9

5

68,955.4

5

68,959.1

5

68,964.6

5

68,973.1

5

68,986.0 (continued)

2.4 EOQ Model with Backordering

109

Table 2.28 (continued) m

ti 2.57413 3.76955 4.97763

si 3.56883 4.77316 6.00000

qi 648.81 654.62 666.86

n

TP(n, si, ti)

Inventory level

py

y

M

t B

T

t1

Time

t

Fig. 2.22 Behavior of the inventory level over time (Rezaei 2005)

every shipment. This results in lower holding, screening, and backordering cost and higher revenue from selling good-quality items, thus increasing profits. When there is no learning, m ¼ 0, the optimal policy is to order equal lots in each shipment (Konstantaras et al. 2012).

2.4.6

EOQ Model for Imperfect-Quality Items

Rezaei (2005) developed a simple EOQ model upon Salameh and Jaber’s work. To avoid any possible confusion, the notations and assumptions utilized in Salameh and Jaber are employed in this article. He assumed that items, received or produced, are not of perfect quality and not necessarily defective; thus, they could be used in another production/inventory situation. Each lot received contains percentage defectives with a known probability density function. Good-quality items have a selling price for per unit, and defective items are sold as a single batch at a discounted price. A 100% screening process of the lot is conducted and shortage is permitted. Based on the above assumptions, a mathematical model is developed that is closer to the real world because of exploitation quality and shortage simultaneously. The behavior of the inventory level is illustrated in Fig. 2.22. With the above

110

2

Imperfect EOQ System

assumptions, the total cost per cycle for the modified EOQ model with backorder for imperfect items is (Rezaei 2005): f ðy, BÞ ¼ Fixed cost of placing an order þ Variable cost of lot size þ Screening cost of lot size þ Holding cost þ Shortage cost:   ½yð1  pÞ  B py2 C t f ðy, BÞ ¼ K þ Cy þ C I y þ h t1 þ þ b 2B ð2:353Þ 2 x 2 Also, the total revenue per cycle is g(y, B) ¼ total sales volume of good quality + total sales volume of imperfect-quality items: gðy, BÞ ¼ syð1  pÞ þ vyp

ð2:354Þ

The total profit per cycle is the total revenue per cycle minus the total cost per cycle, π(y, B) ¼ g(y, B)  f(y, B), and it is given as: 9 8 Shortage > > > > Inspection > > Fixed > > > > cost > > > > cost cost Revenue zfflffl}|fflffl{ >   = < z}|{ zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ > 2 z}|{ ½yð1  pÞ  B py Cb B t2 π ðy,BÞ ¼syð1  pÞ þ vyp  K þ Cy þ CI y þh þ t1 þ |{z} 2 2 x > > > > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > Purchasing > > > > Holding cost > > > > ; : cost

ð2:355Þ ½pÞ By dividing the total profit per cycle by the cycle length, T ¼ yð1E , replacing t1 D yð1E ½pBÞ by and t2 by B/D in (2.355), the total profit per unit time can be written as: D

     hy hy K 1 π U ðy, BÞ ¼ D s  v þ þ D v   C  CI  x x y 1p     y ð 1  pÞ ð C b þ hÞ 2  h B þ B 2 2yð1  pÞ       hy hy K 1 E ½π U ðy, BÞ ¼ D s  v þ þ D v   C  CI  E x x y 1p     yð1  E ½pÞ ð C b þ hÞ  h B þ B2 2 2yð1  E ½pÞ

ð2:356Þ

ð2:357Þ

In order to solve this nonlinear programming problem, take the first partial derivatives of Eπ U(y, B) with respect to y, B, respectively. One obtains:

2.4 EOQ Model with Backordering

111

     ∂E ½TPUðy, BÞ D hð1  E ½pÞ 1 DK 1 ¼ 1E þ 2 E  2 x 1p 1p y ∂y þ

ðh þ C b Þ B2 2yð1  E ½pÞ

ð2:358Þ

∂E½TPUðy, BÞ ðh þ C b Þ ¼h B yð1  E ½pÞ ∂B By setting Eqs. (2.356)–(2.358) equal to zero, one obtains: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   u 1 u 2DKE 1p u     h ii y ¼ t h Cb 1 h ð1  E ½pÞ Cb þh  2D 1  E x 1p B ¼

y ð1  E ½pÞh h þ Cb

ð2:359Þ

ð2:360Þ

From Fig. 2.22 for maximum inventory level, one obtains: M ¼yB

ð2:361Þ

On the other hand, by replacing Eqs. (2.359) and (2.360) in Eq. (2.361), one obtains: M ¼

y ð1  E ½pÞC b h þ Cb

ð2:362Þ

In order to examine the second-order sufficient conditions (SOSC) for a maximum value, they first obtained the Hessian matrix H as follows (Rezaei 2005): 2

3 2 2 ∂ E½π U ðy, BÞ ∂ E ½π U ðy, BÞ 6 7 ∂y2 ∂y∂B 6 7 H¼6 2 7 4 ∂ E½π U ðy, BÞ ∂2 E ½π U ðy, BÞ 5 ∂B∂y ∂B2

ð2:363Þ

where 2

∂ E ½π U ðy, BÞ ¼ ∂y2

 h i  1  2DKE 1p ð1  E ½pÞ þ ðh þ C b ÞB2 ð1  E½pÞy3

ð2:364Þ

112

2

Imperfect EOQ System

2

∂ E½π U ðy, BÞ ðh þ C b Þ ¼ B ð1  E½pÞy2 ∂y∂B

ð2:365Þ

2

∂ E½π U ðy, BÞ ðh þ C b Þ ¼ B ð1  E½pÞy2 ∂B∂y

ð2:366Þ

2

∂ E½π U ðy, BÞ ðh þ C b Þ ¼ ð1  E½pÞy ∂B2

ð2:367Þ

Then they proceeded by evaluating the principal minor determinants of H at point (y*, B*). The first principal minor determinant of H is (Rezaei 2005):

j H 11 j¼

  h i 2 1  2DKE 1p ð 1  E ½ p Þ þ B  ð h þ C b Þ h

j H 22 j¼

2DKE

i

1 1p

ð1  E ½pÞy3

0

ð2:369Þ

Therefore, the Hessian matrix H is negative definite at point (y*, B*) which implies that there exist unique values. Example 2.17 Rezaei (2005) presented an example using D ¼ 50,000 units/year, K ¼ 100/cycle, h ¼ $5/unit/year, x ¼ 1 unit/min (175,200 units/year), CI ¼ $0.5/ unit, C ¼ $25/unit, s ¼ $50/unit, and v ¼ Cb ¼ $20/unit, and the percentage defective random variable, p, is uniformly distributed with its p.d.f.:  f ð pÞ ¼

25

0
p
0:04

0

otherwise

So the maximum values of y and B that maximize Eq. (2.357) are y* ¼ 1601.58, B* ¼ 313.9, and M* ¼ 1287.68.

2.5 2.5.1

EOQ Model with Partial Backordering EOQ Model of Imperfect-Quality Items

Roy et al. (2011) developed an imperfect inventory system under partial backordering. Similar to previous models, the ordering lot size is y. Among these, (1  p)y is of perfect-quality and py of imperfect-quality products. The fraction p follows a probability density function. Generally, it follows a uniform distribution function. The inventory cycle starts with shortages and it continues up to time t1. In the beginning of the cycle, shortages may occur due to lead time, the time gap

2.5 EOQ Model with Partial Backordering Table 2.29 Notations of a given problem

113

E(.) Qs(t) Ql(t) Qi(t)

Expected value operator The level of negative inventory at time t (unit) The lost sale quantity at time t (unit) On-hand inventory at time t (unit)

between placing and receiving of an order, and problems with labor staff, management systems, etc. In order to model the presented problem, some new notations which are specifically  used are  shown in Table 2.29. B y  1p units of total units y are used for the period [t1, t1 + t2] and the B units are used for the period [0, t1]. During the period [t1, t1 + t2], remaining 1p the total demand Dt2 is adjusted with [y(1  p)  B] units. The screening rate x per unit time is always greater than D. During the time span [t1, t1 + t2], E(1  p)x  D must be held to avoid shortages. Also, to meet the total shortage B at time t1, the screening rate x must be satisfied such that E(1  p)x  B + D. Hence, to avoid shortage within the screening time, E(1  p)x  Max (B + D, D), i.e., E(1  p) x  B + D must be satisfied. During stock-out [0, t1], the demand Deδðt1 tÞ at  period δðt 1 t Þ time t is met, and the rest of the demand, D 1  e , remains unsatisfied. Here (t1  t) is the waiting time up to the replenishment at time t1 and is a positive constant (Roy et al. 2011). The differential equation of the level of negative inventory at any time t is (Roy et al. 2011):

dQs ðt Þ ¼ Deδðt1 tÞ , dt

0
t
t 1 with Qs ð0Þ ¼ 0

ð2:370Þ

And its solution will be: Qs ðt Þ ¼ 

  D δðt1 tÞ e  eδt1 , δ

0
t
t1

The lost sale  quantity attime t is: Q1 ðt Þ ¼ D 1  eδðt1 tÞ , 0
t
t 1 Here the maximum backorder level is: B ¼ Qs ðt 1 Þ ¼

 D 1  eδt1 δ

ð2:371Þ

The total backordering and lost sale costs are shown in Eqs. (2.372) and (2.373):

114

2

Imperfect EOQ System

Zt1 BC ¼ Cb

ðQs ðt ÞÞdt 0

Zt1   Cb D eδðt1 tÞ  eδt1 dt ¼ δ 0

C D ¼ b eδt1 δ

Zt1



 eδt  1 dt

ð2:372Þ

0

  Cb D δt1 eδt ¼ e t δ δ   C D ¼ b2 eδt1 eδt  δt 1  1 δ  Cb D  ¼ 2 1  δt 1 eδt1  eδt1 δ Zt1 Q1 ðt Þdt ¼ C 1 D

LSC ¼ C 1 0

Zt1 

 1  eδðt1 tÞ dt

0

   eδðt1 tÞ C D ¼ 1 δt 1  1 þ eδt1 ¼ C1 D t  δ δ

ð2:373Þ

The differential equation of the inventory level at any time t is: dQi ðt Þ ¼ D, dt

t 1
t
t 1 þ t 2 with Qi ðt 1 Þ ¼ ð1  pÞy  B

ð2:374Þ

The solution of Eq. (2.374) is: Qi ðt Þ ¼ ½ð1  pÞy  B  Dðt  t 1 Þ,

t1
t
t1 þ t2

ð2:375Þ

Considering (1  p)y  B ¼ Dt2, the total cycle length is (Roy et al. 2011): T ¼ t1 þ t2 ¼

ð1  pÞy  B þ t1 D

ð2:376Þ

The inventory holding cost HC is:  h

ðð1  pÞy  BÞ2 py2 þ 2D x

 ð2:377Þ

2.5 EOQ Model with Partial Backordering

115 Holding cost

Fixed cost

z}|{ TCðy, t 1 Þ ¼ K þ

Purchasing cost

z}|{ Cy

þ

Screening cost

z}|{ CI y

þ

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   ðð1  pÞy  BÞ2 py2 þh þ 2D x

 b  Cb D  πD  1  δt 1 eδt1  eδt1 þ δt 1  1 þ eδt1 2 δ δ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Backordering cost

Lost sale cost

ð2:378Þ The expected total cost per cycle is (Roy et al. 2011):  E ½TCðy, t 1 Þ ¼ K þ Cy þ C I y þ h

Eðð1  pÞy  BÞ2 E ½py2 þ 2D x



 b  C D πD  þ b2 1  δt 1 eδt1  eδt1 þ δt 1  1 þ eδt1 δ δ

ð2:379Þ

The expected cyclic total revenue is: E½TRðy, t 1 Þ ¼ sð1  EðpÞÞy þ vE ðpÞy

ð2:380Þ

And the expected cycle length is: E ðT Þ ¼

ð1  E ðpÞÞy  B þ t1 D

ð2:381Þ

The expected average profit per cycle using renewal–reward theorem changes to: E ½TRðy, t 1 Þ  E ½TCðy, t 1 Þ E ðT Þ 1 ¼ ½sð1  E ðpÞÞy þ vEðpÞy  K  Cy ð1  E ðpÞÞy  B þ t1 D   E ðð1  pÞy  BÞ2 E½py2 þ CI y  h 2D x   b  Cb D  πD  δt 1 δt 1 δt 1  2 1  δt 1 e  δt 1  1 þ e e δ δ   1 hB he y2 ¼ e1 y þ ðv  C  CI Þy  1 svþ D 2D ey  B þ t1 D      1  2D2   2KD þ hB2 þ 2 ðC b  b π δÞ 1  eδt1 δt 1 C b eδt1  b πδ Þ 2D δ ð2:382Þ

E½TPðy, t 1 Þ ¼

where

116

2

Imperfect EOQ System

e1 ¼ 1  E ðpÞ > 0 h

i

e2 ¼ E ð1  pÞ2 þ

2E ðpÞD 2E ðpÞD ¼ e1 2 þ varðpÞ þ >0 x x

because E ð1  pÞ2 ¼ E ð1  m  p þ mÞ2 where h i m ¼ EðpÞ ¼ E ð1  mÞ2  2ð1  mÞðp  mÞ þ ðp  mÞ2 h i ¼ ð1  mÞ2  2ð1  mÞðE ðpÞ  mÞ þ E ðp  mÞ2 ¼ ð1  EðpÞÞ2 þ varðpÞ ¼ e1 2 þ varðpÞ In order to determine the optimal values of decision variable lot size y and shortage period t1, which maximize the expected average profit, Roy et al. (2011) ½TP  ½TP set ∂E∂y ¼ 0 ¼ ∂E∂t½TP . Now ∂E∂y ¼ 0 gives: 1 he1 e2 y2 þ 2he2 γ ðt 1 Þy  ½αðt 1 Þγ ðt 1 Þ þ e1 βðt 1 Þ ¼ 0

ð2:383Þ

where     hB þ ðv  C  C I Þ αðt 1 Þ ¼ 2D e1 s  v þ D     2D2  βðt 1 Þ ¼ 2KD þ hB2 þ 2 ðCb  b π δÞ 1  eδt1  δt 1 C b eδt1  b πδ δ γ ðt 1 Þ ¼ Dt 1  B The equation can be written as: Fy2 þ Gy  H ¼ 0 The solution of the above equation is: y ð t 1 Þ ¼ where

h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 G þ G2 þ 4FH 2F

ð2:384Þ

2.5 EOQ Model with Partial Backordering

117

F ¼ he1 e2 > 0 G ¼ 2he2 γ ðt 1 Þ > 0 H ¼ αðt 1 Þγ ðt 1 Þ þ e1 βðt 1 Þ  For a positive value of y, G must be positive. Now ∂E∂t½TP ¼ 0 gives: 1

y ðt 1 Þ ¼

h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 M þ M 2 þ 4LN 2L

ð2:385Þ

where L ¼ e1

  ∂αðt 1 Þ ∂γ ðt 1 Þ þ he2 ¼ Dh 2e21 eδt1 þ 1  eδt1 e2 > 0 ∂t 1 ∂t 1

∂βðt 1 Þ ∂γ ðt 1 Þ ∂αðt1 Þ þ α ðt 1 Þ  γ ðt 1 Þ ∂t 1 ∂t 1 ∂t   1      2h δt ðv  s  C I Þ h 2 δt1 2 δt1 1e e 1 þsvþ ¼ 2D e1 þ 1e þb π þ t 1 eδt1 ðCb  hÞ g δ e1 M ¼ e1

∂βðt 1 Þ ∂γ ðt 1 Þ N ¼ γ ðt 1 Þ  βðt 1 Þ ∂t1 ∂t1          K ht 1 eδt1  C b þ heδt1  h  3 2 δt1 δt 1 2 δt 1 3 ¼ 2D t 1 C b e   1  e  1  eδt1  1  e D δ 2δ2 δ2

From Eqs. (2.384) and (2.385), they got for feasibility of their model (Roy et al. 2011): h h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 1 G þ G2 þ 4FH ¼ M þ M 2 þ 4LN 2F 2L

ð2:386Þ

Solving Eq. (2.386), and the intersecting point of the functions y*(t1) and y**(t1), they obtained the optimum value t1*, and hence, using either Eq. (2.384) or Eq. (2.385), they got the optimum value of the lot size y, i.e., y*. Example 2.18 Roy et al. (2011) supposed that the imperfect-quality items in each lot follow a uniform distribution with the following probability density function:  f ðpÞ ¼

10 0

0
p
0:1 otherwise

K ¼ $400/lot, D ¼ 16,000 units/year, h ¼ $4 unit/time, Cb ¼ $6 unit/time, x ¼ 60,000 units/year, CI ¼ $0.5/unit, C ¼ $40/unit, s ¼ $62/unit, b π ¼ $23/unit, and v ¼ $30/unit. Using the above parameters, they had e1 ¼ 0.95, Var( p) ¼ 0.0025/ 3, e2 ¼ 0.93, and the optimum solutions y* ¼ 2082.47 units, t1* ¼ 0.0264187 ¼ 0.03, and E[TP]* ¼ $328,690 because the Hessian matrix H is negative definite.

118

2

Imperfect EOQ System

Example 2.19 Roy et al. (2011) considered that all the parameters of Example 2.18 are the same except b π ¼ $8. Then, the optimal solution is y* ¼ 2100.06 units, t1* ¼ 0.03 units, and E[TP] ¼ $328,849. Since at the derived solution values, the Hessian matrix H is negative definite, all are optimal solutions.

2.5.2

Screening

Wang et al. (2015) developed an imperfect inventory system with partial backordering. They assumed that the replenishment rate is infinite and all items are screened 100% with a known screening rate. At the end of the screening period, the imperfect-quality items are removed from the stock. Furthermore, to meet demand, similar to previous case (1  p)x  D > 0. The screening and demand proceed simultaneously. The shortages are backordered constrained by the screening rate. During the backordering period, it is assumed that the existed backorders are filled before the new incoming orders are met. In order to model the presented problem, some new notations which are specifically used are shown in Table 2.30. Since the new demands cannot be filled immediately, it is also assumed that only a fraction of the new orders would wait during the backordering period, i.e., the demand rate is βD during the backordering period (see Fig. 2.23). The following condition C + CI < (1  p)s + pv must be satisfied. The purchase and screening cost for each product is C + CI. Since (1  p)  100% good and p  100% imperfect items are sold, (1  p)s + pv is the average unit price. This constraint is to ensure the retailer has the benefit from selling products. According to Fig. 2.23 during the shortage period t1, a portion of demand with a rate of βD is backordered. At the beginning of screening interval t2 + t3, products start being screened upon being delivered to the retailer. The products are screened to remove the good ones from imperfect items. Since the delivered products contain p rate imperfect items, the rate of good-quality items is 1  p. Since the screening rate is x, the good items with a rate of (1  p)x are used to meet the demand and eliminate backorders during the period t2. Since the demand rate during the backordering period t2 is βD, during the period t2, one can see that backorders are eliminated at a rate of (1  p)x  βD, and the stock decreases at a rate of (1  p)x. During the period t3 and t4, all incoming demands are filled; the stock decreases at a rate of D. The screening process terminates at the end of the period t3; the imperfect items are subtracted from the stock and these are sold at a discounted price (Wang et al. 2015).

Table 2.30 Notations of a given problem

t2 t t3

Time to eliminate backorders (decision variable) (time) t2 + βt3 (years) (decision variable) (time) Time to screen after eliminating backorders (time)

2.5 EOQ Model with Partial Backordering Fig. 2.23 Graphic representation of the inventory system (see online version for colors) (Wang et al. 2015)

119

Inventory z4 - ( 1-p)x z3

-D

(1-p)x-βD z2 z1

px(t2+t3) -D Time

-B -βD

t1

t2

t3

t4

Wang et al. (2015) first used t2 and t3 to obtain the equations for related expressions. Referring to Fig. 2.23, the maximal backordering level is given by: B ¼ ðð1  pÞx  βDÞt 2

ð2:387Þ

B ¼ βDt 1 ¼ ðð1  pÞx  βDÞt 2

ð2:388Þ

using,

in Fig. 2.23 holds, the shortage interval is t 1 ¼ ðð1  pÞx  βDÞt 2 =βD Since during t2 + t3 the order size is screened at a rate of x, one has (Wang et al. 2015): z 4 ¼ xð t 2 þ t 3 Þ

ð2:389Þ

According to Fig. 2.23, we have: z3 ¼ z4  ð1  pÞxt 2 ¼ ðpt 2 þ t 3 Þx

ð2:390Þ

z2 ¼ z3  Dt 3 ¼ ðpt 2 þ t 3 Þx  Dt 3

ð2:391Þ

z1 ¼ z2  pðt 2 þ t 3 Þx ¼ ðð1  pÞx  βDÞt 3

ð2:392Þ

120

2

t4 ¼

Imperfect EOQ System

z1 ðð1  pÞx  βDÞt 3 ¼ D D

ð2:393Þ

T ¼ t 1 þ t 2 þ t 3 þ t 4 ¼ ð1  pÞxðt 2 þ βt 3 Þ=βD

ð2:394Þ

The total inventory cost per cycle is (Wang et al. 2015): Holding cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ z}|{ t ðz þ z Þ t ðz þ z Þ t z TCðt 2 , t 3 Þ ¼Cxðt 2 þ t 3 Þ þ C I xðt 2 þ t 3 Þ þ K þ h 2 4 3 þ 3 3 2 þ 4 1 2 2 2 C ðt þ t ÞB þ b 2 1 þb π ð1  βÞDðt 2 þ t 1 Þ 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Purchasing cost

Backordering cost

Screening cost

Fixed cost

Lost sale cost

Dx½ð1 þ pÞt 22 þ 2ð1 þ pÞt 2 t 3 þ 2p23 þ x2 ð1  pÞ2 t 23 ¼ Cxðt 2 þ t 3 Þ þ CI xðt 2 þ t 3 Þ þ K þ h 2D " # 2 2 2 2 ð1  pÞ x t 2 ð1  pÞxt 2 ð1  pÞð1  βÞxt 2 þCb  þb π 2βD 2 β ð2:395Þ Since it is not possible to derive the closed-form optimal solution of (t2, t3), Wang et al. (2015) developed an alternative variable transformation approach where t2 + βt3 is replaced by t in Eq. (2.395) and derived the optimal solution of (t2, t): TCðt 2 , t ¼ t 2 þ βt 3 Þ ¼

Cxðt  ð1  βÞt 2 Þ C I xðt  ð1  βÞt 2 Þ þ þK β β

2 ð1  pÞ2 x2 ðt  t 2 Þ2 xð2pðt  t 2 Þ þ 2ð1 þ pÞðt  t 2 Þt 2 β þ ð1 þ pÞβ2 t 22 þh þ 2 2β D 2β2 ! ð1  pÞ2 x2 t 22 ð1  pÞxt 22 ð1  pÞð1  βÞxt 2 þCb  þb π 2βD 2 β

!

ð2:396Þ The total revenue per cycle, TR, is comprised of sales revenue of good items and salvage value of imperfect items: TRðt 2 , t Þ ¼ sð1  pÞxðt 2 þ t 3 Þ þ vpxðt 2 þ t 3 Þ ð1  pÞsxðt  ð1  βÞt 2 Þ pvxðt  ð1  βÞt 2 Þ þ ¼ β β From Eq. (2.395), the ordering cycle can be rewritten as:

ð2:397Þ

2.5 EOQ Model with Partial Backordering

T ¼ t1 þ t2 þ t3 þ t4 ¼

ð1  pÞxðt 2 þ βt 3 Þ ð1  pÞxt ¼ β βD

121

ð2:398Þ

The expected total profit per unit time is given as:    E½TR  E ½TC  t2 vE 1 D vE 1 D t2 E ½TPUðt 2 , t Þ ¼  ð1  β Þ þ ¼ sD  sDð1  βÞ t E2 E2 t E ½T      CD CD t CI D CI D t KDβ   ð1  β Þ 2   ð1  βÞ 2  E2 E2 E2 E2 E 2 xt t t   2 hð2E1 D þ E3 xÞ hð1 þ E 1 ÞDt 2 hð1 þ E1 ÞDð2  βÞt 2 2 t  þ t 1 2  2E2 β t E2 2E 2 t Cb ðE 3 x  E 2 DβÞ t 2 t2  b π D ð1  β Þ 2E2 t t ð2:399Þ where E1 ¼ E[p], E2 ¼ E[1  p], and E3 ¼ E[(1  p)2]. Taking the expectation value for the constraint of (1  p)x  D > 0 yields E2x  D > 0. After algebraic manipulation, Eq. (2.399) is written as (Wang et al. 2015): 

 vE 1 C C I   E ½TPUðt 2 , t Þ ¼ s þ D E2 E2 E2 3 2   KDβ hð2E 1 D þ E 3 xÞ t 2 2 hð1 þ E 1 ÞDt 2 þ þ t 1  7 6 E 2 xt 2E 2 β t E2 7 6 7 6 hð1 þ E ÞDð2  βÞt 2 C ðE x  E DβÞ t2 7 6 1 2 b 3 2  6 þ 7 2E2 2E2 t t 7 6  7 6  5 4 vE C C t π Dð1  βÞ 2 þ sþ 1  I þb E2 E2 E2 t ð2:400Þ Their objective was to maximize E[TPU(t2, t)] subject to t > 0, t2  0, and t  t2. According to Fig. 2.23, it can be seen t3  0. Since t ¼ t2 + βt3, one has t 3 ¼ ttβ 2  0, i.e., t  t2. If t3 < 0, it shows t4 in Eq. (2.393) and z1 in Eq. (2.392) are less than zero. This implies that the model cannot fill all demand and backorders during an ordering cycle. It is obvious that (Wang et al. 2015): 8 8 Max E ½TPUðt 2 , t 3 Þ Max E ½TPUðt 2 , t 3 Þ > > > > > > > > > > > > < s:t: < s:t: are equivalent. and t > 0 t2 þ t3 > 0 > > > > > > t2  0 t2  0 > > > > > > : : t3  0 t  t2

122

2

Imperfect EOQ System

  CI C 1 Since s þ vE E 2  E 2  E 2 D is a constant with respect to the decision variables, t2 and t, Wang et al. (2015) maximized E[TPU(t2, t)] by minimizing the expression in the square bracket in Eq. (2.400), which is defined as:   KDβ hð2E1 D þ E3 xÞ t 2 hð1 þ E1 ÞDt 2 hð1 þ E1 ÞDð2  βÞt 2 2  þ t 1 2 þ E 2 xt t E2 2E 2 t 2E2 β   Cb ðE 3 x  E 2 DβÞ t 2 vE 1 C CI t2 þ   þb π Dð 1  β Þ þ sþ 2E2 t E2 E2 E2 t

E ½TVCUðt 2 , t Þ ¼

ð2:401Þ For further ease of notation, Wang et al. (2015) defined: r 0 ¼ sDð1  βÞ

ð2:402Þ

vE 1 Dð1  βÞ E2

ð2:403Þ

r2 ¼

CDð1  βÞ E2

ð2:404Þ

r3 ¼

C I D ð1  β Þ E2

ð2:405Þ

KDβ E2 x

ð2:406Þ

hð2E1 D þ E 3 xÞDβ 2E2 β

ð2:407Þ

hð1 þ E 1 ÞD E2

ð2:408Þ

hð1 þ E1 ÞDð2  βÞ 2E 2

ð2:409Þ

CsðE 3 x  E 2 DβÞ 2E 2

ð2:410Þ

r1 ¼

r4 ¼ r5 ¼

r6 ¼ r7 ¼

r8 ¼

π D ð1  β Þ r9 ¼ b

ð2:411Þ

Therefore, Eq. (2.401) is written as (Wang et al. 2015): E ½TVCUðt 2 , t Þ ¼ ðr 0 þ r 1  r 2  r 3 þ r 9 Þ  r1

t2 2 t 2 þ r8 2 t t

  t2 r4 t 2 þ þ r5 t 1  2 þ r6 t2 t t t ð2:412Þ

From Eq. (2.389), they saw if t2 ¼ 0, then t1 ¼ 0 (or t1 + t2 ¼ 0) which implies backorders are not allowed. If the time t is given, Lemma 2.1 provides a criterion to

2.5 EOQ Model with Partial Backordering

123

decide whether a shortage period is greater than zero or not. Therefore, Lemma 1 (1) (a) and (2) (a) of Wang et al. (2015) show that if the time t is less than a specific value, the optimal policy is to fill all demand without backorders. Lemma 1 (1) (b) and (2) (b) of Wang et al. (2015) imply that if the time t is greater than a particular value, the optimal policy is to allow shortage period. Since β > 0 and the definition t ¼ t2 + βt3, they saw t2 ¼ t yields t3 ¼ 0, and t4 in Eq. (2.394) is zero. Therefore, Lemma 1 (1) (c) of Wang et al. (2015) indicates that the remaining backorders are fulfilled at once when the order lot finishes the screening process. Note that if an optimal t is given, the conditions for the optimal t2 in Lemma 1 of Wang et al. (2015) still hold. Let t2+ and t+ be solutions that satisfy the first-order condition of E[TVCU(t2, t)]. Taking the first-order derivatives of E[TVCU(t2, t)] with respect to t2 and t, one has: ∂½TVCUðt 2 , t Þ ðr 0 þ r 1  r 2  r 3 þ r 9 Þ  ð2r 5  r 6 Þt þ 2ðr 5  r 7 þ r 8 Þt 2 ¼ t ∂t 2 ð2:413Þ ∂½TVCUðt 2 , t Þ r 4  r 5 t 2 þ ðr 0 þ r 1  r 2  r 3 þ r 9 Þt 2 þ 2ðr 5  r 7 þ r 8 Þt 2 2 ¼ t2 ∂t ð2:414Þ ðt 2 , t Þ ðt 2 , t Þ Letting ∂½TVCU ¼ 0 and ∂½TVCU ¼ 0, then ∂t 2 ∂t

ðr 0 þ r 1  r 2  r 3 þ r 9 Þ þ ð2r 5  r 6 Þt 2ð r 5  r 7 þ r 8 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 4 þ ðr 0 þ r 1  r 2  r 3 þ r 9 Þt 2 þ ðr 5  r 7 þ r 8 Þt 22 t ðt 2 Þ ¼ r5 t 2 ðt Þ ¼

ð2:415Þ ð2:416Þ

Substituting t2 in Eq. (2.415) into Eq. (2.416), one can derive the solution of t as (Wang et al. 2015): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r 4 ðr 5  r 7 þ r 8 Þ  ðr 0 þ r 1  r 2  r 3 þ r 9 Þ2 tþ ¼ 4r 5 ðr 6  r 7 þ r 8 Þ  r 26

ð2:417Þ

Substituting t+ in Eq. (2.416) into Eq. (2.417), the solution of t2 is given as (Wang et al. 2015): tþ 2 ¼

ðr 0 þ r 1  r 2  r 3 þ r 9 Þ þ ð2r 5  r 6 Þt þ 2ð r 5  r 7 þ r 8 Þ

ð2:418Þ

2 r 3 þr 9 , it can be seen that t2+ in Eq. (2.418) is greater than or equal If t þ  r0 þr12rr5 r 6 to zero.

124

2

Imperfect EOQ System

Let the optimal t2 and t be t2* and t*, respectively. If shortage period is not allowed, substituting t2 as zero into E[TVCU(t2, t)] of Eq. (2.412), one can derive the optimal t as (Wang et al. 2015): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi r4 2DK t ¼ ¼β r5 xhð2E1 þ E3 xÞ 

ð2:419Þ

The t* in Eq. (2.419) can also be confirmed by substituting t2 as zero into Eq. (2.416). The optimal expected variable cost per unit time without backorders is (Wang et al. 2015):  rffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DhK ð2E 1 D þ E3 xÞ r4 pffiffiffi ¼ E TVCU 0, r5 E2 x 

ð2:420Þ

If t* results in t ¼ 0, qffiffiffithey obtained that t* in Eq. (2.419) must be less than 2 r 3 þr 9 Thus, if rr45 < r0 þr12rr5 r , then t2* is derived, i.e., the optimal 6  qffiffiffi solution is 0, rr45 .

r0 þr1 r2 r3 þr9 . 2r 5 r 6

If β ¼ 1 (i.e., complete backordering), it results in (Wang et al. 2015): rffiffiffiffi r4 r0 þ r1  r2  r3 þ r9 ð,r 0 þ r 1  r 2  r 3 þ r 9 ¼ 0Þ  r5 2r 5  r 6 and 2r 7  2r 8  r 6 ¼

Cb ðE3 x  E 2 DÞ
0ð E 3 x  E 2 D > 0Þ E2

If (t2**, t) is (t2++, t), substituting t+ in Eq. (2.417) into Eq. (2.398), and taking the expectation value, the optimal cycle is: T ¼

E2 xt þ βD

ð2:421Þ

and the optimal ordering size is:   t þ  ð1  βÞt 2 y ¼ z4 ¼ x t 2 þ t 3 ¼ x β

ð2:422Þ

The optimal cycle length and the optimal ordering size without backorders are derived as (Wang et al. 2015):

2.5 EOQ Model with Partial Backordering

125

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi E 2 xt þ r 4 E 2 x 2KD 2Kx ¼ E2 β T ¼ ¼ βD r 5 βD hxðE 3 x  2E 1 DÞ hDðE 3 x  2E 1 DÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v u x r4 2KDx 2KD u  i  y ¼ ¼ ¼t  h β r5 hðE3 x þ 2E 1 DÞ h E ð1  pÞ2 þ 2E ½pD=x 

ð2:423Þ ð2:424Þ

Subsequently, to compare with the existed EOQ models, they adapted Theorem 1 of Wang et al. (2015) using the critical value of β as β* (see, e.g., Montgomery et al. 1973; Rosenberg 1979; Park 1982; Pentico and Drake 2009) to determine the optimal policy. Let β* be the solution of β derived from: rffiffiffiffi r4 r0 þ r1  r2  r3 þ r9 ¼0  r5 2r 5  r 6 It gives (Wang et al. 2015): pffiffiffiffiffiffi β ¼ 1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Kh½E3 xð1E1 ÞD pffiffiffiffiffiffi ¼ β  β means DxðE 3 xþ2E 1 DÞðE 2 ðsþb π ÞþE 1 vCCI Þ 2Khð1þE 1 ÞD backlogging unfilled demand would have more cost-efficiency than holding stock throughout the cycle. If β 0 and t þ < r0 þr2r17r 2r 8 r , let (t2*, t*) ¼ (t2 , t ).

126

2

Imperfect EOQ System

+ + 2 r 3 þr 9 If 2r7  2r8  r6 > 0 and t þ  r0 þr2r17r 2r 8 r , let (t2*, t*) ¼ (t2 , t ). + + If 2r7  2r8  r6
0, let (t2*, t*) ¼ (t2 , t ).   Substitute (t2, t) by (t2*, t*) into Eq. (2.412) to derive E TVCU t 2 , t  .     If E TVCU t 2 , t π Þ þ E 1 v  C  C I Þ ED2 , the optimal policy is < ðE 2 ðs þ b to meet the demand with partial backordering resulting in an optimal solution (t2*, t*).   π Þ þ E 1 v  C  C I Þ ED2 , the optimal policy is (f) If E TVCU t 2 , t   ðE 2 ðs þ b to lose all sales.

(b) (c) (d) (e)

Example 2.20 Wang et al. (2015) presented an example using:  f ð pÞ ¼

25 0

0
p
0:04 otherwise

K ¼ $500/lot, D ¼ 1000 units/year, h ¼ $10 unit/year, Cb ¼ $5 unit/year, x ¼ 4000 units, b π ¼ $1/unit, CI ¼ $1/unit, C ¼ $40/unit, s ¼ $50/unit, and v ¼ $10/unit. Using the above parameters, E1 ¼ E[p] ¼ 0.02, E2 ¼ E [1  p] ¼ 0.98, and E3 ¼ E[(1  p)2] ¼ 0.960533 and β ¼ 0.72531. If a given β ¼ 0.5 < β* ¼ q 0.72531, 2 should be applied to derive the optimal solution. qStep ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi r4  2DK from Eq. (2.241). Since Calculate t ¼ r5 ¼ β xhð2E1 þE3 xÞ ¼ 0:072222 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DhK ð2E 1 DþE 3 xÞ pffi ¼ 3178:92 < ðE 2 ðs þ b π Þ þ E 1 v  C  CI Þ ED2 ¼ 9367:35, the optixE 2   mal solution is t 2 , t ¼ ð0, 0:072222Þ. But for a given β ¼ 0.9  β* ¼ 0.72531, Step 3 should be applied. So r0 ¼ 5000, r1 ¼ 20.41, r2 ¼ 4081.63, r4 ¼ 114.80, r5 ¼ 22, 007.56, r6 ¼ 10, 408.16, r7 ¼ 5724.49, r8 ¼ 7551.36, r9 ¼ 100, and t  ¼   0:10194, t 2 ¼ 0:05222 . Since   2r7  2r8  r6 ¼  14, 061.9    
0, ðt 2 , t Þ ¼  . Since E TVCU t 2 , t ¼ $2732:4 < t 2 ¼ 0:05222, t ¼ 0:10194 D ðE 2 ðs þ b π Þ þ E 1 v  C  CI Þ E2 ¼ $9367:35 , the optimal policy is to meet the   demand with partial backordering. The optimal solution is t2 , t ¼ ð0:05222, 0:10194Þ with total variable cost per year of $2732.04. Using   þ 2 xt ¼ 0:44401, y ¼ z4 ¼ x t 2 þ t 3 ¼ Eqs. (2.421) and (2.422), T  ¼ EβD x

t þ ð1βÞt 2 β

2.5.3

¼ 157:699, B ¼ ((1  p)x  βD)t2 ¼ 157.699.

Reparation of Imperfect Products

In this section, the work of Taleizadeh et al. (2016b) is presented. Consider the situation where there exists a purchaser which buys the products from a supplier that is located far away. Due to process failure or due to the mishandling of products during transportation, it is possible that the lot contains some imperfect products. By doing an inspection process, the buyer detects imperfect products, and these must be

2.5 EOQ Model with Partial Backordering

127

Table 2.31 Notations of a given problem ti tR tT x K Ks h2 C1 m Decision variables T F Q

Inspection time of products (time/unit) Transportation, repair, and return time of imperfect products (time/unit) Total transportation time of imperfect products (time/unit) Inspection rate (units/time unit) Buyer’s ordering cost ($/order) Repair setup cost ($/setup) Holding cost at the repair facility ($/unit/time unit) Material and labor cost to repair a product ($/unit) Markup percentage by the repair shop (%) Cycle time (time unit) Percentage of duration in which inventory level is positive (%) Order quantity (units)

replaced by perfect products. Due to the fact that the lead time is high because the supplier is far away, the buyer cannot make an additional order to the same supplier with the purpose of substituting the imperfect products. In some situations, the imperfect products have a significant value and they must be repaired. Here, it is considered that the imperfect products are repaired. The demand is constant and known. Shortages are allowed and partially backordered. After the batch is received, firstly, the backordered demand is satisfied and then products are inspected in order to find imperfect products. The imperfect products can be repairable. It is assumed that after the repair process, the products are as good as new. So at the end of the screening period, the imperfect products are withdrawn and sent to a local repair shop in order to be repaired. After their reparation, they are added to inventory. There exists a repair cost and the holding cost of repaired products is higher than the initial holding cost. The total cost at the repair shop consists of fixed and variable cost. The fixed cost is comprised of the repair setup cost and round trip fixed cost to the repair shop. The variable cost consists of the unit transportation cost, unit material and labor cost, and unit holding cost at the repair facility. Additionally, it is assumed that in the repair shop, the repair process is always in control and all the imperfect products can be repaired. In order to model the presented problem, some new notations which are specifically used are shown in Table 2.31. According to the time when the repaired products are added to inventory, four cases are identified and studied. These are the following: Case I. The repaired products are received when the inventory level is positive. Case II. The repaired products are received when the inventory level is zero. Case III. The repaired products are received when the shortage quantity is equal to imperfect products’ quantity. Case IV. The repaired products are received when shortage still remains.

128

2

Imperfect EOQ System

The main goal of the inventory model is to obtain the optimal cycle time (T ), the percentage of the cycle time (F), and the lot size (Q) in order to maximize the total profit. In order to identify imperfect products, the whole lot is inspected at rate x where the inspection time is ti ¼ Imax/x. The buyer must send imperfect products to the repair shop in order to convert them into perfect products. The inspection rate is greater than the demand rate (x > D). Both demand and inspection rates are constant and known. The proportion of products which is imperfect (ρ) and its probability density function are given and known. At the end of the screening period (ti), the imperfect products are withdrawn and sent to a local repair shop. Repaired products return after tR units of time that includes repair and transportation times; here it is assumed that ti + tR
T where T is the cycle duration. The repair process at the repair shop is in control. The fixed cost in the repair shop is determined with KR + 2KS where KR is the repair setup cost and KS is the transportation fixed cost. The variable cost per imperfect product is calculated with C1 + 2CT + h2tR where C1 is the material and labor cost to repair a product, CT is the transportation cost per unit, and h2 is the unit holding cost at the repair store. The time tR consists of repair time at rate R and total transportation time tT of imperfect products. The unit holding cost of repaired products is h1. Here, h1 is greater than the initial unit holding cost (h). FT is the time in which inventory level is positive and (1  F)T is the time in which the shortage occurs. Case I At the beginning of cycle time (T ), the maximum inventory level is Imax ¼ FTD. The products are screened and the inspection time is ti ¼ FTD/x. During the screening process, a ρ percent of products is identified as imperfect products (ρFTD). At the end of inspection time, imperfect products are withdrawn from inventory and sent to the repair shop. After the repair time, the repaired products are added to inventory. Here, it is considered that when the repaired products are received, the inventory level is still positive. Therefore, the inventory level increases ρFTD units when the repaired products arrive. When the inventory level reaches zero, then the shortages start to occur; the shortage quantity is determined by (1  F)TD. The β(1  F)TD is the shortage backordered quantity and the rest is lost sales quantity (1  β)(1  F)TD. Figure 2.24 illustrates the behavior of inventory for Case I. Repair duration is tR ¼ ρFTD/R + tT, and the total cost in the repair shop is KR + 2KS + ρFTD(C1 + 2CT + h2tR), and m margin per unit is claimed as repair charge. The unit repair cost charged to the buyer is therefore (Taleizadeh et al. 2016b):    K R þ 2K S CR ðFTDÞ ¼ ð1 þ mÞ þ ðC 1 þ 2C T þ h2 t R Þ ρFTD

ð2:425Þ

The order quantity of products per cycle is Q ¼ FTD + β(1  F)TD. The total holding cost (HC) per time unit is given by (Taleizadeh et al. 2016b):

2.5 EOQ Model with Partial Backordering

129

Inventory Imax=FTD

ρ FTD

ti

tR

t

β (1-F)TD FT

(1-F)T

(1- β)(1-F)TD

T

Fig. 2.24 Inventory level in Case I (Taleizadeh et al. 2016b)



 ð1  ρÞ2 F 2 TD ρT ðFDÞ2 þ HC ¼ h 2 x     FTD ρFTD F 2 TD þ hR ρF 2 TD  ρFD þ þ t T  ρ2 x R 2

ð2:426Þ

where h is the holding cost of perfect products and hR is the holding cost of repaired products, and the shortage cost (SC) per time unit is (Taleizadeh et al. 2016b): SC ¼ C b

βð1  F Þ2 TD þ gð1  βÞð1  F ÞD 2

ð2:427Þ

where Cb is the backordered cost and g is lost sales cost. β is the percentage of backordered demand. Obviously, the main goal of the buyer is to maximize his/her own profit per time unit (TP). The total profit (TP) per time unit is equal to the total revenue per time unit less the total cost per time unit. Thus, TP(T, F) is expressed as (Taleizadeh et al. 2016b): h K TPðT,F Þ ¼ sðFD þ βð1  F ÞDÞ  þ C ðFD þ βð1  F ÞDÞ þ C I FD T        ð1  ρÞ2 F 2 TD ρT ðFDÞ2 K R þ 2K S ρFTD þðρFDÞð1 þ mÞ þh þ þ C 1 þ 2C T þ h2 þ tT 2 ρFTD x R    2   2 βð1  F Þ TD FTD ρFTD F TD þ gð1  βÞð1  F ÞD þhR ρF 2 TD  ρFD þ þ t T  ρ2 þ Cb 2 x R 2

ð2:428Þ

130

2

Imperfect EOQ System

where s is the unit price of sales, K is the fixed ordered cost, C is the unit product cost, and CI is the unit inspection cost. The optimal value for the percentage of duration of the cycle period (F) and the optimal value for length duration (T ) are given below (Taleizadeh et al. 2016b): C βT  ðC I þ ρð1 þ mÞðC 1 þ 2CT þ h2 t T Þ  hR t T ρ  b π ð1  β ÞÞ F ¼ b ρ D

ρD ρ D

hþCb β  ð2:429Þ h ð1þmÞ 2 2 ρ D R þ ρh 2 þ x  1  ρhR R þ 2 þ x  1 þ 2 T vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi u h i h i ρ D ρD ρ D h þ Cb β u h ð1 þ mÞ þ ρh þ  1  ρhR þ þ 1 þ u ðK þ ðK R þ 2K S Þð1 þ mÞÞ ρ2 D R 2 x R 2 x 2 u u u D 2 u  ðCI þ ρð1 þ mÞðC1 þ 2C T þ h2 t T Þ  hR t T ρ  b π ð1  βÞÞ u 4  T ¼t



Dh h ð 1þm Þ Cb β ρ D ρ2 D2 2 R þ ρDh ρ2 þ Dx  1  ρDhR ρD 2 R þ2þ x 1 þ 2

ð2:430Þ where b π is given by b π ¼ s þ g  C. Note that the expected value of ρ and ρ2 must be substituted in Eqs. (2.429) and (2.430). The expected value of Eq. (2.428) is given as: h K ETPðT, F Þ ¼ sðFD þ βð1  F ÞDÞ  þ C ðFD þ βð1  F ÞDÞ þ C I FD T   E ðρ2 Þh2 D2 F 2 T K R þ 2K S þ ð1 þ m Þ þ EðρÞFDðC 1 þ 2CT þ h2 t T Þ þ R T   E ð1  ρÞ2 hF 2 TD E ðρÞD2 hTF 2 βð1  F Þ2 TD þ þ gð1  βÞð1  F ÞD þ Cb þ 2 x 2   E ðρÞD2 F 2 T Eðρ2 ÞD2 F 2 T E ðρ2 ÞDF 2 T 2   EðρÞFDt T  þhR E ðρÞF TD  x R 2 ð2:431Þ As a result, Eqs. (2.430) and (2.431) are rewritten as: C b βT  ðCI þ EðρÞð1 þ mÞðC1 þ 2C T þ h2 t T Þ  EðρÞh1 t T  b π ð1  β ÞÞ h 2 i h 2 i F¼ 

 E ðρ Þ E ðρÞD Eðρ ÞD Eðρ2 Þ EðρÞD h ð1þmÞ bβ 2 T 2 Eðρ ÞD R þ h 2 þ x  EðρÞ  h1 R þ 2 þ x  E ðρÞ þ hþC 2

ð2:432Þ

2.5 EOQ Model with Partial Backordering

131

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u ðE ðρ2 ÞDh2 ð1 þ mÞ E ðρ2 Þ E ðρÞD u ½K þ ðK R þ 2K S Þð1 þ mÞ½ þh þ  E ðρÞ u R 2 x u   u 2 2 E ðρ ÞD E ðρ Þ E ðρÞD h þ Cb β u þ þ  E ðρÞ þ hR u R 2 x 2 u u u D u π ð1  βÞÞ2 u   4 ðC I þ E ðρÞð1 hþ mÞðC 1 þ 2C T þ h2it T Þ  hRht T E ðρÞ  b  i T ¼t 2 2 2 Þ Cb β E ðρ2 ÞD2 h2 ð1þm þ Dh Eð2ρ Þ þ EðρxÞD  E ðρÞ  DhR EðρR ÞD þ Eð2ρ Þ þ EðρxÞD  E ðρÞ þ Dh 2 R 2

ð2:433Þ where b π is given by b π ¼ s þ g  C. Appendix A of Taleizadeh et al. (2016b) shows that the denominator of Eq. (2.433) is positive. The numerator of Eq. (2.433) must be positive. Thus (Taleizadeh et al. 2016b):   3 ðE ðρ2 ÞDh2 ð1 þ mÞ Eðρ2 Þ E ðρÞD þ h þ  E ð ρ Þ 7 6 R 2 x 7 6 7 6 w1 ¼ ½K þ ðK R þ 2K S Þð1 þ mÞ6   7 2 2 4 E ðρ ÞD E ðρ Þ EðρÞD h þ Cb β 5 hR þ þ  E ð ρÞ þ R 2 x 2 2

D  ðC I þ E ðρÞð1 þ mÞðC 1 þ 2C T þ h2 t T Þ  hR t T EðρÞ  b π ð1  β ÞÞ2 > 0 4 If w1 > 0, then T* is equal to T in Eq. (2.433). If w1
0, then T* is equal zero. E ðρ2 ÞD E ðρ2 Þ Also, if further condition R þ 2 þ EðρxÞD
E ðρÞis not satisfied, then T* is equal to zero. Case II In this case also, at beginning of the cycle, the inventory level is FTD. The products are screened at rate x; consequently, the inspection duration is equal to ti ¼ FTD/x. A ρ percent of products is found imperfect (ρFTD). The imperfect products are withdrawn and sent to a repair shop. The repair duration is tR ¼ ρFTD/ R + tT. The repair charge per unit is determined as in Case I with the following expression (Taleizadeh et al. 2016b):  ð1 þ m Þ

K R þ 2K S ρFTD



 þ ðC1 þ 2CT þ h2 t R Þ

Additionally, this case assumes that the repaired products arrive to the buyer’s store when the inventory level is exactly zero. Therefore, after adding the repaired products to inventory, then the inventory level reaches ρFTD units. At the end of the cycle, the shortage level is (1  F)TD. Figure 2.25 illustrates the behavior of inventory of Case II. The order quantity per cycle is Q ¼ FTD + β(1  F)TD. The holding cost (HC) per time unit is calculated as:

132

2

Imperfect EOQ System

Inventory Imax=FTD

tR

ti

t

FT

(1-F)T T

β (1-F)TD (1- β)(1-F)TD

ρ FT

Fig. 2.25 Inventory level for Case II (Taleizadeh et al. 2016b)



 ð1  ρÞ2 F 2 TD ρT ðFDÞ2 ðρF Þ2 DT þ HC ¼ h þ hR 2 2 x

ð2:434Þ

The shortage cost per time unit is given as in Case I: SC ¼ C b

βð1  F Þ2 TD þ gð1  βÞð1  F ÞD 2

ð2:435Þ

The total profit per time unit is equal to the total revenue per time unit less the total cost per time unit, and it is expressed below: TPðT, F Þ ¼ sDðF þ βð1  F ÞÞ 2  3 ð1  ρÞ2 F 2 TD ρT ðFDÞ2 K þ 7 6 T þ C ðFD þ βð1  F ÞDÞ þ CI FD þ h 2 x 6     7 7 6   7 6  6 þðρFDÞð1 þ mÞ K R þ 2K S þ C 1 þ 2C T þ h2 ρFTD þ t T 7 7 6 R ρFTD 7 6 5 4 2 2 ðρF Þ DT βð1  F Þ TD þ Cb þ gð1  βÞð1  F ÞD þhR 2 2 ð2:436Þ In Case II, the percentage of duration which inventory level is positive (F) is obtained as:

2.5 EOQ Model with Partial Backordering

F¼

Cb βT  ðCI þ ρð1 þ mÞ½C1 þ 2C T þ h2 t T   b π ð1  β ÞÞ  2  2h2 ρ Dð1þmÞ 2 2ρhD 2 þ h ð 1  ρ Þ þ þ h ρ þ C β T R b x R

133

ð2:437Þ

The ngth duration (T ) is: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi u 2 2 ρ D ð 1 þ m Þ h ð 1  ρ Þ u ρhD ρ C β u ðK þ ð1 þ mÞðK R þ 2K S ÞÞ h þ þ hR þ b þ u R x 2 2 2 u u u  D ðC I þ ρð1 þ mÞ½C 1 þ 2CT þ h2 t T Þ  b π ð1  β ÞÞ2 u 4 h i T ¼t hDð1ρÞ2 C b β h2 ρ2 D2 ð1þmÞ ρhD2 ρ2 D þ þ þ h R 2 2 x 2 R ð2:438Þ where b π is given by b π ¼ s þ g  C. The expected value of Eq. (2.438) is given as: 3 K þ C ðFD þ βð1  F ÞDÞ þ C I FD 7 6T  7 6 2  3 7 6 E ð1  ρÞ2 F 2 TD E ðρÞD2 F 2 T 7 6 5 7 6 þh4 þ 7 6 x 2 7 6 E ðTPðT, F ÞÞ ¼ sDðF þ βð1  F ÞÞ  6 7   7 6 2 2 2 6 E ðρ Þh2 D F T 7 K þ 2K S 7 6 þð 1 þ m Þ R þ E ðρÞFDðC 1 þ 2C T þ h2 tT Þ þ 7 6 R T 7 6 5 4 2 Eðρ2 ÞDF 2 T βð1  F Þ TD þ Cb þ gð1  βÞð1  F ÞD þhR 2 2 2

ð2:439Þ Hence, Eqs. (2.438) and (2.439) are re-expressed as: F¼

C b βT  ðC I þ E ðρÞð1 þ mÞ½C 1 þ 2C T þ h2 t T   b π ð1  β ÞÞ    2h2 DEðρ2 Þð1þmÞ 2E ðρÞhD 2 2 þ hE ð 1  ρ Þ þ h E ð ρ Þ þ C β T þ R b R x

ð2:440Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   0 1ffi u u hE ð1  ρÞ2 2 2 u E ð ρ ÞD ð 1 þ m Þ E ð ρ ÞhD E ð ρ Þ C β u ðK þ ð1 þ mÞðK R þ 2K S ÞÞ@h þ þ þ hR þ b A u 2 R x 2 2 u u u u  D ðC I þ E ðρÞð1 þ mÞ½C 1 þ 2C T þ h2 t T Þ  b π ð1  βÞÞ2 u 4   T ¼u 2 t E ðρ2 ÞD C b β h2 E ðρ2 ÞD2 ð1þmÞ hDE ðð1ρÞ Þ E ðρÞhD2 þ þ þ h R 2 2 R 2 x

ð2:441Þ It is easy to see that the denominator of Eq. (2.427) is always positive. The numerator of Eq. (2.427) must be greater than zero. Thus:

134

2

Imperfect EOQ System

Inventory

ρ FTD

(1−ρ) FTD

FTD

FTD

tR

t1

βρFD (1−β)ρFDT

F2T

F1T

t

β (1-F)TD

FT

(1-F)T

(1- β)(1-F)TD

T

Fig. 2.26 Inventory level for Case III (Taleizadeh et al. 2016b)

0 h E ðρ ÞDð1 þ mÞ w2 ¼ ðK þ ð1 þ mÞðK R þ 2K S ÞÞ@ 2 þ R 2

  hE ð1  ρÞ2 2

1 E ðρÞhD E ð ρ2 Þ C b β A þ þ hR þ x 2 2

D  ðC I þ E ðρÞð1 þ mÞ½C 1 þ 2C T þ h2 t T Þ  b π ð1  βÞÞ2 > 0 4

If w2 > 0, then T* is equal to T in Eq. (2.441). Otherwise, if w2
0, then T* is equal zero. Case III The behavior of inventory level for Case III is illustrated in Fig. 2.26. F represents the fraction of cycle time in which the inventory level (without considering the imperfect products) is positive. It is equal to F1 + F2. These are given by F1 ¼ (1  ρ)F and F2 ¼ ρF. Fi (i ¼ 1, 2) is within the interval [0, 1]. The buyer sends the imperfect products to the hrepair shop.  The repair cost(Taleizadeh i ðK R þ2K S Þ et al. 2016b) per unit is given by ð1 þ mÞ þ C 1 þ 2CT þ h2 ρFTD ρFTD R þ tT .

While imperfect products are being repaired, the inventory system faces shortages. Then, the repaired products arrive when the shortage level is exactly the same as the repaired products’ quantity. Now, consequently, the inventory level is equal to zero. In this case, the order quantity per cycle is Q ¼ F1TD + βF2TD + β(1  F) TD ¼ F1TD + β(1  F1)TD. The holding cost (HC) is equal to: HC ¼ h and the shortage cost (SC) is:

  ð1  ρÞ2 F 2 TD ρF 2 TD2 þ 2 x

ð2:442Þ

2.5 EOQ Model with Partial Backordering

135

βð1  F Þ2 TD βρ2 F 2 TD þ gð1  βÞF 2 D þ gð1  βÞð1  F ÞD þ Cb 2 2 βð1  F Þ2 TD βρ2 F 2 TD þ gð1  βÞð1  F 1 ÞD SC ¼ Cb þ Cb 2 2 ð2:443Þ

SC ¼ Cb

Thus, the total profit per time unit is defined as: TPðT, F Þ ¼ sðF 1 D þ βð1  F 1 ÞDÞ 3 2   ð1  ρÞ2 F 2 TD ρF 2 TD2 K þ 7 6 T þ CDðF 1 þ βð1  F 1 ÞÞ þ h 2 x 7 6 6      7 7 6 ð K þ 2K Þ ρFTD R S  6 þρFDð1 þ mÞ þ C 1 þ 2C T þ h2 þ tT 7 7 6 ρFTD R 7 6   5 4 C b βTD 2 2 2 þC I FD þ ρ F þ ð1  F Þ þ gð1  βÞð1  F 1 ÞD 2 ð2:444Þ The optimal values for decision variables, the length duration of cycle time (T ) and the percentage of cycle time in which the inventory is positive (F), are obtained as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi u 2 u ρhD Cb βρ2 C b β h ρ D ð 1 þ m Þ hð 1  ρÞ u ðK þ ð1 þ mÞðK R þ 2K S ÞÞ þ þ þ þ u R x 2 2 2 u u D u  ðCI þ ρð1 þ mÞðC 1 þ 2CT þ h2 t T Þ  b π ð1  ρÞð1  βÞÞ2 u  2 2  T ¼t 4 2 C b β h2 ρ D ð1þmÞ hð1ρÞ D ρhD2 C b βρ2 D þ þ þ R 2 x 2 2 ð2:445Þ F¼

Cb βT  ðCI þ ρð1 þ mÞðC1 þ 2CT þ h2 t T Þ  b π ð1  ρÞð1  βÞÞ  2  2h2 ρ Dð1þmÞ 2 2ρhD þ hð1  ρÞ þ x þ C b βρ2 þ C b β T R

ð2:446Þ

The expected value of Eq. (2.443) is given as: 3  0  1 E ð1  ρÞ2 F 2 TD EðρÞF 2 TD2 7 6 K þ CDðF þ βð1  F ÞÞ þ h@ A þ 1 1 7 6T x 2 7 6 7 6 7 6   2 2 2 ETPðT,F Þ ¼ sðF 1 D þ βð1  F 1 ÞDÞ  6 7 E ð ρ Þh D F T K þ 2K 2 R S 7 6 þð1 þ mÞ þ E ð ρ ÞFD C ð þ 2C þ h t Þ þ 1 T 2 T 7 6 R T 7 6     5 4 C b βTD 2 2 2 þC I FD þ E ρ F þ ð1  F Þ þ gð1  βÞð1  F 1 ÞD 2 2

ð2:447Þ Thus, Eqs. (2.445) and (2.446) are rewritten as:

136

2

Imperfect EOQ System

Inventory

F2TD

F1TD

FTD

F2T

F 2T

F3 T

tR t1 F1T

(1- β)(1-F)TD

F4 T

(F2+F3+F4)T

t β (1-F)TD

T

Fig. 2.27 Inventory level for Case IV (Taleizadeh et al. 2016b) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   0 1ffi u 2 u hE ð 1  ρ Þ 2 2 u Eðρ Þh2 Dð1 þ mÞ EðρÞhD Cb βEðρ Þ C b βA u ðK þ ð1 þ mÞðK R þ 2K S ÞÞ@ þ þ þ þ u 2 R x 2 2 u u u u  D ðC I þ EðρÞð1 þ mÞðC 1 þ 2C T þ h2 tT Þ  b π ð1  EðρÞÞð1  βÞÞ2 u 4   T ¼u 2 t 2 Cb β E ðρ2 Þh2 D2 ð1þmÞ E ðð1ρÞ ÞhD E ðρÞhD2 b βD þ þ x þ Eðρ ÞC 2 R 2 2

ð2:448Þ F¼

C b βT  ðC I þ E ðρÞð1 þ mÞðC 1 þ 2C T þ h2 t T Þ  b π ð1  E ðρÞÞð1  βÞÞ     2h2 E ðρ2 ÞDð1þmÞ 2E ðρÞhD 2 þ hE ð1  ρÞ þ x þ Cb βE ðρ2 Þ þ C b β T R ð2:449Þ

The denominator of Eq. (2.449) is positive. On the other hand, its numerator could be positive, zero, or negative:   0 1 hE ð1  ρÞ2 2 2 E ð ρ Þh D ð 1 þ m Þ E ð ρ ÞhD C βE ð ρ Þ C β 2 b þ w3 ¼ ðK þ ð1 þ mÞðK R þ 2K S ÞÞ@ þ þ þ b A 2 R x 2 2 D  ðC I þ E ðρÞð1 þ mÞðC1 þ 2CT þ h2 tT Þ  b π ð1  E ðρÞÞð1  βÞÞ2 4

If w3 > 0, then T* is equal T in Eq. (2.449). Otherwise, if w3
0, then T* is equal zero. Case IV The inventory pattern for Case IV is shown in Fig. 2.27. In Case IV, the duration length of cycle time is divided into four parts. Each part is represented by Fi and it is in the interval [0, 1] where i ¼ 1, 2, 3, 4. Again, F denotes the fraction of

2.5 EOQ Model with Partial Backordering

137

length duration of the cycle time in which the inventory level is positive (Taleizadeh et al. 2016b). Here, F is equal to F1 + F2 and (1  F) is equal to F3 + F4. The inventory level at the beginning of the cycle is FTD. The products are inspected by rate x with duration of ti (ti ¼ FTD/x). At the end of inspection duration, the imperfect products 2 ÞTD þ (ρFTD ¼ F2TD) are sent to the buyer’s store. Repair duration is t R ¼ ρðF1 þF R h i  ρFTD K R þ2K S t T ¼ R þ t T and repair cost per unit is ð1 þ mÞ þ ðC1 þ 2C T þ h2 t R Þ . ρFTD Notice that when repaired products arrive at the buyer’s store, the shortage level is (F2 + F3)TD. After adding them, the inventory level is still negative. It means that the shortage remains until the end of the cycle; at this time, the shortages reach to (F2 + F3 + F4)TD units. In this case, the order quantity per cycle is Q ¼ (F1 + F2) TD + β(F3 + F4)TD ¼ FTD + β(1  F)TD. The holding cost (HC) of products is computed as:  HC ¼ h

ð1  ρÞ2 F 2 TD ρðF Þ2 TD2 þ 2 x

 ð2:450Þ

and the shortage cost (SC) is determined as: SC ¼ C b

ðF 2 þ F 3 þ F 4 ÞðF 3 þ F 4 ÞβTD þ gð1  βÞðF 3 þ F 4 ÞD 2

ð2:451Þ

The total profit is calculated as follows: the total revenue per time unit less the total cost per time unit. It is illustrated as: TPðT, F Þ ¼ sDðF þ βð1  F ÞÞ 3 2   ð1  ρÞ2 F 2 TD ρðF Þ2 TD2 K þ þ CD ð F þ β ð 1  F Þ Þ þ h 7 6T 2 x 7 6 7 6 7 6 þCI ðF 1 þ F 2 ÞD 6       7 6 7 6 þρDF ð1 þ mÞ K R þ 2K S þ C1 þ 2CT þ h2 ρFTD þ t T 7 7 6 R ρFTD 7 6 5 4 ðF 2 þ F 3 þ F 4 ÞðF 3 þ F 4 ÞβTD þ gð1  βÞð1  F ÞD þCb 2 ð2:452Þ It is clear that F + β(1  F) ¼ 1  1 + F + β(1  F) ¼ 1  (1  β)(1  F) and F2 + F3 + F4 ¼ 1  F1 ¼ 1  (1  ρ)F. Thus, the profit function is simplified as:

138

2

Imperfect EOQ System

    3 K K R þ 2K S ρFTD þ tT 7 þ C 1 þ 2C T þ h2 6 T þ C I DF þ ρDF ð1 þ mÞ R ρFTD 7 6 7 6   2 2 2 2 7 6 ð 1  ρ Þ F TD ρF TD TPðT,F Þ ¼ ðs  C ÞD  b π Dð1  βÞ  6 þh 7 þ 7 6 2 x 7 6 5 4 C b βTD þð1  ð1  ρÞF Þð1  F Þ b π Dð1  βÞF 2 2

ð2:453Þ The optimal values for F and T are given below: F¼

Cb βð2  ρÞT þ 2b π ð1  βÞ  2C I  2ρð1 þ mÞðC 1 þ 2C T þ h2 t T Þ   ð2:454Þ ð1þmÞh2 ρ2 D ð1ρÞ2 h C b βð1ρÞ ρhD 4 þ þ þ T x R 2 2

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   u u ð1 þ mÞh2 ρ2 D ð1  ρÞ2 h ρhD C b βð1  ρÞ u 4ðK þ ð1 þ mÞðK R þ 2K S ÞÞ þ þ þ u R 2 2 x u u 2 π ð1  βÞ þ ρð1 þ mÞðC1 þ 2C T þ h2 t T ÞÞ u DðC I  b   T ¼t 2 2 2 2 C b βDρ2 2C b β ð1þmÞhR 2 ρ D þ ð1ρ2Þ hD þ ρhD  x 8 ð2:455Þ The expected value of Eq. (2.453) is given as: ETPðT, F Þ ¼ ðs  C ÞD  b π Dð1  βÞ   3 2 E ðρ2 Þh2 D2 F 2 T K K R þ 2K S DF þ ð 1 þ m Þ ð þ 2C þ h t Þ þ þ E ð ρ ÞDF C þ C I 1 T 2 T 7 6T T R 7 6 7 6 0   1 6 7 2 2 E ð1  ρÞ F TD E ðρÞF 2 TD2 7 6 C βTD b 4 þh@ A þ ð1  ð1  E ðρÞÞF Þð1  F Þ b π Dð1  βÞF 5 þ x 2 2

ð2:456Þ Consequently, Eqs. (2.454) and (2.455) are re-expressed as: F¼

C b βð2  E ðρÞÞT þ 2b π ð1  βÞ  2C I  2E ðρÞð1 þ mÞðC1 þ 2C T þ h2 t T Þ   2 hE ðð1ρÞ2 Þ E ðρÞhD C b βð1E ðρÞÞ 4 ð1þmÞERðρ ÞDh2 þ þ þ T 2 x 2 ð2:457Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   0 1 u u E ð1  ρÞ2 h EðρÞhD C βð1  E ðρÞÞ 2 u ð 1 þ m ÞE ð ρ Þh D 2 b A u 4ðK þ ð1 þ mÞðK R þ 2K S ÞÞ@ þ þ þ u 2 R x 2 u u u π ð1  βÞ þ EðρÞð1 þ mÞðC1 þ 2C T þ h2 t T ÞÞ2 u DðC I  b   T ¼u t 2 0 2 Eðð1ρÞ2 ÞhD E ðρÞhD2 Eðρ2 ÞC b βD 2Cb β ð1þmÞERðρ Þh D þ þ  8 x 2

ð2:458Þ Notice that Eq. (2.458) is ever positive if its numerator becomes positive. Thus:

2.5 EOQ Model with Partial Backordering

139

Table 2.32 Data for the numerical example (Salameh and Jaber 2000; Taleizadeh et al. 2016b) Description Demand rate Screening rate Selling price Order cost Unit purchase cost Holding cost Unit inspection cost Percentage of defectives Probability density function

Symbol D x s K C h CI ρ f(ρ)

Value 50,000 175,200 50 100 25 5 0.5 U ~ (0, 0.04) 1/(0.04–0)

Units Units/year Units/year $/unit $/order $/unit $/unit/year $/unit

Table 2.33 Data for the numerical example (Salameh and Jaber 2014; Taleizadeh et al. 2016b) Description Repair setup cost Transportation fixed cost Unit transportation cost Unit material and labor cost Unit holding cost in repair shop Repaired rate Total transport time Holding cost of repaired product Markup percentage

Symbol KR KS CT C1 h2 R tT hR m

Value 100 200 2 5 4 50,000 2/220 6 20%

Units $/setup $/trip $/unit $/unit $/unit/year Units/year Year $/unit/year Percent

Table 2.34 Additional data for the numerical example (Taleizadeh et al. 2016b) Description Backorder cost Lost sales cost Percentage of backordered demand

Symbol Cb g β

Value 20 0.5 97%

Units $/unit/year $/unit/year Percent

  1 hE ð1  ρÞ2 2 ð 1 þ m ÞE ð ρ Þh D E ð ρ ÞhD C β ð 1  E ð ρ Þ Þ 2 b A þ w4 ¼ 4ðK þ ð1 þ mÞðK R þ 2K S ÞÞ@ þ þ 2 R x 2 0

DðC I  b π ð1  βÞ þ E ðρÞð1 þ mÞðC 1 þ 2C T þ h2 t T ÞÞ2

must be positive. If w4 > 0, then the optimal cycle time T* is determined with Eq. (2.458). Otherwise, if w4
0, then the cycle time T* is equal to zero. AdditionE ðð1ρÞ2 Þh C b βE ðρ2 Þ ally, the following condition must be satisfied: þ EðρxÞhD  > 0. 2 2 Example 2.21 This section presents a numerical example. The data for the numerical example is shown in Tables 2.32, 2.33, and 2.34. Table 2.32 summarizes the data from Salameh and Jaber (2000), Table 2.33 contains data from Jaber et al.

140

2

Imperfect EOQ System

(2014), and Table 2.34 presents new data for the numerical example (Taleizadeh et al. 2016b). The percentage of imperfect items follows a uniform distribution (U ~ (0, 0.04)). Thus, the expected value of the defective products (ρ) is: Zb

Zb

E ð ρÞ ¼

ρf ðρÞdρ ¼ a

ρ

1 b þ a 0:04 þ 0 dρ ¼ ¼ ¼ 0:02 ba 2 2

a

The E(ρ2) and (1  E(ρ2))2 are given by:   ð b  aÞ 2 ð0:04Þ2 E ρ2 ¼ varðρÞ þ E 2 ðρÞ ¼ þ E 2 ð ρÞ ¼ þ ð0:02Þ2 ¼ 0:0005333333 12 12   ð1  EðρÞÞ2 ¼ 1  2E ðρÞ þ E ρ2 ¼ 1  2ð0:02Þ þ 0:0005333333 ¼ 0:9605333355 Firstly, for each profit function, the wi must be greater than zero. w1 ¼ 8519.089532 > 0, w2 ¼ 8464.869573 > 0, w3 ¼ 8482.853463 > 0, w4 ¼ 8327.949572531 > 0. Additionally, for Cases I and IV, the following conditions must be satisfied: E ðρ2 ÞD E ðρ2 Þ þ 2 þ EðρxÞD  EðρÞ R E ðð1ρÞ2 Þh πβE ðρ2 Þ 4: þ EðρxÞhD  2 2

Case I: condition 1:

¼ 0:0135
0.

Case IV: condition

¼ 2:42 > 0.

Note that if all the conditions are satisfied, then the problem can be optimized. Table 2.35 shows the optimal results for the decision variables and the expected total profit (Taleizadeh et al. 2016b). According to Table 2.35, the optimal policy in Case I happens when the cycle time is T ¼ 0.083598 year, the percent of cycle time duration when the inventory level is positive is F ¼ 81.83131%, the order quantity is Q ¼ 4157.11965 units, and the expected total profit is 1,197,016.99277. The optimal policy in Case II occurs when T ¼ 0.084679 year, F ¼ 82.28241%, and Q ¼ 4211.44917 units and the expected total profit is 1,197,197.0042. The optimal policy for Case III is when T ¼ 0.084706 year, F ¼ 81.51335%, and Q ¼ 4209.78066 units and expected total profit is 1,196,560.33681. The optimal policy in Case IV is when T ¼ 0.084019 year, F ¼ 82.83673%, and Q ¼ 4179.32228 units and the expected total profit is 1,197,083.77763. With these results, it is easy to see that the highest expected profit is obtained in Case II and the lowest profit in Case III. It is important to remark that on the one hand the holding cost of repaired products in Case II is less than Case I. On the other hand, the shortage cost in Case II is less than Cases III and IV, and the shortage cost in Case III is more than other cases (Taleizadeh et al. 2016b).

Case I Case II Case III Case V

TP 1,197,016.99277 1,197,197.00424 1,196,560.33681 1,197,083.77763

T 0.083598053 0.084679076 0.084706825 0.084019058

Table 2.35 Optimal results for each case (Taleizadeh et al. 2016b) F (%) 81.83131 82.28241 81.51335 82.83673%

F1 – – 79.88309% 81.17999%

F2 – – 1.630267% 1.65673%

Q 4157.119653 4211.449176 4209.780665 4179.322286

2.5 EOQ Model with Partial Backordering 141

142

2.5.4

2

Imperfect EOQ System

Replacement of Imperfect Products

Taleizadeh et al. (2018b) developed their previous work (Taleizadeh et al. 2016b) by considering replacement instead of reparation. In order to model the presented problem, some new notations which are specifically used are shown in Table 2.36, and other parameters are the same as those used in models of the previous section as presented in Table 2.31. They have considered three cases including the following: Case I. The reordered items are received when the inventory level is zero. Case II. The reordered items are received when the backordered quantity is equal to the imperfect items’ quantity. Case III. The reordered items are received when shortage still remained. All these cases are presented in Figs. 2.28, 2.29, and 2.30, respectively. For the first case, they derived the total profit function as:

Table 2.36 Notations of a given problem cE hE ti

Unit purchasing cost of an emergency order ($/unit) Holding cost of emergency purchased unit ($/unit/time unit) Inspection time of products (time unit)

(1-β)(1-F)DT

β(1-F)DT

Fig. 2.28 Inventory level for Case I (Taleizadeh et al. 2018b)

2.5 EOQ Model with Partial Backordering

143

ρ FTD

(1-p)FTD

FTD

Inventory

F2

β (1-F)DT

βρ FDT

t1 F1 F

(1–F)

T

Fig. 2.29 Inventory level for Case II (Taleizadeh et al. 2018b)

F2TD

F1TD

F2T

β(1-F1)TD

ti

F3T

FT (1-F)T T

F4T

(1-β)(1-F1)TD

F1T

t

(F2+F3+F4)TD

Imax=FTD

F2TD

Inventory

Fig. 2.30 Inventory level for Case III (Taleizadeh et al. 2018b) Revenue

Purchasing cost

Emergency ordering cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Fixed cost zfflfflffl}|fflfflffl{ K zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ TPðT, F Þ ¼sðFD þ βð1  F ÞDÞ þ vρFD ½ z}|{ þ C ðFD þ βð1  F ÞDÞ þ CI FD þ CE ρFD T  2 2 2 2 2 ð1  ρÞ F TD ρT ðFDÞ ðρF Þ DT βð1  F Þ TD þh þ hE þ gð1  βÞð1  F ÞD þ þ Cb x 2 2 ffl} 2 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflffl |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Lost sale cost Backordering cost Holding cost for Holding cost emergency order

ð2:459Þ

144

2

Imperfect EOQ System

And after some simplifications and algebra, they derived similar to the previous case presented in Sect. 2.5.3: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u hð1EðρÞÞ2 EðρÞhD hE Eðρ2 Þ Cb β D uK þ x þ 2 þ 2  4 ðCI þ ðCE  vÞE ðρÞ  ðs þ g  CÞð1  βÞÞ2 2 u   T ¼t 2 2 2 Cb β hDð1EðρÞÞ þ EðρÞhD þ hE Eð2ρ ÞD þ Cb2βD 2 x 2

ð2:460Þ Cb β  C I  ðE E  vÞEðρÞ þ ðs þ g  C Þð1  βÞ  F ðT Þ ¼  hð1  EðρÞÞ2 þ 2EðρxÞhD þ hE E ðρ2 Þ þ C b β T  w1 ¼ K

Take

hð1E ðρÞÞ2 2

þ EðρxÞhD þ

ð2:461Þ

h E E ðρ 2 Þ 2

 þ C2b β 

þ ðCE  vÞE ðρÞ  ðs þ g  CÞð1  βÞÞ2 . If w1 > 0, T* is equal T in Eq. (2.460). On the other hand, if w1
0, T* is equal to zero, meaning the optimal value of shortage is infinite and no inventory level exists. Also for the second case, the total profit is defined as:

D 4 ðC I

Revenue

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ TPðT, F Þ ¼sðF 1 D þ βð1  F 1 ÞDÞ þ vρFD 2

3 Holding cost zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Purchasing cost Inspection cost Emergency order cost   6 Fixed cost zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 7 2 zfflffl}|fflffl{ zfflfflffl}|fflfflffl{ ð1  ρÞ F 2 TD ρF 2 TD2 7 6 z}|{ K 6 7 þ þ C ðFD þ βð1  F ÞDÞ þ C I FD þ C E ρFD þh T 6 7 2 x 7 6 6 7 2 2 2 6 7 β ð 1  F Þ TD βρ F TD 6 þC b 7 þ Cb þ gð1  βÞð1  F 1 ÞD 4 5 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflffl ffl } |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Backordering cost

Lost sale cost

ð2:462Þ And after some simplifications and algebra: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u hð1EðρÞÞ2 EðρÞhD Cb βEðρ2 Þ Cb β D uK þ x þ 2 þ 2  4 ðC I þ ðC E  vÞEðρÞ  ðs þ g  CÞð1  βÞÞ2 2 u   T ¼t 2 2 2 Cb β hð1EðρÞÞ D þ EðρÞhD þ Cb βE2ðρ ÞD 2 x 2

ð2:463Þ and F ðT Þ ¼

C b βT  ðC I þ ðC E  vÞEðρÞ  ðs þ g  C ÞDð1  βÞÞ   hð1  E ðρÞÞ2 þ 2EðxρÞD þ C b βEðρ2 Þ þ C b β T

ð2:464Þ

2.5 EOQ Model with Partial Backordering

145

 Considering

w2 ¼ K

hð1E ðρÞÞ2 2

þ EðρxÞhD þ

C b βE ðρ2 Þ 2

 þ C2b β 

þ ðCE  cs ÞE ðρÞ  ðs þ g  C Þð1  βÞÞ2 , if w2 > 0, then T* is equal T in Eq. (2.463). If w2
0, then T* is equal zero, meaning the optimal value of shortage is infinite, and no inventory cycle exists. And for the third case, similarly, the total profit is defined as:

D 4 ðC I

Revenue

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ TPðT, F Þ ¼sDðF þ βð1  F ÞÞ þ vρFD Inspection

2

3 Fixed 6 7 Holding cost 6 7 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ cost Purchasing cost Emergency order cost 6 cost 7   2 2 zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 2 2 7 zfflffl}|fflffl{ zfflfflffl}|fflfflffl{ 6 ð 1  ρ Þ F TD ρ ð F Þ TD 6 z}|{ K þ CDðF þ βð1  F ÞÞ þ C I FD þ 7 CE ρFD þh þ 6 7 T 6 x 2 7 6 7 6 7 6 þC ð1  F 1 Þð1  F ÞβTD þ gð1  βÞð1  F ÞD 7 b 4 5 2 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Backordering cost

Lost sale cost

ð2:465Þ After some algebra: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u ð1EðρÞÞ h EðρÞhD Cb βð1EðρÞÞ 2 u4K þ þ ð ð  v ÞE ð ρ Þ  ð s þ g  C Þ ð 1  β Þ Þ  D d þ C E 2 x 2 u   T ¼t 2 2 Cb β ð1E ðρÞÞ hD E ðρÞhD2 C b βDðE ðρÞÞ 4 2 þ  x 2 2 ð2:466Þ F ðT Þ ¼

Cb βð2  E ðρÞÞT þ 2ðCE  vÞð1  βÞ  2C I  2ðs þ g  CÞE ðρÞ   ð1E ðρÞÞ h E ðρÞhD C b βð1E ðρÞÞ 4 þ þ T 2 x 2 ð2:467Þ

Considering

w3 ¼ K



ð1E ðρÞÞ2 h 2

 ðρÞÞ þ EðρxÞhD þ Cb βð1E  2

þ ðCE  vÞE ðρÞ  ðs þ g  CÞð1  βÞÞ2 , if w3 > 0, then T* is given by Eq. (2.466). If w3
0, then T* is equal zero, meaning the optimal value of shortage is infinite and no inventory cycle exists. Also additional condition of Case III is ð1ρÞ2 h Cb βρ2 þ ρhD x > 2 , so if this condition is violated, this means the backordered cost 2 is very huge, and in this situation, shortage is meaningless and economic order quantity is (Jaber et al. 2014):

D 4 ðC I

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD   Q ¼ h 1  2E ðρÞ þ E ðρ2 Þ þ 2EðρÞ Dx þ hE E ðρ2 Þ

ð2:468Þ

146

2

Imperfect EOQ System

Table 2.37 Data taken from Salameh and Jaber (2000) Symbol D x s K C

Value 50,000 175,200 50 100 25

Table 2.38 Date extracted from Jaber et al. (2014)

Units Units/year Units/year $/unit $/order $/unit

Symbol v h CI ρ f(ρ)

Symbol cE hE

Value 20 5 0.5 U ~ (0, 0.04) 1/(0.04–0)

Value 40 8

Units $/unit $/unit/year $/unit

Units $/unit $/unit/year

Table 2.39 Additional data (Taleizadeh et al. 2018b) Description Backordered cost per unit Lost sales cost per unit Percentage of backordered demand

Symbol Cb g β

Value 20 0.5 97%

Units $/unit $/unit

Example 2.22 This section illustrates the use of the three cases with a numerical example. The optimal values of the decision variables and the lot size are calculated. Table 2.37 provides the data taken from Salameh and Jaber (2000). Table 2.38 presents data taken from Jaber et al. (2014). Table 2.39 shows additional data for the numerical example. According to Table 2.37, the distribution of percentage of imperfect items is assumed that follows a uniform distribution [U ~ (0, 0.04)]. Thus, the expected value of the defective items (ρ) is: Zb E ð ρÞ ¼

Zb ρf ðρÞdρ ¼

a

ρ

1 b þ a 0:04 þ 0 dρ ¼ ¼ ¼ 0:02 ba 2 2

a

Moreover, E(ρ2) is:   ðb  aÞ2 ð0:04Þ2 E ρ2 ¼ varðρÞ þ E 2 ðρÞ ¼ þ E 2 ð ρÞ ¼ þ ð0:02Þ2 ¼ 0:0005333333 12 12   ð1  E ðρÞÞ2 ¼ 1  2E ðρÞ þ E ρ2 ¼ 1  2ð0:02Þ þ 0:000533 ¼ 0:9605333355 Firstly, the significance of each profit functions is verified: w1 ¼ 985.3880 > 0, w2 ¼ 931.1284 > 0, w3 ¼ 965.7747 > 0.

References Table 2.40 Optimal value for the decision variables and the economic order quantity

147

Case I Case II Case III

TP 1,200,732.887 1,200,277.629 1,200,667.453

T 0.0289 0.0281 0.0286

F (%) 60.70 57.88 60.70

Q 1428.138 1385.718 1414.757

2 πβE ðρ2 Þ For Case III, the following constraint ð1E2ðρÞÞ h þ EðρxÞhD  2 ¼ 2:4247 > 0 must be satisfied. Therefore, all of the conditions are satisfied. Hence, the optimal value for the decision variables and the economic order quantity for each case are given in Table 2.40.

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148

2

Imperfect EOQ System

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Chapter 3

Scrap

3.1

Introduction

As stated in Chap. 2, the economic order quantity (EOQ) model was first introduced in 1913. Seeking to minimize the total cost, the model generated a balance between holding and ordering costs and determined the optimal order size. Later, the EPQ model considered items produced by machines inside a manufacturing system with a limited production rate, rather than items purchased from outside the factory. Despite their age, both models are still widely used in major industries. Their conditions and assumptions, however, rarely pertain to current real-world environments. To make the models more applicable, different assumptions have been proposed in recent years, including random machine breakdowns, generation of imperfect and scrap items, and discrete shipment orders. The assumption of discrete shipments using multiple batches can make the EPQ model more applicable to real-world problems. The EPQ inventory models assume that all the items are manufactured with high quality and defective items are not produced. However, in fact, defective items appear in the most of manufacturing systems; in this sense, researchers have been developing EPQ inventory models for defective production systems. In these production systems, defective items are of two types: scrapped items and reworkable items. Chung (1997) investigated bounds for production lot sizing with machine breakdown conditions. Rosenblatt and Lee (1986) proposed an EPQ model that deals with imperfect quality. They assumed that at some random point in time, the process might shift from an in-control to an out-of-control state. Chiu et al. (2007) investigated an EPQ model with scrap, rework, and stochastic machine breakdowns to determine the optimal run time and production quantity. Sarkar et al. (2014) developed an EPQ model with a random defective rate, rework process, and backorders for a single-stage production system. Chiu et al. (2010) presented a robust

© Springer Nature Switzerland AG 2021 A. A. Taleizadeh, Imperfect Inventory Systems, https://doi.org/10.1007/978-3-030-56974-7_3

153

154

3 Scrap

production–inventory model with an imperfect rework process for imperfect products. Machine breakdown can occur in that model, and failure time is considered a random variable. Chiu et al. (2011a) developed the EPQ model with discrete shipment and generation of imperfect-quality items. They assumed that a portion of imperfect items are scrapped and that the rest can be repaired or salvaged. At the end of the repair run, repaired items are delivered to customers in discrete batches. In that research, the objective function was to determine the economic production quantity that would minimize the total cost. Chiu et al. (2011b) investigated an EPQ model with discrete shipment and generation of imperfect and scrap items. During a production and repair run, the initial delivery of batches was considered to satisfy demand, and the rest were discretely delivered to the customer at the end of the machine repair time. Chiu et al. (2012) provided an EPQ model with random machine breakdowns and the generation of imperfect and scrap items. Akbarzadeh et al. (2015, 2016) developed EPQ models with scrapped items using vendormanaged inventory systems. The main aim of this chapter is providing a comprehensive framework of analyzing the EPQ models considering the scrap. Hence, some important EPQ inventory models in which scrapped items are categorized in two sections including EPQ models with shortage and without shortage are presented. The shortage sections are divided to two subsections which are partial and full backordering shortage. Also, the models based on the delivery policies including discreet and continuous deliveries are categorized. Therefore, two key features, EPQ models and scrapped items, are the main goals that are investigated in this chapter. In this chapter, a comprehensive review of some major EPQ models in which scrapped items is considered is presented. First, a brief introduction and literature review of mentioned problem is provided. Then, a framework of problem and mathematical model is presented, and a numerical example for each problem is solved. According to above description, three categorizes, no shortage, partial backordering, full backordering, are provided as shown in Fig. 3.1. All categories are investigated in the next sections. All categorized problems in Fig. 3.1 are investigated separately. The common notations of EPQ model considering scrap are shown in Table 3.1. To integrate the report, these notations for all models are used. The main decision variables of this field on inventory are Q and B, but in some studies, other decision variables are considered too.

3.2 No Shortage

155

Continuous delivery

Discrete delivery

Partial backordered

Continuous delivery

EPQ models + Scrap

No shortage

Fully backordered

Continuous delivery

Fig. 3.1 Categories of EPQ model considering scrap

3.2

No Shortage

3.2.1

Continuous Delivery

In this section, the EPQ models with scrap and continuous delivery policy without shortage are presented.

3.2.1.1

Considering Steady Production Rate

Chiu et al. (2003) studied the effect of the steady production rate of scrap items on the economic production quantity (EPQ) model. Many research efforts on the EPQ model assumed that the manufacturing facility functions perfectly during a production run. But in most practical settings, defective items may be generated, by an imperfect production process due to process deterioration or other factors. Chiu et al. (2003) extended the work of Hayek and Salameh (2001) and studied the effect of the steady production rate of the scrap items on the classical finite production model. The imperfect production process may generate x percent of imperfect-quality items. In this problem, it is assumed that the imperfect-quality items are all scrap items and have a defective production rate d0 . Since shortage is not allowed, backordering all demands must be satisfied at all times. Hence, the replenishment policy is to restart a new production run (cycle) whenever on-hand inventories run out. The production rate P is constant and is much larger than demand rate D. The defective production rate d of imperfect-quality items could be expressed as the product of the production rate times the percentage of defective items produced.

156

3 Scrap

Table 3.1 Notations Q P T ts H, I, I1, H1 I(t) f(x) X D H B Cb b π K KS d C s v Cd CT CI CM h h1 g, tr TCU TC E(.)

Production lot size per cycle (unit) Production rate per unit time (units per unit time) Length of production inventory cycle (time) Setup time to produce item (time) Maximum level of on-hand inventory in units (unit) On-hand inventory of perfect-quality items at time t (unit) Probability density function of x The percentage of the imperfect-quality items produced, x may be a random variable for some models with known probability of density function Demand rate (units per unit time) Holding cost per unit per unit time ($/unit/unit time) Size of the backorders (unit) Backorder cost per item per unit time ($/unit/unit time) Unit lost sale cost ($/unit) Setup cost ($/setup) Fixed delivery cost per shipment ($/shipment) Production rate of scrap items (units per unit time) Production cost per item ($/unit) The selling price per unit for good-quality items ($/unit) The selling price per unit for defective items ($/unit) Disposal cost for each scrap item ($/unit) Unit delivery cost ($/unit) Inspection cost per item ($/unit) Machine repair cost per breakdown ($/breakdown) Holding cost per unit per unit of time ($/unit/unit time) Holding cost of defective items per unit per unit of time ($/unit/unit time) Time needed to repair and restore the machine after breakdown Total inventory costs per unit time ($) Total inventory costs per cycle ($) Denotes the expected value operator

Therefore, the defective production rate d of imperfect-quality items can be written as (Chiu et al. 2003): d ¼ P  x,

ð3:1Þ

The production rate of perfect-quality items must always be greater than or equal to the sum of the demand rate and the rate at which defective items are produced. Therefore,

3.2 No Shortage

157

I (t)

H1

P-d-D

-D

Time

t2

t1 T

Fig. 3.2 On-hand inventory of non-defective items (Chiu et al. 2003)

P  d  D  0,   D 1  x  0, P

ð3:2Þ

For the following derivation, the solution procedures are those used by Hayek and Salameh (2001), referring to Fig. 3.2 (Chiu et al. 2003): T ¼ t 1 þ t 2 and T ¼

Q ð1  x Þ D

ð3:3Þ

The production uptime t1 needed to build up H1 units of perfect-quality items is (Chiu et al. 2003): t1 ¼

Q P

ð3:4Þ

and   D H 1 ¼ ðP  d  DÞt 1 ¼ Q 1  x  , P

ð3:5Þ

The production downtime t2 needed to consume the maximum on-hand inventory H1 is (Chiu et al. 2003): t2 ¼

  H1 1 x 1   , ¼Q D D P D

ð3:6Þ

The imperfect-quality items which build up during production uptime t1 as shown in Fig. 3.3 are:

158

3 Scrap

Fig. 3.3 On-hand inventory of defective items (Chiu et al. 2003)

Id (t)

dt1

d

Time

t1

dt 1 ¼ xQ,

ð3:7Þ

In real-life situations, the percentage of imperfect-quality items may be a random variable, with a known probability density function. For instance, if x follows the uniformly distributed over the range [Xu, Xl), the probability density function f(x) is (Chiu et al. 2003): 8
0 Q d2 Q

ð3:12Þ ð3:13Þ

ðQÞ From dE½TCU ¼ 0, one obtains (Chiu et al. 2003): dQ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD   Q ¼ D h 1  P  2h 1  DP EðxÞ þ hE ðx2 Þ 

ð3:14Þ

Suppose that no defective items are produced, then x ¼ 0, and the same equation as that of the classical finite production rate model will be derived (Nahmias and Cheng 2005): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD  Q ¼ h 1  DP 

ð3:15Þ

Example 3.1 Chiu et al. (2003) presented a manufactured product which has experienced a relatively flat demand of 4000 units per year. This item is produced at a rate of 10,000 units per year. The accounting department has estimated that it costs $450 to initiate a production run, each unit costs the company $2 to manufacture, the cost of holding is $0.6 per item per year, and the disposal cost is $0.3 for each scrap item. The production rate of defective items is uniformly distributed over the interval [0, 0.1). Thus, P ¼ 10,000 units per year, D ¼ 4000 units per year, x ¼ uniformly distributed over the interval [0, 0.1], K ¼ $450 for each production run, C ¼ $2 per item, Cs ¼ $0.3 for each scrap item, and h ¼ $0.6 per item per unit time. The optimal production lot size, computed from Eq. (3.14), is Q* ¼ 3323 units (Chiu et al. 2003).

160

3 Scrap

The expected annual inventory cost, from Eq. (3.10), is E[TCU(Q)] ¼ $9625 per year (Chiu et al. 2003). If no defective items are produced, that is, when x ¼ 0, the model is changed to the classical finite production rate model. From Eq. (3.15) it is obtained that Q* ¼ 3162, and annual inventory costs, E[TCU(Q)] ¼ $9138 per year comparing these values, it could be noticed that in the case of scrap items which are produced in EPQ model, the optimal lot size must be determined using the new equations (Chiu et al. 2003).

3.2.1.2

Multi-product Multi-machine

Nobil et al. (2016) considered a multi-product problem with nonidentical machines. This manufacturing system consists of various machine types with different production capacities, production costs, setup times, production rates, and failure rates. In real-world problems, there are different options to purchase machines, considering different factors like production rate, floor space limitation for production, budget constraint, and so forth. As a result, production managers cope with decisions about minimizing machine utilization expenditure and inventory costs simultaneously. Moreover, when factory produces more than one item, factory may need to buy more than one machine. This section investigates this problem. Nobil et al. (2016) considered a multi-machine production system producing m different items. The problem is an EPQ problem with unrelated parallel machine in which utilized production machines are considered. It is an extension of the single-machine multi-product EPQ problem with defective items, in which the determination of machine’s number and items allocation is considered simultaneously. Each machine has a particular failure rate based on the characteristics of product. The proposed inventory model minimizes total cost of the inventory system, including utilization, setup, production, holding, and disposal costs. The following notations presented in Table 3.2 are used for machines i ¼ 1, 2, . . ., n and items j ¼ 1, 2, . . ., n to model the problem. The inventory problem under study is shown in Fig. 3.4 (Nobil et al. 2016). It is evident from Fig. 3.4 that the maximum on-hand inventory of the jth item produced by ith machine is determined by Eq. (3.16): Ij ¼



Qj 1  xij Pij  D j Pij

ð3:16Þ

In Fig. 3.4, the cycle length of jth item produced by ith machine consists of two periods: the uptime production period denoted by Tpij and downtime period Tdij. The lengths of these periods are calculated as follows:

3.2 No Shortage

161

Table 3.2 Notations of given problem (Nobil et al. 2016) N M Kij αij fi ri Cij Ni Tpij Tdij BC FM CP CU CH CK CD yi zij

Number of machines Number of items Setup cost of ith machine to produce jth item ($/setup) Binary parameter, αij ¼ 1 if (1  xij)Pij  Dj; otherwise, αij ¼ 0 The fixed cost of utilizing for ith machine ($/use) Required space of ith machine (square meter) Production cost of jth item per unit on machine i ($/unit) Number of cycles per unit time for ith machine Uptime of the jth item produced by ith machine (time) Downtime of the jth item produced by ith machine (time) Maximum available budget ($) Maximum available space (square meter) Total production cost of all items ($) Total utilization cost of all machines ($) Total holding cost of all items ($) Total setup cost of all items ($) Total disposal cost of all items ($) yi ¼ if machine i is utilized; otherwise, yi ¼ 0 decision variables zij ¼ 1 if jth item produced by ith machine; otherwise, zij ¼ 0 decision variables On-hand inventory

Ij (1− xij)Pij − Dj

Dj

Time Tpij

Tdij Ti

Fig. 3.4 The cycle length of the on-hand inventory of jth item produced by ith machine (Nobil et al. 2016)

162

3 Scrap

Qj Ij ¼ 1  xij Pij  D j Pij  Q j 1  xij Pij  D j Ij Tdij ¼ ¼ Pij D j Dj Tpij ¼ 

ð3:17Þ ð3:18Þ

Consequently, the length of a cycle for ith machine is: 

1  xij Q j T i ¼ Tpij þ Tdij ¼ Dj

ð3:19Þ

Hence, Qj ¼ 

D jT i 1  xij

ð3:20Þ

Besides, all items are manufactured on each machine with a limited capacity and a common cycle length of Ti1 ¼ Ti2 ¼ . . . ¼ Tim. The total inventory system cost consists of the following costs, utilization, setup, production, holding, and disposal, as shown in Eq. (3.21): TC ¼ CU þ CK þ CP þ CH þ CD

ð3:21Þ

In what follows, all the components of Eq. (3.20) are derived. The total utilization cost is calculated by Eq. (3.22): CU ¼

n X

yi f i

ð3:22Þ

i¼1

As the setup cost of ith machine to produce jth item in a cycle is Kij and there are Ni cycles in a specific period of time (i.e., a year), then the total setup cost is given by: CK ¼

n X m X i¼1

N i zij K ij

ð3:23Þ

j¼1

In addition, based on the joint production policy, N i ¼ 1=T i . Hence, CK ¼

n X m X zij K ij Ti i¼1 j¼1

ð3:24Þ

3.2 No Shortage

163

The total production cost of the inventory system based on the production cost of the jth item per unit per cycle on ith machine (Cij) is computed using Eq. (3.25) as: CP ¼

n X m X

n X m X xij Cij N i zij C ij Q j ¼ Ti j¼1 i¼1 j¼1

i¼1

! DT  j i 1  xij

n X m X z C D  ij ij j ¼ i¼1 j¼1 1  d ij

ð3:25Þ

Based on Fig. 3.4, the total holding cost of the inventory system under study is determined as: CH ¼

  n X m X zij h j I j ðT i Þ Ti 2 i¼1 j¼1

ð3:26Þ

Inserting Ij from Eq. (3.16) results in: CH ¼

  n X m X Qj zij h j T i  1  xij Pij  D j Ti 2 Pij i¼1 j¼1

 n X m X zij h j D j Dj  1  xij  ¼ T Pij i i¼1 j¼1 2 1  xij

ð3:27Þ

The total disposal cost of the inventory system based on the disposal cost of the jth item per unit per cycle (Cdj) is calculated by Eq. (3.28) as: CD ¼

n X m X i¼1

¼

N i zij C d j xij Q j

j¼1

n X m X xij C d j zij i¼1

j¼1

Ti

! DT  j i 1  xij

n X m X xij C d j zij D j  ¼ i¼1 j¼1 1  xij

ð3:28Þ

Finally, the total cost is: TC ¼ CU þ CK þ CP þ CH þ CD ¼ "

n X i¼1

yi f i þ

n X m X i¼1

j¼1

zij

#

 C ij D j þ C d j xij D j K ij h jD j Dj 1  xij    þ  T þ T i 2 1  xij Pij i 1  xij

ð3:29Þ

Because of characteristics of the proposed problem, a constrained problem is modeled, as shown below:

164

3 Scrap n X

(

αij zij ¼ 1

ð3:30Þ

i¼1

 αij ¼ 1 1  xij Pij  D j  0 αij ¼ 0 Otherwise

zij  yi

i ¼ 1, 2, . . . , n; j ¼ 1, 2, . . . , m: n X

ð3:31Þ ð3:32Þ

f i yi  BC

ð3:33Þ

r i yi  F M

ð3:34Þ

i¼1 n X i¼1 m X  zij Tpij þ tsij  T i ;

i ¼ 1, 2, . . . , n

ð3:35Þ

j¼1

Inserting Tpij from Eq. (3.17) results in: 0 B B B @

m P

1

1 zij tsij

j¼1 m P j¼1

zD j

ð1xij ÞPij

C C C  Ti A

i ¼ 1, 2, . . . , n

ð3:36Þ

Equation (3.30) shows the constraint that every item must only be allocated to a machine, where αij permits jth item to be assigned to machine i such that Eq. (3.32) shows the constraint that every item can be produced by a machine if and only if the machine is utilized. In the proposed inventory model, the capital required for machines utilization must be smaller than or equal to its maximum available budget (see Eq. 3.33). The space required for all the machines must be smaller than or equal to its maximum available space (see Eq. 3.34). According to Eq. (3.35), the sum of the production and setup times for all items produced by ith machine cannot be greater than the common cycle length of ith machine, Ti. In short, the mathematical formulation of the MINLP problem that minimizes the total inventory system cost under constraints is (Nobil et al. 2016): Min TC ¼

n X

yi f i þ

i¼1



m X j¼1

" zij

n X i¼1

#

 C ij D j þ C d j xij D j K ij h jD j Dj 1  xij   þ  T þ T i 2 1  xij Pij i 1  xij ð3:37Þ

s.t.

3.2 No Shortage

165 n X

αij zij ¼ 1,

j ¼ 1, 2, . . . , m

ð3:38Þ

i¼1

xij  yi ,

i ¼ 1, 2, . . . , n; j ¼ 1, 2, . . . , m: n X

ð3:39Þ

f i yi  BC

ð3:40Þ

r i yi  F M

ð3:41Þ

i¼1 n X i¼1

0 B B B @

m P

1

zij tsij

j¼1 m P



j¼1

1

zD j



C C C  Ti A

i ¼ 1, 2, . . . , n

ð3:42Þ

1  xij Pij

Ti > 0 yi , xij 2 f0, 1g

i ¼ 1, 2, . . . , n

i ¼ 1, 2, . . . , n; j ¼ 1, 2, . . . , m

ð3:43Þ ð3:44Þ

Example 3.2 Nobil et al. (2016) presented several examples solved using three different solution procedures and an optimization software. Input parameters of these problems are chosen randomly from Table 3.3, and results of three solution approaches are represented in Table 3.4. Each cell of Table 3.5 is an average of ten solutions of each problem with different sizes. It is worth mentioning that in the proposed HGA, by combining derivatives method and GA optimal Ti using derivative based on the randomly generated yi and xij are obtained. The conventional GA is utilized to evaluate efficiency of proposed HGA in large-scale problems. With regard to obtained results represented in Table 3.4, the proposed HGA finds the optimal solution acceptable for small-sized problems. Whereas the solution of the proposed HGA for medium-sized problems in all iterations is the same, it can be concluded that these solutions are near optimal solutions. Moreover, the proposed HGA has significant efficiency in comparison with conventional GA, because it finds better solutions with less computational effort. Finally, to evaluate the quality of HGA solutions for small-sized problems, the ten problems were solved using DICOPT, a GAMS solver for MINLP models. To do so, it was considered the case that there are no limitations for machines to produce the items. According to Table 3.5, it can be said this solver obtains a poor feasible solution in comparison with proposed HGA (Nobil et al. 2016).

166

3 Scrap

Table 3.3 Input parameters of MINLP problem (Nobil et al. 2016) Pij ~ U(15,000, 25,000); xij ~ U(0.001, 0.007); Kij ~ U(100, 300); Cij ~ U(40, 70); Sij ~ U(0.03, 0.08); ri ~ U(5, 20); BC ~ U(500,000, 800,000); FM ~ U(4000, 90,000); fi ~ U(120,000, 220,000); Dj ~ U(1000, 3000); hj ~ U(10, 20); dj ~ U(20, 40)

3.2.2

Discrete Delivery

In this section, EPQ models with scrap items and discrete delivery policy are considered where shortage is not permitted.

3.2.2.1

Multi-delivery

Chiu et al. (2011) employed a mathematical modeling and algebraic approach to derive the optimal manufacturing batch size and number of shipment for a vendor– buyer integrated economic production quantity (EPQ) model with scrap. Chiu et al. (2011) assume there is an x portion of defective items produced randomly at a production rate d during regular production time. All produced items are screened, and inspection cost per item is included in the unit production cost C. All non-conforming items are assumed to be scrap and will be discarded at the end of production. Under regular supply (not allowing shortages), the constant production rate P must be larger than the sum of demand rate D and production rate of scrap items d. That is, (P  d  D) > 0. The production rate of scrap items d can be expressed as d ¼ Px. A multi-delivery policy is considered in this study, and it is also assumed that the finished items can only be delivered to customers if the whole lot is quality assured at the end of production process. Fixed-quantity n installments of finished batch are delivered to customers at a fixed interval of time during the production downtime t2 (see Fig. 3.5). Some notations which are specifically used to model this problem are shown in Table 3.6. TC(Q, n), the total production–inventory–delivery costs per cycle consists of (1) setup cost, (2) variable production costs, (3) variable scrap disposal costs, (4) fixed delivery cost, (5) variable delivery costs, (6) variable holding costs at the supplier side for all items produced (defective and perfect-quality items) in t1 and all items waiting to be delivered in t2, and (7) holding cost for finished goods stocked at customer’s end. Therefore, TC(Q, n) is (Chiu et al. 2011):

1 2 3 4 5 6 7 8 9 10

Size n  m 2  5 (S) 3  5 (S) 4  6 (S) 4  8 (S) 5  10 (S) 6  15 (M) 7  20 (M) 10  20 (L) 15  30 (L) 20  40 (L)

Proposed HGA Total cost 721,778.12 812,287.30 1,916,984.20 2,189,670.23 2,931,095.37 3,854,269.34 4,649,783.06 5,072,974.86 7,454,053.68 9,370,322.75 CPU time (s) 34.78 35.51 41.23 49.22 62.80 90.63 147.97 220.82 602.63 1020.63

Table 3.4 Comparison of algorithm for two measures (Nobil et al. 2016) GA Total cost 722,412.07 813,145.36 1,920,841.59 2,193,949.74 3,075,214.55 4,097,709.70 4,973,744.20 5,404,672.06 8,001,922.28 10,198,622.85 CPU time (s) 52.13 67.51 99.12 130.97 201.26 270.45 369.60 506.25 826.34 1573.59

Enumeration Total cost 721,778.12 812,287.30 1,916,984.20 2,189,670.23 2,931,095.37 Complex Complex Complex Complex Complex

CPU time (s) 9.48 28.25 56.12 99 250.98 More than 500

3.2 No Shortage 167

168

3 Scrap

Table 3.5 Comparison of HGA and GAMS software (Nobil et al. 2016)

1 2 3

Size n  m 25 35 5  10

GAMS software Total cost 427,493.95 503,550.26 2,971,438.38

Fig. 3.5 On-hand inventory of perfect-quality items in the proposed EPQ model with scrap and a multiple shipment policy (Chiu et al. 2011)

HGA Total cost 418,114.23 472,946.93 2,541,907

CPU time (s) 0.313 0.706 0.986

CPU time (s) 31.78 33.25 60.98

I(t)

H P−d

–D

•

•

•

tn

Time

t1

t2

t1

T

Table 3.6 Notation of given problem (Chiu et al. 2011) n t1 t2 tn h2

Number of fixed-quantity installments of the finished batch to be delivered to customers, a decision variable to be determined for each cycle The production uptime (time) Time required for delivering all finished products (time) A fixed interval of time between each installment of finished products delivered during production downtime t2 (time) Holding cost for finished goods stocked at customer’s end per unit per unit of time ($/unit/ unit of time)

TCðQ, nÞ ¼ K þ CQ þ C d ðxQÞ þ nK S þ C T ½Qð1  xÞ     h i H þ dt 1 n1 h H þh Ht 2 þ 2 t 2 þ T ðH  Dt 2 Þ ðt 1 Þ þ 2n 2 2 n ð3:45Þ Figure 3.6 shows supplier’s inventory holding during delivery time t2. The variable holding costs for finished products kept by the supplier in delivery time t2 are (Chiu et al. 2011): 1. When n ¼ 1, total holding cost in delivery time ¼ 0. 2. When n ¼ 2, total holding costs in delivery time become (see Fig. 3.7):

3.2 No Shortage

169

Fig. 3.6 On-hand inventory of the finished items kept by supplier during t2 (Chiu et al. 2011)

h

   H t2 1  ¼ h 2 Ht 2 2 2 2

ð3:46Þ

3. When n ¼ 3, total holding costs in delivery time are:

   2H t 2 1H t 2 2þ1  þ  ¼h Ht 2 h 3 3 3 3 32

ð3:47Þ

4. When n ¼ 4, total holding costs in delivery time become:

   3H t 2 2H t 2 1H t 2 3þ2þ1  þ  þ  ¼h Ht 2 h 4 4 4 4 4 4 42

ð3:48Þ

Therefore, the following general term for total holding costs during t2 can be obtained (as shown in Eq. (3.45)): ! n1     X   nðn  1Þ 1 1 n1 Ht 2 ¼ h Ht 2 i Ht 2 ¼ h 2 h 2 2 2n n n i¼1

ð3:49Þ

Taking randomness of scrap rate into consideration and employing the expected values of it, and with further derivations, the long-run average costs per unit time for the proposed EPQ model can be derived as follows (refer to a similar derivation procedure in Chiu et al. 2009):

170

3 Scrap

I (t) H1 q P−d

q

x 2x

q

T 2T 3T 4T n n n n

(n−1)T T n

(k−1)T kT n n

t1

time

td T

Fig. 3.7 The vendor’s on-hand inventory of perfect-quality items when machine breakdown does not occur (Taleizadeh et al. 2017)

E ½TCðQ, nÞ E ðT Þ   E ð xÞ ðK þ nK S ÞD CD 1 ¼ þ Cd þ CT D þ Q 1  E ð xÞ 1  E ð xÞ 1  E ð xÞ    hQD 1 n  1 hQð1  E ðxÞÞ hQD   þ þ 2 2P n 2P 1  E ð xÞ h    i h2 Q 1 n1 D þ ð 1  E ð xÞ Þ þ 2 n n P ð3:50Þ

E ½TCUðQ, nÞ ¼

This study employs algebraic approach to derive the optimal production–shipment policies, instead of using differential calculus on E[TCU(Q, n)] with the need of proving its optimality (Grubbström and Erdem 1999; Chiu 2008; Chiu et al. 2010). In Eq. (3.50), both Q and n are decision variables; by rearranging terms in Eq. (3.50) as the constants Q1, Q, nQ1, and Qn1, one has (Chiu et al. 2011):    E ½TCUðQ, nÞ ¼ β1 þ β2 ðQÞ þ β3 Q1 þ β4 nQ1 þ β5 Qn1 where β1, β2, β3, β4, and β5 denote the following (Chiu et al. 2011):

ð3:51Þ

3.2 No Shortage

171



 E ð xÞ CD þ Cd β1 ¼ þ CT D 1  E ð xÞ 1  E ð xÞ

hð1  E ðxÞÞ ðh  h2 ÞD hD  β2 ¼ þ 2 2P 2Pð1  E ðxÞÞ

 β5 ¼

ð3:52Þ ð3:53Þ

β3 ¼

KD 1  E ð xÞ

ð3:54Þ

β4 ¼

KSD 1  E ð xÞ

ð3:55Þ

 D ð 1  E ð xÞ Þ  ð h  h2 Þ 2 2P

ð3:56Þ

With further rearrangements, Eq. (3.51) becomes (Chiu et al. 2011): i    h  2 ð3:57Þ E ½TCUðQ, nÞ ¼ β1 þ Q1 β2 Q2 þ β3 þ Qn1 β4 nQ1 þ β5  pffiffiffiffiffi pffiffiffiffiffii  pffiffiffiffiffi 2 pffiffiffiffiffi2 β2 Q þ β3 þ Qn1 β3  2 β2 Q E½TCUðQ, nÞ ¼ β1 þ Q1   pffiffiffiffiffi 1 2 pffiffiffiffiffi2 pffiffiffiffiffipffiffiffiffiffi β4 nQ þ β5  2 β4 β5 nQ1 h pffiffiffiffiffi pffiffiffiffiffi  i h pffiffiffiffiffipffiffiffiffiffi β3 Þ þ n1 Q 2 β4 β5 nQ1 β2 Q þQ1 2 E ðTCUðQ, nÞÞ ¼ β1 þ Q1

ð3:58Þ

hpffiffiffiffiffi  pffiffiffiffiffii2 β2 Q  β3

 pffiffiffiffiffi 1 pffiffiffiffiffii2 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi β4 nQ  β5 þ2 þ Qn β2 β3 þ β4 β 5 

1

ð3:59Þ It is noted that if the following square terms (Eqs. 3.60 and 3.61) are equal to zero, then Eq. (3.59) will be minimized (Chiu et al. 2011): 1

Q 

Or



pffiffiffiffiffii2 pffiffiffiffiffi β2 Q  β3 ¼ 0

 pffiffiffiffiffi 1 pffiffiffiffiffii2 β4 nQ  β5 ¼ 0 Qn 1

ð3:60Þ ð3:61Þ

172

3 Scrap

rffiffiffiffiffi β3 Q ¼ β2

ð3:62Þ

rffiffiffiffiffi β5  Q n ¼ β4

ð3:63Þ



And 

Substituting Eqs. (3.53) and (3.54) in Eq. (3.62), the optimal replenishment lot size Q* can be obtained (Chiu et al. 2011): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD Q ¼ hD 2 þ h ð 1  E ð x Þ Þ  DP ðh  h2 Þð1  E ðxÞÞ P 

ð3:64Þ

Substituting Eqs. (3.55), (3.56), and (3.64) in Eq. (3.63), the optimal number of shipments is (Chiu et al. 2011): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi D  u K ð h  h Þ ð 1  E ð x Þ Þ  ð 1  E ð x Þ Þ u 2 P i n ¼ t h 2 D K S hD þ h ð 1  E ð x Þ Þ  P P ð h  h2 Þ ð 1  E ð x Þ Þ

ð3:65Þ

One notes that Eq. (3.65) is identical to what was obtained by using the conventional differential calculus method on E[TCU(Q, n)] (Chiu et al. 2009). Further, from Eq. (3.51) the optimal cost function E[TCU(Q*, n*)] is (Chiu et al. 2011): EðTCUðQ , n ÞÞ ¼ β1 þ 2

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi β2 β3 þ 2 β4 β 5

ð3:66Þ

Example 3.3 Chiu et al. (2011) considered a product with P ¼ 60,000 and D ¼ 3400 units per year. Random scrap rate follows a uniform distribution over the interval [0, 0.3]. In addition, the following values of related variables are considered: C ¼ $100 per item; Cd ¼ $20, per scrap item; h ¼ $20 per item per year; h2 ¼ $80 per item kept at the customer’s end per unit time; K ¼ $20,000 per production run; KS ¼ $4350 per shipment, and CT ¼ $0.1 per item delivered. From Eq. (3.65), one obtains the optimal number of delivery n* ¼ 3. By plugging n back into Eq. (3.51) and resolving the algebraic solution, one finds the optimal production batch size Q* ¼ 2652. Calculating Eq. (3.66) one obtains the long-run average cost E  [TCU(Q*, n)] ¼ $512,047 (Chiu et al. 2011). It is noted that n* should practically be an integer number, but Eq. (3.65) gives a real number. In order to obtain the optimal integer value of n, one should compute the E[TCU(Q, n)] for both integers that are adjacent to real number n*, respectively (for instance, in this example, Eq. (3.65) gives n ¼ 3.1733, so both n ¼ 3 and n ¼ 4

3.2 No Shortage

173

must be plugged in E[TCU(Q, n)]), and select the one with minimum cost as our optimal n* (Chiu et al. 2011).

3.2.2.2

Random Machine Breakdown

Taleizadeh et al. (2017) developed an integrated inventory model to determine the optimal lot size and production uptime while considering stochastic machine breakdown and multiple shipments for a single buyer and single vendor. The proposed model considers a manufacturing system that generates imperfect products such that x percent of manufactured products are defective and that portion is generated randomly at a production rate of d ¼ Px. The defective products are not repairable, and at the end of the production uptime, they will be discarded. The constant production rate of items, P, is greater than the annual demand rate D. They assumed that stochastic machine breakdown can occur, and the number of breakdowns in a year is a random variable, λ, that follows the Poisson probability distribution function. When a breakdown occurs, the production system follows the NR policy. In other words, immediately after a breakdown, the repair will be done, and production will not be started until all of the on-hand inventory is depleted. They assumed the machine repair time to be constant, and to prevent shortages, they considered safety stock. In this manufacturing system, both batch quantity and the distance between two shipments are identical. Also, the cost of transportation is paid by the buyer. Some notations which are specifically used to model this problem are shown in Table 3.7. t1 is the production uptime, and t denotes the time before a breakdown occurs. They investigated two cases, t < t1 and, t  t1 separately, because machine breakdowns can occur randomly during the production uptime.

The First Case—t  t1 In this case, a machine breakdown does not occur during the production uptime. For this case, the on-hand inventories of perfect and defective vendor items are shown in Figs. 3.10 and 3.11, respectively. Moreover, Fig. 3.7 represents the buyer’s inventory level when a machine breakdown does not occur.

Vendor’s Cost According to Fig. 3.7, the setup, variable production, and disposal costs are SCV (1) ¼ K, PC (1) ¼ CPt1, and dCV (1) ¼ CSPt1x, respectively, because x percent of all produced items are scrapped. According to Fig. 3.8, the holding cost for defective items is:

174

3 Scrap

Table 3.7 Notation of given problem (Taleizadeh et al. 2017) H1 H2 K1 h2 T T0 Tu t td td0 t1 TC(t, Q) TC(t1, Q) TCU(t1, Q)

Maximum level of on-hand inventory when machine breakdown does not occur (unit) Maximum level of on-hand inventory when machine breakdown occurs (unit) Fixed ordering cost of buyer ($/order) Holding cost of buyer ($/unit/unit time) Cycle length when breakdown does not occur (time) Cycle length when breakdown occurs (time) Cycle length for integrated case (time) Production time before a random breakdown occurs (time) Time required to deplete all available perfect-quality items when machine breakdown does not occur (time) Time required to deplete all available perfect-quality items when machine breakdown occurs (time) Production uptime when a breakdown does not occur (time) Total inventory costs per cycle when machine breakdown occurs ($) Total inventory costs per cycle when machine breakdown does not occur ($) Total inventory costs per unit time for integrated case ($)

HdCVð1Þ ¼

h1 d ð t 1 Þ 2 2

ð3:67Þ

The holding cost of the safety stock during each cycle is: HSCVð1Þ ¼ hDt r T

ð3:68Þ

To calculate the holding cost of the perfect items, the calculation of number of perfect products per shipment is needed. Thus, the number of perfect items is calculated and added. To determine the holding cost of perfect items, it first requires determining the average inventory of perfect items in the production uptime which is as follows: AIPU ¼

ðQ þ xÞT ðQ þ x þ 2xÞT ðQ þ ð2k  1ÞxÞT þ þ⋯þ 2n 2n 2n kQT xT ¼ þ ð1 þ 3 þ 5 þ ⋯ þ ð2k  1ÞÞ 2n 2n

 kQT xT kð1 þ 2k  1Þ kQT k2 xT ¼ þ þ ¼ 2 2n 2n 2n 2n

The average inventory in the production downtime is:

ð3:69Þ

3.2 No Shortage

175

I (t) H1 H2 P–d

x

q

q

2x

time

(n´–1)T´ T´ n

T´ 2 T´ 3 T´ (k–1)T´ kT´ n n n n n t t1

t2 t´4

T´

T Fig. 3.10 The vendor’s on-hand inventory of perfect-quality items when machine breakdown occurs (Taleizadeh et al. 2017) Fig. 3.11 The vendor’s on-hand inventory of defective items when machine breakdown occurs (Taleizadeh et al. 2017)

I (d)

dt

d

t

AIPD ¼

t+tr

t1

Time

kxT ðkx  QÞT ðkx  2QÞT QT ðn  kÞ þ ⋯þ ðQ þ kxÞT ð3:70Þ þ ¼ n n 2n n n

According to Eqs. (3.69) and (3.70), the average inventory of perfect items is (Taleizadeh et al. 2017):

176

3 Scrap

Fig. 3.8 The vendor’s on-hand inventory of defective items when machine breakdown does not occur (Taleizadeh et al. 2017)

Id (t)

dt1

d

t1

  kQT k 2 xT QT kxT þ þ ðn  k Þ þ 2n 2n 2n 2n

 QT kxT QT kxT ¼ þ ðk þ n  k Þ ¼ þ 2n 2n 2 2

Time

AIPI ¼

ð3:71Þ

Figure 3.7 shows that: x¼

ðP  dÞT Q n nQ T¼ D

ð3:72Þ ð3:73Þ

Using Eqs. (3.72) and (3.73) in Eq. (3.71) gives:     nQ nQ

 k ðP  dÞT D D Q HCPI ¼ þ n 2 2

 nQ2 nkQ ðP  dÞnQ ¼ Q þ nD 2D 2D

 nQ2 nkQ2 ðP  dÞ ¼ 1 þ D 2D 2D 

 ðP  d Þ nQ2 ¼ 1 1þk D 2D Q

According to Figs. 3.7 and 3.8,

ð3:74Þ

3.2 No Shortage

T¼

177

ðP  d Þt 1 k ðP  d Þ T ðP  d Þ n ¼ )n¼ ) ¼ D D D t1 k

ð3:75Þ

Þ Replacing n in Eq. (3.74) with kðPd D , as shown in Eq. (3.75), gives the average 2 inventory as: nQ 2D ½1 þ n  k . Thus, the holding cost of perfect items is:



nQ2 HCPI ¼ h ð1 þ n  k Þ 2D

 ð3:76Þ

Therefore, the vendor’s total cost is (Taleizadeh et al. 2017): TCVðt 1 , QÞ ¼ K þ CPt 1 þ C S Pt 1 x þ

h1 d ð t 1 Þ 2 hnq2 þ hDt r T þ 2 2D

 ð1 þ n  k Þ

ð3:77Þ

Buyer’s Cost According to Figs. 3.7 and 3.9, the transportation and fixed ordering costs are SCB (1) ¼ nCt and OCB (1) ¼ K1, respectively. Moreover, the holding cost is: HCBð1Þ ¼ h2



 QT QT nQ2 n ¼ h2 ¼ h2 2n 2 2D

ð3:78Þ

Therefore, the buyers total cost is (Taleizadeh et al. 2017): TCBðt 1 , QÞ ¼ K 1 þ nCT þ h2

nQ2 2D

ð3:79Þ

From Eqs. (3.77) and (3.79), the total cost is (Taleizadeh et al. 2017): TCðt 1 , QÞ ¼ K þ K 1 þ nCT þ CPt 1 þ C S Pt 1 x þ þhDt r T þ

hnQ2 nQ2 ð 1 þ n  k Þ þ h2 2D 2D

h1 dt 1 2 2

ð3:80Þ

According to Figs. 3.7 and 3.9, T¼

ðP  d Þt 1 D

ð3:81Þ

178

3 Scrap

I (tb)

Fig. 3.9 The buyer’s inventory level when machine breakdown does not occur (Taleizadeh et al. 2017)

Q

T/n

Ti

Dt 1 Q

ð3:82Þ

ðP  d Þt 1 Q

ð3:83Þ

k¼ n¼

Time

As the defective rate x, is a random variable with a known probability density function, its expected value can be used. Using all related parameters from Eqs. (3.80) to (3.83), the expected production–inventory cost per cycle, E[TC(t1, Q)], is (Taleizadeh et al. 2017): C ðP  dÞt 1 h1 PE ðxÞt 1 2 E ½TCðt 1 , QÞ ¼ K þ K 1 þ CPt 1 þ CS Pt 1 E ðxÞ þ T þ q 2

 ðP  d Þt 1 nQ2 ðP  dÞt 1  þhDg þ h2 Q 2D Q



 2 ðP  dÞt 1 hQ ðP  dÞt 1 Dt 1 þ 1þ  2D Q Q Q ð3:84Þ This can be simplified to:

C T ð 1  E ð xÞ Þ hQð1  EðxÞÞ þ hgð1  E ðxÞÞ þ Q 2D



2 h Qð1  E ðxÞÞ h PE ðxÞ hP2 ð1  EðxÞÞ hPð1  E ðxÞÞ 2 þ 2 þ 1 þ t 1 þ fK þ K 1 g  2D 2 2 2D E½TCðt 1 , QÞ ¼ Pt 1 C þ C S EðxÞ þ

ð3:85Þ where tr ¼ g is the fixed repair time.

3.2 No Shortage

179

The Second Case—t < t1 In this case, a machine failure occurs during the production uptime, and the NR policy is assumed. When a machine breakdown occurs, the machine will immediately be repaired, and production will only restart when the inventory level reaches zero. The vendor’s on-hand inventories of perfect-quality and defective items are shown in Figs. 3.10 and 3.11, respectively. Moreover, Fig. 3.12 depicts the buyer’s inventory level in case of a breakdown.

Vendor’s Cost According to Fig. 3.10, the setup, variable production, and disposal costs are SCV (2) ¼ K, PC (2) ¼ CPt1, and dCV (1) ¼ CsPtx, respectively. According to Fig. 3.11, the holding cost of defective items is: HdCVð2Þ ¼

h1 dt 1 2 þ h1 tdt r 2

ð3:86Þ

And the holding cost of safety stock during each cycle is: HSCVð2Þ ¼ hDt r T 0

ð3:87Þ

Similar to holding cost of safety stock during each cycle is: HCPI ¼ h

 0 2  nQ ð 1 þ n0  k 0 Þ 2D

ð3:88Þ

Moreover, the machine repair cost is assumed to be M, Therefore, the vendor’s total cost is:

Fig. 3.12 The buyer’s inventory level when machine breakdown occurs (Taleizadeh et al. 2017)

I (tb) Q

T' /n'

Time T'

180

3 Scrap

TCVðt, QÞ ¼ K þ C M þ CPt þ C S Ptx þ

h1 dt 2 hn0 Q2 þ hDt r T 0 þ 2 2D

 ð1 þ n0  k 0 Þ þ h1 tdt r

ð3:89Þ

Buyer’s Cost According to Figs. 3.10 and 3.12, the transportation and fixed ordering costs are SCB (2) ¼ nCT and OCB (2) ¼ K1, respectively. Moreover, the holding cost is:

HCBð2Þ ¼ h2

QT 0 0 n 2n0

 ¼ h2

QT 0 n0 Q2 ¼ h2 2 2D

ð3:90Þ

n0 Q 2 2D

ð3:91Þ

Therefore, the buyers total cost is: TCBðt, QÞ ¼ K 1 þ n0 CT þ h2 From Eqs. (3.89) and (3.90), the total cost is: TCðt, QÞ ¼ K þ K 1 þ CM þ CPt þ CS Ptx þ

h1 dt 2 þ h1 tdt r 2

hn0 Q2 n0 Q 2 þhDt r T þ ð 1 þ n0  k 0 Þ þ h2 þ n0 C T 2D 2D

ð3:92Þ

0

According to Figs. 3.10 and 3.12, ðP  d Þt D Dt k0 ¼ Q

T0 ¼

ð3:94Þ

ðP  dÞt Q

ð3:95Þ

T T0 T ) 0¼ n n n

ð3:96Þ

n0 ¼ T 0 ¼ n0

ð3:93Þ

As the defective rate, x, is a random variable with a known probability density function, this expected value can be used. Using all related parameters from Eqs. (3.93) to (3.96), the expected production–inventory cost per cycle, E[TC (t, Q)], is (Taleizadeh et al. 2017):

3.2 No Shortage

181



h1 E ðxÞ hPð1  EðxÞÞ2 hð1  E ðxÞÞ þ  2 2 2D



CT hQ h2 Q þ C þ CS E ðxÞ þ h1 gEðxÞ þ hg þ þ ð1  EðxÞÞ Pt þ PQ 2D 2D

E½TCðt, QÞ ¼ K þ K 1 þ CM þ Pt 2

ð3:97Þ where tr ¼ g is the fixed repair time. Now, Eqs. (3.85) and (3.97) can be rewritten as (Taleizadeh et al. 2017): E ½TCðt 1 , QÞ ¼ ðK þ K 1 Þ þ S1 t 1 þ S2 t 1 2

ð3:98Þ

E ½TCðt, QÞ ¼ ðK þ K 1 þ M Þ þ ðS1 þ h1 gPE ðxÞÞt þ S2 t 2

ð3:99Þ

where S1 ¼ CP þ CS PE ðxÞ þ þ

CT Pð1  EðxÞÞ þ hgPð1  E ðxÞÞ Q

hQPð1  EðxÞÞ h2 Qð1  E ðxÞÞ þ 2D 2D S2 ¼

ð3:100Þ

h1 PE ðxÞ hP2 ð1  E ðxÞÞ2 hPð1  EðxÞÞ þ  2 2 2D

ð3:101Þ

Integrating the EPQ Models with and Without Breakdowns Because the defective rate and the number of breakdowns are random variables, the cycle length of the proposed model is not constant. Thus, the expected production– inventory cost per unit time, E[TCU(t1, Q)], can be obtained as (Taleizadeh et al. 2017): t1 R E½TCUðt 1 , QÞ ¼

E ½TCðt, QÞf ðt Þdt þ

R1

E ½TCðt 1 , QÞf ðt Þdt

t

0

ð3:102Þ

E ðT U Þ

and Zt1 E ðT U Þ ¼

0

Z1

E ðT Þf ðt Þdt þ 0

EðT Þf ðt Þdt t

ð3:103Þ

182

3 Scrap

The authors assume that the number of breakdowns per unit time is a random variable that follows a Poisson probability distribution function. Therefore, the time between two breakdowns follows the exponential probability distribution function with parameter λ. According to Eq. (3.103) (Taleizadeh et al. 2017), Zt1 E ðT U Þ ¼

Pð1  E ðxÞÞt f ðt Þdt þ D

Z1

Pð1  E ðxÞÞt 1 f ðt Þdt D

ð3:104Þ

t1

0

Zt1 Given that

f ðt Þdt ¼ F ðt 1 Þ ¼ 1  eλt1

ð3:105Þ

0

Then, Zt1

1 1 t  f ðt Þdt ¼ t 1 eλt1  eλt1 þ λ λ

ð3:106Þ

0

Substituting Eqs. (3.105) and (3.106) into Eq. (3.104) gives (Taleizadeh et al. 2017)

E ðT U Þ ¼

8 t 0 which can be simplified to:

ð3:115Þ

184

3 Scrap

 2 1  eλt1 ðK þ K 1 Þλ t1  þ h1 gPE ðxÞλ þ 2S2 λð1 þ eλt1 Þ

ð3:116Þ

Hence, E[TCU(t1, q)] is convex if and only if (Taleizadeh et al. 2017):  2 1  eλt1 ðK þ K 1 Þλ 0  t1  þ ¼ w ðt 1 Þ h1 gPE ðxÞλ þ 2S2 λð1 þ eλt1 Þ

ð3:117Þ

To determine the optimal values of t1 and q, the first derivatives of E[TCU(t1, Q)] with respect to t1 and Q should be made equal to zero, which gives (Taleizadeh et al. 2017): ∂E½TCUðt 1 , QÞ C D h h ¼ 0 )  T2 þ þ 2 ¼ 0 ) Q ¼ 2 2 ∂Q Q

rffiffiffiffiffiffiffiffiffiffiffiffiffi 2C T D h þ h2

 ∂E ½TCUðt 1 , QÞ Deλt1 ¼ ðK þ K 1 Þλ2 2 ∂t 1 ð1  eλt1 Þ Pð1  EðxÞÞ   þ 1  eλt1  λt 1 fh1 gPE ðxÞλ þ 2S2 g ¼ 0 Since

Deλt1 2 ð1eλt1 Þ Pð1E ðxÞÞ

ð3:118Þ

ð3:119Þ

is greater than zero, Eq. (3.119) can be rewritten as

(Taleizadeh et al. 2017):  ðK þ K 1 Þλ2 þ eλt1 1 þ λt 1 fh1 gPE ðxÞλ þ 2S2 g ¼ 0 ) eλt1 1 þ λt 1 ¼

ðK þ K 1 Þλ2 þ h1 gPE ðxÞλ þ 2S2 ðK þ K 1 Þλ2 ¼ þ1 h1 gPE ðxÞλ þ 2S2 h1 gPE ðxÞλ þ 2S2 ð3:120Þ

ðKþK 1 Þλ Assuming y ¼ h1 gPE ðxÞλþ2S2 2

eλt1 1 þ λt 1 ¼ λ2 y þ 1

ð3:121Þ

To find the optimal run time, it can be supposed that: t 1L

pffiffiffiffiffi ¼ 2y ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð K þ K 1 Þ 2

ðxÞÞ h1 gPE ðxÞλ þ h1 PE ðxÞ þ ð1E  hPð1  E ðxÞÞ D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λy þ λ2 y2 þ λy t 1U ¼ 2 hP

ð3:122Þ

ð3:123Þ

3.2 No Shortage

185

Theorem 3.1 The optimal run time must follow the relation t 1L < t 1 < t 1U . Proof It is proved in Appendix B of Taleizadeh et al. (2017). Theorem 3.2 The total cost function TCU(t1, Q) is convex. Proof According to Eq. (3.117), E[TCU(t1, Q)] is convex if and only if 0  t 1  ðKþK 1 Þλ h1 gPEðxÞλþ2S 2 λt 1

1þe

þ

2ð1eλt1 Þ λð1þeλt1 Þ

¼ wðt 1 Þ because both λ and t are positive, and 1 

 2. Thus (Taleizadeh et al. 2017),  1  eλt1 ðK þ K 1 Þλ þ wðt 1 Þ > λ h1 gPE ðxÞλ þ 2S2

ð3:124Þ

ðKþK 1 Þ if v ¼ h1 gPE ðxÞλþ2S2 , Eq. (3.124) becomes (Taleizadeh et al. 2017)



1  eλt1 wðt 1 Þ ¼ vλ þ λ

ð3:125Þ

ðt 1 , QÞ Also, given that ∂E½TCU ¼ 0, ∂t 1

ðK þ K 1 Þλ2 þ1 h1 gPE ðxÞλ þ 2S2   1  eλt1 2 λt 1 ) t 1 ¼ vλ þ Or λt 1 ¼ vλ þ 1  e λ eλt1 þ λt 1 ¼

ð3:126Þ ð3:127Þ

Combining Eqs. (3.126) and (3.127) gives (Taleizadeh et al. 2017): 

1  eλt1 wðt 1 Þ > vλ þ ¼ t1 λ

ð3:128Þ

Thus, the total cost function TCU(t1, Q) is convex. Example 3.4 Taleizadeh et al. (2017) presented an example to illustrate the applicability of our proposed model. Assume that the production and demand rates are P ¼ 10,000 and D ¼ 4000 units per year, respectively. x percent of the items produced during the production time could be defective following a uniform probability distribution function over the interval [0, 0.2]. Machine breakdowns might occur during the production uptime. The number of machine failures follows a Poisson distribution function with β ¼ 0.5. The other parameters are as follows: K ¼ $450 per production run, K1 ¼ $150 per shipment, h ¼ $0.6 per item per unit time, h1 ¼ $0.8 per defective item per unit time, h2 ¼ $0.9 per item per unit time, C ¼ $2 per item, Cs ¼ $0.3 per scrapped item, CT ¼ $0.3 per item, CM ¼ $500 per each breakdown, and tr ¼ 0.018 per year (Taleizadeh et al. 2017).

186

3 Scrap

From Eq. (3.118), the optimal batch size for each delivery is Q* ¼ 730.2967. According to Eqs. (3.108) and (3.123), E[TCU(t1L*, Q*)] ¼ $11,657.06 and t1L* ¼ 0.3985. Also, from Eqs. (3.108) and (3.123), t1U* ¼ 0.4188 and E[TCU (t1U*, Q*)] ¼ $11,656.51. Given that [t1L* ¼ 0.3985, t1U* ¼ 0.4188] E [0, (t1U*) ¼ 0.4569], obviously the expected total cost function, E[TCU(t1, Q)], is convex, meeting the required condition. Using Newton’s method and the upper and lower bounds (t1L*, t1U*) as two initial points, the optimal production uptime will be equal to t1* ¼ 0.4122. The optimal expected total cost is E[TCU(t1*, Q*)] ¼ $11,656.35.

3.3

Fully Backordered

3.3.1

Continuous Delivery

In this section, EPQ models with scrap which considered continuous delivery policy with shortage (Fully backordered) are presented.

3.3.1.1

Proposing an Arithmetic–Geometric Mean Inequality Method

Shyu et al. (2014) proposed an arithmetic–geometric mean inequality method to simplify the algebraic method of completing perfect square established by Huang (2006) to find the optimal solution under which the expected annual cost minimized. First consider that the shortage is not permitted. From Chiu (2006), the expected annual cost can be expressed as:  E ½TCUðQÞ ¼ D

   E ð xÞ C KD 1 hQ D 1 þ CS þ 1 þ Q 1  E ð xÞ 2 P 1  E ð xÞ 1  E ð xÞ 1  E ð xÞ   E ð xÞ D hQ E ðx2 Þ hQ 1  þ P 1  E ð xÞ 2 1  E ð xÞ F ¼ E þ þ GQ, Q ð3:129Þ

where the constants:  E¼D

E ðxÞ F þ CS 1  E ð xÞ 1  E ðxÞ

 ð3:130Þ

3.3 Fully Backordered

187

F¼

Kλ 1  E ð xÞ

ð3:131Þ

and

1 1  E ð xÞ

G¼

      2  h D D 1 þE x E ð xÞ h 1 2 P P

ð3:132Þ

By using the arithmetic–geometric mean inequality, it can easily be obtained that: EðTCUðQÞÞ ¼ E þ

pffiffiffiffiffiffiffi F þ GQ  E þ 2 FG: Q

ð3:133Þ

when the equality F ¼ GQ, Q

ð3:134Þ

Holds, E[TCU(Q)], has a minimum. Therefore, rffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2KD   Q ¼ ¼ : D G h 1  P  2h 1  DP E ðxÞ þ hE ðx2 Þ 

ð3:135Þ

Hence, the minimum value of E[TCU(Q)] is as follows (Hsu and Hsu 2016): pffiffiffiffiffiffiffi E ½TCUðQ Þ ¼ E þ 2 FG   E ð xÞ F 1 ¼D þ CS þ 2KD 1  E ð xÞ 1  E ð xÞ 1  E ð xÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :    u    D D uh 1 þ E x2  h 1  E ð xÞ t P P  2 KD

ð3:136Þ

Now under backordering, the problem will be remodeled. Some notations which are specifically used to model this problem are shown in Table 3.8. The expected annual cost can be expressed as:

Table 3.8 Notation of given problem (Hsu and Hsu 2016) t1 t2 t3 t4

Production uptime (time) Production downtime (time) Time shortage permitted (time) Time needed to satisfy all the backorders by the next production (time)

188

3 Scrap



 E ð xÞ C KD 1 þ CS E½TCUðQ, BÞ ¼ D þ Q 1  E ðxÞ 1  E ð xÞ 1  E ðxÞ

0 1  i ð C þ hÞ B 1  x C B 2 h D 1 þ 1 Q  2B þ b E@ DA 2Q 2 P 1  E ð xÞ 1  E ð xÞ 1x P h   i E ðxÞ D hQ Eðx2 Þ Q þh B  1  þ P 2 1  E ð xÞ 1  E ðxÞ 2 F IB ¼ E þ þ GQ þ þ JB Q Q   h i F I JQ 2 J2 ¼Eþ þ þ G Bþ Q Q Q 2F 4I h

ð3:137Þ where the constants E, F, and G are the same as before case and the constants I and J are given as: I¼

 ðC b þ h Þ 1x and J ¼ h E 2ð 1  E ð x Þ Þ 1  x  DP

G Q to get Equation (3.138) implies that when Q is given, B can be set as B ¼  2F the minimum value of E[TCU(Q, B)] as follows:

   F J2 Q E TCUðQ, BðQÞÞ ¼ E þ þ G  Q 4I

ð3:138Þ

By using the arithmetic–geometric mean inequality, one easily obtains that: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     C G2 J2 E TCUðQ, BðQÞÞ ¼ A þ þ D  QEþ2 F D Q 4F 4I "

ð3:139Þ

when the equality   F J2 ¼ G Q Q 4I

ð3:140Þ

Holds E[TCU(Q, B(Q))], has a minimum. Therefore, using Eq. (3.140), Q* is:

3.3 Fully Backordered

189

sffiffiffiffiffiffiffiffiffiffiffiffiffi F Q ¼ 2 G  J4I vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD ¼u 2  u  2 1E ð x Þ f g   2h 1  D EðxÞ þ hEðx2 Þ th 1  DP  C hþh  P b 

E

ð3:141Þ

1x 1xD P

and the optimal allowable backorder level is: B ¼ 

J  Q 2I

ð3:142Þ

Therefore, the minimum value of the expected annual cost is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     E ð xÞ J2 F   E ½TCUðQ , B Þ ¼ E þ 2 F G  þ CS ¼D 4I 1  E ð xÞ 1  E ð xÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ffi 2 u u u 7 6 u    7 6 u D D 7 6 2 u þ E ðx Þ E ð xÞ 1 1 7 1  E ð x Þ h u 2hKD 6 P P 6 2 3  2 þu B7 7 6 u1  E ð x Þ 6 Cb þ h 1  E ð xÞ 1  E ð xÞ 7 u 6 6 1x 7 7 u 5 4 E4 5 t D 1x P ð3:143Þ

3.3.1.2

Random Defective Rate

Hsu and Hsu (2016) developed economic production quantity (EPQ) models to determine the optimal production lot size and backorder quantity for a manufacturer under an imperfect production process. They considered EPQ models with shortages backordered. It is assumed that all customers are willing to wait for a new supply when there is a shortage. The production process is imperfect, and the fraction of defective items at the time of production is x. Once an item is produced, it is inspected immediately. The inspection time is negligible in comparison to the time taken to produce each item. Some notations which are specifically used to model this problem are shown in Table 3.9.

190

3 Scrap

They consider three cases regarding the time taken to sell the defective items (Hsu and Hsu 2016): Case I. The defective items may be sold to a secondary market (when vd > 0) or scrapped (i.e., vd ¼ 0) at the time identified and are not counted into inventory (see Fig. 3.13). Case II. The defective items are kept in stock and sold at the end of the production period within each cycle (see Fig. 3.14). Case III. The defective items are sold at the end of the production cycle or at the beginning of the next production run (see Fig. 3.15). They considered the above three cases because different industries will dispose of the defective items at different timings; for example, pharmaceutical companies will scrap the defective products and not count them into inventory, and the furniture industry will sell the defective items at a discounted price as a secondary market. Moreover, the timing of disposing of the defective items may be different for different companies (which should be negotiated between the company and its secondary market customers): B , P ð 1  xÞ  D

ð3:144Þ

Q B  , P Pð1  xÞ  D

ð3:145Þ

t1 ¼ t2 ¼ t3 ¼

ðPð1  xÞ  DÞQ B  , PD D B t4 ¼ , D Q ðt 1 þ t 2 Þ ¼ P

T ¼ ðt 1 þ t 2 þ t 3 þ t 4 Þ ¼

Q ð 1  xÞ D

ð3:146Þ ð3:147Þ ð3:148Þ ð3:149Þ

Since the defective rate x is constant, Production cost ¼ C

D , ð 1  xÞ

ð3:150Þ

Inspection cost ¼ CI

D , ð 1  xÞ

ð3:151Þ

Setup cost ¼ K

D , Qð1  xÞ

ð3:152Þ

3.3 Fully Backordered

191

Table 3.9 Notation of given problem (Hsu and Hsu 2016) TCC TCj(Q, b) TPj(Q, b)

The total cost per cycle ($/cycle) The total annual cost for case j ( j ¼ I, II, III) ($/year) The total annual profit for case j ( j ¼ I, II, III) ($/year)

Inventory Level [P(1–x) – D]Q –B P

Time

–B

t1

Q P

t2

t3

t4

Q (1–x) D

Fig. 3.13 Behavior of the inventory level over time for Case I (Hsu and Hsu 2016)

  ðt þ t Þ 1 Q 3 Holding cost ¼ h ðPð1  xÞ  DÞ  B 2 T 2 P   2 ! Q 1  x  DP  B 1  ¼ h , 2 Q 1  x  DP ðt þ t 4 Þ 1 1 B2 ¼ Cb  Shortage cost ¼ C b B 1 T 2 2 Q 1  x  DP

ð3:153Þ ð3:154Þ

TCI ðQ, BÞ ¼ T P þ T i þ T K þ T h þ T b 0

1   2 D CD CI D KD 1 B Q 1xP B C C B2 þ þ þ h@ ¼   Aþ  b  D D ð1  xÞ ð1  xÞ Qð1  xÞ 2 Q 1x 2Q 1  x  P P ð3:155Þ Note that since the total annual production and the inspection (screening) costs are independent of Q and B, the relevant costs to determining the optimal Q and B include only the setup, holding, and backordering costs. However, since both the

192

3 Scrap Inventory Level [P(1–x)– D]Q –B P

Qx

Time

–B

t1

t Q 2 P

t3

t4

Q (1–x) D

Fig. 3.14 Behavior of the inventory level over time for Case II (Hsu and Hsu 2016)

Inventory Level [P(1–x)– D]Q P

–B

xQ

Time

–B

t1

t Q 2 P

t3

t4

Q (1–x) D

Fig. 3.15 Behavior of the inventory level over time for Case III (Hsu and Hsu 2016)

3.3 Fully Backordered

193

total annual production and inspection costs are functions of the defective rate x, these costs are included in the total annual cost for the purpose of sensitivity analyses. To obtain the total revenue per cycle, note that for each production lot of size Q, Q (1  x) units are non-defective with a selling price of s per unit, and Qx units are defective, which can be sold at v per unit. Dividing the total revenue per Þ Dx cycle by the cycle time T ¼ Qð1x D , the total annual revenue is equal to sD þ v ð1xÞ. That is, if the defective rate is x, then in order to produce D units of good-quality D Dx items, a total of ð1x Þ units will be produced, among which ð1xÞ units are defective. The total annual profit is the total annual revenue less the total annual cost and is given as follows (Hsu and Hsu 2016): xD D D D C  CI K TPI ðQ, BÞ ¼ sD þ v ð 1  xÞ ð 1  xÞ ð 1  xÞ Q ð 1  xÞ 0 1   2 D 1 B Q 1xP B C 1 B2  h@   A  Cb   D D 2 2 Q 1x Q 1x P P

ð3:156Þ

Since the revenue is also independent of the production lot size and the backorder quantity, maximizing the total annual profit is equivalent to minimizing the total annual cost. By taking the second derivative of TC1(Q, B) with respect to Q and B, one obtains: 2

∂ TCI ðQ, BÞ ðh þ Cb ÞB2 2KD þ 3 ¼ 3 2 ∂Q Q ð1  xÞ Q 1  x  DP

ð3:157Þ

2

∂ TCI ðQ, BÞ ðh þ C b Þ ¼  ∂B2 Q 1  x  DP 2

ð3:158Þ

2

∂ TCI ðQ, BÞ ∂ TCI ðQ, BÞ ðh þ C b Þ B ¼ ¼ 2 ∂Q∂B ∂B∂Q Q 1  x  DP

ð3:159Þ

and 2

∂ TCI ðQ, BÞ ∂Q2 ¼

!

2

∂ TCI ðQ, BÞ ∂B2

2BDðh þ C b Þ  Q ð1  xÞ 1  x  DP 4

!

2



∂ TCI ðQ, BÞ ∂Q∂B

!2

ð3:160Þ

Since the production process is imperfect with a defective rate of x, the effective annual production rate P0 ¼ P(1  x). Note that to make sure that the production process has enough capacity to satisfy the customers’ demand, the assumption that P

194

3 Scrap

(1  x) > D (i.e., x < 1  DP ) should be held. If x < 1  DP , then 2

∂ TCI ðQ, BÞÞ ∂B2

2

∂ TCI ðQ, BÞ ∂Q2

> 0,

> 0 and which implies that the total annual cost is a convex function of Q and B (see, e.g., Ghorpade and Limaye 2010, Proposition 3.71 on page 137). Another way to prove the convexity of TC1(Q, B) is to use the Hessian matrix equations (see, e.g., Rardin and Rardin 1998) and verify the existence of the following equation: 0

½Q

2

∂ TCI ðQ, BÞ B ∂Q2 B B B @ ∂2 TC ðQ, BÞ I

∂Q∂B

1 2 ∂ TCI ðQ, BÞ C Q  ∂Q∂B C > 0, Q, B 6¼ 0: C 2 ∂ TC ðQ, BÞ A B I

∂B2

Solving the elements of the Hessian matrix, one obtains: 0

1 2 ∂ TCI ðQ, BÞ ∂2 TCI ðQ, BÞ B C" # ∂Q2 ∂Q∂B B C Q B C ½ Q B B C @ ∂2 TCI ðQ, BÞ ∂2 TCI ðQ, BÞ A B ∂Q∂B ∂B2 3 2 ðh þ C b ÞB2  ðh þ C b Þ 2KD   B7 6 Q 3 ð 1  xÞ þ 3  D D 7" # 6 Q 1x Q2 1  x  7 Q 6 P P 7 ¼ ½ Q B 6 7 6 ðh þ C b Þ ðh þ C b Þ 7 B 6 4  B   5 D D 2 Q 1x Q 1x P P  " # Q 2KD 2KD 0 ¼ > 0 for Q, B 6¼ 0 ¼ Q2 ð1  xÞ Q ð 1  xÞ B Hence, TC1(Q, B) is a strictly convex function for all Q and B different from zero. By taking the first derivative of TC1(Q, B) with respect to Q and B and setting the results to zero, the optimal production lot size Q*CI and the maximum backorder quantity B*CI for Case I are given as follows (Hsu and Hsu 2016): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh þ C b Þ 2DK  ¼ h Cb ð1  xÞ 1  x  DP   D h BCI ¼ QCI 1  x  P ðh þ C b Þ

QCI

ð3:161Þ ð3:162Þ

3.3 Fully Backordered

195

Figure 3.16 depicts the behavior of the inventory over time for Case II. Note that the only difference between Cases I and II is that in Case II, the defective items are held until the end of the production period, so one obtains (Hsu and Hsu 2016): ðt þ t 2 Þ 1 1 QxD TCII ðQ, BÞ ¼ TCI ðQ, BÞ þ hQx 1 ¼ TCI ðQ, BÞ þ h T 2 2 Pð1  xÞ 0 1   2 D Q 1  x   B KD 1 B C B2 hQxD C P ¼ þ h@   Aþ  b þ D D Q ð 1  xÞ 2 2P ð1  xÞ Q 1x 2Q 1  x  P P ð3:163Þ and the optimal solution for Case II is given as: QCII

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh þ C b Þ 2DK    ¼ h Cb ð1  xÞ 1  x  DP þ ðh þ Cb Þ DP   D h BCII ¼ QCII 1  x  P ðh þ C b Þ

ð3:164Þ ð3:165Þ

Figure 3.17 depicts the behavior of the inventory over time for Case III. Note that the only difference between Cases II and III is that in Case III, the defective items are held until the beginning of the next production run, so one obtains:

 ðt 3 þ t 4 Þ D ¼ TCII ðQ, BÞ þ hQx 1  TCIII ðQ, BÞ ¼ TCII ðQ, BÞ þ hQx T Pð1  xÞ 0 1   2 D KD 1 B Q 1xP B C ¼ þ h@   A D Q ð 1  xÞ 2 Q 1x P

 2 1 B 1 QxD D þ Cb  þ hQx 1  þ h D 2 2 P ð 1  xÞ Pð1  xÞ Q 1x P ð3:166Þ and the optimal solution for Case III is given as (Hsu and Hsu 2016): QCIII

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh þ C b Þ 2DK     ð3:167Þ ¼ h C b ð1  xÞ 1  x  DP þ ðh þ Cb Þ 2xð1  xÞ  x DP   D h ð3:168Þ BCIII ¼ QCIII 1  x  P ðh þ C b Þ

196

3 Scrap

Fig. 3.16 On-hand inventory of perfect-quality items in EPQ model with scrap and breakdown occurring in inventorystacking period (Chiu et al. 2008)

H3

–D

H1

–D

H2 P–d–D

Time

B

t

t4

t2

t3

t r T 1 –t 4 –t T

If x is a random variable with a probability density function f(x), then Hsu and Hsu (2016) derived production, inspection, setup, holding, shortage, and total cyclic costs as presented in Eqs. (3.169)–(3.174): CQ,

ð3:169Þ

C I Q,

ð3:170Þ

K,         1 D 1 D h Q 1x  B ðt 2 þ t 3 Þ ¼ h Q 1  x  B 2 P 2 P 

Bð1  xÞ Q ð 1  xÞ   D D 1  x  DP

B 2 ð 1  xÞ 1 1 C b B ðt 1 þ t 4 Þ ¼ C b  2 2 Q 1  x  DP     1 D B TCC ¼ CQ þ C I Q þ K þ h Q 1  x  2 0 1 P B ð 1  xÞ C C b ð 1  xÞ BQ  @ ð 1  xÞ    þ   B2 D A D D D 1x 2Q 1  x  P P From Eq. (3.149), the expected cycle length would be:

ð3:171Þ ! ð3:172Þ ð3:173Þ

ð3:174Þ

3.3 Fully Backordered

197

Fig. 3.17 On-hand inventory of scrap items in EPQ model with scrap and breakdown occurring in inventory-stacking period (Chiu et al. 2008)

d•T1 ⎛ ⎜ ⎝

⎛

d• ⎜⎝t4+ t

d d Time

d•t4

t4

t

t2

t3

tr T1–t4 –t T E ðT Þ

Q ð 1  E ð xÞ Þ D

ð3:175Þ

Using the renewal reward theorem, the expected total annual cost would be (Hsu and Hsu 2016): EðTCI ðQ, BÞÞ ¼

with A1 ¼ E

EðTCC Þ CD D KD þ CI þ ¼ 1  E ð xÞ 1  E ðxÞ Qð1  EðxÞÞ E ðT Þ   ðh þ C Þ hE ð1  xÞ2 h DQ B2 b þ  2B þ Qþ  A1 2 P 2ð1  E ðxÞÞ 1  EðxÞ 2Qb



ð3:176Þ



1x 1xDP

, and

EðTPI ðQ, BÞÞ ¼ sD þ v

DE ðxÞ  EðTCI ðQ, BÞÞ ð 1  E ð xÞ Þ

ð3:177Þ

By taking the second derivative of E[TC1(Q, B)] with respect to Q and B, one obtains: 2

∂ E ðTCI ðQ, BÞÞ ðh þ Cb ÞB2 A1 2KD þ 3 ¼ 3 2 ∂Q Q ð 1  E ð xÞ Þ Q ð 1  E ð xÞ Þ

ð3:178Þ

2

∂ E ðTCI ðQ, BÞÞ ðh þ Cb ÞA1 ¼ 2 Q ð 1  E ð xÞ Þ ∂B

ð3:179Þ

2

∂ EðTCI ðQ, BÞÞ Bðh þ Cb ÞA1 ¼ 2 ∂Q∂B Q ð 1  E ð xÞ Þ And

ð3:180Þ

198

3 Scrap 2

∂ E ðTCI ðQ, BÞÞ ∂Q2 ¼

!

2

∂ E ðTCI ðQ, BÞÞ ∂B2

!

2



∂ E ðTCI ðQ, BÞÞ ∂Q∂B

!2

2KDðh þ Cb ÞA1 Q4 ð1  E ðxÞÞ

ð3:181Þ

From Eqs. (3.178)–(3.181), one can see that if A1 > 0 and E(x) < 1, then  2  2  2 ∂ E ðTCI ðQ, BÞÞ ∂ E ðTCI ðQ, BÞÞ I ðQ, BÞÞ > 0 , ∂ EðTC > 0 and  2 2 2 ∂B ∂B ∂Q  2 2 ∂ E ðTCI ðQ, BÞÞ > 0, which implies that the expected total annual cost is a convex ∂Q∂B 2

∂ E ðTCI ðQ, BÞÞ ∂Q2

function of Q and B and that exist unique values of Q and B that minimize Eq. (3.176) and maximize Eq. (3.177).  D If x is uniformly distributed between 0 and 1 P D b, then we have A1 ¼ 1 þ bP in 1Db (see the derivation in Hsu and Hsu 2016). P

For A1 to be real, b (the maximum possible value of x) should be less than 1  DP ; otherwise, ln 1  DP  b is undefined. If x follows a beta distribution with shape parameters p ¼ q ¼ 2, then b (the maximum possible value of x) should also be less than 1  DP. Consider the following Hessian matrix (Hsu and Hsu 2016): 0

2

∂ EðTCI ðQ, BÞÞ ∂2 EðTCI ðQ, BÞÞ B ∂Q2 ∂Q∂B B ½ Q B B B 2 @ ∂ EðTCI ðQ, BÞÞ ∂2 EðTCI ðQ, BÞÞ ∂Q∂B ∂B2 2 ðh þ C b ÞB2 A1 2KD þ 6 Q3 ð1  EðxÞÞ Q3 ð1  E ðxÞÞ 6 ¼ ½ Q B 6 4 Bðh þ C b ÞA1  ¼

2KD Q2 ð1  E ðγ ÞÞ

1 C" # C Q C C A B 3 Bðh þ C b ÞA1 " # Q2 ð1  EðxÞÞ 7 7 Q 7 ðh þ C b ÞA1 5 B

Qð1  E ðxÞÞ Q2 ð1  EðxÞÞ " # Q 2KD 0 ¼ > 0 for Q, B 6¼ 0 Q ð 1  E ð xÞ Þ B

Since it is assumed that the maximum possible value of x is less than 1  DP , E (TCI(Q, B)) is a strictly convex function for all nonzero Q and B. By taking the first derivative of E[TP1(Q, B)] or E[TC1(Q, B)] with respect to Q and B setting the results to zero, the optimal production lot size Q*rI and the maximum backorder quantity B*rI for Case I are given as follows (Hsu and Hsu 2016): QrI

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD  ¼u t  2 D h E ð1  xÞ  P ð1  E ðxÞÞ  ðhþChb ÞA1 f1  EðxÞg2

ð3:182Þ

3.3 Fully Backordered

199

BrI ¼ QrI ð1  E ðxÞÞ

h ðh þ C b ÞA1

ð3:183Þ

Note that if the defective rate x is constant, Eqs. (3.182) and (3.183) reduce to Eqs. (3.167) and (3.168), respectively. The expected cost per year and the optimal solution for Case II are given as (Hsu and Hsu 2016): 1 D E½TCII ðQ, BÞ ¼ E ½TCI ðQ, BÞ þ hQE ðxÞ 2 Pð1  EðxÞÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD   QrII ¼ u t  2 D h E ð1  xÞ  P ð1  2E ðxÞÞ  ðhþChb ÞA1 f1  EðxÞg2 BrII ¼ QrII ð1  EðxÞÞ

h ðh þ C b ÞA1

ð3:184Þ ð3:185Þ

ð3:186Þ

If γ is constant, Eqs. (3.185) and (3.186) reduce to Eqs. (3.164) and (3.165), respectively. For Case III, one obtains (Hsu and Hsu 2016):

EðxÞ  E ðx2 Þ  EðxÞ DP E½TCIII ðQ, BÞ ¼ E ½TCII ðQ, BÞ þ hQ ð1  E ðxÞÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD  QrIII ¼ u t  2 D h E ð1  xÞ  P 2ðEðxÞ  Eðx2 ÞÞ  ðhþChb ÞA1 f1  E ðxÞg2 BrIII ¼ QrIII ð1  EðxÞÞ

h ðh þ Cb ÞA1

ð3:187Þ ð3:188Þ

ð3:189Þ

If x is constant, Eqs. (3.188) and (3.189) reduce to Eqs. (3.167) and (3.168), respectively (Hsu and Hsu 2016). Example 3.5 Hsu and Hsu (2016) presented an example in which a designer window treatments manufacturer in Taiwan is a subsidiary of an interior decorating company. The manufacturer supplies all the window curtains needed by the interior decorating company. Since the window curtains are tailored designs, customers are willing to wait when a shortage occurs. Those window curtains produced which are not of perfect quality are sold to night market retailers at a discounted price. It should be noted that there are many famous night markets in Taiwan where people can buy cheaper products. During the production downtimes, the manufacturer will use the capacity to produce other products such as designer cushions, designer slipcovers, or designer bedding sets. Suppose that the manufacturer has the following parameters: P ¼ 3000 units/year, D ¼ 1000 units/year, K ¼ $400/lot, C ¼ $100/unit, CI ¼ $2/ unit, h ¼ $5/unit/year, Cb ¼ $10/unit/year, s ¼ $300/unit, and v ¼ $80/unit.

200

3 Scrap

First, it is assumed that the defective rate γ has a constant value and obtains the optimal solution of the three cases given in Table 3.10. Table 3.11 shows the optimal solutions when x is uniformly distributed between 0 and b (Hsu and Hsu 2016). If the defective rate x follows a uniform distribution with the probability density function (Hsu and Hsu 2016), ( f ðxÞ ¼

1 , b 0,

0xb otherwise

then Zb E ½ x ¼

x b dx ¼ , b 2

0

  E x2 ¼

Zb

Zb x f ðxÞdx ¼ 2

0

h i E ð 1  xÞ 2 ¼

x2 b2 dx ¼ , b 3

0

Zb

Zb 2

ð1  xÞ f ðxÞdx ¼ 0

ð1  xÞ2 b2 dx ¼ 1  b þ , b 3

0

2

A1 3

D D 6 1P 7 ln 4 ¼1þ 5: D bP 1 b P

3.3.1.3

Random Breakdown

Chiu et al. (2008) are concerned with determination of optimal lot size for an economic manufacturing quantity model with backordering, scrap, and breakdown occurring in inventory-stocking period. Also, they investigated the optimal manufacturing lot size for EMQ model with scrap, backlogging, and random breakdown occurring in inventory-stacking period. Some notations which are specifically used to model this problem are shown in Table 3.12. Let t denote production time before a breakdown taking place in the inventorystacking period, and let the constant machine repair time tr ¼ g. From Fig. 3.16, one can obtain the following (Chiu et al. 2008):

3.3 Fully Backordered

201

H 1 ¼ ðP  d  DÞt

ð3:190Þ

H 2 ¼ H 1  t r D ¼ H 1  gD

ð3:191Þ

T1 ¼

Q P

ð3:192Þ

H 3 ¼ H 2 þ ðP  d  DÞðT 1  t 4  t Þ

ð3:193Þ

T ¼ T 1 þ t2 þ t3 þ tr ,

ð3:194Þ

H3 , D B t3 ¼ D B t4 ¼ PdD

ð3:195Þ

t2 ¼

ð3:196Þ ð3:197Þ

where d ¼ Px. Total scrap items produced during production uptime T1 are (see Fig. 3.17): d  T1 ¼ Q  x

ð3:198Þ

Total production–inventory cost per cycle is (Chiu et al. 2008): TCðT 1 , BÞ ¼ K þ C  ðP  T 1 Þ þ C S  ðP  x  T 1 Þ þ CM h i H H þ H2 H þ H3 H þh 1 ðt Þ þ 1 ðt r Þ þ 2 ðT 1  t 4  t Þ þ 3 ðt 2 Þ 2 2 2 2  : d ðt 4 þ t Þ ðt þ t Þ þ dT 1 þh ðt 4 þ t Þ þ ðt 4 þ t Þt r þ 4 ðT 1  t 4  t Þ 2 2 h i B B þC b ðt 4 Þ þ ðt 3 Þ 2 2 ð3:199Þ Substituting all related parameters from Eqs. (3.190) to (3.198) in Eq. (3.199), one obtains: TCðT 1 , BÞ ¼ K þ C  ðP  T 1 Þ þ CS  ðP  x  T 1 Þ þ C M  h

P T Bð1  xÞ D 1

C b ð 1  xÞ hg   B2 þ   ðB þ gDÞ  hPgT 1 ð1  xÞ þ hPgt D D 2D 1  x  1x P P h i B B þC b ðt 4 Þ þ ðt 3 Þ 2 2 ð3:200Þ

þ

0.01 607.6 133 196,447.82 605.27 132.49 196,442.71 596.37 130.54 196,422.79

0.02 615.39 132.65 196,224.51 610.59 131.62 196,214.08 592.98 127.82 196,174.38

623.4 132.3 195,996.6 615.96 130.72 195,980.63 589.82 125.17 195,921.3

0.03

0.04 631.61 131.94 195,763.96 621.37 129.8 195,742.21 586.88 122.59 195,663.4 640.06 131.57 195,526.43 626.82 128.85 195,498.65 584.15 120.08 195,400.52

0.05 685.99 129.58 194,259.79 654.65 123.66 194,197.76 573.38 108.31 194,005.3

0.1

0.15 739.25 127.32 192,844.5 683.21 117.66 192,740.07 566.95 97.64 192,457.57

1, Q*cI; 2, B*cI; 3, TPI(Q*cI, B*cI); 4, Q*cII; 5, B*cII; 6, TPII(Q*cII, B*cII); 7, Q*cIII; 8, B*cIII; 9, TPIII(Q*cIII, B*cIII)

x 1 2 3 4 5 6 7 8 9

Table 3.10 The optimal solutions when x is constant (Hsu and Hsu 2016) 801.78 124.72 191,252.78 712.07 110.77 191,095.64 564.43 87.8 190,728.31

0.2

0.25 876.36 121.72 189,449.51 740.66 102.87 189,226.5 565.69 78.57 188,781.05

202 3 Scrap

0.04

0.02 615.28 132.61 196,224.26 610.48 131.58 196,213.84 593.06 127.82 196,174.55

0.02

0.01 607.57 132.99 196,447.76 605.25 132.48 196,442.65 596.39 130.54 196,422.83

0.03 623.13 132.21 195,996.04 615.7 130.63 195,980.07 589.98 125.18 195,921.68

0.06 0.04 631.12 131.77 195,762.93 620.9 129.64 195,741.19 587.16 122.59 195,664.07

0.08 0.05 639.25 131.3 195,524.78 626.07 128.59 195,497.03 584.58 120.07 195,401.57

0.1

0.2 0.1 682 128.32 194,252.19 651.18 122.52 194,190.51 574.96 108.18 194,009.57 0.15 727.93 123.93 192,824.69 674.24 114.79 192,721.74 570.29 97.09 192,467.3

0.3

0.4 0.2 776 117.34 191,211.35 693.83 104.91 191,058.72 570.05 86.2 190,745.78

1, E[γ]; 2, Q*rI; 3, B*rI; 4, E[TPI(Q*rI, B*rI)]; 5, Q*rII; 6, B*rII; 7, E[TPII(Q*rII, B*rII)]; 8, Q*rIII; 9, B*rIII; 10, E[TPIII(Q*rIII, B*rIII)]

b 1 2 3 4 5 6 7 8 9 10

Table 3.11 The optimal solutions when x is uniformly distributed between 0 and b (Hsu and Hsu 2016) 0.5 0.25 823.58 107 189,371.51 707.97 91.98 189,160 573.84 74.56 188,807.84

3.3 Fully Backordered 203

204

3 Scrap

Table 3.12 Notation of given problem (Chiu et al. 2008) T1 t tr t2 t3 t4 H1 H2 H3 TCU(T1, B) TC(T1, B)

The optimal production time to be determined (time) Production time before a random breakdown occurs (time) Time required for repairing and restoring the machine (time) Time required for depleting all available perfect-quality on-hand items (time) Shortage permitted time (time) Time required for filling backorder quantity (time) Level of on-hand inventory when machine breakdown occurs (unit) Level of on-hand inventory when machine is repaired and restored (unit) The maximum level of on-hand inventory for each production cycle (unit) Total production–inventory costs per unit time Total production–inventory costs per cycle

The production cycle length is not constant due to the assumption of random scrap rate, and a uniformly distributed breakdown is assumed to occur in the inventory-stacking time. Thus, to take randomness of scrap and breakdown into account, one can use the renewal reward theorem in inventory cost analysis to cope with variable cycle length and use integration of TC(T1, B) to deal with the random breakdown happening in inventory-stacking time. The expected total production– inventory costs per unit time can be calculated as follows (Chiu et al. 2008):

T E

 TCðT 1 , BÞ  f ðt Þdt

0

E ½TCUðT 1 , BÞ ¼ E ¼

R

1 t 4

T 1 t4 R

E ðT Þ  TCðT 1 , BÞ  1=t4 dt

0 T 1 Pð1E ðxÞÞ=D

 ð3:201Þ

Substituting Eqs. (3.190) through (3.200) in Eq. (3.201), one obtains (Chiu et al. 2008):  E ½TCUðT 1 , BÞ ¼ D

 E ð xÞ ðK þ C M ÞD C 1 þ CS þ PT 1 1  E ð xÞ 1  E ð xÞ 1  E ð xÞ  h i h D 1 hPT 1 E ðx2 Þ þ 1 PT 1  2B þ 2 P 2 1  E ð xÞ 1  E ð xÞ 0 1   h i E ð xÞ B2 ðCb þ hÞ B 1  x C D PT 1 þh B 1 þ E@ A D P 2PT 1 1  EðxÞ 1  E ð xÞ 1x P 0 1

þ

E ð xÞ hgD 1 1 hgD 1 B C þ hgD ðB þ gDÞE@   D A 1  E ð xÞ 2PT 1 2 1  E ðxÞ 1  E ð x Þ 1x P ð3:202Þ

3.3 Fully Backordered

205

E ð x2 Þ E ðxÞ 1 Let E0 ¼ 1E1 ðxÞ ; E 1 ¼ 1E ðxÞ ; E 2 ¼ 1E ðxÞ ; E 3 ¼ 1E ðxÞ E   1 1 1E ðxÞ E 1xD



1x 1xDP



; E4 ¼

P

Then Eq. (3.202) becomes (Chiu et al. 2008):  h i ðK þ CM ÞD h D 1 PT 1  2B E 0 E0 þ PT 1 2 P   h i 2 hPT 1 B D þ ðC b þ hÞE 3 þ h B  1  PT 1 E 1 E2 þ P 2 2PT 1 hgD hgD E þ ðB þ gDÞE4 þ hgDE 1  2PT 1 2 0 ð3:203Þ

E½TCUðT 1 , BÞ ¼ D½CE 0 þ C S E1  þ

The optimal inventory operating policy can be obtained by minimizing the expected cost function. For the proof of convexity of E[TC(T1, B)], one can utilize the Hessian matrix equation in Rardin and Rardin (1998) and verify the existence of the following: 0

½ T1

2

∂ E½TCUðT 1 , BÞ B ∂T 1 2 B B B @ ∂2 E½TCUðT , BÞ 1

∂T 1 ∂B

1 2 ∂ E½TCUðT 1 , BÞ C T  ∂T 1 ∂B C 1 >0 C 2 ∂ E½TCUðT , BÞ A B

ð3:204Þ

1

∂B2

E[TC(T1, B)] is strictly convex only if Eq. (3.204) is satisfied, for all T1 and B different from zero. By computing all the elements of the Hessian matrix equation, one obtains (Chiu et al. 2008): 0

2

∂ E ½TCUðT 1 , BÞ ∂2 E ½TCUðT 1 , BÞ B ∂T 1 2 ∂T 1 ∂B B ½ T 1 B B B 2 @ ∂ E ½TCUðT 1 , BÞ ∂2 E ½TCUðT 1 , BÞ ∂T 1 ∂B ∂B2 2 2 2DðK þ M Þ hg D ¼ E0 þ E >0 PT 1 PT 1 4

1 C" # C T1 C C A B

ð3:205Þ

Equation (3.205) is positive, because all parameters are positive. Hence, E[TC(T1, B)] is a strictly convex function. It follows that for optimal production uptime T1 and maximal backorder level B, one can differentiate E[TC(T1, B)] with respect to T1 and with respect to B and solve linear systems of Eqs. (3.206) and (3.207) by setting these partial derivatives equal to zero:

206

3 Scrap

h  i ∂E½TCUðT 1 , BÞ ðK þ CM ÞD h D hP E0 þ E2 ¼ E0 þ P 1  2 2 P 2 ∂T 1 PT 1   B2 D hgDB hg2 D2 E1  ðCb þ hÞE3  hP 1  E4  E4 2 2 P 2PT 1 2PT 1 2 2PT 1 ∂E ½TCUðT 1 , BÞ B hgD ¼ hE 0 þ ðC þ hÞE 3 þ hE 1 þ E PT 1 b 2PT 1 4 ∂B vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi u u 2DðK þ C ÞE þ hD2 g2 E 1  hE4 M 0 4 4ðC b þhÞE 3 1u T 1 ¼ t   2 D h D P h 1  E0  P ðC b þhÞE 3  2h 1  P E 1 þ hE 2 B ¼

  h gD E4 PT 1  2 ðC b þ hÞE 3

ð3:206Þ

ð3:207Þ

ð3:208Þ ð3:209Þ

Plugging E0, E2, E3, and E4 in Eqs. (3.208) and (3.209), the optimal production run time and optimal backordering quantity become (Chiu et al. 2008):  312 2 hD 2 hE ð1=ð1xD=PÞÞ g 2D ð K þC Þþ E ð 1= ð 1xD=P Þ Þ 1 M 1E ð x Þ ð þh ÞE ½ ð 1x Þ= ð 1xD=P Þ  4 C 1 b 5 T 1 ¼ 4  D  D h2 ð1E ðxÞÞ2 P 2 h 1 P 2h 1 P E ðxÞþhE ðx Þ ðCb þhÞE½ð1xÞ=ð1xD=PÞ 

h  Pð1  EðxÞÞ  T 1  Dgh 2 E   B ¼ 1x ðC b þ hÞ  E 1x D 

1 1xDP



ð3:210Þ ð3:211Þ

P

From Eq. (3.193) to Eqs. (3.208) and (3.209), one can obtain the optimal lot size Q* and optimal backorder level B* as follows (Chiu et al. 2008): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi u u 2DðK þ C ÞE þ hD2 g2 E 1  hE4 M 0 4 u 4ðC b þhÞE 3 Q ¼ t   D h2 D h 1  P E 0  ðCb þhÞE3  2h 1  P E 1 þ hE 2 B ¼

  h gD E4 Q  2 ðC b þ hÞE3

ð3:212Þ ð3:213Þ

Plugging E0, E2, E3, and E4 in Eqs. (3.212) and (3.213), the optimal production lot size and optimal backordering quantity become (Chiu et al. 2008):

3.3 Fully Backordered

207

2

 312 hD 2 hE ð1=ð1xD=PÞÞ 2DðK þ CM Þ þ 1EgðxÞ  E ð1=ð1  x  D=PÞÞ 1  4ðCb þh ÞE ½ð1xÞ=ð1xD=PÞ 5 Q ¼ 4  D  D h2 ð1E ðxÞÞ2 2 h 1  P  2h 1  P E ðxÞ þ hE ðx Þ  ðCb þhÞE½ð1xÞ=ð1xD=PÞ

B ¼

h  Pð1  E ðxÞÞ  Q  Dgh 2 E   1x ðCb þ hÞE 1x D





1 1xDP

ð3:214Þ ð3:215Þ

P

Chiu et al. (2008) supposed that machine breakdown factor is not an issue to be considered, then the cost and time for repairing failure machine M ¼ 0 and g ¼ 0, Eqs. (3.214) and (3.215) become the same equations as were given by: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD Q ¼u 2  u  2 1E ð x Þ f g D h th 1      2h 1  D EðxÞ þ hEðx2 Þ P P C b þh 

E

B ¼

ð3:216Þ

1x 1xD P

1  E ð xÞ  h  Q ðC b þ hÞ E 1x 1xD

ð3:217Þ

P

Further, suppose that regular production process produces no defective items, i.e., x ¼ 0, then Eqs. (3.216) and (3.217) become the same equations as were presented by the classic EPQ model with backordering permitted (Hillier and Lieberman 1995; Silver et al. 1998): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD Cb þ h  Q ¼ D Cb h 1P     h D   Q B ¼ 1 Cb þ h P 

ð3:218Þ ð3:219Þ

Numerical 3.6 Chiu et al. (2008) assumed annual production rate of a manufactured item is 18,000 units and demand of this item is 3000 units per year. The percentage of random scrap items produced x and follows a uniform distribution over the range [0, 0.15). Other parameters used are as follows (Chiu et al. 2008): K ¼ $240 for each production run, Cs ¼ $1.00 disposal cost for each scrap item, C ¼ $2.00 per item, CM ¼ $500 repair cost for each breakdown, h ¼ $0.6 per item per unit time, Cb ¼ $0.8 per item backordered per unit time, and g ¼ 0.018 years, time needed to repair and restore the machine. Applying Eqs. (3.214), (3.215), and (3.202), one can obtain the optimal production time T1* ¼ 0.2345 years, the optimal lot size Q* ¼ 4221, the backorder B* ¼ 1368, and E [TCU(Q*, B*)] ¼ $7851.30 (Chiu et al. 2008).

208

3 Scrap

Numerical 3.7 Chiu et al. (2008) assumed another manufactured item can be produced at a rate of 44,000; its annual demand is 11,000 units. A random percentage of scrap items produced x follows a uniform distribution over the interval [0, 0.20]. Other parameters used are as follows (Chiu et al. 2008): K ¼ $350 for each production run, Cs ¼ $0.80 disposal cost for each scrap item, C ¼ $2.40 per item, CMM ¼ $600 repair cost for each breakdown, h ¼ $1.00 per item per unit time, Cb ¼ $1.40 per item backordered per unit time, and g ¼ 0.036 years, time needed to repair and restore the machine. Applying Eqs. (3.210) and (3.211), one obtains the optimal T1* ¼ 0.1711 years or 8.90 weeks, and backorder level B* ¼ 2008. From Eq. (3.203), the long-run average costs E[TCU(T1*, B*)] ¼ $28,324. The optimal production run time T1* can be used to determine a multi-item production schedule. From Eq. (3.212), one also obtains optimal lot size Q* ¼ 7526 (Chiu et al. 2008).

3.3.1.4

Integrated Procurement–Production–Inventory Model

Nobil et al. (2018) derived an integrated procurement–production–inventory system for a single product and its raw materials without/with shortage. This system considers that the manufacturing process fabricates both perfect and defective finished products. The defective products are considered as scrapped items. The products are fabricated at P rate where x% of these are not useful, so non-defective items are produced with (1  x)P rate. In fact, the inventory level of finished product increases with (1  x)P  D rate, where D is demand rate of items. The finished product needs n type of raw materials to produce it; these are provided from outside suppliers. Thus, producer has to consider cost of ordering and purchasing raw materials in the inventory system costs in addition to cost of producing the item. Here, producer defines the quantity of raw material j that needs to be ordered for some producing periods (Mj) and store in his/her stock for beginning of production. Their model is developed without/with shortage for final finished product. For the shortage case is supposed that items in each cycle can have shortage up to B units (Nobil et al. 2018). Some notations which are specifically used to model this problem are shown in Table 3.13. First consider that the shortage is not permitted. In this case, the on-hand inventory graph of the raw material j and the finished product for the proposed problem without shortage are shown in Fig. 3.18. From Fig. 3.18, the following equations are expressed (Nobil et al. 2018): tP ¼

Q P

I ¼ ½ð1  xÞP  D

ð3:220Þ Q P

ð3:221Þ

3.3 Fully Backordered

209

 ð1  xÞP  D I Q, td ¼ ¼ DP D

 ð1  xÞP  D ð1  xÞQ Q T ¼ tP þ td ¼ þ Q¼ , DP D P

ð3:222Þ ð3:223Þ

Then, T j ¼ M jT ¼

ð1  xÞM j Q D

ð3:224Þ

The total cost consists of the following costs: production cost, the disposal cost of scrapped items, the setup cost for manufacturing the finished product, holding cost of finished product, the ordering cost of raw materials, the purchasing cost of raw materials, and the holding cost for raw materials. Thus, these costs are obtained as follows (Nobil et al. 2018): The production cost per cycle and the number of cycles per unit time are equal to CQ and T1, respectively. Therefore, the production cost per unit time is calculated by: CQ T DCQ DC ¼ ¼ Qð1  xÞ ð1  xÞ

Production cost ¼

ð3:225Þ ð3:226Þ

C S xQ DC S xQ DC S x ¼ ¼ T Q ð 1  xÞ ð 1  x Þ

 K DK DK 1 Set up cost ¼ ¼ ¼ T Qð1  xÞ ð1  xÞ Q

Disposal cost ¼

Holding cost for finished product ¼

H ½I  T  2T

ð3:227Þ ð3:228Þ ð3:229Þ

Substituting I and T from Eqs. (3.221) and (3.223) into Eq. (3.229), hence: Holding cost for finished product ¼

H 2T   Q Qð1  xÞ ð3:230Þ  ðð1  xÞP  DÞ  D P

Thus, using Eq. (3.223) in above relation, thus (Nobil et al. 2018):

210

3 Scrap

Table 3.13 Notation of given problem (Nobil et al. 2018) N αj I Oj C Raw j

Number of raw materials Amount of raw material j required to produce one finished product (amount of raw material j/unit of finished product) Maximum on-hand inventory of finished product (time) Ordering cost of raw material j per order (time) Purchasing cost of raw material j per unit ($/unit)

hj T Tj Mj

Holding cost of raw material j per item per unit time ($/unit/unit time) Cycle length of the finished product (time) Cycle length of the raw material j (time) Number of cycles for raw material j

(1–x)P–D

Finished product

Cycle 1

Cycle (Mj-1)

Cycle 2

Cycle Mj

I

–D

tp

Mj a j Q

td

tp

td

tp

td

–ajP

(Mj–1)ajQ

–ajP

Raw material j

(Mj–2)ajQ

2ajQ a jQ

T

T

T

T

Tj=(MjT)

Fig. 3.18 The on-hand inventory graph for the problem without shortage (Nobil et al. 2018)

H ½ð1  xÞP  DÞ Q 2P   H D ¼ 1x Q 2 P

Holding cost for finished product ¼

ð3:231Þ

3.3 Fully Backordered

211

The ordering cost of the raw material j for Mj cycles and the number of cycles per unit time are equal to Oj and T1j, respectively. Thus, total ordering cost per unit time is obtained by: Ordering cost ¼

 n n X Oj X DO j 1 ¼ Tj ð 1  xÞ M j Q j¼1 j¼1

ð3:232Þ

The purchasing cost of the raw material j for Mj cycles is equal to Mj Rj αj Q that αj is proportion of per unit of finished product. Thus, total purchasing cost per unit time is computed by: Raw material purchasing cost ¼

n M C Raw α Q n C Raw α D X X j j j j j ¼ T ð 1  x Þ j j¼1 j¼1

ð3:233Þ

From Fig. 3.18, the area of the raw material in this figure is equal to:  

2M j  3 α j Qt P 2M j  1 α j Qt P α j Qt P 3α j Qt P 5α j Qt P þ þ þ⋯þ þ 2 2 2 2 2     þ α j Qt d þ 2α j Qt d þ 3α j Qt d þ ⋯ þ M j  2 α j Qt d þ M j  2 α j Qt d

ð3:234Þ Based on Appendix A of Nobil et al. (2018), the area of the raw material j can easily be determined as follows:

  α j α j ð1  xÞP  D 2 2 αj Q Mj  þ DP 2P 2 2

 ð1  xÞP  D 2  Q Mj DP

 Area of raw material j ¼

ð3:235Þ

Thus, total holding cost per unit time for all the raw materials is obtained as follows:

  α j ð1  xÞP  D 2 2 Q Mj DP T j 2P 2 j¼1

  α j ð1  xÞP  D 2 Q Mj  DP 2

   n X Dh j α j α j ð1  xÞP  D ¼ QM j þ DP ð1  xÞ 2P 2 j¼1

 Dh j α j ð1  xÞP  D Q  DP 2ð 1  x Þ

Holding cost of raw materials ¼

 n X hj αj

þ

ð3:236Þ

212

3 Scrap

Now, based on Eqs. (3.226)–(3.228), (3.231)–(3.233), and (3.236), the total cost for the IPPI model without shortage, denoted by TC, is written as follows:

TC ¼

 8  n 0 M j  1 & integer; j ¼ 1, 2, . . . , n   D C Raw α j þ CS x þ C j Δ1j ¼ > 0; j ¼ 1, 2, . . . , n ð 1  xÞ Δ2 ¼

DK >0 ð 1  xÞ

DO j > 0; j ¼ 1, 2, . . . , n ð 1  xÞ 

 Dh j α j α j ð1  xÞP  D > 0; j ¼ 1, 2, . . . , n þ Δ4j ¼ DP ð1  xÞ 2P 2

   Dh j ð1  xÞP  D h D Δ5j ¼ ; j ¼ 1, 2, . . . , n 1x  DP 2 P 2ð 1  x Þ Δ3j ¼

ð3:238Þ

ð3:239Þ ð3:240Þ ð3:241Þ ð3:242Þ ð3:243Þ

Now consider that shortage is permitted and is fully backordering. The on-hand inventory graph for the raw material j and the finished product for the model with shortage are shown in Fig. 3.19. From Fig. 3.19, the following equations are deduced: t1 ¼

B ð1  xÞP  D

ð3:244Þ

3.3 Fully Backordered

213

Q B P I Q B t2 ¼ ¼  , ð1  xÞP  D P ð1  xÞP  D I ¼ ½ð1  xÞP  D

ð3:245Þ ð3:246Þ

Q tP ¼ t1 þ t2 ¼ , P

 ð1  xÞP  D I B Q t3 ¼ ¼ DP D D

ð3:247Þ

B t4 ¼ D

 ð1  xÞP  D Q td ¼ t3 þ t4 ¼ DP

ð3:249Þ

T ¼ tP þ td ¼

ð3:248Þ

ð3:250Þ

 ð1  xÞP  D ð1  xÞQ Q Q¼ þ DP D P

ð3:251Þ

ð1  xÞM j Q D

ð3:252Þ

So, T j ¼ M jT ¼

The total cost is comprised of the following costs: production cost, the disposal cost of scrapped items, the setup cost for producing the finished product, the holding cost for finished product, the backorder cost, the ordering cost of raw materials, the purchasing cost of raw materials, and holding cost of raw materials. Thus, these costs are obtained as follows (Nobil et al. 2018): The production cost per time unit for the finished product is calculated by: Production cost ¼

CQ DCQ DC ¼ T Q ð 1  xÞ ð 1  x Þ

ð3:253Þ

The disposal cost per time unit for the scrapped items is determined by: Disposal cost ¼

C S xQ DC S xQ DC S x ¼ ¼ T Q ð 1  xÞ ð 1  x Þ

The setup cost per time unit is obtained with:

ð3:254Þ

214

3 Scrap

 K DK DK 1 ¼ Setup cost ¼ ¼ T Qð1  xÞ ð1  xÞ Q

ð3:255Þ

Based on Fig. 3.19, the holding cost per unit of finished product is given by: Holding cost for finished product ¼

h ½ I  ðt 2 þ t 3 Þ 2T

ð3:256Þ

Substituting I, t2, t3, and T from Eqs. (3.245), (3.246), (3.248), and (3.251), respectively, thus (see Appendix C of for detail calculations), Holding cost for finished product ¼

h 2



2  Q P B  ðð1  xÞP  DÞ þ P ðð1  xÞP  DÞ Q  hB ð3:257Þ Based on Fig. 3.19, the backorder cost per unit of finished product is computed by: Backordering cost ¼

Cb ½B  ðt 1 þ t 4 Þ 2T

ð3:258Þ

Substituting t1 and t4 from Eqs. (3.244) and (3.249) respectively, hence, Backordering cost ¼



 Cb B B þ B 2T ð1  xÞP  D D

ð3:259Þ

From Eq. (3.251), 

 Cb D B B B þ 2ð1  xÞQ ð1  xÞP  D D

2 Cb P B ¼ 2ðð1  xÞP  DÞ D

Backordering cost ¼

ð3:260Þ

The total ordering cost per for time unit is determined by:

 n n X Oj X DO j 1 ¼ Ordering cost ¼ Tj ð 1  xÞ M j Q j¼1 j¼1 The total purchasing cost per time unit is calculated by:

ð3:261Þ

3.3 Fully Backordered

215

Finished product

(1–x)P–D

Cycle 1

Cycle 2

Cycle Mj

I –D

–D

B´

tr

t2

t3

t4

M j aj Q Raw material j

–ajP

(Mj –1)aj Q –ajP

(Mj –2)aj Q

aj Q

tp

td

tp

T

td

tp

T

td

T Tj =(Mj T)

Fig. 3.19 The on-hand inventory graph for the problem with shortage (Nobil et al. 2018)

Raw material purchasing cost ¼

n M C Raw α Q n C Raw α D X X j j j j j ¼ T ð 1  x Þ j j¼1 j¼1

ð3:262Þ

From Fig. 3.19, the area of the raw material j in this figure is equal to:  

2M j  3 α j Qt P 2M j  1 α j Qt P α j Qt P 3α j Qt P 5α j Qt P þ þ þ⋯þ þ 2 2 2 2 2     þ α j Qt d þ 2α j Qt d þ 3α j Qt d þ ⋯ þ M j  2 α j Qt d þ M j  2 α j Qt d

ð3:263Þ Because the uptime period and downtime period, denotedby tP andtd, are same as ÞPD the model without shortage, in other words, t P ¼ QP and t d ¼ ð1xDP Q. Therefore, the area of the raw material j in Fig. 3.19 is obtained as follows:

216

3 Scrap

  α j α j ð1  xÞP  D 2 2 αj Q Mj  þ Area of raw material j ¼ DP 2P 2 2

 ð1  xÞP  D 2  Q Mj DP 

ð3:264Þ

Thus, the total holding cost per time unit is computed as follows:

   n X Dh j α j α j ð1  xÞP  D QM j þ Holding cost of raw materials ¼ DP ð1  xÞ 2P 2 j¼1

 Dh j α j ð1  xÞP  D Q  DP 2ð 1  x Þ ð3:265Þ Now, based on Eqs. (3.253)–(3.255), (3.257), (3.260)–(3.262), and (3.265), the total cost for the IPPI model with shortage is given by:  8  n 0 M j  1 & integer; j ¼ 1, 2, . . . , n ð3:267Þ

3.3 Fully Backordered

Δ1j ¼

  D CRaw α þ C x þ C j S j ð 1  xÞ

217

> 0; j ¼ 1, 2, . . . , n

DK >0 ð1  xÞ DO j ¼ > 0; j ¼ 1, 2, . . . , n ð1  xÞ 

 Dh j α j α j ð1  xÞP  D > 0; j ¼ 1, 2, . . . , n ¼ þ DP ð1  xÞ 2P 2

   Dh j ð1  xÞP  D h D ; j ¼ 1, 2, . . . , n 1x  ¼ DP 2 P 2ð 1  x Þ ðh þ C b ÞP ¼ >0 2ðð1  xÞP  DÞ

Δ2 ¼ Δ3j Δ4j Δ5j Δ6

ð3:268Þ

They have used metaheuristic algorithms to solve their models, so for more detailed information, readers can refer to Nobil et al. (2018).

3.3.1.5

Service Level Constraint

Chiu (2006) studied the effect of service level constraint on EPQ model with random defective rate. In the realistic inventory control and management, due to certain internal orders of parts/materials and other operating considerations, the planned backlogging is the strategy to effectively minimize overall inventory costs. While allowing backlogging, abusive shortage in an inventory model, however, may cause an unacceptable service level and turn into possible loss of future sales (because of the loss of customer goodwill). Therefore, the maximal allowable shortage level per cycle is always set as an operating constraint of the business in order to achieve minimal service level while deriving the optimal lot size decision (Chiu 2006). Some notations which are specifically used to model this problem are shown in Table 3.14. The EPQ model assumes that the production rate P must always be greater than or equal to the demand rate D. The production rate of perfect-quality items must always be greater than or equal to the sum of the demand rate and the production rate of defective items (P  d  D  0 or 1  x  DP  0). Figure 3.20 depicts the on-hand inventory level and allowable backorder level for the EPQ model with backlogging permitted. For the following derivation, they employ the solution procedures used by Hayek and Salameh (2001). From Fig. 3.20, one can obtain the cycle length T, production uptime t1, the maximum level of on-hand inventory H1, production downtime t2, shortage permitted time t3, and t4 as follows (Chiu 2006):

218

3 Scrap

T¼

4 X i¼1

ti ¼

Qb ð1  xÞ , D

H1 , PdD Q H 1 ¼ ðP  d  DÞ b  B, P H1 t2 ¼ , D B t3 ¼ , D B t4 ¼ , PdD Q ðt 1 þ t 4 Þ ¼ : P t1 ¼

ð3:269Þ ð3:270Þ ð3:271Þ ð3:272Þ ð3:273Þ ð3:274Þ ð3:275Þ

The scrap items built up randomly during production uptime (t1 + t4) are (Chiu 2006): d  ðt 1 þ t 4 Þ ¼ x  Q:

ð3:276Þ

The inventory cost per cycle is (Chiu 2006): h i H TCðQ, BÞ ¼ CQ þ CS xQ þ K þ h 1 ðt 1 þ t 2 Þ 2   h i d ðt 1 þ t 4 Þ B þC b ðt 3 þ t 4 Þ þ h ðt 1 þ t 4 Þ : 2 2

ð3:277Þ

Since scrap items are produced randomly during a regular production run, the cycle length T is a variable (see Fig. 3.20). One may employ the renewal reward theorem (Zipkin 2000) to cope with the variable cycle length. By substituting variables from Eqs. (3.269) to (3.276) in Eq. (3.277), the expected cost E ½TCUðQ, BÞ ¼ E½TCEððQTbÞ, BÞ can be obtained as follows (Chiu and Chiu 2003):  E ½TCUðQ, BÞ ¼ D

 E ð xÞ C KD 1 þ CS þ Q 1  E ð xÞ 1  E ð xÞ 1  E ð xÞ 0 1  h i ðC þ hÞ B 1  x C B2 h D 1 þ 1 Qb  2B þ b E@ DA 2Q 2 P 1  E ð xÞ 1  E ð xÞ 1x P h   i E ð xÞ 2 D hQ E ðx Þ Q þh B  1  þ P 2 1  E ð xÞ 1  E ð xÞ

ð3:278Þ

3.3 Fully Backordered

219

Table 3.14 Notation of given problem (Chiu 2006) t1 t2 t3 t4

Production uptime (time) Production downtime (time) Time shortage permitted (time) Time needed to satisfy all the backorders by the next production (time) I (t) Q H1

P –d–D

Fig. 3.20 On-hand inventory of the EPQ model with random defective rate and backlogging permitted (Chiu 2006)

t1

–D P –d–D

–D

Time

t2

B

t3

T

t4

T

For the proof of convexity of E[TCU(Q, B)], one can utilize the Hessian matrix equation (Rardin and Rardin 1998): 0

½Q

2

∂ E½TCUðQ, BÞ B ∂Q2 B B B @ ∂2 E½TCUðQ, BÞ ∂Q∂B

¼

2KD 1 > 0: Q 1  E ð xÞ

1 2 ∂ E½TCUðQ, BÞ C Q  ∂Q∂B C C 2 ∂ E½TCUðQ, BÞ A B ∂B2 ð3:279Þ

Equation (3.279) is positive, because all parameters are positives. Hence, the expected inventory cost function E[TCU(Q, B)] is a strictly convex function for all Q and B different from zero (Chiu 2006). Hence, it follows that for the optimal production lot size Q and the maximal level of backorder B, one can differentiate E[TCU(Q, B)] with respect to Q and with respect to B and solve the linear system of Eq. (3.280) by letting these partial derivatives equal to zero (Chiu 2006):

220

3 Scrap

  ∂E½TCUðQ, BÞ KD 1 h D 1 ¼ 2 þ 1 P 1  E ð xÞ ∂Q Q 1  E ð xÞ 2 0 1   E ð xÞ 2 ð C þ hÞ B D h E ðx2 Þ B 1x C  2 b E@ þ ¼0 Ah 1 λ P 1  E ð xÞ 2 1  E ð xÞ 2Q 1  EðxÞ 1x P ð3:280Þ

 ∂E ½ðTCUðQ, BÞÞ 1 B ðC b þ hÞ 1x ¼ h þ E 1  E ð xÞ Q 1  E ð xÞ ∂B 1  x  DP þh

E ð xÞ 1  E ð xÞ

¼0

ð3:281Þ

Hence, one derives the optimal production policy, Q*b and B*, as shown below (Chiu 2006): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD Q ¼u 2  u  2 f 1E ð x Þ g   2h 1  D EðxÞ þ hEðx2 Þ th 1  DP  C hþh  P b 

E

B ¼

ð3:282Þ

1x 1xD P

1  E ð xÞ  h  Q ðC b þ hÞ E 1x D 1x

ð3:283Þ

P

Now for the EPQ model with random defective rate when shortage is not permitted, the cycle length T ¼ t1 + t2 (see Fig. 3.20). The expected annual cost ðQÞÞ E ½TCUðQÞ ¼ EðTC E ðT Þ can be obtained as follows (Chiu et al. 2003):  E ½TCUðQÞ ¼ D

   E ð xÞ C KD 1 hQ D 1 þ CS þ 1 þ Q 1  E ð xÞ 2 P 1  E ð xÞ 1  E ð xÞ 1  E ð xÞ   E ð xÞ D hQ E ðx2 Þ hQ 1  þ P 1  E ð xÞ 2 1  E ð xÞ ð3:284Þ

Differentiating E[TCU(Q)] with respect to Q twice, we find that E[TCU(Q)] is convex, and by minimizing the expected annual cost E[TCU(Q)], one can derive the optimal production quantity Q* as shown in Eq. (3.285) (Chiu 2006): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD   Q ¼ : D h 1  P  2h 1  DP E ðxÞ þ hE ðx2 Þ 

ð3:285Þ

Now effects of backlogging and service level constraint on the EPQ model will be considered.

3.3 Fully Backordered

221

The expected annual cost per when backlogging is not permitted is always greater than or equal to that of the EPQ model with allowed backlogging. That is, E(TCU (Q))  E(TCU(Q, B)), for any given Q ¼ Qbacklogging (Chiu 2006), because: E ðTCUðQÞÞ  EðTCUðQ, BÞÞ ¼

 ð C þ hÞ hB 1x E  b 1  E ð xÞ 1  E ð xÞ 1  x  DP 

E ð xÞ B2 h B 2Q 1  E ð xÞ

ð3:286Þ

Substituting B, one has E ðTCUðQÞÞ  EðTCUðQ, BÞÞ ¼

1  E ð xÞ h2  Q  0 2ðCb þ hÞ E 1x D 1x

ð3:287Þ

P

Since parameters h and Cb are nonnegative numbers, the random defective rate x and, 1  x  DP  0 and the production lot size Q  0, hence Eq. (3.287) 0. So it is better (in terms of total inventory costs) to permit shortage and have them backordered for the EPQ model with random defective rate. While allowing backlogging, abusive shortage in an inventory model, however, may cause an unacceptable service level and turn into possible loss of future sales. Hence, the maximal allowable shortage level per cycle is always set as an operating constraint for the business in order to attain the minimal service level. Suppose that set α to be the maximum proportion of shortage permitted per cycle (i.e., the service level ¼(1  α)%), then (Chiu 2006): t3 þ t4 , T α t þ t4 : ¼ 3 1  α t1 þ t2 α¼

ð3:288Þ ð3:289Þ

Substituting t1, t2, t3, and t4, one obtains (Chiu 2006): α B , ¼ 1α 1  x  DP Q  B

ð3:290Þ

Substituting B, one has the following (Chiu 2006):

  ð C b þ hÞ 1 1x D 1 E , ¼ 1  x  h P α 1  E ð xÞ 1  x  DP

ð3:291Þ

222

3 Scrap

( " )

1 # 1  E ðxÞ 1 1x 1 : Cb ¼ h  E α ð1  x  D=PÞ 1  x  D=P

ð3:292Þ

Assume that (Chiu 2006): ( " )

1 # 1 1  E ðxÞ 1x  E 1 : f ðα, xÞ ¼ h  α 1  x  DP 1  x  DP

ð3:293Þ

Equation (3.293) represents the relationship between the imputed backorder cost f (α, x) and the maximum proportion of shortage permitted time α. In other words, when the service level (1  α)% of the EPQ model is set, the corresponding imputed backorder cost f(α, x) can be obtained. Hence, one can utilize this information to determine   whether or not the service level is achievable. For the computation of 1x E 1x D , one can refer to Chiu (2006). P

Let Cbi be the tangible backorder cost per item. If Cbi > f(α, x), then the service level (1  α)% is achievable. Otherwise, the tangible backorder cost should be increased to f(α, x), and then use it to derive the new optimal operating policy (in terms of Q* and B*), so that the overall inventory costs can be minimized and the service level constraint will be attained. Letting Cbi be the adjustable intangible backorder cost (per item per unit time), then bi should satisfy the following condition in order to attain the f(α, x) service level (Chiu 2006): bi  ½ f ðα, xÞ  C bi :

ð3:294Þ

Therefore, by using Cb ¼ f(α, x), one can derive the new optimal production lot size Q* and the optimal backorder level B* that minimizes the expected annual inventory costs as well as achieves the minimal service level (1  α)% (Chiu 2006). Example 3.8 Chiu (2006) considered a company which produces a product for several regional clients. He assumed D ¼ 4000 units per year, P ¼ 10,000 units per year, Cs ¼ $0.3 per scrap item, K ¼ $450 per setup, C ¼ $2 per item, x U[0, 0.1], Cbt ¼ $0.2 per item backordered per unit time, α ¼ 0.3, and h ¼ $0.6 per item per year. First let Cb ¼ Cbt. From Eqs. (3.278), (3.282), and (3.283), one obtains the overall costs E[TCU(Qb*, B*)] ¼ $9087, the optimal production quantity Q* ¼ 6284, and the optimal backorder level B* ¼ 2589. For EPQ model with backlogging not allowed, from Eqs. (3.286) to (3.287), the total cost E[TCU(Q*)] ¼ $9625 and the optimal production quantity Q* ¼ 3323 are obtained. One notices that the EPQ model with backlogging permitted has a lower overall cost than that of the EPQ model with no shortage allowed. Another research to investigate the effects of service level constraint on order and shortage quantities is performed by Shyu et al. (2009). Readers for more detailed information can refer to this work.

3.4 Partial Backordered

3.4

223

Partial Backordered

3.4.1

Continuous Delivery

In this section, the EPQ models with scrap and continuous delivery policy with partial backordered shortage are presented.

3.4.1.1

Multi-product Single-Machine System

Taleizadeh et al. (2010) developed a multi-product single-machine production system under economic production quantity (EPQ) model in which the existence of only one machine causes a limited production capacity for the common cycle length of all products, the production defective rates are random variables, shortages are allowed and take a combination of backorder and lost sale, and there is a service rate constraint for the company. Imperfect production processes, due to process deterioration or some other factors, may randomly generate X percent of defective items at a rate d. The inspection cost per item is involved when all items are screened. All defective items are assumed to be scrapped; i.e., no rework is allowed. The annual constant production rate (P) is much larger than the annual constant demand rate (D) as the basic assumption of the finite production model. In other words, the expected production rate of the scrapped items θ can be expressed as d ¼ PE[X]. Also, Taleizadeh et al. (2010) assumed that there is a real constant production capacity limitation on a single machine on which all products are produced and that the setup cost is nonzero. Some notations which are specifically used to model this problem are shown in Table 3.15. Index j ¼ 1, 2, . . ., n refers to the number of products. The production rate Pj is always assumed to be greater than or equal to the demand rate Dj. Furthermore, the production rate of the perfect-quality items is assumed to be greater than or equal to the sum of the demand rate and the production D rate of defective items. In other words, Pj  Dj  dj  0 or 1  E ðxÞ  P jj  0:. Figure 3.21 depicts the on-hand inventory level and allowable backorder level of the EPQ model with permitted backlogging. To model the problem, a part of modeling procedure used in Hayek and Salameh (2001) is applied. Since all products are manufactured on a single machine with a limited capacity, the cycle length for all of them is equal (T1 ¼ T2 ¼ . . . ¼ Tn ¼ T). Then, based on Fig. 3.21, for j ¼ 1, 2, . . ., n, one obtains (see Appendix A of Taleizadeh et al. 2010): T¼

4 X i¼1

t ij

   QBj 1  E x j þ 1  ξ j B j ¼ Dj

ð3:295Þ

224

3 Scrap

t 1j ¼

I 1j Pj  Dj  dj

 Qj  ξ jB j I 1j ¼ P j  D j  d j Pj

ð3:297Þ

I 1j Dj

ð3:298Þ

ξ jB j B j ¼ ξ jD j D j

ð3:299Þ

ξ jB j Pj  Dj  dj

ð3:300Þ

t 2j ¼ t 3j ¼ t 4j ¼

ð3:296Þ

t 1j þ t 4j ¼

Qj Pj

ð3:301Þ

The objective function of the model is the summation of the expected annual production, holding, shortage, disposal, and setup costs as: Z ¼ Production cost þ Holding cost þ Backordering cost þ Disposal cost þ Setup cost

ð3:302Þ

In the following subsections, different parts of the objective function are described. The production cost per unit and the production quantity per period of the jth product are Cj and Qj, respectively. Hence, the production cost of the jth product per period is CjQj. While the total annual production cost of the jth product in a disjoint production policy (each product is ordered separately) is NCjQj, this cost for the C Q joint policy (all products have a unique ordering cycle) is Tj j . Furthermore, since the shortages are in combinations of backorders and lost sales, based on Eq. (3.307), one obtains:   T  Dj  1  ξj Bj T  Dj  1  ξj Bj   Qj ¼ ¼ E 1  xj 1  E xj

ð3:303Þ

Hence, the expected annual production cost will be:  n Cj X j¼1

TD j ð1ξ j ÞB j E ð1x j Þ

T

 ¼

n X j¼1

"  # n X Cj 1  ξj Bj C jD j    T 1  E xj 1  E xj j¼1

ð3:304Þ

3.4 Partial Backordered

225

Table 3.15 Notation of given problem (Taleizadeh et al. 2010)  The probability density function of xj fxj xj ξj α I 1j N t 1j

The fraction of jth product shortage that is backordered The safety factor of total allowable shortages The maximum units of on-hand inventory level, when the regular production process stops, (unit) The number of cycles per year The production uptime of the jth product, (time)

t 2j

The production downtime of the jth product, (time)

t 3j

The permitted shortage time of the jth product, (time)

t 4j

The time needed to satisfy all backorders in the next production of the jth product, (time) The annual expected total costs, ($/year)

Z

Fig. 3.21 A production– inventory cycle (Taleizadeh et al. 2010)

I Qj –ζjBj

I1j

Pj –Dj–Bj –Dj –ζjDj

t 3j

t 4j t

ζjBj

t1

t 2j

j

(1 –ζj)Bj

Pj –Dj–dj

T

The holding cost per unit of the jth product per unit time for both the healthy and the scrapped items is hj. According to Fig. 3.21, the total holding costs of healthy items per cycle and per year are shown in Eqs. (3.305) and (3.306), respectively: "

 I 1j  1 t j þ t 2j hj 2 j¼1

n X

"

#

 I 1j  1 hj t j þ t 2j N 2 j¼1 n X

ð3:305Þ # ð3:306Þ

However, Eq. (3.306) for the joint production policy in which N ¼ T1 becomes:

226

3 Scrap

" # n  I 1j  1 1X 2 h t þtj T j¼1 j 2 j

ð3:307Þ

Finally, the expected total annual holding cost of healthy items is (see Appendix B of Taleizadeh et al. 2010): " 

    P j  D j  d j 1  ξ j þ ξ jP j 1  E x j P j D j d j D j Bj  2   2 T   2   2 2 P j 1E x j P j 1E x j j¼1  2  2      2   2  2 # P j  D j  d j 1  ξ j þ 2ξ j 1  ξ j P j 1  E x j P j  D j  d j þ ξ j P j 1  E x j Bj þ  2   2  T 2D j P j 1  E x j P j D j d j

n X  h j P j d j

ð3:308Þ Since the scrap items of each product are assumed to be held until the end of its production time, based on Fig. 3.21, the total holding costs of the scrapped items per cycle and per year are shown in Eqs. (3.309) and (3.310), respectively:  2  3  d j t 1j þ t 4j  h j4 t 1j þ t 4j 5 2 j¼1

n X

 2  3  d j t 1j þ t 4j  t 1j þ t 4j 5 h j4 N 2 j¼1 n X

ð3:309Þ

ð3:310Þ

Again, for the joint production policy, Eq. (3.310) becomes:  2  3 " # 1 4  n n  dj Qj 2 1 X 4d j t j þ t j 1X 1 4 5 hj hj tj þtj ¼ T j¼1 T j¼1 2 Pj 2

ð3:311Þ

Hence, the expected total annual holding cost of scrapped items according to Eq. (3.303) is: n X j¼1

" h jd j

!# 2   2 B2j D j T  2D j 1  ξ j B j 1  ξj  2   2 þ  2   2 T 2 Pj 1  E xj 2 Pj 1  E xj

ð3:312Þ

Finally, the expected total annual holding cost of healthy and scrapped items is:

3.4 Partial Backordered

227

" 

    P j  D j  d j 1  ξ j þ ξ jP j 1  E x j P j D j d j D j Bj  2   2 T   2   2 2 P j 1E x j P j 1E x j j¼1  2  2      2   2  2 # P j  D j  d j 1  ξ j þ 2ξ j 1  ξ j P j 1  E x j P j  D j  d j þ ξ j P j 1  E x j Bj þ  2   2  T 2D j P j 1  E x j P j D j d j " 2 !#   2 2 n X Bj D j T  2D j 1  ξ j B j 1ξ j þ h jd j  2   2 þ  2   2 T 2 P j 1E x j 2 P j 1E x j j¼1

n X  h j P j d j

ð3:313Þ Based on Fig. 3.21, the backordered and lost sale costs per cycle are shown in Eqs. (3.314) and (3.315), respectively: 

 Bj  3 C bj ξ j t j þ t 4j 2 j¼1

n X

n X



 b πj 1  ξj Bj

ð3:314Þ ð3:315Þ

j¼1

These costs for a year become: N

n X

Cbj ξ j

j¼1

N

n X

   Bj 3 t j þ t 4j 2

 b πj 1  ξj Bj

ð3:316Þ ð3:317Þ

j¼1

Because of the joint production policy, Eqs. (3.316) and (3.317) will change to Eqs. (3.318) and (3.319), respectively:   n  Bj 3 1X t j þ t 4j C bj ξ j T j¼1 2

ð3:318Þ

n 1X  b πj 1  ξj Bj T j¼1

ð3:319Þ

Finally, the expected annual backordered and lost sale costs are (see Appendix C of Taleizadeh et al. 2010): n 1X C ξ T j¼1 bj j

"

  2 # Pj  1  ξj Dj  dj Bj  2D j P j  D j  d j

ð3:320Þ

228

3 Scrap n 1X  b π 1  ξj Bj T j¼1 j

ð3:321Þ

Since theP quantity of scrapped items is E(xj)Qj, the expected total disposal cost per cycle is nj¼1 Cs j E x j Q j . This quantity per year becomes: N

n X

 Cs j E x j Q j

ð3:322Þ

j¼1 n  1X Cs j E x j Q j T j¼1

Since Q j ¼

TD j ð1ξ j ÞB j 1E ðx j Þ

ð3:323Þ

the annual expected total scrapped item cost is:

 n  T  Dj  1  ξj Bj 1X  Cs E x j T j¼1 j 1  E xj    n n X Cs j E x j 1  ξ j B j Cs j E x j D j X    ¼ T 1  E xj j¼1 1  E x j j¼1

ð3:324Þ

The cost of a setup is K which occurs N times per year. So, the annual setup cost will be: NK ¼

K T

ð3:325Þ

As a result, the objective function of the model becomes (Taleizadeh et al. 2010):

3.4 Partial Backordered

229

"  # n n X X C j 1ξ j B j C jD j    MinZ ¼ T j¼1 1  E x j j¼1 1  E x j "      n X  P j D j d j D j P j  D j  d j 1  ξ j þ ξ jP j 1  E x j hj Pj d j Bj þ  2   2 T   2   2 P j 1E x j 2 P j 1E x j j¼1  2  2      2   2  2 # P j  D j  d j 1  ξ j þ 2ξ j 1  ξ j P j 1  E x j P j  D j  d j þ ξ j P j 1  E x j Bj þ  2   2  T 2D j P j 1  E x j P j D j d j " 2 !#   2 n X B2j D j T  2D j 1  ξ j B j 1ξ j h jd j þ  2   2 þ  2   2 T 2 P j 1E x j 2 P j 1E x j j¼1 "   2 # n n X P j  1ξ j D j d j B j 1 1X   þ C ξ b π j 1ξ j B j þ T j¼1 bj j T 2D j P j  D j  d j j¼1    n n X Cs j E x j 1  ξ j B j K Cs j E x j D j X    þ þ T T 1E x j j¼1 1  E x j j¼1 "  2   n X h jd j 1  ξ j C bj ξ j P j  1  ξ j D j  d j  ¼  2   2 þ 2D j P j  D j  d j 1E x j j¼1 2 P j  2  2      2   2  2 # P j  D j  d j 1  ξ j þ 2ξ j 1  ξ j P j 1  E x j P j  D j  d j þ ξ j P j 1  E x j Bj þ  2   2  T 2D j P j 1  E x j P j D j d j " #         n X h jd jD j 1  ξ j h j P j  d j P j  D j  d j 1  ξ j þ ξ jP j 1  E x j  Bj  2   2 þ  2   2 2 P j 1E x j 1E x j j¼1 2 P j 0 

þ

n X j¼1

"

  n B X B C j þb π j þ Cs j E x j 1  ξ j B !    2 B n  B j X h j P j d j P j D j d j D j h jd j D j j¼1 @ þ þ  2  1E x j  2   2  2 T T 2 P j 1E x j 2 P j 1E x j j¼1

 # C j þ Cs j E x j D j K  þ T 1E x j

ð3:326Þ

To make sure that all of the n products will be produced by a single machine, a capacity limitation should be considered as explained in the next subsection. The maximum capacity of the single machine and the minimum service rate are the two constraints of the model that are described in the two following subsections. Since t 1j þ t 4j and tsj are the production time and setup time of the jth product, respectively, the summation  Pof the total production and setup time (for all products) Pn  1 n 4 will be j¼1 t j þ t j þ j¼1 ts j in which it should be smaller or equal to the period length (T ). So the capacity constraint of the model is (Taleizadeh et al. 2010): n  n  X X t 1j þ t 4j þ ts j  T j¼1

ð3:327Þ

j¼1

Then, based on the derivation in Appendix D of Taleizadeh et al. (2010), one obtains:

230

3 Scrap n P

ts j 

j¼1

1

n P j¼1

n P j¼1

ð1ξ j ÞB j

P j ð1E ðx j ÞÞ

T

ð3:328Þ

Dj P j ð1E ðx j ÞÞ

Since the shortage quantity of the jth product per period is Bj, the annual demand of the jth product is Dj, the number of periods in each year is N, and the safety factor of allowable shortage is α, the service rate constraint becomes: n X N  Bj T Dj j¼1

ð3:329Þ

According to Appendix D of Taleizadeh et al. (2010), the service rate constraint is: n P

T SL ¼

j¼1

α

C 2j 2C 1j D j

n P j¼1

C3j 2C1j D j

T

ð3:330Þ

Based on the objective function in Eq. (3.326) and the constraints in Eqs. (3.328) and (3.330), the final model is:

3.4 Partial Backordered

231

"   2   n X b π j 1ξj h jd j 1  ξ j C bj ξ j P j  1  ξ j D j  d j  Min Z ¼ þ  2  þ  2 2D j 2D j P j  D j  d j 2 Pj 1E xj j¼1 "   2   n X b π j 1ξj h jd j 1  ξ j C bj ξ j P j  1  ξ j D j  d j  þ  2  þ Min Z ¼  2 2D j 2D j P j  D j  d j 2 Pj 1E xj j¼1  2  2      2   2 #  2 P j  D j  d j 1  ξ j þ 2ξ j 1  ξ j P j 1  E x j P j  D j  d j þ ξ j P j 1  E x j Bj þ  2   2  T 2D j P j 1  E x j Pj Dj d j "        # n X h jd jD j 1  ξ j h j P j  d j P j  D j  d j 1  ξ j þ ξ jP j 1  E x j  Bj  2   2 þ  2   2 2 P 2 Pj 1E xj 1  E x j¼1 j j   n X ðC j þ Cs j E x j 1  ξ j B j   T 1E xj j¼1 ! "    2  # n n X X hj Pj d j Pj Dj d j Dj h jd j D j C j þ Cs j E x j D j K  þ  2  þ þ  2   2  2 T þ T 1E xj 2 Pj 1E xj 2 Pj 1E xj j¼1 j¼1  n n P P 1ξj Bj   ts j  j¼1 j¼1 P j 1  E x j s:t: : T n P Dj   1 P 1  E x j¼1 j j n P

T SL ¼

C 2j

1 j¼1 2C j D j n C 3j P α 1 j¼1 2C j D j

T

T,B j  0 8j,j ¼ 1, 2, . ..,n:

ð3:331Þ Example 3.9 Consider a multi-product inventory control problem with five products in which their general and specific data are given and in Tables 3.16 and 3.18, respectively. Two numerical examples are given. In the first example, the probability distribution of xj is uniform, and in the second example, the distribution for Xj is normal. The setup cost is K ¼ $100,000, and the safety factor of total allowable shortages is α ¼ 0:35. Based on data of Table 3.17, the problem is solved using the proposed algorithm, and the optimal results are given in Tables 3.18 and 3.19 for the uniform and normal distributions, respectively.

Table 3.16 General data (Taleizadeh et al. 2010) Product 1 2 3 4 5

Dj 800 900 1000 1100 1200

Pj 10,000 11,000 12,000 13,000 14,000

tsj 0.01 0.015 0.02 0.025 0.03

ζj 0.75 0.80 0.85 0.90 0.95

^π j 1000 900 800 700 600

Cj 500 400 300 200 100

hj 15 12 9 6 3

Cbj 350 300 250 200 150

Csj 80 70 60 50 40

232

3 Scrap

Table 3.17 Specific data (Taleizadeh et al. 2010) Product 1 2 3 4 5

xj ~ U[aj, bj] aj bj 0 0.1 0 0.15 0 0.2 0 0.25 0 0.3

E[xj] 0.05 0.075 0.1 0.125 0.15

dj 500 825 1200 1625 2100

xj ~ N[μj σ j2] μj ¼ E[xj] 0.25 0.28 0.33 0.38 0.42

σ j2 0.01 0.02 0.03 0.04 0.05

dj 2500 3080 3960 4940 5880

Table 3.18 The optimal results of Example 1 (uniform distribution) (Taleizadeh et al. 2010) Product 1 2 3 4 5

Uniform TMin

TSL

T

T*

0.1578

3.0897

1.9841

3.0897

Bj 268.5 254.6 223.6 172.6 98.9

QjB 2531.2 2951.2 3395.8 3864.5 4356.2

Z

1,625,500

Table 3.19 The optimal results of Example 2 (normal distribution) (Taleizadeh et al. 2010) Product 1 2 3 4 5

3.5

Uniform TMin

TSL

T

T*

0.2044

4.1222

1.6771

4.1222

Bj 349.3 334.7 302.5 239.2 137.2

QjB 4280.6 5059.8 6084.9 7275.1 8517

Z

2,246,700

Conclusion

This report provided a comprehensive review of all EPQ models which considered a scrapped item. A brief introduction and literature review of mentioned problem is provided. For each problem the framework and mathematical model are presented separately, and then a numerical example of each study is reported. In total, ten studies were reviewed and analyzed. The difference between reviewed works was the stochastic parameters which were considered in those problems. Finally, these studies can be extended by incorporating other features in future studies.

References

233

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Hsu, L. F., & Hsu, J. T. (2016). Economic production quantity (EPQ) models under an imperfect production process with shortages backordered. International Journal of Systems Science, 47 (4), 852–867. Huang, Y.-F. (2006). Algebraic improvement on effects of random defective rate and imperfect rework process on Economic Production Quantity model. Journal of Applied Sciences, 6(5), 1082–1084. Nobil, A. H., Sedigh, A. H. A., & Cárdenas-Barrón, L. E. (2016). A multi-machine multi-product EPQ problem for an imperfect manufacturing system considering utilization and allocation decisions. Expert Systems with Applications, 56, 310–319. Nahmias, S., & Cheng, Y. (2005). Production and operations analysis (Vol. 6). New York: McGraw-Hill. Nobil, A. H., Cárdenas-Barrón, L. E., & Nobil, E. (2018). Optimal and simple algorithms to solve integrated procurement-production-inventory problem without/with shortage. RAIRO-Operations Research, 52(3), 755–778. Rosenblatt, M. J., & Lee, H. L. (1986). Economic production cycles with imperfect production processes. IIE Transactions, 18(1), 48–55. Rardin, R. L., & Rardin, R. L. (1998). Optimization in operations research (Vol. 166). Upper Saddle River, NJ: Prentice Hall. Sarkar, B., Cárdenas-Barrón, L. E., Sarkar, M., & Singgih, M. L. (2014). An economic production quantity model with random defective rate, rework process and backorders for a single stage production system. Journal of Manufacturing Systems, 33(3), 423–435. Shyu, M. L., Hsu, K. H., Tu, Y. C., & Huang, Y. F. (2009). The EPQ model with random defective rate under service constraint without calculus. Journal of Information and Optimization Sciences, 30(2), 245–251. Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and scheduling (Vol. 3, p. 30). New York: Wiley. Shyu, M. L., Hsu, K. H., Tu, Y. C., & Huang, Y. F. (2014). The EPQ model with random defective rate under service constraint without calculus. Journal of Information and Optimization Sciences, 30, 245–251. Taleizadeh, A. A., Niaki, S. T. A., & Najafi, A. A. (2010). Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint. Journal of Computational and Applied Mathematics, 233(8), 1834–1849. Taleizadeh, A. A., Samimi, H., Sarkar, B., & Mohammadi, B. (2017). Stochastic machine breakdown and discrete delivery in an imperfect inventory-production system. Journal of Industrial & Management Optimization, 13(3), 1511–1535. Zipkin, P. H. (2000). Foundations of inventory management (1st ed.). Boston, MA: McGraw-Hill/ Irwin.

Chapter 4

Rework

4.1

Introduction

Scrap and rework costs are a manufacturing reality impacting organizations across all industries and product lines. Scrap and rework costs are caused by many things— when the wrong parts are ordered, when engineering changes are not effectively communicated, or when designs are not properly executed on the manufacturing line. No matter why scrap and rework occurs, its impact on an organization is always the same—wasted time and money. And while no one, especially an operations manager, wants to admit it, these expenses add up quickly and negatively impact the bottom line. Although it is near impossible to eliminate scrap and rework completely, you can reduce the amount of scrap and rework in your organization by optimizing the way you document product data, review manufacturing processes, and communicate manufacturing and engineering changes throughout your supply chain. If priority is given to evaluating and improving your manufacturing processes, it becomes much easier to reduce the amount of scrap and rework in your organization. When imperfect-quality items are produced in the finite production model, as described in real-life situations, one cannot depend on the classical EPQ model to determine the optimal replenishment policy. The effect of defective items on the finite production model must be studied in order to minimize overall inventory costs. This chapter considers the EPQ model with the rework process of imperfect-quality items and the assumption that not all of the defective are repairable; a portion θ of them are scrap and will not be reworked. Mathematical modeling and analysis is employed in this chapter, and the disposal cost for each scrap item and the repairing and holding costs for each reworked items are included in the cost analysis. The renewal reward theorem is utilized to deal with the variable cycle length, and the optimal lot size that minimizes the overall costs for the imperfect-quality EPQ model

© Springer Nature Switzerland AG 2021 A. A. Taleizadeh, Imperfect Inventory Systems, https://doi.org/10.1007/978-3-030-56974-7_4

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4 Rework

No shortage

Partial backordering

EPQ + Rework

236

Backordering

Multi-delivery

Fig. 4.1 Categories of EPQ model considering rework

is derived where backorders are permitted. A numerical example has been used to illustrate the proposed methodology. For future research, one interesting and realistic consideration will be that when the rework process itself is imperfect. This chapter studied an imperfect production system with a set of new working assumptions such as process compressibility, reworking, and inspection simultaneously. Also regular production process and rework process can be carried out with different process rates in their corresponding upper and lower bounds by paying their related costs and different stochastic percent of defectives. Next an integrated model is presented which simultaneously determines production lot size, backlog, rate of regular production, and rate of rework with the objective of minimizing the total costs. While the model is nonlinear and could not be easily solved within a closedform solution, a simple algorithm is developed to obtain optimal solution. The mentioned problem based on the several features is organized. So four categorizes, no shortage, partial backordering, full backordering, and multi-delivery, are provided which is shown in Fig. 4.1. All categories are investigated in the next sections. The common notations of EPQ models are shown in Table 4.1. To integrate the model of this chapter, these notations are considered for all models.

4.2

Literature Review

In today’s manufacturing environment, most firms are confronted with fierce competition both domestic and offshore, in terms of quality, on-time delivery, and price. Customers demand quality and expect delivery on time in full (DOTIF) and usually negotiate for yearly price decreases. Consequently, the primary goal of firms today is to reduce costs while improving overall quality. In fact, reducing the inventory level is the most effective way of controlling product costs, quality, and delivery time.

4.2 Literature Review

237

Table 4.1 Notations i P, P1 or P1i, Pi P2 or P2i Q or Qi D or Di Rs C CJ CR Cd Cq CT CI d θ I, I1, H, H1 K KS x B Cb bb C

Index of product, i ¼ 1, 2, . . . Production rate of product or product i (units per unit time)

b π β 1β ts SL F T QT or QTi s v N h h1

Cost of lost sale per unit ($/unit) Proportion of backordering (%) Proportion of lost sale (%) Setup time of machine to produce product (time) Safety factor of total allowable shortages (%) Proportion of time during which inventory level is positive (%) Period of time (time) Number of end product or product i to be transported in each shipment (unit)

E[]

Rework rate of non-conforming item or item i (units per unit time) Production quantity of product or product i (unit) Demand rate of product or product i (units per unit time) Rate of screening (unit per year) Manufacturing or production cost per unit ($/unit) Reject cost per unit ($/unit) Rework cost per unit ($/unit) Disposal cost per unit ($/unit) Quality improvement cost per unit ($/unit) Delivery cost per unit ($/unit) Inspection or screening cost per unit ($/unit) Defective rate (units per unit time) Portion of the imperfect-quality items cannot be reworked and are scrapped (%) Inventory level (unit) Fixed cost ($/setup) Fixed delivery cost per shipment ($/shipment) Proportion of defective (%) Size number of backordered (unit) Cost of backordered per unit per time ($/unit/unit time) Cost of backordered per unit ($/unit)

Unit selling price ($/unit) Unit discounted selling price of defective items ($/unit) Number of cycle Holding cost per unit per time for healthy item ($/unit/unit time) Holding cost for each imperfect-quality items being reworked or not reworked per unit time ($/item/unit time) Expected value operator

The classical EPQ model has been used for a long time and is widely accepted and implemented. Regardless of its simplicity, the EOQ and EPQ model is still applied industrywide today (Osteryoung 1986; Zipkin 2000). However, finding an economic order quantity has been based on some unrealistic assumptions. One of

238

4 Rework

unrealistic assumption in EPQ model is that all produced items are healthy. The classical EPQ model shows that the optimal lot size will generate minimum manufacturing cost, thus producing minimum total setup cost and inventory cost. However, this is only true if all manufactured products are of perfect quality. In reality this is not the case; therefore, it is necessary to look at and allow cost for carrying imperfect-quality items, because this cost can influence the decision for selecting the economic lot size. A number of works have been published to address this unrealistic assumption. A brief discussion of these works is given below. Porteus (1986) was one of a group of people who formulated the relationship between process quality improvement and setup cost reduction. In the model, he found that there is a significant relationship between them. Lowering the setup cost alone makes the lot size smaller and the defective items fewer and produces lower annual cost. He illustrated that the annual cost can be further reduced when a joint investment in both process quality improvement and setup reduction is optimally made. From his model, it is assumed that once the process is out of control, it continues to produce defective items until the next setup is adjusted. However, this particular assumption is not realistic in the case of dynamic process control and when the product type is expensive. Many researchers have extended Porteus’ studies; for example, Chand (1989) validates Porteus’s model by including the learning effects on setup frequency and process quality. Tapiero et al. (1987) have presented a theoretical framework to examine the trade-offs between pricing, reliability, design, and quality control issues in manufacturing operations. Cheng (1989) has proposed an EPQ model with a flexible and imperfect process. A geometric programming (GP) approach has been developed for solving this model. The investment costs of this model tend toward infinity when the setup cost is close to zero. Cheng (1991) proposed an EOQ model with demand-dependent unit cost and imperfect production processes. He formulated the optimization problem as a GP, and it is solved to obtain a closed-form optimal solution. Salameh and Jaber (2000) hypothesized a production–inventory situation where items are not of perfect quality. The imperfect-quality items could be used in another production–inventory situation. Their work also considered that the imperfect items can be sold as a single batch at a lower price by the end of 100% inspection. It shows that the economic lot size quantity tends to increase as the average percentage of imperfect-quality items increases. However, it does not include the impact of the reject and the rework on their model and ignore the factor of when to sell. Furthermore, their work only considered the EOQ model. Goyal and Cárdenas-Barrón (2001) presented a simple approach for determining the economic production quantity for an item with imperfect quality. It is suggested that this simple approach is comparable to the optimal method of Salameh and Jaber. Some related research can be found in Chan et al. (2003), Cárdenas-Barrón et al. (2012, 2015), Taleizadeh and Noori-Daryan (2016), Taleizadeh and Heydaryan (2017), Taleizadeh et al. (2019, 2020), Shafiee-Gol et al. (2016), Moshtagh and Taleizadeh (2017),

4.3 No Shortage

239

Keshavarz et al. (2019), Rosenblatt & Lee (1986), Aggarwal & Aneja (2016), Glock & Jaber (2013), Tersine & Tersine (1988), Krishnamoorthi & Panayappan (2012), Taft (1918), Taleizadeh et al. (2015) and Alizadeh-Basbam and Taleizadeh (2020).

4.3 4.3.1

No Shortage Imperfect Item Sales

Chan et al. (2003) developed an imperfect inventory system that a process produces a single product in a batch size of Q. In addition, storage and withdrawals are uniform and continuous. The demand rate for the product is deterministic and constant over a planning horizon of 1 year. The production process produces this item with finite production rate P units per year. The manufacturing cost of each unit is C$, and the inventory holding cost per unit per year and the setup cost per batch are donated by h and K, respectively. In this problem, it is assumed that each lot produced contains p1 percent imperfect-quality items, with a known probability distribution function f( p1). Items of imperfect quality detected by the inspection process are sold at a lower price. It is assumed that the inspection cost for each unit is a fixed constant and the detection of defectives is achieved by nondestructive and error-free testing. The lot also contains a percentage of defectives, p2, with a known probability distribution function f( p2). It is assumed that these defective items can be reworked instantaneously at a cost and kept in stock. After the rework process, these items are assumed to be of good quality. Each lot reworked also contains a percentage of defectives, p3, with a known probability distribution function f( p3). These units are rejected with an associated cost. Three different approaches are developed to comply with three different situations/cases as below. Some new notations which are specially used for this problem presented in Table 4.2. Case I Imperfect-quality items are sold at a discounted price when identified and are not counted into the inventory. The production rate of good items per year (including the items that can be reworked) is represented as Pp ¼ P[1  p1  p3]. The amount of good-quality items added to stock, as the result of a single production run, is (Pp  D)(Q/P) ¼ Q[1  p1  p3  (D/P)] (Chan et al. 2003) (see Fig. 4.2). Case II After the inspection process, the imperfect items are kept in stock and sold at the end of the production period within each cycle as a single batch at a reduced price per unit. As a result, the production rate during the production period will be expressed as Pg ¼ P[1  p3] (Chan et al. 2003) (see Fig. 4.3). Case III These imperfect-quality items are kept in stock and sold at the end of the cycle (just before next production run). The production rate is the same as Case II. The main difference between Case II and Case III is the time factor for the sale of the imperfect-quality items (Chan et al. 2003) (see Fig. 4.4).

240

4 Rework

Table 4.2 New notations of given problem Pp p1 p2 p3 pg Q QI QII QIII Q0

Production rate of good items per year (including the items that can be reworked) is represented as Pp ¼ P[1  p1  p3] (units per unit time) Percentage of imperfect-quality items, with a known probability distribution function f( p1) (units per unit time) Percentage of rework items, with a known probability distribution function f( p2) (units per unit time) Percentage of reject items from rework process, with a known probability distribution function f( p3) (units per unit time) Production rate of good items and imperfect-quality items per unit time (include the rework items) (units per unit time) Lot size in number of units per lot (optimal value for the new model) (unit) Lot size in number of units per lot for Case I (unit) Lot size in number of units per lot for Case II (unit) Lot size in number of units per lot for Case III (unit) Optimum lot size in number of units per lot for the classical EPQ model (unit)

Fig. 4.2 The behavior of the inventory level per cycle for Case I (Chan et al. 2003)

Define E(Pp) ¼ [1  E(P1)  E(P3)] as expected proportion of good items used. The amount of good items available for use in the lot per each single production run is Q[E(Pp) – D/P]. To avoid shortage, it is assumed that:   E Pp  D=P

ð4:1Þ

Consider a firm having an expected total profit (ETP) as follows: ETPðQÞ ¼ ETRðQÞ  ETCðQÞ

ð4:2Þ

4.3 No Shortage

241

Fig. 4.3 The behavior of the inventory level per cycle for case II (Chan et al. 2003)

Fig. 4.4 The behavior of the inventory level per cycle for Case III (Chan et al. 2003)

where ETR(Q) and ETC(Q) are the expected total revenue and expected total cost per year. The magnitude of these is obtained as follows: ETRðQÞ ¼ ðGood quality itemsÞ  full price per item þ ðimperfect quality itemsÞ lower price per item

242

4 Rework

¼ Ds þ

D  E ð p1 Þ  v   ¼ Ds þ Dvξ E Pp

where:   ξ ¼ E ðp1 Þ=E Pp

ð4:3Þ

ETC(Q) is the sum of setup cost, rejection cost, rework cost, inspection cost, manufacturing cost, and inventory holding cost, in which only the inventory holding cost varies on the different cases. The inventory holding cost is obtained as the average inventory times holding cost per item per year, giving for Case I, Case II, and Case III, respectively. For the purposes of brevity, the derivation of the inventory holding cost is omitted. The detail derivation is, however, available from the authors:     hQ E Pp  D=P hQ , 2 2 n   o D  E P p  ½1  ξ , P

n   o D hQ E Pp þ 2E ðp1 Þ  ½1 þ ξ , P

The total number of units produced per year is D/E(Pp); the number of setups is the number of units produced per year divided by the lot size, Q. Therefore, the annual setup cost is given as DK/[E(Pp)Q]. The manufacturing cost and the inspection cost per unit is constant, and then the annual manufacturing cost and the inspection cost can be written as CD/E(Pp) and CID/E(Pp), respectively. In addition, an average of [D/E(Pp)]E(P3) units are rejected per year with an associated cost; CJ and [D/E(Pp)]E(P) units are reworked per year with a cost CR; the annual rejection cost and rework cost can be represented as DCJE( p3)/E(Pp) and DCRE( p2)/E(Pp), respectively. The values of E(P1), E(P2), and E(P3) follow from the assumed distribution of the quality characteristic being measured. By substituting all the cost components into Eq. (4.2), the ETP(Q) for the three different approaches can be expressed as: Case I Imperfect items sold as identified which are not kept in stock (see Fig. 4.2) (Chan et al. 2003): 9 8 > > > > > > " # Revenue > 6zfflfflfflfflffl  =

6 K D p I 6   þ C J E ðp3 Þ þ C R E ðp2 Þ þ C I þ C ETPðQÞ ¼6Ds þ Dvξ  þ 2 Q > E Pp > > > 4 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> ; : Holding Cost 2

Setup, Rejection, Rework, Inspection and Production Costs

ð4:4Þ

4.3 No Shortage

243

Case II Imperfect items sold at the end of production period (see Fig. 4.3) (Chan et al. 2003): Revenue

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ ETPðQÞII ¼ ½Ds þ Dvβ



8 >
> #> > =

   " D z}|{ hQ E Pp  ðD=PÞð1  βÞ þ K þ C J Eðp Þ þ C R E ðp Þ þ C I þ C   3 2 2 Q > E Pp > > : > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> ;  Holding  Cost

Setup, Rejection, Rework, Inspection and Production Costs

ð4:5Þ Case III Imperfect items sold just before the next batch (see Fig. 4.4) (Chan et al. 2003): ETPðQÞIII ¼

9 8 Holding Cost > > > > > > > > z}|{ hQ½EðPp Þþ2Eðp1 ÞðD=PÞð1þβÞ > > Revenue 2 > > = " # zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ <   ½Ds þ Dvβ  K D   > > þ þ C J E ðp3 Þ þ CR E ðp2 Þ þ CI þ C > > > E Pp > > > Q > > > ; : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > Setup, Rejection, Rework, Inspection and Production Costs

ð4:6Þ Differentiating and equating d[ETP(Q)]/dQ ¼ 0, the optimal Q* (lot size) which generates minimum expected total cost can be given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK       Q ¼ h E Pp  ðD=PÞ E Pp sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK II        Q ¼ h E Pp  DP ½1  ξ E Pp sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK        ¼ h E Pp þ 2Eðp1 Þ  DP ½1 þ ξ E Pp I

QIII

ð4:7Þ

ð4:8Þ

ð4:9Þ

The second derivates of Eqs. (4.4)–(4.6) give 00 00 00 ETP (Q)I ¼ ETP (Q)II ¼ ETP (Q)III ¼  2DK/{[1  E( p1)  E( p3)]Q3}  0 and are negative for all values of positive Q, which implies that there exist unique values of Q*I, Q*II, and Q*III that maximize Eqs. (4.4)–(4.6), respectively. Note that when P1 ¼ P1 ¼ P1 ¼ 0 (i.e., all products produced are of perfect quality),

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4 Rework

Table 4.3 Optimal values (Chan et al. 2003) Case I II II

Order size 1850 1801 1607

Average inventory cost $1696.08 $1742.13 $1951.36

Setup cost $1694.68 $1740.79 $1950.94

ETP(Q) $777,166.60 $777,074.45 $776,655.06

Eqs. (4.7)–(4.9) reduce to the classical EPQ model, Q0 ¼ (2DCd/h(1  D/P))1/2. If the time for producing a batch of units is zero, or approaches zero, the arrival rate is infinite. When this occurs, the batch model reverts to the classical EOQ model, Q0 ¼ (2DCd/h)1/2. Example 4.1 Chan et al. (2003) developed a numerical illustration which is provided to illustrate the usefulness of the models developed in previous section. Suppose that an electronic company producing high-voltage transformers for Model: XTA with a production capacity of 960 units per 8-h shift. Items in range of 93–107 V are used directly by the company for Model: XTA. Items outside this range and in the range of 90–110 V may be used for various activities or sold to another producer. If the quality characteristic exceeds the upper limit of 110 V, the transforms are rejected; if it is lower than the lower limit of 90 V, the manufacturer can adjust the voltage in the plant by changing a resistor at a cost and kept in stock instantaneously. Thus, the parameters needed for analyzing for this situation are given as: C J ¼ $15 per unit, C R ¼ $8 per unit, D ¼ 60,000 units per year, h ¼ $3 per unit per year C I ¼ $0:5 per unit, C ¼ $15 per unit, K ¼ $45 per lot, s ¼ $30 per unit, v ¼ $12 per unit

Since the plant works one shift per day, 5 days a week, and 50 weeks a year, the annual production capacity is P ¼ 960(5)(50) ¼ 240,000 units/year. Then, the expected value of E( p1), E( p2), and E( p3) can be obtained from the normal distribution, x N(μ ¼ 100, σ ¼ 5), as 0.1160, 0.0228, and 0.0228, respectively. Substituting these values into Eqs. (4.7)–(4.9), the optimal values of Q*I, Q*II, and Q*III can be obtained. Table 4.3 illustrates the optimal lot sizes, average inventory cost, setup cost, and the expected total profit for the three different cases (Chan et al. 2003).

4.3.2

Rework Policy

Chiu and Chiu (2003) studied the effect of the reworking of repairable defective items on EPQ model. They assumed that x percent of defective items were generated randomly by an imperfect process, at a production rate d, and not all of the defective items produced are repairable. There is a portion θ% of the imperfect-quality items cannot be repaired, are scrap items. Furthermore, the proposed EPQ model does not

4.3 No Shortage

245

allow backorders when excessive demand occurred, the optimal production quantity derived, must be able to satisfy demand at all times. The production rate P is constant and is much larger than the demand rate D. The production rate d of the defective items could be expressed as the production rate times the defective rate: d ¼ P  x. All repairable defective items are reworked at a steady rate P1, and the following notations are used in our analysis: The production rate of perfect-quality items must always be greater than or equal to the sum of the demand rate and the production rate of defective items (Chiu and Chiu 2003). The production rate of perfect-quality items must always be greater than or equal to the sum of the demand rate and the production rate of defective items, so P  d  D  0 must be satisfied:

 D 0x 1 P

ð4:10Þ

For the following derivation, they employed the solution procedures used by Hayek and Salameh (2001), referring to Fig. 4.5: T¼

Q ð 1  θ  xÞ D

ð4:11Þ

where 0  θ  1 and θ  x  Q are scrap items randomly produced by the regular production process. Hence, the cycle length T is a variable, not a constant: T ¼ t1 þ t2 þ t3 The production uptime t1 is:

Fig. 4.5 On-hand inventory of perfect-quality items (Chiu and Chiu 2003)

ð4:12Þ

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4 Rework

Fig. 4.6 On-hand inventory of defective items (including scrap items) (Chiu and Chiu 2003)

t1 ¼

Q P

ð4:13Þ

and

 D H 1 ¼ ðP  d  DÞt 1 ¼ Q 1  x  P

ð4:14Þ

The total defective items produced during the regular production uptime t1, as illustrated in Fig. 4.6, are: d  t1 ¼ x  Q

ð4:15Þ

The repairable portion (1  θ)% of imperfect-quality items are reworked immediately when the regular production ends. The time t2 needed for the rework is computed in (4.16), and the maximum level of on-hand inventory when rework process finished is calculated in (4.17): x  Qð1  θÞ d  Qð1  θÞ ¼ P1 P1 P

D Dd ð1  θÞ dθ H ¼ H 1 þ ðP1  DÞt 2 ¼ Q 1    P1 P P P t2 ¼

ð4:16Þ ð4:17Þ

The production downtime t3 is:

H 1 P1 þ dð1  θÞ dθ   t3 ¼ ¼ Q P1 P D D DP Solving the inventory cost per cycle, TC(Q) is (Chiu and Chiu 2003):

ð4:18Þ

4.3 No Shortage

247

Setup Cost

z}|{ TCðQÞ ¼ K þ

Production Cost

z}|{ CQ

Disposal Cost

Rework Cost

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ þ C R xQð1  θÞ þ Cd ðθxQÞ

Holding Cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   ðH 1 þ H Þ H 1 þ dt 1 H þh ðt 1 Þ þ ðt 2 Þ þ ðt 3 Þ þ 2 2 2

P t h1 1 2 ðt 2 Þ |fflfflfflfflfflfflffl2{zfflfflfflfflfflfflffl} Holding Cost of Reworked Item

ð4:19Þ In this model, and as in real-life situation, the percentage of defective items is considered to be a random variable with a known probability density function. Thus, to take the randomness of imperfect production quality into account, one can utilize the expected values of x in the inventory cost analysis. Since θxQ are scraps produced randomly during a regular production run, it follows that the cycle length is a variable. They employed the renewal reward theorem approach to cope with the variable cycle length, that is, to compute the E[T] first. Then in the expected annual inventory cost E[TCU(Q)] ¼ E[TC(Q)]/E[T], from Eqs. (4.11) through (4.19), one obtains that (Chiu and Chiu 2003):  E ½TCUðQÞ ¼D C

 E ½ x E ½x 1 þ C R ð1  θ Þ þ Cd θ 1  θE ½x 1  θE ½x 1  θE ½x

 DQð1  θÞ2 KD 1 hQ D 1 þ 1 þ þ Q 1  θE ½x 2 P 1  θE ½x 2P1

 2 E ½x  E ½x D hQθ2 E ½x2   ð h1  h Þ  hQθ 1  þ P 1  θE ½x 2 1  θE½x 1  θ  E ½x ð4:20Þ

Suppose that: E0 ¼

1 ; 1  θE ½x

E1 ¼

E ½ x ; 1  θE ½x

E2 ¼

E ½ x 2 1  θE½x0

Then, Eq. (4.20) becomes (Chiu and Chiu 2003):

 KD hQ D E0 þ 1 E Q 2 P 0

 DQð1  θÞ2 D hQθ2 E1 þ þ ðh1  hÞE2  hQθ 1  E2 P 2P1 2 ð4:21Þ

E ½TCUðQÞ ¼D½CE 0 þ C R ð1  θÞE 1 þ Cd θE1  þ

Differentiating E[TCU(Q)] with respect to Q, the first and the second derivatives of E[TCU(Q)] are shown in Eqs. (4.22) and (4.23):

248

4 Rework

 dE ½TCUðQÞ KD Dð1  θÞ2 h D ¼ 1  E E þ þ ðh1  hÞE 2  h 0 0 dQ 2 P 2P1 Q2

 D h  θ2 θ 1 E1 þ E2 P 2 d2 E½TCUðQÞ 2KD ¼ 3 E0 Q dQ2

ð4:22Þ ð4:23Þ

From Eq. (4.23), since E0, K, D, and Q are all positive numbers, the second derivative of E[TCU(Q)] with respect to Q is greater than zero; the expected total inventory cost function E[TCU(Q)] is a convex, for all Q is different from zero. The optimal production quantity Q* can be obtained by setting the first derivative of E [TCU(Q)] equal to zero; refer to Eq. (4.24) (Chiu and Chiu 2003): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD  

 Q ¼u t

2 D ð 1θ Þ h 1  Dp þ p ðh1  hÞE ½x2   2hθ 1  Dp E½x þ hθ2 E½x2  

1

ð4:24Þ Supposing that all of the defective items are not repairable, meaning that they are all scrap items (θ ¼ 1), from Eq. (4.24), one obtains: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD  

 Q ¼u t

h 1  Dp  2h 1  Dp E ½x þ hE½x2  

ð4:25Þ

Further, if the process produces all perfect-quality items, i.e., x ¼ 0, it follows that Eq. (4.24) will give the same result as that of the classical EPQ model (Nahmias and Cheng 2005; Silver et al. 1998; Tersine 1994) as below: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD  Q ¼u t

h 1  Dp 

ð4:26Þ

Example 4.2 Chiu and Chiu (2003) presented an example for a local company which manufactures a product for several regional industrial clients. It has experienced a relatively flat demand of 4000 units per year. This item is produced at a rate of 10,000 units per year. The accounting department has estimated that it costs $450 to initiate a production run, each unit costs the company $2 to manufacture, and the cost of holding is $0.6 per item per year. The defective items are reworked at a rate of 600 units per year, each repairable defective item costs the company $0.5 to rework, and there is a disposal cost of $0.3 for each scrap item inspected and identified prior to starting the reworking process. In addition, there is a holding cost of $0.8 per year,

4.3 No Shortage

249

per unit of the items being reworked. Supposing that the percentage of defective items produced is uniformly distributed over the interval [0, 0.2], and not all of the imperfect-quality items are repairable, there is a θ % of them that are scrap items. Chiu and Chiu (2003) used the following parameters as P ¼ 10,000 units per year, D ¼ 4000 units per year, P1 ¼ 600 units per year, x ¼ Uniform [0. 0.2], θ ¼ 0.3 (scrap rate out of the imperfect-quality items), K ¼ $450 for each production run, C ¼ $2 per item, CR ¼ $0.5 repaired cost for each item reworked, Cd ¼ $0.3 disposal cost for each item reworked, h ¼ $0.6 per item per unit time, and h1 ¼ $0.8 per item reworked per unit (Chiu and Chiu 2003). The optimal production lot size can be computed from Eq. (4.24). For example, when θ ¼ 0.1, the value of Q* ¼ 3162 units, and if θ ¼ 0.3, Q* ¼ 3200 units. One notices that as x increases, the value of Q* decreases. For different θ values, if θ increases, then Q* increases. The optimal total inventory costs can also be obtained from Eq. (4.21). For example, when θ ¼ 0.1, the value of E[TCU(Q*)] ¼ $9281, and if θ ¼ 0.3, then E[TCU(Q*)] ¼ $9354. One notices that as x increases, E[TCU(Q*)] increases. For different θ values, if θ increases, then E[TCU(Q*)] increases.

4.3.3

Imperfect Rework

Chiu et al. (2004) examined the EPQ model with the random defective rate and an imperfect rework process. Consider that a practical production process generates randomly x percent of imperfect-quality items at a production rate P. The basic assumption of the finite production model with imperfect-quality items produced is that the production rate P must always be greater than or equal to the sum of the demand rate D and the production rate of defective items d. In order to model the problem, the following specific notations are used for this problem as presented in Table 4.4. Hence, the following condition must hold (Chiu et al. 2004): PdD0

 D 0x 1 P

ð4:27Þ

For the following derivation, the solution procedures are those used by Hayek and Salameh (2001). The production cycle length (T ) is the summation of the production uptime (t1), the reworking time (t2), and the production downtime (t3), referring to Fig. 4.7 (Chiu et al. 2004): T¼

3 X i¼1

ti

ð4:28Þ

250

4 Rework

Table 4.4 New notations of given problem (Chiu et al. 2004) θ1 d1

The proportion of reworked items that fail (become scraps), θ1 is assumed to be a random variable with known probability density function The production rate of scrap items (during the rework process), in units per unit time

Fig. 4.7 On-hand inventory of perfect-quality item (Chiu et al. 2004)

I (t) Q

P1 – d1 – D H H1

P1 – d1 – D –D P – d –D

–D

P – d –D t1

t2

Time

t3 T

T

The production uptime tl needed to accumulate Hl units of perfect-quality items is (Chiu et al. 2004): t1 ¼

Q H1 ¼ P PdD

ð4:29Þ

and H 1 ¼ ðP  d  DÞ 

Q P

ð4:30Þ

The total defective items produced during the regular production uptime tl, as illustrated in Fig. 4.8, are (Chiu et al. 2004): d  t1 ¼ x  Q

ð4:31Þ

The repairable portion (1  θ) of defective items is reworked right after the regular production process ends. The time t2 needed for the reworking is computed in Eq. (4.32). The maximum level of on-hand inventory H is obtained in Eq. (4.33) (Chiu et al. 2004):

4.3 No Shortage

251

Fig. 4.8 On-hand inventory of defective items (Chiu et al. 2004)

Id(t)

θ% of defectives are scrap items

dt1 dt1(1 _ θ )

d

_P

d

_P

1

1

Time

t1

t2

t3

T

t2 ¼

T

xQð1  θÞ dQð1  θÞ ¼ P1 P1 P

H ¼ H 1 þ ðP1  d1  DÞt 2

dθ DðP1 þ d Þ d 1 d d 1 dθ Ddθ  þ þ ¼Q 1  P1 P P P1 P P1 P P1 P

ð4:32Þ

ð4:33Þ

Referring to Fig. 4.9, since the rework process itself is assumed to be imperfect either, a random portion θ1 of the reworked items becomes scrap items. The production rate d1 in producing scrap items during the rework process can be written as: d 1 ¼ P1  θ 1 ,

where 0  θ1 < 1

ð4:34Þ

The total scrap items produced when the rework process ends can be computed in Eq. (4.35): d1  t 2 ¼ θ1 ½xð1  θÞQ

ð4:35Þ

The production downtime “t3” is:

H H dθ ðP1 þ dÞ d1 d d1 dθ dθ    þ þ t3 ¼ ¼ Q P1 P D D DP DP1 P DP1 P P1 P

ð4:36Þ

The cycle length T can be obtained (Chiu et al. 2004): T¼

Q½1  θx  ð1  θÞxθ1  Qf1  x½θ þ ð1  θÞθ1 g ¼ D D

ð4:37Þ

where 0 < θ < 1 and θxQ are scrap items randomly produced during the regular production process and [(1  θ)xθ1]Q are scrap items produced during the rework

252

4 Rework Is(t )

[θ + (1 _ θ )θ 1 ] ⫻ Q

θ

⫻

Q

d1 d1

dθ

dθ

t1

t2

t3

Time

t2

T

T

Fig. 4.9 On-hand inventory level of scrap items (Chiu et al. 2004)

process. Hence, the cycle length T is not a constant. Letting `φ’ denotes the overall scrap rate, then φ ¼ [θ + (1  θ)θ1]. Then, the cycle length can be rewritten as: T¼

3 X

ti ¼

i¼1

Q ½ 1  φ  x D

ð4:38Þ

Solving the total inventory cost per cycle, TC(Q) is (Chiu et al. 2004): Setup Cost

Production Cost

Rework Cost

Disposal Cost

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ z}|{ z}|{ TC ¼ K þ CQ þ CR ½xð1  θÞQ þ Cd ðφxQÞ   H 1 þ d  t1 H1 þ H H P t þh ðt 1 Þ þ ðt 2 Þ þ ðt 3 Þ þ h1 1 2 ðt 2 Þ 2 2 2 |fflfflfflfflfflfflffl2{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Holding Cost

Holding Cost of Reworked Items

ð4:39Þ In this model, as in real life, the proportion of imperfect-quality items is considered to be a random variable. Thus, to take the randomness of imperfect production quality into account, one can utilize the expected values of x, θ, and θ1 in the inventory cost analysis. Let E[x], θ, φ, and θ1 represent the expected values of x, θ, φ, and θ1, respectively. And to cope with the variable cycle length, the renewal reward theorem approach is employed to compute E[T]. Hence, it follows that in the expected total inventory cost E[TCU(Q)] ¼ E[TC(Q)]/E[T], from Eqs. (4.29) through (4.39), one obtains (Chiu et al. 2004):

4.3 No Shortage

253

 E ½TCUðQÞ ¼D C

E ½ x E ½ x 1 þ C R ð1  θ Þ þ Cd φ 1  φE ½x 1  φE ½x 1  φE½x

 KD 1 hQ D 1 þ þ 1 Q 1  φE ½x 1 P 1  φ  E ½ x

DQð1  θÞ2 E ½ x2  ½ h1  hð 1  θ 1 Þ  2P1 1  φE ½x

 E ½ x D hQφ2 E½x2   hQφ 1  þ P 1  φE ½x 2 1  φE ½x



ð4:40Þ

þ

where: φ ¼ ½θ þ ð1  θÞθ1  For simplicity assume; E0 ¼

1 ; 1  φE ½x

E1 ¼

E ½ x ; 1  φE½x

E2 ¼

E ½x2  1  φE ½x

Then, Eq. (4.40), the expected inventory cost per unit time becomes:

 KD hQ D E0 þ 1 E Q 1 P 0

 DQð1  θÞ2 D hQφ2 þ ½h1  hð1  θ1 ÞE2  hQφ 1  E1 þ E2 P 2P1 2 ð4:41Þ

E ½TCUðQÞ ¼D½CE 0 þ CR ð1  θÞE 1 þ C d φE1  þ

The optimal production lot size can be obtained by minimizing the cost function E[TCU(Q)]. Differentiating E[TCU(Q)] with respect to Q, the first and the second derivatives of E[TCU(Q)] are shown in Eqs. (4.42) and (4.43):

 dE ½TCUðQÞ KD D ð1  θ Þ2 h D ¼ 2 E0 þ 1  E0 þ ½h1  hð1  θ1 ÞE 2 dQ 2 P 2P1 Q ð4:42Þ

 D hφ2 E þ E  hφ 1  P 1 2 2 d2 E½TCUðQÞ 2KD ¼ 3 E0 Q dQ2

ð4:43Þ

Equation (4.43) is positive, because E0, K, D, and Q are all positive. The second derivative of E[TCU(Q)] with respect to Q is greater than zero; hence, the expected inventory cost function E[TCU(Q)] is a convex function for all Q different from zero (Chiu et al. 2004). The optimal production quantity can be obtained by setting the

254

4 Rework

first derivative of E[TCU(Q)] equal to zero (referring to Eq. (4.42)), from: dE ½TCUðQÞ ¼0 dQ Q ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KD   Dð1θÞ2   D h 1  P þ P1 ½h1  hð1  θ1 ÞE ½x2   2hφ 1  DP E½x þ hφ2 E ½x2  ð4:44Þ

Example 4.3 Chiu et al. (2004) proposed an example for a regional firm which produces a product for several industrial clients. This item has experienced a relatively flat demand of 4000 units per year, and it is produced at a rate of 10,000 units per year. The accounting department has estimated that it costs the company $450 to initiate a production run, and each unit costs $2 to manufacture, in which inspection cost per item is included. The cost of holding is $0.6 per item per year, and the disposal cost is $0.3 for each scrap item. The defective rate x is uniformly distributed over the interval [0, 0.2]. The rate of rework is 600 units per year. Each defective item costs the company $0.5 to repair plus an additional holding cost of $0.8 per item reworked per year. The scrap rates, 0 and 9, are both assumed to follow the uniform distribution over the range [0, 0.1]. Summary of parameters used is as follows: P ¼ 10,000 units per year, D ¼ 4000 units per year, P1 ¼ 600 units per year, x ~ Uniform [0, 0.2], θ ~ Uniform [0, 0.1], θ1 ~ Uniform [0, 0.1], K ¼ $450 for each production run, C ¼ $2 per item, CR ¼ $0.5 repaired cost for each item reworked, Cd ¼ $0.3 disposal cost for each item reworked, h ¼ $0.6 per item per unit time, and h1 ¼ $0.8 per item reworked per unit. Hence, it follows that the overall scrap rate, φ ¼ [θ + (1  θ) θ1] ¼ 0.0975, and the optimal production lot size can be obtained from Eq. (4.44). For example, if φ ¼ 0.1, then the value of optimal lot size Q* ¼ 3113. One notices that as x increases, the value of Q* decreases, and for different φ values, as φ increases, the value of Q* increases. The optimal inventory costs can be calculated from Eq. (4.40). For example, if φ ¼ 0.1, then the value of E[TCU(Q*)] ¼ $9453. One notices that as x increases, the value of E[TCU(Q*)] increases, and for different φ values, as φ increases, the optimal cost function E[TCU(Q*)] increases (Chiu et al. 2004).

4.3.4

Quality Screening

Moussawi-Haidar et al. (2016) studied a production quantity model, in which production occurs at a rate P and demand occurs at rate D units per unit time, P > D. The inventory builds up at the rate P  D. During production, a random proportion P of defective items is produced, with a known probability density function f(P). Demand during production is met from non-defective items only, which requires the units demanded to be screened before they are sold to customers. During this process, if an item is found to be defective, it is replaced with a

4.3 No Shortage

255

non-defective item. The number of defective items accumulated when production stops, and before screening is conducted, is equal to the total number of items screened from the total demand. As soon as production stops, screening the remaining units of the produced lot is conducted at the rate x per unit per unit time, where x > D. The screening cost during production is higher than that after production, i.e., CI1 > CI2. In order to model the problem, the following specific notations are used for this problem as presented in Table 4.5. They analyzed two models that differently address the defective items identified during production and screening. The first model assumes that defective items are sold at a discount at the end of the production cycle. The second model assumes that defective items are reworked at a constant rate.

4.3.4.1

Salvaging of Defective Items

The defective items accumulated at the end of the screening period are sold at a discounted price v. The following notation will be used to develop the mathematical model (Moussawi-Haidar et al. 2016). According to Fig. 4.10, demand during production is met using good items only. Therefore, in [0, t1], a number of units are screened before they are sold to customers. To be able to satisfy demand from good items only, more than the demand is screened. The total number of units screened can be computed as follows:   D þ Dp þ Dp2 þ ⋯ t 1 ¼

D t 1p 1

ð4:45Þ

At the end of t1, the number of defective items identified is the total number of units screened during the interval [0, t1], as given in (4.45), less the demand during this period. This is written as: 

 D pD y  D t1 ¼ 1p 1p P

ð4:46Þ

The on-hand inventory not screened at the end of t1 is equal to the maximum inventory level, y(1  D/P), less the number of defective items identified at the end of t1, as given in (4.46). This is depicted in Fig. 4.10:

Table 4.5 New notations of given problem (Moussawi-Haidar et al. 2016) CI1 CI2 y

Screening cost per item during production ($/item) Screening cost per item after production stops ($/item) Total number of items produced during a production cycle (item)

256

4 Rework

Fig. 4.10 The behavior of the inventory level over a production cycle (Moussawi-Haidar et al. 2016)

 D pD y  y 1 P 1p P

ð4:47Þ

At t1, the on-hand inventory not screened in [0, t1] is screened at the rate x. It can be easily checked that the total number of defective items in a cycle, yp, is the summation of the defective items found during the [0, t1],ipP/(1  p)y/P, h  interval  pDy D and those found during the screening period t: p y 1  P  ð1p ÞP . Two conditions are required. First, to avoid shortages during production, the number of good items produced should meet demand during production, i.e., N (y, p)  Dt1, which implies the following condition on p: p  1  D=P

ð4:48Þ

The on-hand inventory not screened at the end of production is expressed in (4.47) and requires t2 units of time to be screened at rate x per unit per unit time. Thus, t2 can be written as: t2 ¼

yð1  D=PÞ  ðpD=ð1  pÞÞðy=PÞ x

ð4:49Þ

They let t3 be the time from when production stops until the end of the cycle, i.e., t3 ¼ T  t1. Then t3 can be written as: t3 ¼

yð1  D=PÞ  yp D

ð4:50Þ

4.3 No Shortage

257

The second condition is a limit on the screening time t2. Naturally, since t2 should be less than t3, after some term arrangement, the following condition on the screening rate, Rs, is derived: Rs >

Dð1  D=PÞ  pD2 =ð1  pÞ 1  D=P  p

ð4:51Þ

Let TR( y) be the total revenue per cycle. TR( y) is the summation of the selling price of good-quality items and the discounted selling price of defective items. Thus, it is written as: TRðyÞ ¼ syð1  pÞ þ υyp

ð4:52Þ

Also, let TC( y) be the total cost per cycle. TC( y) is the summation of the production setup cost, unit production cost, screening cost during and after production, and inventory holding cost. The number of units screened at the end of production, i.e., at time t1, is equal to the inventory level at t1 less the total number of defectives identified during production and given in (4.46). To compute the holding cost expression, they refer to Fig. 4.10, in which the average inventory is the summation of the three areas, ABC, CDEF, and BGF. From Fig. 4.10, the cycle time T can be found as T ¼ y(1  P)/D. Computing the areas of the three triangles, the total cost per cycle, TC( y), can be written as follows (Moussawi-Haidar et al. 2016): Screening cost after production

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{  z}|{ z}|{ pD TCðyÞ ¼ K þ Cy þ þ C I 2 y ð1  D=PÞ  Pð 1  p Þ

3 2 pD 2 y p 1  D=P  6y2 ð1  D=P  pÞ2 y2 ð1  D=PÞ Pð1  pÞ 7 7 þ þ þ h6 4 5 Rs 2P 2D Setup Cost

Screening cost during production

Production Cost

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ D y CI 1 ð1  pÞ P

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Holding Cost

ð4:53Þ The total profit per cycle is the total revenue less the total cost and is given as:  TPðyÞ ¼syð1  pÞ þ υyp  K þ Cy þ CI 1

  D y pD þ C I 2 y ð1  D=PÞ  ð 1  pÞ P Pð1  pÞ

33 2 pD 2 y p 1  D=P  6y2 ð1  D=P  pÞ2 y2 ð1  D=PÞ 7 Pð1  pÞ 7 77 þh6 þ þ 4 5 5 2P Rs 2D ð4:54Þ

258

4 Rework

Taking the expected value of the total profit per cycle ETP( y) with respect to p, they got the following expression for the expected total profit per cycle ETP( y): 



 1 y D p ETPðyÞ ¼syð1  E ðpÞÞ þ υyE ðpÞ  K þ Cy þ C I1 DE þ C I2 y ð1  D=PÞ  E 1p P P 1p

33 2 h i D p 2 2 y2 E ðpÞ 1  D=P  E 7 6y E ð1  D=P  pÞ P 1p 7 y2 ð1  D=PÞ 77 6 þ þ þh4 55 2D 2P Rs

ð4:55Þ Using the renewal reward theorem (see Ross 1996, Theorem 4.6.1), they found the expected profit per unit time as follows: ETPUðyÞ ¼

ETPðyÞ , E ðT Þ

where the expected duration of the production cycle is E(T ) ¼ [y(1  E( p))]/D. This gives the following expression for the expected profit per unit time, ETPU( y) (Moussawi-Haidar et al. 2016):



vDE ðpÞ CI1 D2 1 CI2 D D p   ETPUðyÞ ¼sD þ E 1  D=P  E P 1p 1  E ðpÞ Pð1  EðpÞÞ 1  p 1  EðpÞ 3 2 n o 2 E ð1  D=P  pÞ ð Þ D 1  D=P 7 6 þ 7 6 2 2P 7 6 CD D y

7 6  K h 7 6 D p 1  E ðpÞ yð1  E ðpÞÞ 1  E ðpÞ 6 7 5 4 DE ðpÞ 1  D=P  P E 1  p þ Rs

ð4:56Þ Setting the unit discounted price v to zero results in a model that allows scrapping the defective items, rather than selling them at discount. Thus, their model in this case reduces to one that allows scrapping defective items. All the mathematical expressions will remain unchanged. The first-order conditions of ETPU( y) with respect to y give the optimal production lot size as (Moussawi-Haidar et al. 2016): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u KD  y ¼u u  2 t E½ð1D=PpÞ  Dð1D=PÞ DEðpÞð1D=PDPEðp=ð1pÞÞÞ h þ þ 2 Rs 2P 

ð4:57Þ

4.3 No Shortage

4.3.4.2

259

Reworking of Defective Items

In this section Moussawi-Haidar et al. (2016) considered that all defective items will be reworked. Using previous models and also Figs. 4.11 and 4.12, the new total profit becomes: Screeningcost duringproduction Revenue

z}|{ ETPUðyÞ ¼ sD 

Production Cost

z}|{ CD

ReworkCost

zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl ffl{

D2 1 E 1p P

SetupCost

zfflffl}|fflffl{ DK  CR pD  z}|{  y

CI 1

HoldingCost of ReworkedItems

zfflfflfflfflfflffl}|fflfflfflfflfflffl{ Dd 2   h1 y 2 2P P1 " #

2



 e DJ DJ e DJ DJ dD DJ Dd dD dJD e e þ þ J  hy þ J  =2 þ 2 þ J 2D x 2x x PP1 x PP1 Px 2P |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Screening cost after production

zfflffl}|fflffl{ CI 2 DJ

Holding Cost

ð4:58Þ h

i p where J ¼ 1  DP 1  E 1p and e J ¼ 1  DP  Pd ¼ 1  DP  E ðpÞ. And after proving the concavity and setting the first derivative respect to y equal to zero and some simplification, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u KD y ¼u u 

2  2 t DeJ DJ e DJ  e DJ 1 J  DEðpÞ DE ðpÞ J  DJ  =2 h 2P þ x J  2x þ J  x þ 2 þ P1 P1 þ h1 DE2Pðp1Þ þ e x P1 

ð4:59Þ

Fig. 4.11 The behavior of the inventory level over a production cycle when defective items are reworked (Moussawi-Haidar et al. 2016)

Inventory level of all items

Z1

P_ D _ d

P–D

D

Z2

D – P1

Z3

Inventory level of good items

0

t1

t2 T

D

t3

t4

Time

260

4 Rework

Inventory level of good items

P_ D_d

D

D _ P1 D

0

t1

t3

t2

t4

Inventory level of all items Time

T

P_D D

Inventory level of defective items

0

t2+ t3+t4

t1

Time

T P1

d

0 t1

t2

t3

t4

Time

T

Fig. 4.12 The inventory of good, defective, and all items (Moussawi-Haidar et al. 2016)

Example 4.4 Moussawi-Haidar et al. (2016) analyzed how the optimal production quantity and optimal profit vary with the model parameters, for each of the models. Note that in each model, the conditions on the expected proportion of defective items and the screening rate should be held. So they developed numerical results similar to those in Hayek and Salameh (2001). This illustrates the application of their model and allows comparing their results with those of Hayek and Salameh (2001) for the model with rework. They set D ¼ 1200 units/year, P ¼ 1600 units/year, P1 ¼ 100 units/year, C ¼ $104, s ¼ $200/unit, v ¼ $80/unit, Rs ¼ 175,200 units/ year, CI 1 ¼ $0.5/unit, CI 2 ¼ $0.6/unit, K ¼ $1500, h ¼ $20/unit/year, h1 ¼ $22/unit/ year, CR ¼ $8/unit, p U[0, 0.1] with probability distribution function:  f ðpÞ ¼ Then,

10

0  p  0:1

0

otherwise

4.4 Backordering

261

E ðpÞ ¼ 0:05

1 E ¼ 1:0536 1p

p ¼ 0:0536 E 1p When the defective items are salvaged, and p ¼ 0, the economic production quantity and related profits are, respectively, 848.52 units and $110,332. For more detailed information, readers can refer to Moussawi-Haidar et al. (2016).

4.4 4.4.1

Backordering Simple Rework

Cárdenas-Barrón (2009) developed an EPQ inventory model in an imperfect production system environment with rework process and planned backorders. In addition to common assumptions of the EPQ model, he assumed that the proportion of defective products is known, the products are 100% screened, and the screening cost is not considered. All defective products are reworked and converted into goodquality products. Scrap is not generated at any cycle; backorders are allowed, and all backorders are satisfied. Production and reworking are done in the same manufacturing system at the same production rate. Two types of backorder costs are considered: linear backorder cost (backorder cost is applied to average backorders) and fixed backorder cost (backorder cost is applied to maximum backorder level allowed). The b b presenting the unit backorder cost new notation which is used in this model is C independent of time. Figure 4.13 shows the inventory behavior for the EPQ inventory model with rework at the same cycle and planned backorders. Using Fig. 4.13, the maximum inventory Imax can be calculated as the sum of I1 + I2 (Cárdenas-Barrón 2009). According to triangle (146), it can be concluded that (Cárdenas-Barrón 2009): tan θ1 ¼ Pð1  xÞ  D ¼

I1 þ B T1 þ T2

Also, it is obvious that T1 + T2 ¼ TP is the production time of manufacturing Q units; therefore T1 + T2 is equal to Q/P. Then according to Fig. 4.13, Q ½Pð1  xÞ  D ¼ I 1 þ B P I 1 ¼ Q½ð1  xÞ  D=P  B

262

4 Rework

Fig. 4.13 Inventory behavior for the EPQ inventory model with rework (Cárdenas-Barrón 2009)

PD¼

I2 T3

where T3 is the production time of manufacturing the defective products (xQ); therefore T3 is equal to xQ/P. Then, xQ ðP  DÞ ¼ I 2 P I 2 ¼ xQð1  D=PÞ Adding I1 and I2, the maximum inventory Imax is obtained (Cárdenas-Barrón 2009): I max ¼ I 1 þ I 2 ¼ Q½ð1  xÞ  D=P  B þ xQð1  D=PÞ I max ¼ Q½1  ð1 þ xÞðD=PÞ  B According to triangle (356), one can obtain T2 as:

4.4 Backordering

263

T2 ¼

Q½ð1  xÞ  D=P  B ½Pð1  xÞ  D

Cárdenas-Barrón (2009) presented that: Area of triangle ð356Þ ¼ Area of triangle ð567Þ ¼ Area of triangle ð679Þ ¼

T 2 I 1 fQ½ð1  xÞ  D=P  Bg ¼ 2 2½Pð1  xÞ  D

2

T 3 I 1 xQfQ½ð1  xÞ  D=P  Bg ¼ 2P 2

T 3 I max xQfQ½1  ð1 þ xÞðD=PÞ  Bg ¼ 2 2P

According to triangle (7910), T4 can be obtained as: T4 ¼

Q½1  ð1 þ xÞðD=PÞ  B D

where T4 is the time needed for maximum consumption at hand inventory level Imax; then, Area of triangle ð7910Þ ¼

T 4 I max fQ½1  ð1 þ xÞðD=PÞ  Bg ¼ 2 2D

2

Then, the inventory average which can be computed summing the area of triangles (356), (567), (679), and (7910) divided by T gives:

I¼

fQ½ð1xÞD=PBg2 2½Pð1xÞD

2

ÞðD=PÞBg þ fQ½1ð1þx2D þ

xQfQ½12xð1þ2xÞðD=PÞBg P

T

After some simplifications,  I¼

2      Q2 þ B2 ð1  xÞ þ Q2 ð1 þ x þ x2 Þ DP2 þ Q2 ðx3  2Þ DP þ 2BQ DP þ ðx  1Þ 2Q½ð1  xÞ  D=P

In order to simplify the mathematical expression, Cárdenas-Barrón (2009)  2 defined Z ¼ 1  x, E ¼ 1  x  D/P, I ¼ ð1 þ x þ x2 Þ DP2 , O ¼ (x3  2)(D/P), U ¼ D/P + x  1. Then,

264

4 Rework

Q2 ðZ þ I þ OÞ þ B2 Z  2BQE 2QE

 Q ZþIþO B2 Z I¼ þ B 2 E 2QE

I¼

After some algebra, I¼

   B 2 ð 1  xÞ Q 1  1 þ x þ x2 ðD=PÞ þ B 2 2Qð1  x  D=PÞ

assume:   L ¼ 1  1 þ x þ x2 ðD=PÞ Finally, the inventory average is given by I¼

Q B2 Z Lþ B 2 2QE

And easily T1 and T5 can be expressed by: T1 ¼

B ½Pð1  xÞ  D T5 ¼

B D

So according to Fig. 4.12 and using T1 and T5, the average backorders J can be computed as below: J¼ J¼

B2 2½Pð1xÞD

2

B þ 2D

T

B ð 1  xÞ 2Q½ð1  xÞ  D=P

ð4:60Þ

B2 Z 2QE

ð4:61Þ

2

J¼

Cárdenas-Barrón (2009) assumed that all defective products have the same manufacturing cost when they are reworked. Consequently, the total cost function (TC) is given as:

4.4 Backordering

265 Shortage Cost

Setup Cost

Holding cost zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ Production Cost z}|{ z}|{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ KD Cb BD b TC ¼ þ hI þ þ C b J þ CDð1 þ xÞ Q Q 2

ð4:62Þ

2

B Z B Z  B and J ¼ 2QE into the second and fourth terms of Substituting I ¼ Q2 L þ 2QE Eq. (4.62), respectively, Eq. (4.62) can be rewritten as:

TCðQ, BÞ ¼

b DB Cb B2 Z C KD hQL hB2 Z þ þ  hB þ b þ þ CDð1 þ xÞ Q 2 2QE Q 2QE

ð4:63Þ

The problem is to find the lot size (Q) and the size of backorders (B) that minimize the total cost inventory system (4.63). Assuming Q and B are continuous, let us take the first partial derivatives with respect to Q and B of Eq. (4.63), with these derivatives expressed by Eqs. (4.64) and (4.65), respectively: b BD C B2 Z ∂TCðQ, BÞ KD hL hB2 Z C ¼ 2 þ  2  b2  b 2 2 2Q E ∂Q Q Q 2Q E

ð4:64Þ

b D C BZ ∂TCðQ, BÞ hBZ C ¼ hþ b þ b QE Q QE ∂B

ð4:65Þ

In order to verify that Eq. (4.63) is convex in Q and B, they must show that the following two well-known conditions hold: 2

2

∂ TCðQ, BÞ ∂ TCðQ, BÞ > 0, >0 2 ∂B2 ∂Q ! ! !2 2 2 2 ∂ TCðQ, BÞ ∂ TCðQ, BÞ ∂ TCðQ, BÞ >0  ∂Q∂B ∂B2 ∂Q2

ð4:66Þ ð4:67Þ

First, they proved condition (4.66). By taking the second partial derivatives with respect to Q and B of Eq. (4.63), it can be easily shown that condition (4.66) is satisfied. The second partial derivatives are given by Eqs. (4.68) and (4.69). Both equations must be greater than zero. Since x is on interval (0, 1), thus K is greater than zero, and E also will be greater than zero if and only if x is less than 1 – D/P. Notice that the former analysis is valid for any finite, nonzero value of Q and B. Therefore, both second derivatives (Eqs. 4.68 and 4.69) are greater than zero (Cárdenas-Barrón 2009): 2 b BD C B2 K ∂ TCðQ, BÞ 2KD hB2 K 2C >0 ¼ 3 þ 3 þ b3 þ b 3 2 ∂Q Q Q E Q Q E

ð4:68Þ

266

4 Rework 2

∂ TCðQ, BÞ hK C b K þ >0 ¼ QE QE ∂Q2

ð4:69Þ

After that, they proved condition (4.67). For the sake of brevity, they provided only the final results. Taking the partial derivatives with respect to Q and B of Eq. (4.66) yields (Cárdenas-Barrón 2009), 2 h i ∂ TCðQ, BÞ 1 BK b bD ¼ 2 ðh þ C b Þ þ C ∂Q∂B Q E

ð4:70Þ

Then, 2

∂ TCðQ, BÞ ∂Q∂B

!2

 ¼



h i2 1 BK b h þ C ð Þ þ C D b b Q2 E

ð4:71Þ

Developing the squared binomial, one obtains (Cárdenas-Barrón 2009):

2 !2 b bD 2 C b b BDK ðh þ Cb Þ ∂ TCðQ, BÞ B2 K 2 ðh þ Cb Þ2 2C þ ¼ þ ∂Q∂B Q4 E Q4 E 2 Q4

ð4:72Þ

On the other hand, !2 !2 2 2 ∂ TCðQ, BÞ ∂ TCðQ, BÞ 2KDðH þ W Þ B2 K 2 ðh þ C b Þ2 þ ¼ 2 2 ∂B ∂Q Q4 E Q4 E2 þ

b b BDK ðh þ Cb Þ 2C Q4 E

ð4:73Þ

Substituting Eqs. (4.72) and (4.73) into (4.67), and after simplifying, one obtains (Cárdenas-Barrón 2009): ! ! !2 2 2 2 ∂ TCðQ, BÞ ∂ TCðQ, BÞ ∂ TCðQ, BÞ  ∂Q∂B ∂B2 ∂Q2  

2 1 bbD ¼ 4 2KDZ ðh þ C b Þ  E C Q E

ð4:74Þ

Equation (4.74) should be greater than zero if and only if the following condition is satisfied (Cárdenas-Barrón 2009):

4.4 Backordering

267

2 b bD > 0 2KDZ ðh þ C b Þ  E C

ð4:75Þ

Finally, it can be concluded that total cost function (4.63) is convex in Q and B, if and only if condition (4.75) is satisfied. Then Cárdenas-Barrón (2009) using the firstorder derivative derived the optimal values as below: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ffi u u2KDZ ðh þ C Þ  E C b D b b t Q¼ h½Z ðh þ C b ÞL  Eh

 b bD E hQ  C B¼ Z ðh þ C b Þ

ð4:76Þ

ð4:77Þ

Substituting Eqs. (4.76) and (4.77) into Eq. (4.63) and after some algebraic steps, TC ¼



 1 Z ðh þ C b Þ (sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 

2  b bD b bD ½Z ðh þ C b ÞL  hE h þ hE C  2KDZ ðh þ Cb Þ  E C þ CDð1 þ xÞ ð4:78Þ

Equation (4.78) also can be written as (Cárdenas-Barrón 2009):  TCðQÞ ¼

n o h b b D þ CDð1 þ xÞ Q½Z ðh þ C b ÞL  hE þ E C Z ðh þ C b Þ

ð4:79Þ

Another alternative mathematical expression for the total cost was proposed by Goyal and Cárdenas-Barrón (2003), which is (Cárdenas-Barrón 2009): TCðQ, BÞ ¼

 D b b B þ CDð1 þ xÞ 2K þ C Q

ð4:80Þ

It is important to mention that when x is equal to 1  D/P, then E is zero. Then, rffiffiffiffiffiffiffiffiffiffi 2KD Q¼ hL B¼0

ð4:81Þ ð4:82Þ

The reader can remember that L is equal to 1  (1 + x + x2) (D/P), and one may obtain a negative value under the radical in Eq. (4.81) when L is less than zero.

268

4 Rework

Therefore, 1  (1 + x + x2) (D/P) must be greater than zero to avoid a negative value for L. Thus, it is easy to see that the valid interval for R can be determined solving the following expression (Cárdenas-Barrón 2009):   1  1 þ x þ x2 ðD=PÞ > 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1  4ð1  P=DÞ x< 2 Then, 0 0 is assumed. Since the production process contains p rate of defective items, P(t2 + t3)p units of defective items are reworked to perfect items during interval t4. Wee et al. (2013) assumed the defective items are 100% reworked to perfect items and then the inventory increases at a rate of P  D during interval t4. Afterward, during interval t5, the inventory is depleted without production. Referring to Fig. 4.14, the backordering level B is as follows: B ¼ ðð1  xÞP  DÞt 2

ð4:84Þ

During the interval t3, the inventory level increases from zero to I1 with a rate of (1  x)P  D. So: I 1 ¼ ðð1  xÞP  DÞt 3

ð4:85Þ

ðð1  xÞP  DÞt 2 D

ð4:86Þ

Since B ¼ kt1, then: t1 ¼

They know that the interval t4 is assumed to rework the defective units generated during the total production time (t2 + t3). Thus,

270 Table 4.7 Solution to Example 4.6 (CárdenasBarrón 2009)

4 Rework x 0 0.01 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

Q 87.97727 1574.694 1583.218 1594.994 1608.08 1622.556 1638.51 1656.044 1675.265 1696.296 1719.262 1744.292 1771.505 1800.982 1832.718 1866.505 1901.682

B 31.052 24.72761 24.7332 24.73314 24.71774 24.67667 24.59631 24.45839 24.23812 23.90122 23.39939 22.66302 21.589 20.01965 17.70373 14.22006 8.813453

TC(Q, B) 2546.884 15,283.11 15,855.13 16,569.7 17,283.76 17,997.29 18,710.28 19,422.72 20,134.6 20,845.89 21,556.59 22,266.68 22,976.14 23,684.99 24,393.21 25,100.85 25,808

Fig. 4.14 Inventory of perfect-quality items for EPQ model (Wee et al. 2013)

t 4 ¼ Pðt 2 þ t 3 Þ and t 4 ¼ xð t 2 þ t 3 Þ From Fig. 4.12, it is shown that the maximum inventory level I2 is:

ð4:87Þ

4.4 Backordering

271

I 2 ¼ I 1 þ ðP  DÞt 4 ¼ ðPð1  xÞ  DÞt 3 þ ðP  DÞt 4 ¼ ðxt 2 þ t 3 ÞP  ðxðt 2 þ t 3 Þ þ t 3 ÞD

ð4:88Þ

Since z2 units is consumed at a rate of k during the interval t5, one has: t5 ¼

I 2 ðxt 2 þ t 3 ÞP  ðxðt 2 þ t 3 Þ þ t 3 ÞD ¼ D D

ð4:89Þ

The total inventory cost per cycle, TC, is as below: Productio n Cost

Rework Cost

Setup Cost

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ z}|{ TC ¼CPðt 2 þ t 3 Þ þ CR Pðt 2 þ t 3 Þx þ K þ Shortage Cost

Holding Cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{     zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ðt 1 þ t 2 ÞB b t ðI þ I 2 Þ t 5 I 2 t I Cb þ Cb B þ h 3 1 þ 4 1 þ 2 2 2 2

ð4:90Þ

From Eqs. (4.86), (4.87), and (4.89), the total cycle time can be rewritten as T¼

5 X i¼1

ti ¼

ðt 2 þ t 3 ÞP D

ð4:91Þ

For ease of notation and analysis, t (t ¼ t2 + t3) is set as decision variable. Moreover, the rework and production unit costs are the same (C ¼ CR). Then the total cost per unit time is expressed as (Wee et al. 2013): b t D TC KD C b t 22 ð1  xÞ C ¼ C ð1 þ xÞD þ þ ðð1  xÞP  DÞ þ b 2 ðð1  xÞP  DÞ T tP tP 2t   h 2 2 2 þ ððt  ð1  xÞt 2 Þ P þ 2t  t 2 D  1 þ x þ x t D  ð1  xÞt 22 DÞ 2t

TCUðt2 , tÞ ¼

ð4:92Þ Obviously, the objective is to minimize TCU(t2, t) subject to t  t2  0. The production time t is greater than zero. Lemma 4.1 For any given t > 0, the following cases that minimize TCU(t2|t) exist (Wee et al. 2013): bb D  Case I. For a given t  ChP , t 2 ¼ 0; Case

II.

b Cb D hPðC b þhÞð1xÞP ,

If

hP  ðCb þ hÞð1  xÞP  0, for a given

b Cb D hP

0, for a given t > hPðCbCþh Þð1xÞP , t 2 > t. Proof See Appendix of Wee et al. (2013). Lemma 4.1 shows that the optimal time to eliminate backorders, t 2, is dependent on the production time, t. The time t 2 ¼ 0 implies that shortage is not allowed. If the production time is predetermined, Lemma 4.1 shows the decision whether to schedbb D , the optimal policy is to ule shortage period. If the production time is greater than ChP allow a shortage period. Lemma 4.1 Case III shows if production time is greater than b Cb D hPðCb þhÞð1xÞP, the optimal time to eliminate backorders is greater than the production time, which is an infeasible solution for Model I (Fig. 4.14). Taking the first derivatives of TCU(t2, t) with respect to t2 and t, one has (Wee et al. 2013):

 b bD ðð1  xÞP  DÞ ððCb þ hÞð1  xÞt 2  ht ÞP þ C

∂ ð4:93Þ TCUðt 2 , t Þ ¼ Pt ∂t 2 ( ) b b Dt 2 ðð1  xÞP  DÞ þ t 2 hPðP  Dð1 þ x þ x2 ÞÞ 2KD  2C ∂ TCUðt 2 , t Þ ¼ ∂t

ðC b þ hÞð1  xÞt 22 Pðð1  xÞP  DÞ 2Pt 2 ð4:94Þ

Let ∂t∂2 TCUðt 2 , t Þ ¼ 0 and

∂ TCUðt 2 , t Þ ∂t

¼ 0, so:

b bD hPt  C ðCb þ hÞð1  xÞP vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi u u2KD þ t 2 ðð1  xÞP  DÞ ðC b þ hÞð1  xÞPt 2 þ 2C b bD t t 2 ðt Þ ¼

t ðt 2 Þ ¼

hPðP  Dð1 þ x þ x2 ÞÞ

ð4:95Þ

ð4:96Þ

  Let bt 2 , bt be the solution satisfying the first-order conditions for TCU(t2, t). Eq. (4.95) intoEq.  (4.96), they derived the production time and time Substituting  bt to eliminate backorders bt 2 as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 Dðð1  xÞP  DÞ 2K ðCb þ hÞð1  xÞP  C b bt ¼ 1 D P h hD þ ð1  xÞC b P  ð1  x3 ÞðCb þ hÞD bt 2 ¼

b bD hPbt  C ðC b þ hÞð1  xÞP

From the denominator of Eq. (4.97),

ð4:97Þ ð4:98Þ

4.4 Backordering

273

  hD þ ð1  xÞC b P  1  x3 ðC b þ hÞD   > ðC b þ hÞD  1  x3 ðC b þ hÞDðð1  xÞP > DÞ ¼ ðCb þ hÞDx3  0 ð4:99Þ Thus, bt and bt 2 in Eqs. (4.97) and (4.98) are real solutions. Based on the above analysis and results, Wee et al. (2013) proposed the following theorem: Theorem 4.1 b 2 DðP  ð1 þ x þ x2 ÞDÞ > 0, i.e., hPbt  C b D > 0, then TCU(t2, t) (i) If 2hKP  C b   b   is convex in t2 and t. If bt  bt 2 the optimal solution t 2 , t of TCU(t2, t) is bt 2 , bt . b D  0, then the optimal b 2 DðP  ð1 þ x þ x2 ÞDÞ  0, i.e., hPbt  C (ii) If 2hKP  C b

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b    2DK solution t 2 , t of TCU(t2, t) is 0, hPðPDð1þxþx2 ÞÞ . Proof See Appendix of Wee et al. (2013).   Substituting (t2, t) by t 2 , t  into Eq. (4.92), the optimal total cost per unit time can be derived, i.e., TCU t 2 , t  . From t* and t 2, the optimal batch size and optimal maximal backordering level are as follows (Wee et al. 2013): Q ¼ Pt  

B ¼ ðð1  xÞP 

ð4:100Þ DÞt 2

ð4:101Þ

b DðP  ð1 þ x þ x2 ÞDÞ > 0 in Theorem 4.1(i) is The constraint of 2hKP  C b equivalent to Eq. (4.101) in Cárdenas-Barrón (2009). Replacing t* and t 2 by bt and bt 2, respectively, it can be concluded that Q* and B* are the same as Eqs. (4.102) and (4.103) in Cárdenas-Barrón (2009). Similar to the result in Cárdenas-Barrón (2009), Theorem 4.1(ii) derives the optimal solution as in Jamal et al. (2004). Substituting t* ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK by hPðPDð1þxþx2 ÞÞ into Eq. (4.100), it is shown that the resulting Q* is identical to 2

Eq. (4.94) in Jamal et al. (2004).

4.4.2.2

Model II: EPQ Model for Production Time Less than the Time to Eliminate Backorders

In this subsection, Model II inventory model is developed. Figure 4.15. describes the inventory behavior for which the production run time is less than the time to eliminate backorders. The production starts at the beginning of interval s2. Here, the reworking period is s3 + s4. They used s2 and s3 to obtain the following expressions for the objective function. From Fig. 4.15, it can be seen that (Wee et al. 2013):

274

4 Rework

B1 ¼ ðP  DÞs3

ð4:102Þ

The maximum backordering level (B2) is: B2 ¼ B1 þ ðð1  xÞP  DÞs2 ¼ ðP  DÞs3 þ ðð1  xÞP  DÞs2

ð4:103Þ

Since B2 ¼ Ds1, s1 ¼

ðP  DÞs3 þ ðð1  xÞP  DÞs2 D

ð4:104Þ

The interval s3 + s4 is the time for reworking the defective units produced during the total production time (s2). Thus, one has P(s3 + s4) ¼ Ps2x and: s4 ¼ xs2  s3

ð4:105Þ

From Fig. 4.14, the maximum inventory level (I) is: I ¼ ðP  DÞs4 ¼ ðP  DÞðxs2  s3 Þ

ð4:106Þ

Since I units with a rate of k is consumed during interval s5, one has: s5 ¼

ðP  DÞðxs2  s3 Þ I ¼ D D

ð4:107Þ

The total inventory cost per cycle, TC0 , is: Holding Cost

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ z}|{ I ðs þ s5 Þ 0 þ TC ¼CPs2 þ CPs2 x þ K þ h 4 2 Setup Cost

Production Cost

Shortage Cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{

 B2 s1 ðB1 þ B2 Þs2 B1 s3 b b B2 Cb þ þ þC 2 2 2

ð4:108Þ

From Eqs. (4.104), (4.105), and (4.107), the total cycle time S is rewritten as: S¼

5 X i¼1

si ¼

s2 P D

ð4:109Þ

For simplicity of notation, they defined s2 and s (s ¼ s2 + s3) as decision variables of the inventory model. Then the total cost per unit time is simplified as:

4.4 Backordering

275

Fig. 4.15 Inventory of perfect-quality items for EPQ model II (Wee et al. 2013)

TCU0 ðs2 , sÞ ¼

TC0 KD h ðs  s2  s2 xÞ2 ðP  DÞ ¼ C ð1 þ xÞD þ þ s2 P 2s2 S h  C þ b ðs  s2 xÞ2 P  ð1  xÞs22 þ 2ð1  xÞðs  s2 Þs2 2s2 b D C þ ðs  s2 Þ2 ÞD þ b ½ðs  s2 xÞP  sD s2 P

ð4:110Þ

Evidently, the objective is to minimize TCU0 (s2, s) subject to s > s2 > 0. Taking the first derivatives of TCU0 (s2, s) with respect to s and s2, one has: h i b b D  Pððh þ ðCb þ hÞxÞs2  ðC b þ hÞsÞ ðP  DÞ C ∂ TCUðs2 , sÞ ¼ ð4:111Þ Ps2 ∂s

 b b DðP  DÞs  h s2  ð1 þ xÞ2 s2 PðP  DÞ 2KD  2 C 2 ∂ TUCðs2 , sÞ ¼ ∂s2 2Ps22  2  Cb xs2 ðxP  DÞ  s2 ðP  DÞ þ 2s22 ð4:112Þ ∂ Using ∂s TCU0 ðs2 , sÞ ¼ 0 and ∂s∂2 TCU0 ðs2 , sÞ ¼ 0, one obtains:

276

4 Rework

b bD ðh þ ðCb þ hÞxÞPs2  C ðC b þ hÞP vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffi u b bD u2KD þ sðP  DÞ sðC b þ hÞP þ 2C u  s2 ðt Þ ¼ t

P hðP  DÞð1 þ xÞ2 þ Cb xðxP  DÞ sðs2 Þ ¼

ð4:113Þ

ð4:114Þ

Let (bs2 , bs ) be the solution satisfying the first-order conditions for TCU0 (s2, s). Substituting Eq. (4.113) into Eq. (4.114), it can be derived that (Wee et al. 2013): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 DðP  DÞ 2K ðC b þ hÞP  C 1 D b bs2 ¼ P Cb hðP  ð1 þ ð1  xÞxÞDÞ  Cb xð1  xÞD bs ¼

b bD ðh þ ðCb þ hÞxÞPbs2  C ðC b þ hÞP

ð4:115Þ ð4:116Þ

Using the previous analysis and results, they proposed the following theorem for Model II: Theorem 4.2 If h(P  (1 + (1  x)x)D)  Cbx(1  x)D > 0 and   b 2 ðCb þhÞD hx2 ðPDÞþCb ð1xÞðð1xÞPÞD 2KPðCb þhÞðhxCb ð1xÞÞ2 > C b b b D > 0, then TCU0 (s2, s) is convex in s2 and s. The i.e., ðhx  Cb ð1  xÞÞPbs2  C optimal solution (s2, s) of TCU0 (s2 , s*) is (bs2 , bs) (Wee et al. 2013). Proof See Appendix of Wee et al. (2013). Substituting (s2, s) by (s2, s*) into Eq. (4.110), the optimal total cost per unit time can be obtained. The optimal lot size and the optimal backordering level can be expressed as: 

Q0 ¼ Ps2



 

B2 ¼ ðð1  xÞP  DÞs2 þ ðP  DÞ s  s2

ð4:117Þ ð4:118Þ

Example 4.7 Wee et al. (2013) to verify their results used the parameter values of Cárdenas-Barrón (2009) as D ¼ 300 unit per year, P ¼ 550 units per year, K ¼ $50 b b ¼ $1 per unit shortage, Cb ¼ $10 per unit per lot size, h ¼ $50 per unit per year, C shortage per year, and C ¼ $7 per unit. Firstly, they applied Model I to find the b 2 DðP  ð1 þ x þ x2 ÞDÞ > 0 for interval p 2 solutions. Since 2hKP  C b optimal300  0, 1  550 , they derived (bt 2 , bt ) from Eqs. (4.97) and (4.98). Table 4.8 illustrates the optimal policy of this example for Model I with different defective rates. When p > 0.20, t3 and I1 are negative values (infeasible optimal solutions for Model I).

4.4 Backordering

277

When p > 0.20, the optimal policy of the example is Lemma 4.1 Case III (Wee et al. 2013). To obtain feasible policies, Wee et al. (2013) applied Model II when Model I leads to infeasible policies. Table 4.9 illustrates the optimal policy for Model II with different defective rates. They had shown that when p ¼ 0.20, Model II results in an infeasible optimal solution. Table 4.9 indicates that the optimal solutions are derived from Model II when p > 0.20.

4.4.3

Random Defective Rate: Same Production and Rework Rates

Sarkar et al. (2014) developed the work of Cárdenas-Barrón (2009) by considering that proportion of defective products in each cycle is a stochastic variable and follows a probability distribution (uniform, triangular, and beta). Other assumptions are as same as Cárdenas-Barrón’s (2009) work. Basically, three different inventory models are developed for three different distribution density functions such as uniform, triangular, and beta. Case A The proportion of defective products follows a uniform distribution. In order to take the randomness of proportion of defective products into account, the expected value of R is used in the development and analysis of inventory model. The inventory behavior through time is represented in Fig. 4.16. According to Fig. 4.16, the maximum inventory Imax is simply computed as the sum of I1 + I2. From to triangle (146), it is easy to see that (Sarkar et al. 2014): tan θ1 ¼ Pð1  E½xÞ  D ¼

I1 þ B T1 þ T2

ð4:119Þ

In this case, it is assumed that the x follows a uniform distribution with range [a, b], 0 < a < b < 1. For a uniform distribution, it is well-known that the expected value for x is given as E[x] ¼ (a + b)/2. Furthermore, the production time of producing Q units is TP ¼ T1 + T2. Therefore, T1 + T2 must be equal to Q/P. Substituting the expected value E[x] and T1 + T2, one obtains:

aþb I þB D¼ 1 P 1 2 T1 þ T2

ð4:120Þ

or 



2ab D aþb D  B¼Q 1  B I1 ¼ Q 2 P 2 P According to triangle (689), it is easy to see that:

ð4:121Þ

p TCU* Q* I 1 I 2 B* t 1 t 2 t 3 t 4 t 5

0.2 2893.37 100.48 0.53 9.66 25.05 0.0835 0.17894 0.00375 0.03654 0.0322

0.25 2983.12 102.06 0.96 10.64 21.83 0.07277 0.19406 0.00850 0.04639 0.03547

0.30 3078.1 101.29 1.88 11.93 17.53 0.05844 0.20626 0.02209 0.05525 0.03978

0.35 3180.86 97.36 2.07 13.42 12.25 0.04082 0.21297 0.03594 0.06196 0.04474

Table 4.8 The results of Example 1 for Model I with different p values (Wee et al. 2013) 0.40 3293.64 90.25 1.46 14.95 6.38 0.02127 0.21275 0.04866 0.06563 0.04983

0.45 3417.97 80.85 0.15 16.39 0.52 0.00172 0.2062 0.05920 0.06615 0.05463

1–300/550 3429.87 79.93 0 16.51 0 0 0.205361 0.06003 0.06606 0.05505

278 4 Rework

p TCU* Q* I* B1 B2 s1 s2 s3 s4 s5

0.2 2893.26 100.29 9.87 0.75 24.78 0.08258 0.18234 0.00301 0.03948 0.0329

0.25 2982.45 102.97 10.07 1.63 22.69 0.07563 0.18722 0.00651 0.04029 0.03358

0.30 3072.88 105.33 10.25 4.11 20.39 0.06797 0.19152 0.01644 0.04101 0.03418

0.35 3164.63 107.29 10.4 6.67 17.88 0.05962 0.19507 0.02667 0.0416 0.03467

Table 4.9 The results of Example 1 for Model II with different p values (Wee et al. 2013) 0.40 3257.76 108.76 10.51 9.26 15.19 0.05065 0.19774 0.03705 0.04205 0.03504

0.45 3352.35 109.67 10.58 11.85 12.35 0.04116 0.19939 0.0474 0.04232 0.03527

1–300/550 3361.02 109.72 10.58 12.08 12.08 0.04028 0.19949 0.04834 0.04234 0.03528

4.4 Backordering 279

280

4 Rework

tan θ2 ¼ P  D ¼ T3 ¼

I2 T3

E ½xQ Q ða þ bÞ ¼ P 2 P

ð4:122Þ ð4:123Þ

where T3 is the production time of producing the defective products. Therefore, T3 must be equal to E[x]Q/P. Thus, I 2 ¼ T 3 ðP  DÞ ¼

Qða þ bÞðP  DÞ Qða þ bÞð1  D=PÞ ¼ 2P 2

ð4:124Þ

Now, the maximum inventory Imax can be obtained as summing I1 and I2; hence (Sarkar et al. 2014),

Qða þ bÞð1  D=PÞ aþb D  Bþ I max ¼ I 1 þ I 2 ¼ Q 1  2 2 P 

 D aþb ¼Q 1 1þ B P 2

ð4:125Þ

From Fig. 4.16, T is the sum of T1, T2, T3, T4, and T5. It is well-known that T is the time between runs. And T2 and T4 are:     D=P  B Q½ð1  E½xÞ  D=P  B Q 1  aþb 2     T2 ¼ ¼ ½Pð1  E ½xÞ  D P 1  aþb D 2      D Q 1  ð1 þ E½xÞ DP  B Q 1  1 þ aþb 2 P B T4 ¼ ¼ D D

ð4:126Þ ð4:127Þ

where T2 is the time needed to build up I1 units in inventory and T4 is the time needed for consumption at hand maximum inventory level Imax, then (Sarkar et al. 2014). As it was stated before, T3 is equal to E[x]Q/P ¼ ((a + b)/2)Q/P. In order to calculate the average inventory, the area of above horizontal line (time) should be calculated:    D 2 P B Q 1  aþb T 2I 1 2     ¼ 2 D 2 P 1  aþb 2 aþb       aþb D T 3I 1 2 Q Q 1 1þ 2 P B ¼ 2P 2     D 2 Q 1  1 þ aþb T 4 I max 2 P B ¼ 2 2D

ð4:128Þ ð4:129Þ ð4:130Þ

Finally, the cyclic inventory average I can be calculated as below (Sarkar et al. 2014):

4.4 Backordering

281

I(t) Imax

9 P-D

I2

θ2

6

8 D

P(1_E [x])_ D

I1

θ1

3

o

1

B

θ1

7

5

D

B

4

2

10

11

12

time T1

T2

T3

T4

T5

TP T

Fig. 4.16 Inventory behavior for the EPQ with rework at the same cycle and planned backorders (Sarkar et al. 2014)

I¼

1 T



 

 3 aþb aþb aþb D   aþb D 2 Q Q 1  1þ B 7 6 Q 1 2 4 4 P P  B 7 6  2  þ 7 6 P 7 6 2 P 1aþb D 6 7 2 7 6 7 6      2 5 4 aþb D Q 1 1þ 2 P B þ 2D ð4:131Þ 2

After some simplifications,

282

4 Rework

"



2 !  2  1 aþb Q2 D2 aþb aþb 2

 Q þB þ 2 þ I ¼  1þ 1 2 2 2 aþb D P 2Q 1   2 P !

3

# Q2 D aþb D aþb þ þ 1  2 þ 2BQ P 2 P 2 ð4:132Þ In order to express the above mathematical equation in a more compact expression, the following symbols were defined: aþb 2 aþb D E ¼1  2 P "

2 # 2 aþb aþb D þ I ¼ 1þ 2 2 P2 " #

3

 aþb D 2 O¼ 2 P γ ¼1

U¼

D aþb þ  1 ¼ E P 2

ð4:133Þ ð4:134Þ ð4:135Þ

ð4:136Þ ð4:137Þ

Then, I¼

 1  2 Q ðγ þ I þ OÞ þ B2 γ  2BQE 2QE

 Q γþIþO B2 γ þ B I¼ 2 E 2QE

ð4:138Þ ð4:139Þ

With further rearrangement: " Q 1 I¼ 2

 

2 ! # B2 1  aþb aþb aþb D 2 B 1þ þ þ  D 2 2 P 2Q 1  aþb 2 P

ð4:140Þ

If L is defined as follows, L¼1



2 !  aþb aþb D þ 1þ 2 2 P

finally, the inventory average is given as follows:

ð4:141Þ

4.4 Backordering

283

I¼

Q B2 γ Lþ B 2 2QE

ð4:142Þ

With regard to inventory average of backorders B, it can be determined by the sum of the area of triangles: under time line and divided by T. T1 and T5 can be calculated as: T1 ¼

B B   ¼  ½Pð1  E ðxÞÞ  D P 1  aþb D 2

ð4:143Þ

B D

ð4:144Þ

T5 ¼

where T1 is the time needed to satisfy the backorder level once production process is started again and T5 is the time needed to build up the backorder level of B units. Thus, for average shortage (Sarkar et al. 2014): T 1B B2   ¼   2 2 P 1  aþb D 2

ð4:145Þ

T 5 B B2 ¼ 2 2D

ð4:146Þ

So cyclic backordering can be determined by dividing the average of shortage by T, as below: " # 1 B2 B2    þ B¼ T 2 P 1  aþb 2D D 2   B2 1  aþb 2   B¼  DP 2Q 1  aþb 2 B¼

B2 γ 2QE

ð4:147Þ ð4:148Þ ð4:149Þ

Therefore, the total cost of the system is (Sarkar et al. 2014): Shortage Cost

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ Production Cost Holding Cost z}|{ zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ z}|{ b b BD C KD TC ¼ þ C b B þ CDð1 þ E½xÞ þ hG þ Q Q Setup Cost

ð4:150Þ

After, substituting the value of above expressions in one obtains (Sarkar et al. 2014):

284

4 Rework

TCðQ, BÞ ¼

b BD Cb B2 γ C KD hQL hB2 γ þ þ  hB þ b þ þ CDð2  γ Þ Q 2 2QE Q 2QE

ð4:151Þ

The cost equation consists of two decision variables as Q and B. In order to derive the optimal values of both decision variables, the second-order Hessian matrix should be positive definite. So one should perform: Necessary conditions ∂TCðQ, BÞ ¼0 ∂Q

and

∂TCðQ, BÞ ¼0 ∂B

ð4:152Þ

Sufficient conditions 2

∂ TCðQ, BÞ >0 ∂Q2

2

and

2

∂ TCðQ, BÞ ∂ TCðQ, BÞ  ∂B2 ∂Q2

>0

!2 2 ∂ TCðQ, BÞ ∂Q∂B ð4:153Þ

For minimization of the cost equation, the condition is  2 2 4 4 b ð2KDðCb þ hÞγ Þ=Q E  Cb D =Q > 0 , i.e., if the expression

2    b D2 =Q4 is greater than 0, the sufficient condition ð2KDðCb þ hÞγ Þ=Q4 E  C b 

of the optimality criteria is satisfied. Therefore, it can be concluded that the cost   ð2KDðC b þ hÞγ Þ=Q4 E  equation is convex when the expression

2  b D2 =Q4 > 0. The optimal values are as follows: C b ∂TC ¼0 ∂Q sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 D2 2KDγ ðC b þ hÞ  EC b Q¼ h½γLðC b þ hÞ  Eh ∂TC ¼0 ∂B

ð4:154Þ

ð4:155Þ ð4:156Þ

or

B¼

 b bD E hQ  C ðC b þ hÞγ

ð4:157Þ

However, the solution (Q, B) does not necessarily exist although TC(Q, B) is convex as it was shown in Chung (2011). He proved that TC(Q, B) exists if and only

4.4 Backordering

285

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 D2L and if 2KDh < C b 2 D2L, then B* ¼ 0 and Q ¼ 2KD=hL . if 2KDh  C b b Substituting the above optimal values in the cost equation and after some simplification, the minimum cost is obtained as follows:

TC ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

 b 2 D2 ðh½γLðC b þ hÞ  EhÞ þ DE C bbh 2KDγ ðC b π þ hÞ  E C b γ ð C b þ hÞ þ CDð2  γ Þ

ð4:158Þ

Case B The proportion of defective products follows a triangular distribution. In this case, it is assumed that s follows a triangular distribution with parameters [a, b, c]; 0 < a < b < c < 1 where parameters a and c are the inferior and superior limits, respectively, and b is the mode of the triangular distribution. For a triangular distribution, it is well-known that the expected value for x is given as E [x] ¼ (a + b + c)/3. For this case, from Fig. 4.16, the maximum inventory Imax is also computed as the sum of I1 + I2. So (Sarkar et al. 2014): tan θ1 ¼ Pð1  E½xÞ  D ¼

I1 þ B T1 þ T2

ð4:159Þ

Similar to previous case, T1 + T2 ¼ Q/P. Substituting the expected value E[x] and T1 + T2, one obtains:

aþbþc I þB D¼ 1 P 1 3 TP

ð4:160Þ

  aþbþc D  B I1 ¼ Q 1  3 P

ð4:161Þ

or

Also similar to previous case: B Pð1  E½xÞ  D     D=P  B Q½ð1  E ½xÞ  D=P  B Q 1  aþbþc 3     T2 ¼ ¼ ½Pð1  E ½xÞ  D D P 1  aþbþc 3 T1 ¼

T3 ¼

E ½xQ Qða þ b þ cÞ ¼ P 3P

ð4:162Þ ð4:163Þ ð4:164Þ

286

4 Rework

    B Q½1  ð1 þ E½xÞD=P  B Q 1  DP 1 þ aþbþc 3 ¼ T4 ¼ D D B T5 ¼ D

ð4:165Þ ð4:166Þ

Since, T3 is equal to E[x]Q/P. Hence,

h i aþbþc D 1 I 2 ¼ T 3 ðP  D Þ ¼ Q 3 P

ð4:167Þ

Therefore, the maximum inventory Imax can be found as: 

I max



 D aþbþc 1þ B ¼ I1 þ I2 ¼ Q 1  P 3

ð4:168Þ

In order to calculate the average inventory, Sarkar et al. (2014) presented the following equations:    2 Q 1  aþbþc  DP  B 1 3     T I ¼ 2 2 1 D 2 P 1  aþbþc 3     aþbþc aþbþc D I1T 3 Q 3 Q 1  3  P  B ¼ 2P 2    aþbþc  aþbþc D B I max T 3 Q 3 Q 1  P 1 þ 3 ¼ 2P 2     2 Q 1  DP 1 þ aþbþc B T 4 I max 3 ¼ 2 2D

ð4:169Þ ð4:170Þ ð4:171Þ ð4:172Þ

Finally, the inventory average ITri can be calculated as (Sarkar et al. 2014): I Tri ¼

D Q

3 

   aþbþc D 2 Q aþbþc Q 1 aþbþc  D B 7 6 Q 1 B 3 3 P 7 6  3 P þ 7 6 2P aþbþc 7 6 D 7 6 2 P 1 6 3 7 7 6  

  7 6 7 6 Q aþbþc Q 1 D 1þ aþbþc B       25 4 D aþbþc B Q 1 P 1þ 3 3 P 3 þ þ 2P 2D ð4:173Þ 2

4.4 Backordering

287

In order to express the above mathematical equation in a more compact expression, the following symbols were defined: aþbþc 3 aþbþc D ETri ¼ 1   3 P

2 ! 2 aþbþc aþbþc D ¼ 1þ þ 3 3 P !

3

 aþbþc D OTri ¼ 2 3 P K Tri ¼ 1 

I Tri

U Tri ¼

D aþbþc þ  1 ¼ ETri P 3

ð4:174Þ ð4:175Þ ð4:176Þ

ð4:177Þ ð4:178Þ

Then, I Tri

  Q K Tri þ I Tri þ OTri B2 K Tri ¼ B þ 2 E Tri 2QE Tri

ð4:179Þ

Assume LTri is defined as follows: LTri ¼ 1 



2 !  aþbþc aþbþc D þ 1þ 3 3 P

ð4:180Þ

Also to calculate the average shortage using Fig. 4.16 (Sarkar et al. 2014), T 1B B2   ¼   aþbþc 2 2 P 1 3 D

ð4:181Þ

T 5 B B2 ¼ 2 2D

ð4:182Þ

Thus, the inventory average of backorders B divided by T results in cyclic average backordering as below: " # 1 B2 B2 B2 K Tri     B¼ ¼ þ T 2 P 1  aþbþc 2D 2QE Tri D 3 Therefore, the total cost of the system is as follows (Sarkar et al. 2014):

ð4:183Þ

288

4 Rework Shortage Cost

Holding Cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ z}|{ b BD C b B2 K Tri hQLTri hB2 K Tri C AD þ TCðQ, BÞ ¼ þ  hB þ b þ þ Q Q 2 2QE Tri 2QE Tri Setup Cost

Production Cost

zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ CDð2  K Tri Þ

ð4:184Þ

In order to derive the optimal values of decision variables, similar to previous case, Necessary conditions ∂TCðQ, BÞ ¼0 ∂Q

and

∂TCðQ, BÞ ¼0 ∂B

ð4:185Þ

Sufficient conditions 2

∂ TCðQ, BÞ >0 ∂Q2

2

and

2

∂ TCðQ, BÞ ∂ TCðQ, BÞ  ∂B2 ∂Q2

!2 2 ∂ TCðQ, BÞ ∂Q∂B

>0

ð4:186Þ

For minimization of the cost function, the condition presented in Eq. 4.186, should be satisfied. Therefore, it can be concluded that the cost equation is convex    2 2 4 4 b when the expression ð2KDðC b þ hÞK Tri Þ=Q E Tri  Cb D =Q > 0 . The optimal values are as follows (Sarkar et al. 2014): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 D2 2KDK Tri ðCb þ hÞ  E Tri C b Q¼ h½K Tri LTri ðCb þ hÞ  E Tri h

 b b D E Tri hQ  C B¼ ðC b þ hÞK Tri

ð4:187Þ

ð4:188Þ

b 2 D2LTri. On the other hand, if It is to be noted that TC(Q, B) exists if 2KDh  C b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p b 2 D2LTri, then B* ¼ 0 and Q ¼ 2KDh < C 2KD=hL Tri . Substituting the above b optimal values in the cost equation and after some simplification, the minimum cost is obtained as follows (Sarkar et al. 2014):

4.4 Backordering

TC ¼

289

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

 2 2 b b bh 2KDK Tri ðC b þ hÞ  ETri C b D ðh½K Tri LTri ðC b þ hÞ  ETri hÞ þ DE Tri C K Tri ðCb þ hÞ þ CDð2  K Tri Þ ð4:189Þ

Case C The proportion of defective products follows a beta distribution. In this case, it is assumed that x follows a beta distribution with range [α, β]; 0 < α < β < 1 where parameters α and β are the inferior and superior limits, respectively, of the beta distribution. For beta distribution, it is well-known that the expected value for x is given as E[x] ¼ α/(α + β). Similar as previous cases, they followed the same procedure to obtain the decision variable Q and B. From Fig. 4.16, the maximum inventory Imax is calculated as the sum of I1 + I2: Pð1  E ½xÞ  D ¼

I1 þ B T1 þ T2

ð4:190Þ

As the production time of producing Q units is TP ¼ T1 + T2. Hence, T1 + T2 ¼ Q/ P. Substituting the expected value E[x] and T1 + T2, one obtains: P 1

α I þB D¼ 1 αþβ TP

ð4:191Þ

or  I1 ¼ Q 1 

 α D  B αþβ P

ð4:192Þ

From Fig. 4.16, one has: B Pð1  E½xÞ  D h i α Q 1  αþβ  DP  B

 T2 ¼ α D P 1  αþβ T1 ¼

T3 ¼

E ½xQ Qα ¼ P Pðα þ βÞ

ð4:193Þ

ð4:194Þ

ð4:195Þ

290

4 Rework

T4 ¼

h

 i α Q 1  DP 1 þ αþβ B D B T5 ¼ D

ð4:196Þ ð4:197Þ

Using the above equations,

h i α D I 2 ¼ T 3 ðP  D Þ ¼ Q 1 αþβ P 

 D α I max ¼ I 1 þ I 2 þ Q 1  1þ B P αþβ

ð4:198Þ ð4:199Þ

In order to calculate the average inventory, h

 i2 α D Q 1    B αþβ P 1 h

 i T I ¼ 2 2 1 α 2 P 1  αþβ  D h

 i α α D T 3 I 1 Q αþβ Q 1  αþβ  P  B ¼ 2 2P

  h

i α α D Q Q 1  þ 1  B αþβ αþβ P I max T 3 ¼ 2P 2 h n

o i2 D α Q 1  1 þ  B P αþβ T 4 I max ¼ 2 2D

ð4:200Þ

ð4:201Þ ð4:202Þ

ð4:203Þ

Finally, as before, the inventory average Ibeta can be calculated summing the area presented in Eqs. (4.200)–(4.203) and divided by T. Hence, one obtains Ibeta as (Sarkar et al. 2014): I beta ¼

D Q

3 

 h

 i2 α α D α D Q Q 1    B 7 6 Q 1  αþβ  P  B αþβ αþβ P 7 6 

þ 7 6 2P α 7 6 D 7 6 2 P 1 6 αþβ 7 7 6  

  6 h n

o i2 7 7 6 Q α Q 1D 1þ α α B 4 Q 1  DP 1 þ αþβ B 5 αþβ P αþβ þ þ 2P 2D ð4:204Þ 2

4.4 Backordering

291

In order to express the above mathematical equation in a more compact expression, the following were defined symbols: K beta ¼ 1 

α αþβ

ð4:205Þ

α D  αþβ P

2 ! 2 α α D I beta ¼ 1 þ þ αþβ αþβ P !

3

 α D Obeta ¼ 2 αþβ P

ð4:206Þ

E beta ¼ 1 

U beta ¼

ð4:207Þ

ð4:208Þ

D α þ  1 ¼ Ebeta P αþβ

ð4:209Þ

Then, I beta

  Q K beta þ I beta þ Obeta B2 K beta ¼ B þ 2 Ebeta 2QE beta

ð4:210Þ

Simplifying, one obtains: " I beta ¼

Q 1 2



2 ! # 2 α B 1  αþβ α α D  1þ þ þ

αþβ αþβ P α 2Q 1  D

αþβ

B

P

ð4:211Þ

If Lbeta is defined as follows, Lbeta ¼ 1 



2 !  α α D þ 1þ αþβ αþβ P

ð4:212Þ

finally, the inventory average is given by: I beta ¼

Q B2 Abeta Lbeta þ B 2 2QE beta

In order to calculate the average shortage, one has:

ð4:213Þ

292

4 Rework

T 1B B2  i ¼ h

2 α 2 P 1  αþβ D

ð4:214Þ

T 5 B B2 ¼ 2 2D

ð4:215Þ

Thus, the inventory average of backorders B can be calculated adding the area presented in Eqs. (4.215) and (4.216), and divided by T, one obtains (Sarkar et al. 2014): 2 B¼

3 2

2

14 B B B2 K beta h

 iþ 5¼ T 2 P 1 α D 2D 2QE beta αþβ

ð4:216Þ

Therefore, the total cost of the system is (Sarkar et al. 2014): Shortage Cost

Holding Cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ z}|{ b BD Cb B2 K beta C KD hQLbeta hB2 K beta þ þ þ  hB þ b þ TCðQ, BÞ ¼ Q Q 2 2QE beta 2QE beta Setup Cost

Production Cost

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ CDð2  K beta Þ

ð4:217Þ

In order to solve the problem, the necessary and sufficient conditions should be evaluated: Necessary conditions ∂TCðQ, BÞ ¼0 ∂Q

and

∂TCðQ, BÞ ¼0 ∂B

ð4:218Þ

Sufficient conditions 2

∂ TCðQ, BÞ >0 ∂Q2 >0 

2

and

2

∂ TCðQ, BÞ ∂ TCðQ, BÞ  ∂B2 ∂Q2

!2 2 ∂ TCðQ, BÞ ∂Q∂B ð4:219Þ

For minimization of the cost equation, the condition is  2 2 4 b D =Q > 0 . That is, if this expression is ð2KDðCb þ hÞK beta Þ=Q4 Ebeta  C b

greater than 0, the sufficient condition of the optimality criteria is satisfied. It can be concluded that the cost equation is convex when the expression is greater than 0. The optimal values are as follows (Sarkar et al. 2014):

4.4 Backordering

293

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2 D2 2KDK beta ðCb þ hÞ  E beta C b Q¼ h½K beta Lbeta ðC b þ hÞ  Ebeta h

 b b D Ebeta hQ  C B¼ ðCb þ hÞK beta

ð4:220Þ

ð4:221Þ

Example 4.8 Sarkar et al. (2014) considered numerical experiments based on Cárdenas-Barrón’s (2009) data as D ¼ 300 units/year, a ¼ 0.03, b ¼ 0.07, b b ¼ $1/unit short, P ¼ 550 units/year, Cb ¼ $10/unit/year, h ¼ $50/unit/year, C K ¼ $50/lot size, and C ¼ $7/unit. Then, the optimal solution is TC ¼ $2634.41/ year, Q ¼ 90.92 units, B ¼ 30.13 units. Example 4.9 The values of the following parameters are to be taken in appropriate units: D ¼ 300 units/year, a ¼ 0.03, b ¼ 0.04, c ¼ 0.07, P ¼ 550 units/year, Cb ¼ $10/unit/year, h ¼ $50/unit/year, C 0b ¼ $1/unit short, K ¼ $50/lot size, and C ¼ $7/ unit. Then, the optimal solution is TC ¼ $2628.63/year, Q ¼ 90.71 units, B ¼ 30.21 units (Sarkar et al. 2014). Example 4.10 The values of the following parameters are to be taken in appropriate units: D ¼ 300 units/year, α ¼ 0.03, β ¼ 0.07, P ¼ 550 units/year, Cb ¼ $10/unit/ b b ¼ $1/unit short, K ¼ $50/lot size, and C ¼ $7/unit. Then, year, h ¼ $50/unit/year, C the optimal solution is TC ¼ $3078.10/year, Q ¼ 101.29 units, B ¼ 17.53 units. Sarkar et al. (2014) compared numerical outcomes of the three models in Table 4.10.

4.4.4

Rework Process and Scraps

Sivashankari and Panayappan (2014) developed different modeling of an EPQ model with shortage, defective items, and reworking. Figure 4.17 shows the inventory behavior for the production inventory model during one cycle with planned backorders in which rework is not presented.

4.4.4.1

Without Rework

Times t1 and t4 are needed to build up B units of items, so: t1 ¼

B ; PDd

t4 ¼

B D

Time t2 is needed to build up Q1 units of items; therefore:

ð4:222Þ

294

4 Rework

Table 4.10 Optimum cost, order quantity and backorder for three distribution functions (Sarkar et al. 2014) Example no. 1 2 3

Total cost (per year) 2634.41 2628.63 3078.1

Order quantity (units) 90.92 90.71 101.29

t2 ¼

Backorder quantity (units) 30.13 30.21 17.53

I Max PDd

ð4:223Þ

The production phase occurs during time: t1 þ t2 ¼

Q Q B ! t2 ¼  P P PDd

ð4:224Þ

Time t3 is needed to consume all units IMax at demand rate D: t3 ¼

I Max D

ð4:225Þ

but, Q1 ¼ ðP  D  dÞt 2  B ¼ ðP  D  dÞ

 Q B P

ð4:226Þ

The production cycle time T (from Eqs. 4.222–4.226) is given by: T¼

4 X i¼1

ti ¼

  Q PDd Q 1þ ¼ ð 1  xÞ P D D

ð4:227Þ

Backordering occurs during time t1 + t4. The average shortage during t1 + t4 is: h i h i 1 1 1 B B B PB2 Bt 1 þ Bt 4 ¼ þ ¼ T 2 2 2T D P  D  d 2QðP  D  dÞ

ð4:228Þ

Positive inventory occurs during the time. Therefore, the average inventory during time is:

4.4 Backordering

295

h i h i ðP  d ÞI 2Max 1 1 1 I I I Max Qt 2 þ I Max t 3 ¼ Max Max þ ¼ T 2 2 2T D PDd 2QðP  D  d Þ  2 QðP  D  dÞ P B ¼ P 2QðP  D  dÞ ¼

ð4:229Þ

QðP  D  dÞ PB2 þ B 2P 2QðP  D  dÞ

The total cost function TC(Q, B) ¼ setup cost + production cost + holding cost + shortage cost is (Sivashankari and Panayappan 2014): Setup Cost

zfflfflfflfflffl}|fflfflfflfflffl{ DK TCðQ, BÞ ¼ þ Qð1  xÞ

Production Cost

zffl}|ffl{ CD 1x

þ

Holding Cost

Shortage Cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ hQðP  D  d Þ hPB2 C b PB2 þ  hB þ 2P 2QðP  D  d Þ 2QðP  D  d Þ

ð4:230Þ

The necessary conditions for having a minimum are: ∂TCðQ, BÞ ¼0 ∂Q

and

∂TCðQ, BÞ ¼0 ∂B

ð4:231Þ

Partially differentiating Eq. (4.230) with respect to Q gives:

Q – Inventory Level

P _ D _ d

N

O

A B

_D

I Max

E

B t1

M

t2

C

t3

D

t4

B

Time (T)

P

Fig. 4.17 Production inventory model in one cycle with defective item (Sivashankari and Panayappan 2014)

296

4 Rework

hð P  D  d Þ ∂TC DK PB2 h  2 þ ¼ 2 2P ∂Q Q ð 1  xÞ 2Q ðP  D  d Þ 

C b PB2 2Q ðP  D  dÞ 2

¼0

ð4:232Þ

So this yields to: 2PDðP  D  d ÞK þ PB2 ðh þ C b Þð1  xÞ hð P  D  d Þ 2 ð 1  x Þ P2 ðh þ Cb ÞB2 2PDK þ PB2 þ ¼ h ð P  D  d Þ 2 ð 1  x Þ hð P  D  d Þ 2

Q2 ¼

ð4:233Þ

Partially differentiating Eq. (4.230) with respect to B gives: ∂TC PBh Cb PB ¼ hþ ¼0 QðP  D  dÞ QðP  D  dÞ ∂B

ð4:234Þ

Therefore, B¼

hQðP  D  dÞ P ðh þ C b Þ

ð4:235Þ

Substituting Eq. (4.235) into (4.233) and simplifying yields to: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2PDK ðh þ CS Þ Q¼ hC b ðP  D  dÞð1  xÞ

ð4:236Þ

Sivashankari and Panayappan (2014) proved that the derived optimal solutions are global one and its related Hessian matrix is positive definite. Example 4.11 Sivashankari and Panayappan (2014) presented an example with parameter values P ¼ 5000 units, D ¼ 4500 units, K ¼ 100, h ¼ 10, C ¼ 100, Cb ¼ h ¼ 10, 100, and x ¼ 0.01. The optimal solutions are Q* ¼ 1421.34, B* ¼ 63.96, and total cost ¼ 455,185.06.

4.4.4.2

Rework Case and Quality Improvement

Sivashankari and Panayappan (2014) and Krishnamoorthi and Panayappan (2012) developed case Sect. 4.4.4.1 under rework policy as presented in Fig. 4.18.

4.4 Backordering

297

H1 represents the quantity of good items remaining after consumption at the end of time t1: t1 ¼

Q P

I 1 ¼ ðP  D  λÞt 1 ¼ ðP  D  d Þ

ð4:237Þ

 Q B P

ð4:238Þ

Time t1 needed to build up Q1 units of item; therefore: t1 ¼

ðP  D  dÞðQ=PÞ  B Q I1 B ¼  ¼ PDd P PDd PDd

ð4:239Þ

Time t2 needed to rework the defective items: t2 ¼

MS OJ  JK xQ  xθQ xQð1  θÞ ¼ ¼ ¼ P P P P

ð4:240Þ

H2 represents the quantity of items that should remain after consumption: I 2 ¼I 1 þ NS ¼ I 1 þ ðP  DÞt 2 ¼ ðP  D  dÞðQ=PÞ  B þ

 ðP  DÞxQð1  θÞ Q ¼ðP  DÞ  Qx  B þ P P

ðP  DÞxQð1  θÞ P

ð4:241Þ Time t3 needed to build up H2 units of items; therefore: t3 ¼

 

 ðP  DÞxQð1  θÞ 1 Q ðP  D Þ  Qx  B þ P D P

ð4:242Þ

Shortages time: B D B t5 ¼ PDd t4 ¼

So the period length is:

ð4:243Þ

298

4 Rework

  ðP  DÞxð1  θÞ Q D xDð1  θÞ ðP  DÞ þ xþ þ T ¼t 1 þ t 2 þ t 3 þ t 4 þ t 5 ¼ P P P D P Q Q ¼ ½1  x  xð1  θÞ ¼ ð1  xθÞ D D ð4:244Þ The average inventory is calculated as: I¼

h i 1 1 1 1 I 1 t 1 þ I 1 t 2 þ ðI 2  I 1 Þt 2 þ t 3 I 2 T 2 2 2

ð4:245Þ

And after some simplifications, I¼

h

i Q Pð1  xθÞ2  D 1 þ x  2xθ þ x2 ð1  θÞ2 2Pð1  xθÞ DB  ½1 þ 2xð1  θÞ 2Pð1  xθÞ

ð4:246Þ

The average shortage is as follows: B¼

h i B 2 ðP  D Þ B2 Pð1  xÞ 1 1 1 Bt 4 þ Bt 5 ¼ ¼ T 2 2 2T ðP  D  d Þ 2QðP  D  dÞð1  xθÞ

ð4:247Þ

Fig. 4.18 On-hand inventory of EPQ model with the rework and shortages permitted (Krishnamoorthi and Panayappan 2012)

4.4 Backordering

299

The total cost of the system TC(Q, B) is the accumulation of the setup cost, production cost, holding cost, shortage cost, reworking cost, rejection cost, and quality cost for defective items (Krishnamoorthi and Panayappan 2012). Quality Improvement ShortageCost Cost Cq Dx C Dxð1  θÞ z}|{ CJ Dxθ Cb PB2 ð1  xÞ DK CD þ z}|{ þ z}|{ R þ þ z}|{ þ z}|{ 1  xθ 1  xθ 1  xθ 1  xθ Qð1  xθÞ 2QðP  D  dÞð1  xθÞ



 hQ hDB þ Pð1  xθÞ2  D 1 þ x  2xθ þ x2 ð1  θÞ2  ð1 þ 2xð1  θÞÞ 2Pð1  xθÞ 2Pð1  xθÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} SetupCost

ReworkCost

ProductionCost

RejectionCost

TC ¼ z}|{

HoldingCost

ð4:248Þ Partially derivative TC(Q, B) with respect to Q and B (Krishnamoorthi and Panayappan 2012),



 2 3 h Pð1  xθÞ2  D 1 þ x  2xθ þ x2 ð1  θÞ2 Cb PB2 ð1  xÞ ∂TC 4 DK 5¼0 þ  2 ¼ 2Pð1  xθÞ ∂Q 2Q ðP  D  d Þð1  xθÞ Q2 ð1  xθÞ

ð4:249Þ Let:

 E ¼ Pð1  xθÞ2  D 1 þ x  2xθ þ x2 ð1  θÞ2

ð4:250Þ

Then, Q2 ¼

2PDðP  D  dÞK þ P2 B2 C b ð1  xÞ hð P  D  d Þ ð E Þ

ð4:251Þ

And: 2BPC b ð1  xÞ ∂TC Dhð1  2xð1  θÞÞ ¼ þ ¼0 2Pð1  θxÞ 2QðP  D  dÞð1  θxÞ ∂B

ð4:252Þ

So, B¼

DQhðP  D  d Þð1 þ 2x  2xθÞ 2P2 C b ð1  xÞ

Substitute the value of (4.253) in (4.251), after simplification:

ð4:253Þ

300

4 Rework

  D2 hðP  D  d Þð1 þ 2x  2xθÞ2 2PDK Q2 1  ¼ hð E Þ 4P2 ð1  xÞCb ðEÞ

ð4:254Þ

Therefore, the optimum lot size is: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 8P3 Dð1  xÞC b K h

i Q¼u u 2 2 2 u 4P ð1  xÞCb h Pð1  xθÞ  D 1 þ x  2xθ þ x2 ð1  θÞ t D2 h2 ðP  D  dÞð1 þ 2x  2xθÞ2

ð4:255Þ

Example 4.12 Krishnamoorthi and Panayappan (2012) to support their proposed model presented an example with parameter values. Let P ¼ 5000 units; D ¼ 4500 units; K ¼ 100; h ¼ 10; CR ¼ 5; CJ ¼ 1; Cq ¼ 5; C ¼ 100; x ¼ 0.01 and θ ¼ 0.1; and Cb ¼ 10 (Krishnamoorthi and Panayappan 2012). The optimal solutions are Q* ¼ 1120.76; B* ¼ 46.67; and total cost ¼ 451,686.71.

4.4.5

Rework and Preventive Maintenance

Chen et al. (2010) considered a single-item production process. At the beginning of a production cycle, the production system is assumed to be in an in-control state in which the process only produces acceptable items. After a period of production time, process may shift to out-of-control state to produce some non-conforming items. The elapsed time for a process to shift is a random variable that follows a general distribution with increasing hazard rate. In practical production systems, some non-conforming items may be reworked. Hence a percentage of the non-conforming items are scrapped items; they are discarded before the rework process starts. Other non-conforming items are reworked, and the rework process starts immediately when the regular production ends. The state of a process is observed by inspection. The process is inspected at time t1, t2, . . ., tm, and PM is carried out right after each inspection. The production cycle ends either when the system is in the out-of-control state or the last inspection is completed. Then to renew a production cycle, extra works on the system are needed and should be ensured that the next cycle begins with in-control state. Therefore, a PM does not be performed in the last inspection when the system is in the out-of-control state (Chen et al. 2010). The behavior of inventory system is presented in Fig. 4.19. Also some new notations which are specifically used for the proposed model are presented in Table 4.11. It should be noticed that unlike common imperfect inventory control systems, in this work order quantity and backordering level are not decision variables. In this model the optimal length of the inspection intervals, the optimal cost of PM, and the number of inspections, based on the integrated model, are decision variables.

4.4 Backordering

301

The expected regular production cycle time is given by (Chen et al. 2010): E ðT Þ ¼

m X i¼1

hi 

i1  Y

1  pj



ð4:256Þ

j¼1

When the general production cycle ends, the total expected number of non-conforming items is E(N ). And in realistic production systems, the imperfectquality items may be reworked; hence the rework process could eliminate waste and affect the cost of manufacturing. Then, a percentage (1  x1) of the non-conforming items are reworked and the number of non-conforming items which can be reworked is (1  x1)E(N ). The rework rate of non-conforming items in units per unit time is Pr. Then, the expected rework cycle time can be obtained as the following formula shows (Chen et al. 2010): E ðT r Þ ¼ ð1  x1 ÞEðN Þ=Pr

ð4:257Þ

The total expected number of non-conforming items per production cycle is given by: E ðN Þ ¼

m X

j1   Y p jE N j  ð 1  pi Þ

j¼1

ð4:258Þ

i¼1

where E(Nj) is the expected number of non-conforming items produced due to out of control during the period (tj1, tj) and is given by: 

Zb j



E Nj ¼ a

  f ðt Þ  dt ¼ θP b j  t  F a j1

j1

8 > <     θP  bj F bj  F a ¼  F a j1 > :

 j1

Zb j  a

j1

9 > = tf ðt Þdt > ;

ð4:259Þ

Then from Fig. 4.19, the expected inventory cycle length is presented as follows (Chen et al. 2010): E ðCycle LengthÞ ¼

1 ½P  E ðT Þ D

ð4:260Þ

The expected inventory holding cost is calculated by multiplying the inventory holding cost per unit of time by the expected inventory over the course of the inventory cycle (area under the curve in Fig. 4.19) from the time at which shortages begin to be filled (t ¼ B/(P  D)) to the time when inventory is depleted (t ¼ E

302

4 Rework I(t)

Pr–D

–D P–D

h1

h2 t1

t

hm tm

t2

B Tr

T CT

Fig. 4.19 Inventory cycle (Chen et al. 2010) Table 4.11 New notations of given problem Pr Tr m hj tj

Rework rate of non-conforming items in units per unit time (units per unit time) The time of reworking non-conforming items for each cycle (time) Number of inspections carried out during each production run Length of the jth inspection interval P Time for the jth inspection t j ¼ mj¼1 h j

θ x1

Non-conforming rate when system is in the out-of-control state The percentage of non-conforming items that are scrapped and will not be reworked, 0  x1  1 The percentage of scrapped items produced during the rework process 0  x2  1 Imperfect factor Imperfectness coefficient at the kth PM Actual age of system instantly before the jth PM Actual age of system instantly after the jth PM with a0 ¼ 0 Cost of the actual PM activities ($/PM activity) Cost of the maximum PM level ($/PM activity) Restoration cost ($/restoration) The conditional probability that the process shifts to the out-of-control state during the time interval (tj1, tj) Number of non-conforming items produced within (tj1, tj) Probability density function of the time to shift Cumulative distribution of f (t), F ðt Þ ¼ 1  F ðt Þ, r ðt Þ ¼ f ðt Þ=F ðt Þ

x2 η γk bj aj C aPM C max PM R(T ) pj Nj f(t) F(t)

4.4 Backordering

303

(CT)  B/D). Hence, the expected inventory holding cost is determined by the following formula (Chen et al. 2010): E ðHolding CostÞ ¼

h 2   P 1 2 2  ½ðP  DÞEðT Þ  B  ½ð1  x1 ÞE ðN Þ Pr DðP  DÞ ð4:261Þ

The integrated model they developed is also allowing shortages for the determination of EPQ. So from Fig. 4.19, the expected shortage cost is given by (Chen et al. 2010): EðShortage CostÞ ¼



 Cb 2 P B 2 ðP  DÞD

ð4:262Þ

When the regular production ends, a percentage x1 of the non-conforming items are scrapped, and the others are reworked. Restated, the quantity that can be reworked is (1  x1)E(N ). And the repair cost for each non-conforming item reworked is CR. Then, the expected reworking cost is given by (Chen et al. 2010): E ðRework CostÞ ¼ C R ð1  x1 ÞEðN Þ

ð4:263Þ

Also, when regular production ends, the number of non-conforming items that are scrapped is x1E(N ). The rework process is assumed to be imperfect; a percentage x2 of the reworked items fail the restoring process and become scrap items. The quantity of the reworked items that become scrap is (1  x1)x2E(N ) when the rework process ends. The disposal cost for each scrapped item is Cd. Therefore, the expected disposal cost for scrap items is given by (Chen et al. 2010): E ðDisposal CostÞ ¼ C d ½x1 þ ð1  x1 Þx2 E ðN Þ

ð4:264Þ

The system cannot be as good as new after implementing PM, but it will be younger, according to the level of PM activities. The reduction in the used age of the equipment is a function of the PM cost. Assume after each PM the system recovers. The parameter is a degradation factor which impacts the influence of PM activities on the used age of the process. Let γ k be the imperfect coefficient at the kth PM, then (Chen et al. 2010): γ k ¼ ηk1

C aPM Cmax PM

ð4:265Þ

304

4 Rework

Ben-Daya (1999) considered both linear and nonlinear relationships between the age reduction and PM cost. Here, they assumed the relationship is linear as follows (Chen et al. 2010): ak ¼ ð1  γ k Þbk

ð4:266Þ

Note that the actual age of a production system at time tj is given by: b 1 ¼ h1 b j ¼ a j1 þ h j

ð4:267Þ

for j ¼ 2, 3, . . . , m

Usually, the PM activity is implemented after each inspection unless the system is ceased, and no PM is being implemented at the end of the production cycle. These give the expected maintenance cost E(PM) in the following lemma. The expected PM cost per regular production cycle is given by (Chen et al. 2010): EðPM CostÞ ¼ C aPM

j m1 Y X

ð1  p i Þ

ð4:268Þ

j¼1 i¼1

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

nPM ¼expected number of PM per production cycle

There might be errors in inspections, but for simplicity, such errors are not considered. The expected number of inspection in the production cycle is one more than the expected number of PM because there is one inspection at the end of the production cycle. This gives: " E ðInspection CostÞ ¼ ðnPM þ 1ÞCI ¼ C I 1 þ

j m1 Y X

# ð 1  pi Þ

ð4:269Þ

j¼1 i¼1

The production system in out-of-control state must be terminated for maintenance. When this happens, the charge for restoration is needed. Therefore, the restoration cost (RC) should be included in the total cost. Assume that the out-ofcontrol state occurs at the time t in the period (aj  1, bj) and the detection delay time is bj  t. According to Ben-Daya (2002), when the restoration cost is assumed to change linearly with the detection delay, RCj ¼ R(bj  t) ¼ r0 + r1(bj  t) where r0 and r1 are constants. The restoration cost per production cycle is given by (Chen et al. 2010):

4.4 Backordering

305

E ðRestoration CostÞ ¼

m X

pj

j¼1



j1 Y

ð 1  pi Þ

i¼1



!     Z  F b j  F a j1 f ðt Þ    dt r0 þ r1 b j  r1 t  F a j1 F a j1 ð4:270Þ

The expected total cost (ETC) per each cycle is summation of setup (setup cost is K ), holding, shortage, preventive maintenance, restoration, disposal, inspection, and rework costs. Moreover, after the production cycle ends, the total amount of production minus the quantity which is disposal leaves quantity that can be sold. Therefore, the expected total revenue is: Expected Total Revenue ¼ S½EðQÞ  ðx1 þ ð1  x1 Þx2 ÞEðN Þ

ð4:271Þ

where E(Q) is the expected total production quantity and E(Q) ¼ PE(T ). Then, the expected total profit per unit time can be obtained obtain given by: ETP ¼ Expected Total Profit ¼

Expected Total Revenue  Expected Total Cost EðCycle LengthÞ

ð4:272Þ

The next problem is optimizing the decision variables for the above integrated profit model. Methods of adjusting the number of inspections and optimizing the solution are also discussed. Using the above formulas, one determines simultaneously the optimal length of the inspection intervals, h1, h2, . . ., hm; the optimal cost of PM, C aPM ; and the number of inspections, m, based on the integrated model. As equipment breaks down, an inspection schedule should be arranged to find the fault quickly and then propose the correct measure. In practice, most of the mechanical malfunctions follow the non-Markov shock model with IFR function. As the production process continues, the optimal inspection interval is progressively reduced. Each inspection interval has the same cumulative hazard rate. Restated (Chen et al. 2010), Zt jþ1

Zt1 r ðt Þdt ¼

t

r ðt Þdt

for j ¼ 2, 3, . . . , m

ð4:273Þ

0

Since the hazard rate is reduced at the end of each inspection interval due to the PM activities, condition (4.301) becomes (Chen et al. 2010):

306

4 Rework

Zb j

Zh1 r ðt Þdt ¼

a

r ðt Þdt

for j ¼ 2, 3, . . . , m

ð4:274Þ

0

j1

Usually researchers assume that the time of process staying in the in-control state is a random variable, following Weibull probability function. So its probability distribution function is given by (Chen et al. 2010): f ðt Þ ¼ λvt v1 eλt

v

t > 0, v  1, λ > 0

ð4:275Þ

So hazard rate of the Weibull distribution is applied in Eq. (4.273) to obtain the length of the inspection intervals (Chen et al. 2010): hj ¼



a

v j1

1 þ ð h1 v Þ v  a

j1

for j ¼ 2, 3, . . . , m

ð4:276Þ

This means, when h1 is determined, so are other hj’s. Therefore, to maximum expected total profit, the value of decision variables h1, . . ., m and C aPM are needed to be determined. They implemented the stepwise partial particularization procedure to achieve the goal. Since the characteristics of the profit function, some modifications to the standard method have to be made to account for the inherent internality constraint on the number of inspections. The optimal value of m  2 could be determined by choosing m that satisfies two inequalities (Chen et al. 2010): ETPðm  1Þ  ETPðmÞ

and

ETPðm þ 1Þ  ETPðmÞ

ð4:277Þ

Therefore, the optimal value of the number of inspections, m*, and the length of the first inspection interval, h1, can be obtained by the following procedure if the PM level is determined (Chen et al. 2010). Step 1: Estimate m0. The maximum number of inspection existed during each production run according to historical experience and the condition of production. Step 2: First setup m ¼ 1. One can search an optimal value h1 and calculate the expected total cost ETπ 1 under this condition. Step 3: Repeat Step 2 for m ¼ 2, 3, . . ., m0. One has the optimal value h1 and the expected total profit form ETP2 to ETPm0 under different m, respectively. Step 4: The optimal values m* and h1 must meet the following condition:     ETP h1 , m , C aPM ¼ Max ETP j , j ¼ 1, 2, . . . , m0 Example 4.13 Chen et al. (2010) presented several examples to illustrate the important aspects of the integrated profit model. The time that the process remains in the in-control state is assumed to follow a Weibull distribution with scale and shape parameters λ ¼ 5 and v ¼ 2.5. The following parameters are fixed D ¼ 500,

4.4 Backordering

307

P ¼ 1000, Pr ¼ 700, K ¼ $150, h ¼ $0.5, Cb ¼ $8, CI ¼ $10, Cd ¼ $20, C Max PM ¼ $30, CR ¼ $5, s ¼ $10, η ¼ 0.99, r0 ¼ 50, r1 ¼ 0.5, and x2 ¼ 0.1. The results show clearly that the expected total profit increases when the actual implemented PM level increases. The optimum PM level when PM CaPM ¼ $30 is obtained when max CMax PM ¼ $30, leading to a total profit of $4728.41 much more than without PM ($4672.17). Using comprehensive numerical analysis, Chen et al. (2010) found the optimal number of inspections, the optimal length of first inspection interval, the EPQ and the expected total profit per unit time under different values of x1 (x1 ¼ 0.1, 0.5 and 1.0), B (B ¼ 0, 50 and 100) and (0.2 and 0.4) as presented in Table 4.12.

4.4.6

Random Defective Rate: Different Production and Rework Rates

Chiu (2003) studies the effect of the reworking of defective items on the finite production model. He assumed production process may generate randomly x percent of defective items at a production rate d. The inspection cost per item is involved when all items are screened. Not all of the defective items produced are reworked. A portion θ of the imperfect-quality items are scrap and must be discarded before the rework process starts. The production rate P is a constant and is much larger than the demand rate D. When regular production ends, the reworking of defective items starts immediately at a constant rate P1. The production rate d of the imperfectquality items can be expressed as the product of the production rate times the percentage of defective items produced. The production rate P must always be greater than or equal to the sum of the demand rate D and the defective rate d. Therefore, the following condition must hold: P  D  d  0;

0x1

D P

ð4:278Þ

For the following derivation, the solution procedures are those used by Hayek and Salameh (2001). Referring to Fig. 4.20, T¼

Qð1  θxÞ D

ð4:279Þ

where 0  θ  1 and θxQ are scrap items randomly produced by the regular production process. Hence, the cycle length T is a variable, not a constant:

308

4 Rework 5 X

Q P

ð4:280Þ

I1 PDd Q I 1 ¼ ðP  D  d Þ  B P

ð4:281Þ

T¼

ti ,

t1 þ t5 ¼

i¼1

Therefore, the production uptime t1 is: t1 ¼

ð4:282Þ

The time t2 needed to rework (1  θ) imperfect-quality items is computed in Eq. (4.283), and the maximum level of on-hand inventory when the rework process finished is calculated in Eq. (4.284): xQð1  θÞ dQð1  θÞ ¼ P1 P1 P

D dθ Dd ð1  θÞ I ¼ I 1 þ ðP1  DÞt 2 ¼ Q 1    B P1 P P P t2 ¼

ð4:283Þ ð4:284Þ

Using Fig. 4.20, I D B t4 ¼ D B t5 ¼ PDd t3 ¼

ð4:285Þ ð4:286Þ ð4:287Þ

The defective items produced during the regular production uptime t1, as depicted in Fig. 4.21, are (Chiu 2003): dðt 1 þ t 5 Þ ¼ xQ

ð4:288Þ

The reworking of (1  θ) imperfect-quality items starts immediately, when the regular production time t1 ends (Chiu 2003). Solving the inventory cost per cycle, TC(Q, B) is (Chiu 2003): Holding Cost of Reowrked Item

Rework Cost Disposal Cost zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ zfflfflffl}|fflfflffl{ z}|{ P t TCðQ, BÞ ¼ CQ þ C R xQð1  θÞ þ C d xQθ þKþ h1 1 2 ð t 2 Þ   2 d ðt 1 þ t 5 Þ I1 I1 þ I I B þ h ðt 1 Þ þ ðt 2 Þ þ ðt 3 Þ þ ðt 1 þ t 5 Þ þ C b ðt 4 þ t 5 Þ 2 2 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} Production Cost

Holding Cost

Shortage Cost

ð4:289Þ

θ ¼ 0.4

θ ¼ 0.2

d1 ¼ 0.1 d1 ¼ 0.5 d1 ¼ 1 d1 ¼ 0.1 d1 ¼ 0.5 d1 ¼ 1

B¼0 m* 3 3 3 3 3 4

h1 0.29 0.26 0.24 0.26 0.24 0.21 Q* 702 668 640 664 625 729

ETP 4741.53 4732.98 4725.49 4731.89 4721.17 4713.48

Table 4.12 Optimal values under various conditions (Chen et al. 2010) B ¼ 50 m* 4 4 4 4 4 4 h1 0.29 0.26 0.24 0.26 0.23 0.21 Q* 845 815 788 810 773 743

ETP 4737.7 4729.45 4722.37 4728.41 4718.31 4709.62

B ¼ 100 m* h1 6 0.29 6 0.26 6 0.24 6 0.25 6 0.24 7 0.22

Q* 1051 1047 1033 1045 1022 1098

ETP 4694.67 4686.05 4678.79 4684.98 4674.68 4666.39

4.4 Backordering 309

310

4 Rework

Fig. 4.20 On-hand inventory of non-defective items (Chiu 2003)

I(t)

P1 – D I I1 –D P – D –d

P – D –d

Time

t1

t2

t3

B

t4 t5

T

T

Using the renewal reward theorem in dealing with the variable cycle length, that is to compute E[T] first. Then the expected annual cost E[TCU(Q, B)] ¼ E[TC (Q, B)]/E[T], so (Chiu 2003): 

E ½x E ½ x 1 E½TCUðQ, BÞ ¼D C þ C R ð1  θÞ þ Cd θ 1  θE ½x 1  θE ½x 1  θE½x h

 i KD 1 h D 1 þ 1 Q  2B þ Q 1  θE ½x 2 P 1  θE½x



DQð1  θÞ2 E ½ x2  ð h1  hÞ 2P1 1  θE ½x

2 B 1x 1 þ ðC b þ hÞE 2Q 1  x  D=P 1  θE½x h

 i E ½ x D hQθ2 E ½x2  þ hθ B  1  Q þ P 2 1  θE ½x 1  θE ½x þ

ð4:290Þ Let: E ½ x 1 ; E1 ¼ ; 1  θE ½x 1  θE ½x

1x 1 ¼E 1  x  D=P 1  θE½x

E0 ¼

E2 ¼

E ½ x2  ; 1  θE ½x

and

E3

Then Eq. (4.290), the expected annual cost, becomes (Chiu 2003):

ð4:291Þ

4.4 Backordering

311

Fig. 4.21 On-hand inventory of defective items (Chiu 2003)

 E½TCUðQ,BÞ ¼D CE0 þ C R ð1  θÞE1 þ C d θ

 h

 i E ½x  KD h D E0 þ 1  Q  2B E0 E1 þ Q 2 P 1  θE ½x

h

 i DQð1  θÞ2 D B2 hQθ2 E2 þ hθ B  1  Q E1 þ ðh1  hÞE 2 þ ðC b þ hÞE3 þ 2P1 2Q 2 P

ð4:292Þ For the proof of convexity of E[TCU(Q, B)], one can utilize the Hessian matrix equation and obtain the following (Chiu 2003): 2

2

∂ E½TCUðQ, BÞ 6 ∂Q2 6 ½Q B6 2 4 ∂ E½TCUðQ, BÞ ∂B∂Q

3 2 ∂ E½TCUðQ, BÞ 7 Q  ∂Q∂B 2KD 7 ¼ E >0 7 2 Q 0 ∂ E½TCUðQ, BÞ 5 B

ð4:293Þ

∂B2

Since the Hessian matrix is positive definite, the first derivative respect to decision variables yields to optimal values:

 Dð1  θÞ2 ∂E ½TCUðQ, BÞ KD h D ¼ 2 E0 þ 1  þ ðh1  hÞE 2 2 P 2P1 ∂Q Q

 B2 D hθ2 E1 þ  2 ðCb þ hÞE 3  hθ 1  E P 2 2 2Q ∂E½TCUðQ, BÞ B ¼ hE 0 þ ðCb þ hÞE 3 þ hθE 1 Q ∂B

ð4:294Þ

ð4:295Þ

Setting partial derivatives presented in Eqs. (4.294) and (4.295) equal to zero yields to:

312

4 Rework

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD h

 i

 Q ¼u u 2 u hð1  D=PÞ þ Dð1  θÞ =p1 ðh1  hÞE½x2   h2 f1  θE½xg2 = t 

fðC b þ hÞE ½ð1  xÞ=ð1  x  D=PÞg  2hθð1  D=PÞE ½x þ hθ2 þ E ½x2 

B ¼



h Cb þ h





1  θE½x 1 h Q ¼ Q E3 C b þ h E ½ð1  xÞ=ð1  x  D=PÞ

ð4:296Þ ð4:297Þ

Example 4.14 Chiu (2003) presented an example in which a company produces a product for several industrial clients under the following parameters: P ¼ 10,000 units per year, D ¼ 4000 units per year, P1 ¼ 600 units per year, x ¼ U[0, 0.2], θ ¼ 0.1, K ¼ $450 for each production run, C ¼ $2 per item (inspection cost per item is included), CR ¼ $0.5 per item, Cd ¼ $0.3 for each scrap item, h ¼ $0.6 per item per unit time, h1 ¼ $0.8 per item reworked per unit time, and Cb ¼ $0.2 per item backordered per unit time. The optimal production lot size can be obtained from Eq. (4.296) as Q* ¼ 5929 units. The optimal backorder quantity can be computed from Eq. (4.297) as optimal B* ¼ 2558 units.

4.4.7

Imperfect Rework Process

Chiu (2007) derived the optimal replenishment policy for imperfect-quality economic manufacturing quantity (EMQ) model with rework and backlogging. He used a random defective rate, and all items produced are inspected, and the defective items are classified as scrap and repairable. A rework process is involved in each production run when regular manufacturing process ends, and a rate of failure in repair is also assumed. This assumption is the main difference between this work and previous ones. Some new notations which are used in this model are presented in Table 4.13. According to previous models and referring to Fig. 4.22, T¼

5 X

ti

ð4:298Þ

i¼1

The production uptime t1 needed to accumulate I1 units of perfect-quality items is (Chiu 2007): t1 ¼

I1 PDd

ð4:299Þ

4.4 Backordering

313

Q B P

I 1 ¼ ðP  D  d Þ

ð4:300Þ

The basic assumption of the imperfect-quality EMQ model is that the production rate P of perfect-quality items must always be greater than or equal to the sum of the demand rate D and the production rate of defective items d. Hence, the following condition must hold: 0x1

D P

ð4:301Þ

The time t2 needed to rework of the repairable defective items is computed as t2 ¼

xQð1  θÞ P1

ð4:302Þ

Since the rework process is assumed to be imperfect, the production rate of the scrap items, d1, can be written as: d 1 ¼ P1  θ 1

ð4:303Þ

The maximum level of on-hand inventory, when the rework process ends, is (Chiu 2007): I ¼ I 1 þ ðP1  d 1  DÞt 2

dθ DðP1 þ dÞ d1 d d1 dθ Ddθ  þ þ B ¼Q 1  P1 P P P1 P P1 P P1 P

ð4:304Þ

Similar to previous model, I D B t4 ¼ D B t5 ¼ PDd t3 ¼

ð4:305Þ ð4:306Þ ð4:307Þ

The defective items produced during the regular production uptime t1 + t5, as illustrated in Fig. 4.23, are: dðt 1 þ t 5 Þ ¼ xQ

ð4:308Þ

Among the defective items, a random portion y of the imperfect-quality items is scrap; the reworking of (1 – θ) of defective items starts immediately, when the

314

4 Rework

Table 4.13 Notations d1 θ1

Defective production rate during rework process (units per unit time) Portion of the defective produced during rework process (%)

Fig. 4.22 On-hand inventory of perfect-quality items (Chiu 2007)

I(t) Q P1 – d1– D I I1

P1 – d1– D –D P – d –D

P – d –D

t1

–D

Time

t2

t3

B T

t4 t5

T

regular production time t1 ends. Since the rework process is assumed to be imperfect either, a random portion y1 of the reworked items fail the repairing and become scrap (refer to Fig. 4.24); they are calculated as follows: d 1 t 2 ¼ θ1 ½xð1  θÞQ

ð4:309Þ

Hence, the cycle length T becomes (Chiu 2007): T¼

Q½1  θ  x  ð1  θÞxθ1  D

ð4:310Þ

where 0  θ  1 and [θxQ] are scrap items randomly produced during the regular production process and [(1  θ)xθ1]Q are scrap items randomly generated during the rework process. Hence, the cycle length T is not a constant. If φ is used to denote the total scrap rate, then φ ¼ [θ + θ1(1  θ)]; one notes that the mean of random variable φ will follow the standard normal distribution (based on the Central Limit theorem). Therefore, Eq. (4.309) can be rewritten as: T¼

Q½1  xφ D

The total inventory cost per cycle, TC, is:

ð4:311Þ

4.4 Backordering

315

Id(t)

θ % of defective items are scrap

d(t1+t5)

d

d(t1+t5)(1-θ )

d –P1

–P1 Time

t1

t2

t3

t4 t5

T

T

Fig. 4.23 On-hand inventory of defective items (Chiu 2007)

Is(t)

[ θ +(1- θ) θ1]xQ

de

d1

t1

t2

d1 de

θx Q

t3 T

Time

t4 t 5

t1 t2

T

Fig. 4.24 On-hand inventory of scrap items (Chiu 2007) Holding Cost of ReworkedItems

Disposal Cost ReworkCost Production Cost zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ z}|{ z}|{ P t TCðQ,BÞ ¼ K þ CQ þ C R ½ x  ð1  θ Þ  Q  þ C d ðφ  x  Q Þ þ h1 1 2 ðt 2 Þ 2   d ðt 1 þ t 5 Þ I1 I1 þ I I B h ðt 1 Þ þ ðt 2 Þ þ ðt 3 Þ þ ðt 1 þ t 5 Þ þ C b ðt 4 þ t 5 Þ 2 2 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Setup Cost

Holding Cost

Backordering Cost

ð4:312Þ Chiu (2007) assumed that the proportion of defective items x and the ratio of scrap items θ and θ1 are random variables with known probability density functions.

316

4 Rework

Thus, to take the randomness of imperfect production quality into account, one can utilize the expected values of x, θ, and θ1 in the inventory cost analysis. Let E[x], θ, and θ1 represent the expected values of x, θ, and θ1, respectively. Since the production cycle length is not a constant, one may employ the renewal theorem approach to cope with the variable cycle length, that is, to compute the E[T] first. Then the expected annual inventory cost function E[TCU(Q, B)]/E[T] becomes (Chiu 2007):  E½TCUðQ, BÞ ¼D C

 E ½x  E ½x 1 KD 1 þ Cd φ þ þ C R ð1  θ Þ Q 1  φE½x 1  φE ½x 1  φE ½x 1  φE ½x h

 i DQð1  θÞ2 E ½x2  h D 1  2B þ Q 1 þ ½h1  hð1  θ1 Þ 2 P 2P 1  φE ½x 1  φE ½x 1 0 1 h

 i E ½x B2 1 D B 1x C þ hφ B  1  Q þ ðC b þ hÞE @ A D 1  φE ½x 2Q P 1  φE ½x 1x P hQφ2 E ½x2  þ 2 1  φE ½x

ð4:313Þ where φ ¼ [θ + (1  θ)θ1], and if E ½ x 1 ; E1 ¼ ; 1  φ  E ½ x 1  φ  E ½ x

1x 1 ¼E 1  x  D=P 1  φE ½x

E0 ¼

E2 ¼

E ½ x2  1  φ  E ½x

and

E3

Then Eq. (4.340), the expected annual cost, becomes (Chiu 2007): E ½TCUðQ, BÞ ¼D½C  E0 þ CR  ð1  θÞ  E 1 þ C d  φ  E1  þ

KD E Q 0

 DQð1  θÞ2 hQ D 1  E0 þ ½ h1  hð 1  θ 1 Þ   E 2 1 P 2P1 h

 i B2 D h  Q  φ2 Q  E1 þ þ ðCb þ hÞE 3 þ h  φ  B  1   E2 P 2Q 2 ð4:314Þ þ

Similar to previous model, to derive the optimal values, using Hessian matrix,

4.4 Backordering

2

317

2

∂ E½TCUðQ, BÞ 6 ∂Q2 6 ½Q B6 2 4 ∂ E½TCUðQ, BÞ ∂B∂Q

3 2 ∂ E½TCUðQ, BÞ 7 Q  ∂Q∂B 2KD 7 ¼ E >0 7 2 Q 0 5 B ∂ E½TCUðQ, BÞ

ð4:315Þ

∂B2

So E[TCU(Q, B)] is strictly convex. Therefore,

 D ð 1  φÞ 2 ∂E½TCUðQ, BÞ KD h D ¼ 2 E0 þ 1  ðh1  hÞE 2 þ 2 P 2P1 ∂Q Q

 B2 D hφ2 E1 þ  2 ðCb þ hÞE 3  hφ 1  E P 2 2 2Q ∂E ½TCUðQ, BÞ B ¼ hE0 þ ðC b þ hÞE 3 þ hφE 1 Q ∂B

ð4:316Þ

ð4:317Þ

Setting partial derivatives presented in Eqs. (4.316) and (4.317) equal to zero yields to: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2KD h

 i

 Q ¼u u 2 u hð1  D=PÞ þ Dð1  φÞ =p1 ðh1  hÞE ½x2   h2 f1  φE½xg2 = t 

fðC b þ hÞE ½ð1  xÞ=ð1  x  D=PÞg  2hφð1  D=PÞE½x þ hφ2 þ E ½x2 

B ¼



h Cb þ h





1  φE ½x 1 h Q ¼ Q E3 C b þ h E ½ð1  xÞ=ð1  x  D=PÞ

ð4:318Þ ð4:319Þ

Example 4.15 Chiu (2007) supposed a supplier produces a product with the parameters values of K ¼ $450 for each production run, C ¼ $2 per item (inspection cost per item is included), h ¼ $0.6 per item per unit time, CR ¼ $0.5 repaired cost for each item reworked, h1 ¼ $0.8 per item reworked per unit time, Cd ¼ $0.3 for each scrap item, Cb ¼ $0.2 per item per unit time, D ¼ 4000 units per year, P ¼ 10,000 units per year, P1 ¼ 600 units per year, x ¼ the proportion of imperfect-quality items produced is uniformly distributed over the interval [0, 0.2], θ is uniformly distributed over the range [0, 0.1]; its expected value 0.05, and θ1 is uniformly distributed within [0, 0.1] where its expected value is 0.05. Hence, from the definition of φ, the overall scrap φ ¼ [θ + θ1(1  θ)] ¼ 0.0975 and the optimal replenishment policy can be calculated from Eqs. (4.318) to (4.319). Then the value of optimal lot size Q* ¼ 5305 and the optimal allowable backorder level is B* ¼ 2175.

318

4.5 4.5.1

4 Rework

Partial Backordering Immediate Rework

In this section, an EMQ model with production capacity limitation, imperfect production processes, immediate rework, and partial backordered quantities in a multi-product single-machine manufacturing system is developed. The defective items of n different types of products are generated at a rate xi; i ¼ 1, 2, . . ., n per cycle. So the good item quantities are (1–xi)Pi. The production and demand rates of the ith product per cycle are Pi and Di, respectively. In this production system, each cycle consists of three parts: production uptime, rework time, and production downtime. Since all of the products are manufactured on a single machine with a limited capacity, a unique cycle length for all items is considered, that is, T1 ¼ T2 ¼  ¼ Tn ¼ T. They assumed that the total scrapped items are reworkable and no imperfect items are produced at the end of the rework process. Also, the producer has to use the same resource for production and rework processes simultaneously. Because a single machine has a limited joint production system capacity, shortage is allowed with a certain fraction of it to be backordered. In this work, they extended Jamal et al. (2004) to consider a more realistic inventory control problem in which multi-product single-machine strategy is used to produce several items under immediate rework, partial backordering, and capacity constraints (Taleizadeh and Wee 2015). Figure 4.25 shows the inventory control problem under study. First, a singleproduct problem which consists of ith product is first developed. The fundamental assumption of an economic manufacturing model with rework process is: ð1  xi ÞPi  Di  0

ð4:320Þ

Figure 4.25 shows that T 1i and T 5i are the production uptimes for non-defective and defective items, respectively. T 2i is the reworking time and T 3i and T 4i are the production downtimes, respectively. Finally, the cycle length is: T¼

5 X

T ij

ð4:321Þ

j¼1

In this model, a part of the modeling procedure is adopted from Jamal et al. (2004). As noted before, since all products are manufactured on a single machine with a limited capacity, the cycle length for all products is equal (T1 ¼ T2 ¼  ¼ Tn ¼ T ), based in Fig. 4.25. One has:

4.5 Partial Backordering

319

T 1i ¼

βi Bi Qi  Pi ð1  xi ÞPi  Di

Qi x Pi i

 1  DPii  xi DPii Qi T 2i ¼

T 3i ¼

Di

ð4:323Þ



β i Bi Di

ð4:324Þ

Bi Di

ð4:325Þ

β i Bi ð1  xi ÞPi  Di

ð4:326Þ

T 4i ¼ T 5i ¼

ð4:322Þ

It is evident from Fig. 4.25 that: I i ¼ ðð1  xi ÞPi  Di Þ

Qi  βi Bi Pi

I 0i ¼ I i þ xi ðPi  Di Þ

Qi Pi

ð4:327Þ ð4:328Þ

Hence, using the equation the cycle length for a single product problem is: T¼

5 X j¼1

T ij ¼

Qi þ ð1  βi ÞBi Di

ð4:329Þ

And the order quantity for the ith product is: Qi ¼ Di T  ð1  βi ÞBi

ð4:330Þ

The elements of the cost function are the setup cost, the holding cost, the processing cost, the rework cost, and the shortage cost which are expressed as: TC ¼ CA þ CP þ C Re þ C H þ C B þ C L Pn Ki C A ¼ i¼1 T

ð4:331Þ ð4:332Þ

The production cost per unit is ci, and the production quantity of ith product per period is Qi. So, the production cost of ith product per period is CiQi. Hence, the annual production cost for ith product is NCiQi, and the following cost is the joint policy cost:

320

4 Rework

Fig. 4.25 On-hand inventory of perfect-quality items (Taleizadeh and Wee 2015)

Pn

i¼1 C i Q

CP ¼

T

ð4:333Þ

The rework cost per unit of ith product is C Ri , and the quantity of ith product that needs to be reworked per period is xiQi. So, the rework cost of ith product per period is CRi xiQi. Hence, the rework cost for ith product per year is NC Ri xiQi, and the annual rework cost for the joint policy is: Pn CRe ¼

i¼1 C Ri xi Qi

T

ð4:334Þ

From Fig. 4.25, Eq. (4.335) shows the inventory holding cost of the system for an independent and joint production policy, respectively: X  I   I i þ I 0   I 0   i 2 hi i t 1i þ t i þ i t 3i 2 2 2   0  X   Ii þ Ii 2 I0   1 I ¼ hi i t 1i þ t i þ i t 3i T 2 2 2

C H ¼N

ð4:335Þ

Also, from Fig. 4.25, Eqs. (4.335) and (4.336) show the annual backordered and the lost sale costs in the joint policy production, respectively:

4.5 Partial Backordering

321

Pn CB ¼

Pn CL ¼

 t 4i þ t 5i Bi 2T

i¼1 C bi βi



π i ð1 i¼1 b

ð4:336Þ

 βi ÞBi

ð4:337Þ

2T

where C bi βiBi and b π i (1  βi)Bi are the backordered and the lost sale cost of ith product per period, respectively. Consequently, one has: TC ¼C A þ C P þ C R þ C H þ C B þ C L Pn Pn Pn X  I i   I i þ I 0   I 0   i¼1 K i i¼1 C i Qi i¼1 C Ri xi Qi i 2 ¼ þ þ þN hi t1 þ t i þ i t 3i T T T 2 i 2 2  4 5 Pn Pn C b β i t i þ t i Bi b π i ð1  βi ÞBi þ i¼1 i þ i¼1 2T 2T ð4:338Þ In the joint production systems with reworks, the total production, rework, and setup should be smaller than the cycle length. In our problem,  P Pn  times 1 2 5 i¼1 t i þ t i þ t i þ i tsi must be smaller or equal to T(T1 ¼ T2 ¼  ¼ Tn ¼ T ). Hence, the capacity constraint is: n  X i¼1

 X t 1i þ t 2i þ t 5i þ tsi  T i

From Eqs. (4.350), (4.351), and (4.354), the capacity constraint model becomes: X ð1 þ xi Þ i

Pi

ðDi T  ð1  βi ÞBi Þ þ

X

tsi  T

ð4:339Þ

i

The final model of the joint production system is: Min : TCðT,Bi Þ ¼

n n n n X X α1 B X B2i X α3i Bi  α4i i þ α5i þ ðC þ C Ri xi ÞDi þ α2 T  T T i¼1 T i¼1 i i¼1 i¼1

Pn ð 1 þ x i Þ ð1  βi ÞBi i¼1 Pi

s:t: : T  ¼ T Production Min Pn ð1 þ xi ÞDi 1  i¼1 Pi T,Bi 8i; i ¼ 1,2, ...,n Pn

i¼1 tsi 

ð4:340Þ where:

322

4 Rework

α1 ¼

n X

Ki > 0

ð4:341aÞ

i¼1

0

1 ðDi Þ2 ðð1  xi ÞPi  Di Þ þ 4xi ðDi Þ2 ðð1 xi ÞPi  Di Þþ 2xi 2 ðDi Þ2 ðPi  Di Þ n B C X 2ð P i Þ 2 B C α2 ¼ hi B C

2 @ A 1 xi ðDi Þ i¼1 þððð1 xi ÞPi  Di Þþ xi ðPi  Di ÞÞ Di   Pi Pi Pi >0 ð4:341bÞ 0

1 ðð1  xi ÞPi  Di ÞDi ð1  βi Þ þ 2βi Pi Di B C 2ðPi Þ2 B C B C B ð1  βi ÞDi ð4xi ðð1  xi ÞPi  Di Þ þ xi 2 ðPi  Di ÞÞ þ xi Pi Di βi þ xi 2 Di ð1  βi ÞðPi  Di Þ C C>0 þ α3i ¼ hi B B C ðPi Þ2 B C

B C @ A ðð1  xi ÞPi  Di Þ þ xi ðPi  Di Þ 1 αi þ þ Di   β i Di Pi Pi Pi Di

ð4:341cÞ



α4i ¼ hi ððð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ Di 

π  C i  CRi xi Þð1  βi Þ 1 xi ðDi ð1  βi ÞÞ ðb >0   i Pi Pi 2 Pi

ð4:341dÞ 0

1 ð1 þ xi Þhi βi ð1  βi Þ h ðð1  xi ÞPi  Di Þð1  βi Þ hi βi 2 þ þ i B C Pi 2ð1  xi ÞPi  2Di B C 2 ðP i Þ2 B C B h ð1  β Þ2 ð2x ðð1  x ÞP  D Þ þ x 2 ðP  D ÞÞ C β ðð1  x ÞP  ð1  β ÞD Þ C B i C b i i i i i i i i i i i i i i þ Bþ C 2 2 1  x ð ð ÞP  D ÞD B C i i i i ð Þ P i α5i ¼ B C>0 B β ð1  β Þh ððð1  x ÞP  D Þ þ β ðP  D Þ þ P  ð1 þ x ÞD Þ C i i i i i i i Bþ i C i i i i B C Pi Di B C

B C 2 @ hi βi 2 A 1 þ xi ð1  β i Þ þ  hi ððð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ Di  Di Pi Pi 2

ð4:341eÞ Since the Hessian matrix of objective function is positive for all nonzero Bi and T, TC(T, Bi) is convex: 3 T 6 7 P 6 B1 7 6 7 2α1 þ ni¼1 α4i Bi 7 ¼ ½T, B1 , B2 , . . . , Bn   H  6 B >0 6 27 T 6 7 4⋮5 Bn 2

ð4:342Þ

4.5 Partial Backordering

323

To derive the optimal values of the decision variables, take the partial differentiations of TC(T, Bi) with respect to T and Bi (for details, see Appendix 2 of Taleizadeh and Wee (2015)):

∂TCðT, Bi Þ ¼ ∂T



Pn

2 i¼1 α5i Bi

 α1 þ

n P i¼1

T2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P 2 u α1  ni¼1 α4i 4α5i u ¼ t

Pn α23i  α2  i¼1 4α5i

α4i Bi þ α2 ! T ð4:343Þ

∂TCðT, Bi Þ 2α5i Bi  α4i α T þ α4i ¼ ð4:344Þ  α3i ! Bi ¼ 3i T 2α5i ∂Bi P α2  P α2  To ensure feasibility, both α1  ni¼1 4α4i5i and α2  ni¼1 4α3i5i should simultaneously be positive or negative. In order to solve the above problem, Taleizadeh and Wee (2015) introduced the following solution procedures: Step 1: Check for feasibility:

P P α2  If ð1  xi ÞPi  Di  0, ni¼1 ð1 þ xi Þ DPii < 1 and both α1  ni¼1 4α4i5i and P α2  α2  ni¼1 4α3i5i be either positive or negative simultaneously, go to Step 2.

Step 2: Find a solution: Using Eqs. (4.343) and (4.344), calculate T and Bi. Step 3: Check the constraints. Step 4: Derive the optimal solution: Based on the derived value of T*, then Bi can be derived from Eq. (4.344). For  Qi ¼ Di T   ð1  βi ÞBi , the optimal values of the order quantity can be obtained. Calculate the objective function using Eq. (4.339), and then go to Step 5. Step 5: Terminate the procedure. Taleizadeh and Wee (2015) considered a production system with production capacity limitation, imperfect production processes, immediate rework, and partial backordered quantity. The defective items of n different types of products are generated at a rate xi; i ¼ 1, 2, . . ., n per cycle. The production and demand rates of the ith item per cycle are Pi and Di, respectively. So, the perfect item quantities are (1–xi)Pi. They assumed that the total scrapped items are reworkable and no imperfect items occur at the end of the rework process. Also, the producer has to use the same resource for production and rework processes simultaneously. Shortage is allowed with certain fraction of it to be backordered because the single machine has limited joint production system capacity. Two multi-product problems with immediate rework and capacity constraint with partial backordering are considered for 15 products.

324

4 Rework

Example 4.16 The general and the specific data of mentioned examples are given in Table 4.14. The best results using the proposed methodology are shown in Table 4.15. Since T ¼ 2.5619 is greater than its lower bound T Production ¼ 0.1159, Min so T* ¼ T ¼ 2.5619. Example 4.17 The general and the specific data of mentioned examples are given in Table 4.16. The best results using the proposed methodology are shown in Table 4.17. Since T ¼ 2.5619 is smaller than T Production ¼ 2.6247, so T* ¼ Min T Production . Min

4.5.2

Repair Failure

Material is considered as one of the most important resources in any production system, and management of inventory is playing an important role in increasing the profitability of an organization. In the last decades, there have been tremendous efforts by industries to reduce the cost of inventory. The primary concern on inventory management is to reduce the costs of setup and holding. Inventory management has direct relationship with maintaining market share since customers may switch to different vendors due to the shortage. When goods are produced internally, the economic production quantity (EPQ) model is employed to determine the optimal production lot size. The traditional EPQ model assumption does not consider defective items. Due to imperfect quality of raw materials and/or production facilities, rework and repair of the defective items are considered in this study. This study is significant because a number of production units such as printed circuit board assembly in the PCBA manufacturing, metal components, and plastic injection molding have rework items (Taleizadeh et al. 2010). As studied previous models presented in this chapter, the imperfect-quality EPQ model considers a manufacturing process with a constant production rate P and demand rate D, where P > D.This process randomly generates x percent of defective items at a rate d. Taleizadeh et al. (2010) assumed that all items produced are screened and the inspection cost per item is included in the unit production cost C. All defective items produced are reworked at a rate of P1 at the end of each production cycle. Also some new notations which are specifically used for the proposed model are presented in Table 4.18. They assume an imperfect rework process where a random portion θ of the items is scrapped. Let d be the production rate of the defective items during the regular manufacturing process (it can be expressed as the product of production rate P and the defective percentage x) where d ¼ Px. Let d1 be the production rate of scrapped items during the rework which could be expressed as the product of the reworking rate and the percentage of scrapped items produced during the rework process with d1 ¼ P1θ. A real constant production capacity limitation on a single machine in which all products are produced and the setup cost is considered to be nonzero are

4.5 Partial Backordering

325

assumed. Since all products are manufactured on a single machine with a limited capacity, the cycle length for each is equal, i.e., T1 ¼ ⋯ ¼ Tn ¼ T. They first presented the problem statement for a single product case and then they changed it to a multi-product case. The basic assumption of EPQ model with imperfect-quality items produced is that Pi must always be greater than or equal to the sum of demand rate Di and the production rate of defective items di. Therefore, one has (Taleizadeh et al. 2010): Pi  Di  d i  0, 0  xi  1 

Di Pi

ð4:345Þ

The production cycle length (see Fig. 4.26) is the summation of the production uptime, the reworking time, the production downtime, and the shortage permitted time: T¼

5 X

t ij

ð4:346Þ

j¼1

The modeling procedure is adopted from Hayek and Salameh (2001). Since all products are manufactured on a single machine with a limited capacity, the cycle length of each product will be equal. One has (Taleizadeh et al. 2010): t 1i ¼

Ii Pi  Di  di

ð4:347Þ

t 5i ¼

βi Bi Pi  Di  di

ð4:348Þ

t 2i ¼

xi Qi P1

ð4:349Þ

t 3i ¼

I Max i Di

ð4:350Þ

Bi Di

ð4:351Þ

t 4i ¼

I i ¼ ðPi  Di  d i Þ I Max i

¼ I i þ ðP1i  d1i 

Di Þt 2i

Qi  βi Bi Pi

ð4:352Þ



Di d 1i d i Di di ¼ Qi 1     βi Bi ð4:353Þ P1i P P1i P P1i Pi

Note that t 1i  t 5i are the production uptimes, t 2i is the reworking time, and t 3i and are the production downtimes. Also t 4i is the permitted shortage time, and t 5i is the time needed to satisfy all the backorders for the next production. During the rework

t 4i

326

4 Rework

Table 4.14 General data for Example 4.16 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Di 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500 11,000 11,500 12,000

tsi 0.0025 0.003 0.0035 0.004 0.0045 0.0025 0.003 0.0035 0.004 0.0045 0.0025 0.003 0.0035 0.004 0.0045

Ki 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

C Ri 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Ci 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6

hi 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

C bi 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

b πi 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

xi 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

βi 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7

Table 4.15 The best results for Example 4.16 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

T Production Min 0.1159

T 2.5619

T 2.5619

Qi 380.1 505.6 630.7 755.4 879.9 1006.40 1130.90 1255.30 1379.60 1503.80 1631.20 1755.50 1879.80 2004.20 2128.10

Bi 8.33 13.59 19.63 26.3 33.5 45.89 54.78 64.02 73.58 83.42 113.48 125.98 138.75 151.77 165.03

TC 932,400

process, the production rate of scrap items can be written as in Eq. (4.354) and calculated as in Eq. (4.355): d1i ¼ P1i θi

ð4:354Þ

d1i t 2i ¼ xi θi Qi

ð4:355Þ

Hence, it follows that the cycle length in single product state is:

4.5 Partial Backordering

327

Table 4.16 General data for Example 1 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Di 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500 11,000 11,500 12,000

tsi 0.0005 0.001 0.0015 0.002 0.0025 0.0005 0.001 0.0015 0.002 0.0025 0.0005 0.001 0.0015 0.002 0.0025

Ki 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

C Ri 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Ci 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6

hi 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

C bi 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

b πi 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

xi 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

βi 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7

Table 4.17 The best results for Example 2 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

T Production Min 2.6247

T 2.6247

T 2.5619

Bi 8.48 13.83 19.98 26.77 34.1 46.79 55.86 65.28 75.03 85.07 115.88 128.65 141.69 154.99 168.55

TC 981,350

Qi ½1  θi  xi  Di

ð4:356Þ

Di T  ð1  βi ÞBi 1  θ i xi

ð4:357Þ

T¼ Qi ¼

Qi 389.5 518 646.2 774 901.6 1031.20 1158.80 1286.20 1413.60 1540.80 1671.30 1798.70 1926.00 2053.30 2180.50

328

4 Rework

Table 4.18 New notations of given problem P1i d1i h1i

Rework rate of non-conforming items in units per unit time ith item (units per unit time) The production rate of scrapped items during the rework process of ith item (units per unit time) Unit holding cost for each scrap of ith item (units per unit time)

Since both the random defective rate and the scrap rate are in [0, 1] and [0, Qiθixi] are the scrap items randomly produced during the imperfect rework process, it follows that the production cycle length T is not a constant. Solving the total inventory cost per year TC(Q, B) yields: Holding Cost of Perfect Quality Items

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ Disposal Cost   Rework Cost  zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{  2  I Max  3  d i t1i þ t 5i  1 5  I   I i þ I Max i ti þ ti þ NC Ri xi Qi þ NC di xi Qi θi þ Nhi i t 1i þ ti þ i ti þ 2 2 2 2  2  P1i t i  2  ð1  βi ÞBi  4  β i Bi  4 5  ti t þ t i þ Nb t i þ NK i Nh1i þ NC bi πi |{z} 2 2 i 2 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} Setup |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Cost

Production Cost

zfflffl}|fflffl{ NC i Qi

TCðQ,BÞ ¼ þ

Holding Costof Imperfect Quality Items

Back Ordered Cost

LostSale Cost

ð4:358Þ The maximum capacity of the single machine and the service rate are two constraints of the model that are described in the following subsections. Since t 1i þ t 2i þ t 5i and Si are the production uptimes, the rework time and the setup time of the ith product, respectively, the summation uptimes, rework and P  of the total production P setup time (for all products) is ni¼1 t 1i þ t 2i þ t 5i þ ni¼1 tsi which is smaller or equal to the period length (T ). Therefore, one has: n  X i¼1

n  X t 1i þ t 2i þ t 5i þ tsi  T

ð4:359Þ

i

n n X X Di ðP1i þ di Þ tsi  T þ Pi P1i ð1  θi xi Þ i i Pn i tsi T Pn D ðP1i þdi Þ ¼ T Min 1  i Pi P1i ð1θi xi Þ

ð4:360Þ

Since the shortage quantity of the ith product per period is Bj, the annual demand of the jth product is Dj, the number of periods in each year is N, and the safety factor of allowable shortage is SL. Therefore, the service level constraint is as follows: n X NBi  SL Di i

4.5 Partial Backordering

329

Fig. 4.26 EPQ inventory system (Taleizadeh et al. 2010)

Pn T

C 2j j 2C 1j D j

SL 

Pn

C 3j j 2C 1j D j

¼ T SL

ð4:361Þ

Finally, the final model is: Min : TCðT, Bi Þ ¼

n X i¼1

Pn

s:t: : T   1

SL 

D ðP1i þ d i Þ

i¼1

n

n

n

n



Pi P1i ð1  θi xi Þ

C2j

j¼1

T

i¼1 tsi

Pn

Pn

X X XK ðBi Þ2 X 2 Bi X 3 i Ci  C i Bi þ C 4i T þ C5i þ  T T i¼1 T i¼1 i¼1 i¼1 i¼1 n

C1i

2C 1j D j

Pn j¼1

C 3j

! ¼ T SL

2C 1j D j

T, Bi  0 8i,i ¼ 1, 2, .. ., n ð4:362Þ In which:

330

4 Rework

! J i ð1  βi Þ2 ð1  βi ÞRi βi 2 ðPi  di Þ ¼hi þ þ ð1  θi xi Þ2 ð1  θi xi Þ 2ðPi  d i  Di Þ

2 !

d i ð1  β i Þ βi Pi  di  ð1  βi ÞDi 1 þ Cbi þ h1i 2 Di ðPi  di  Di Þ 2P1i ð1  θi xi ÞPi

ð1  βi Þb π i Pi  di  ð1  βi ÞDi þ ð4:363aÞ 2 Di ðPi  di  Di Þ





C Ri x i ð 1  β i Þ C d i xi θ i ð 1  β i Þ Ci ð1  βi Þ C 2i ¼ þ þ >0 ð4:363bÞ ð 1  θ i xi Þ ð 1  θ i xi Þ ð 1  θ i xi Þ

2J i Di ð1  βi Þ Di Ri 3 C i ¼ hi þ ð1  θi xi Þ ð 1  θ i xi Þ 2

! 2Di ð1  βi Þ 1 di þ h1i ð4:363cÞ 2P1i Pi ð 1  θ i xi Þ 2

2 ! 2 J D 1 d D i i i i >0 ð4:363dÞ þ h1i C 4i ¼ hi 2P1i Pi ð 1  θ i xi Þ ð 1  θ i xi Þ 2 C1i

C 5i ¼ μi ¼ 1  ¼

C Ri xi Di C d xi θ i D i C i Di þ þ i >0 ð1  θi xi Þ ð1  θi xi Þ ð1  θi xi Þ

Di d i ðP1i  d1i  Di Þ  , 2P1i Pi Pi

ðPi  Di Þ μi di U þ þ i , P1i Pi 2Di 2Pi 2

Ui ¼ 1 

Ri ¼

Di di ðd1i þ Di Þ  , P1i Pi Pi

βi d i βU β þ i iþ i P1i Pi Di Pi

ð4:363eÞ Ji ð4:363fÞ

Since the Hessian  P matrix ofPobjectivefunction is positive for all nonzero Bi and T, X  H  X T ¼ 2 ni¼1 K i þ ni¼1 C 2i Bi =T  0, TC(T, Bi) is convex. To derive the optimal values of the decision variables, take the partial differentiations of TC(T, Bi) with respect to T and Bi (for details, see Appendix 2 of Taleizadeh and Wee (2015)): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   uP Pn ðC2i Þ2 u n u i¼1 K i  i¼1 4C1 i u ∂TCðT, Bi Þ   ¼0!T¼u 2 3 t P P ∂T ðC i Þ n n 4 C  i¼1 i i¼1 4C 1

ð4:364Þ

i

C 3 T  þ C2i ∂TCðT, Bi Þ ¼ 0 ! Bi ¼ i ∂Bi 2C 1i Then,

ð4:365Þ

4.6 Multi-delivery

331

Qi ¼

Di T  ð1  βi ÞBi , ð 1  θ i xi Þ

0  θi  1

ð4:366Þ

To ensure production of all products by a machine and satisfy service level constraint of each, the following steps to derive T*, Bi , and Qi must be performed (Taleizadeh et al. 2010): Step 1. Check for feasibility. If

Pn

Di ðP1i þd i Þ i¼1 Pi P1i ð1θi xi Þ

< 1 and

Pn

C 3i i¼1 2C 1i Di

< SL, go to step 2;

else the problem is infeasible. Step 2. Calculate T using Eq. (4.364). If T  0, go the Step 3; else the problem is infeasible. Step 3. Calculate TSL using Eq. (4.358). Step 4. Calculate TMin using Eq. (4.360). Step 5. If T  Max {TMin, TSL}, then T* ¼ T; else T ¼ Max {TMin, TSL}. Step 6. Calculate Bi , 8i ¼ 1, 2, . . . , n using Eq. (4.365). Step 7. Calculate Qi , 8i ¼ 1, 2, . . . , n using Eq. (4.366). Step 8. Terminate procedure. Examples 4.18 and 4.19 Taleizadeh et al. (2010) considered five-product inventory control problem where the general and the specific data are given in Tables 4.19, 4.20, and 4.21. They considered two numerical examples with uniform and normal distributions for xi and θi. Tables 4.22 and 4.23 show the best results for the two numerical examples. The safety level, SL, is 30%

4.6 4.6.1

Multi-delivery Multi-delivery Policy and Quality Assurance

Chiu et al. (2009) developed a multi-delivery policy and quality assurance into an imperfect economic production quantity (EPQ) model with scrap and rework. A portion of non-conforming items produced is considered to be scrap, while the other is assumed to be repairable and is reworked in each cycle when regular production ends. Finished items can only be delivered to customers if whole lot is quality assured after rework. Fixed quantity multiple installments of finished batch are

332

4 Rework

delivered by request to customers at a fixed interval of time. Expected integrated cost function per unit time is derived. This paper incorporates a multi-delivery policy and quality assurance into an imperfect EPQ model with scrap and rework. Consider that during regular production time and x portion of defective items is produced randomly, at a production rate d. Among defective items, a portion is assumed to be scrap and the rest can be reworked and repaired at a rate P1, in each cycle after a production run (see Fig. 4.27). Similar to previous case, (P  d  D) > 0 or (1  x  D/P) > 0, where d ¼ Px (Chiu et al. 2009). Also some new notations which are specifically used for this proposed model are presented in Table 4.24. It is assumed that finished items can only be delivered to customers if whole lot is quality assured at the end of rework. Fixed quantity of n installments of finished batch is delivered by request to customers, at a fixed interval of time during production downtime t3 (Fig. 4.27): T ¼ t1 þ t2 þ t3

ð4:367Þ

Q H1 ¼ P Pd

ð4:368Þ

t1 ¼

xQð1  θÞ P1

ð1  θxÞ 1 xð1  θÞ t 3 ¼ nt n ¼ T  ðt 1 þ t 2 Þ ¼ Q   D P1 P t2 ¼

H 1 ¼ ðP  dÞt 1 ¼ ðP  dÞ

Q ¼ ð1  xÞQ P

H ¼ H 1 þ P1 t 2 ¼ ð1  θxÞQ

ð4:369Þ ð4:370Þ ð4:371Þ ð4:372Þ

where T, cycle length; H, maximum level of on-hand inventory when regular production process ends; H1, maximum level of on-hand inventory when rework process finishes; Q, production lot size; t1, production uptime for proposed EPQ model; t2, time for reworking of defective items; t3, time for delivering all quality assured finished products; tn, fixed interval of time between each installment of finished products delivered during t3; I(t), on-hand inventory of perfect-quality items at time t; and Id(t), on-hand inventory of defective items at time t (Chiu et al. 2009). On-hand inventory of defective items during production uptime t1 and reworking time t (Fig. 4.28) shows that maximum level of on-hand defective items is dt1 and (Chiu et al. 2009): d  t 1 ¼ P  x  t 1 ¼ xQ

ð4:373Þ

4.6 Multi-delivery

333

Table 4.19 General data for Examples 4.18 and 4.19 (Taleizadeh et al. 2010) P 1 2 3 4 5

Di 600 700 800 900 1000

Pi 4000 4500 5000 5500 6000

P1i 2000 2500 3000 3500 4000

tsi 0.003 0.004 0.005 0.006 0.007

Ki 500 450 400 350 300

βi 0.3 0.4 0.5 0.6 0.7

Ci 15 12 10 8 6

hi 5 4 3 2 1

h1i 2 2 2 2 2

C di 3 3 3 3 3

C bi 10 8 6 4 2

b πi 12 10 8 6 4

C Ri 1 2 3 4 5

Table 4.20 Specific data for Example 4.18 (Taleizadeh et al. 2010) Product 1 2 3 4 5

Xi U[ai, bi] ai bi 0 0.05 0 0.1 0 0.15 0 0.2 0 0.25

E[Xi] 0.025 0.05 0.075 0.1 0.125

di ¼ PiE[Xi] 100 225 375 550 750

θi U[ai, bi] ai bi 0 0.15 0 0.2 0 0.25 0 0.3 0 0.35

E[θi] 0.075 0.1 0.125 0.15 0.175

d1i ¼ P1iE[θi] 150 250 375 525 700

E[θi] 1 2 3 4 5

d1i ¼ P1iE[θi] 0.01 0.02 0.03 0.04 0.05

Table 4.21 Specific data for Example 4.19 (Taleizadeh et al. 2010) Product 1 2 3 4 5

Xi U[ai, bi] ai bi 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05

E[Xi] 40 90 150 220 300

di ¼ PiE[Xi] 0.1 0.15 0.2 0.25 0.3

θi U[ai, bi] ai bi 0.01 200 0.02 375 0.03 600 0.04 875 0.05 1200

A θ portion among non-conforming items is assumed to be scrap (Eq. 4.374). Other reparable portion (1  θ) is reworked right after production uptime t1 ends (Chiu et al. 2009): θd  t 1 ¼ θ  P  x  t 1 ¼ θxQ

ð4:374Þ

Total costs per cycle TC(Q) consist of setup cost, variable production cost, variable rework cost, disposal cost, fixed and variable delivery cost, holding cost during t1 and t2, variable holding cost for items reworked, and holding cost for finished goods during delivery time t3 where n fixed-quantity installments of finished batch are delivered by request to customers at a fixed interval of time. Cost for each delivery is (Chiu et al. 2009): K S þ CT

 H n

Total delivery costs for n shipments in a cycle are:

ð4:375Þ

334

4 Rework

Table 4.22 The best results for Example 4.18 (Taleizadeh et al. 2010) Uniform TMin 0.2617

Product 1 2 3 4 5

T 0.5748

TSL 0.2574

T* 0.5748

Bi 25.14 13.68 9.02 6.8 6.52

Qi 327.92 396.15 459.66 522.46 585.68

Z 46,998

Qi 329.34 397.37 460.43 522.46 584.54

Z 46,017

Table 4.23 The best results for Example 4.19 (Taleizadeh et al. 2010) Uniform TMin 0.1576

Product 1 2 3 4 5

T 0.5778

TSL 0.2585

T* 0.5778

Bi 25.2 13.74 9.08 6.86 6.59

h

i H ¼ nK S þ C T H ¼ nK S þ CT Qð1  θxÞ n K S þ CT n

ð4:376Þ

Therefore, TC(Q) (Appendix A of Chiu et al. (2009)) is: Production Cost Setup Cost z}|{ z}|{ TCðQÞ ¼ K þ CQ þ

P t h1 1 2 ðt 2 Þ 2 |fflfflfflfflfflffl{zfflfflfflfflfflffl}

Fixed Delivery Cost Rework Cost zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ zfflfflfflffl}|fflfflfflffl{ zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ z}|{ þ C R ½xð1  θÞQ þ C d ½xθQ þ C T ½Qð1  θxÞ þ nK S  

 H 1 þ dt 1 H1 þ H n1 Ht 3 þh ðt 1 Þ þ ðt 2 Þ þ h 2 2 2n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Holding Cost of Reworked Items

Disposal Cost

Delivery Cost

Holding Cost

ð4:377Þ Since x of defective items is assumed to be a random variable with a known probability density function, for randomness of defective rate, one can use values of x in inventory cost analysis. Substituting all related parameters from Eqs. (4.367) to (4.377) in TC(Q), one obtains expected production–inventory–delivery cost per unit time, E[TCU(Q)] (Appendix B of Chiu et al. (2009)) as:

4.6 Multi-delivery

335

Fig. 4.27 On-hand inventory of perfect-quality items in EPQ model with multi-delivery policy, scrap, and rework (Chiu et al. 2009) Table 4.24 New notations of given problem P1 t1 t2 n tn

Rework rate of non-conforming items in units per unit time (units per unit time) Production uptime for product The rework time for non-conforming product Number of fixed quantity installments of the finished batch to be delivered to customers in each cycle, it is assumed to be a constant for all products A fixed interval of time between each installment of finished products delivered.

E½TCUðQÞ ¼

E ½TCðQÞ ðK þ nK S Þ C E ½xð1  θÞD CD þ ¼ þ R 1  θE ½x Qð1  θE ½xÞ E ½T  ð1  θE ½xÞ Cd E½xθD h ðE ½xÞ2 QDð1  θÞ2 hQD þ 1 þ CT D þ 2Pð1  θE ½xÞ 2P1 ð1  θE½xÞ ð1  θE ½xÞ h

 i hQD 2 þ 2E ½x  ðE ½xÞ  θðE½xÞ2 ð1  θÞ 2P1 ð1  θE ½xÞ 

 n  1 hQð1  θE ½xÞ hQD hQE ½xð1  θÞD  þ  2 2P1 n 2P

þ

ð4:378Þ Optimal production lot size can be obtained by minimizing expected cost function E[TCU(Q)]. Differentiating E[TCU(Q)] with respect to the Q gives first and second derivative as (Chiu et al. 2009):

336

4 Rework

Fig. 4.28 On-hand inventory of defective items in EPQ model with multidelivery policy, scrap and rework (Chiu et al. 2009)

dE ½TCUðQÞ KD nK S D hD ¼ 2 þ þ dQ Q ð1θE½xÞ Q2 ð1θE ½xÞ 2Pð1θE ½xÞ h

 i hD þ 2E ½x ðE ½xÞ2 θðE ½xÞ2 ð1θÞ 2P1 ð1θE½xÞ  2 2

 n1 hð1θE½xÞ hD hE ½xð1θÞD h1 ðE ½xÞ Dð1θÞ þ þ   n 2P 2 2P1 2P1 ð1θE½xÞ ð4:379Þ d 2 E ½TCUðQÞ 2ðK þ nK S ÞD ¼ 3 Q ð1  θE½xÞ dQ2

ð4:380Þ

Equation (4.380) is positive because K, n, KS, D, Q, and (1 – θE[x]) are all positive. Second derivative of E[TCU(Q)] with respect to Q (Eq. 4.380) is greater than zero, and hence E[TCU(Q)] is a convex function for all Q different from zero. Optimal production lot size Q* can be obtained by setting first derivative (Eq. 4.379) of E[TCU(Q)] equal to zero (Chiu et al. 2009): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2ðK þ nK S ÞD 9 Q ¼u i u8 hD hD h 2 2 u> > 2E ½x  ðE ½xÞ  θðE ½xÞ ð1  θÞ þ > u> < = P P 1 u 

 u 2 2

 t> h ðE ½xÞ Dð1  θÞ > > n1 > D E ½xð1  θÞD :þ ; hð1  θE ½xÞ2  h ð1  θE½xÞ þ 1 þ P1 P1 n P 

ð4:381Þ Example 4.20 Chiu et al. (2009) presented an example in which it is assumed that a product can be manufactured at a rate of 60,000 units per year and this item has experienced a flat demand rate of 3400 units per year. During production uptime, random defective rate is assumed to be uniformly distributed over the interval [0, 0.3]. Among defective items, a portion θ ¼ 0.1 is considered to be scrap, and other portion can be reworked and repaired, at a rate P1 ¼ 2100 units per year. Additional parameters considered by this example are given as follows: CR ¼ $60 per item

4.6 Multi-delivery

337

reworked; Cd ¼ $20 per scrap item; C ¼ $100 per item; K ¼ $20,000 per production run; h ¼ $20 per item per year; h1 ¼ $40 per item reworked per unit time (year); n ¼ 4 installments of finished batch are delivered per cycle; KS ¼ $4350 per shipment, a fixed cost; and CT ¼ $0.1 per item delivered. Optimal batch size Q* ¼ 3495 can be obtained from Eq. (4.381) and long-run average production–inventory delivery costs per year E[TCU(Q*)] ¼ $448,390 from Eq. (4.378) (Chiu et al. 2009).

4.6.2

Multi-delivery and Partial Rework

Chiu et al. (2012) studied an extended EPQ model which incorporated quality assurance issue and a multi-delivery policy into the classic EPQ model. The quality assurance issue is in regard to the production system which has an x portion of random defective items produced at a production rate d, and among defective items, a θ portion is assumed to be scrap, and the other (1  θ) portion can be reworked and repaired at a rate P1, within the same cycle when regular production ends, while the multi-delivery policy is with regard to fixed quantity of n installments of the finished batch which are delivered to customer at a fixed interval of time during the production downtime (i.e., when the whole lot is quality assured at the end of rework) (Chiu et al. 2012). Figure 4.29 depicts the on-hand inventory of perfect-quality items of the proposed model. Figure 4.30 illustrates the expected reduction in inventory holding costs (in yellow/shade areas) of the proposed model (in blue). Based on the description of the proposed model and Fig. 4.29, the following expressions can be derived accordingly (Chiu et al. 2012): Q P

ð4:382Þ

xQð1  θÞ P1

ð4:383Þ

t1 ¼ t2 ¼

t 3 ¼ nt n ¼ T  ðt 1 þ t 2 Þ T ¼ t1 þ t2 þ t3 ¼ t¼

Q ð1  θxÞ D

Dðt 1 þ t 2 Þ PD

ð4:384Þ ð4:385Þ ð4:386Þ

H ¼ ðP  DÞt ¼ Dðt 1 þ t 2 Þ

ð4:387Þ

H 1 ¼ Qð1  xÞ  Dðt 1 þ t 2 Þ

ð4:388Þ

H 2 ¼ H 1 þ P1 t 2

ð4:389Þ

The on-hand inventory of defective items during production uptime t1 and reworking time t2 is illustrated in Fig. 4.31. It is noted that the maximum level of

338 Fig. 4.29 On-hand inventory of perfect-quality items in EPQ model with (n + 1) delivery policy and partial rework (Chiu et al. 2012)

4 Rework

I(t) P1

H2 H1

–D

P–d

H

t

t

t1–t t1

t2

tn

t3

Time

T

Fig. 4.30 Expected reduction in inventory holding costs (in yellow) of the proposed model in comparison with model of Chiu et al. (2012)

defective items is dt1. A θ portion among non-conforming items is assumed to be scrap items as shown in Eq. (4.390). Other repairable portion (1  θ) is reworked right after the production uptime t1 ends (Chiu et al. 2012): θDt 1 ¼ θPxt 1 ¼ θxQ

ð4:390Þ

Total production–inventory–delivery costs per cycle TC(Q) consist of setup cost, variable production cost, variable rework cost, disposal cost, fixed and variable delivery cost, holding cost for perfect-quality items during production uptime t1 and reworking time t2, holding cost for defective items during uptime t1, variable holding cost for items reworked during t2, and holding cost for finished goods during

4.6 Multi-delivery

339

Fig. 4.31 On-hand inventory of nonconforming items in EPQ model with (n + 1) delivery policy and partial rework (Chiu et al. 2012)

the delivery time t3 where n fixed-quantity installments of the finished batch are delivered to customers at a fixed interval of time (for computation of the last term refer to Appendix of Chiu et al. (2009): Fixed Delivery

Production

Cost Cost Disposal Cost Delivery Cost ReworkCost zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflffl}|fflfflfflfflfflffl{ z}|{ z}|{ TCðQ, BÞ ¼ K þ CQ þ C R ½xð1  θÞQ þ C d ðθxQÞ þ ðn þ 1ÞK S þ C T ½xð1  θÞQ   D ðt 1 Þ H H1 þ H2 H n1 þh ðt Þ þ ðt 2 Þ þ 1 ðt 1  tÞ þ ðt 1 Þ þ h ðH:t 3 Þ 2 2 2 2 2n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Setup Cost

HoldingCost

Dt ð1  θÞ þh1 1 ðt 2 Þ 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Holding Cost of Reworked Items

ð4:391Þ Because x proportion of defective items is assumed to be a random variable with a known probability density function, one could use the expected values of x in the related cost analysis: E½TCUðT Þ ¼ E½TCðT Þ=E½T 

ð4:392Þ

Similar to previous case, Chiu et al. (2012) showed that the cost function is convex, so the root of first derivative of total cost with respect to Q yields to optimal values of production quantity as presented in Eq. (4.432):

340

4 Rework

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2½ðn þ 1ÞK S þ K D 9   Q ¼ u u 8 



 2Dð1  θÞ 2 ð 1  θ Þ u > > D 2D 1 1 4D x x > > > u >  þ E E  2E ð x Þ þ E > > > u > P1 P1 1x P P2 1  x P P 1x > > > u > > > > u >   < = 2 2 u u h  DE ðxÞ ð1  θÞ 1 þ D þ ½1  θE ðxÞ2  D½1  2θE½x u > > P1 P1 P > > u > > u >    > > h i > > u > > 1 > DE ð x Þ ð 1  θ Þ DE ð x Þ ð 1  θ Þ D D 2D 2 > u > > > ½ 1  θE ð x Þ   2  2 ½ 1  θE ð x Þ  þ þ  þ : ; u P P n P P P 1 1 u u t h DE ðxÞ2 ð1  θÞ2 þ 1 P1

ð4:393Þ Example 4.21 Chiu et al. (2012) used again the data of their previous work (Chiu et al. 2009) here to demonstrate the aforementioned results derived by presented alternative approach. Using Eq. (4.393) the optimal production quantity is Q ¼ 4219 and E[TCU(Q)] ¼ $435,712. It is noted that the overall reduction in production–inventory–delivery costs amounts to $12,678, or 13.25% of total other related costs.

4.6.3

Multi-delivery Single Machine

Chiu et al. (2015) developed a multi-item EPQ model with scrap, rework, and multiple deliveries. Consider that L products are made in turn on a single machine with the purpose of maximizing the machine utilization. All items made are screened, and inspection cost for each item is included in the unit production cost Ci. During production process for each product i (where i ¼ 1, 2, . . ., L ), an xi portion of non-conforming items is produced randomly at a rate di. Among these non-conforming items, a θi portion is considered to be scrap items, and the other portion can be reworked and repaired at a rate of P2i right after the end of regular production process in each cycle with an additional cost CRi. Under the normal operation, the constant production rate P1i for product i must satisfy (P1i  di  Di) > 0, where Di is the demand rate for product i per year and di can be expressed as di ¼ xiP1i. Unlike classic EPQ model which assumes a continuous issuing policy for meeting product demands, this study adopts a multi-delivery policy. It is assumed that finished goods for each product i can only be delivered to customers if whole production lot is quality assured in the end of rework process for each product i. Fixed quantity n installments of the finished batch are delivered at a fixed interval of time during delivery time t3i (refer to Fig. 4.32). Also some new notations which are specifically used for this proposed model are presented in Table 4.25. One can obtain the following formulas directly from Figs. 4.32 and 4.33 (Chiu et al. 2015):

4.6 Multi-delivery

341

t 1i ¼

Qi H 1i ¼ P1i P1i  di

ð4:394Þ

xi Qi ð1  θi Þ P2i

ð4:395Þ

t 2i ¼

t 3i ¼ nt ni ¼ T  ðt 1i þ t 2i Þ T ¼ t 1i þ t 2i þ t 3i ¼

ð4:396Þ

Qi ð1  θi xi Þ Di

ð4:397Þ

H 1i ¼ ðP1i  d i Þt ii

ð4:398Þ

H 2i ¼ H 1i þ P2i t 2i

ð4:399Þ

d i t 1i ¼ xi Qi

ð4:400Þ

Total delivery cost for product i (n shipments) in a cycle is: nK Si þ CTi Qi ð1  θi xi Þ

ð4:401Þ

Holding costs for finished products during the t3, where n fixed-quantity installments of the finished batch are delivered to customers at a fixed interval of time, is:

I (t)i P22

H22 H12 H21 H21 H1

P21

P12-d2

P2L P1L-dL

P11-d1

P1L-d1

–λ 1

tn1

t11

t21

t12

Time

t31

t22

t1L

idle

t2L time

t11 t21

T Fig. 4.32 On-hand perfect-quality inventory for product i in the proposed multi-item EPQ model under a common cycle policy (Chiu et al. 2015)

342

4 Rework

Table 4.25 New notations of given problem P1i P2i K Si

The production rate of ith item(units per unit time) Rework rate of non-conforming items in units per unit time ith item (units per unit time) Fixed delivery cost per shipment for product I ($/shipment)

C Ti t1i t2i n

Unit shipping cost for product I ($/units)

tni h1i

Production uptime for product i in the proposed EPQ model (time) The rework time for product i in the proposed EPQ model (time) Number of fixed quantity installments of the finished batch to be delivered to customers in each cycle, it is assumed to be a constant for all products A fixed interval of time between each installment of finished products delivered during t2i, for product i (time) Unit holding cost for each reworked item ($/units per unit time)

Fig. 4.33 On-hand inventory of defective items for product i in the proposed multi-item EPQ model under a common cycle policy (Chiu et al. 2015)

ID(t)i d12t12 d11t11

q 2 portion q1

d11

t11

d12

-P22

-P21

t21

d1L

q L portion

-P2L

t12 t22

t1L t2L idle time

Time

T*

hi

 n1 H 2i t 3i 2n

ð4:402Þ

Total production–inventory–delivery costs per cycle TC(Qi) for L products consist of the variable production cost, setup cost, rework cost, fixed and variable delivery cost, holding cost during production uptime t1i and rework time t2i, and holding cost for finished goods kept during the delivery time t3. Therefore, total TC (Qi) for L products are (Chiu et al. 2015):

4.6 Multi-delivery L X

TCðQi Þ ¼

i

343

L X i¼1

9 8 Disposal Cost Rework Cost Setup Cost Production Cost > > zfflfflfflfflfflfflfflfflfflfflfflffl ffl }|fflfflfflfflfflfflfflfflfflfflfflffl ffl { zfflfflfflfflfflffl}|fflfflfflfflfflffl{ > > zffl}|ffl{ z}|{ > > > > > > C Q þ K þ C ½ x ð 1  θ ÞQ  þ C ½ x θ Q  i i R i i d i i > > i i i i i > > > > > > Holding Cost of Reworked Items > > Fixed Delivery Cost Shipment Cost > > zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ = < z}|{ zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ h i P t 2i 2i S T  þ nK i þ C i ½ Q i ð1  θ i x i Þ þ h1 i ðt 2i Þ > > 2 > > > > > > h i > > H þ D t H þ H n  1 > > 1i i 1i 1i 2i > > þhi ðt 1i Þ þ ðt 2i Þ þ ðI 2i t 3i Þ > > > > 2n 2 2 > > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl } ; : Holding Cost

ð4:403Þ To take the randomness of defective rate x into account, by applying the expected values of x in the cost analysis and substituting all variables, the following expected E[TCU(Q)] can be obtained: " E ½TCUðQÞ ¼E

L X

# TCðQi Þ

i¼1

1 E ½T 

8   9 E ½xi  E ½ xi  1 KD 1 > > > > Di C i þ C Ri ð1  θi Þ  þ C di θi þ C Ti Di þ i i > > > Qi 1  θi  E ½xi  > 1  θi  E ½xi  1  θ i  E ½ xi  1  θi  E ½xi  > > > > > > > > > >  

L 2 2 < = S X nK i Di h1i Qi Di ð1  θi Þ E ðxi Þ hi Qi Di 1  θi 1  θi 1 ½  þ þ þ E x þ ¼ i 2 2 P nP 2 Q ½ 1  θ E ð x Þ  1  θ  E ½ x  E ð x Þ 1  θ > Pi 2i 2i i i i i i i > > i > i¼1 > > > > >

2

 > >  >

 > > θ θ ð 1  θ Þ 1 1 1 1 θ 2θ 1  θ > > i i 2 2 i i i i > > þ E ½xi   E ½xi   þ þ 1  E ½xi   þ : ; Di nDi nP1i n P1i Di Di P2i nP2i

ð4:404Þ where E[T] ¼ Qi[1  θiE(xi)]/Di. Replacing Qi with T yields to: E½TCUðT Þ ¼

L X i¼1

9 8 CRi ð1  θi ÞE ½xi  C di θ i C i D i C i Di T > > > D þ þ þ C > i i > > > > 1  θ i  E ½ xi  1  θi  E ½xi  1  θ i  E ½ xi  > > > > > >

> > 2 2 S 2 > > nK h TD ð 1  θ Þ E ð x Þ K > > 1 i i i i i i > > þ þ > > > > 2 > >T T P2i 2½1  θi E ðxi Þ = < " #  2 θi E ½xi  1 > > > > þ hi TDi 1  1 þ þ > > > 2 > D D 2 n > > P n ½ 1  θ E ½ x   i i 1i i i n ½ 1  θ E ½ x   P > > 1i i i > > > > " # > > > > 2 > > > > h TD ð 1  θ ÞE ½ x  ½ 1  E ½ x   ð 1  θ ÞE ½ x  i i i i i i > >þ i > > þ ; : 2 2 2 P2i n½1  θi E ½xi  P2i n½1  θi E ½xi  ð4:405Þ Let E 0i , E1i denote the following:

344

4 Rework

E 0i ¼

1 1  θ i  E ½ xi 

E 1i ¼

E ½ xi  1  θ i  E ½ xi 

ð4:406Þ

Equation (4.405) becomes (Chiu et al. 2015): E ½TCUðT Þ ¼

L X i¼1

9 8 KD 1 > > Ci Di E 0i þ CRi ð1  θi ÞE 1i þ C di θi E 1i þ CTi Di þ i i  > > > Qi 1  θi  E ½xi  > > > > > > > > >   2 > > 2 2 1 S > > h TD ð 1  θ Þ E > > nK K 1i i i i i > > i > > þ þ =

> hi TDi 1 E θi E i E i 1 > > > > þ  þ i þ > > > > D D 2 n P n P n i i 1i 1i > > > > > >   > > 2 1 0 1 > > h TD ð 1  θ ÞE ð 1  θ Þ ½ 1  E ½ x  E E > > i i i i i i i i > > ; :þ þ 2 P2i n P2i n ð4:407Þ Chiu et al. (2015) developed an algebraic derivation using the optimal cycle length derived as below: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  P  u 2 Li¼1 K i þ nK Si  u 0  1 T ¼u E 0i θi E 0i E 1i ð1  θi ÞE 1i ð1  θi Þ½1  E ½xi E 0i E 1i 1 u 2 1  þ þ þ þ u L B h i Di C uP B Di Di n P1i n P1i n P2i n P2i n C u B C   u @ 2 2 A ti¼1 h1i TD2i ð1  θi Þ E 1i þ 2P2i

ð4:408Þ Example 4.22 Chiu et al. (2015) considered a production schedule is to produce five products in turn on a single machine using a common production cycle policy. Production rate P1i for each product is 58,000, 59,000, 60,000, 61,000, and 62,000, respectively, and annual demands Di for five different products are 3000, 3200, 3400, 3600, and 3800, respectively. Random defective rates xi during production uptime for each product follow the uniform distribution over the intervals of [0, 0.05], [0, 0.10], [0, 0.15], [0, 0.20], and [0, 0.25], respectively. Among the defective items, di portion is scrap items where Di for five different products are 0, 0.025, 0.050, 0.075, and 0.100, respectively, and additional disposal costs are $20, $25, $30, $35, and $40 per scrapped item. The other portion of non-conforming items is assumed to be repairable at the reworking rates P2i of 1800, 2000, 2200, 2400, and

4.6 Multi-delivery

345

2600, respectively, with additional reworking costs of $50, $55, $60, $65, and $70 per reworked item. Other parameters used include: Ci ¼ Unit manufacturing costs are $80, $90, $100, $110, and $120, respectively. hi ¼ Unit holding costs are $10, $15, $20, $25, and $30, respectively. Ki ¼ Production setup costs are $3800, $3900, $4000, $4100, and $4200, respectively. h1i ¼ Unit holding costs per reworked are $30, $35, $40, $45, and $50, respectively. K Si ¼ The fixed delivery costs per shipment are $1800, $1900, $2000, $2100, and $2200. T C i ¼ Unit transportation costs are $0.1, $0.2, $0.3, $0.4, and $0.5, respectively. n ¼ Number of shipments per cycle, in this study it is assumed to be a constant 4. The optimal common production cycle time T* ¼ 0.6066 (years) can be computed by Eq. (4.408), and applying Eq. (4.407), one obtains the expected production–inventory–delivery costs per unit time for L products, E[TCU (T* ¼ 0.6066)] ¼ $2,015,921.

4.6.4

Multi-product Two Machines

Chiu et al. (2018) developed a multi-product two-machine imperfect inventory system with discrete delivery. Assumed L diverse products (where i ¼ 1, 2, . . ., L ) sharing a mutual part are to be produced using a two-machine fabrication scheme. Machine one (i.e., the stage 1) solely produces the common parts for all end products at a rate of P1,0 (see Fig. 4.34). Then, machine two (i.e., the stage 2) fabricates L diverse products at annual rate of P1,i, using a common cycle length strategy (see Fig. 4.35). Also some new notations which are specifically used for this proposed model are presented in Table 4.26. The objectives of the production–distribution plan are to meet annual demand rates Di, shorten fabrication cycle length, and minimize overall relevant costs. The main purpose of this model is to determine the optimal values of number of shipments transported to sales offices per cycle and period length. Under quality screening, random defective rate xi is observed in both production processes (where i ¼ 0, 1, 2, . . ., L; and i ¼ 0 stands for its status of stage 1 when all common parts were produced by machine one). Defective items are produced at a rate of d1,i. It is assumed that all defective items can be repaired by a follow-up rework process, at a rate of P2,i, right after the end of regular production processes (see Figs. 4.34, 4.35, and 4.36). To disallow shortages, this study assumes (P1,i – d1, i – Di) > 0. The proposed two-machine multi-product fabrication scheme with postponement aims at releasing the production workload of common parts from machine two. Therefore, the proposed scheme should have a more efficient result on fabricating

346

4 Rework

customized end products in the second stage. The proposed solution process starts with determining the optimal common production cycle time for machine two and then applying the obtained cycle length to machine one for production of all common parts in advance (see both Figs. 4.34 and 4.35). The following prerequisite condition must satisfy to ensure that machine two has sufficient capacity to fabricate and rework all L products under a common cycle length discipline: L X

ðt 1,i þ t 2,i Þ < T

or

i¼1

L X

 Qi

i¼1 L X

 Di

i¼1

 E ½ xi  1 þ

> > > > > > > > > > Reworked Items > > > > > > Fixed Delivery Cost Delivery Cost zfflfflfflfflfflfflfflfflfflfflfflffl ffl }|fflfflfflfflfflfflfflfflfflfflfflffl ffl { Setup Cost Rework Cost Production Cost > >   > > z}|{ zfflffl ffl }|fflffl ffl { zfflfflfflffl ffl }|fflfflfflffl ffl { > > zffl}|ffl{ z}|{ > > P2,i t 2,i S T > > > C i Q i þ K i þ C Ri ½ x i Q i  þ nK i þ Ci ½Qi  þ hR,i ðt 2,i Þ > > > > > 2 > > > > L < = X Holding Cost TC2 ðT,nÞ ¼ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   > > > i¼1 > I 1,i þ Di t 1,i I þI Q n1 > > > > ðt 1,i Þ þ 1,i 2,i ðt 1,2,i Þ þ ðI 2,i t 3,i Þ þ i t 1,i > > > þ h1,i > 2n 2 2 2 > > > > > >     > > T > > > > n Q  I t nðn þ 1Þ n nI i ðt 1,i þ t 2,i Þ i n,i > > i > > I þ þh t þ þ h x TQ > > 1,i i S,i i i n,i > > 2 2 2 |fflfflfflffl ffl {zfflfflfflffl ffl } > > > > > > : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Safety Stock Holding Cost ; HoldingCost at Customer Side

ð4:411Þ Substitute Eqs. (B.1) to (B.12) from Appendix B of Chiu et al. (2018) in Eq. (4.411) and take into account randomness of xi by using the expected values of x, and with further derivation, E[TCU2(T, n)] can be obtained as follows: E ½TCU2 ðT, nÞ ¼E ½TC2 ðT, nÞ=E½T  3 2   γ 2,i  K i þ nK Si h1,i TD2i

T þ C Ri Di E½xi  þ γ 1,i  L 6 C i þ C i Di þ 7 X T 2 n 7 6 ¼ 7 6

2 2 2 5 4 γ h TD ð E ½ x  Þ h TD E ½ x  1 R,i i 3,i i 1,i i¼1 i i þ þ hS,i E½xi TDi þ   2P2,i 2 n γ 1,i γ 2,i

 γ 1i ¼

E ½ xi  1 1   P2,i Di P1,i



ð4:412Þ

4.6 Multi-delivery

349

Fig. 4.37 On-hand inventory level of common parts waiting to be fabricated into customized end products in stage 2 of the proposed study (Chiu et al. 2018)

 γ 2i ¼

2

E ½xi  1 E ½ xi  1  þ þ P2,i Di P1,i P2,i

 ð4:413Þ

In stage 1, machine one has to make enough common parts in advance for the fabrication of L diverse end products. Hence, machine one must start producing common parts (t1,0 + t2,0) ahead of time (see Fig. 4.35). The basic formulas displayed in Appendix C of Chiu et al. (2018) can also be observed directly from Figs. 4.35, 4.37, and 4.38. Similarly, machine one must have sufficient capacity to produce and rework all common intermediate parts. That is, the following prerequisite condition must satisfy (Chiu et al. 2018): 

 Q0 E ½x0  ðt 1,0 þ t 2,0 Þ < T or þ

> > > > t C Q þ K þ C ½ x Q  þ h ð Þ 0 0 R 0 R,0 2,0 0 0 0 > > = L < 2 X " # TC1 ðT, nÞ ¼ X > > i¼1 > > > þh1,0 I 1,0 þ D0 t 1,0 ðt 1,0 Þ þ I 1,0 þ I 2,0 ðt 2,0 Þ þ I i ðt 1,i þ t 2,i Þ þ hS,0 x0 TQ0 > > > ; : 2 2 i

ð4:416Þ Substitute previous in Eq. (4.416) and take into account the random defective rate x0 by using the expected values of x0, and with further derivation, E[TCU1(T, n)] can be derived as follows (Chiu et al. 2018): E ½TCU1 ðT, nÞ ¼ E½TC1 ðT, nÞ=E ½T 

ð4:417Þ

Therefore, total relevant cost per unit time for the proposed study, E[TCU(T, n)], is (Chiu et al. 2018):

4.6 Multi-delivery

351

E ½TCUðT, nÞ ¼ E ½TCU1 ðT, nÞ þ E½TCU2 ðT, nÞ

ð4:418Þ

To determine the optimal production–distribution policy, one must first prove the convexity of E[TCU2(T, n)]. Hessian matrix equations are employed to show if the following holds (see Appendix D of Chiu et al. 2018 for details). Since  condition  T PL 2K i ½T, nH ¼ i¼1 T  0, setting the partial derivative equal to zero gives: n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  u PL  S u i¼1 K i þ nK i  u T ¼uL n n o o h i γ 1i tP h D2i 1 γ 1i hR,i D2i ðE ½xi Þ2 h3,i D2i E ½xi  1 γ  þ þ þ D E ½ x  þ þ h S,i i i i P1,i 2 n 2P2,i 2 n P2,i i¼1

ð4:419Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi i u  PL PL hD2i u 1 u i¼1 K i i¼1 2 ðh3,i  h1,i Þγ i o i n ¼ u tPL S nPL h1,i D2i 2 hR,i D2i ðE½xi Þ2 h3,i D2i h 1 γ1 þ 2 P1,i þ EP½2,ixi  þ ni þ hS,i Di E ½xi  i¼1 K i i¼1 2 γ i þ 2P2 2,i

ð4:420Þ Example 4.23 Chiu et al. (2018) presented an example using the following values for parameters in stage 1: K0 ¼ $8500, C0 ¼ $40, CR,0 ¼ $25, h1,0 ¼ hS,0 ¼ $5, hR,0 ¼ $15, and x0 uniformly over the range [0, 0.04]. Consequently, the following values for parameters in stage 2, Ci ¼ $80, $70, $60, $50, and $40; CR,i ¼ $45, $40, $35, $30, and $25; xi over the ranges [0,0.21], [0, 0.16], [0, 0.11], [0, 0.06], and [0, 0.01]; and Ki ¼ $10,500, $10,000, $9500, $9000, and $8500, respectively, are considered. P1,i ¼ 128,276, 124,068, 120,000, 116,066, and 112,258 (which also are based on the similar 1/α relationship between P1,i and P1,0; i.e., P1,i ¼ 1/(1/P1,i – 1/P1,0)) and P2,i ¼ 102,621, 99,254, 96,000, 92,852, and 89,806 (similarly they are calculated by P2,i ¼ 1/(1/P2,i – 1/P2,0)), respectively. Also, K Si ¼ $2200, $2100, $2000, $1900, and $1800; h1,i ¼ $30, $25, $20, $15, and $10; hR,i ¼ $50, $45, $40, $35, and $30; C Ti ¼ $0.5, $0.4, $0.3, $0.2, and $0.1; h3,i ¼ $90, $85, $80, $75, and $70; and hS,i ¼ $30, $25, $20, $15, and $10, respectively. Finally the optimal fabrication-distribution decisions n* ¼ 3, T* ¼ 0.4453, and E[TCU(T*, n*)] ¼ $2,145,825 can be obtained.

4.6.5

Shipment Decisions for a Multi-product

Chiu et al. (2016) developed an imperfect inventory system to simultaneously determine the production and shipment decisions for a multi-item vendor–buyer integrated inventory system with a rework process. Fabricating multi-products on a

352

4 Rework

single machine with the aim of maximizing machine utilization is an operating goal of most manufacturing firms. In the proposed multi-product intra-supply chain system, the production rate is P1i per year and the annual demand rate is Di, where i ¼ 1, 2, . . ., L. All products made are checked for their quality, and the unit screening cost is included in the unit production cost Ci. It is also assumed that the production process can randomly produce xi portion of non-conforming items at a rate di, where di can be expressed as di ¼ xiP1i, and (P1i – di – Di) > 0 must be satisfied in order to sustain regular operations (i.e., avoid the occurrence of shortage). All defective items produced are reworked and fully repaired at the rate of P2i at the end of each production cycle, with additional rework cost CRi per item. After the rework process, the entire quality assured lot of each product i is transported to sales offices/customers under a multi-delivery policy, in which n fixed quantity installments of the lot are shipped at fixed intervals of time in t3i (Chiu et al. 2016). The schematic process of above description is presented in Figs. 4.39 and 4.40. Also some new notations which are specifically used for this proposed model are presented in Table 4.26 in Sect. 4.6.4. The on-hand inventory of product i stored at the sales offices/customers’ side is illustrated in Fig. 4.41. Accordingly, the sales offices’ holding costs along with delivery cost for all L products are included in the proposed cost analysis (Chiu et al. 2016). Using Figs. 4.39 and 4.41, the following formulas can be obtained (Chiu et al. 2016): H 1i ¼ ðP1i  di Þt 1i

ð4:421Þ

H 2i ¼ H 1i þ P2i t 2i

ð4:422Þ

t 1i ¼ t 2i

Qi P1i

xi Q i P2i

ð4:423Þ ð4:424Þ

t 3i ¼ nt ni ¼ T  ðt 1i þ t 2i Þ

ð4:425Þ

T ¼ t 1i þ t 2i þ t 3i

ð4:426Þ

d i t 1i ¼ xi Qi

ð4:427Þ

Total delivery cost of n shipments of product i at t3i is (Chiu et al. 2016): nK Si þ C Ti Qi

ð4:428Þ

From Fig. 4.28, the holding cost of the finished items of product i at t3 is (Chiu et al. 2016):

4.6 Multi-delivery

353

Fig. 4.39 On-hand inventory level of perfect-quality product i at time t in the proposed system (Chiu et al. 2016)

Fig. 4.40 On-hand inventory level of defective product i at time t in the proposed system (Chiu et al. 2016)

hi

 n1 H 2i t 3i 2n

ð4:429Þ

According to the proposed multi-delivery policy, when n fixed quantity (i.e., D) installments of finished lot of product i are transported to sales offices at a fixed time interval tni, the following formulas are obtained (Chiu et al. 2016):

354

4 Rework

Fig. 4.41 On-hand inventory level of product i stored at the sales offices at time t in the proposed system (Chiu et al. 2016)

t 3i n H QTi ¼ 2i n

ð4:430Þ

I i ¼ QTi  Di t ni

ð4:432Þ

t ni ¼

ð4:431Þ

The sales offices’ stock holding cost of product i is (Chiu et al. 2016):  h2i

QT  I i nðn þ 1Þ nI I i t ni n i t ni þ i ðt 1i þ t 2i Þ þ 2 2 2

 ð4:433Þ

Therefore, TC(Qi, n) for i ¼ 1, 2, . . ., L, which comprises the variable fabrication cost; setup cost; variable reworking cost; production units’ inventory holding cost during the periods t1i, t2i, and t3i (including holding cost of non-conforming items in

4.6 Multi-delivery

355

t1i); inventory holding cost of reworked items in t2i; fixed and variable transportation costs; and the stock holding cost from the sales offices/customers, is: 9 8 Holding Cost > > > > Fixed Delivery Shipment Rework Production > > > > > > > > of ReworkedItems > > > > Cost Cost Cost > > zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ Cost Setup Cost > >   z}|{ zfflffl ffl }|fflffl ffl { > > zfflfflfflfflffl}|fflfflfflfflffl{ zffl}|ffl{ z}|{ > > d t > > 1i 1i S T > nK i þ C i ½Qi  þ h1i C i Qi ðt2i Þ > þ K i þ C Ri ½xi Qi  þ > > > > 2 > > > > >   = < L L > X X H þ d t H þ H n  1 1i i 1i 1i 2i TCðQi , nÞ ¼ ðt1i Þ þ ðt 2i Þ þ ðH 2i t 3i Þ þhi > > 2n 2 2 > i¼1 i¼1 > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > > > > > > Holding Cost > > > >     > > T > > > > n Qi  I i t ni nðn þ 1Þ nI ð t þ t Þ i 1i 2i > > > > þh2i I þ t þ > > i ni > > 2 2 2 > > > > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ; : Sales Offices’ Stock Holding

ð4:434Þ Substituting relevant parameters from Eq. (4.433) in Eq. (4.434), using the expected values of x to take randomness of defective rate into account, and applying the renewal reward theorem, E[TCU(Qi, n)] is obtained as follows (Chiu et al. 2016):  9 8 QD KD D > > > C i Di þ i i þ C Ri ½E ðxi ÞDi  þ nK Si i þ C Ti ½Di  þ h1i i i E ðxi Þ2 > > > > > Qi Qi 2P2i > > > > > >   L < = 2 X Eðxi Þ 1 E ðxi Þ E ðxi Þ hi Qi Di 1 1 E ½TCUðQi , nÞ ¼ þ þ þ þ þ þ > > 2 P2i P2i Di n P1i n nP2i Di > i¼1 > > >   > > > > > > E ð x Þ E ð x Þ h Q D 1 1 1 i i > > ; : þ 2i i i   þ þ 2 Di n P1i n nP2i P1i P2i

ð4:435Þ Since Qi ¼ TDi, E ½TCUðT,nÞ ¼

L X i¼1

8  2 9 nK Si TDi > Ki 2 > T > > > C i Di þ þ CRi ½Eðxi ÞDi  þ ðE ðxi ÞÞ > þ C i ½Di  þ h1i > > > > T T 2P2i > > > > > >   = < 2 2 hi TDi 1 E ð xi Þ 1 E ð xi Þ ð E ð xi Þ Þ 1  þ þ þ þ þ þ > > P2i Di n P1i n nP2i Di 2 P2i > > > > > >   > > 2 > > h TD E ð x Þ E ð x Þ 1 1 1 > > 2i i i i > > þ þ   ; :þ P2i Di n P1i n nP2i P1i 2 ð4:436Þ Chiu et al. (2016) used matrix to prove the convexity of the total cost  Hessian  T PL 2K i function. Since ½T, nH ¼ i¼1 T  0 therefore, E[TCU(T, n)] is strictly n

356

4 Rework

convex for all T and n not equal to zero, and E[TCU(T, n)] has a minimum value. Then they set the first derivatives of E[TCU(T, n)] with respect to T and with respect to n equal to zeros and solved the linear system and derived: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  P u 2 i K i þ nK Si u n o h i h i o T ¼ tP n h D2 E½x 2 D2 E ½xi  E ½xi 2 2 1 þ h2 D2i P11i þ EP½x2ii  þ 1i Pi 2i i þ ni D1i  P11i  EP½x2ii  ðh2i  hi Þ i hi Di Di þ P2i  P2i 

ð4:437Þ ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i u P PL 2 1 E ½xi  1 u 2 ð K Þ  D   ð  h Þ h i 2i i i¼1 i Di i P1i P2i u h i o n ¼ tP  PL n 2 n 1 E½xi  E½xi 2 o h1i D2i E ½xi 2 E ½xi  L S 2 1 K h D þ  D þ þ h þ  i i Di 2 i P1i i¼1 i i¼1 P2i P2i P2i P2i ð4:438Þ Example 4.24 Chiu et al. (2016) considered a five-product example being manufactured in sequence on a machine under the common cycle time policy in a multi-product inventory system with a rework process. Their annual production rates P1i are 58,000, 59,000, 60,000, 61,000, and 62,000, respectively, and their annual demand rates Di are 3000, 3200, 3400, 3600, and 3800, respectively. For each product, the production units have experienced the random non-conforming rates that follow the uniform distribution over intervals of [0, 0.05], [0, 0.10], [0, 0.15], [0, 0.20], and [0, 0.25], respectively. All non-conforming products are assumed to be repairable and are reworked at the end of the regular production, at annual rates P2i of 46,400, 47,200, 48,000, 48,800, and 49,600, respectively. Additional costs for rework are $50, $55, $60, $65, and $70 per non-conforming product, respectively. Other values of system variables used in this example are listed below: Ki ¼ $17,000, $17,500, $18,000, $18,500, and $19,000. Ci ¼ $80, $90, $100, $110, and $120. hi ¼ $10, $15, $20, $25, and $30. h1i ¼ $30, $35, $40, $45, and $50. K Si ¼ $1800, $1900, $2000, $2100, and $2200. h2i ¼ $70, $75, $80, $85, and $90. C Ti ¼ $0.1, $0.2, $0.3, $0.4, and $0.5. First, in order to determine the number of deliveries, using Eq. (4.437), n* ¼ 4.4278. Practically, n* should be an integer number only, and to find the integer value of n*, one can plug n+ ¼ 5 and n ¼ 4 in Eq. (4.436) and obtain (T ¼ 0.6666, n+ ¼ 5) and (T ¼ 0.6193, n ¼ 4). Then, using Eq. (4.435) with these two different values to obtain E[TCU(0.6666, 5)] ¼ $2,229,865 and E[TCU(0.6193, 4)] ¼ $2,229,658, respectively. By choosing a policy with minimum cost, the optimal production–shipment policy for the proposed system is determined as n* ¼ 4, T* ¼ 0.6193, and E[TCU(T*, n*)] ¼ $2,229,658.

4.6 Multi-delivery

4.6.6

357

Pricing with Rework and Multiple Shipments

Consider a situation in which a manufacturing system produces perfect and defective items. The perfect items are ready to cover the customer’s demand. On the other hand, the defective items can be reworked after finishing the regular production process. At the end of the rework process, the manufacturer will deliver n equal size shipments to the customers during a specific time such that time between two consecutive deliveries during production downtime is equal (see Fig. 4.42). In addition to the aforementioned, it is important to point out that the manufacturing system randomly produces an x portion of defective items with a production rate d. Consequently, the production rate of defective items d can be expressed as d ¼ Px. All items manufactured are screened, and inspection cost per item is included in the unit manufacturing cost. Furthermore, it is assumed that all defective items are reworkable at the end of the regular production. The reworkable items are recovered at a rate of P1 in each cycle. Nonetheless, during the rework process, a θ1 portion of reworked items fails and becomes scrap. If d1 represents the production rate of scrap items during the rework process, then d1 is calculated as d1 ¼ p1θ1. The behavior of inventory level through time of proposed manufacturing problem is shown in Fig. 4.41. In the next section, the mathematical inventory model of manufacturing problem is presented (Taleizadeh et al. 2016). This problem reexamines the research work of Chiu et al. (2014). According to Fig. 4.42, the following equations can be derived: H 1 ¼ ðP  dÞt 1 ¼ ð1  xÞQ

ð4:439Þ

H ¼ H 1 þ ðP1  d1 Þt 2 ¼ ð1  θ1 xÞQ

ð4:440Þ

Q P xQ t2 ¼ P1

ð 1  θ 1 xÞ 1 x t3 ¼ T  t1  t2 ¼ Q   D P P1

ð4:441Þ

dt 1 ¼ Pxt 1 ¼ xQ

ð4:444Þ

θ1 dt 1 ¼ θ1 Pxt 1 ¼ θ1 xQ

ð4:445Þ

t1 ¼

ð4:442Þ ð4:443Þ

Thus, the profit function for each cycle production is given by (Taleizadeh et al. 2016):

358

4 Rework

Fig. 4.42 Inventory level of perfect-quality items in the EPQ model with a multi-delivery policy and rework (Taleizadeh et al. 2016) 2

3 Disposal Cost Delivery Cost Rework Cost Fixed Delivery Cost zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ 7 z}|{ z}|{ þ K þ C R ½xQ þ nK S þ Cd ½Qxθ1  þ C T Qð1  xθ1 Þ 7 7 h i h i 7 P1 t 2 H 1 þ Dt 1 H1 þ H n1 7 h1 ðt 2 Þ þh ðt 1 Þ þ ðt 2 Þ þ h ðHt 3 Þ 5 2n 2 2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl2{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}

Production Cost

6 Sale z}|{ 6 6 CPðs,QÞ ¼ sQ 6 6þ 4

z}|{ CQ

SetupCost

HoldingCost of ReworkedItems

HoldingCost

ð4:446Þ Interested readers may see in Chiu et al. (2014) the details of the derivation of the second term in the above profit function. Dividing by T, the annual profit function is obtained as follows: CPðs,QÞ T 2 3 CQ þ K þ C R ½xQ þ Cd ½Qxθ1  þ nK S þ CT Qð1  xθ1 Þ sQ 1 4 5 h i h i ¼   P t H þ Dt 1 H þH n1 T T þh1 1 2 ðt 2 Þ þ h 1 ðt 1 Þ þ 1 ðt 2 Þ þ h ðHt 3 Þ 2 2n 2 2

APðs, QÞ ¼

ð4:447Þ To incorporate the price in the above inventory model, the demand is considered as a linear function of price, and it is expressed as D ¼ a  bs. Substituting D with a – bs and E(T ) with Q(1  θ1E(X)), then the average annual profit becomes:

4.6 Multi-delivery

E ½APðs,QÞ ¼

359

E ½CPðs, QÞ E ½T 

  3 ða  bsÞ E ½x2 Q K þ nK S þ C C þ E ½ x  þ C θ E ½ x  þ h R d 1 1 6 1  θ 1 E ðx Þ 7 Q 2P1 6 7 6 7

 7 sða  bsÞ 6 hQða  bsÞ hQða  bsÞ 2 2 7 6 ¼ þ 2E ð x Þ  E ð x Þ  θ E ð x Þ þ 1 7 1  θ 1 E ðx Þ 6 ð E ð x Þ Þ ð 1  θ E ð x Þ Þ 2P 1  θ 2P 1 1 1 6 7 6 7

4 5 n  1 1  θ1 E ðxÞ a  bs E ðxÞða  bsÞ  þ C T Qða  bsÞ  þhQ 2n 2P 2 2P1 2

ð4:448Þ From calculus, it is well-known that E [AP(S, Q)] is concave if and only if the following equations are satisfied: 2

∂ E½APðs, QÞ 2ðK þ nK S Þða  bsÞ

> > > > P > hi ðDi ÞðPi Þ2 þ ðDi Þ2 ðPi Þ  2ðDi Þ3 n n h ðD Þ3 P > > i i > > 2 4 5 > > T þ þ T σ > > i 2 2 : i¼1 ; 2ðP Þ i¼1 2ðP Þ i

i

ð5:62Þ The capacity of the single machine is the constraint of the EPQ inventory model. This constraint is formulated as shown in Eq. (5.63). The sum of the production,

5.3 No Shortage

389

rework, and setup times for all products cannot be greater than the common cycle length T. Hence: n  X

n  X t pi þ t ri þ tsi  T

i¼1

ð5:63Þ

i¼1

Substituting t pi and t ri from Eqs. (5.50) and (5.49), respectively, results in: !

Pn

i¼1 tsi

T

1

Pn

i¼1

ð1þσ i ÞDi Pi

ð5:64Þ

¼ T Min

The mathematical formulation of the inventory problem that aims to minimize the total inventory cost is shown as below: Min :

Z¼

n X Ki i¼1

T

þ

n X

χ 1i þ

i¼1

n X

χ 2i ðT Þ 

i¼1

n X

χ 3i ðσ i T Þ þ

i¼1

n X

n   X χ 4i σ 2i T þ χ 5i ðσ i Þ

i¼1

i¼1

s:t: T  T Min 0  σ i  αi ; i ¼ 1, 2, . . . , n ð5:65Þ where χ 1i , χ 2i , χ 3i , χ 4i , and χ 5i are the objective function coefficients. In order to show the convexity of the nonlinear programming problem presented in Eq. (5.65), the determinant of Hessian matrix for the objective function is derived as below: Hessian ¼

2

P

iKi

T

þ

! X X 2 3 2 2 Δi σ i þ 6 Δi σ i T i

ð5:66Þ

i

As T is positive and all parameters are nonnegatives, the Hessian is greater than zero. Hence, taking into account that all the constraints in (5.65) are linear, the problem becomes a convex nonlinear programming problem (CNLPP). Consequently, the optimal solution can be first obtained using the derivatives. Then, the global minimum solution is determined by obtaining the optimal solution (Nobil and Taleizadeh 2016). To find the optimal cycle length and the optimal proportion of produced reworkable for each product, the partial derivatives of the objective function Z are taken with respect to T and xi. Setting the equations obtained by taking the derivatives, the optimal cycle length and proportion are given by:

390

5

Multi-product Single Machine

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi uP Pn ðχ 5 Þ u u i K i  i¼1 4χi 4 i t P 2 T¼ iχi σi ¼

χ 3i T  χ 5i ; 2χ 4i T

i ¼ 1, 2, . . . , n

ð5:67Þ ð5:68Þ

Here, a heuristic algorithm with nine steps is proposed to solve the EPQ problem under consideration. Step 1. Calculate χ 1i , χ 2i , χ 3i , χ 4i , and χ 5i and go to Step 2.  2 Pn P ðχ 5 Þ Step 2. If K i  i¼1 4χi 4 > 0, then go to Step 3. Otherwise, the solution is i

i

infeasible; go to Step 9. Step 3. Calculate T using Eq. (5.67) and σ i using Eq. (5.68). Then, go to Step 4. Step 4. If σ i  0, then σ i ¼ 0; else, if σ i  αi, then σ i ¼ αi; then go to Step 5. P Step 5. If ni¼1 ð1þσPii ÞDi < 1, then go to Step 6. Otherwise, the solution is infeasible; go to Step 9. Step 6. Obtain the lower bound of cycle length, i.e., TMin, using Eq. (5.64) and go to Step 7. Step 7. If T  TMin, then set T ¼ T. Otherwise, T ¼ TMin, and go to Step 8. Step 8. Based on the value of T, calculate σ i , Qi , and Z using Eqs. (5.68), (5.53), and (5.65), respectively, and go to Step 9. Step 9. Terminate the procedure. Example 5.4 Nobil and Taleizadeh (2016) presented an example for an imperfect single-machine production system with three products. The general input data of this numerical example is shown in Table 5.10. Then, the optimal solution is obtained based on the proposed solution procedure as follows: Step 1. The values for χ 1i , χ 2i , χ 3i , χ 4i , and χ 5i are calculated and they are shown in Table 5.11.  2 P P ðχ 5 Þ Step 2. As i K i  ni¼1 4χi 4 ¼ 690:0595 > 0, the initial feasibility is checked. i

Then, go to Step 3. Step 3. Using Eq. (5.67), T is obtained as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ffi uP P3 ðχ 5 Þ u rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u i K i  i¼1 4χi 4 i t 690:0595 P 2 T¼ ¼ ¼ 0:5029 2728:2112 χ i i Moreover, based on Eq. (5.68), σ is are calculated and are shown in Table 5.12.

5.3 No Shortage

391

Table 5.10 General data for example (Nobil and Taleizadeh 2016) Item 1 2 3

Pi 1000 1100 1300

Di 300 350 320

αi 0.10 0.09 0.08

tsi 0.010 0.015 0.020

Ci 100 90 80

C Ri 50 40 30

li 54 42 28

Ki 3000 3500 4000

hi 4 5 6

Table 5.11 The objective function coefficient (Nobil and Taleizadeh 2016) Item 1 2 3

χ 1i 31,620 32,823 26,419.2

Table 5.12 Values of σ i (Nobil and Taleizadeh 2016)

Table 5.13 The range of each σ i (Nobil and Taleizadeh 2016)

χ 2i 672 976.2396 1079.9715

Item σi

Item Range σi

χ 3i 666 1025.0516 887.0627

1 28.2595

1 [0–0.10] 0.10

χ 4i 54 88.5847 58.1680

χ 5i 1200 700 640

2 13.6417

3 3.3135

2 [0–0.09] 0.09

3 [0–0.08] 0.00

Step 4. The range of each σ i is checked and they are shown in Table 5.13. Step 5. The second feasibility condition P As ni¼1 ð1þσPii ÞDi ¼ 0:923 < 1, the initial feasibility is checked. Then, go to Step 6. Step 6. The lower bound is equal to 0.5842. Then, go to Step 7. Step 7. Check the second optimality. As T < TMin, then (T  ¼ TMin ¼ 0.584203359055833), and go to Step 8. Step 8. Calculate the optimal values and terminate the procedure. Based on T ¼ 0.5842, σ i and Qi are computed and they are shown in Table 5.14. As result, the minimum annual inventory cost is Z ¼ $110,051.754897060.

5.3.5

Scrapped

Shafiee-Gol et al. (2016) studied pricing and production decisions in multi-product single-machine manufacturing system considering discrete delivery and rework. They solved that problem in two situations: first, when the condition is satisfied and, second, when it is not satisfied. In this study, the finished products can only be delivered to customers if the whole lot is quality assured at the end of rework. In addition, in many circumstances, the delivery of the products is not continuous. All items produced are screened such that

392

5

Table 5.14 Values of σ i , Qi σ i Qi

Multi-product Single Machine

1

2

3

0.10 175.26

0.09 204.47

0.00 186.94

Table 5.15 New notations for given problem (Shafiee-Gol et al. 2016) bi φi t oi

Price sensitivity of demand The fail portion of reworking item that scrapped Preparation time for ith product Time required for delivering all quality assured finished products

t di ρ tni AP(T, Q)

Number of fixed-quantity installments of the finished batch to be delivered to customers in each cycle, which is assumed to be a constant for all products A fixed interval of time between each installment of finished products delivered during t ri for ith product Total production–inventory–delivery profits per unit time

inspection cost is included in unit production cost. After the end of regular process, all defective products are reworked with an additional reworking cost. During the rework, a portion φi of reworked products fails and becomes scrap at an additional disposal cost CSi. They considered Pi  d1i  Di > 0 (or 1  xi  Di/Pi > 0) to prevent occurring the shortage. When the whole lot is quality assured at the end of the rework, the finished items of each product can be delivered to customers (Shafiee-Gol et al. 2016). In addition to notations introduced in Sect. 5.1, the presented one in Table 5.15 is used to model the on-hand problem. The following equations can be obtained from Fig. 5.5: I i ¼ ðPi  d1i Þt pi ¼ ðPi  d1i Þ

Qi ¼ ð1  xÞQi Pi

H i ¼ I i þ ðP2i  d2i Þt ri ¼ Qi ð1  φi xÞ t pi ¼

Qi Ii ¼ Pi Pi  d1i

xQi P2i   ð 1  φi x Þ 1 x t di ¼ ρt ni ¼ T  t pi  t ri ¼ Qi   ai  bi si Pi P2i t ri ¼

ð5:69Þ ð5:70Þ ð5:71Þ ð5:72Þ ð5:73Þ

d1i t pi ¼ Pi xt pi ¼ xQi

ð5:74Þ

φi d1i t pi ¼ φi Pi xt pi ¼ φi xQi

ð5:75Þ

One of the necessary conditions in the multi-item production system is as follows:

5.3 No Shortage

393

Fig. 5.5 On-hand inventory level of perfect-quality product i in the proposed multi-item production system under the common cycle time policy (Chiu et al. 2013) n X i¼1

tsi þ

n  X  t pi þ t ri  T

ð5:76Þ

i¼1

or n X i¼1

tsi  T 

n  X Q

xQ þ i Pi P2i i

i¼1

 ð5:77Þ

For any given production cycle, the total production–inventory–delivery profit for all n products (AP(T, Q)), which is equal to sales revenue minus variable production costs, setup cost, cost of reworking defective items, disposal cost, fixed and variable delivery costs, holding cost during production uptime and rework time, and holding cost for finished goods during the delivery time, can be obtained as expressed in Eq. (5.83) (Shafiee-Gol et al. 2016).

394

5

Multi-product Single Machine

inspection n X

revenue

X zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ APðS, QÞ ¼ si Qi ð1  φi xÞ 

cost zffl}|ffl{ C Ii Qi

production reworking cost

disposal cost

zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{  C Ri ðxQi Þ  C Si ðxQi φi Þ 

setup cost z}|{ Ki

i

i¼1

holding cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ X H 1i þ d 1i t 1i H þ H 2i n1 hi ðt 1i Þ þ 1i ðt2i Þ þ ðH 2i t 3i Þ  2 2 2n i holding cost of reworked item



zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ h i P t h1i 1i 2i ðt 2i Þ 2

shipping cost

fixed cost

X zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{ zffl}|ffl{  C Ti ½Qi ð1  φi xÞ  nK Si i

ð5:78Þ i xÞQi Because T and every si and Qi obey T ¼ ð1φ ai bi si , 8i, T and Qi can be selected as decision variables. So:

si ¼

ai ð1  φi xÞQi  bi Tbi

ð5:79Þ

Equation (5.80) is the natural result of Eqs. (5.78) and (5.79). If constraint (5.77) is satisfied, they are optimal feasible solutions. Otherwise, Lagrangian relaxation method is used: AAPðT, QÞ ¼ ¼

APðT, QÞ "T  n 1 X ai

 ð1  φi xÞQi ð1  φi xÞQi  C Ti ½Qi ð1  φi xÞ T i¼1 bi Tβi "  # 2 P xQi  ðnK Si þ K i þ CIi Qi þ C Ri ðxQi Þ þ C Si ðxQi φi ÞÞ þ h1i 1i 2 P2i

2      Qi Q ð2  φi x  xÞ xQi Qi xQi n1 Qi ð1  φi xÞ T  hi þ i þ þ 2n 2P1i P2i P1i P2i 2 

ð5:80Þ In summary, the function that should be maximized is as follows: AAPðT, QÞ ¼

XAi Q2 i

i

where:

T2

Bi Q2i Di Qi Gi þ þ þ J i Qi þ T T T

 ð5:81Þ

5.3 No Shortage

395

Ai ¼  Bi ¼

ð 1  φ i xÞ 2 bi

ð5:82Þ

h x ð 2  φi x  x Þ hi ð n  1Þ ð 1  φ i x Þ h1i P1i x2 h  i  i þ 2P2i 2P1i 2P1i 2P22i þ

hi ðn  1Þð1  φi xÞx 2P2i

Di ¼

ð5:83Þ

ai ð 1  φ i x Þ þ Ci þ C Ri x þ C Si xφi þ C Ti ½ð1  φi xÞ bi Gi ¼ nK Si  K i Ji ¼

ð5:84Þ ð5:85Þ

hi ð n  1Þ ð 1  φi x Þ 2n

ð5:86Þ

Constraint (5.77) can be regarded as (5.82). Therefore, this function should be maximized according to a linear constraint of decision variables: Z

X N i Qi  T  0

ð5:87Þ

i

From calculus, it is well known that F(T, Q) is concave if and only if the following equations are satisfied: 2

2

∂ F ðT, QÞ 6 ∂Q 2 6 1 6 2 6 ∂ F ðT, QÞ 6 6 ∂Q ∂Q 6 2 1 6 H¼6 ⋮ 6 2 6 ∂ F ðT, QÞ 6 6 ∂Q ∂Q 6 n 1 6 2 4 ∂ F ðT, QÞ 2

∂T∂Q1

2K 1 B1 6 2 þ2 T 6 T 6 6 6 0 6 6 6 H¼6 ⋮ 6 6 6 0 6 6 6 2 4 ∂ F ðT, QÞ ∂Q1 ∂T

2

∂ F ðT, QÞ ∂Q1 ∂Q2 2

∂ F ðT, QÞ ∂Q2 2 ⋮ 2 ∂ F ðT, QÞ ∂Qn ∂Q2

2

∂ F ðT, QÞ ... ∂Q1 ∂Qn 2

∂ F ðT, QÞ ∂Q2 ∂Qn ⋱ ⋮ 2 ∂ F ðT, QÞ ... ∂Q2n ...

2

2

∂ F ðT, QÞ ∂T∂Q2

...

0

...

0

2K 2 B þ2 2 2 T T ⋮

...

0

⋱

⋮

0

...

2K n B þ2 n T T2

∂ F ðT, QÞ ∂Q2 ∂T

...

∂ F ðT, QÞ ∂Qn ∂T

2

∂ F ðT, QÞ ∂T∂Qn

2

3 2 ∂ F ðT, QÞ ∂Q1 ∂T 7 7 7 2 ∂ F ðT, QÞ 7 7 ∂Q2 ∂T 7 7 7 7 ⋮ 7 2 ∂ F ðT, QÞ 7 7 ∂Qn ∂T 7 7 7 2 ∂ F ðT, QÞ 5

ð5:88Þ

∂T 2

3 2 ∂ F ðT, QÞ 7 ∂T∂Q1 7 7 2 ∂ F ðT, QÞ 7 7 ∂T∂Q2 7 7 7 7 ⋮ 7 2 ∂ F ðT, QÞ 7 7 ∂T∂Qn 7 7 7 2 ∂ F ðT, QÞ 5 ∂T 2

ð5:89Þ

396

5

½Q1

Q2 . . . Qn

T   H  ½Q1

Multi-product Single Machine

T T < 0

Q2 . . . Qn

ð5:90Þ

Above equation can be changed as below: n X n X i¼1

2

2Qi Q j

j¼1

2

2

∂ F ðT, QÞ ∂ F ðT, QÞ ∂ F ðT, QÞ þ 2TQi þ Q2i ∂Qi ∂Q j ∂T∂Qi ∂Q2i

2

þ T2

∂ F ðT, QÞ ∂T 2

0 2ðPi Þ2 ð1  E ðX i ÞÞ2

 ðC i þ C Si E ðX i ÞÞDi ψ 4i ¼ >0 1  E ðX i Þ

ð5:148Þ ð5:149Þ

Theorem 5.2 The objective function Z in (5.143) is convex (Taleizadeh et al. 2010c). Proof To prove the convexity of Z, one can utilize the Hessian matrix equation as: ½T, B1 , B2 , . . . , Bn   H  ½T, B1 , B2 , . . . , Bn T ¼

2K 0 T

ð5:150Þ

Then, the objective function is strictly convex. The expected cost function Z is strictly convex for all nonzero T and Bi. Hence, it follows that to find the optimal production period length and the optimal level of backorder Bi, one can partially differentiate Z with respect to T and Bi and solve the resulted system of equations

408

5

Multi-product Single Machine

obtained by letting the partial derivatives be equal to zero; thus, the systems of equations become: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ∂Z K

 ¼0!T ¼u uP ∂T t n 3 Pn ðψ 2i Þ2 ψ  i¼1 i i¼1 4ψ 1

ð5:151Þ

ψ 2T  ψ2 ∂Z ¼ 0 ! Bi ¼ i 1 ¼ i 1 T  ∂Bi 2ψ i 2ψ i

ð5:152Þ

i

Then: Qi

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u D jT  Di K u

 ¼ ¼ u 1  E ðX i Þ 1  E ðX i Þ tPn 3 Pn ðψ 2i Þ2 ψ  i¼1 i i¼1 4ψ 1

ð5:153Þ

i

To handle the constraint, the optimal solution in Eq. (5.144) should be checked. If the constraint is not satisfied, then TMin as the minimum value of T will be considered as the optimal point: T Min ¼

1

Pn tsi Pn i¼1 Di

ð5:154Þ

i¼1 Pi ð1E ½X i Þ

To ensure the possibility and acceptability of production of all products by a machine, the steps involved in the algorithm of finding the optimal and possible  values of T  , Bi , QBi must be performed as follows: P Di Step 0. If ni¼1 Pi ð1E ½X i Þ  1, then go to Step 2; else the problem will be infeasible. Step 1. Calculate T by Eq. (5.151). Step 2. Calculate TMin by Eq. (5.40). Step 3. If T  TMin, then T ¼ T; else T ¼ TMin. Step 4. Calculate Bi by Eq. (5.152). Step 5. Calculate Qi by Eq. (5.153). Examples 5.7 and 5.8 Taleizadeh et al. (2010c) considered a multi-product inventory control problem with five products in which their general and specific data are given in Tables 5.20 and 5.21, respectively. They considered two numerical examples with uniform and normal probability distributions for Xi. The setup cost is considered K ¼ $450, and Tables 5.22 and 5.23 show the best results for the two numerical examples.

5.4 Backordering Table 5.20 General data for Examples 5.7 and 5.8 (Taleizadeh et al. 2010c)

Table 5.21 Specific data for Examples 5.7 and 5.8 (Taleizadeh et al. 2010c)

409 Item 1 2 3 4 5

Product 1 2 3 4 5

Di 200 300 400 500 600

Pi 1800 2500 3000 3500 4500

tsi 0.001 0.002 0.003 0.004 0.005

Xi ~ U [ai, bi] E[Xi] ai bi 0 0.1 0.05 0 0.15 0.075 0 0.2 0.1 0 0.25 0.125 0 0.3 0.15

Ci 15 12 10 8 6

di 90 187.5 300 437.5 675

hi 5 4 3 2 1 X i ~N μi 0.25 0.28 0.33 0.38 0.42

Cbj 10 8 6 4 2

μi , σ 2i σ 2i 0.01 0.02 0.03 0.04 0.05

CSj 1 0.8 0.6 0.4 0.2

di 450 700 990 1330 1890

Table 5.22 The best results for Example 5.7 (uniform distribution) (Taleizadeh et al. 2010c) Product 1 2 3 4 5

5.4.2

Uniform TMin 0.0526

T 0.5608

T 0.5608

Bi 33.02 48.80 63.70 78.21 94.57

Qi 118.06 181.88 249.24 320.46 395.86

Z 21,614

Multidefective Types

Pasandideh et al. (2015) extend two works of Taleizadeh et al. (2010c, 2013a) by taking into account reworkable items of various types that require different rework rates to become perfect-quality items. Note that rework is not considered in Taleizadeh et al. (2010c), while in Taleizadeh et al. (2013a), no scrap is assumed. This single-machine imperfect production problem assumes that perfect and imperfect-quality products are produced at certain percentages on a single machine. Furthermore, all imperfect products are classified as reworked and scrapped. The following notations which are presented in Table 5.24 are used to model the problem on hand: In this inventory problem, the annual constant production rate of item i in a regular production time, (Pi), is assumed to be greater than to the annual constant demand rate of product (Di), where the annual constant imperfect production rate is xiPi. Mathematically speaking, (1  xi)Pi > Di or ai ¼ (1  xi)Pi  Di > 0. In addition, the xi parameter considers two types of parameters, the proportion of reworkable products (αij) and the proportion of scrapped items (θi). After termination of the regular production, scrapped items are disposed, and the rework process starts

410

5

Multi-product Single Machine

Table 5.23 The best results for Example 5.8 (normal distribution) (Taleizadeh et al. 2010c) Product 1 2 3 4 5

Normal TMin 0.5796

T 0.5777

T 0.5796

Bi 32.91 48.30 61.90 74.34 89.27

Qi 154.56 241.50 346.02 467.41 599.57

Z 29,286

Table 5.24 New notations for given problem (Pasandideh et al. 2015) i j αij θi vij vij Pi μi Hm i Ii H ij fi δi Fi H ij

Index of product Index of defective type The proportion of produced ith product with jth defective type The proportion of ith scrapped items The ratio of the rework rate of ith item with jth defective type to the ith item production rate (vij  1) Rework rate of ith defective item type j Space required per unit and the maximum on-hand inventory of ith item The maximum on-hand inventory of ith item The maximum on-hand inventory of ith item, based on which the regular production process stops The maximum on-hand inventory of the ith item, based on which the rework process stops for jth defective type The unit warehouse construction cost of ith item per unit space Aisle space for ith item which is a percentage of its required storage space Total required space of ith product The maximum on-hand inventory of the ith item, based on which the rework process stops for jth defective type

with the v1i Pi , v2i Pi , . . . and vm i Pi rates, where it is assumed no scrapped item is produced during the rework process. As the rework process of a product usually does not require more time compared to its corresponding regular production time, the rework rate is greater than or equal to the regular production rate for all products, i.e., vij  1. As a result, the rework production rate vij Pi of the product i is greater than or equal to the demand rate (Di). In other words, vij Pi > Di or yij ¼ vij Pi  Di > 0. Additionally, the following conditions are assumed to model the problem (Pasandideh et al. 2015): • For each item, there are m types of failures that require rework. • The number of reworkable items with percentage rework rate of α1i is less than the quantity of reworkable items with the percentage rework rate of α2i and so on. In m1 other words, αm  ⋯  α1i . i  αi • The rework rates are proportions of the regular production rate (vij Pi ).

5.4 Backordering

411

• The items with the percentage rework rate of α2i require less processing time than the ones with the percentage rework rate of α1i and so on. In other words, vm i  1 vm1  ⋯  v  1. i i All production systems with the above conditions can benefit from the modeling and the solution procedure provided in the next sections. The proposed approach in this model enables production managers to determine the optimal period length, the lot size, and the allowable shortage of each product so as the total cost, including setup, production, warehouse construction, holding, shortage, reworking, and disposal, which is minimized (Pasandideh et al. 2015). In order to develop an inventory model that is even more applicable to real-world inventory problems, machine capacity, limited budget, and service level requirement are imposed. Figure 5.7 shows the on-hand inventory and shortages of product i per cycle. In this figure, t 1i and t mþ4 are the production uptimes, t 2i , t 3i , . . . , t mþ1 are i i mþ3 mþ3 rework periods, and t i and t i are production downtimes. Based on what was stated in Sect. 5.2, these periods in each cycle of product i are easily obtained using Eqs. (5.155)–(5.161) as follows: t 1i ¼

Qi Bi  Pi ðð1  xi ÞPi  Di Þ

ð5:155Þ

H 1i  I i Q ¼ α1i 1 i v1i Pi  Di vi P i

ð5:156Þ

t 2i ¼ t 3i ¼ α2i

Qi Qi , : . . . , t mþ1 ¼ αm i i vm P 2 vi Pi i i

ð5:157Þ

t mþ2 ¼ i

Hm i Di

ð5:158Þ

t mþ3 ¼ i

Bi Di

ð5:159Þ

Bi ð1  xi ÞPi  Di

ð5:160Þ

t mþ4 ¼ i

T¼

mþ4 X

t ei

ð5:161Þ

e¼1

In addition, the inequality αi ¼ (1  xi)Pi  Di > 0 must hold in the production uptimes t 1i and t mþ4 of a cycle. Besides, it is assumed that the time at which rework i 2 period t i starts is the time at which the regular process periods t 1i ends, t 3i after t 2i , and so on. Moreover, scrapped items are disposed at the time the regular process stops. Then, based on Fig. 5.7, one has (Pasandideh et al. 2015):

412

5

Multi-product Single Machine

Inventory m

Hi

yim

Him-1 . . .

2

Hi

Ii

-Di

yi2

1 Hi

1

yi ai ti1

ti2

ti3

tim+1

Bi

tim+3

tim+2

tim+4

time

ai

T

Fig. 5.7 The inventory of a product in a cycle (Pasandideh et al. 2015)

I i ¼ αi

  Qi  Bi Pi

ð5:162Þ

and: H 1i ¼ I i þ α1i y1i

  y1i ¼ v1i Pi  Di  0

Qi ; v1i Pi

ð5:163Þ

therefore: Hm i

¼

H m1 i

þ

m Qi αm i yi vm P i i

Q ¼ Ii þ i Pi

m X αij yij j j¼1 vi

! ð5:164Þ

Moreover, the cycle length is: T¼

ð1  θi ÞQi Di

ð5:165Þ

Di T ð1  θ i Þ

ð5:166Þ

Qi ¼

The total costs of all items TC is the sum of total setup cost CA, total production cost CP, total rework cost CR, total holding cost CH, total backorder cost CB, total disposal cost CD, and total warehouse construction cost CC of all items. In other words:

5.4 Backordering

413

TC ¼ CA þ CP þ CR þ CH þ CB þ CD þ CC

ð5:167Þ

The annual setup cost of all items is: CA ¼

N X

NAi ¼

i¼1

N X Ki T i¼1

ð5:168Þ

The annual production cost of all items is easily obtained as: CP ¼

n X

ð5:169Þ

NC i Qi

i¼1

For N reworks, each with a quantity of αij Qi and a cost of CRi per unit of item i, the annual rework cost is derived as: CR ¼

n X i¼1



Di CRi ðxi  θi Þ ð1  θ i Þ

 ð5:170Þ

Based on Fig. 5.7, the annual holding cost of the inventory system under consideration is:    1 

 n 1 2 X  mþ1  H m   H m1 þ H m hi I i  1  I i þ H i  2  H i þ H i  3  i þ i t mþ4 ti þ ti CH ¼ ti þ ti þ ⋯ þ i T 2 2 2 i 2 2 i¼1

ð5:171Þ Using Fig. 5.7, the annual backorder cost of the inventory system is shown in Eq. (5.172): CB ¼

n X  Cbi Bi  mþ3 t i þ t mþ4 i 2T i¼1

ð5:172Þ

For N disposals, each at a cost of Cdi per unit of item i, the annual disposal cost of all items becomes: CD ¼

n X i¼1



Di Cdi θi ð1  θ i Þ

 ð5:173Þ

As the space required per unit and the maximum on-hand inventory of ith item are m μi and H m i , respectively, the required storage space of product i is μi H i , of which an m additional δi μi H i is needed for its aisle space. Thus, the total required space of ith item becomes (Pasandideh et al. 2015):

414

5

Multi-product Single Machine

F i ¼ μi H m i ð1 þ δi Þ

ð5:174Þ

Then, the total warehouse construction cost is obtained by (Pasandideh et al. 2015): CC ¼

n X

f i μi H m i ð1 þ δ i Þ

ð5:175Þ

i¼1

The total cost TC reduces to (Pasandideh et al. 2015): TC ¼ Z ¼

n X i¼1

 Δ1i

B2i T

 þ

n X

Δ2i ðT Þ 

i¼1

n X i¼1

Δ3i ðBi Þ þ

n X i¼1

Δ4i þ

n X

Δ5i

i¼1

  1 T

ð5:176Þ

The capacity of the single machine, the budget, and the service level of each item are the three constraints of the model described as below (Pasandideh et al. 2015). The summation of the total production, rework, and setup times for all products should be smaller than the cycle length. Therefore (Pasandideh et al. 2015): n  X

n  X t 1i þ t 2i þ t 3i þ ⋯ þ t m1 tsi  T ! T þ i

i¼1

Pn

i¼1 tsi

 1

Pn

þ

Pn

i¼1

Bi i¼1 ðð1xi ÞPi Di Þ

Di i¼1 ðð1θi ÞPi Þ

  ¼ T min P αj 1 þ mj¼1 v ij

ð5:177Þ

i

As there is a total available budget of W to produce Qi items for each product i with a production cost of Ci, to rework αij Qi items for each product i with a rework cost of CRi, to dispose θiQi items for each product i with a disposal cost of Cdi, and to construct the warehouse for all items, the budget constraint is obtained as (Pasandideh et al. 2015): 1 f i μi αi Di ð1 þ δi Þ C i Qi þ þ⋯þ þ C di θi Qi þ TC n B Pi ð1  θi Þ X C B ! CW B m j j X C B αi yi f i μi Di ð1 þ δi Þ A i¼1 @ þ μ ð 1 þ δ ÞB T  f i i i i j Pi ð1  θi Þ v j¼1 i 0

C Ri α1i

which can be simplified to:

CRi αm i Qi

5.4 Backordering 0

T

415



Di C i þ C di θi þ C Ri B Pn B ð1  θi Þ B i¼1 B @ Δ3 f i μi ð1 þ δi Þ i 1 2Δi

W !! 1 ¼ T Budget  f μ D ð1 þ δ Þ Xm α j y j i j i i i i i αi þ α þ C j¼1 i j¼1 v j Pi ð1  θi Þ C i C C A

Xm

ð5:178Þ Based on the total shortage quantity and the safety factor of ith item, Bi and SLi, respectively, the service level constraint is derived as (Pasandideh et al. 2015): N  Bi Bi  SLi ! T  ¼ T SL i Di SLi Di

ð5:179Þ

The final EPQ model of the problem under investigation is presented as follows (Pasandideh et al. 2015): Z¼

n X i¼1

s:t: :

 Δ1i

B2i T

T T

 þ

n X i¼1

Δ2i ðT Þ 

n X

Δ3i ðBi Þ þ

n X

i¼1

Δ4i þ

i¼1

Min

i¼1

Δ5i

  1 T ð5:180Þ

T  T Budget T  T SL i Bi  0,

n X

T>0

The objective function of the nonlinear mathematical model presented in (5.180) is convex (for more details, see Pasandideh et al. (2015)). To show this, they proved that the Hessian matrix of the objective function is positive. Besides, the constraints in Inequalities (5.177)–(5.179) all are given in linear forms, and hence all are convex as well. As a result, the mathematical model in (5.180) is a convex nonlinear programming, and the local minimum is the global solution. To find the optimal common cycle length and the optimal backorder level of each product, the partial derivatives of the objective function Z are taken with respect to T and Bi. Setting the equations obtained by taking the derivatives, the optimal cycle length and backorder level are given as (Pasandideh et al. 2015): ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 5 u Δ u i i  T ¼ u t P 2 Pn ðΔ3i Þ2 Δ  ¼τ 1 i i i¼1 4Δ

ð5:181Þ

i

Bi ¼

Δ3i T 2Δ1i

ð5:182Þ

416

5

Multi-product Single Machine

In short, the proposed eight-step solution procedure flows (Pasandideh et al. 2015): Step 1. Initial feasibility. If τ > 0, ξ  0 and ai ¼ (1  xi)Pi  Di > 0 for all products, then go to Step 2. Otherwise, the EPQ inventory model is infeasible; then go to Step 8. Step 2. Value of the cycle length. Calculate T using Eq. (5.181). Calculate Bi using Eq. (5.182). Pn Pn ðBi Þ and 1  Step 3. Secondary feasibility. If i¼1 tsi  i¼1 ðPi Di xi Pi Þ    Pn Pm α i j Di are simultaneously either positive or negative, i¼1 Pi ð1θi Þ 1 þ j¼1 v j i

then go to Step 4. Otherwise, the EPQ inventory model is infeasible. Go to Step 8. Step 4. Checking the shortage level. Calculate T si by Eq. (5.179), and calculate T SL max  SL SL  SL using T SL . max ¼ Max T 1 , T 2 , . . . , T n Step 5. Calculate TMin using Eq. (5.177) of Pasandideh et al. (2015),calculate TBudget by Eq. (5.178), calculate T Min using T Min ¼ Max T min , T SL max , and calculate T Max using T Max ¼ T Budget . Step 6. Checking the constraints. This step involves four conditions to determine the optimal values of the decision variables as follows: Condition 1. If T max < T min, then the EPQ inventory model is infeasible and then go to Step 8. Condition 2. If T max  T  T min , then T ¼ T and go to Step 7. Condition 3. If T  T max , then T  ¼ T max and go to Step 7. Condition 4. If T  T min , then T  ¼ T min and go to Step 7. Step 7. Finding the optimal solution. Based on the values of T and B, obtain Qi using Eq. (5.166) and Z by Eq. (5.180). Step 8. Terminating the proposed solution procedure. Example 5.9 Pasandideh et al. (2015) considered an imperfect single-machine production system with 10 products, with the total available budget of $4,000,000,000 per period. Also, let the general data on each product be the one listed in Tables 5.25 and 5.26. Then, the optimal solution is obtained based on the proposed solution procedure as follows: Step 1. Initial feasibility. In order to check whether the problem has a feasible solution space, all ai , ξi , Δoi and τ are first calculated, and they are shown in Table 5.27. As the conditions for initial feasibility hold, the problem has a feasible solution space and we go to Step 2. Step 2. Value of the cycle length. Using Eq. (5.181), The cycle length is obtained as:

5.4 Backordering

417

Table 5.25 General data (Pasandideh et al. 2015) Item 1 2 3 4 5 6 7 8 9 10

Pi 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500

Di 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

xi 0.135 0.12 0.111 0.102 0.092 0.082 0.073 0.063 0.054 0.047

α1i 0.005 0.005 0.006 0.006 0.005 0.005 0.007 0.007 0.008 0.008

α2i 0.05 0.045 0.04 0.036 0.032 0.027 0.021 0.016 0.011 0.009

α3i 0.08 0.07 0.065 0.06 0.055 0.05 0.045 0.04 0.035 0.03

θi 0.005 0.005 0.004 0.003 0.003 0.003 0.002 0.002 0.001 0.001

v1i 2 2 2 2 2 3 3 3 3 3

fi 50 55 60 65 70 75 80 85 90 95

tsi 0.0004 0.0004 0.0005 0.0005 0.0006 0.0006 0.0007 0.0007 0.0008 0.0008

v2i 3 3 3 3 3 4 4 4 4 4

v3i 4 4 4 4 4 5 5 5 5 5

Table 5.26 General data (continued) (Pasandideh et al. 2015) Item 1 2 3 4 5 6 7 8 9 10

Ki 400 500 600 700 800 900 1000 1100 1200 1300

hi 8 10 12 14 16 18 20 22 24 26

Ci 46 44 42 40 38 36 34 32 30 28

CRi 25 24 23 22 21 20 19 18 17 16

Cdi 20 20 20 20 16 16 16 16 12 12

Cbi 5 8 11 14 17 20 23 26 29 32

μi 2 2 2 3 3 3 4 4 4 5

δi 3 3 3 3 2 2 2 2 4 4

εi 0.09 0.09 0.085 0.085 0.08 0.08 0.075 0.075 0.07 0.07

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1X 5 8500 ¼ 0:0259588149281040 T¼ Δ ¼ τ i i 12, 613, 894:7041 Moreover, based on Eq. (5.182), shortage quantity is calculated and they are shown in Table 5.28. Pn Pn ðBi Þ Step 3. Secondary feasibility. If and 1  i¼1 tsi  i¼1 ðPi Di xi Pi Þ < 0    Pn Pm α i j Di < 0 , then the problem is feasible, and go to i¼1 Pi ð1θi Þ 1 þ j¼1 v j i

Step 4: TMin ¼ 0.000314130978925651. Step 4. Checking the shortage level. Using Eq. (5.178), T SL i s are obtained, and they are presented in Table 5.29, based on which T SL max ¼  SL SL  SL SL Max T 1 , T 2 , . . . , T n ¼ T 1 ¼ 0:0045533. Step 5. Constraints’ boundaries. The constraints’ boundaries are calculated, and they are given in Table 5.30.

Item 1 2 3 4 5 6 7 8

ai 4190 4620 5023 5435 5864 6303 6743 7201.5

xi 0.074284 0.080169 0.077814 0.076581 0.07477 0.07556 0.069894 0.070498

Δ1i 12.97995 20.80563 28.52828 36.27206 44.10994 52.02743 59.97305 68.07829

Δ2i 333,798.2 403,082.4 478,290.4 838,341.9 729,072.5 836,971.3 1,265,413 1,427,115

Table 5.27 Values of ai , ξi , Δoi for all products (Pasandideh et al. 2015) Δ3i 407.9886 449.9916 491.9818 793.97 645.9734 692.996 979.9814 1041.988

Δ4i 49,597.99 51,805.03 53,663.86 55,074.62 56,051.96 56,611.84 56,723.05 56,433.87

Δ5i 400 500 600 700 800 900 1000 1100

τ 12,613,895

418 5 Multi-product Single Machine

5.4 Backordering

419

Table 5.28 Values of Bi (Pasandideh et al. 2015)

B1 ¼ 0.407971433593460 B2 ¼ 0.280723316550802 B3 ¼ 0.223835193381573 B4 ¼ 0.284110174629736 B5 ¼ 0.190078540956709 B6 ¼ 0.172883367260570 B7 ¼ 0.212088244897210 B8 ¼ 0.198659291969368 B9 ¼ 0.310626962203211 B10 ¼ 0.369785917528195

Table 5.29 Values of T SL i (Pasandideh et al. 2015) TS1L 0.0045

TS2L 0.0028

TS3L 0.0022

TS4L 0.0026

TS5L 0.0017

TS6L 0.0014

TS7L 0.0018

TS8L 0.0016

TS9L 0.0025

TS10 L 0.0028

Table 5.30 Constraints’ boundaries (Pasandideh et al. 2015) Value

TMin 0.000314

T SL Max 0.004533

TBudget 0.034185

T Min 0.004533

T Max 0.042731

Step 6. Checking the constraints. As the third condition holds, i.e., T Max  T  T Min , and T ¼ T ¼ 0.0259588. Step 7. Finding the optimal solution. Based on T ¼ 0.02596, Bi and Qi are computed, and they are shown in Table 5.31. Moreover, the minimum annual inventory cost is Z ¼ $1,201,220.69088020. Step 8. Terminate the procedure solution.

5.4.3

Interruption in Manufacturing Process

An EPQ inventory model with interruption in process, scrap and rework is analyzed by Taleizadeh et al. (2014). The shortages are permitted and fully back-ordered. In this EPQ inventory model, the decision variables are cycle length and backordered quantities of each product, and the main objective is to minimize the expected total cost. As it was discussed before, there are many real work situations in which the manufactured imperfect-quality products should be reworked or repaired with an additional cost. This model considers a manufacturing system that generates imperfect products. Furthermore, these defective products are repairable, and interruption in manufacturing process will occur. Moreover, it is assumed that there is no

Bi Qi

0.4079 26.08

1

2 0.2807 28.69

0.2238 31.27

3

Table 5.31 Values of Bi , Qi (Pasandideh et al. 2015) 0.2841 33.85

4

5 0.1901 36.45 0.1729 39.05

6 0.2121 41.62

7

0.1987 44.22

8

9 0.3106 46.77

10 0.3698 49.37

420 5 Multi-product Single Machine

5.4 Backordering

421

Ii

IiMax P1i – d1i – Di Ii –Di

Pi – di – Di ti1

ti2

ti3

ti7 t

Ii1 Ii2

ti4

ti5

ti6

Pi – di – Di

Bi T

Fig. 5.8 Inventory level of perfect-quality items when interruption occurs during the shortage cycle (Taleizadeh et al. 2014)

interruption during the rework process. Since there is only a single machine, the source of regular production and rework, obviously, is the same. In this problem the following basic assumption of EPQ inventory model with random defective rate, P  D  d > 0 or 1  x  DP > 0, is considered. Also, both x and θ are generated randomly between [0, 1]. Owing to some realistic reasons such as finishing the raw material, regular power failure, lubrications, breakdown (corrective maintenance), re-setup for each product, preventive maintenance, predictive maintenance, and maintenance schedule, among many other relevant reasons, then an interruption in production process may occur. Consequently, according to above reasons, manufacturer needs to have an interruption during production up time. It is important to point out that the interruption in the production process can happen when each product is being manufactured. Thus, two cases can be occurred: (1) interruption in the backorder-filling stage and (2) interruption when there are no shortages. When an interruption in the production process takes place, then the machine cannot work and should stay so until the state is changed. Figures 5.8 and 5.9 show the level of on-hand inventory of perfect-quality products in the proposed EPQ inventory model for the two proposed cases. They assumed that certainly interruption is occurred and they can control it. This therefore would suggest us that firstly the best time of the interruption should be determined. Naturally, based on the time the interruption occurs, the inventory system cost will be different. In other words, this cost depends on the length of

422

5

Multi-product Single Machine

Ii

IiMax

P1i – d1i – Di

Ii

Pi – di – Di

Ii1 Ii2

–Di –Di

t ti2

Pi – di – Di

ti7

ti1 ti3

ti4

ti5

ti6

Bi T

Fig. 5.9 Inventory diagram when interruption occurs during the non-shortage cycle (Taleizadeh et al. 2014)

time before interruption occurs. According to Fig. 5.8, if the interruption occurs when there are shortages, only the backordering cost and carrying cost of defective products will influence the inventory system cost. While if the interruption occurs when there are no shortages, only the carrying cost of healthy and defective products will affect the inventory system cost. So based on the time when the interruption occurs, they discussed the two following possible cases: (1) interruption when there are shortages and (2) interruption when there are no shortages. The previous assumption simplifies the problem because one only needs to determine the optimal value for the cycle length (T). For simplicity and without loss of generality, at first, the problem is modeled for a single-product case. After, the problem is changed to a multi-product case. The basic assumption of EPQ inventory model with imperfect-quality products manufactured is that Pi must always be greater than or equal to the sum of demand rate Di and the production rate of defective items di. Thus, one has:   Di Pi  d i  Di > 0 ∴ 0  xi < 1  P1

ð5:183Þ

The production cycle length is the summation of the production uptime, the reworking time, the production downtime, the machine restoring time, and the shortage permitted time:

5.4 Backordering

423

T¼

7 X

t ij

ð5:184Þ

j¼1

If think that Fig. 5.8 shows a possible case to which manufacturer can face, then: t 1i þ t 3i ¼

Bi þ t 2i Di Pi  Di  di

ð5:185Þ

This yields to (Taleizadeh et al. 2014): t 3i ¼

Bi þ t 2i Di  t 1i Pi  Di  d i

ð5:186Þ

and the inventory levels I 1i and I 2i are given by: I 1i ¼ Bi  ðPi  d i  Di Þt 1i I 2i

¼

I 1i

þ

Di t 2i

ð5:187Þ ð5:188Þ

As a result, the shortage cost is: 

 Bi þI 1i 1 I 1i þI 2i 2 I 2i t 3i Bi 7 ti þ ti þ þ ti 2 2 2 2 8    2 2 1 3 9 2 1 1 2 1 2 1 > > 1 > = < Bi t i  2 ðPi Di di Þ t i þBi t i  ðPi Di di Þt i t i þ 2 Di t i þ 2 Bi t i > ¼C bi 2 > > > >  1 ðP D d Þt 1 t 3 þ 1 D t 2 t 3 þ Bi ; : i i i i i i i 2 2 2Di ð5:189Þ

CSi ¼C bi

After some simple algebra, one has: 8 9  1 2 1 1  2 2 Bi 2 2 1 2 > > 1 > > B t  ð  D  d Þ t þ B t  ð P  D  d Þt t þ t þ P D i i i i i i i i i < ii 2 i = i 2Di 2 i i     SCi ¼ C bi   2 2 > > Bi þ ti Di Bi þ ti Di 1 > : þ 1 Bi  1 ðPi  Di  d i Þt1i ;  t1 þ Di t2i  t1 > Pi  D i  d i i Pi  D i  d i i 2 2 2

ð5:190Þ Also the holding cost of the defective products (HCD) is:

424

5

Multi-product Single Machine

9 8    1 2 > > Bi þDi t 2i 1 1 2 1 > > > > Pi xi t i þPi xi t i t i þPi xi t i > > =

>  1 2 1  1 2 > > Bi þDi t 2i Bi þDi t 2i 1 > > 1 > > P x t þ x þ x t P x t P P : i i i i i i i i i i i 2 2 Pi Di di Pi Di di ; (  2 ) Bi þDi t 2i 1 2 1 ¼h1i Pi xi t i t i þ Pi xi 2 Pi Di di ð5:191Þ Combining the shortage cost and holding cost of defective items, one has: (

)  2 2 Bi Di t 2i 1  2 2 1 Bi þ Di t 2i Bi 2 Z i ¼C bi  ðP i  þ Di t i þ þ þ 2 2 Pi  Di  di Pi  Di  di 2Di (  2 ) Bi þ Di t 2i 1 1 2 þ h1i Pi xi t i t i þ Pi xi 2 Pi  Di  di Bi t 2i

di Þt 1i t 2i

ð5:192Þ Then the first derivative of Zi respect to t 1i gives:   ∂Z i ¼ C bi ðPi  d i Þt 2i þ h1i Pi xi t 2i ¼ ðCbi ðPi  d i Þ þ h1i Pi xi Þt 2i 1 ∂t i

ð5:193Þ

According to the above equation, it is clear that the summation of shortage cost and holding cost of defective products will be increasing on t 1i if: C bi ð1  xi Þ < h1i xi

ð5:194Þ



Meaning t 1i ¼ 0. Otherwise, it is easy to see that the summation of shortage cost and holding cost of defective products will be decreasing on t 1i if Cbi(1  xi) > h1ixi. More precisely, based on previous discussion, it implies that the t 1i is: 

t 1i ¼

Bi Pi  Di  di

ð5:195Þ

In the case interruption placed when there is no shortage, as mentioned before, only the carrying cost of healthy and defective products affects the inventory system cost. According to Fig. 5.9, one has: t 4i ¼

Qi Bi   t 2i Pi Pi  d i  Di

and the inventory levels I 1i , I 2i , Ii are given by:

ð5:196Þ

5.4 Backordering

425

I 1i ¼ ðPi  di  Di Þt 2i

ð5:197Þ

I 2i ¼ ðPi  di  Di Þt 2i  Di t 3i

ð5:198Þ

I i ¼ I 2i þ ðPi  Di  d i Þt 4i ¼ ðPi  Di  d i Þt 2i  Dt 3i þ ðPi  Di  di Þ

  Qi Bi   t 2i ð5:199Þ Pi Pi  Di  di

In this situation, since only the interruption can occur when the rework process is  not started, consequently just the carrying cost during t 1i þ t 3i þ t 4i , similar to the case, should be compared. The holding cost of healthy products during first  t 1i þ t 3i þ t 4i is determined as: 8 9  2 2 1   1 > > 2 3 3 > > ð  D  d Þ t þ ð  D  d Þt  D t P 2 P t i i i i i i i

i 3 2 i > >  t2 >  : þ ðPi  Di  d i Þ i  Bi  2Di t i þ ðPi  Di  d i Þt i ; 2 Pi Pi ðPi  Di  d i Þ i

ð5:200Þ This can be simplified to: 9 8 Q2i > 1  3 2 1 > 2 3 > > > = < ðPi  di Þt i t i  2 Di t i þ 2 ðPi  Di  d i Þ P2 > i CHi ¼ hi  2  > > Di Qi t 3i > > Bi Qi 1 Bi þ 2Di Bi t 3i > > þ  ; : 2 Pi  Di  d i Pi Pi

ð5:201Þ

Also holding cost of defective products is given by:  2  2 o 1 1 Pi xi t 2i þ Pi xi t2i t3i þ Pi xi t 2i t 4i þ Pi xi t 4i 2( 2    2 )   1 2 2 2 3 2 Qi Bi 1 Qi Bi 2 2 ¼h1i Pi xi  t þ  t t þ ti ti þ ti Pi Pi  D i  d i i 2 i 2 Pi Pi  D i  d i i

CH1i ¼h1i

n

ð5:202Þ Yielding: ( CH1i ¼ h1i

xi Q2i Pi xi B2i xi Bi Qi Pi xi t 2i t 3i þ þ  2Pi 2ðPi  Di  d i Þ2 Pi  Di  di

)

So the summation holding costs of healthy and defective products is:

ð5:203Þ

426 Fig. 5.10 On-hand inventory of defective items (Taleizadeh et al. 2014)

5

Multi-product Single Machine

I Pi xi (ti1+ ti3+ ti4) Pi xi (ti1+ ti3 ) 1

Pi xi ti

t ti1

ti2

ti3

ti4

ti5

    Q2 B Q 1 B2i þ 2Di Bi t3i Di Qi t 3i 1  2 1 Z i ¼hi ðPi  d i Þt 2i t 3i  Di t 3i þ ðPi  Di  d i Þ 2i  i i þ  Pi Pi 2 2 2 Pi  D i  d i Pi ( ) 2 2 xi Q i Pi xi Bi xi Bi Q i þ  þ h1i Pi xi t2i t 3i þ 2Pi 2ðPi  Di  d i Þ2 Pi  Di  d i

ð5:204Þ Then the first derivative of Zi in respect to t 2i gives:   ∂Z i ¼ hi ðPi  d i Þt 3i þ h1i Pxi t 3i 2 ∂t i

ð5:205Þ

Meaning the summation of the holding costs of healthy and defective products is  increasing on t 2i . Thus, the best time for considering the interruption is t 2i ¼ 0. It is worth mentioning that the best interruption time for the second case is equal to zero  (t 2i ¼ 0) which is as same as the best interruption time of the first case shown in Eq. (5.191). It is important to point out that the inventory behavior of these two cases is similar, and it is shown in Fig. 5.9. Thus, in the situation under which ever Cbi(1  xi) > h1ixi, the best value of interruption time in both cases (interruption when there are shortages and interruption when there are no shortages) is the same. Consequently, if for each product always Cbi(1  xi) > h1ixi is considered, then a unique model can be used to solve the problem on hand. Since the main issue to solve is the determination of the best time of interruption, then in both cases (i.e., when shortages are not permitted and are permitted), only the different terms of objective functions of those cases are separately studied. This is due to the fact that their same parts are independent from t 1i and t 2i . They did it in this way in order to simplify the mathematical calculations. Obviously, is not necessary to calculate the total cost function because only the best time of interruption which depends on the value of t 1i and t 2i should be determined. Since t 1i and t 2i are only used in the production and interruption periods, there is no need to determine the other terms of cost function. For the situation when the best time of interruption is determined, they have showed that for both backordering and no backordering

5.4 Backordering

427 I

Fig. 5.11 On-hand inventory of scrapped items (Taleizadeh et al. 2014)

P1i qi ti5= Pi xi qi ti5 = Pi xi {ti1+ ti3+ ti4} qi

t 5

ti

cases the time of interruption is the same. Then for this condition all terms of the total objective cost function of the inventory model must be expressed and determined (Taleizadeh et al. 2014) (Figs. 5.10 and 5.11). According to Fig. 5.9, one has: t 1i ¼

Bi Pi  di  Di

ð5:206Þ

t 2i ¼ t m

ð5:207Þ

I 1i

t 3i ¼

Pi  di  Di

t 4i ¼

Ii Pi  di  Di

ð5:208Þ ð5:209Þ

 1 The3 total  defective products manufactured during the production uptime t i þ t i þ t 4i can be computed as below:     d i t 1i þ t 3i þ t 4i ¼ Pi xi t 1i þ t 3i þ t 4i ¼ xi Qi

ð5:210Þ

and the total scrap products generated during the rework process are (Taleizadeh et al. 2014):   d 1i t 5i ¼ P1i θi t 5i ¼ d i θi t 1i þ t 3i þ t 4i ¼ xi Qi θi

ð5:211Þ

Then, one has: t 5i ¼

xi Qi P1i

ð5:212Þ

428

5

t 6i ¼

Multi-product Single Machine

I Max i Di

ð5:213Þ

Bi Di

ð5:214Þ

t 7i ¼ Likewise, one has:

I 1i ¼ Di t 2i ¼ Di t m

ð5:215Þ

In order to determine the maximum level of on-hand inventory when regular production process stops, Ii, one can subtract the backordered quantity and demand during the interruption time from the total healthy manufactured products which yields: I i ¼ ðPi  di  Di Þ

Qi  Bi  Di t m Pi

ð5:216Þ

and the maximum inventory level is given by: ¼ I i þ ðP1i  d1i  Di Þt 5i I Max i ¼ ðPi  di  Di Þ

Qi xQ  Bi  Di t m þ ðP1i  d1i  Di Þ i i Pi P1i

The cycle length is equal to T ¼ all times as follows: T¼

7 X j¼1

t ij

P7

j j¼1 t i

ð5:217Þ

which can be verified easily summing

  Qi di d1i ¼ 1  þ xi 1  Di Pi P1i

ð5:218Þ

This yields to: Qi ¼

Di T Ji

ð5:219Þ

where:

  di d1i J i ¼ 1  þ xi 1  Pi P1i The total cost function per cycle is:

ð5:220Þ

5.4 Backordering

429 Production Cost

DisposalCost

Rework Cost

Setup Cost

zffl}|ffl{ zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ z}|{ TC ðQi , Bi Þ ¼ C i Qi þ C Ri xi Qi þ CSi xi Qi θi þ K i  





  I1    5  I Max  6 Pi xi θi t 1i þ t 3i þ t 4i  5  B I   I i þ I Max i ti þ i ti t i þ hi i t 4i þ þCbi i t 1i þ t 7i þ i t 2i þ t 3i þ h1i 2 2 2 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 0

Back Ordered Cost

Holding Cost of Scrap Product

Holding Cost of Perfect Quality Product

     Pi xi t3i þ t4i  3 4    t5 Pi xi t1i  1  þhi ti þ Pi xi t 1i t2i þ t3i þ t4i þ ti þ ti þ Pi xi t 1i þ t 3i þ t 4i i 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Holding Cost of Defective Product

ð5:221Þ Equation (5.221) can be simplified to as below (Taleizadeh et al. 2014): 0 0



Bh B N 2x N x 2 M TC0 ðQi , Bi Þ ¼ @ i @ 2i þ i i þ i 2i þ 2 Pi Pi P1i ðP1i Þ

 þ

Cbi 2Di

Ni xi M i Pi þ P1i

Di

1

2

1

x2 C h1i x2i θi C 2 x þ i þ i Aþ AQi Pi P1i 2P1i

      Pi  d i h 1 1 J þ þ i Bi 2  hi i Bi Qi þ ½hi t m J i þ Ci þ CRi xi þ CSi xi θi Qi Ni 2 N i Di Di  2    hi Di þ Di N i t 2m Cbi Di ðt m Þ2 Pi  d i Pi þhi t m Bi þ K i þ þ Ni 2N i 2 Ni

ð5:222Þ However, Eq. (5.222) for the annual joint production policy using Eq. (5.219) becomes (Taleizadeh et al. 2014): 0 TCðT,Bi Þ ¼

0

0



TC ðQi , Bi Þ Bhi Di B N i 2xi N i xi M i þ þ ¼@ 2 @ 2þ T Pi P1i ðP1i Þ2 2J i Pi 2

2

Ni xi M i Pi þ P1i

Di

2

1

1

x2 C h1i x2i θi Di 2 C x þ i þ i Aþ AT Pi P1i 2P1i J i 2

     2 C bi Pi  d i h 1 1 Bi D þ þ þ i  hi Bi þ ½C i þ C Ri xi þ C Si xi θi  hi t m J i  i 2Di Ni 2 N i Di T Ji  2 2     2 h D þ D N t C D ð t Þ ht P B Pi  di 1 i i i i m þ i m i i þ Ki þ þ bi i m Ni T 2N i 2 Ni T

ð5:223Þ Remember that the existence of only one machine in the manufacturing system results in limited production capacity. In some sense, the maximum capacity of the single machine is the only constraint of the model. This constraint is described in the following: Since t 1i þ t 3i þ t 4i , t 5i , and tsi are the production uptimes, rework time and setup time of the ith product, respectively, and t 2i is the interruption time when machine is producing ith product, then the summation of the total production uptimes, rework repairing time, setup time (for all products) is  1 machine  Pand Pntime, n 2 3 4 5 expressed as þ t þ t þ t þ t þ t ts i i i i i¼1 i i¼1 i . Evidently, this should be smaller or equal to the period length (T ). In general, a necessary condition to guarantee feasibility is (Taleizadeh et al. 2014):

430

5

Multi-product Single Machine

n  n X  X t 1i þ t 2i þ t 3i þ t 4i þ t 5i þ tsi  T i¼1

ð5:224Þ

i¼1

This can be simplified to: 5 X j¼1



t ij

1 x ¼ t m þ Qi þ i Pi P1i

 ð5:225Þ

Thus: n  X

t m þ Qi

i¼1

  1 x þ i1 þ tsi  T Pi Pi

ð5:226Þ

Using Eq. (5.219), one has: Pn i¼1 ðtsi þ t m Þ i ¼ T Min T Pn D i h 1 1  i¼1 J i Pi þ Px1ii

ð5:227Þ

Finally, from Eqs. (5.223) and (5.227), the multi-product single-machine model with scrap, rework, interruption in process, and backlogging situation becomes: TCðT, Bi Þ ¼ s:t: :

n X

X X 1 X Bi 2 X 2 Bi X 3 ψi þ ψi T  ψ 4i Bi þ ψ 5i þ ψ 6i þ T T T i¼1 i¼1 i¼1 i¼1 i¼1 n

ψ 1i

i¼1

n

n

n

n

Pn

i¼1 ðtsi þ t m Þ  ¼ T Min T Pn Di 1 x þ i 1  i¼1 J i Pi P1i Bi  0 i ¼ 1, 2, . . . , n

ð5:228Þ where:  ψ 1i

¼

C bi 2Di

    Pi  d h1i 1 1 þ þ >0 Ni 2 N i Di ψ 2i ¼

hi t m P i >0 Ni

ð5:229Þ ð5:230Þ

5.4 Backordering

ψ 3i

hi D2i ¼ 2J 2i þ

431

!  2 x2i Ni 2xi N i xi 2 M i N i xi M i 1 xi þ þ þ þ þ þ Di Pi P1i Pi P1i ðPi Þ2 Pi P1i ðP1i Þ2

h1i ðxi Þ2 θi D2i 2P1i J 2i

>0

ð5:231Þ ψ 4i ¼ hi > 0 ψ 5i ¼ K i þ ψ 6i ¼

h

ð5:232Þ

D 1 ðt m Þ2 ðPi  d i Þðhi þ Cbi Þ > 0 2N i

ð5:233Þ

Di ðC þ C Ri xi þ C Si xi θi Þ  hi t m Di Ji i

ð5:234Þ

It should be noted that Ni ¼ Pi  di  Di, Mi ¼ P1i  d1i  Di, and J i ¼  i 1  Pdii þ xi 1  Pd1i1i .

Theorem 5.3 The objective function F ¼ TC(T, Bi) in (5.228) is convex (Taleizadeh et al. 2014). Proof To prove the convexity of F ¼ TC(T, Bi), one can utilize the well-known Hessian matrix equation as in (5.235). Then, according to Appendix F of Taleizadeh et al. (2014), the objective function is strictly convex. Therefore, the expected total cost function F ¼ TC(T, Bi) is strictly convex for all nonzero T and Bi. Hence, it follows that to find the optimal production period length (T) and the optimal level of backorders (Bi), one can partially differentiate F ¼ TC(T, Bi) with respect to T and Bi and solve the resulted system of equations obtained by equating the partial derivatives to zero (see Appendix G of Taleizadeh et al. (2014)): Pn T

½T, B1 , B2 , . . . , Bn   H  ½T, B1 , B2 , . . . , Bn  ¼

5 i¼1 ψ i

T

0

ð5:235Þ

According to Appendix G of Taleizadeh et al. (2014), the optimal solutions of decision variables are: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! u n 2 n uX 4ψ 1 ψ 5  ðψ 2 Þ2 X ðψ 4i Þ i i i 3 t T¼ ψi  = 4ψ 1i 4ψ 1i i¼1 i¼1 Bi ¼

ψ 4i T   ψ 2i 2ψ 1i

ð5:236Þ

ð5:237Þ

To determine the feasible and optimal solution, the following solution procedure is proposed by Taleizadeh et al. (2014):

432

5

Multi-product Single Machine

Table 5.32 General data for both examples (Taleizadeh et al. 2014) Product 1 2 3 4 5

Pi 4000 4500 5000 5500 6000

P1i 2000 2500 3000 3500 4000

tsi 0.003 0.004 0.005 0.006 0.007

Ki 500 450 400 350 300

Ci 15 12 10 8 6

hi 5 4 3 2 1

h1i 2 2 2 2 2

CSi 3 3 3 3 3

Cbi 10 8 6 4 2

CRi 1 2 3 4 5

Table 5.33 Specific data for Example 5.10 (Taleizadeh et al. 2014) Products 1 2 3 4 5

Di 600 700 800 900 1000

tm 0.003 0.003 0.003 0.003 0.003

Step 1. Calculate O1 ¼

Xi ai 0 0 0 0 0

~ U [ai, bi] bi E[Xi] 0.05 0.025 0.1 0.05 0.15 0.075 0.2 0.1 0.25 0.125

Pn

4ψ 1i ψ 5i ðψ 2i Þ i¼1 4ψ 1i

di ¼ PiE[Xi] 100 225 375 550 750 2

, O2 ¼



θi ~U a0i , b0i a0i b0i E[θi] 0 0.15 0.075 0 0.2 0.1 0 0.25 0.125 0 0.3 0.15 0 0.35 0.175

Pn

3 i¼1 ψ i 

ðψ 4i Þ

d1i ¼ P1iE[θi] 150 250 375 525 700

2

4ψ 1i

, and TMin.

Step 2. If O1O2 > 0, calculate period length T using Eq. (5.234). Otherwise, there is no feasible solution; terminate the procedure. Step 3. If T is less than TMin, then T ¼ TMin, else T ¼ T. Step 4. Calculate Bi for i ¼ 1, 2, . . ., n using Eq. (5.235). Step 5. Calculate Qi for i ¼ 1, 2, . . ., n using Eq. (5.219). Step 6. Terminate the procedure. Examples 5.10 and 5.11 This section provides two numerical examples to illustrate the use of solution procedure of the presented inventory model. Consider the five product inventory control problems where the general data for both examples are given in Table 5.32. Furthermore, they considered that in both numerical examples, the defective and scrap portions (Xi and θi) follow a uniform distribution. Tables 5.33 and 5.34 show an additional specific data for both examples. The use of the solution procedure with Example 5.10 is illustrated as follows (Taleizadeh et al. 2014) From Step 1, O1, O2 and TMin are obtained as follows:  2 n X 4ψ 1i ψ 5i  ψ 2i O1 ¼ ¼ 2000:155877, 4ψ 1i i¼1 ¼ 3112:546773,

O2 ¼

n X i¼1



ψ 3i

2 ψ 4i  4ψ 1i

and T Min ¼ 0:418778052:

From Step 2, since both O1 and O2 are positive, then O1O2 > 0, the period length (T ) can be determined with Eq. (5.236), and its value is T ¼ 0.80163. From Step 3, T is not less than TMin, so T ¼ T ¼ 0.80163. The results of Steps 4 and 5 are shown in Table 5.35.

5.4 Backordering

433

Table 5.34 Specific data for Example 5.11 (Taleizadeh et al. 2014)



θi ~U a0i , b0i

Xi ~ U [ai, bi] Products 1 2 3 4 5

Di 630 735 840 945 1050

tm 0.005 0.005 0.005 0.005 0.005

ai 0 0 0 0 0

bi 0.055 0.11 0.165 0.22 0.275

E[Xi] 0.0275 0.055 0.0825 0.11 0.1375

di ¼ PiE [Xi] 110 247.5 412.5 605 825

a0i 0 0 0 0 0

b0i 0.15 0.2 0.25 0.3 0.35

E[θi] 0.075 0.1 0.125 0.15 0.175

d1i ¼ P1iE [θi] 150 250 375 525 700

Table 5.35 The best result for Example 5.10 (Taleizadeh et al. 2014) Tmin 0.418778

T 0.80163

T 0.80163

Bi 135.0451 155.6826 175.9271 195.7637 215.17

Qi 481.8815 563.9608 647.3731 732.4538 819.5579

Z 45,364.04

Table 5.36 The best result for Example 5.11 (Taleizadeh et al. 2014) Tmin 1.291679

T 0.785506

T 1.291679

Bi 227.289 261.7311 295.4154 328.3045 360.3449

Qi 815.4394 954.6343 1096.316 1241.115 1389.702

Z 48,306.31

In similar way, Example 5.11 is solved and its results are shown in Table 5.36.

5.4.4

Immediate Rework Process

Two joint production systems in a form of multi-product single machine with and without rework are studied in Taleizadeh et al. (2011) where shortage is allowed and back-ordered. For each system, the optimal cycle length and the backordered and production quantities of each product are determined such that the cost function is minimized. Jamal et al. (2004) developed an EPQ model to determine the optimum production quantity of an item where rework is performed in two different situations. The objective function of their model was to minimize the total inventory cost of the production system under consideration. Taleizadeh et al. (2011) extended the model of Jamal et al. (2004) to be more realistic inventory control problem in which several items and several constraints are available.

434

5

Multi-product Single Machine

Ii

IiMax= Ii + xi (Pi – Di)

Qi Pi Pi – Di – Di

Q Ii = ((1–xi )Pi – Di ) i - Bi Pi (1–xi )Pi – Di

Ti4 Ti1

Ti2

Ti5

Ti3

bi T

Fig. 5.12 On-hand inventory of perfect-quality items for MP-SM with rework (Taleizadeh et al. 2011)

Figure 5.12 depicts a cycle for the inventory control problem under study. In order to model the problem, a single-product problem consisting of the ith product is first developed, and then the model will extend to include several products. The basic assumption in EPQ model with rework process is that (1  xi)Pi must always be greater than or equal to the demand rate Di. As a result, one has: ð1  xi ÞPi  Di  0

ð5:238Þ

The production cycle length (see Fig. 5.12) is the summation of the production uptimes, the rework time, and the production downtimes, i.e.: T¼

5 X

T ij

ð5:239Þ

j¼1

where T 1i and T 5i are the production uptimes (the periods in which perfect and defective items are produced), T 2i is the reworking time, and T 3i and T 4i are the production downtimes. In this model, a part of the modeling procedure is adopted from Jamal et al. (2004) to model the problem. Based on Fig. 5.12, one has:

5.4 Backordering

435

I i ¼ ðð1  xi ÞPi  Di Þ

Qi  Bi Pi

ð5:240Þ

and: ¼ I i þ xi ðPi  Di Þ I Max i

Qi Q Q ¼ ðð1  xi ÞPi  Di Þ i þ xi ðPi  Di Þ i  Bi ð5:241Þ Pi Pi Pi

It is obvious from Fig. 5.12 that: T 1i ¼

Qi Bi  Pi ð1  xi ÞPi  Di T 2i ¼ xi

 T 3i ¼

Qi Pi

ð5:243Þ

 1  DPii  xi DPii Qi Di



Bi Di

ð5:244Þ

Bi Di

ð5:245Þ

Bi ð1  xi ÞPi  Di

ð5:246Þ

T 4i ¼ T 5i ¼

ð5:242Þ

Hence, using Eq. (5.239), the cycle length for a single-product problem is: T¼

5 X

T ij ¼

j¼1

Qi Di

ð5:247Þ

or: Qi ¼ Di T

ð5:248Þ

The total cost of the system includes setup, processing, rework, shortage, and inventory carrying costs. Although the processing and rework costs are constants and do not affect the optimal solution, they are used in the objective function in order to determine the annual total cost. Defective items are produced in every batch and they are reworked within the same cycle. During the rework process, some processing and inventory holding costs are incurred for reworked quantities. Then, the total cost per year, TC(Q, B), is obtained as:

436

5 Production Cost

zfflffl}|fflffl{ NC i Qi

TCðQ, BÞ ¼

Setup Cost

Multi-product Single Machine

Rework Cost

zfflfflfflfflffl}|fflfflfflfflffl{ z}|{ þ NK i þ NC Ri xi Qi Shortage Cost

Holding Cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{  

 2  I Max  3  NC bi Bi T 4i þ T 5i I i  1  I i þ I Max i i þ Nhi T þ Ti þ Ti þ 2 2 i 2 2 ð5:249Þ Finally, the objective function of the joint production system (MP-SM with rework) using Eq. (5.247) becomes: TCðT, BÞ ¼

n X

Pn

i¼1 K i

C i Di þ

T

i¼1

þ

n X

CRi xi Di

i¼1

 

 X n n X  2  I Max  3 C bi Bi T 4i þ T 5i I i  1  I i þ I Max i i þ hi T þ Ti þ Ti þ 2T 2T i 2T 2T i¼1 i¼1 ð5:250Þ Since the production plus rework times of the ith product is T 1i þ T 2i þ T 5i and the of the production, reworking, and setup time (for all setup time is tsi, thePsummation   P products) will be ni¼1 T 1i þ T 2i þ T 5i þ ni¼1 tsi . It is clear that this summation must be smaller or equal to the cycle length (T). Hence, the capacity constraint of the model becomes: n  X

n  X T 1i þ T 2i þ T 5i þ tsi  T

i¼1

ð5:251Þ

i¼1

Then, based on Eqs. (5.240), (5.241), and (5.244), one has: n X

X Di Tþ tsi  T Pi i¼1 n

ð 1 þ xi Þ

i¼1

ð5:252Þ

From Eqs. (5.238) to (5.246), (5.248), and (5.250), the final model of the joint production system is obtained as follows: Min :

TCðT, BÞ X X α3 X B2i X α4i  α5i Bi  α6i Bi T þ þ T T i¼1 i¼1 i¼1 i¼1 n

¼ α1 T 2 þ α2 T þ

 ðCi þ xi CRi ÞDi

n

n

n

ð5:253Þ

5.4 Backordering

437

s:t: :

T 

Pn 1

Pn

i¼1 tsi

i¼1 ð1

þ xi Þ DPii

 ¼ T Lower

ð5:254Þ

where α1 ¼

 n X hi ðð1 þ xi Þðð1  xi ÞPi  Di ÞÞD2i þ xi ðPi  Di ÞD2i >0 2 Pi i¼1

α2 ¼

 n X hi 3ðð1  xi ÞPi  Di ÞDi 2 ðPi  ð1 þ xi ÞDi ÞDi þ >0 2 Pi Pi i¼1 α3 ¼

n X

Ki > 0

ð5:255Þ ð5:256Þ ð5:257Þ

i¼1



 ð1  xi ÞPi hi ðð1  xi ÞPi  2Di Þ C þ bi >0 2 ðð1  xi ÞPi  Di ÞDi 2 ðð1  xi ÞPi  Di ÞDi   h D ðð1  xi ÞPi  Di Þ xi ðPi  Di Þ þ α5i ¼ i 1 þ 2xi i þ >0 Pi Pi 2 Pi

 h ð1 þ xi ÞDi α6i ¼ i >0 2 Pi

α4i ¼

ð5:258Þ ð5:259Þ ð5:260Þ

P Note that since ni¼1 ðC i þ xi CRi Þ Di is constant, it has been removed from the objective function. In Sect. 5.4, a solution method is given for the developed model. In this case, according to Fig. 5.13, the objective function of the joint production system is obtained as follows: Shortage Cost

Holding Cost   zfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ h zfflffl}|fflffl{ z}|{  NC bi Bi T 3i þ T 4i Ii  1 2 TCðQ, BÞ ¼ NC i Qi þ NK i þ Nhi T þ Ti þ 2 2 i   P n n n n h i X X X Cbi Bi T 4i þ T 5i  Ii  1 2 i¼1 K i ¼ C i Di þ hi þ T þ Ti þ 2T T 2T i i¼1 i¼1 i¼1 Production Cost

Setup Cost

ð5:261Þ where: T 1i ¼

Qi BI  Pi ð1  xi ÞPi  Di

ð5:262Þ

438

5

Multi-product Single Machine

Ii

Ii = ((1– xi )Pi – Di )

Qi Pi

– Bi – Di

(1–xi )Pi – Di

Ti3

Ti4 t

Ti1

bi

Ti2

T

Fig. 5.13 On-hand inventory of perfect-quality items for MP-SM without rework (Taleizadeh et al. 2011)

 T 2i ¼

 1  DPii  xi DPii Qi Di

Bi Di

ð5:263Þ

Bi Di

ð5:264Þ

Bi ð1  xi ÞPi  Di

ð5:265Þ

T 3i ¼ T 4i ¼



The capacity constraint in this case will be: n  X

n  X T 1i þ T 4i þ tsi  T

i¼1

ð5:266Þ

i¼1

Hence, the final model of MP-SM EPQ without rework becomes: X γ 2 X B2i X γ 3i  γ 4i Bi þ γ 5i þ T T i¼1 i¼1 i¼1 n

Min :

TCðT, BÞ ¼ γ 1 T þ

n

n

ð5:267Þ

5.4 Backordering

439

Pn

s:t: :

i¼1 tsi T Pn Di  ¼ T Lower 1  i¼1 Pi

ð5:268Þ

where: γ1 ¼

 n X hi ððð1  xi ÞPi  Di ÞÞD2i >0 2 Pi i¼1 γ2 ¼

n X

ð5:269Þ

Ki > 0

ð5:270Þ

i¼1

 

 2ð1  xi ÞPi hi þ C bi γ 3i ¼ >0 2 ðð1  xi ÞPi  Di ÞDi    ðð1  xi ÞPi  Di Þ hi Di γ 4i ¼ 1  ð 1 þ xi Þ þ >0 Pi 2 Pi   h D ðð1  xi ÞPi  Di ÞDi γ 5i ¼ C i Di þ i 1  ð1 þ xi Þ i 2 Pi Pi 2

ð5:271Þ ð5:272Þ ð5:273Þ

In order to derive an optimal solution for the final model, a proof of the convexity of the objective function is first provided. Then, a classical optimization technique using partial derivatives is utilized to derive the optimal solution. Theorem 5.4 The objective function TC(T, B) is convex (Taleizadeh et al. 2011). Proof Using Eq. (5.274), since the Hessian matrix, results in positive values for all nonzero Bi and T, TC(T, B) ¼ Z are convex: ½T, B1 , B2 , . . . , Bn   H  ½T, B1 , B2 , . . . , Bn T X 2α3 2 α6i Bi T > 0 T i¼1 n

¼ 2α1 T 2 þ

ð5:274Þ

To find the optimal production period length T and the optimal backorder quantities bis, partial differentiations of Z with respect to T and Bi are obtained in Eqs. (5.275) and (5.276): ∂Z α ¼ 2α1 T þ α2  32  ∂T T ! 2α1 T þ 3

α2 

n X i¼1

Pn

2 i¼1 α4i Bi 2

T !



n X

α6i Bi ¼ 0

i¼1

α6i bi T  2

α3 þ

n X i¼1

! α4i B2i

ð5:275Þ ¼0

440

5

Multi-product Single Machine

∂Z 2α B ¼ 4i i  α5i  α6i T ¼ 0 ! 2α4i Bi  α5i T  α6i T 2 ¼ 0 ! Bi T ∂Bi ¼

α5i T þ α6i T 2 2α4i

ð5:276Þ

Then the following solution procedure is used to solve and ensure the feasibility of the problem: P Step 1. Check for feasibility. If (1  xi)Pi  Di  0 and ni¼1 ð1 þ xi ÞDi =Pi < 1, go to Step 2. Otherwise, the problem is infeasible. Step 2. Find a solution point. Find the optimal solution using Eqs. (5.275) and (5.276) and by an iterative approach. To do this, start with Bi ¼ 0 and insert Bi ¼ 0 into Eq. (5.242). Then, the new real positive values of T are obtained. This iterative search will continue until the difference between two consecutive values of T is smaller than a given Δ. Step 3. Check the constraint. Check the constraint based on the obtained value of T. If T > TLower, then T ¼ T. Otherwise, T ¼ TLower, and go to Step 4. Step 4. Obtain the optimal solution. Based on the obtained value of T and using QI ¼ Di T  , Bi will be derived by Eq. (5.276). Then calculate the objective function value and go to Step 5. Step 5. Terminate the procedure. The objective function of the MP-SM without rework model is convex too. To find the optimal production period length T and the optimal backorder quantities Bis, partial differentiations of Z with respect to T and Bi are obtained as are given in Eqs. (5.277) and (5.278): γ ∂Z ¼ γ 1 T  22  ∂T T

Pn

2 i¼1 γ 3i Bi 2

T

¼0!T ¼ 2

γ2 þ

Pn

2 i¼1 γ 3i Bi

γ1

2γ B γ ∂Z ¼ 3i i  γ 4i ¼ 0 ! Bi ¼ 4i T T 2γ 3i ∂Bi

ð5:277Þ ð5:278Þ

By substituting Eq. (5.277) in Eq. (5.276), one has: T¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ2 Pn ðγ4i Þ2 γ 1  i¼1 4γ

ð5:279Þ

γ 4i T 2γ 3i

ð5:280Þ

3i

Bi ¼

In this case, the following solution procedure is used to solve and ensure the feasibility of the problem (Taleizadeh et al. 2011):

5.4 Backordering

441

Table 5.37 General data for Example 5.12 (Taleizadeh et al. 2011) Product 1 2 3 4 5 6 7 8 9 10

Di 100 150 200 250 300 350 400 450 500 550

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500

tsi 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Ki 700 650 600 550 500 450 400 350 300 250

Ci 24 22 20 18 16 14 12 10 8 6

CRi 24 22 20 18 16 14 12 10 8 6

hi 10 9 8 7 6 5 4 3 2 1

Cbi 20 18 16 14 12 10 8 6 4 2

xi 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

P Step 1. Check for feasibility. If (1  xi)Pi  Di  0 and ni¼1 DPii < 1 and γ 1 > Pn ðγ4i Þ2 i¼1 4γ 3i , go to Step 2. Otherwise, the problem is infeasible. Step 2. Find a solution point. Using Eqs. (5.279) and (5.280), find the optimal solution. Step 3. Check the constraint. Check the constraint based on the obtained value of T. If T > TLower, then T ¼ T. Otherwise, T ¼ TLower and go to Step 4. Step 4. Obtain the optimal solution. Based on the obtained value of T and using Qi ¼ Di T  , Bi will be obtained by Eq. (5.280). Then calculate the objective function value and go to Step 5. Step 5. Terminate the procedure. Examples 5.12 and 5.13 Consider two multi-product EPQ problems consisting of ten products with breakdown and capacity constraint. In these examples, the demand, production, and proportion of defective rates of each product are considered constant. Furthermore, the production rate of non-defective items is considered constant and is greater than the demand rate for each product. For each example, both the immediate rework and no rework situations are considered. Moreover, there are no scrapped or defective items during the rework process. The production and rework are accomplished using the same resource at the same speed, and shortages are allowed as backorders (Taleizadeh et al. 2011). The general and the specific data of these examples are given in Tables 5.37 and 5.38, respectively. Tables 5.39 and 5.40 show the best results for the both examples considering immediate rework using the first solution procedure. It should be noted that a value of Δ ¼ 1012 is assumed in the solution procedure. Using the second solution procedure, Tables 5.41 and 5.42 show the best results for both examples when rework is not performed (Taleizadeh et al. 2011). Comparison study based on the results given in Tables 5.39, 5.40, 5.41 and 5.42 shows that lower optimum total costs are obtained for both examples in which reworking is allowed. Furthermore, the optimum cycle length obtained for the systems where rework is permitted is greater than those of the non-reworking systems (Taleizadeh et al. 2011).

442

5

Multi-product Single Machine

Table 5.38 General data for Example 5.13 (Taleizadeh et al. 2011) Product 1 2 3 4 5 6 7 8 9 10

Di 100 200 300 400 500 600 700 00 900 1000

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500

tsi 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Ki 700 650 600 550 500 450 400 350 300 250

Ci 24 22 20 18 16 14 12 10 8 6

CRi 24 22 20 18 16 14 12 10 8 6

hi 10 9 8 7 6 5 4 3 2 1

Cbi 20 18 16 14 12 10 8 6 4 2

xi 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Table 5.39 The best results for Example 5.12 with immediate rework (Taleizadeh et al. 2011) Product 1 2 3 4 5 6 7 8 9 10

TLower 0.1246

T 0.7794

T 0.7794

Qi 77.9442 116.9163 155.8884 194.8605 233.8362 272.8047 311.7768 350.7489 389.7210 428.6931

Bi 12.9534 19.4380 25.9523 32.5130 39.1414 45.7818 52.7732 60.0117 68.1049 79.7102

Z 63,610

Table 5.40 The best results for Example 5.13 with immediate rework (Taleizadeh et al. 2011) Product 1 2 3 4 5 6 7 8 9 10

TLower 0.8931

T 0.6014

T 0.8931

Qi 89.3094 178.6178 267.9281 357.2374 446.5468 535.8562 625.1655 714.4749 803.7843 893.936

Bi 9.9875 19.9476 29.9332 39.9866 50.1512 60.4890 71.1199 82.3392 95.09655 114.3833

Z 101,890

5.4 Backordering

443

Table 5.41 The best results for Example 5.12 without rework (Taleizadeh et al. 2011) Product 1 2 3 4 5 6 7 8 9 10

TLower 0.1294

T 7.2282

T 0.1294

Qi 12.9417 19.4125 25.8834 32.3542 38.8251 45.2959 51.7668 58.2376 64.7085 71.1793

Bi 0.7356 1.0236 1.2616 1.4511 1.5934 1.6893 1.7397 1.7452 1.7501 1.7589

Z 77,215

Table 5.42 The best results for Example 5.13 without rework (Taleizadeh et al. 2011) Product 1 2 3 4 5 6 7 8 9 10

5.4.5

TLower 0.0779

T 3.6794

T 0.0779

Qi 7.7880 15.5761 23.3641 31.1521 38.9402 46.7282 54.5162 62.3043 70.0923 77.8804

Bi 0.4427 0.8034 1.0892 1.3051 1.4549 1.5417 1.6483 1.7173 1.7839 1.8369

Z 127,020

Repair Failure

Taleizadeh et al. (2013b) developed a multi-product single-machine EPQ model with production capacity limitation and random defective production rate and failure during repair by considering shortage backordering. The objective was to determine the optimal period lengths, backordered quantities, and order quantities. All items produced are screened, and the inspection cost per item is included in the unit production cost. All defective items produced can be reworked, and rework starts when the regular production process ends. During the regular production time, defective items may be produced randomly. The random fraction of defective items is reworked during the rework process, and complete backordering is allowed (Taleizadeh et al. 2013b). Initially the problem is modeled as a single-product case, and then it is modified as a multi-product case. The basic assumption of EPQ model with imperfect-quality items produced is that Pi must always be greater than or equal to the sum of demand rate Di and the production rate of defective items is di. One has (Taleizadeh et al. 2013b):

444

5

Multi-product Single Machine

Ii

IiMax (Pi – di – Di )

Qi Pi

P1i – d 1i – Di

– Bi

– Di

Pi – di – Di 4

ti 1

2

ti

5

ti

t

3

ti

ti

Pi – di – Di

Bi T

Fig. 5.14 On-hand inventory of perfect-quality items (Taleizadeh et al. 2013b)

Pi  d i  Di  0

  Di ∴0  xi  1  Pi

or

1  E ½X i  

Di 0 Pi

ð5:281Þ

The production cycle length (see Fig. 5.14) is the summation of the production uptime, the reworking time, the production downtime, and the shortage permitted time (Taleizadeh et al. 2013b): T¼

5 X

t ij

ð5:282Þ

j¼1

where the production uptime is t 1i and t 5i , reworking time is t 2i , and production downtime is t 3i and t 4i . Also t 4i is the time shortage permitted, and t 5i is the time needed to satisfy all the backorders by the next production. To model the problem, a part of the modeling procedure is adopted from Hayek and Salameh (2001). Since all products are manufactured on a single machine with a limited capacity, based on Fig. 5.14: Ii Pi  λi  Di

ð5:283Þ

E ½X i   Qi di  Qi ¼ P1i P1i Pi

ð5:284Þ

t 1i ¼ t 2i ¼ t 3i

  I Max 1 P1i þ di d 1i d i B i ¼ ¼ Qi    i Di Di P1i Pi P1i Pi Di Di

ð5:285Þ

5.4 Backordering

445

Bi Di

ð5:286Þ

Bi Pi  di  Di

ð5:287Þ

t 4i ¼ t 5i ¼

I i ¼ ðPi  di  Di Þ 

I Max i

¼ I i þ ðP1i  d 1i 

Di Þt 2i

Qi  Bi Pi

D d λ dD ¼ Qi 1  i  1i i  i i Pi P1i Pi P1i Pi

ð5:288Þ   Bi

ð5:289Þ

During rework process, the production rate of scrapped items is presented in Eqs. (5.290) and (5.291): d1i ¼ P1i E½θi , where 0  θi  1     Pi E ½X i Qi di  Qi 2 d1i  t i ¼ ðP1i E ½θi Þ ¼ E ½θ i  ¼ E ½θi   E ½X i   Qi P1i Pi Pi

ð5:290Þ ð5:291Þ

Hence, the cycle length for a single-product state is (Taleizadeh et al. 2013b): T¼

Qi E½1  θi xi  , Di

where 0  θi  1

ð5:292Þ

or: Qi ¼

Di T , E ½ 1  θ i xi 

where 0  θi  1

ð5:293Þ

During the imperfect rework process, the random defective rate has a range of [0, 1], and the scrap rate has a range of [0, Qi E[θi]E[xi]]. The total inventory cost per year TC(Q, B) is (Taleizadeh et al. 2013b): Production Cost

TCðQ, BÞ ¼

zfflffl}|fflffl{ NC i Qi

Rework Cost

Disposal Cost

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ þ NC Ri E ½X i Qi þ NC Si E ½X i Qi E ½θi  Holding Cost of Perfect Quality Items

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ 1 

5  Max   Max      d t þ t I þ I I I i i i i i t 1i þ t 5i þ Nhi i t 1i þ t 2i þ i t 3i þ 2 2 2 2

  P1i t 2i  2  B  þ Nh1i þ NC bi i t 4i þ t 5i þ NK i ti |{z} 2 2 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Setup Cost Holding Cost of Imperfect Quality Items

Shortage Cost

ð5:294Þ

446

5

Multi-product Single Machine

The multi-product single-machine cost from Eq. (5.291) becomes (Taleizadeh et al. 2013b): TCðQ, BÞ ¼

n X Ci Q

T

i¼1

i

þ

n X CRi E½X i Q i¼1

T

i

þ

n X CSi E ½X i Q E½θi  i

T

i¼1

 

 n X  2  I Max  3  d i t 1i þ t 5i  1  hi I i  1  I i þ I Max 5 i i þ ti þ ti t þ ti þ ti þ 2 T 2 i 2 2 i¼1

 X n n n X  X h1i P1i t 2i  2  B  Ki þ ti þ C bi i t 4i þ t 5i þ T 2 2T T i¼1 i¼1 i¼1 ð5:295Þ Since t 1i þ t 2i þ t 5i are the production and rework times and tsi is the setup time for ith product, similar to previous case, the constraint of the model is (Taleizadeh et al. 2013b): n  X

n  X t 1i þ t 2i þ t 5i þ tsi  T

i¼1

ð5:296Þ

i¼1

Then, based on Eqs. (5.283)–(5.285) and (5.288), one has (Taleizadeh et al. 2013b): n X i¼1

X Di ðP1i þ di Þ tsi  T Tþ Pi P1i E ½1  θi X i  i¼1 n

ð5:297Þ

From Eqs. (5.283) to (5.289) and Eq. (5.293), TC(Q, B) in Eq. (5.294), and constraint in Eq. (5.296), one can formulate the problem as (Taleizadeh et al. 2013b): Min : ¼

TCðQ, BÞ n X

C 1i

i¼1

s:t: :

Th 1

n n n n X X X ðBi Þ2 X 2 Ki C i Bi þ C 3i T þ C4i þ  T T i¼1 i¼1 i¼1 i¼1 Pn i¼1 tsi i T, Bi  0 8i, i ¼ 1, 2, . . . , n P n D ðP1i þd i Þ i¼1 Pi P1i E ½1θi X i 

ð5:298Þ ð5:299Þ

where: C1i ¼

Cbi ðPi  λi Þ hi h þ  i >0 Di ðPi  di  Di Þ 2ðPi  di  Di Þ 2Di

ð5:300Þ

5.4 Backordering

C2i ¼

447

hi D i hi di Di hi þ þ Pi E ½1  θi X i  P1i Pi E ½1  θi X i  E ½1  θi X i    Di d1i di d i Di   1  Pi P1i Pi P1i Pi

>0 C3i ¼

ð5:301Þ

h1i d2i D2i 2

2

þ

hi ðPi  di  Di ÞD2i 2

2

þ

hi ðPi  d i  Di Þd i D2i

2P1i ðPi Þ ðE ½1  θi X i Þ 2ðPi Þ ðE ½1  θi X i Þ 2P1i ðPi Þ2 ðE ½1  θi X i Þ2 " #  di D2i Di Di d 1i λi d i Di þ hi   >0 1  þ Pi Pi P1i Pi P1i 2E ½1  θi X i  2Pi P1i ðE ½1  θi X i Þ2 ð5:302Þ

C 4i ¼

½C i þ CRi E½X i  þ CSi E½X i E ½θi Di d i Di þ >0 E ½1  θ i X i  2Pi E½1  θi X i 

ð5:303Þ

In order to derive the optimal solution of the final model, a proof of the convexity of the objective function is provided. A classical optimization technique using partial derivatives is performed to derive the optimal solutions (Taleizadeh et al. 2013b). Theorem 5.5 The objective function TC(Q, B) in (5.297) is convex (Taleizadeh et al. 2013b). Proof To prove the convexity of TC(Q, B) ¼ Z, the following Hessian matrix is developed (Taleizadeh et al. 2013b): 1 X 2 K 0 T i¼1 i n

½T, B1 , B2 , . . . , Bn   H  ½T, B1 , B2 , . . . , Bn T ¼

ð5:304Þ

From Appendix 1 of Taleizadeh et al. (2013b), the objective function for all nonzero T and Bi is shown to be strictly convex. T and Bi are solved by letting the partial derivatives equal to zero. One has (Taleizadeh et al. 2013b): ∂Z ¼0!T¼ ∂T

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn i¼1 K i Pn 3 Pn  2 2 =4C1i i¼1 C i  i¼1 C i

  ∂Z ¼ 0 ! Bi ¼ C2i =2C1i T ∂Bi then:

ð5:305Þ ð5:306Þ

448

5

Multi-product Single Machine

Table 5.43 General data for examples (Taleizadeh et al. 2013b) P 1 2 3 4 5

Di 300 400 500 600 700

Pi 3000 3500 4000 4500 5000

P1i 2000 2500 3000 3500 4000

tsi 0.003 0.004 0.005 0.006 0.007

Ki 500 450 400 350 300

Ci 15 12 10 8 6

hi 5 4 3 2 1

h1i 2 2 2 2 2

CSi 3 3 3 3 3

Cbi 10 8 6 4 2

CRi 1 2 3 4 5

Table 5.44 Specific data for Example 5.14 (Taleizadeh et al. 2013b) Items 1 2 3 4 5

Xi ~ U [ai, bi] ai bi 0 0.05 0 0.1 0 0.15 0 0.2 0 0.25

di ¼ PiE[Xi] 75 175 300 450 625

E[Xi] 0.025 0.05 0.075 0.1 0.125

Qi ¼

θi ~ U [ai, bi] ai bi 0 0.15 0 0.2 0 0.25 0 0.3 0 0.35

Di T E ½1  θ i X i 

E[θi] 0.075 0.1 0.125 0.15 0.175

d1i ¼ P1iE[θi] 150 250 375 525 700

ð5:307Þ

The constraint below must be satisfied; otherwise, the minimum value of T will be considered as the optimal point (Taleizadeh et al. 2013b): T Min ¼ h

1

Pn

1

D ðP1i þd i Þ i¼1 Pi P1i E ½1θi X i 

i

n X

tsi

ð5:308Þ

i¼1

To ensure feasibility, the following solution procedure must be performed (Taleizadeh et al. 2013b): P þdi Þ < 1, go to Step 2; else the problem Step 1. Check for feasibility. If ni¼1 Pi DP1ii ðEP½1i1θ iXi  will be infeasible. Step 2. Calculate T by Eq. (5.305). Step 3. Calculate TMin by Eq. (5.308). Step 4. If T  TMin, then T ¼ T; else T ¼ TMin. Step 5. Calculate Bi by Eq. (5.304). Step 6. Calculate Qi by Eq. (5.307). Step 7. Terminate procedure. Examples 5.14 and 5.15 Consider a multi-product inventory control problem with five products where the general and specific data are given in Table 5.43. Specific data for Examples 5.14 and 5.15 are presented in Tables 5.44 and 5.45, respectively. They considered two numerical examples with uniform and normal probability distributions for Xi and θi. Tables 5.46 and 5.47 show the optimal results for the two numerical examples (Taleizadeh et al. 2013b).

5.5 Partial Backordering

449

Table 5.45 Specific data for Example 5.15 (Taleizadeh et al. 2013b)



X i ~N μi , σ 2i θi ~N μi , σ 2i 2 σi di ¼ PiE[Xi] μi ¼ E[θi] μi ¼ E[Xi] Items 1 0.1 0.01 150 0.15 2 0.15 0.02 350 0.18 3 0.2 0.03 600 0.21 4 0.25 0.04 900 0.24 5 0.3 0.05 1250 0.27

σ 2i 0.01 0.02 0.03 0.04 0.05

d1i ¼ P1iE[θi] 300 450 630 840 1080

Table 5.46 The best results for Example 5.14 (uniform distribution) (Taleizadeh et al. 2013b) Items 1 2 3 4 5

Uniform TMin 0.7909

T 0.9183

T* 0.9183

Bi 77.72 101.31 124.49 147.34 169.91

Qi 276 369.16 463.49 559.36 657.17

Z 32,129

Table 5.47 The best results for Example 5.15 (normal distribution) (Taleizadeh et al. 2013b) Items 1 2 3 4 5

5.5

Normal TMin 1.0822

T 0.9060

T* 1.0822

Bi 91.858 119.92 147.36 174.02 199.78

Qi 327.11 440.81 558.69 682.05 812.37

Z 34,656

Partial Backordering

In this subsection, five different problems under partial backordering shortage are presented.

5.5.1

Rework

Taleizadeh et al. (2013a) developed a multi-product single-machine EPQ model with partial backordering, imperfect production, rework, budget, and service level constraints. Their objective was to minimize the joint total cost of the system subject to service level and budget constraints. Each product cycle consists of three time periods: production uptime, reworking time, and production downtime. They assumed that the total quantity of imperfect-quality items can be reworked, and no

450

5

Multi-product Single Machine

Ii

Hi

Q = H i + αi (Pi – Di ) i Pi

Max

Pi – Di – Di

Q Hi = ((1– αi )Pi – Di ) i – β i B i Pi

ti4

(1– αi )Pi – Di

ti5 t

β i Bi

2

ti

1 ti

3 ti

– β i Di

Bi

T

Fig. 5.15 On-hand inventory for perfect-quality items (Taleizadeh et al. 2013a)

scrap will be left at the end of the rework period. For the joint production system, capacity and budget are limited, and a fraction of the shortage is backordered (Taleizadeh et al. 2013a). Taleizadeh et al. (2013a) extended Jamal et al.’s (2004) study by considering a more realistic inventory control problem wherein a joint production strategy with a single machine is used to produce several items under a limited capacity and service level and budget constraints. The inventory control problem under study is shown in Fig. 5.15. A singleproduct problem (defined as the ith product) is initially developed in order to model the problem. The model is then further modified to extend for multiple products. The fundamental assumption of the EPQ model with rework process is that the rate of production minus defectives must always be greater than or equal to the demand. With this, one has (Taleizadeh et al. 2013a): ð1  αi ÞPi  Di  0

ð5:309Þ

The production cycle length (see Fig. 5.15) is the sum of the production uptimes for the good and defective items, t 1i and t 5i , respectively; the reworking time, t 2i ; and the production downtimes, t 3i and t 4i . Therefore, one has total production cycle length of: T¼

5 X j¼1

t ij

ð5:310Þ

5.5 Partial Backordering

451

As noted before, since all products are manufactured on a single machine with a limited capacity, the cycle length for all products is illustrated on Fig. 5.15. Therefore, one has: t 1i ¼

β i Bi Qi  Pi ð1  αi ÞPi  Di

Qi Pi   1  DPii  αi DPii Qi t 2i ¼ αi

t 3i ¼

Di

ð5:312Þ



β i Bi Di

ð5:313Þ

βi Bi Bi ¼ βi Di Di

ð5:314Þ

βi Bi ð1  αi ÞPi  Di

ð5:315Þ

t 4i ¼ t 5i ¼

ð5:311Þ

It is evident from Fig. 5.15 that: H i ¼ ðð1  αi ÞPi  Di Þ

Qi  βi Bi Pi

ð5:316Þ

and: Qi Pi Q Q ¼ ðð1  αi ÞPi  Di Þ i þ αi ðPi  Di Þ i  βi Bi Pi Pi

¼ H i þ αi ðPi  Di Þ H Max i

ð5:317Þ

Hence, from Eq. (5.310), the cycle length for a single product is: T¼

5 X j¼1

t ij ¼

Qi þ ð1  βi ÞBi Di

ð5:318Þ

or: Qi ¼ Di T  ð1  βi ÞBi

ð5:319Þ

The total cost function of the model is the sum of setup, processing, rework, shortage, and inventory carrying costs. One has: TC ¼ CA þ CP þ CR þ CH þ CB þ CL For N setups at $ Ki per setup, the annual setup cost is:

ð5:320Þ

452

5

CA ¼

n X

Multi-product Single Machine

ð5:321Þ

NK i

i¼1

For a joint policy N ¼ 1/T, one has: Pn

i¼1 K i

CA ¼

ð5:322Þ

T

The total production cost is the summation of the product of production cost per unit and the quantity per period for all ith products which are Ci and Qi, respectively. The annual production cost is obtained by multiplying CiQi with N. The cost for this joint policy is: Pn

i¼1 C i Qi

CP ¼

ð5:323Þ

T

The total rework cost is the summation of the product of the rework cost per unit of the ith product, and the quantities of the ith product that is to be reworked are CRi and αiQi, respectively. The annual rework cost is obtained by multiplying the total rework cost with N. The cost for this joint policy is: Pn CR ¼

i¼1 C Ri αi Qi

ð5:324Þ

T

From Fig. 5.15, the holding cost of inventory system in independent, and joint production policies are shown in Eqs. (5.325) and (5.326), respectively. One has:

 n X  2  H Max  3 H   H i þ H Max i hi i t 1i þ ti þ i ti 2 2 2 i¼1

ð5:325Þ

 n  2  H Max  3 1 X H i  1  H i þ H Max i hi ti þ ti þ i ti T i¼1 2 2 2

ð5:326Þ

CH ¼ N and: CH ¼

The total backorder cost is the product of the backorder cost per unit of the ith item and the backorder quantity of the ith product which are Cbi and βiBi, respectively. The annual backorder cost of the system for the joint policy is: Pn CB ¼

i¼1 C bi βi Bi

2T



t 4i þ t 5i

 ð5:327Þ

5.5 Partial Backordering

453

The lost sale cost of ith product per period is b π i ð1  βi ÞBi . The annual lost sale cost of the system for the joint policy is: Pn CL ¼

π i ð1 i¼1 b 2T

 βi ÞBi

ð5:328Þ

where b π i and (1  βi)Bi are the lost sale cost per unit of ith product and lost sale quantity of ith product per period, respectively. Therefore, one has (Taleizadeh et al. 2013a): Pn ¼

i¼1 K i þ T

TC ¼ CA þ CP þ CR þ CH þ CB þ CL Pn

 n  2  H Max  3 C Q C Ri αi Qi 1 X H   H i þ H Max i¼1 i i i hi i t 1i þ þ i¼1 þ ti þ i ti T T 2 2 2 T i¼1  4 5  Pn Pn C bi βi Bi t i þ t i b π i ð1  βi ÞBi þ i¼1 þ i¼1 2T 2T

Pn

ð5:329Þ The objective function of the joint production system (multi-product single machine) is: TC ¼

n X i¼1

Pn ðC i þ αi C Ri ÞDi þ 0

i¼1 K i

T

1 ðDi Þ ðð1  αi ÞPi  Di Þ þ 4αi ðDi Þ2 ðð1  αi ÞPi  Di Þ þ 2αi 2 ðDi Þ2 ðPi  Di Þ B C B C 2ðPi Þ2 CT þ hi B   B C 2 @ A 1 αi ðDi Þ þððð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ Pi   Pi Pi Pi 0 1 0 1 ðð1  αi ÞPi  Di Þ þ αi ðPi  Di Þ B C C P i Di B ðð1  αi ÞPi  Di ÞDi ð1  βi Þ þ 2βi Pi Di B C B C B C β þ D B C i   i 2 B C @ A 2ðPi Þ P D  1  αi B C þ i i  hi B CBi Pi B C B C B C 2 2 @ ð1  βi ÞDi ð4αi ðð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ þ αi βi Pi Di þ αi Di ð1  βi ÞðPi  Di Þ A þ 2 ðPi Þ     π  C i  αi C Ri Þð1  βi Þ Bi 1 αi ðDi ð1  βi ÞÞ ðb  hi ððð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ Di    i Pi Pi Pi 2 T 0 1 2 2 ð1 þ αi Þhi βi ð1  βi Þ h ðð1  αi ÞPi  Di Þð1  βi Þ hi βi þ þ i B C Pi 2ð1  αi ÞPi  2Di 2ðPi Þ2 B C B C B C 2 B hi ð1  βi Þ ð2αi ðð1  αi ÞPi  Di Þ þ αi 2 ðPi  Di ÞÞ C bi βi ðð1  αi ÞPi  ð1  βi ÞDi Þ C Bþ C 2 þ 2 B C Bi 2ðð1  αi ÞPi  Di ÞDi ðPi Þ C þB B C T B βi ð1  βi Þhi ððð1  αi ÞPi  Di Þ þ αi ðPi  Di Þ þ Pi  ð1 þ αi ÞDi Þ C Bþ C B C P D i i B C B C   2 @ hi β 2 A 1 þ αi ð1  βi Þ i þ  hi ððð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ Di  Di Pi Pi 2

ð5:330Þ

454

5

Multi-product Single Machine

In the joint production systems having rework, the total production, rework, and setup should be smaller than the cycle length. In our problem,  P Pn  times n 1 2 5 i¼1 t i þ t i þ t i þ i¼1 tsi must be less than or equal to T. Hence, the model with capacity constraint is (Taleizadeh et al. 2013a): n  X

n  X t 1i þ t 2i þ t 5i þ tsi  T

i¼1

ð5:331Þ

i¼1

From Eqs. (5.311), (5.312), and (5.315), the capacity constraint model becomes: n n X X ð1 þ αi Þ ðDi T  ð1  βi ÞBi Þ þ tsi  T Pi i¼1 i¼1

ð5:332Þ

Since the production quantity is Qi, the total available budget is W, and αiQi is the number of the ith product which need rework, the budget constraint then becomes (Taleizadeh et al. 2013a): n X

C i Qi þ CRi αi Qi  W

ð5:333Þ

i¼1

From Eqs. (5.319) and (5.333), one has: n X

ðC i þ αi C Ri ÞðDi T  ð1  βi ÞBi Þ  W

ð5:334Þ

i¼1

For the service level constraint, the ith product shortage quantity per period, the annual demand of the ith product, the number of periods in each year, and the safety factor of allowable shortage are Bi, Di, N, and SL, respectively. With this, the service level constraint becomes: n X N  Bi  SL Di i¼1

ð5:335Þ

The service level constraint is: Pn T  SL

λ4i i¼1 2λ5i Di Pn λ3i   i¼1 2λ5i Di

Level ¼ T Service Min

ð5:336Þ

From Eqs. (5.330), (5.332), and (5.334), the final model of the joint production system is:

5.5 Partial Backordering

Min :

455

TCðT, BÞ X X Bi X B2 X λ1 þ λ2 T  λ3i Bi  λ4i þ λ5i i þ T T T i¼1 i¼1 i¼1 i¼1 n

¼

n

n

n

 ðC i þ αi C Ri ÞDi s:t: :

ð5:337Þ

Pn ð1 þ αi Þ ð1  βi ÞBi i¼1 Pi   T ¼ T Production Min Pn ð1 þ αi ÞDi 1  i¼1 Pi Pn W þ i¼1 ðC i þ αi C Ri Þð1  βi ÞBi Pn T ¼ T Budget Max ð C þ α C ÞD i i Ri i i¼1 Pn λ4i Pn

i¼1 tsi



T 

i¼1 2λ5i Di

SL  T, Bi

Pn

λ3i i¼1 2λ5i Di

ð5:338Þ

ð5:339Þ

Level  ¼ T Service Min

ð5:340Þ

8i; i ¼ 1, 2, . . . , n

ð5:341Þ

where: λ1 ¼

n X

Ki > 0

ð5:342Þ

i¼1

1 ðDi Þ2 ðð1  αi ÞPi  Di Þ þ 4αi ðDi Þ2 ðð1  αi ÞPi  Di Þ þ 2αi 2 ðDi Þ2 ðPi  Di Þ n C B X 2ðPi Þ2 C B λ2 ¼ hi B C>0   2 A @ 1 αi ðDi Þ i¼1 þððð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ Di   Pi Pi Pi 0

ð5:343Þ 0

1 ðð1  αi ÞPi  Di Þ þ αi ðPi  Di Þ B ðð1  α ÞP  D ÞD ð1  β Þ þ 2β P D B C C P i Di B C B C i i i i i i i i þB   B C β i Di C 2 @ B A C P D  1  α 2 ð P Þ i i i i C>0 þ λ3i ¼ hi B B C Pi B C B C 2 2 @ ð1  βi ÞDi ð4αi ðð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ þ αi βi Pi Di þ αi Di ð1  βi ÞðPi  Di Þ A 2 ðP i Þ 0

1



λ4i ¼ hi  >0



ð5:344Þ

ððð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ 1 α Di   i Di ð1  βi Þ Pi Pi Pi ðb π i  Ci  αi CRi Þð1  βi Þ 2 ð5:345Þ

456

λ5i ¼

5

Multi-product Single Machine

ð1 þ αi Þhi βi ð1  βi Þ h ðð1  αi ÞPi  Di Þð1  βi Þ2 hi β i 2 þ þ i Pi 2ð1  αi ÞPi  2Di 2 ð Pi Þ 2 þ

hi ð1  βi Þ2 ð2αi ðð1  αi ÞPi  Di Þ þ αi 2 ðPi  Di ÞÞ C bi βi ðð1  αi ÞPi  ð1  βi ÞDi Þ þ 2ðð1  αi ÞPi  Di ÞDi ð Pi Þ 2

βi ð1  βi Þhi ððð1  αi ÞPi  Di Þ þ αi ðPi  Di Þ þ Pi  ð1 þ αi ÞDi Þ hi βi 2 þ Pi D i Di   2 1 þ α i ð1  β i Þ  hi ððð1  αi ÞPi  Di Þ þ αi ðPi  Di ÞÞ Di  >0 Pi Pi þ

ð5:346Þ Firstly, they proved the convexity of the objective function using the Hessian matrix. The roots of the objective function are then derived applying principles in differential calculus (Taleizadeh et al. 2013a). Theorem 5.6 The objective function TC(T, Si) in (5.337) is convex (Taleizadeh et al. 2013a). Proof In Eq. (5.347), since the Hessian matrix is positive definite for all nonzero Bi and T, therefore TC(T, Bi) is convex (Taleizadeh et al. 2013a): ½T, B1 , B2 , . . . , Bn   H  ½T, B1 , B2 , . . . , Bn T ¼

2λ1 þ

Pn

i¼1 λ4i Bi

T

>0

ð5:347Þ

To derive the optimal values of the decision variables, they took the partial differentiations of TC(T, Bi) with respect to T and Bi. One has (Taleizadeh et al. 2013a): Pn Pn ∂TC  i¼1 λ5i B2i  λ1 þ i¼1 λ4i Bi ¼ þ λ2 ! T ∂T T2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u P   u λ1  ni¼1 λ 2 4λ5i u ¼ t Pn λ3i 2  λ2  i¼1 4λ5i

ð5:348Þ

∂TC 2λ5i Bi  λ4i λ T þ λ4i  λ3i ! Bi ¼ 3i ¼ ð5:349Þ T 2λ5i ∂Bi  P P  where both λ1  ni¼1 λ4i 2 =4λ5i and λ2  ni¼1 λ3i 2 =4λ5i should be either positive or negative simultaneously. This is to ensure that a feasible solution exists. In order to solve the above problem, they introduced the following solution procedure to ensure that all possible conditions are considered. All constraints are checked to affirm that only feasible optimal solution is obtained (Taleizadeh et al. 2013a):

5.5 Partial Backordering

457

P Step 1. Check for initial feasibility. If (1  αi)Pi  Di  0, SL > ni¼1 λ3i =2λ5i Di ,   Pn P and both λ1  i¼1 λ4i 2 =4λ5i and λ2  ni¼1 λ3i 2 =4λ5i are simultaneously either positive or negative, then proceed to Step 2. Otherwise, the problem is infeasible. Step 2. Find a solution point. Use Eqs. (5.348) and (5.349) to calculate values for T and Si. Step 3. Check for secondary feasibility condition: P P Pn ð1þαi ÞDi iÞ Condition 1. If both ni¼1 tsi  ni¼1 ð1þα are i¼1 Pi Pi ð1  β i ÞBi and 1  either positive or negative, go to Step 4. Otherwise, the problem is not feasible; then go to Step 6.   Level , go to Step 4. Otherwise, Condition 2. If T Budget  Max T Production , T Service Max Min Min the problem is not feasible; then go to Step 6. Service Level , T Budget from Step 4. Check the constraint. Calculate T Production Max , T Min Min Eqs. (5.338) to (5.340). They introduced the following conditions to arrive in determining the optimal values to be considered:    Level  T  T Budget , T Service Condition 3. If Max T Production Max , then T ¼ T. Min Min Budget  Condition 4. If T  T Budget Max , then T ¼ T Max .  Level Condition 5. If T  Max T Production , T Service Min Min  Production Service Level  , T Min Max T Min .

,

then

T ¼

Step 5. Derive the optimal solution. Based on the derived value of T, they derived si using Eq. (5.349). For Qi ¼ DiT  (1  βi)Bi, the optimal values of the order quantity can be obtained. Calculate the objective function using Eq. (5.337), and then go to Step 6. Step 6. Terminate the procedure. Examples 5.16, 5.17, and 5.18 Consider a production system having imperfect production processes. All of the parameters are considered constants in each cycle. Taleizadeh et al. (2013a) assumed that the total imperfect-quality items produced are reworkable and no items will be left as scrap. The manufacturer uses the same resource for both the production and the rework processes, and due to joint production system, there is production capacity limitation and budget constraint. Shortage is allowed and a fraction of these will be backordered. Three multi-product EPQ problems with imperfect items, immediate rework, and capacity and budget constraints with partial backordering are considered for 15 products. The general and the specific data for two examples are given in Tables 5.48, 5.49, and 5.50. For the examples, it is assumed that the values for the safety factor of total allowable shortages and the available budget per period are SL ¼ 0.90, W ¼ 400,000; SL ¼ 0.90, W ¼ 200,000; and SL ¼ 0.99, W ¼ 400,000, for Examples 5.16, 5.17, and 5.18, respectively. The optimal results obtained using the proposed methodology is shown in Tables 5.51, 5.52, and 5.53. 5.16, since lies between the lower-bound  In  Example  T ¼ 2.5619  Service Level Max T Production ¼ 1:2673 and the upper-bound (T Budget , T ¼ 3:1557), Max Min Min

458

5

Multi-product Single Machine

Table 5.48 General data for Example 5.16 (Taleizadeh et al. 2013a) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Di 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500 11,000 11,500 12,000

tsi 0.0025 0.0030 0.0035 0.0040 0.0045 0.0025 0.0030 0.0035 0.0040 0.0045 0.0025 0.0030 0.0035 0.0040 0.0045

Ki 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

CRi 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Ci 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6

hi 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Cbi 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

bi π 1 3 5 7 9 11 13 157 17 19 21 23 25 27 29

αi 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

βi 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7

Cbi 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

bi π 1 3 5 7 9 11 13 157 17 19 21 23 25 27 29

αi 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

βi 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7

Table 5.49 General data for Example 5.17 (Taleizadeh et al. 2013a) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Di 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500 11,000 11,500 12,000

tsi 0.0005 0.0010 0.0015 0.0020 0.0025 0.0005 0.0010 0.0015 0.0020 0.0025 0.0005 0.0010 0.0015 0.0020 0.0025

Ki 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

CRi 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Ci 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6

hi 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

therefore the condition for the feasibility of the solution algorithm is satisfied. However in Example 5.17, the obtained value is T ¼ 2.5619 and is greater than   Service Level both the Max T Production , T ¼ 1:6023. And in this ¼ 1:2673 and T Budget Max Min Min situation, the second condition on the third step is satisfied; therefore, they considered the upper bound as the optimal value for the cycle length which will then be T  ¼ T Budget Max ¼ 1:6023. Finally inExample 5.18, since T ¼ 0.9386 is smaller than Level ¼ 1:3277 and T Budget , T Service ¼ 2:7035 , the third both Max T Production Max Min Min

5.5 Partial Backordering

459

Table 5.50 General data for Example 5.18 (Taleizadeh et al. 2013a) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Di 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500 11,000 11,500 12,000

tsi 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060 0.0065 0.0070 0.0075 0.0080

Ki 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

CRi 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Ci 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6

hi 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Cbi 5 7 9 11 13 157 17 19 21 23 25 27 29 31 33

b πi 1 3 5 7 9 11 13 157 17 19 21 23 25 27 29

αi 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

βi 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7

Table 5.51 The optimal results for Example 5.16 (Taleizadeh et al. 2013a) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

T Production Min 0.1159

Level T Service Min 1.2673

T Budget Max 3.1557

T 2.5619

T 2.5619

Qi 380.10 505.60 630.70 755.40 879.90 1006.40 1130.90 1255.30 1379.60 1503.80 1631.20 1755.50 1879.80 2004.20 2128.10

Bi 8.33 13.59 19.63 26.30 33.50 45.89 54.78 64.02 73.58 83.42 113.48 125.98 138.75 151.77 165.03

Z 932,400

condition on the third step is satisfied; therefore, the lower limit of the cycle length  Level ¼ 1:3277 is the optimal value for the cycle length. All Max T Production , T Service Min Min three provided examples present all the possible cases that may occur for the T (see the third step on the solution procedure). It is also important to note that if the upper bound of the cycle length T Budget is smaller than its lower-bound Max  Production  Level , T Service Max T Min , the problem becomes infeasible as stated as the Min feasibility condition provided in the first step of the proposed algorithm.

460

5

Multi-product Single Machine

Table 5.52 The optimal results for Example 5.17 (Taleizadeh et al. 2013a) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

T Production Min 0.1159

Level T Service Min 1.2673

T Budget Max 1.6023

T 2.5619

T 1.6023

Qi 237.30 315.50 393.40 471.10 548.60 628.10 705.70 783.30 860.70 938.10 1018.50 1096.10 1173.60 1251.10 1328.50

Bi 6.06 9.89 14.27 19.10 24.31 32.09 38.29 44.72 51.37 58.21 76.71 85.13 93.72 102.48 111.40

Z 473,730

Table 5.53 The optimal results for Example 5.18 (Taleizadeh et al. 2013a) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

T Production Min 0.0857

5.5.2

Level T Service Min 1.3277

T Budget Max 2.7935

T 0.9386

T 1.3277

Qi 260.70 325.00 389.00 453.00 516.80 583.10 647.10 711.10 775.00 838.90 906.20 970.30 1034.40 1098.40 1162.40

Bi 9.74 13.94 18.53 23.43 28.59 35.88 41.74 47.76 53.94 60.26 77.17 84.76 92.47 100.30 108.26

Z 390,040

Repair Failure

A multi-product single-machine EPQ model with limited production capacity, random defective production rate, and repair failure is developed by Taleizadeh et al. (2010b). The aim of this model is to minimize the expected total annual cost by optimizing the period length, the backordered quantities, and the rework items. The

5.5 Partial Backordering

461

main difference between this work and what presented in Sect. 5.5.1 is considering repair failure in rework process. This model is presented in Chap. 4, Sect. 4.5.2.

5.5.3

Scrapped

Taleizadeh et al. (2010a) extended a multi-product single-machine EPQ model with the production capacity limitation and random defective production rate. The shortage was assumed to occur in combination of backorder and lost sale, and there was a limitation on the service level. Imperfect production processes, due to process deterioration or some other factors, may randomly generate X percent of defective items at a rate d. The inspection cost per item is involved when all items are screened. All defective items are assumed to be scrapped; i.e., no rework is allowed. The expected production rate of the scrapped items d can be expressed as d ¼ PE[X] (Taleizadeh et al. 2010a). Figure 5.16 depicts the on-hand inventory level and allowable backorder level of the EPQ model with permitted backlogging. To model the problem, they employed a part of modeling procedure used by Hayek and Salameh (2001). Then, based on Fig. 5.16, for i ¼ 1, 2, . . ., n: T¼

4 X j¼1

t ij ¼

QBi ð1  E ðX i ÞÞ þ ð1  βi ÞBi Di

t 1i ¼

I 1i Pi  Di  di

I 1i ¼ ðPi  Di  d i Þ

ð5:352Þ ð5:353Þ

βi Bi Bi ¼ βi Di Di

ð5:354Þ

βi Bi Pi  Di  di 1  Q t i þ t 4i ¼ i Pi

t 4i ¼

ð5:351Þ

I 1i Di

t 2i ¼ t 3i ¼

Qi  β i Bi Pi

ð5:350Þ

ð5:355Þ ð5:356Þ

The objective function of the model is the summation of the expected annual production, holding, shortage, disposal, and setup costs as (Taleizadeh et al. 2010a):

462

5

Multi-product Single Machine

I

Qi – β i Bi

Ii1 Pi – Di – di – Di – β i Di Ti3

Ti4 t

β i Bi

ti1

(1– β i )Bi

ti2

Pi – Di – di

T

Fig. 5.16 A production–inventory cycle (Taleizadeh et al. 2010a)

Z ¼ CP þ CH þ CB þ CL þ CS þ CA

ð5:357Þ

The production cost per unit and the production quantity per period of the ith product are Ci and Qi, respectively. Hence, the production cost of the ith product per period is CiQi. While the total annual production cost of the ith product in a disjoint production policy (each product is ordered separately) is N  C i QBi , this cost for the C QB joint policy (all products have a unique ordering cycle) is iT i . Furthermore, since the shortages are in combinations of backorders and lost sales, based on Eq. (5.350) one has: QBi ¼

T  Di  ð1  βi ÞBi T  Di  ð1  βi ÞBi ¼ E ð1  X i Þ 1  E ðX i Þ

ð5:358Þ

Hence, the expected annual production cost is:

CP ¼

n Ci X i¼1

h

TDi ð1βi ÞBi 1E ðX i Þ

T

i ¼

n X i¼1

 n  X C i ð1  β i Þ B i C i Di  1  EðX i Þ i¼1 1  E ðX i Þ T

ð5:359Þ

5.5 Partial Backordering

463

The holding cost per unit of the ith product per unit time for both the healthy and the scrapped items is hi. According to Fig. 5.16, the total holding costs of healthy items per cycle and per year are shown in (5.360) and (5.361), respectively:

1  n X  Ii  1 2 hi t þ ti 2 i i¼1

1  n X  Ii  1 2 t þ ti hi N 2 i i¼1

ð5:360Þ ð5:361Þ

However, Eq. (5.361) for the joint production policy in which N ¼ 1/T becomes:

 n  1 X I 1i  1 2 h t þ ti T i¼1 i 2 i

ð5:362Þ

Finally, the expected total annual holding cost of healthy items is (see Appendix 2 of Taleizadeh et al. (2010a)): n X i¼1

" hi ðPi  d i Þ

ðPi  Di  d i ÞD j 2ðPi Þ2 ð1  EðX i ÞÞ2

T

½ðPi  Di  d i Þð1  βi Þ þ βi Pi ð1  EðX i ÞÞ Bi ðPi Þ2 ð1  EðX i ÞÞ2

h i 3 ðPi  Di  di Þ2 ð1  βi Þ2 þ 2βi ð1  βi Þð1  E ðX i ÞÞPi ðPi  Di  di Þ þ ðβi Pi Þ2 ð1  E ðX i ÞÞ2 ðB Þ2 i 5 þ T 2Di ðPi Þ2 ð1  E ðX i ÞÞ2 ðPi  Di  d i Þ

ð5:363Þ Since the scrapped items of each product is assumed to be held until the end of its production time, based on Fig. 5.16, the total holding costs of the scrapped items per cycle and per year are shown in (5.364) and (5.365), respectively: 

1  n X  di t i þ t 4i  1 4 ti þ ti hi 2 i¼1 

1  n X  di t i þ t 4i  1 4 ti þ ti hi N 2 i¼1

ð5:364Þ ð5:365Þ

Again, for the joint production policy, Eq. (5.365) becomes: "   # 

1  2 n n 4 X  1 X di ti þ ti  1 1 d Qi t i þ t 4i ¼ hi hi i 2 T i¼1 T i¼1 2 Pi

ð5:366Þ

Hence, the expected total annual holding cost of scrapped items (unit holding cost of health and scrapped items are equal) is:

464

5 n X i¼1

"

Multi-product Single Machine

! # B2i ð1  β i Þ2 2 2 T 2ð1  E ðX i ÞÞ ðPi Þ

ðDi Þ2 T  2Di ð1  βi ÞBi hi d i þ 2ð1  EðX i ÞÞ2 ðPi Þ2

ð5:367Þ

Finally, the expected total annual holding cost of healthy and scrapped items is: n X

"

ðPi  Di  d i ÞDi ½ðP  Di  di Þð1  βi Þ þ βi Pi ð1  E ðX i ÞÞ T i Bi 2 2 2 ð P Þ ð 1  E ð X Þ Þ ðP i Þ2 ð1  E ðX i ÞÞ2 i i i¼1 h i 3 ðPi  Di  d i Þ2 ð1  βi Þ2 þ 2βi ð1  βi Þð1  EðX i ÞÞPi ðPi  Di  di Þ þ ðβi Pi Þ2 ð1  EðX i ÞÞ2 ðB Þ2 i 5 þ T 2ðDi ÞðPi Þ2 ð1  E ðX i ÞÞ2 ðPi  Di  di Þ " ! # n X B2i ðDi Þ2 T  2Di ð1  βi ÞBi ð1  β i Þ2 þ hi d i þ 2 2 2 2 2ð1  E ðX i ÞÞ ðP i Þ 2ð1  EðX i ÞÞ ðPi Þ T i¼1 CH ¼

hi ðPi  di Þ

ð5:368Þ Based on Fig. 5.16, the backordered and lost sale costs per cycle are shown in (5.369) and (5.370), respectively: n X

C bi βi

j¼1 n X

h

 Bi  3 t þ t 4i 2 i

i

ð5:369Þ

b π i ð1  βi ÞBi

ð5:370Þ

i¼1

These costs for a year become: CB ¼ N

n X

C bi βi

i¼1

CL ¼ N

n X

h

 Bi  3 t i þ t 4i 2

i

b π i ð1  βi ÞBi

ð5:371Þ ð5:372Þ

i¼1

Because of the joint production policy, Eqs. (5.371) and (5.372) will change to (5.373) and (5.374), respectively: CB ¼

n h  i 1 X B C bi βi i t 3i þ t 4i T i¼1 2

ð5:373Þ

n 1X b π ð1  βi ÞBi T i¼1 i

ð5:374Þ

CL ¼

Finally, the expected annual backordered and lost sale costs are:

5.5 Partial Backordering

CB ¼

465

 n ðPi  ð1  βi ÞDi  di ÞðBi Þ2 1 X C bi βi T i¼1 2Di ðPi  Di  di Þ

ð5:375Þ

n 1X b π ð1  βi ÞBi T i¼1 i

ð5:376Þ

CL ¼

The disposal cost per unit of the scrapped item of the jth product is CSi, and the quantity P of scrapped items is E(Xi)Qi. Hence, the expected total disposal cost per cycle is ni¼1 C Si E ðX i ÞQi . This quantity per year becomes: N

n X

CSi EðX i ÞQi

ð5:377Þ

i¼1

Because of the joint production policy, Eq. (5.377) changes to: 1 X C E ðX i ÞQi  T i¼1 Si n

ð5:378Þ

i ð1βi ÞBi , the annual expected total scrapped item cost is: Since Qi ¼ TD1E ðX i Þ

 n T  Di  ð1  βi ÞBi 1 X CS ¼  C E ðX i Þ T i¼1 Si 1  E ðX i Þ   n n X CSi E ðX i ÞDi X C Si EðX i Þð1  βi Þ Bi ¼  T 1  E ðX i Þ 1  E ðX i Þ i¼1 i¼1

ð5:379Þ

The cost of a setup is K which occurs N times per year. So, the annual setup is: CA ¼ N  K ¼

K T

As a result, the objective function of the model becomes:

ð5:380Þ

466

5

Min Z ¼ CP þ CH þ CB þ CL þ CS þ CA ¼ n X

"

Multi-product Single Machine

 n  X C i ð1  βi Þ Bi C i Di  1  E ðX i Þ i¼1 1  E ðX i Þ T i¼1

n X

ðPi  Di  di ÞDi ½ðP  Di  di Þð1  βi Þ þ βi Pi ð1  EðX i ÞÞ T i Bi 2 2 2 ð P Þ ð 1  E ð X Þ Þ ðPi Þ2 ð1  E ðX i ÞÞ2 i i i¼1 h i 3 ðPi  Di  di Þ2 ð1  βi Þ2 þ 2βi ð1  βi Þð1  E ðX i ÞÞPi ðPi  Di  di Þ þ ðβi Pi Þ2 ð1  EðX i ÞÞ2 ðB Þ2 i 5 þ T 2Di ðPi Þ2 ð1  E ðX i ÞÞ2 ðPi  Di  d i Þ " ! # n n X B2i ðDi Þ2 T  2Di ð1  βi ÞBi ð1  β i Þ2 1X þ hi d i þ b π ð1  βi ÞBi þ 2 2 2 2 T i¼1 i 2ð1  E ðX i ÞÞ ðPi Þ 2ð1  E ðX i ÞÞ ðPi Þ T i¼1

 X  n n n  ðPi  ð1  βi ÞDi  d i ÞðBi Þ2 C Si E ðX i ÞDi X C Si E ðX i Þð1  βi Þ Bi K 1X þ  þ C bi βi þ T T T i¼1 2Di ðPi  Di  di Þ 1  E ðX i Þ 1  E ðX i Þ i¼1 i¼1 " n X hi θ i ð 1  β i Þ 2 C β ðP  ð1  βi ÞDi  di Þ þ bi i i ¼ 2 2 2Di ðPi  Di  di Þ i¼1 2ð1  E ðX i ÞÞ ðPi Þ h i3 2 2 ðPi  Di  di Þ ð1  βi Þ þ 2βi ð1  βi Þð1  E ðX i ÞÞPi ðPi  Di  di Þ þ ðβi Pi Þ2 ð1  EðX i ÞÞ2 ðB Þ2 5 i þ T 2Di ðPi Þ2 ð1  E ðX i ÞÞ2 ðPi  Di  d i Þ ! n X hi di 2Di ð1  βi Þ h ðP  di Þ½ðPi  Di  di Þð1  βi Þ þ βi Pi ð1  E ðX i ÞÞ þ i i  Bi 2ð1  E ðX i ÞÞ2 ðPi Þ2 ðPi Þ2 ð1  E ðX i ÞÞ2 i¼1  n  X ðC i þ b π i þ C Si E ðX i ÞÞð1  βi Þ Bi  T 1  E ðX i Þ i¼1 ! n n 2 X ðC i þ C Si E ðX i ÞÞDi  K X hi ðPi  d i ÞðPi  Di  di ÞDi hi di ðDi Þ þ þ T þ þ T 1  E ðX i Þ 2ðPi Þ2 ð1  E ðX i ÞÞ2 2ð1  E ðX i ÞÞ2 ðPi Þ2 i¼1 i¼1 þ

hi ðPi  d i Þ

ð5:381Þ The maximum capacity of the single machine and the minimum service rate are the two constraints of the model that are described in the following: Since t 1i þ t 4i and tsi are the production time and setup time of the ith product, respectively, the summation of  the Pntotal production and setup time (for all products) will be Pn  1 4 i¼1 t i þ t i þ i¼1 tsi in which it should be smaller or equal to the period length (T ). So the capacity constraint of the model is: n  n X  X t 1i þ t 4i þ tsi  T i¼1

ð5:382Þ

i¼1

Then, based on the derivation: Pn T

P i ÞBi  ni¼1 Piðð1β 1E ðX i ÞÞ Pn Di

i¼1 tsi

1

i¼1 Pi ð1E ðX i ÞÞ

ð5:383Þ

5.5 Partial Backordering

467

Since the shortage quantity of the ith product per period is Bi, the annual demand of the ith product is Di, the number of periods in each year is N, and the safety factor of allowable shortage is SL, the service rate constraint becomes: n X N  Bi  SL Di i¼1

ð5:384Þ

Finally: Pn

C 2i i¼1 2C 1i Di

SL T Pn C3i  ¼ T SL  i¼1 2C1 D i

ð5:385Þ

i

Based on the objective function in (5.381) and the constraints in (5.383) and (5.385), the final model becomes (Taleizadeh et al. 2010a): ! " n X b π i ð1  βi Þ hi d i ð1  βi Þ2 C β ðP  ð1  βi ÞDi  di Þ Min Z ¼ þ þ bi i i 2 2 2D 2Di ðPi  Di  di Þ i 2 ð 1  E ð X Þ Þ ð P Þ i i i¼1 h i3 2 2 ðPi  Di  d i Þ ð1  βi Þ þ 2βi ð1  βi Þð1  E ðX i ÞÞPi ðPi  Di  di Þ þ ðβi Pi Þ2 ð1  E ðX i ÞÞ2 ðB Þ2 5 i þ T 2Di ðPi Þ2 ð1  E ðX i ÞÞ2 ðPi  Di  di Þ     n X C j þ C Si E ðX i Þ ð1  βi Þ Bi  T 1  E ðX i Þ i¼1 ! n X hi di 2Di ð1  βi Þ hi ðPi  d i Þ½ðPi  Di  d i Þð1  βi Þ þ βi Pi ð1  Eðβi ÞÞ  Bi þ 2ð1  E ðX i ÞÞ2 ðPi Þ2 ðPi Þ2 ð1  E ðX i ÞÞ2 i¼1 i1 0 h  n n

hi ðPi  di ÞðPi  Di  di ÞDi þ θi ðDi Þ2 X X ðCi þ C Si E ðX i ÞÞDi K @ AT þ þ þ T 1  E ðX i Þ 2ðPi Þ2 ð1  EðX i ÞÞ2 i¼1 i¼1 s:t: :

Pn

Pn

ð1  βi ÞBi Pi ð1  E ðX i ÞÞ T P Di 1  ni¼1 Pi ð1  E ðX i ÞÞ Pn C 2i i¼1 2C 1i Di T   P C3 SL  ni¼1 1i 2C i Di T,Bi  0 8i, i ¼ 1, 2, .. ., n i¼1 tsi 

i¼1

ð5:386Þ In order to find the optimal solution of model (5.386), they first provided a proof of the convexity of the objective function. Then, a derivative approach to find the optimal point of the objective function is presented. As stated before, for a feasibility requirement, the production rate of the perfect-quality items is assumed to be greater

468

5

Multi-product Single Machine

than or equal to the sum of thePdemand rate and the production rate of defective Di items. In multi-product model, ni¼1 Pi ð1E ½X i Þ  1. Next, the presented constraints are checked to see if the solution satisfies them. These steps will then be applied using an algorithm that is presented after the convexity proof (Taleizadeh et al. 2010a). To prove the convexity of the objective function, let us rewrite the model as (Taleizadeh et al. 2010a): Min s:t: :

Z¼

n X i¼1

Pn

X X ðBi Þ2 X 2 Bi X 3 K  Ci  Ci Bi þ C4i T þ C 5i þ T T T i¼1 i¼1 i¼1 i¼1 n

C 1i

n

n

n

Pn

ð1  βi ÞBi Pi ð1  E ðX i ÞÞ T P Di 1  ni¼1 Pi ð1  EðX i ÞÞ Pn C 2i i¼1 2C 1i Di T  P C3 SL  ni¼1 1i 2Ci Di T, Bi  0 8i, i ¼ 1, 2, . . . , n i¼1 tsi



i¼1

ð5:387Þ In which: C 1i h þ

" b π ð1  βi Þ hi d i ð1  βi Þ2 C β ðP  ð1  βi ÞDi  di Þ ¼ i þ þ bi i i 2Di 2Di ðPi  Di  d i Þ 2ð1  E ðX i ÞÞ2 ðPi Þ2

ðPi  Di  d i Þ2 ð1  βi Þ2 þ 2βi ð1  βi Þð1  E ðX i ÞÞPi ðPi  Di  di Þ þ ðβi Pi Þ2 ð1  E ðX i ÞÞ2 2Di ðPi Þ2 ð1  E ðX i ÞÞ2 ðPi  Di  d i Þ

C2i ¼ "

C 3i



ðCi þ C Si E ðX i ÞÞð1  βi Þ 0 1  E ðX i Þ

i3 5>0

ð5:388Þ ð5:389Þ

# hi d i Di ð1  βi Þ hi ðPi  di Þ½ðPi  Di  di Þð1  βi Þ þ βi Pi ð1  E ðX i ÞÞ ¼ þ ð1  E ðX i ÞÞ2 ðPi Þ2 ðPi Þ2 ð1  EðX i ÞÞ2 >0 ð5:390Þ

5.5 Partial Backordering

469

i3 2 h hi ðPi  di ÞðPi  Di  di ÞDi þ di ðDi Þ2 5>0 C 4i ¼ 4 2ðPi Þ2 ð1  E ðX i ÞÞ2

C 5i

 ðC i þ CSi E ðX i ÞÞDi ¼ >0 1  E ðX i Þ

ð5:391Þ

ð5:392Þ

Theorem 5.7 The objective function Z in (5.387) is convex (Taleizadeh et al. 2010a). Proof To prove the convexity of Z, one can utilize the Hessian matrix equation. Then, based on Appendix 6 of Taleizadeh et al. (2010a), the objective function is strictly convex (Taleizadeh et al. 2010a). Since the objective function is convex, the constraints being in linear forms are convex too, the model in (5.387) is a convex nonlinear programming problem (CNLPP), and its local minimum is the global minimum. Hence, it follows that for the optimal production period length and optimal level of backorder Bj, one can partially differentiate Z with respect to T and Bj and solve the system of Eqs. (5.393) and (5.394). These equations are derived based on simultaneously letting the partial derivatives zero: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i u P h  uK  ni¼1 C2i 2 =4C 1i u ∂Z i ¼ 0 ! T ¼ t P h  2 n 3 ∂T =4C 1i i¼1 C i C3 T  þ C2i ∂Z ¼ 0 ! Bi ¼ i ∂Bi 2C 1i

ð5:393Þ

ð5:394Þ

i P h  2 Pn 4 where to ensure feasibility both K  ni¼1 C 2i =4C1i and i¼1 C i  i Pn h 3 2 1 =4C i are either positive or negative simultaneously. Then: i¼1 C i Qi ¼

Di T   ð1  βi ÞBi 1  E ðX i Þ

ð5:395Þ

Now: Pn T

Min

¼

P i ÞBi  ni¼1 Piðð1β 1E ðX i ÞÞ Pn Di

i¼1 tsi

1

i¼1 Pi ð1E ðX i ÞÞ

ð5:396Þ

470

5

Multi-product Single Machine

 ¼ P C3 SL  ni¼1 2C1iD

ð5:397Þ

Pn T

SL

C 2i i¼1 2C 1i Di

i

where to have positive TSL the constraint SL >

Pn

i

C 3i i¼1 2C1i Di

should be held.

To ensure the possibility and acceptability of producing all products on a single machine and satisfying the service level constraint, the steps involved in the algorithm of finding the optimal and possible values of T  , Bi , Qi must be performed as follows (Taleizadeh et al. 2010a):

 Pn Pn C3i Pn ðC2i Þ2 Di Step 1. If > 0 , and i¼1 Pi ð1E ½X i Þ  1 , SL > i¼1 2C 1i Di , K  i¼1 4C 1i

 2 Pn 4 Pn ðC3i Þ > 0 (or the first two inequalities hold and the last two i¼1 C i  i¼1 4C 1 i

inequalities are both negative), then go to Step 2. Else the problem is infeasible. Step 2. Calculate T using Eq. (5.393). Step 3. Calculate TSL by Eq. (5.397). Step 4. Calculate Bi, 8 i, i ¼ 1, 2, . . ., n using Eq. (5.394). Step 5. Calculate TMin by Eq. (5.396). Step 6. If T  Max {TMin, TSL}, then T ¼ T; else T ¼ Max {TMin, TSL}.. Step 7. Calculate Qi by Eq. (5.395), Bi by Eq. (5.394), and Z by Eq. 5.387. In the next section, two numerical examples are given to illustrate the applications of the proposed method in cases of uniform and normal distribution functions for f X i ðxi Þ (Taleizadeh et al. 2010a). Examples 5.19 and 5.20 Consider a multi-product inventory control problem with five products in which their general and specific data for two examples are given in Tables 5.54 and 5.55, respectively. In Example 5.19, the probability distribution of Xi is uniform, and in Example 5.20, the distribution for Xi is normal. The setup cost is K ¼ $100,000, and the safety factor of total allowable shortages is SL ¼ 0.35. Based on the available data of Tables 5.54 and 5.55, the problem is solved using the proposed algorithm, and the optimal results are given in Tables 5.56 and 5.57 for the uniform and normal distributions, respectively

5.5.4

Immediate Rework

Taleizadeh and Wee (2015) developed a multi-product single-machine manufacturing system with rework, production capacity constraint, and partial backordering. The defective items of n different types of products are generated xi: i ¼ 1, 2, . . ., n percent per cycle. So the good item quantities are (1  di)Pi. The production and demand rates of the ith product per cycle are Pi and Di, respectively (Taleizadeh and Wee 2015). In this production system, each cycle consists of three parts: production

5.5 Partial Backordering

471

Table 5.54 General data (Taleizadeh et al. 2010a) Product 1 2 3 4 5

Di 800 900 1000 1100 1200

Pi 10,000 11,000 12,000 13,000 14,000

βi 0.75 0.80 0.85 0.90 0.95

tsi 0.01 0.015 0.02 0.025 0.03

b πi 1000 900 800 700 600

Table 5.55 Specific data (Taleizadeh et al. 2010a) Product 1 2 3 4 5

Xi ~ U [ai, bi] bi ai 0 0.1 0 0.15 0 0.2 0 0.25 0 0.3

E[Xi] 0.05 0.075 0.1 0.125 0.15

di 500 825 1200 1625 2100

Ci 500 400 300 200 100



X i ~N μi , σ 2i μi ¼ E[Xi] 0. 25 0.28 0.33 0.38 0.42

hi 15 12 9 6 3

Cbi 350 300 250 200 150

σ 2i 0.01 0.02 0.03 0.04 0.05

CSi 80 70 60 50 40

di 2500 3080 3960 4940 5880

uptime, rework time, and production downtime. They assumed that the total scrapped items are reworkable and no imperfect items are produced at the end of the rework process. Also, the producer has to use the same resource for production and rework processes simultaneously. Because a single machine has a limited joint production system capacity, shortage is allowed with a certain fraction of it to be backordered (Taleizadeh and Wee 2015). Figure 5.17 shows the inventory control problem under study. In the modeling, a single-product problem consisted of ith product is first developed. The fundamental assumption of an economic manufacturing model with rework process is (Taleizadeh and Wee 2015): ð1  xi ÞPi  Di  0

ð5:398Þ

Figure 5.17 shows that T 1i and T 5i are the production uptimes for non-defective and defective items, respectively. T 2i is the reworking time and T 3i and T 4i are the production downtimes, respectively. Finally, the cycle length is: T¼

5 X

T ij

ð5:399Þ

j¼1

T 1i ¼

βi Bi Qi  Pi ð1  xi ÞPi  Di T 2i ¼ xi

Qi Pi

ð5:400Þ ð5:401Þ

472

5

Multi-product Single Machine

Table 5.56 The optimal results of Example 5.19 (uniform distribution) (Taleizadeh et al. 2010a) Product 1 2 3 4 5

Uniform TMin 0.1578

TSL 3.0897

T 3.0897

T 1.9841

Bi 268.5 254.6 223.6 172.6 98.9

Qi 2531.2 2951.2 3395.8 3864.5 4356.2

Z 1,625,500

Table 5.57 The optimal results of Example 5.20 (normal distribution) (Taleizadeh et al. 2010a) Product 1 2 3 4 5

Normal TMin 0.2044

TSL 4.1222

T 4.1222

T 1.6771

T 3i ¼

  1  DPii  xi DPii Qi Di



β i Bi Di

Qi 4280.6 5059.8 6084.9 7275.1 8517

Z 2,246,700

ð5:402Þ

β i Bi Bi ¼ β i Di Di

ð5:403Þ

β i Bi ð1  xi ÞPi  Di

ð5:404Þ

T 4i ¼ T 5i ¼

Bi 349.3 334.7 302.5 239.2 137.2

It is evident from Fig. 5.17 that: I i ¼ ðð1  xi ÞPi  Di Þ

Qi  βi Bi Pi

ð5:405Þ

and: I 0i ¼ I i þ xi ðPi  Di Þ

Qi Q Q ¼ ðð1  xi ÞPi  Di Þ i þ xi ðPi  Di Þ i  βi Bi ð5:406Þ Pi Pi Pi

Hence, using Eq. (5.399), the cycle length for a single-product problem is (Taleizadeh and Wee 2015):

5.5 Partial Backordering

473 Ii

I'i = Ii + xi (Pi – Di)

Qi Pi Pi – Di

Qi

Ii = ((1–xi )Pi – Di )

Pi

– Di

– β i Bi

(1– xi )Pi – Di

Ti4 Ti1

β i Bi

Ti2

Ti5 t

Ti3 – β i Di

Bi

T

Fig. 5.17 On-hand inventory of perfect-quality items (Taleizadeh and Wee 2015)

T¼

5 X

T ij ¼

j¼1

Qi þ ð1  βi ÞBi Di

ð5:407Þ

and the order quantity for the ith product is: Qi ¼ Di T  ð1  βi ÞBi

ð5:408Þ

The elements of the cost function are the setup cost, the holding cost, the processing cost, the rework cost, and the shortage cost which are expressed as (Taleizadeh and Wee 2015): TC ¼ CP þ CH þ CB þ CL þ CR þ CA

ð5:409Þ

In the following, different elements of the objective function are described (Taleizadeh and Wee 2015): The cost of a setup is Ki which occurs N times per year. So, the annual setup cost P is ni¼1 NK i . For a joint policy, N ¼ 1/T, one has: Pn CA ¼

i¼1 K i

T

ð5:410Þ

The production cost per unit is Ci, and the production quantity of ith product per period is Qi. So, the production cost of ith product per period is CiQi. The annual production cost for ith product is NCiQi, and the following cost is the joint policy cost:

474

5

Multi-product Single Machine

Pn

i¼1 C i Qi

CP ¼

ð5:411Þ

T

The rework cost per unit of ith product is CRi and the quantity of ith product that needs to be reworked per period is xiQi. So, the rework cost of ith product per period is CRixiQi. Hence, the rework cost for ith product per year is NCRixiQi, and the annual rework cost for the joint policy is: Pn

i¼1 C Ri xi Qi

CR ¼

ð5:412Þ

T

From Fig. 5.17, Eqs. (5.413) and (5.414) show the inventory holding cost of the system for an independent and joint production policy, respectively: CH ¼ N

 n X I   I i þ I 0i  2  I 0i  3  hi i t 1i þ ti þ ti 2 2 2 i¼1

 n 1 X I i  1  I i þ I 0i  2  I 0i  3  h t þ ti þ ti CH ¼ T i¼1 i 2 i 2 2

ð5:413Þ ð5:414Þ

Also, from Fig. 5.17, Eqs. (5.415) and (5.416) show the annual backordered and the lost sale costs in the joint policy production, respectively: Pn

i¼1 C bi βi Bi

CB ¼

t 4i þ t 5i

2T

Pn

π i ð1 i¼1 b

CL ¼



 βi ÞBi

2T

 ð5:415Þ ð5:416Þ

where CbiβiBi and b π i ð1  βi ÞBi are the backordered and the lost sale cost of ith product per period, respectively. Consequently, one has: Pn

i¼1 K i

TC ¼

T Pn þ

Pn þ

i¼1 C i Qi

T 

Pn

4 5 i¼1 C bi β i Bi T i þ T i

2T

i¼1 C Ri xi Qi

þ

T Pn

 þ

þ

 n 1 X I i  1  I i þ I 0i  2  I 0i  3  Ti þ Ti þ Ti hi T i¼1 2 2 2

π i ð1  βi ÞBi i¼1 b 2T ð5:417Þ

The objective function of the joint production system (multi-product single machine) becomes:

5.5 Partial Backordering

TC ¼

n X

475

Pn ðC i þ C Ri xi ÞDi þ

i¼1 K i

T 1 ðDi Þ2 ðð1  xi ÞPi  Di Þ þ 4di ðDi Þ2 ðð1  xi ÞPi  Di Þ þ 2xi 2 ðDi Þ2 ðPi  Di Þ B C 2 n X B C 2ðPi Þ CT hi B þ   B C 2 A i¼1 @ 1 xi ðDi Þ þððð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ Di   Pi Pi Pi 0 1    ðð1  xi ÞPi  Di ÞDi ð1  βi Þ þ 2βi Pi Di ðð1  xi ÞPi  Di Þ þ xi ðPi  Di Þ 1 α βi Di C þ þ Di   i B 2 n Pi Pi Pi Di X B 2ðPi Þ C CBi  hi B B C 2 2 A i¼1 @ ð1  βi ÞDi ð4xi ðð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ þ xi Pi Di βi þ xi Di ð1  βi ÞðPi  Di Þ 2 ðPi Þ    n  X π  C i  xi C Ri Þð1  βi Þ Bi 1 x ðDi ð1  βi ÞÞ ðb  hi ððð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ Di   i  i Pi Pi Pi 2 T i¼1 1 0 ð1 þ xi Þhi βi ð1  βi Þ hi ðð1  xi ÞPi  Di Þð1  βi Þ2 hi βi 2 þ þ C B Pi 2ð1  xi ÞPi  2Di 2ðPi Þ2 C B C B C B 2 2 B hi ð1  βi Þ ð2xi ðð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ C bi βi ðð1  xi ÞPi  ð1  βi ÞDi Þ C C 2 B þ þ n B 2 X C Bi ð ð ÞP  D ÞD 2 1  x i i i i ðPi Þ C B þ C T B C B i¼1 C B þ βi ð1  βi Þhi ððð1  xi ÞPi  Di Þ þ αi ðPi  Di Þ þ Pi  ð1 þ xi ÞDi Þ C B P i Di C B C B   2 2 A @ hi β ð 1  β Þ 1 þ x i i þ i  hi ððð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ Di  Di Pi Pi i¼1

0

ð5:418Þ In the joint production systems with reworks, the total production, rework, and setup should beP smaller than the cycle length. In our problem, Pn  times n 1 2 5 T þ T i i þ Ti þ i¼1 i¼1 tsi must be smaller or equal to T. Hence, the capacity constraint is: n  X

n  X T 1i þ T 2i þ T 5i þ tsi  T

i¼1

ð5:419Þ

i¼1

From Eqs. (4.400), (4.401) and (4.404), the capacity constraint model becomes: n n X X ð1 þ d i Þ ðDi T  ð1  βi ÞBi Þ þ tsi  T Pi i¼1 i¼1

ð5:420Þ

The final model of the joint production system is (Taleizadeh and Wee 2015): Min :

TCðT, Bi Þ X X Bi X B2 X α1 α3i Bi  α4i þ α5i i þ þ α2 T  T T T i¼1 i¼1 i¼1 i¼1 n

¼

 ðCi þ C Ri xi ÞDi

n

n

n

ð5:421Þ

476

5

s:t: : T

Pn

Multi-product Single Machine

Pn ð1 þ xi Þ ð1  βi ÞBi i¼1 Pi   ¼ T Production Min Pn ð1 þ xi ÞDi 1  i¼1 Pi

i¼1 tsi



T, Bi

ð5:422Þ

8i; i ¼ 1, 2, . . . , n

ð5:423Þ

where α1 ¼ 0

n X

Ki > 0

ð5:424Þ

i¼1

1 ðDi Þ2 ðð1xi ÞPi Di Þþ4di ðDi Þ2 ðð1 xi ÞPi Di Þþ2xi 2 ðDi Þ2 ðPi Di Þ n B C X 2ðPi Þ2 B C α2 ¼ hi B C   2 @ A 1 xi ðDi Þ i¼1 þððð1xi ÞPi  Di Þþ xi ðPi  Di ÞÞ Di   Pi Pi Pi >0 ð5:425Þ 0

1 ðð1  xi ÞPi  Di ÞDi ð1  βi Þ þ 2βi Pi Di B C 2ðPi Þ2 B C B C B ð1  βi ÞDi ð4xi ðð1  xi ÞPi  Di Þ þ xi 2 ðPi  Di ÞÞ þ d i Pi Di βi þ xi 2 Di ð1  βi ÞðPi  Di Þ C B C>0 α3i ¼ hi B þ C ðPi Þ2 B C   B  C @ A ðð1  xi ÞPi  Di Þ þ xi ðPi  Di Þ 1 αi β i Di þ þ Di   Pi Di Pi Pi

ð5:426Þ     π  C i  xi CRi Þð1  βi Þ 1 x ðDi ð1  βi ÞÞ ðb α4i ¼ hi ððð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ Di   i  i >0 Pi 2 Pi Pi

ð5:427Þ

1 ð1 þ xi Þhi βi ð1  βi Þ hi ðð1  xi ÞPi  Di Þð1  βi Þ hi β i 2 þ þ C B Pi 2ð1  xi ÞPi  2Di C B 2ðPi Þ2 C B B h ð1  β Þ2 ð2x ðð1  x ÞP  D Þ þ x 2 ðP  D ÞÞ C β ðð1  x ÞP  ð1  β ÞD Þ C C B i i i i i i i i bi i i i i i i þ C Bþ 2 ð ð ÞP  D ÞD 2 1  x C B i i i i ð Þ P i α5i ¼ B C>0 C B β ð1  β Þh ððð1  x ÞP  D Þ þ α ðP  D Þ þ P  ð1 þ x ÞD Þ i i i i i i i i i C Bþ i i i C B D P i i C B   C B 2 A @ hi β i 2 1 þ xi ð1  β i Þ þ  hi ððð1  xi ÞPi  Di Þ þ xi ðPi  Di ÞÞ Di  Di Pi Pi 0

2

ð5:428Þ

5.5 Partial Backordering

477

Firstly, they proved the convexity of the objective function using the Hessian matrix. The roots of the objective function are then derived using differential calculus (Taleizadeh and Wee 2015). Theorem 5.8 The objective function TC(T, Bi) in (5.421) is convex (Taleizadeh and Wee 2015). Proof Since the Hessian matrix is positive for all nonzero Bi and T, TC(T, Bi) is convex (Taleizadeh and Wee 2015): ½T, B1 , B2 , . . . , Bn   H  ½T, B1 , B2 , . . . , Bn T ¼

2α1 þ

Pn

i¼1 α4i Bi

T

> 0 ð5:429Þ

To derive the optimal values of the decision variables, take the partial differentiations of TC(T, Bi) with respect to T and Bi: Pn Pn ∂TCðT, Bi Þ  i¼1 α5i B2i  α1 þ i¼1 α4i Bi ¼ þ α2 ! T ∂T T2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P α1  ni¼1 α4i 2 =4α5i  P ¼  α2  ni¼1 α3i 2 =4α5i ∂TCðT, Bi Þ 2α5i Bi  α4i α T þ α4i ¼  α3i ! Bi ¼ 3i T 2α5i ∂Bi

ð5:430Þ ð5:431Þ

P P To ensure feasibility, both α1  ni¼1 α4i 2 =4α5i and α2  ni¼1 α4i 2 =4α5i should simultaneously be positive or negative. In order to solve the above problem, they introduced the following solution procedures: P Step 1. Check for feasibility. If (1  xi)Pi  Di  0, ni¼1 ð1 þ xi ÞDi =Pi < 1, and n P P both α1  ni¼1 α4i 2 =4α5i and α2  α4i 2 =4α5i be either positive or negative i¼1

simultaneously, go to Step 2. Otherwise, the problem is infeasible. Step 2. Find a solution point. Using Eqs. (5.430) and (5.431), calculate T and Bi. Step 3. Calculate T  ¼ T Production from Eq. (5.432). If T  T Production , then T ¼ T; Min Min . else, T  ¼ T Production Min Step 4. Derive the optimal solution. Based on the derived value of T, they derived Bi . For Qi ¼ Di T   ð1  βi ÞBi , the optimal values of the order quantity can be obtained. Calculate the objective function and then go to Step 5. Step 5. Terminate the procedure. Examples 5.21 and 5.22 Consider a production system with production capacity limitation, imperfect production processes, immediate rework, and partial backordered quantity. The general and the specific data of these examples are given in Tables 5.58 and 5.59. The best results using the proposed methodology are shown in Tables 5.60 and 5.61 (Taleizadeh and Wee 2015). In Example 5.21,

478

5

Multi-product Single Machine

Table 5.58 General data for Example 5.21 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Di 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500 11,000 11,500 12,000

tsi 0.0025 0.0030 0.0035 0.0040 0.0045 0.0025 0.0030 0.0035 0.0040 0.0045 0.0025 0.0030 0.0035 0.0040 0.0045

Ki 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

CRi 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Ci 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6

hi 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Cbi 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

b πi 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

xi 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

βi 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7

b πi 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

xi 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25

βi 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.7

Table 5.59 General data for Example 5.22 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Di 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900

Pi 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10,000 10,500 11,000 11,500 12,000

tsi 0.0005 0.0010 0.0015 0.0020 0.0025 0.0005 0.0010 0.0015 0.0020 0.0025 0.0005 0.0010 0.0015 0.0020 0.0025

Ki 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

CRi 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Ci 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6

hi 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Cbi 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

since T ¼ 2.5619 is greater than its lower-bound T Production ¼ 0:1159 , so Min T ¼ T ¼ 2.5619 . In Example 5.22, T ¼ 2.5619 is smaller than T Production ¼ Min  Production 2:6247, so T ¼ T Min ¼ 2:6247.

5.5 Partial Backordering

479

Table 5.60 The best results for Example 5.21 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

T Production Min 0.1159

T 2.5619

T 2.5619

Qi 380.10 505.60 630.70 755.40 879.90 1006.40 1130.90 1255.30 1379.60 1503.80 1631.20 1755.50 1879.80 2004.20 2128.10

Bi 8.33 13.59 19.63 26.30 33.50 45.89 54.78 64.02 73.58 83.42 113.48 125.98 138.75 151.77 165.03

TC 932,400

Table 5.61 The best results for Example 5.22 (Taleizadeh and Wee 2015) Product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

5.5.5

T Production Min 2.6247

T 2.5619

T 2.6247

Qi 389.5 518.0 646.2 774.0 901.6 1031.2 1158.8 1286.2 1413.6 1540.8 1671.3 1798.7 1926.0 2053.3 2180.5

Bi 8.48 13.83 19.98 26.77 34.10 46.79 55.86 65.28 75.03 85.07 115.88 128.65 141.69 154.99 168.55

TC 981,350

Preventive Maintenance

There are many instances in which the produced imperfect-quality items should be reworked or repaired with additional costs (Haji et al. 2009). It is assumed that during the rework process, there is no interruption. In each production run, all repairable defective items are reworked, right after the regular production process

480

5

Multi-product Single Machine

ends (Chiu and Chang 2014; Chiu 2010). They assumed that because of some controllable realistic reasons such as lubrications, re-setup for each product, programming on machine, cleaning the machine, checking the machine, and many other reasons, an interruption can happen when each product is being produced. On the other hand, they assumed that only one interruption during producing each type of product as preventive maintenance is sufficient to achieve the mentioned goal. In Sect. 5.4.3, a multi-product single-machine economic production quantity model with preventive maintenance, scrap and rework, and full backordering is presented (Taleizadeh et al. 2014). In this section, partial backordering case from Taleizadeh (2018) is presented. He assumed that preventive maintenance can occur when the inventory level is positive or negative similar to previous case. Similar to Taleizadeh et al. (2014), generally, two cases can be studied: interruption in the backorder-filling stage (see Fig. 5.18) and interruption when there is no shortage (see Fig. 5.19). When an interruption in the production process happens, the machine cannot work and should stay until the state is changed. Figures 5.18 and 5.19 show the level of on-hand inventory of perfect-quality items in the proposed EPQ model for two proposed cases. Since the preventive maintenance occurs in the manufacturing process, firstly the best time of the interruption should be determined. Based on the time of preventive maintenance, the total inventory cost will be different. On the other word, this cost depends on the length of time before interruption occurs. According to Fig. 5.18, if the interruption occurs when inventory level is negative, it only affects the backordering cost and carrying cost of defective items, while if the interruption occurs during positive inventory level, only the carrying cost of healthy and defective items will be affected. So based on the above explanation, about the time of interruption, the two following possible cases in determining the best time of preventive maintenance will be discussed (Taleizadeh 2018): Pi  Di  di  0 ∴ 0  xi 

  D 1 i Pi

ð5:432Þ

The summation of the production uptime, the production downtime, the reworking time, and the preventive maintenance length is equal to production cycle length. So one has: T¼

7 X

t ij

ð5:433Þ

j¼1

According to Fig. 5.18 showing the first case, an interruption during negative inventory level, one has:

5.5 Partial Backordering

481

Ii

Hi Max

P1i – Di – d1i Hi – di

Pi – Di – di ti1 ti2

7

ti3

ti

t – β i Di

ti4

H i1 H i2

ti5

ti6

– (1– β i)Di ti2

β i Bi

– β i Di

β i Bi Pi – Di – di

(1 – β i ) Bi T

Fig. 5.18 Inventory diagram when preventive maintenance occurs during the shortage cycle (Taleizadeh 2018) Ii

Hi Max

P1i – Di – di

Hi H i1 H i2 Pi – Di – di

Pi – Di – di – Di – Di

ti1

7

ti

t ti2

ti3 ti4

ti5

ti6

– β i Di

β i Bi

β i Bi (1 – β i ) Bi T

Fig. 5.19 Inventory diagram when interruption occurs during the non-shortage cycle (Taleizadeh 2018)

482

5

Multi-product Single Machine

βi Bi þ t 2i βi Di Pi  Di  di

ð5:434Þ

βi Bi þ t 2i βi Di  t 1i Pi  Di  d i

ð5:435Þ

t 1i þ t 3i ¼ This gives: t 3i ¼ In continuation:

H 1i ¼ βi Bi  ðPi  di  Di Þt 1i

ð5:436Þ

H 2i ¼ H 1i þ βi Di t 2i

ð5:437Þ

So the cyclic backordering cost is the area of region that is finished at t 3i (as shown in Fig. 5.18) multiplied by the backordering unit cost as below (Taleizadeh 2018): 

 βi Bi þ H 1i 1 H 1i þ H 2i 2 H 2i t 3i ti þ ti þ 2 2 2 8 9    2 1 2 1 1 > < βi Bi t 1i  ðPi  Di  di Þ t 1i þ βi Bi t 2i  ðPi  Di  d i Þt 1i t 2i þ βi Di t 2i þ βi Bi t 3i > = 2 2 2 ¼C bi > 1 > 1 : ;  ðPi  Di  di Þt 1i t 3i þ βi Di t 2i t 3i 2 2 9 8     > > βi Bi t 1  1 ðPi  Di  di Þ t 1 2 þ βi Bi t 2  ðPi  Di  d i Þt 1 t 2 þ 1 βi Di t 2 2 > > i i i i i i = < 2 2     ¼C bi   2 2 β i B i þ t i β i Di 1 β Bi þ t i βi Di 1 > > 1 1 1 > ; : þ βi Bi  ðPi  Di  di Þt 1i  t i þ βi Di t 2i i  ti > 2 2 2 P i  Di  d i P i  Di  d i

CBi ¼C bi

ð5:438Þ Then, Eq. (5.438) can be simplified to Eq. (5.439) as below:      2 β   βi Bi þ t 2i βi Di 1  t1i t 2i βi Di CBi ¼ C bi βi Bi t 2i  ðPi  Di  d i Þt 1i t2i þ βi Di t 2i þ i Bi þ Di t 2i 2 Pi  Di  d i 2

ð5:439Þ Also the lost sale cost is:   CLi ¼ b π i ð1  βi ÞDi t 2i þ ð1  βi ÞBi and the holding cost of the imperfect items is:

ð5:440Þ

5.5 Partial Backordering

483

9 8    1 2  1 2 > > βi Bi þ t 2i βi Di 1 1 2 1 > > > > Pi xi t i þ Pi xi t i t i þ Pi xi t i  Pi xi t i > > =

> 1 2  1 2 > > βi Bi þ t 2i βi Di 1 > 1 β i Bi þ t i βi Di > > > þ x þ x t  P x t P P : 2 i i P D d i i i i i i 2 Pi  Di  d i ; i i i (  2 ) βi Bi þ t 2i βi Di 1 1 2 ¼h1i Pi xi t i t i þ Pi xi 2 Pi  Di  d i ð5:441Þ So the summation of backordering and lost sale costs and holding cost of imperfect items in this case for the ith product is:      2 β   βi Bi þ t 2i βi Di 1 ψ 1i ¼C bi βi Bi t 2i  ðPi  Di  d i Þt 1i t 2i þ βi Di t 2i þ i Bi þ Di t 2i  t 1i t 2i βi Di 2 Pi  Di  d i 2 (  2 ) 2   β Bi þ t i β i D i 1 þ h1i Pi xi t 1i t 2i þ Pi xi i þb π i ð1  βi ÞDi t 2i þ ð1  βi ÞBi 2 Pi  Di  d i

ð5:442Þ Then the first derivative of ψ 1i with respect to t 1i gives: ∂ψ 1i ¼ C bi ðPi  ð1  βi ÞDi  d i Þt 2i þ h1i Pi xi t 2i ∂t 1i

ð5:443Þ

According to Eq. (5.443), the summation of backordering and lost sale costs and holding cost of imperfect items is increasing on t 1i if: C bi ðPi ð1  xi Þ  ð1  βi ÞDi Þ < h1i Pi xi

ð5:444Þ



It means t 1i ¼ 0 which is meaningless, because before starting the production of each item, there is setup time during which required operations can be done and there is no need for additional preventive maintenance operation. In the other way, the summation of backordering and lost sale costs and holding cost of imperfect items is decreasing on t 1i if Cbi(Pi(1  xi)  (1  βi)Di) > h1iPixi, meaning: 

t 1i ¼

β i Bi Pi  Di  di

ð5:445Þ

So the inventory control diagram of optimal case, in respect to the value of t 1i , shown in Eq. (5.445), can be shown in Fig. 5.19. In the next section, the second possible case is studied. In the case, preventive maintenance occurs when there is no shortage; the preventive maintenance cost only affects the holding cost of healthy and imperfect items. According to Fig. 5.19:

484

t 4i ¼

5

Multi-product Single Machine

βi Bi Qi   t 2i Pi Pi  Di  di

ð5:446Þ

and: H 1i ¼ ðPi  Di  di Þt 2i

ð5:447Þ

H 2i ¼ ðPi  Di  di Þt 2i  Di t 3i

ð5:448Þ

H i ¼H 2i þ ðPi  Di  di Þt 4i

  ð5:449Þ Q Bi ¼ðPi  Di  di Þt 2i  Dt 3i þ ðPi  Di  di Þ i   t 2i Pi Pi  Di  d i

In this case, since only interruption  can occur when the rework process is not started, only the holding cost during t 1i þ t 3i þ t 4i similar to  the first case  should be investigated. The carrying cost of perfect products during t 1i þ t 3i þ t 4i is:  1   2    2 H i þ H 2i 3 Hi þ Hi 4 1 ðPi  Di  d i Þ t 2i þ ti þ ti 2 2 2 9 8  2 1   1 > > ðPi  Di  d i Þ t 2i þ 2ðPi  Di  di Þt 2i  Di t 3i t 3i > > > > 2 2 > > > > = < 0 1 Qi 3   ¼hi P ð  D  d Þ  β B  2D t i i i i i i i C Q > > βi Bi 1 Pi > > >þ B  t 2i > @ A i > > > > 2 P ð P  D  d Þ i i i i ; : þðPi  Di  d i Þt 2i 

CHi ¼hi

ð5:450Þ After some factorizations: ( CHi ¼ hi

D ðPi  di Þt 2i t 3i  i 2

   )  3 2 ðPi  Di  d i Þ Qi 2   βi Bi βi bi þ 2Di t 3i 3 Qi  βi Bi þ Di t i þ ti þ 2 Pi Pi 2 Pi  Di  d i

ð5:451Þ Also the holding cost of defective items is: n  2  2 o 1 1 Pi xi t 2i þ Pi xi t 2i t 3i þ Pi xi t 2i t 4i þ Pi xi t 4i 2 2 (    2 )   β B βi Bi 1 2 2 2 3 2 Qi 1 Qi 2 2 i i ¼ h1i Pi xi t þ ti ti þ ti  t þ  t 2 i 2 Pi Pi  Di  d i i Pi Pi  Di  d i i CH1i ¼ h1i

ð5:452Þ After some simplifications (Taleizadeh 2018):

5.5 Partial Backordering

( CH1i ¼ h1i

Pi xi t 2i t 3i

485

)  2 βi Bi xi βi Qi Bi 1 Q2i 1 þ xi þ Px  2 Pi 2 i i Pi  Di  di Pi  Di  di ð5:453Þ

So the summation of holding costs of perfect and imperfect items is: (

 2  )   Q β B β Bi þ 2Di t3i Di  3 2 ðPi  Di  di Þ Qi  βi Bi þ Di t3i i þ i i i ti þ 2 Pi Pi 2 Pi  Di  di 2 ( )  2 2 βi Bi xβQB 1 Q 1 þ h1i Pi xi t2i t3i þ xi i þ Pi xi  i i i i P i  Di  d i Pi  Di  di 2 Pi 2

ψ 2i ¼hi ðPi  d i Þt 2i t3i 

ð5:454Þ Then the first derivative of ψ 2i in respect to t 2i gives: ψ 2i ¼ hi ðPi  di Þt 3i þ h1i Pi xi t 3i ∂t 2i

ð5:455Þ

Equation (5.455) means that the summation of the holding costs of perfect and imperfect items are increasing on t 2i and the best time for considering the interruption   is t 2i ¼ 0. The best time for preventive maintenance of the second case, t 2i ¼ 0, is as same as the best time for the preventive maintenance of the first case shown in Eq. (5.455), and the inventory control diagrams of these two cases are similar, as is shown in Fig. 5.20. So as long as Cbi(Pi(1  xi)  Di) > h1iPixi, the best times for the preventive maintenance in both cases are similar. So if we consider always for each product Cbi(Pi(1  xi)  Di) > h1iPixi, then a unique model can be used to solve the problem on hand (Taleizadeh 2018). It is considered in Fig. 5.20 as the final inventory control diagram. So according to Fig. 5.20, one has: βi Bi Pi  di  Di

ð5:456Þ

t 2i ¼ t m

ð5:457Þ

t 3i ¼

H 1i Pi  di  Di

ð5:458Þ

t 4i ¼

Hi Pi  di  Di

ð5:459Þ

t 1i ¼

  Total imperfect items produced during the production uptime, t 1i þ t 3i þ t 4i , are:

486

5

Multi-product Single Machine

Ii Hi Max

P1i – Di – d1i Hi Pi – Di – di

ti1

2

ti

ti3

7

ti ti4

1

Hi

β i Bi

– Di

ti5

– β i Di

ti6

t

β i Bi

– Di Pi – Di – di

(1 – β i ) Bi T

Fig. 5.20 Inventory diagram if preventive maintenance occurs at the best time (Taleizadeh 2018)

    d i t 1i þ t 3i þ t 4i ¼ Pi xi t 1i þ t 3i þ t 4i ¼ xi Qi

ð5:460Þ

and total scrapped items produced during the rework process are:   φ1i t 5i ¼ P1i θi t 5i ¼ d i θi t 1i þ t 3i þ t 4i ¼ xi Qi θi

ð5:461Þ

So: t 5i ¼

xi Qi P1i

ð5:462Þ

t 6i ¼

H Max i Di

ð5:463Þ

βi Bi Bi ¼ βi Di Di

ð5:464Þ

t 7i ¼ Also:

H 1i ¼ Di t 2i ¼ Di t m

ð5:465Þ

In order to determine the maximum level of on-hand inventory when regular production process stops, Hi, one can subtract the back-ordered quantity and demand during the preventive maintenance time from the total healthy produced items which gives:

5.5 Partial Backordering

487

H i ¼ ðPi  di  Di Þ

Qi  βi Bi  Di t m Pi

ð5:466Þ

and the maximum inventory level is: ¼ H i þ ðP1i  d1i  Di Þt 5i H Max i ¼ ðPi  di  Di Þ

Qi xQ  βi Bi  Di t m þ ðP1i  d 1i  Di Þ i 1 i Pi Pi

ð5:467Þ

Obviously, the cycle length is equal to the first case, as below: T¼

7 X

t ij

j¼1

  ð1  βi ÞBi Qi di d 1i ¼ 1  þ xi 1  þ Di Pi P1i Di

ð5:468Þ

This gives: Qi ¼

Di T  ð1  βi ÞBi Wi

ð5:469Þ

where:

  di d 1i W i ¼ 1  þ xi 1  Pi P1i

ð5:470Þ

The total cyclic cost function including production, rework, disposal, setup, backordered, holding of defective, perfect and scrapped items, and lost sale costs is (Taleizadeh 2018): BackOrderedCost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ DisposalCost SetupCost ReworkCost

 zffl}|ffl{ zfflfflffl}|fflfflffl{ zfflfflfflfflffl}|fflfflfflfflffl{ z}|{  H1   βB TCðQi ,Bi Þ ¼ Ci Qi þ CRi xi Qi þ C Si xi qi θi þ K i þ Cbi i i t 1i þ t 7i þ i t 2i þ t 3i 2 2  



 Max    

Pi xi θi t 1i þ t 3i þ t 4i  5  H H i  4  H i þ H Max i t i þ hi t þ t 5i þ i t 6i þ b þh1i π i ð1  βi ÞDi t 2i þ ð1  βi ÞBi 2 2 i 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} LostSaleCost ProductionCost

HoldingCostof ScrapItems

HoldingCostof PerfectQualityItems

     Pi xi t 3i þ t 4i  3 4  Pi xi  1 3 4  5 Pi xi t 1i  1  ti þ ti þ t i þ Pi xi t 1i t 2i þ t 3i þ t 4i þ ti þ ti þ ti ti þhi 2 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

HoldingCostof DefectiveItems

ð5:471Þ After some algebra and simplifications, Eq. (5.471) can be simplified to (Taleizadeh 2018):

488

5

Multi-product Single Machine

!  2 ! h1i θi ðxi Þ2 hi X i 2xi X i xi xi 2 xi 2 U i 1 X i xi U i þ Qi 2 TCðQ,BÞ ¼ þ þ þ þ þ þ 1 2 Pi 2 Pi P1i Pi P1i P1 2 Di Pi P1i 2P i i       2 C bi βi Y i hi βi 1 1 1 xi X xU þ þ þ þ þ iþ i i B2i  hi βi Qi Bi X i Di Pi P1i 2Di X i 2 Pi P1i    Di þ X i xi ðDi þ U i Þ þ C i þ CRi xi þ CSi xi θi  hi t m þ Qi P1i Pi     X þ Di þ Pi xi þ hi β i t m i þb π i ð1  βi Þ Bi Xi  2  hi Di þ Di X i t m 2 C bi Di ðPi  φi Þðt m Þ2 þ Ki þ þ þb π i ð1  βi ÞDi t m 2X i 2X i ð5:472Þ However, Eq. (5.472) for the annual joint production policy (multi-product single machine) using Eqs. (5.468)–(5.470) changes to: !  2 ! h1i Di 2 θi ðxi Þ2 hi Di 2 X i 2xi X i xi xi 2 xi 2 U i 1 X i xi U i þ T þ þ þ þ þ þ 2W i 2 Pi 2 Pi P1i Pi P1i ðP1i Þ2 Di Pi P1i 2W i 2 P1i !  2 ! ð1  β i Þ2 h1i θi ðxi Þ2 Bi 2 X i 2xi X i xi xi 2 xi 2 U i 1 X i xi U i hi þ þ þ þ þ þ þ þ P1i T 2W i 2 Pi 2 Pi P1i Pi P1i ðP1i Þ2 Di Pi P1i    2    2 2 Bi h β ð1  β i Þ 1 x i C bi βi Y i hi βi 1 1 X i xi U i Bi þ þ þ þ þ þ þ i i Wi Pi P1i T 2Di X i 2 T Pi P1i X i Di 0  2 ! 1 2 2 Di ð1  βi Þhi X i 2xi X i xi xi x U 1 X i xi U i þ þ þ þ i iþ þ C B W i2 Pi 2 Pi P1i Pi P1i ðP1i Þ2 Di Pi P1i C B CB i B þB C     2 A @ D h θ ð x Þ ð1  β i Þ h β D 1 xi X i xi U i  i 1i i i2  i i i þ þ þ Pi P1i Wi Pi P1i W i P1i       ð1  β i Þ X þ Di þ Pi xi D i þ X i x i ðD i þ U i Þ Bi  þ hi β i t m i C i þ C Ri xi þ C Si xi θi  hi t m þ Xi Pi T Wi P1i  2  2   2 hi Di þ Di X i t m C bi Di ðPi  di Þðt m Þ 1 þ Ki þ þ þb π i ð1  βi ÞDi t m þb π i ð1  β i Þ 2X i 2X i T    D D þ X i x i ðD i þ U i Þ þ þ i C i þ CRi xi þ C Si xi θi  hi t m i Wi Pi P1i ATCðT,BÞ ¼

ð5:473Þ The existence of only one machine results in limited production capacity. Since t 1i þ t 3i þ t 4i , t 5i , and tsi are the production uptimes, rework time, and setup time of the ith product, respectively, and t 2i is the interruption time when the machine is producing ith product, the summation of them should be smaller or equal to the period length (T ). So the capacity constraint of the model is:

5.5 Partial Backordering

489

n  n X  X t 1i þ t 2i þ t 3i þ t 4i þ t 5i þ tsi  T i¼1

ð5:474Þ

i¼1

using: 5 X



t ij

j¼1

1 x ¼ t m þ Qi þ i Pi P1i

 ð5:475Þ

Equation (5.474) can be simplified to (Taleizadeh 2018): n  X

t m þ Qi

i¼1

  1 x þ i þ tsi  T Pi P1i

ð5:476Þ

and finally using Eq. (5.475), one has (Taleizadeh 2018): T 

Pn 1

þ tm Þ i¼1 ðtsi h i Di 1 xi þ i¼1 W i Pi P1i

Pn

¼ T Min

ð5:477Þ

Also for the service level constraint, the shortage quantity of ith product per period is Bi + Ditm, the annual demand of the ith product is Di, the number of periods in each year is N, and the safety factor of allowable shortages is SL. So the service level constraint is:   n X Bi þ Di t m N  1  SL Di i¼1

ð5:478Þ

Using Eq. (5.478), one has: Pn T

i¼1

2Di λ1i t m λ2i 2Di λ1i

ð1  SLÞ 

Pn

λ3i i¼1 2Di λ1i

The final multi-product single-machine model becomes:

ð5:479Þ

490

5

TCðT, bi Þ ¼

n X i¼1

s:t: :

X X 1 X Bi 2 X 2 Bi X 3 λi þ λi T  λ4i Bi þ λ5i þ λ6i þ T T T i¼1 i¼1 i¼1 i¼1 i¼1 n

λ1i

Multi-product Single Machine

n

n

n

n

Pn

i¼1 ðtsi þ t m Þ  ¼ T Capacity T  Min Pn Di 1 xi þ 1  i¼1 W i Pi P1i

Pn 2Di λ1i t m  λ2i i¼1 2Di λ1i Level T ¼ T Service Min Pn λ3i ð1  SLÞ  i¼1 2Di λ1i T, bi  0 i ¼ 1, 2, . . . , n ð5:480Þ where: λ1i

  ð1  βi Þ2 h1i θi ðxi Þ2 h β ð1  β i Þ ¼ hi Oi þ Ai þ i i 2 Wi P 2W i 1i    C bi βi Y i hi βi 2 1 1 þ þ þ 2Di X i 2 X i Di

>0   ð1  β i Þ X i þ Di þ Pi xi 2 λ i ¼ hi β i t m Gi þb π i ð1  β i Þ  Wi Xi h1i Di 2 θi ðxi Þ2 hi D i 2 O þ >0 i 2W i 2 2W i 2 P1i   D ð1  βi Þ h1i θi ðxi Þ2 hβD h O þ λ4i ¼ i þ i i i Ai > 0 i i P1i Wi W i2  2  2 hi D i þ d i X i t m C D ðP  di Þðt m Þ2 þ bi i i þb π i ð1  βi ÞDi t m λ5i ¼ K i þ 2X i 2X i λ3i ¼

>0

ð5:481Þ ð5:482Þ ð5:483Þ ð5:484Þ

ð5:485Þ λ6i ¼

Di G Wi i

ð5:486Þ

It should be noted that (Taleizadeh 2018): X i ¼ Pi  d i  Di ,

U i ¼ P1i  d 1i  Di ,

Y i ¼ Pi  d i  ð1  βi ÞDi ,

5.5 Partial Backordering

491



  di d1i W i ¼ 1  þ xi 1  , Pi P1i    D i þ X i xi ð D i þ U i Þ þ Gi ¼ C i þ CRi xi þ CSi xi θi  hi t m , P1i Pi   1 xi X i xi U i þ þ Ai ¼ þ Pi P1i Pi P1i and: Oi ¼

 2 X i 2xi X i xi xi 2 xi 2 U i X i xi U i þ þ þ þ þ þ Di 1 : Pi P1i Pi 2 Pi P1i Pi P1i ðP1i Þ2

Theorem 5.9 The objective function F ¼ TC(T, Bi) shown in Eq. (5.480) is convex (Taleizadeh 2018). Proof To prove the convexity of F ¼ TC(T, Bi), one can utilize the Hessian matrix. Equation (5.487) shows the objective function is strictly convex (Taleizadeh 2018): Pn ½T, B1 , B2 , . . . , Bn   H  ½T, B1 , B2 , . . . , Bn T ¼

5 i¼1 λi

T

0

ð5:487Þ

So the solutions are (Taleizadeh 2018): ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n   ! n  4 2 ! uX 4λ1 λ5  λ2 2 X λ i i i T ¼t = λ3i  i 1 1 4λ 4λi i i¼1 i¼1 λ4i T  λ2i 2λ1i

ð5:489Þ

Di T  ð1  βi ÞBi   1  Pdii þ xi 1  Pd1i1i

ð5:490Þ

Bi ¼ Qi ¼

ð5:488Þ

In order to obtain feasible T, the necessary condition is having both Pn 4λ1i λ5i ðλ2i Þ2 Pn 3 ðλ4i Þ2 and either positive or negative and having both i¼1 i¼1 λi  4λ1i 4λ1 Pn λ3i Pn 2Di λ1i tmi λ2i and ð1  SLÞ  i¼1 2D λ1 either positive or negative. So in order i¼1 2d λ1 i i

i i

to obtain the feasible and optimal solution, the following solution procedure is developed by Taleizadeh (2018):

492

Step 1. Calculate L1 ¼

5

Pn i¼1

4λ1i λ5i ðλ2i Þ 4λ1i

2

, L2 ¼

Pn

3 i¼1 λi

Multi-product Single Machine



ðλ4i Þ

2

4λ1i

Level , T Service , and Min

T Capacity . Min Step 2. If L1L2 < 0 or L2 ¼ 0, there is no feasible solution, so terminate the procedure. It means there is no feasible solution for the set of parameters. Step 3. If L1L2 > 0, meaning feasible region and optimal solution exist, calculate the period length using Eq. (5.494). n o Level , T Capacity 4. If T is less than Max T Service Min Min n o Capacity  Service Level Max T Min , T Min ; else T ¼ T.

Step

,

then

T ¼

Step 5. Calculate Bi for i ¼ 1, 2, . . ., n using Eq. (5.489) and T obtained from Step 4. Step 6. Calculate Qi for i ¼ 1, 2, . . ., n using Eq. (5.490), T and Bi obtained from Steps 4 and 5 respectively. Step 7. Terminate the procedure. Examples 5.23 and 5.24 Consider a turning manufactory with only one CNC (computer numerical control) machine used to lathe metal plates to different sizes. Its customers are some factories needing metal plates in different sizes. The total demand rate of each size is deterministic, while because of the different quality of each plate, their unit costs are different. Moreover, the production rates, setup time for each product because of programming on machine, and other parameters are different too. According to the history of the factory, the manufacturer has realized that some of the customers wait to receive the product if the manufacturer could not satisfy their demands as when as they want, while some of them go to another factory to satisfy their demands. The aim of the manufacturer is to determine the best time of preventive maintenance, production and shortage quantities, and period length such that its total cost is minimized. In order to provide numerical examples for this factory, consider a production system with five products where the general data for two examples are given in Table 5.62 and the lower limit of service level is 0.9. They considered two numerical examples with uniform distribution for Xi and θi. Tables 5.63 and 5.64 show the specific data of both examples (Taleizadeh 2018). Tables 5.65 and 5.66 show the best results of numerical examples. In Example 5.23, according to the solution procedure, after calculating coefficients, T should be calculated using Eq. (5.496) which is equal to 0.8642. Since T ¼ 0:8642 > n o Capacity Level Max T Service ¼ 0:5965, T ¼ 0:6745 , the optimal value of period length Min Min is 0.8642 (see the flowchart of the proposed solution procedure), and other decision variables should be calculated based on the optimal value of period length as shown in Table 5.65. But in Example 5.24 since T ¼ 0:9124 < n o Level ¼ 0:8954, T Capacity ¼ 1:3945 , the optimal value of period length Max T Service Min Min n o Level ¼ 0:8954, T Capacity ¼ 1:3945 ¼ 1:3954 , and optimal is T  ¼ Max T Service Min Min order and shortage quantities should be calculated based on T ¼ 1.3954 as shown in Table 5.66 (Taleizadeh 2018).

5.5 Partial Backordering

493

Table 5.62 General data for both examples (Taleizadeh 2018) Product 1 2 3 4 5

Pi 10,000 11,000 12,000 13,000 14,000

P1i 5000 6000 7000 8000 9000

tsi 0.005 0.01 0.015 0.02 0.025

Ki 100 200 300 400 500

Ci 10 10 10 10 10

hi 3 3 3 3 3

h1i 1 1 1 1 1

CSi 2 2 2 2 2

Cbi 5 5 5 5 5

b πi 6 6 6 6 6

βi 0.7 0.7 0.7 0.7 0.7

CRi 2 2 2 2 2

Table 5.63 Specific data for Example 5.23 (Taleizadeh 2018)



~ e0i , f 0i θi U

Xi~U [ei, fi] Items 1 2 3 4 5

Di 1600 1700 1800 1900 2000

tm 0.005 0.005 0.005 0.005 0.005

ei 0 0 0 0 0

fi 0.01 0.015 0.02 0.025 0.03

E[Xi] 0.005 0.0075 0.01 0.0125 0.015

di ¼ PiE[Xi] 50 82.5 120 162.5 210

e0i 0 0 0 0 0

Table 5.64 Specific data for Example 5.24 (Taleizadeh 2018) Items 1 2 3 4 5

Di 2000 2500 3000 3500 4000

tm 0.007 0.007 0.007 0.007 0.007

Xi~U [ei, fi] ei fi 0 0.055 0 0.1 0 0.155 0 0.20 0 0.255

E[Xi] 0.025 0.05 0.075 0.1 0.125

di ¼ PiE[Xi] 250 550 900 1300 1750

~ θi U e0i 0 0 0 0 0

f 0i 0.05 0.1 0.2 0.25 0.3



e0i , f 0i f 0i 0.15 0.2 0.25 0.3 0.35

E[θi] 0.0255 0.05 0.15 0.125 0.155

d1i ¼ P1iE [θi] 250 600 1400 2000 2700

E[θi] 0.075 0.1 0.125 0.15 0.175

d1i ¼ P1iE[θi] 375 600 875 1200 1575

Table 5.65 The best results for Example 5.23 (Taleizadeh 2018) T Capacity Min 0.6745

Level T Service Min 0.5965

T 0.8642

T 0.8642

Bi 125.2145 141.9812 189.0294 212.7183 248.0982

Qi 1354.2324 1498.1943 1552.6183 1654.1735 1801.6512

TC 98,458.2

Table 5.66 The best results for Example 5.24 (Taleizadeh 2018) T Capacity Min 1.3945

Level T Service Min 0.8954

T 0.9124

T 1.3945

Bi 271.5423 293.0621 325.5423 360.4520 394.2689

Qi 2452.8542 2842.3956 3412.4125 3965.7163 4698.0254

TC 109,241.1

494

5.6

5

Multi-product Single Machine

Conclusion

To provide a comprehensive introduction about the multi-product single-machine EPQ inventory management research status, this chapter presented the recent studies in relevant fields. The literature review framework in this chapter provided a clear overview of the multi-product single-machine study field, which can be used as a starting point for further study. This chapter presents a framework to classify different types of multi-product single machine in terms of economic production quantity problem and reviews the literature based on the framework. In this chapter, several multi-product single-machine EPQ problems are discussed in details, and the models are classified into three categories in terms of the inclusion or noninclusion of the shortage and its type. The first category includes several models in which shortage is not allowed. In the second category of models, shortage is allowed and is back-ordered. Finally, the multi-product single-machine EPQ models with partial backordering are examined.

References Chiu, S. W. (2010). Robust planning in optimization for production system subject to random machine breakdown and failure in rework. Computers & Operations Research, 37, 899–908. Chiu, Y. S. P., & Chang, H.-H. (2014). Optimal run time for EPQ model with scrap, rework and stochastic breakdowns: a note. Economic Modelling, 37, 143–148. Chiu, S. W., Pai, F. Y., & Wu, W. K. (2013). Alternative approach to determine the common cycle time for a multi-item production system with discontinuous deliveries and failure in rework. Economic Modelling, 35, 593–596. Eilon, S. (1985). Multi-product batch production on a single machine—a problem revisited. Omega, 13, 453–468. Goyal, S. (1984). Determination of economic production quantities for a two-product single machine system. The International Journal of Production Research, 22, 121–126. Hayek, P. A., & Salameh, M. K. (2001). Production lot sizing with the reworking of imperfect quality items produced. Production Planning & Control, 12, 584–590. Haji, B., Haji, R., & Haji, A. (2009). Optimal batch production with rework and non-zero setup cost for rework. In International Conference on Computers & Industrial Engineering, 2009 (CIE 2009) (pp. 857–862). Piscataway, NJ: IEEE. Jamal, A., Sarker, B. R., & Mondal, S. (2004). Optimal manufacturing batch size with rework process at a single-stage production system. Computers & Industrial Engineering, 47, 77–89. Nobil, A. H., & Taleizadeh, A. A. (2016). A single machine EPQ inventory model for a multiproduct imperfect production system with rework process and auction. International Journal of Advanced Logistics, 5, 141–152. Pasandideh, S. H. R., & Niaki, S. T. A. (2008). A genetic algorithm approach to optimize a multiproducts EPQ model with discrete delivery orders and constrained space. Applied Mathematics and Computation, 195, 506–514. Pasandideh, S. H. R., Niaki, S. T. A., Nobil, A. H., & Cárdenas-Barrón, L. E. (2015). A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost. International Journal of Production Economics, 169, 203–214.

References

495

Rogers, J. (1958). A computational approach to the economic lot scheduling problem. Management Science, 4, 264–291. Shafiee-Gol, S., Nasiri, M. M., & Taleizadeh, A. A. (2016). Pricing and production decisions in multi-product single machine manufacturing system with discrete delivery and rework. Opsearch, 53, 873–888. Taleizadeh, A. A. (2018). A constrained integrated imperfect manufacturing-inventory system with preventive maintenance and partial backordering. Annals of Operations Research, 261, 303– 337. Taleizadeh, A. A., Niaki, S. T. A., & Najafi, A. A. (2010a). Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint. Journal of Computational and Applied Mathematics, 233, 1834–1849. Taleizadeh, A. A., Wee, H. M., & Sadjadi, S. J. (2010b). Multi-product production quantity model with repair failure and partial backordering. Computers & Industrial Engineering, 59, 45–54. Taleizadeh, A., Jalali-Naini, S. G., Wee, H. M., & Kuo, T. C. (2013a). An imperfect multi-product production system with rework. Scientia Iranica, 20, 811–823. Taleizadeh, A. A., Wee, H. M., & Jalali-Naini, S. G. (2013b). Economic production quantity model with repair failure and limited capacity. Applied Mathematical Modelling, 37, 2765.2774. Taleizadeh, A. A., Cárdenas-Barrón, L. E., & Mohammadi, B. (2014). A deterministic multi product single machine EPQ model with backordering, scraped products, rework and interruption in manufacturing process. International Journal of Production Economics, 150, 9–27. Taleizadeh, A. A., & Wee, H. M. (2015). Manufacturing system with immediate rework and partial backordering. International Journal of Advanced Operations Management, 7, 41–62. Taleizadeh, A. A., Sadjadi, S. J., & Niaki, S. T. A. (2011). Multiproduct EPQ model with single machine, backordering and immediate rework process. European Journal of Industrial Engineering, 5, 388–411. Taft, E. W. (1918). The most economical production lot. Iron Age, 101(18), 1410–1412. Taleizadeh, A., Najafi, A., & Niaki, S. A. (2010c). Economic production quantity model with scrapped items and limited production capacity. Scientia Iranica, Transaction E: Industrial Engineering, 17, 58. Taleizadeh, A., Cárdenas-Barrón, L., Biabani, J., & Nikousokhan, R. (2012). Multi products single machine EPQ model with immediate rework process. International Journal of Industrial Engineering Computations, 3, 93–102. Taleizadeh, A. A., Sarkar, B., & Hasani, M. (2017). Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering. Journal of Industrial and Management Optimization, 13(5), 1–24. Wee, H., Widyadanab, G., Taleizadeh, A., & Biabanid, J. (2011). Multi products single machine economic production quantity model with multiple batch size. International Journal of Industrial Engineering Computations, 2, 213–224.

Chapter 6

Quality Considerations

6.1

Introduction

Traditional economic order quantity (EOQ) models offer a mathematical approach to determine the optimal number of items a buyer should order to a supplier each time. One major implicit assumption of these models is that all the items are of perfect quality (Rezaei and Salimi 2012). However, presence of defective products in manufacturing processes is inevitable. There is no production process which can guarantee that all its products would be perfect and free from defect. Hence, there is a yield for any production process. Basic and classical inventory control models usually ignore this fact. They assume all output products are perfect and with equal quality; however, due to the limitation of quality control procedures, among other factors, items of imperfect quality are often present. So it has given researchers the opportunity to relax this assumption and apply a yield to investigate and study its impact on several variables of inventory models such as order quantity and cycle time. As aforementioned, defective products in manufacturing processes are inevitable. A common assumption in the EPQ inventory literature with defectives is that the rework of a defective item is followed immediately after it is identified (MoussawiHaidar et al. 2016). This assumption of continuous screening during production complicates the analysis and is not practical for most production systems, especially when the fraction of defective items is low and the production rate is high, which makes continuous screening during production very expensive. To simplify the analysis and the computation of the average inventory, it was assumed in the literature that the products are lumped into two groups, good products and to be reworked products. Some other researchers assumed that some of these defective items are sold to the secondary market with lower price.

© Springer Nature Switzerland AG 2021 A. A. Taleizadeh, Imperfect Inventory Systems, https://doi.org/10.1007/978-3-030-56974-7_6

497

6 Quality Considerations

Partial backordering

Withreturn

Backordering

Noshortage

Backordering

Partial backordering

Withoutreturn

Noshortage

Fig. 6.1 Categories EOQ and EPQ models

EOQandEPQmodelswithquality consideration

498

One of the assumptions in the EOQ inventory literature stated that the received items are of perfect quality. This problem has received considerable attention for the last 30 years. Porteus (1986) first considered investments for the quality improvement problem and setup cost reduction. He believed that the number of defective items in a lot depends on the probability of the production process (machine) becoming out of control, and he concluded that the quality of items can be improved by reducing lot sizes. Some researches can be found in Taleizadeh et al. (2015, 2019a, b), Mohammadi et al. (2015) Salameh & Jaber (2000), and Taleizadeh & Moshtagh (2019). In this chapter, the EOQ and EPQ problems with quality considerations are investigated. The reviewed models are classified into two main categories including “with return” and “without return” and three subcategories related to shortage consideration. The first category includes several models which in some of them shortage is not allowed and others shortage was considered as backordering and partial backordering. Likewise, the second category consists of the same subcategories for the studies which were not considered return policy. This grouping is provided for EOQ and EPQ models separately. To provide a comprehensive introduction about the mentioned research status, in this chapter, the recent studies in relevant fields are reviewed. The literature review framework in this chapter provides a clear overview of the EOQ and EPQ models with quality considerations, which can be used as a starting point for further study. The classification is shown in Fig. 6.1. The common notations of investigated studies are shown in Table 6.1. To integrate the presented models in this chapter similar to the previous chapters, these notations are used. The main decision variables of this field on inventory are Q and T, but in some studies other decision variables are considered too.

6.1 Introduction

499

Table 6.1 Notations D h h1 K KS Kd P P1 Rs P1 C CR CI Cr Cfa Cfr R CR s v Cb g b π β Ti T1 T2 x n m1 m2 Q B y θ F TP ETP ATP f(x) f(m1) f(m2) E[.]

Annual demand (item/year) Holding cost per unit and per unit of time ($/item/unit of time) Holding cost of a defective item kept in inventory ($/item/unit of time) Fixed order cost ($/lot) Setup cost ($/setup) Transport cost of defective lots back to supplier ($/lot) Production rate per year (item/year) Rework rate (item/year) Rate of screening or inspection (item/year) Rework rate (item/year) Purchasing cost per unit of product ($/item) Unit rework cost per item ($/item) Inspection cost per unit of product ($/item) Cost for returning a defective item ($/item) Cost of accepting a defective item ($/lot) Cost of rejecting a non-defective item ($/lot) The cost that is paid because of the wrong rejection by the vendor ($/item) Unit rework cost ($/item) Selling price per unit of product ($/item) Selling price per imperfect unit ($/item) Backordering cost per unit of demand and per unit of time ($/item/unit of time) The goodwill loss on a unit of unfilled demand ($/item) Lost sale cost per unit of unfilled demand b π ¼ s  C þ g ($/item) Rate of backordering (%) Length of cycle (i) (year) Length of cycle in which the inventory level is more than or equal to zero (year) Length of cycle in which the inventory level is less than zero (year) Proportion of defective items (%) Size of the sample (integer) Probability of type I error (classifying a non-defective item as defective) Probability of type II error (classifying a defective item as non-defective) Order quantity (item) The maximum backordering quantity (item) Number of perfect items in a lot (item) Number of defective products in a sample of n items (item) Fill rate or the percentage of cycle time that inventory level is positive (%) Total profit per cycle ($) Expectation of total profit per cycle ($) Annual total profit ($) Function of probability density of defective rate Probability density function of m1 Probability density function of m2 Expectation operator

500

6.2

6 Quality Considerations

Literature Review

Inventory models with imperfect-quality items have received significant attention in the literature. Muhammad and Alsawafy (2011) developed the economic order quantity model with imperfect-quality items. They considered that the incoming lot has a fraction of scrap and reworkable items, and the lot will go through a 100% inspection. Rezaei and Salimi (2012) formulated and solved a problem to determine the maximum purchasing price a buyer is willing to pay to a supplier to avoid receiving imperfect items under two conditions. First, they assumed that the buyer’s selling price is independent of the buyer’s purchasing price, while under the second condition, they assumed that changing the buyer’s purchasing price influences the buyer’s selling price and customer demand. Cheikhrouhou et al. (2018) developed an economic order quantity model for a sampling, sample quality inspection, and a returned policy of defective items. Their research filled the research gap in the literature in providing a model, which took into account the impairment loss resulting from the sample inspection. It also highlighted the strong link between order sizes, sample sizes, and lot sizes. Khan et al. (2011) determine an inventory policy for imperfect items received by a buyer. They adopted a realistic approach of screening. That is, an inspector may classify a non-defective item to be defective (type I error), and he may also classify a defective item to be non-defective (type II error). The defective items classified by the inspector and those returned from the market are accumulated and sold at a discounted price at the end of each procurement cycle. Konstantaras et al. (2012) assume planned shortages to occur in each cycle. Two models were developed. The first model assumes an infinite planning horizon for which the optimal replenishment policy was determined. The second model assumes that the planning horizon consists of unequal cycles in each of which the percentage of imperfect-quality items reduces with every shipment following a learning curve. For this model, a closed-form solution of the total profit was derived in terms of the cost parameters and the relevant decision variables which are the replenishment points, the points where the inventory level becomes zero and the number of the cycles. Hauck and Vörös (2015) considered the traditional lot-sizing problem when after arrival, the quality of each item in a lot is checked. The percentage of defective items is probability variable, and investments could be made to increase the screening rate. The variable screening rate especially implicates cases where unplanned backlogs may develop. Aslani et al. (2017) extend an EOQ model with partial backordering when the supplier’s production process has a random yield. They introduce order quantity and fill rate as decision variables. Moreover, they proposed a solution algorithm based on recursive method to solve the model and find optimal values of decision variables. In order to make improvement in yield, they investigate the effect of investing money to improve the mean and variability of yield by logarithmic functions. The results show that sometimes investment to improve the yield rate causes a reduction of costs. The word “sometimes” refers to the cost required to be invested for

6.3 EOQ Model with No Return

501

improvement plans. Taleizadeh and Zamani-Dehkordi (2017) define three levels of defective items in each sample according to their numbers. If this number is less than α1, it is not necessary to inspect all the items; else if this number is between α1 and α2, all the items should be inspected, and if this number is more than α2, the order is rejected and another order without any defective items is received. The rate of imperfect items in each order is p, and regarding the number of defective items, it has three levels, lower than p1, between p1 and p2, and higher than p2. Cheng (1991) proposed a simple equation to model the relationship between unit production cost and process capability and quality assurance expenses for the EPQ problem. The optimal solution is then derived using differential calculus, which yields a simple closed-form expression for the optimal value of both production quantity and expected fraction acceptable. He also presented a sensitivity analysis of the impacts of the cost parameters on the optimal solution, followed by a discussion of the problems associated with cost estimation. Tsou et al. (2012) developed an EPQ model with continuous quality characteristic, rework and reject. The findings revealed that there is an optimal lot size, which generates minimum total cost in their model. It was also found that if the percentage of imperfect-quality items and rejected items is zero, or approaches zero, the optimal lot size of our model is equal to the classical EPQ model. In addition, it was shown that Taguchi’s cost does not affect the model. Moussawi-Haidar et al. (2016) considered the realistic case where defective items undergo quality control by the consumer or seller during the purchasing process. As soon as production is completed, a cheaper and faster screening process identifies all the defective items. They investigated two realistic cases where defective items are scrapped and when defective items are reworked. The equations to calculate the optimal total profit per unit time and order quantities were presented. Haji et al. (2009) considered an imperfect production system in which a constant percentage of defective items are produced. In their model, all the defective items produced in each cycle are reworked in the same cycle immediately after normal production ends. They assumed that a 100% inspection takes place in both the normal production and rework processes. They also assumed that type 1 errors (i.e., perfect items incorrectly rejected) and type 2 errors (i.e., imperfect items incorrectly accepted) will be committed. For this system, they obtained the optimal production quantity that minimizes the total cost of the system, which is the sum of the normal and rework processing costs, inspection costs, inventory holding costs, and inspection error costs due to inspection errors. Features of reviewed studies are given in Table 6.2.

6.3

EOQ Model with No Return

In this subsection, three models in which return policy is not considered are presented. The model development is investigated. Then, the solution procedure to solve the optimization problem is presented. Also, numerical examples have been reviewed to illustrate the implementation of the proposed method, if there is any.

502

6 Quality Considerations

Table 6.2 Features of reviewed studies Reference Muhammad and Alsawafy (2011) Cheikhrouhou et al. (2018) Khan et al. (2011) Konstantaras et al. (2012) Hauck and Vörös (2015) Aslani et al. (2017) Taleizadeh and Zamani-Dehkordi (2017) Cheng (1991) Tsou et al. (2012) Moussawi-Haidar et al. (2016) Haji et al. (2009) Al-Salamah (2016)

6.3.1

EOQ ✓ ✓ ✓ ✓ ✓ ✓ ✓

EPQ

Shortage

Return

Partial backordering

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓

✓

No Return Without Shortage

Rezaei and Salimi (2012) mathematically modeled the relationship between the buyer and supplier with regard to conducting the inspection, resulting in a change of the buyer’s economic order quantity and purchasing price. They formulated and analyze the problem under two conditions: (1) assuming there is no relationship between the buyers’ selling price, buyer’s purchasing price, and customer demand and (2) assuming there is relationship between the buyers’ selling price, buyer’s purchasing price, and customer demand. The mathematical model of on-hand problem is presented in Sect. 2.3.5.1.

6.3.2

Two Quality Levels with Backordering

Muhammad and Alsawafy (2011) developed an economic order quantity of imperfect-quality items where the incoming lot has fractions of scrap and reworkable items. These fractions are considered to be random variables with known probability density functions. The demand is satisfied from perfect items and reworked items, whereas the scrap items are sold in a single batch at the end of the cycle with a salvage cost. The notations which are specially used in this problem are presented in Table 6.3. Figure 6.2 represents the model where the lot of size Q is received with purchasing price of C per unit and the fixed ordering cost K. It is assumed that each order contains a probabilistic fraction of scrap and reworkable items ps and pr with known probability density functions f( ps) and f( pr), respectively. Good and reworked items

6.3 EOQ Model with No Return

503

Table 6.3 New notations for given problem (Muhammad and Alsawafy 2011) ps pr p Z1 Z2 Z3 Z4

Percentage of scrap items (%) Percentage of reworkable items (%) Percentage of scrap and reworkable items (%) Inventory level after the inspection period (item) Inventory level after the selling of the scrap items and return reworked items (item) Inventory level just before receiving the reworked items (item) Inventory level just after receiving the reworked items (item)

Fig. 6.2 Behavior of inventory level over time (Muhammad and Alsawafy 2011)

Inventory level y Z4 Z1 Py Z2

Z3

PRY

t1

t2

t3

Time

T

can be sold. On the other hand, scrap items will be sold in a batch at the end of the cycle with salvage (discount) per unit. The optimal order quantity is found by taking the difference between the total revenue and total cost, the latter of which consists of four types: procurement cost, inspection cost, rework cost, and inventory carrying cost. Revenues come from selling of good items and scrap items. Shortage is not allowed and inspection and rework processes are error-free (Muhammad and Alsawafy 2011). Since shortage is not allowed, to avoid shortage, the number of good items is at least equal to the demand during inspection time: ð1  ps  pr ÞQ  Dt 1

ð6:1Þ

Since ps and pr are coming from a probability density functions, they will be limited as below:

504

6 Quality Considerations

E ð ps Þ þ E ð pr Þ  1 

D , Rs

for

Rs  D

ð6:2Þ

The time t1 needed to inspect the lot is: t1 ¼

Q Rs

ð6:3Þ

The expected total revenue is the summation of sales of the good items and scrap items and it is given as: E ½TRðQÞ ¼ pð1  E ðps ÞÞQ þ vQE ðps Þ

ð6:4Þ

The expected total comprises four different costs. The first cost is the procurement cost: PCðQÞ ¼ K þ CQ

ð6:5Þ

The expression for the expected total cost is (Muhammad and Alsawafy 2011): E½TCðQÞ ¼ K þ CQ þ CR E ðPr ÞQ þ C I Q     Eð1  ps ÞQE ðT Þ E ðps ÞQ2 E p2r Q2 þ þh  2 Rs P1

ð6:6Þ

The expected total profit equals the expected total revenues minus the expected total cost (Muhammad and Alsawafy 2011): E ½TPðQÞ ¼ E½TRðQÞ  E ½TCðQÞ ¼ sð1  Eðps ÞÞQ þ vQE ðps Þ      Eð1  ps ÞQE ðT Þ E ðps ÞQ2 E p2r Q2 þ   K þ CQ þ C R E ðpr ÞQ þ C I Q þ h 2 Rs P1 ð6:7Þ The expected cycle period is given by: E ðT Þ ¼

E ð1  ps ÞQ D

ð6:8Þ

The expected total profit per unit time is: E ½TPUðQÞ ¼

E ½TPðQÞ E ðT Þ

ð6:9Þ

6.3 EOQ Model with No Return

505

To find the optimal order quantity, the first derivative of E[TPU(Q)] is taken, set to zero, and solved for Q:   ∂E½TPUðQÞ 1 ¼ 1  E ðps Þ ∂Q  2 2   2  33 2 E ð 1  p Þ s E pr D55 E ð p ÞD KD s þ  4 2  h4  2 R P Q s 1

ð6:10Þ

From Eq. (6.10), they found the expression of the economic order quantity:

    2DE ðp Þ 2DE p2  12 2 r s EOQ2 ¼ 2KD= h E ð1  ps Þ þ  P1 Rs

ð6:11Þ

The second derivative is equal to: 2

∂ E ½TPUðQÞ 2KD ¼ 2 0 Q Q ð1  Eðps ÞÞ

ð6:12Þ

Since the second derivative is always negative, this means that there exists a unique value of Q* that maximizes objective function. Maximizing Eq. (6.7) is equal to minimizing Eq. (6.13) (Muhammad and Alsawafy 2011):  2   2  39 2 E ð 1  p Þ s E pr D5= E ð p ÞD 1 KD s 4 ECðQÞ ¼ ð6:13Þ  hQ  þ Rs P1 2 1  E ð ps Þ : Q 2 ; 8
2KDðF þ β  βF Þ2 > u >  > > i > th a : Q ð F Þ ¼ > s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > 2 > > 2 2 < hF þ Cb βð1  F Þ2 ðσ 2 þ μ2 Þ > 2Kh ð σ þ μ Þ > < if β > 1  : > Dμ2 π 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > βðβhC b ÞQðσ 2 þ μ2 Þ½ψ 3 Q  ψ 2 þ ψ 1 Q  β½ψ 3 Q  ψ 2 þ ψ 1 Q > > > > : b : F  ðQ Þ ¼ > > ð 1  β Þ ½ψ 3 Q  ψ 2 þ ψ 1 Q  > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :   2 2 Otherwise : c : F ¼ 1, and Q ¼ 2KD=hðσ þ μ Þ

ð6:31Þ where:   ψ 1 ¼ ðh þ βCb Þ μ2 þ σ 2

ð6:32Þ

ψ 2 ¼ 2Dgð1 þ βÞ2 μ

ð6:33Þ

ψ 3 ¼ 2KDð1 þ βÞ2

ð6:34Þ

512

6.3.4.1

6 Quality Considerations

Yield Improvement Models

In this section, making improvement in the yield rate is investigated. In the long term, one may consider the mean and the variance of the yield to be functions of capital expenditures. In this section, we seek to improve yield by investing to increase the mean of yield or to decrease its variance. Generally, doing investments to improve the yield needs cost analysis. If the investment is found affordable, then it can lead to a variety of advantages such as cost reduction and establishing a longterm relationship. When the yield improves, the total quantity of products for satisfying a constant demand decreases. Another advantage is that because of the decline in the size of each order, the inspection also decreases. Please note that we do not mean the inspection will be skipped; by “decrease in inspection,” we mean because the number of products decreases, the number of inspections will decrease too; meanwhile the strategy of 100% inspection is still carried out. Elimination of the inspection is the ultimate aim, if after implementation of several improvements, the production process becomes a process without producing any imperfect item. In addition to the above points, reaching a desired situation and improving the yield to a proper state will turn the buyer into a long-term customer for the supplier. In this section, they sought to improve yield by investing to increase the mean of yield (μ). Aslani et al. (2017) used the companion yield parameter, γ ¼ 1 for 1  γ  1/Lx. Please note that as μ increases from Lx to 1, γ approaches 1 from 1/ Lx. Therefore, they assumed that γ follows a logarithmic investment function as follows: αðγ Þ ¼ a  bLnγ

ð6:35Þ

where a and b are positive parameters given by (Aslani et al. 2017): a¼

Lnγ 0 φ

and

b¼

1 φ

ð6:36Þ

α(γ) represents the investment cost required to change the companion yield parameter from γ 0 to γ, as suggested by Porteus (1986), and is a strictly decreasing and convex function of γ. The target is to improve the companion yield to a desired value in order to improve the mean of yield. Corollary 6.1 Since the yield would improve, hence the optimal order quantity and fill rate, (Q and F), would change. Substituting Qimp and Fimp into Eq. (6.25), and considering the investment cost required to create improvements, the total cost after the improvement plan is presented in Eq. (6.37) (Aslani et al. 2017):     ATCimp Qimp , F imp ¼ iαðγ Þ þ ATC Qimp , F imp

ð6:37Þ

Also, Aslani et al. (2017) sought to improve yield by investing to decrease the variance of yield (σ 2). In other words, they examined reducing σ 2 as variability of the

6.3 EOQ Model with No Return

513

yield to an acceptable value. There is a cost per year ρ(σ 2) of reducing yield variance to an acceptable value:     ρ σ 2 ¼ m  nLn σ 2

ð6:38Þ

where m and n are positive constants and ρ(σ 2) is a convex and strictly decreasing function of σ 2 given by (Aslani et al. 2017):   Ln σ 20 1 m¼ and n ¼ τ τ

ð6:39Þ

Corollary 6.2 Due to the change in variance of yield rate, it is obvious that the optimal order quantity and fill rate, (Q and F), will change. Substituting Q0imp and F 0imp into Eq. (6.25), a new annual total cost (ATC(Q0 imp  F0 imp)) will be calculated. Finally, by considering the investment cost mandatory to create improvements, the total cost after the improvement plan will be:     ATC0imp Q0imp , F 0imp ¼ iαðγ Þ þ ATC Q0imp , F 0imp If the condition β > 1 

ð6:40Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Khðσ 2 þ μ2 Þ=Dμ2 C 2b is established in Proposition 6.2,

the optimal functions are not closed-form; therefore, a specific procedure should be used to solve the problem and find the optimal values. To solve the problem, they proposed a solution method as follows, in Discussion 6.1: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Discussion 6.1 When β > 1  2Khðσ 2 þ μ2 Þ=Dμ2 C 2b, then a recursive algorithm is implemented to find the optimal solutions. At the first step, insert F1 ¼ 1 (or any acceptable value) into the function Q*(F) (Eq. 6.31a), calculate the result, and call it Q1. Then, at the second step, substitute the value of Q1 into the function F*(Q) (Eq. 6.31b), and calculate F2. At the third step, insert the value of F2 into the function Q*(F) and calculate Q2. Following this procedure to the next steps, they progressively approached the optimal values for Q and F. They defined two parameters which represent the appropriate time to stop the algorithm. Assume that εF and εQ are the indexes for ending the algorithm. Whenever the result of subtraction between two consecutive values of a decision variable becomes less than its index, the recursive algorithm ends. Thus, whenever ΔF ¼ |Fi  Fi  1| < εF and ΔQ ¼ | Qi  Qi  1| < εQ both take place, the algorithm will stop proceeding, and Fi and Qi are the optimal values of decision variables. The results indicate that this algorithm is quite efficient. Discussion 6.2 Since shortage can be written as B ¼ D(1  F)T, and also cycle time is T ¼ xQ/D(F + β  βF), it can be easily observed that the service constraint, βB/ Q  Lx, will be satisfied if and only if the following condition is held:

514

6 Quality Considerations

β  Lx F=ð1  F Þðu  Lx Þ,

ðF 6¼ 1Þ

ð6:41Þ

As can be seen, it must be F 6¼ 1. Therefore, if the optimal value of fill rate is calculated to be F ¼ 1, in order to remove this error, F ¼ 1  ε is used where ε is an infinitesimal amount. Note that this issue does not reduce the generality of the solution algorithm. If the value of decision variables does not meet the service constraint, it means the supplier is unable to answer all backordered shortages. In these circumstances, the buyer can decide whether to continue the cooperation with this supplier or not. In this problem, the buyer is very sensitive to the service constraint, and if the supplier cannot meet the service constraint, then the buyer will switch to other suppliers (Aslani et al. 2017). Example 6.2 Consider a company that purchases its required product from a supplier. Each received batch from this supplier includes a random proportion of defective items. This means supplier’s production process works to a random yield. Assume that the yield is between 0.4 (Lx ¼ 0:4) and 1 and is a continuous random variable. Also the mean and variance of yield are known. The mean of the yield is μ ¼ 0.7 and its variance is σ 2 ¼ 0.01. The buyer needs 500 units of the product each year. Other data used is sequenced as β ¼ 0.5, g ¼ 2 ($/item), h ¼ 4 ($/item/ year), K ¼ 120 ($/lot), εF ¼ 0.001 and εQ ¼ 0.1, Cb ¼ 2 ($/unit/year) (Aslani et al. 2017). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Step 1. Is β > 1  2Khðσ 2 þ μ2 Þ=Dμ2 C 2b met? ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðσ 2 þμ2 Þ ðσ 2 þμ2 Þ 1  2KhDμ ¼ 0:300146 ) βð¼ 0:5Þ > 1  2KhDμ 2 C2 2 C2 b

b

Yes. Step 2. Calculating optimal values. Since condition β >1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Khðσ 2 þ μ2 Þ=Dμ2 C 2b is met, the recursive algorithm must be employed. As it can be seen in Table 6.5, at the seventh iteration, △F  εF(¼0.001) and △Q  εQ(¼0.1). Therefore, F7 ¼ F* and Q7 ¼ Q*.  Step 3. Is β  ð1FLÞxðFμLx Þ established?

Lx F  ð1F  ÞðμLx Þ



¼ 0:879859 ) βð¼ 0:5Þ  ð1FLxÞFðμLx Þ

Yes. Step 4. F* and Q* are optimal. According to the solution, the optimal values are Q* ¼ 379.509, T* ¼ 0.719179, B* ¼ 216.633, and ATC* ¼ 616.870.

6.4 EOQ Model with Return

515

Table 6.5 The process of achieving optimal values through the recursive method (Aslani et al. 2017) Iteration F Q ΔF ΔQ ATC

6.4

1 1 244.949 – – 699.854

2 0.5069 344.292 0.4931 99.343 621.702

3 0.4206 371.895 0.0863 27.603 617.115

4 0.4022 377.971 0.0184 6.076 616.880

5 0.3984 379.211 0.0038 1.240 616.870

6 0.3977 379.459 0.0008 0.248 616.870

7 0.3975 379.509 0.0001 0.05 616.870

EOQ Model with Return

In this subsection, three problems that take into account the return policy are presented. The model development is investigated. Then, the solution procedure to solve the optimization problem is presented. Also, numerical examples have been reviewed to illustrate the implementation of the proposed method, if there is any.

6.4.1

Inspection and Sampling

Cheikhrouhou et al. (2018) developed an inventory model with lot inspection policy. With the help of lot inspection, even though all products need not to be verified, still the retailer can decide the quality of products during inspection. If the retailer found that the products have imperfect quality, the products are sent back to the supplier. As it is lot inspection, misclassification errors (type I error and type II error) are introduced to model the problem. This model emphasizes a sample inspection to avoid maximum possibility of type I and type II errors of defective items. The mathematical model analyzes two ways how to reduce defective lots. The retailer chooses whether, after inspection of defective items, they are immediately sent back to the supplier or wait for the next shipment from the supplier to send back the defective items. The model mainly stands on two major factors: order size (Q*) and sample size (n*), which are related to each other (Cheikhrouhou et al. 2018). The notations which are specially used in this problem are presented in Table 6.6. In this section, the mathematical model is introduced. Let Q represents the order quantity size of lots, and a new constant L is introduced, which represents the number of items per lots. In order to satisfy the demand D of items per year, a shipment containing several lots is received every T unit of time. In a perfect model and without defective items, the relation D ¼ QL/T is established in the stationary state, where T represents the cycle time. To calculate the total cost of the model, the following costs had to be calculated (Cheikhrouhou et al. 2018):

516

6 Quality Considerations

Table 6.6 New notations for given problem (Cheikhrouhou et al. 2018) n Ks TI L xe Clot ATP(Q, n)

Number of items inspected per lot (integer) Transport cost of defective lots back to the supplier (for the first subcase model) ($) Inspection time for a shipment (time) Number of items per lot (integer) Percentage of defective items perceived by the retailer (%) Lot purchase cost ($/lot) Annual total profit ($/year)

TC ¼ Inspection cost þ Inspection error cost þ Holding cost þ Transportation cost þ Purchasing cost þ Fixed order cost

ð6:42Þ

Upon receiving an order, an inspection based on a sample is applied. Let n represents a sample size, satisfying constraint 1  n  L and n 2 N. It is also considered that the inspection time for a shipment TI depends on the order size and on the number of samples used for inspection. Thus, one can obtain relation T I ¼ nQ Rs and the inspection cost is (Cheikhrouhou et al. 2018): Inspection cost ¼ CI nQ

ð6:43Þ

As described in the model by Khan et al. (2011), their screening process is assumed to be error-free. But it is quite realistic to account for type I and type II errors committed by inspectors as it is offline inspection by the inspectors; thus, there is a chance of accepting imperfect products as perfect and rejecting perfect product as imperfect. For this reason, the increased number of samples permits to reduce the risk of falsely qualifying a lot. Depending on the sample size n, inspectors classify some non-defective lots as defectives, i.e., (1  x)mn1, while some defective units as non-defectives, i.e., xmn2. For this reason, the percentage of defective items perceived by the retailer xe is different from the actual one x. Thus, the fraction of defective units perceived can be obtained as (Cheikhrouhou et al. 2018): E ½xe  ¼ ð1  E ½xÞE½m1 n þ E ½xð1  E½m2 n Þ

ð6:44Þ

It is considered a cost of misclassification due to an insufficient number of samples to qualify the quality of the lot. One can assume that the number of items, which goes into type I error (false rejection), is dependent on n. The inspectors falsely reject a lot if the inspectors have a type I error, which happens on each non-defective item from the sample (Cheikhrouhou et al. 2018): Inspection falsely rejecting cost ¼ Qð1  E½xÞE½m1 n

ð6:45Þ

6.4 EOQ Model with Return

517

The number of items, which goes into type II error (false acceptation), is dependent on n. The inspectors falsely accept a lot if they have a type II error, which happens on each defective item from the sample: Inspection falsely accepting cost ¼ QE ½xE ½m2 n

ð6:46Þ

Let Cfr and Cfa, respectively, be the cost of rejecting a non-defective lot (type I error) and the cost of accepting a defective lot (type II error). In case of critical products, such as food, medical, or parts of an aircraft, the cost of acceptance ca is much more than that of a false rejection (see for reference Raouf et al. 1983). Costs of inspection error per cycle can be expressed as (Cheikhrouhou et al. 2018): Costs of inspection error ¼ C fa QE ½xE ½m2 n þ C fr Qð1  E ½xÞE½m1 n

ð6:47Þ

Figure 6.8 represents the behavior of the inventory level. It can be noticed that the number of items that are withdrawn from the inventory is: xe QL þ nQð1  xe Þ

ð6:48Þ

which represents the number of defective lots plus the number of items used for the inspection in the accepted lots. Under this condition, the inventory cycle T is determined as (Cheikhrouhou et al. 2018): T¼

QL  ðxQL þ nQð1  xe ÞÞ Qð1  xe ÞðL  nÞ ¼ D D

ð6:49Þ

The remainder of the model is subdivided into two cases. The first case (Case I) considers a special transport is organized to send the defective lots back to the supplier. The model has an additional transport costs Ks, but the defective products are no longer stored in the warehouse to avoid additional holding costs. The second case (Case II) considers that for the sake of convenience, the supplier can take back the defective items in the next shipment. These two cases are the more efficient possibilities as it is logically more expensive to send a transportation cost not immediately after the screening process (because of the holding costs of defective items).

6.4.1.1

Case I: Defective Items Are Immediately Sent Back to the Supplier with an Additional Transportation Cost

As h is the holding cost per unit per item, the holding costs per cycle for the first case can be determined from Fig. 6.5 as (Cheikhrouhou et al. 2018):

518

6 Quality Considerations

Fig. 6.5 Behavior of the inventory model subcase (1) (Cheikhrouhou et al. 2018)

I QL

QL – DTI

xeQL – nQ(1–xe) T1 t

T

2  Q ð1  xe Þ2 ðL  nÞ2 Q2 nLðxe L þ nð1  xÞÞ Holding cost ¼ h þ Rs 2D

ð6:50Þ

In this case, the defective lots are sent back to the supplier with an additional cost for transportation cost Ks. The purchasing cost per cycle is determined as (Cheikhrouhou et al. 2018): Purchasing cost ¼ K s þ Qð1  xe ÞC lot

ð6:51Þ

The total revenue in a cycle is: Total revenue ¼ sðQL  xe QL  ð1  xe ÞnQÞ ¼ sQððL  nÞð1  xe ÞÞ

ð6:52Þ

The total profit is the total revenue per cycle minus the total cost per cycle divided by the cycle time and is given as follows (Cheikhrouhou et al. 2018): TP1 ðQ, nÞ ¼

1 T

2

3 sQððL  nÞð1  xe ÞÞ  ðK s þ Qð1  xe ÞC lot Þ  K  CI nQ  6 2 7 Q ð1  xe Þ2 ðL  nÞ2 Q2 nLðxe L þ nð1  xe ÞÞ 6 7  6 h þ 7 Rs 2D 4 5 ðC fa QE ½xe E ½m2 n þ Cfr Qð1  E ½xe ÞE ½m1 n Þ ð6:53Þ

And the annual total profit is:

6.4 EOQ Model with Return

ATP1 ðQ, nÞ ¼

519

D Qð1  xe ÞðL  nÞ 2 3 sQððL  nÞð1  xe ÞÞ  ðK s þ Qð1  xe ÞClot Þ  K  C I nQ  6 2 7 Q ð1  xe Þ2 ðL  nÞ2 Q2 nLðxe L þ nð1  xe ÞÞ 6 7  6 h þ 7 Rs 4 5 2D ðC fa QE ½xe E½m2 n þ Cfr Qð1  E½xe ÞE ½m1 n Þ ð6:54Þ

For maximization of the profit, by taking partial derivatives with respect to Q and n, one can obtain (Cheikhrouhou et al. 2018): ∂ATP1 ðQ, nÞ ðK þ K s ÞD ð1  E½xe ÞðL  nÞ ¼ 2 h 2 ∂Q Q ð 1  E ½ x e  Þ ð L  nÞ h

DLnðE ½xe L þ nð1  E ½xe ÞÞ Rs ð1  E ½xe ÞðL  nÞ

ð6:55Þ

And: ðK þ K s ÞD ∂ATP1 ðQ, nÞ DC lot C I DL ¼   2 2 ∂n ð L  nÞ ð1  E½xe ÞðL  nÞ Qð1  E½xe ÞðL  nÞ2   DC fa E ½xE ½m2 n 1 þ ln ðE½m2 Þ  ð1  E½xe ÞðL  nÞ ðL  nÞ   DC fr ð1  E½xÞE½m1 n Qð1  E½xe Þ 1  þ ln ðE½m1 Þ þ h 2 ð 1  E ½ x e  Þ ð L  nÞ ð L  nÞ   DLQ E ½xe L2 þ 2nLð1  E ½xe Þ  n2 ð1  E½xe Þ h Rs ð1  E ½xe ÞðL  nÞ2 ð6:56Þ For sufficient condition, the second-order derivative can be obtained as follows: All terms of Hessian matrix are obtained as follows: 2

∂ ATP1 ðQ, nÞ 2ðK þ K s ÞD α2

The first case is the one that the number of imperfect products in a chosen sample is more than the upper limit that is α2 and determined by the buyer and the supplier. In this situation, all the items are rejected, and a new order is received that has not any imperfect items. The revenue of this case in a cycle is computed as the following: sD [F + β(1  F)], the amount of money that is received by selling products per unit of K I time; nC T , the cost of inspection of a sample per unit of time; T , the ordering cost per 2 unit of time; CD[F + β(1  F)], cost of buying products per unit of time; hDTF 2 , holding cost per unit of time; gD(1  β)(1  F), shortage cost that is related to lost 2 sales per unit of time; and βCb DT2ð1FÞ , shortage cost that is related to backorders per unit of time. Also there is a situation that buyers reject the lot wrongly; in other words, the buyer rejects the lot while the number of defective items is less than α2, so the particular amount of money is considered for this situation that the buyer should pay to the supplier. The probability of this situation is computed as ψ ¼ P (x  x2|θ  α2), and the penalty that the buyer should pay per cycle is computed as ψCT fr . So the total revenue per unit of time is obtained from the formulations above (Taleizadeh and Zamani-Dehkordi 2017): TP1 ¼ sD½F þ βð1  F Þ 2 3 K hDTF 2 nC I 6 T þ CD½F þ βð1  F Þ þ 2 þ T 7 7 6 4 5 2 βCb DT ð1  F Þ ψC fr þgDð1  βÞð1  F Þ þ þ 2 T

ð6:72Þ

528

6 Quality Considerations

The received batch

The chosen sample

First situation: the number of defective products in a sample is lower than α1

Second situation: the number of defective items in a sample between α1 and α2

Third situation: the number of defective items in a sample is more

The decision is that the batch is accepted with full inspection; the defective products are differentiated from perfect items and sold with a lower price

The decision is that the batch is not accepted and a new batch is received that has no defective item

The decision is that the batch is accepted without full inspection, customer can return the defective items and received a particular amount of money instead of it

than α2

The lot that is provided to the customers after making a decision according to the inspection process

Defective items

Perfect items

Patient customers that wait in the queue to buy products

Impatient customers that do not wait in the queue to buy products and cause partial backordering

Fig. 6.9 Process of the system (Taleizadeh and Zamani-Dehkordi 2017)

6.4.3.2

Case II: α1  θ  α2

The second case is the one that the number of imperfect products in a chosen sample is between the lower limit and upper limit (α1 and α2); in this situation, all the items should be inspected to separate the defective items from the perfect items. Then he sells the perfect items with the particular price within the cycle and sells the defective

6.4 EOQ Model with Return

529

items after the inspection time with the lower price than the perfect items. The revenue of this case is computed as the following: sD(1  x)[F + β(1  F)], the amount of money that is received per unit of time by selling the perfect units that are (1  x) percentage of all the units that are purchased; vxDF, the amount of money that is received per unit of time by selling the imperfect items that are x percentage of all the units that are purchased; KT , the ordering cost per unit of time; CD [F + β(1  F)], cost of buying products per unit of time; CIDF, cost of screening all units per unit of time (cost of inspection of the sample is computed in this part of 2 2 total cost); hDð1x2 Þ TF , holding cost of the perfect units per unit of time; hxD2TF2/RS, holding cost of the imperfect units per unit of time; and gD(1  β)(1  F), shortage cost that is related to lost sales per unit of time. So the total revenue per unit of time is obtained from the formulation below (Taleizadeh and Zamani-Dehkordi 2017): TP2 ¼ sDð1  xÞ½F þ βð1  F Þ þ vxDF " # K=T þ CD½F þ βð1  F Þ þ CI DF þ hDð1  xÞ2 TF 2 =2  þhxD2 TF 2 =RS þ gDð1  βÞð1  F Þ þ βCb DT ð1  F Þ2 =2

6.4.3.3

ð6:73Þ

Case III: θ  α1

Finally, the third situation is the one that the number of imperfect products in a chosen sample is lower than the lower limit (α1) that is expected. In this situation, none of the items is inspected, and all the products are given to the customers as their orders with the price that is considered for the perfect products. Customers can receive the particular amount of money instead of the defective items that they return to the vendor. The revenue of this situation is computed as the following: sD [F + β(1  F)], the amount of money that is received per unit of time by selling the perfect units that are (1  x) percentage of all the units that are purchased; K/T, the ordering cost per unit of time; CD[F + β(1  F)], cost of buying products per unit of time; hD(1  x)2TF2/2, holding cost of the perfect units per unit of time; nCI/T, the cost of inspection of a sample per unit per time; hxD2TF2/RS, holding cost of the imperfect units per unit of time; gD(1  β)(1  F), shortage cost that is related to lost sales per unit of time; and βCbDT(1  F)2/2, shortage cost that is related to backorders per unit of time. The total number of returned item is computed as (x + x2 + x3 + ⋯), and we know that lim ðx þ x2 þ x3 þ ⋯ þ xm Þ ¼ x=ð1  xÞ, so m!1

we have CrDFx/(1  x) as a return cost of imperfect units per unit of time that the customers return them to a vendor. So the total revenue per unit of time is obtained from the formulation above:

530

6 Quality Considerations

TP3 ¼ sDð1  xÞ½F þ βð1  F Þ þ vxDF 2 3 2 K nC I hDð1  xÞ TF 2 6 T þ CD½F þ βð1  F Þ þ T þ 7 ð6:74Þ 2 7 6 4 5 βC b DT ð1  F Þ2 hxD2 Tφ2 x þ þ gDð1  βÞð1  F Þ þ þ C r DF 1x x 2 In order to obtain the optimal values of decision variables, Taleizadeh and Zamani-Dehkordi (2017) first derived the expected value of the total profit for each case. Each case is done with a certain probability computed according to the intervals related to the number of defective items in each sample and subsequently the numbers of defective items in each lot. So according to these probabilities and related case for each one, the total profit per year that includes all three cases is (Taleizadeh and Zamani-Dehkordi 2017): ETP1 ¼ sD½F þ βð1  F Þ 2 3 K hDTF 2 nCI 6 T þ CD½F þ βð1  F Þ þ 2 þ T 7 7 6 4 5 2 βC b DT ð1  F Þ ψC fr þgDð1  βÞð1  F Þ þ þ 2 T

ð6:75Þ

For the second case, one has: TP2 ¼ sDE 2 ð1  xÞ½F þ βð1  F Þ þ vE2 ðxÞDF 2 3 hDE 2 ð1  xÞ2 TF 2 K 6 T þ CD½F þ βð1  F Þ þ CI DF þ 7 2 6 7 6 7 4 hE 2 ðxÞD2 TF 2 βCb DT ð1  F Þ2 5 þ þ gDð1  βÞð1  F Þ þ RS 2

ð6:76Þ

And finally for the third case: ETP3 ¼ sDðE3 ð1  xÞÞ½F þ βð1  F Þ þ vðE 3 ðxÞÞDF    2 3 2    hD E ð 1  x Þ TF 2 3 K nC x I 6 þ CD½F þ βð1  F Þ þ 7 þ þ C r DF E3 6 1x 7 T 2 6 T 7 4 5 βC b DT ð1  F Þ2 þgDð1  βÞð1  F Þ þ 2 ð6:77Þ To access a general optimum order quantity, Taleizadeh and Zamani-Dehkordi (2017) combined the three cases, so the expected for x is calculated in three cases as below. For the first case, they had formulated the expectation of the number of defective products in an order received by the vendor, while the number of defective

6.4 EOQ Model with Return

531

items in the sample is more than α2, E1 ðxÞ is computed as below (Taleizadeh and Zamani-Dehkordi 2017): E1 ðxÞ ¼ Prðθ  α2 \ x  x1 ÞE xx1 ½x þ Prðθ  α2 \ x1  x  x2 ÞE x1 xx2 ½x þPrðθ  α2 \ x2  xÞE xx2 ½x ð6:78Þ For the second case, because the number of defective products in the sample is between α1 and α2, one will have (Taleizadeh and Zamani-Dehkordi 2017): E 2 ðxÞ ¼ Prðα1  θ  α2 \ x  x1 ÞE xx1 ½x þ Prðα1  θ  α2 \ x1  x  x2 ÞEx1 xx2 ½x þPrðα1  θ  α2 \ x2  xÞE xx2 ½x ð6:79Þ And if the number of defective products in the random sample is lower than α1, one will have (Taleizadeh and Zamani-Dehkordi 2017): E3 ðxÞ ¼ Prðθ  α1 \ x  x1 ÞE xx1 ½x þ Prðθ  α1 \ x1  x  x2 ÞE x1 xx2 ½x þPrðθ  α1 \ x2  xÞE xx2 ½x ð6:80Þ Finally, the total revue is: ETP ¼ Prðθ  α2 ÞðsD½F þ βð1  F Þ þ Prðα1  θ  α2 ÞðsDðE 2 ð1  xÞÞ½F þ βð1  F Þ       þPrðθ  α1 ÞðsD E 3 ð1  xÞ ½F þ βð1  F Þ þ v E 3 ðxÞ DF þ v E 2 ðxÞ DF   βC b DT ð1  F Þ2 ψC fr K hDTF 2 C I n  þ CD½F þ βð1  F Þ þ þ þ þ gDð1  βÞð1  F Þ þ 2 2 T T T 2 3      2 TF 2 h E 2 ðxÞ D2 TF 2 hD E 2 ð1  xÞ 6 K þ CD½F þ βð1  F Þ þ C DF þ 7 þ I 6T 7 2 RS 7 6 6 7 2 4 5 βC b DT ð1  F Þ þgDð1  βÞð1  F Þ þ 2    2 3    hD E 3 ð1  pÞ2 TF 2 K C n x 6 þ CD½F þ βð1  F Þ þ I þ 7 þ C r DF E 3 6 2 T 1x 7 6 T 7 4 5 βC b DT ð1  F Þ2 þgDð1  βÞð1  F Þ þ 2

ð6:81Þ Taleizadeh and Zamani-Dehkordi (2017) proved that the ETP is a concave function. To access optimum F, they set T as constant and set ∂ETP ¼ 0 and then ∂F derived:

532

6 Quality Considerations

F  ðT Þ ¼

B1  B2 ðT Þ 2A1 ðT Þ

ð6:82Þ

where B1, B2(T ), andA1(T) are computed as below: B1 ¼ Prðθ  α2 Þðs  sβ  C þ Cβ þ gð1  βÞÞD þ Prðα1  θ  α2 ÞðsDðE2 ð1  xÞÞ sDðE2 ð1  xÞÞβ þ vðE 2 ðxÞÞD  CD þ CDβ  C I D þ gDð1  βÞÞ ð6:83Þ B2 ðT Þ ¼ ½Prðθ  α2 Þ þ Prðα1  θ  α2 Þ þ Prðθ  α1 ÞβC b DT ð6:84Þ

 hDT βC b DT βC DT þ  ½Prðα1  θ  α2 Þ þ Prðθ  α1 Þ b A1 ðT Þ ¼ Prðθ  α2 Þ 2 2 2       hD E2 ð1  xÞ2 T hD E 3 ð1  xÞ2 T Prðα1  θ  α2 Þ  Prðθ  α1 Þ 2 2 ð6:85Þ Moreover, to get the optimum T by solving ∂ETP ¼ 0, they obtained: ∂T 

  hD βCb D βC D þ  ðPrðα1  θ  α2 Þ þ Prðθ  α1 ÞÞ b C 3 B1 2  4 Prðθ  α2 Þ 2 2 2       2 2 hD E2 ð1  xÞ hD E3 ð1  xÞ  Prðθ  α1 Þ Prðα1  θ  α2 Þ 2 2 T ¼ 2ðððPrðθ  α1 Þ þ Prðα1  θ  α2 ÞÞβCb D  Prðθ  α1 ÞβC b DT Þ2    0 1



 hD E 3 ð1  pÞ2 hD βC b D βC D b A þ Prðθ  α1 Þ @  8C2 Prðθ  α2 Þ   2 2 2 2

ð6:86Þ where: βCb D ðPrðθ  α2 Þ þ Prðα1  θ  α2 Þ þ Prðθ  α1 ÞÞ 2

ð6:87Þ

C 3 ¼ ðK þ nC I þ ψC fr ÞðPrðθ  α2 ÞÞ  K ðPrðα1  θ  α2 ÞÞ  ðK þ nC I ÞPrðθ  α1 ÞÞ

ð6:88Þ

C2 ¼ 

Example 6.5 Taleizadeh and Zamani-Dehkordi (2017) provided some numerical results according to their real case. They considered n ¼ 20, α1 ¼ 1, and α2 ¼ 4, and also the buyer and the vendor should define the maximum limit that is considered as

6.4 EOQ Model with Return

533

0.15, and the buyer defines the minimum limit as 0.06. According to this information (Taleizadeh and Zamani-Dehkordi 2017), 1 E ð xÞ ¼ ba

Zb xdx ¼ a

  E x2 ¼

1 ba

Zb x2 dx ¼ a

b2  a2 bþa ¼ 2 2ðb  aÞ

b3  a3 b2 þ a2 þ ab ¼ 3 3ð b  aÞ

Zb  ð ln ð1  aÞ þ aÞ  ð ln ð1  bÞ þ bÞ x 1 x E ¼ dx ¼ ba 1x ba 1x 

ð6:89Þ

ð6:90Þ

ð6:91Þ

a

1 E½x  0:06 ¼ 0:06  0

Z0:06 xdx ¼

0:062  02 0:06 þ 0 ¼ 0:03 ¼ 2 2ð0:06  0Þ

ð6:92Þ

0

1 E ½0:06  x  0:15 ¼ 0:15  0:06

Z0:15 xdx ¼

0:152  0:062 2ð0:15  0:06Þ

0:06

0:15 þ 0:06 ¼ 0:105 ¼ 2 1 E ½0:15  x  0:25 ¼ 0:25  0:15

Z0:25 xdx ¼

ð6:93Þ 0:252  0:152 2ð0:25  0:15Þ

0:15

¼

0:25 þ 0:15 ¼ 0:20 2

ð6:94Þ

A sample of 20 products is chosen randomly, and the probabilities of different situations according to the number of defective items in the chosen selection are obtained (Taleizadeh and Zamani-Dehkordi 2017): (a) If the number of defective products θ is more or equal to α2 ¼ 4, the buyer rejects the order and receives another lot instead of it that does not have any defective items. (b) If the number of defective products θ is equal to 2 or 3, all the items should be inspected. (c) If the number of defective products θ is lower or equal to α1 ¼ 1, no item is inspected. The probability of being θ defective items in a sample is calculated as (Taleizadeh and Zamani-Dehkordi 2017):

534

6 Quality Considerations

Table 6.9 The probability of different situations (Taleizadeh and Zamani-Dehkordi 2017) θ  1.00 1.00  θ  4.00 θ  4.00 Table 6.10 The optimal values

b  0.0600 0.796 0.237 0.004 β 0.1 0.2 0.3 0.4 0.5

0.0600  b  0.1500 0.186 0.460 0.032 F 0.74 0.63 0.49 0.32 0.23

0.1500  b  0.2500 0.018 0.304 0.963 T 0.1454 0.0787 0.559 0.0441 0.0366

 n θ f ðθ; n, λÞ ¼ x ð1  xÞnθ θ

ETP 75,559 28,321 7393.1 1972.0 514.72

ð6:95Þ

For instance, they calculated the probability of the situation that θ  1 and x ¼ 0.01 (Taleizadeh and Zamani-Dehkordi 2017): f ð1; 20, 0:010Þ ¼ Prðθ ¼ 0Þ þ Prðθ ¼ 0Þ ! ! 20 20 20 ¼ 0:0100 ð1  0:010Þ þ 0:0101 ð1  0:010Þ19 ¼ 0:983 0 1

ð6:96Þ

The probabilities of different situations are provided in Table 6.9 (Taleizadeh and Zamani-Dehkordi 2017). Also the other parameters are D ¼ 50 units per year, v ¼ 20 $/unit, x ¼ 1 unit/min, CI ¼ 0.5 $/unit, Cr ¼ 15 $/unit, Cfr ¼ 70 $/unit, s ¼ 50 $/unit, C ¼ 25 $/unit, Cb ¼ 1 $/unit, g ¼ 2 $/unit, and K ¼ 10 $/order. The results are presented in Table 6.10.

6.5

EPQ Model Without Return

In this subsection, two models of EPQ models without return policy are presented. Quality assurance and screening are the main topics of these models.

6.5 EPQ Model Without Return

535

Table 6.11 New notations for a given problem (Cheng 1991) ε P(ε) ¼ a/(1  ε)b

6.5.1

Expected fraction of products found to be acceptable in a production run (%) Unit cost of production ($/item)

Quality Assurance Without Shortage

In this subsection, the study of Cheng (1991) has been presented. His problem considers an EPQ model with imperfect production processes and quality-dependent unit production cost. Also, discussion of the procedure for determining the optimal solution is presented. Finally, the problems associated with cost estimation are addressed. Consider the case where a company employs a production process with a certain level of capability to manufacture a single product, the quality of which is monitored by some quality assurance programmer. The capability of the process and the effectiveness of the quality assurance programmer depend on a great variety of factors such as production technology, machine capability, jigs and fixtures, work methods, use of online monitoring devices, skill level of the operating personnel, and inspection, maintenance, and replacement policies. A high level of process capability in conjunction with a stringent quality assurance system will result in products of an acceptable level of quality being more consistently produced, hereby reducing the subsequent costs of scrap and rework of substandard products and wasted materials and labor hours. There is also an array of intangible benefits resulting from adopting advanced production technology and improving quality assurance, which include improvement in yields, process flexibility, product variety, development and delivery lead times, profit margin, market share, and customer goodwill (Cheng 1991). However, superior process capability and effective quality management can be achieved only through substantial investment in plant, machinery, equipment, and employee training. Evidently this will increase the production overhead that will inevitably be apportioned to the individual products by the costing system, so that a higher unit cost of production will result. In addition, the unit direct cost of production, which consists of three cost components (direct labor, direct material, and direct expenses), also goes up. This is so because a more capable process calls for more skillful labor and more expensive tools, which will result in higher direct labor and direct expenses. Furthermore, a more effective quality assurance programmer means the use of better-quality material and more stringent quality assurance policies, which will increase the direct material and direct expenses (Cheng 1991). The notations which are specially used in this problem are presented in Table 6.11. While it is hard to quantify accurately the various intangible benefits, an increase in production overhead will, as stated above, invariably push up the unit cost of production. Thus, it seems rational to hypothesize that the unit (tangible) cost of production is an increasing function of process capability and quality assurance expenses. However, this is not to deny that the increased (tangible) costs can often be offset by the increased (intangible) benefits as witnessed by such manufacturers as Ford,

536

6 Quality Considerations

Firestone, Toyota, Xerox, and many others who have invested heavily in modern production technologies and quality assurance systems. They report that for every dollar they spend in quality control, they save two dollars in total (intangible) production costs and costs related to product returns (an intangible benefit) (Cheng 1991): Total relevant cost per production run ¼ K S þ pðεÞQ þ

hε2 Q2 2D

ð6:97Þ

where a, b > 0 are constant real numbers chosen to provide the best fit of the estimated cost function. The objective is to minimize the total relevant cost per unit time subject to the constraint that the expected fraction of products found to be acceptable in a production run cannot exceed 100%. So the function which should be minimized is as below (Cheng 1991):   Total relevant cost per production run D hε2 Q2 C ðQ, εÞ ¼ K þ pðεÞQ þ ¼   Qε Qε S 2D D s:t:0  ε  1 ð6:98Þ To solve the constrained minimization problem, Cheng (1991) took the first partial derivative of C(Q, ε) with respect to Q and ε and proved that the following equations are the optimal values of decision variables: Q ¼ ð1 þ bÞ ε ¼

6.5.2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2K S D=h

1 1þb

ð6:99Þ ð6:100Þ

Quality Screening and Rework Without Shortage

Moussawi-Haidar et al. (2016) explicitly integrated the inspection time into the economic production model with rework and demonstrated the significant effect that the inspection time has on the results. They considered a manufacturing process with random supply and a screening process conducted during and at the end of production. They analyzed two scenarios for dealing with the defective items produced: selling at a discount and reworking. For each scenario, the demand during production is met using non-defective items only. The expected profit functions are developed using the renewal reward theory, and closed-form expressions for the optimal production lot size are derived. This problem is considered in Sect. 4.3.4.

6.6 EPQ Model with Return

6.6

537

EPQ Model with Return

In this subsection, two models of EPQ models with return policy are presented. Continuous quality characteristic, inspection errors, and 100% quality screening are the main topics of these models.

6.6.1

Continuous Quality Characteristic Without Shortage

Tsou et al. (2012) extended the traditional EPQ model by considering imperfectquality items and continuous quality characteristic. In many manufacturing industries such as textiles, if products do not enjoy some precise and rather complete dimension and quality properties, they are sold as second or third rate. However, sometimes, it is possible to disguise the defect through some rework (reprocessing) such as re-dyeing or surface repairing. In case the defect is not repairable, e.g., an item lacks minimum required durability, the product is discarded as useless. It must be noted that this study assumes an item produced follows a general distribution pattern. As shown in Fig. 6.1, the produced items are divided into three categories: they are perfect, imperfect, or defective. An item is perfect if the quality characteristic is inside the internal specification limits. An item is imperfect if the quality characteristic goes beyond the internal specification limits and yet inside the external specification limits. An item is defective if its quality characteristics depart from external specification limits (see Fig. 6.10) (Tsou et al. 2012). It is assumed that perfect items are kept in stock and sold at a price, imperfect items are sold at a discounted price when imperfection is identified, and defective item can be reworked or rejected. If the quality characteristic of an item is above the upper specification level (USL), it can be reworked. But an item must be rejected if its quality characteristic is below the lower specification level (LSL). For example, if the quality characteristic is length of a wire above the USL after the process, it can be reworked, but if it is low after the process, it must be rejected (Tsou et al. 2012). Fig. 6.10 The distribution of product quality characteristic (Tsou et al. 2012) Imperfect Defective (P1) (P3)

LSL2

LSL1

Good Quality



Imperfect (P1)

USL1

Defective (P2)

USL2

538

6 Quality Considerations

Table 6.12 New notations for a given problem (Tsou et al. 2012) LSL1 LSL2 USL1 USL2 μ p1 p2 p3 x f0(x) f1(x)

Lower specification limit for perfect items Lower specification limit for imperfect items Upper specification limit for perfect items Upper specification limit for imperfect items Target quality characteristic Percentage of imperfect-quality items (%) Percentage of reworked items (%) Percentage of rejected items (%) Random variable which represents the actual value of the quality characteristic (%) Qualitative probability density function (general distribution) Qualitative probability density function for an item that has been reworked (general distribution)

After the rework process, each item is assumed to be good. The distribution of reworked item is assumed to follow a general distribution within the internal specification limits. A perfect-quality item can be sold at a price, $s, with a discount of Taguchi’s cost of poor quality (COPQ) as its major quality characteristic departs from the target value. The imperfect-quality items could be used in low-end production situations or sold to a particular purchaser at a discounted price, $v, also with a discount of Taguchi’s COPQ as its major quality characteristic departs from the target value. Quality characteristics outside the external specification limits are considered to be defective and will be scrapped directly. In the next section, the mathematical model is developed (Tsou et al. 2012). The model developed in this study can be used to determine the optimal lot size in different manufacturing companies, textile industries, or ornamental products manufacturing industries; it must be borne in mind that this model suffers from some limitations, which are basically associated with the underpinning assumptions. Moreover, the model is limited to the deterministic (non-stochastic) models of cost parameters, similar production, and rework rates (Tsou et al. 2012). The notations which are specially used in this problem are presented in Table 6.12. The study centers on the following assumptions: A process produces a single product in a batch size of Q; storage and withdrawals are uniform and continuous; the demand rate for the product is deterministic and constant; and the factor of shortages or backorders is ignored. Figure 6.10 shows the distribution of product quality with different quality characteristic ranges. Different ranges for product quality distribution have been listed in Table 6.13. Figure 6.11 is the roadmap for our decision-making to handle products with different quality levels (Tsou et al. 2012). Tsou et al. (2012) assumed that the product quality distribution follows the general distribution function, f0(x). After the rework process, a defective item is assumed to be of good quality such that the actual value of the quality characteristics

6.6 EPQ Model with Return

539

Table 6.13 Different ranges for product quality distribution LSL1  x  USL1 LSL2  x  LSL1 or USL1  x  USL2 USL2  x x  LSL2

Item is good and kept in stock after inspection and sold at a full price with a discount of Taguchi’s COPQ Item is imperfect and sold at a lower price with a discount of Taguchi’s COPQ Defective items can be reworked instantaneously at a cost, and then kept in stock. After the rework process, item is assumed to be of good quality with distribution, f1(x) Item is defective and not rework able. So, it is rejected with a cost

Perfect Rework Production (Q units)

Imperfect (p1 percent)

Perfect (p2 percent) Scrap (p3 percent)

Defective (p2 + p3 percent) Reject

Fig. 6.11 Roadmap for decision-making on imperfect and defective products (Tsou et al. 2012)

has a general distribution with probability density function, f1(x), and quality characteristic value between LSL1 and USL1: Zþ1 f 0 ðxÞdx ¼ 1

ð6:101Þ

f 1 ðxÞdx ¼ 1

ð6:102Þ

1 USL Z 1

LSL1

A perfect-quality item sold at a price, $s, with a discount of Taguchi’s COPQ as its quality characteristic departs from the target value μ. The imperfect item could be sold to a low-end buyer at a discounted price, $v, and a discount of Taguchi’s COPQ as its major quality characteristic departs from the target value. Figure 6.12 illustrates the behavior of the inventory level per cycle. As shown in Fig. 6.12, the Tp requires that Q items go to the storage if the production rate is P and without producing the imperfect and defective production. From Fig. 6.12, TpD items are sold, thus decreasing the final warehouse. Defective products have the similar effect on the final inventory of stock (Tsou et al. 2012). From the definition above, p1 and p2 can be defined as (Tsou et al. 2012):

540

6 Quality Considerations

Fig. 6.12 The behavior of the inventory level per cycle (Tsou et al. 2012)

Inventory level Q

TpD

P

Q (p1+p3)

P−D P (1–p1–p3)–D

–D

Time Tp T

LSL Z 1

p1 ¼

USL Z 2

f 0 ðxÞdx þ LSL2

f 0 ðxÞdx

ð6:103Þ

USL1

Zþ1 p2 ¼

f 0 ðxÞdx

ð6:104Þ

f 0 ðxÞdx

ð6:105Þ

USL2 LSL Z 2

p3 ¼ 1

The proportion of perfect item is: 1  ð p1 þ p3 Þ

ð6:106Þ

The cycle time, T, can be calculated as: Qð1  p1  p3 Þ ¼ TD ! T ¼

Qð1  p1  p3 Þ D

ð6:107Þ

And the production time, Tp, can be calculated as: Q ¼ TP ! T p ¼

Q P

ð6:108Þ

They defined the total revenue and the total cost per cycle as TR(Q) and TC(Q), respectively. The total profit per cycle, TP(Q), is the total revenue per cycle minus the total cost per cycle. It is given as:

6.6 EPQ Model with Return

541

TPðQÞ ¼ TRðQÞ  TCðQÞ

ð6:109Þ

The total revenue per cycle, TR(Q), is the total sales volume of product within the specification limits. Therefore, one has (Tsou et al. 2012): USL Z 1

TPðQÞ ¼

LSL1 Z

f 0 ðxÞQðs  L0 ðxÞÞdx þ LSL1

LSL2

Zþ1

USL Z 2

f 0 ðxÞQðv  L1 ðxÞÞdx þ

þ

f 0 ðxÞQðv  L1 ðxÞÞdx

USL1

USL Z 1

f 0 ðxÞdx LSL1

USL2

2

6 ¼ Q4sð1  p1  p3 Þ þ vp1 

USL Z 1

ð6:110Þ

LSL Z 1

f 0 ðxÞL0 ðxÞdx  LSL1

f 0 ðxÞL1 ðxÞdx LSL2

USL Z 1

USL Z 2

f 0 ðxÞL1 ðxÞdx  p2



f 1 ðxÞQðs  L0 ðxÞÞdx

USL1

3

7 f 1 ðxÞL0 ðxÞdx5

LSL1

The total revenue per unit time can be written as (Tsou et al. 2012): TRYðQÞ ¼ TRðQÞ=T 2

USL Z 1

6 TPðQÞ ¼ D4sð1  p1  p3 Þ þ vp1 

f 0 ðxÞL0 ðxÞdx LSL1

USL Z 1

USL Z 2



f 0 ðxÞL1 ðxÞdx  p2 USL1

LSL Z 1

3

f 0 ðxÞL1 ðxÞdx LSL2

7 f 1 ðxÞL0 ðxÞdx5=ð1  p1  p3 Þ

LSL1

ð6:111Þ It can be seen that total revenue per unit time for any distribution is independent of Q. Therefore, the maximization of total profit per unit time is the same as minimization of the total cost per unit time. The total cost per cycle can be found as (Tsou et al. 2012):   1 D TCðQÞ ¼ CQ þ C R Qp2 þ K S þ C I Q þ h Q ð1  p1  p3 Þ  T 2 P The total cost per unit time can be written as (Tsou et al. 2012):

ð6:112Þ

542

6 Quality Considerations

TCYðQÞ ¼ TCðQÞ=T TCYðQÞ ¼ ðCQ þ C R Qp2 þ 1Þ

  D KSD 1 D þ þ h Q 1  p1  p3  1  p1  p3 Qð1  p1  p3 Þ 2 P ð6:113Þ

By differentiating and equating d(TCY(Q))/dQ ¼ 0, the optimal lot size, Q*, which generates the minimum expected total cost, can be obtained by (Tsou et al. 2012): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DK S   Q ¼ h 1  p1  p3  DP ð1  p1  p3 Þ 

ð6:114Þ

The second derivatives of Eq. (6.113) are positive for all positive Q, which implies that there exists a unique Q* that minimizes Eq. (6.113). Example 6.6 To verify the usefulness of the previous model, Tsou et al. (2012) considered a production system with the following parameters: KS ¼ $125/cycle; C ¼ $0.1/unit; CR ¼ $0.05/unit; h ¼ $15/unit/year; D ¼ 15,000 units/year; P ¼ 20,000 units/year; CI ¼ $0.02/unit; p1 ¼ 15%; and p2 ¼ 10% and p3 ¼ 5% (Tsou et al. 2012). Using Eqs. (6.114) and (6.113), the optimal lot size and the minimum total relevant cost are Q* ¼ 2500 and TCY(Q*) ¼ 4219.

6.6.2

Inspections Errors Without Shortage

Haji et al. (2009) provided a framework to integrate the existence of products with imperfect-quality items, inspection errors, rework, and scrap items into a single economic production quantity (EPQ) model. To achieve this objective, a suitable mathematical model is defined, and the optimal production lot size that minimizes the total cost is obtained. Sensitivity analysis is carried out for this model. The sensitivity analysis results indicate that the model is very sensitive to defective proportions and type 1 errors of inspection. Nowadays, given the progress made, inspection errors are often ignored. But the findings of this study show that the values of EPQ and total cost are very sensitive to type I error of inspection. If the existence of such errors is ignored, then the obtained results will differ considerably from the optimal outcome. This will impose additional costs to the system. Haji et al. (2009) considered a production system in which imperfect-quality items are passed to the rework process; after the rework process, they are classified as either good items or scraps. These important issues must be addressed when dealing with imperfect production and rework processes: • Imperfect-quality items must be separated so that they are not passed to stock.

6.6 EPQ Model with Return

543

Table 6.14 New notations for a given problem (Haji et al. 2009) β α e1 e2 Q Qi Qr v1 v2 v3

Proportion of defects in the production process in each cycle (%) Proportion of scraps in the rework process in each cycle (%) Proportion of good items that are incorrectly rejected in each cycle (%) Proportion of bad items that are incorrectly accepted in each cycle (%) Net batch quantity needed per cycle to satisfy demand (item) Input batch quantity required to be processed per cycle (item) The number of reworkable items (item) The cost per imperfect item incorrectly accepted in the production process ($/item) The cost per perfect item incorrectly rejected in the rework process ($/item) The cost per scrap item incorrectly accepted in the rework process ($/item)

• Errors may be committed while screening products during the production and rework processes to separate imperfect-quality items and scraps. Imperfectquality items or scraps may be incorrectly accepted (type II error), and good items may be incorrectly rejected (type I error). Assuming the rework process, scrap production, and inspection errors (type I and type II errors), a suitable mathematical model is defined, and then the optimal production lot size that minimizes the total system cost is obtained. The notations which are specially used in this problem are presented in Table 6.14. Consequently, these two operations involve two types of costs: setup cost and processing cost. Since the production rate (P) is greater than the demand rate (D), inventory is accumulated during the production period. Having enough inventory sometimes helps satisfy demand during the period wherein production is stopped due to various reasons. They further assumed that the net production rate, after excluding the defective items, is constant and greater than the demand rate. Figure 6.13 shows that inventory increases in the first phase of production, i.e., during the normal production time tp at rate R1, which is equal to: R1 ¼ P½ð1  βÞð1  e1 Þ þ βe2   D

ð6:115Þ

At the end of this phase, the rework of defective items starts. During the rework phase, inventory increases at rate R2, which is equal to: R2 ¼ P½ð1  αÞð1  e1 Þ þ αe2   D

ð6:116Þ

Since Qi stands for the input quantity in each cycle, then the required processing time for this quantity (the production time), tp, is: tp ¼

Qi P

ð6:117Þ

It is assumed that the quality of the output of the production process is not perfect. At each cycle of the production process, a fixed fraction β of non-conforming items

544

6 Quality Considerations

P

Q

Q

D

R2 D' R1

h2 h1

tp

tr

td T

Fig. 6.13 A comparison of inventories with defective and non-defective products (Haji et al. 2009)

Sent

D

Raw Materials Q1

e ef

Production Process Q 1

ct

iv

Type II error

es 1 βQ

Pe (1

-β )

Se

nt t βQ o Re w 1 (1 – e ork Pr 2) o ce

rfe

Pro ork ew R β)e 1 to (1nt Q1 Se

ct

s

Type I error

use

reho

to wa

ce

ss

Reworkable Items

ss

Qr

Sent

to wa reho use Q (1 1 -β)( I-e ) 1

Fig. 6.14 The product flow diagram in the regular production process (Haji et al. 2009)

may be produced. Therefore, the number of non-conforming items produced at the production process is equal to βQi and the number of perfect-quality items at the end of the production process is (1  β)Qi. The perfect items at the production process are inspected and put in the inventory to be used when necessary, and the non-conforming items are screened for the rework process. It is assumed that raw materials and input products are of perfect quality. Since the inspection process is error-prone, types I and II errors may be committed. That is, non-conforming items may be incorrectly accepted and conforming items may be incorrectly rejected. From type I errors, the amount of correctly accepted items passed to inventory is (1  β) Qi (1  e1). From type II errors, the quantity of incorrectly accepted items is βQi e2 (see Figs. 6.14 and 6.15). Hence, the number of items recognized as good items, Q, is:

6.6 EPQ Model with Return

545

Sent

Type II error S

Reworkable items

a cr

ps

Sc re

αQ

r αQ

r

en

ed

(1–

sc

2)

Rework process

Qr

as

e

to wa

rap

rehou

se

s

Sent to Scraps Area

Q r

Pe (1

rfe c –α ts )

s ap scr ne α)e 1 ree (1– Sc Qr s da

Type I error

Sent to wa rehou se Q (1– r α)(1– e1 )

Fig. 6.15 The product flow diagram in the rework process (Haji et al. 2009)

Q ¼ Qi ð1  βÞð1  e1 Þ þ Qi βe2

ð6:118Þ

The number of reworkable items (denoted by Qr) consists of the number of incorrectly rejected good items and correctly accepted defective items for rework, computed as: Qr ¼ ½βð1  e1  e2 Þ þ e1 Qi

ð6:119Þ

The total system cost TC(Q) can be obtained as follows: Rework cost

Inspection cost

zfflfflffl}|fflfflffl{ zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ K D CD D D TCðQÞ ¼z}|{ S þ z}|{ þ CR ω þ C I ð 1 þ ωÞ þ Q φ φ φ Setup cost

Production cost

Inspection error cost

z}|{ D γ φ

Holding cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{   hDQ ω P ðL  M Þ þ ðL þ MωÞ  1  ω þ 2Pφ φ D ð6:120Þ where:     λ ¼ ðα þ βÞ e1 e2 þ e21  e1 þ αβ 2e1 þ 2e2  2e1 e2  e21  e22  1 ð6:121Þ

þ 1  e21 φ ¼ λ  e2 ðβ þ αωÞ

ð6:122Þ

L ¼ ð1  βÞð1  e1 Þ þ βe2

ð6:123Þ

546

6 Quality Considerations

Table 6.15 A comparison of the results of changing the β values (Haji et al. 2009)

Β 0.05 0.1 0.15 0.2 0.25 0.3 0.36 0.37

Q* 115.74 122.36 131.02 142.79 159.83 187.18 259.35 283

TC (Q*) 100,070.18 101,871.91 103,660.92 105,431.03 107,172.30 108,866.79 110,778.05 111,071.71

M ¼ ð1  αÞð1  e1 Þ þ αe2

ð6:124Þ

γ ¼ v1 βe2 þ v2 ωð1  αÞe1 þ v3 ωαe2 :

ð6:125Þ

Haji et al. (2009) showed that TC(Q) is a convex function of Q. Therefore, setting the first derivative of objective function equal to zero will yield an optimum value as below: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2K S Pφ i Q ¼u t h ω h φ ðL  M Þ þ DP ðL þ MωÞ  1  ω 

ð6:126Þ

Example 6.7 Haji et al. (2009) presented an example with the following data: D ¼ 3000, item/year; KS ¼ 60, $/setup; P ¼ 5000, item/year; h ¼ 80, $/item/year; α ¼ 0.01; β ¼ 0.05; e1 ¼ 0.05; e2 ¼ 0.01; v1 ¼ 10, $/item; v2 ¼ 8, $/item; v3 ¼ 7, $/item; C ¼ 30, $/item; CR ¼ 10, $/item; and CI ¼ 1, $/item. The results (see Table 6.15) show that the model is very sensitive to parameters e1 and β, whereas it is much less sensitive to parameters α and e2. In this case, the changes in TC are directly related to the changes in all parameters. The changes in Q are directly related to the changes in β and e1, whereas these changes are inversely related to the changes in e2.

6.7

Conclusion

In this chapter, 11 inventory models with quality factors are presented. These models consider the traditional lot-sizing problem when after arrival, the quality of each item in a lot is checked. All models are categorized in four sections about EOQ and EPQ models with and without returns. Learning in inspections, sampling plans, inspection errors, different quality levels, quality assurance, quality screening, continues quality characteristics and different types of shortages are the main topics investigated and differ models.

References

547

References Al-Salamah, M. (2016). Economic production quantity in batch manufacturing with imperfect quality, imperfect inspection, and destructive and non-destructive acceptance sampling in a two-tier market. Computers & Industrial Engineering, 93, 275–285. Aslani, A., Taleizadeh, A. A., & Zanoni, S. (2017). An EOQ model with partial backordering with regard to random yield: Two strategies to improve mean and variance of the yield. Computers & Industrial Engineering, 112, 379–390. Cheikhrouhou, N., Sarkar, B., Ganguly, B., Malik, A. I., Batista, R., & Lee, Y. H. (2018). Optimization of sample size and order size in an inventory model with quality inspection and return of defective items. Annals of Operations Research, 271, 445–467. Cheng, T. C. E. (1991). EPQ with process capability and quality assurance considerations. Journal of the Operational Research Society, 42(8), 713–720. Haji, A., Sikari, S. S., & Shamsi, R. (2009). The effect of inspection errors on the optimal batch size in reworkable production systems with scraps. International Journal of Product Development, 10(1–3), 201–216. Hauck, Z., & Vörös, J. (2015). Lot sizing in case of defective items with investments to increase the speed of quality control. Omega, 52, 180–189. Khan, M., Jaber, M. Y., & Bonney, M. (2011). An economic order quantity (EOQ) for items with imperfect quality and inspection errors. International Journal of Production Economics, 133(1), 113–118. Konstantaras, I., Skouri, K., & Jaber, M. Y. (2012). Inventory models for imperfect quality items with shortages and learning in inspection. Applied Mathematical Modelling, 36(11), 5334– 5343. Mohammadi, B., Taleizadeh, A. A., Noorossana, R., & Samimi, H. (2015). Optimizing integrated manufacturing and products inspection policy for deteriorating manufacturing system with imperfect inspection. Journal of Manufacturing Systems, 37, 299–315. Moussawi-Haidar, L., Salameh, M., & Nasr, W. (2016). Production lot sizing with quality screening and rework. Applied Mathematical Modelling, 40(4), 3242–3256. Muhammad, A., & Alsawafy, O. (2011). Economic order quantity for items with two types of imperfect quality. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 2(1), 73–82. Pentico, D. W., & Drake, M. J. (2011). A survey of deterministic models for the EOQ and EPQ with partial backordering. European Journal of Operational Research, 214, 179–198. Porteus, E. L. (1986). Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research, 34(1), 137–144. Raouf, A., Jain, J. K., & Sathe, P. T. (1983). A cost-minimization model for multicharacteristic component inspection. IIE Transactions, 15, 187–194. Rezaei, J., & Salimi, N. (2012). Economic order quantity and purchasing price for items with imperfect quality when inspection shifts from buyer to supplier. International Journal of Production Economics, 137(1), 11–18. Salameh, M. K., & Jaber, M. Y. (2000). Economic production quantity model for items with imperfect quality. International Journal of Production Economics, 64(1), 59–64. Taleizadeh, A. A., & Zamani-Dehkordi, N. (2017). Economic order quantity with partial backordering and sampling inspection. Journal of Industrial Engineering International, 13, 331–345. Taleizadeh, A. A., Tavassoli, S., & Bhattacharya, A. (2019a). An inventory system for a deteriorating product with inspection policy under multiple prepayments and delayed payment and linked-to order quantity. Annals of Operations Research, 287, 403–437. Taleizadeh, A. A., & Moshtagh, M. S. (2019). A consignment stock scheme for closed loop supply chain with imperfect manufacturing processes, lost sales, and quality dependent return: Multi levels Structure. International Journal of Production Economics, 217, 298–316.

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Taleizadeh, A. A., Yadegari, M., & Sana, S. S. (2019b). Production models of multiple products using a single machine under quality screening and reworking policies. Journal of Modelling in Management, 14(1), 232–259. Taleizadeh, A. A., Kalantari, S. S., & Cárdenas-Barrón, L. E. (2015). Determining optimal price, replenishment lot size and number of shipments for an EPQ model with rework and multiple shipments. Journal of Industrial & Management Optimization, 11(4), 1059–1071. Tsou, J.-C., Hejazi, S. R., & Barzoki, M. R. (2012). Economic production quantity model for items with continuous quality characteristic, rework and reject. International Journal of Systems Science, 43(12), 2261–2267.

Chapter 7

Maintenance

7.1

Introduction

The role of the equipment condition in controlling quality and quantity is well known (Ben-Daya and Duffuaa 1995). Equipment must be maintained in top operating conditions through adequate maintenance programs. Despite the strong link between maintenance production and quality, these main aspects of any manufacturing system are traditionally modeled as separate problems. Few attempts have been made to integrate them in a single model that captures their underlying relationships. There exists an extensive literature addressing the issue of production planning and an equally broad literature tackling maintenance planning questions. Production planning models seek typically to balance the costs of setting up the system with the costs of production and material holding, while maintenance models attempt typically to balance the costs and benefits of sound maintenance plans in order to optimize the performance of the production system. In both domains, issues of production modeling and maintenance modeling have experienced an evident success both from theoretical and applied viewpoints. In this chapter, the EPQ problems considering maintenance policy under different situations are presented. The presented models are classified into three main categories. The first category includes several models in which shortage is not allowed. The second and third categories include models in which shortage was considered as backordering and partial backordering. To provide a comprehensive introduction about the mentioned research status, the studies in relevant fields are reviewed. The classification is shown in Fig. 7.1. The common notations of presented models are shown in Table 7.1. To integrate the chapter, the presented notations in Table 7.1 are used for all models. The main decision variables of this field on inventory are Q and T, but in some studies, other decision variables are considered too.

© Springer Nature Switzerland AG 2021 A. A. Taleizadeh, Imperfect Inventory Systems, https://doi.org/10.1007/978-3-030-56974-7_7

549

550

7 Maintenance

Fig. 7.1 Category of presented EPQ models

7.2 7.2.1

Fully backordered

Prevenve maintenance

Prevenve maintenance

Paral backordered

Mul-product single-machine

Imperfect maintenance

Prevenve maintenance

No shortage

No Shortage Preventive Maintenance

Ben-Daya and Makhdoum (1998) considered a production process producing a single item. A production cycle begins with a new system which is assumed to be in an in-control state, producing items of acceptable quality. However, after a period of time in production, the process may shift to an out-of-control state. The elapsed time for the process to be in the in-control state, before the shift occurs, is a random variable assumed to follow a general distribution with increasing hazard rate. The process is inspected at times tl, t2, . . ., tn to assess its state. In fact, the output quality of the product is monitored by an x-control chart. PM activities are coordinated with quality control inspection and are carried out periodically at a subset of the above time epochs according to one of the three PM policies. The production cycle ends either with a true alarm signaling that the system is out of control or after m inspection intervals, whichever occurs first. The number m is a decision variable. The process is then restored to the in-control state and to as good as new condition by maintenance and/or replacement. The usual assumptions of the classical EPQ model apply here. In particular, the demand is constant and continuous, and all demands must be met (Ben-Daya and Makhdoum 1998). The specific notations which are used for this problem is presented in Table 7.2.

7.2 No Shortage

551

Table 7.1 Notations D h K KS P C CPM Cm PM yj wj Lj tj f(t) F(t) r(t) m η CI Cb Bn T Q n f(t) ETC

Demand rate (item/year) Holding cost per unit and per unit of time ($/unit/unit of time) Fixed order cost ($/order) Setup cost ($/setup) Production rate (item/year) Purchasing cost per unit of product ($/unit) Cost of preventive maintenance ($/period) Maximum cost of preventive maintenance ($/period) Actual age of the process right before the jth PM (time) Actual age of the process right after the jth P (time) Length of the ith sampling (inspection) interval (time) Time at the end of jth interval or at jth inspections (time) Probability density function of the time to shift distribution Cumulative time to shift distribution function F ðt Þ ¼ 1  F ðt Þ Hazard function r ðt Þ ¼ f ðt Þ=F ðt Þ Number of inspection interval or inspection undertaken during each production run (integer) Imperfectness factor Inspection cost ($/unit) Backordering cost per unit of demand and per unit of time ($/unit/unit of time) The maximum backordering quantity in units for the nth cycle (item) Length of production cycle (time) Order quantity (item) Sample size (integer) Probability density function of the time to shift distribution Expected total cost ($)

It is obvious that the shift pattern to the out-of-control state of a preventively maintained system is not the same as that of a system which is not maintained. The effect of PM on the system is modeled as follows: whenever a PM is carried out, the age of the system is reduced by a certain amount. This is equivalent to a reduction in the shift rate of the system to the out-of-control state (Ben-Daya and Makhdoum 1998). The joint production, maintenance, and quality model is based on the following assumptions: 1. The duration of the in-control period is assumed to follow an arbitrary probability distribution, f(t), having an increasing hazard rate r(t) and a cumulative distribution function F(t). 2. The process is inspected at times tl, t2, . . ., to determine its state, and the output quality of the product is monitored by an x-control chart. The inspection interval

552

7 Maintenance

Table 7.2 Specific notations (Ben-Daya and Makhdoum 1998) w1 rmax k Z max 1 Z1 Z2 Cf Ca Cin Cout CS CR(τ) Kn Cn α β nPM γj pi I(t) E (PM) E (HC) E (QC) ETC

Warning limit coefficient of the xi-control chart Failure rate threshold beyond which PM activities should be performed Control limit coefficient of the xi-control chart The expected time to perform the highest level of a PM (time) The expected time to perform a PM (time) The expected time to repair the process if a failure is detected (time) Cost per false alarm ($/alarm) Cost to locate and repair the assignable cause ($) Quality cost per unit time while producing in control ($/unit of time) Quality cost per unit time while producing out of control ($/unit of time) Cost incurred by producing a non-conforming item ($/unit) Restoration cost ($) Fixed sampling cost ($/sample) Cost per unit sample ($/unit of sample) Pr (exceeding control limits one process in control) Pr (not exceeding control limits one process out of control) Expected number of PMs during the production cycle (integer) Fraction used to compute the reduction in the process age at time γ j (%) The conditional probability that the process shifts to the out-of-control state during the jth interval given that it was in the in-control state at the beginning of the jth interval The function of the inventory level (item) Expected preventive maintenance cost per production cycle ($) Expected inventory holding cost per inventory cycle ($) Expected quality control cost per production cycle including the repair cost ($) Expected total cost ($)

lengths Ll, L2, . . ., are chosen such that the integrated hazard rate over each interval is the same. 3. Quality inspection activities are carried out at the end of each interval. If a PM criterion is met at the end of the interval, production ceases for an amount of time Z1 to carry out preventive maintenance activities. 4. If any inspection shows that the state of the process is out of control, production ceases until the accumulated on-hand inventory is depleted to zero. If the process is found to be in control, production continues until the next sampling is due or the predetermined level of inventory is accumulated. 5. The production cycle begins with a new system and ends either with a true alarm or after a specified number m of inspection intervals, whichever occurs first. In other words, if no true alarm is observed by time tm1, then the cycle is allowed to continue for an additional time Lm. At time tm, necessary maintenance work is carried out. Therefore, there is no cost of sampling and charting during the mth sampling interval. The process is brought back to the in-control state by repair

7.2 No Shortage

553

and/or replacement. Thus, a renewal occurs at the end of each cycle. This type of renewal process (Ross 2013) has the property that the expected cost per unit time can be expressed as the ratio of the expected cost per cycle to the expected length of the cycle. Other important assumptions can be found in Ben-Daya and Makhdoum (1998). They developed the main cost functions corresponding to the three PM policies. For each policy, they derived the quality control, inventory holding, and preventive maintenance costs. Policy 7.1 The process is inspected at times tl, t2, . . ., to assess the state of the process. At times tn, t2n, t3n, . . ., where n is a decision variable, the process is shut down, and both quality control inspections and PM tasks are carried out in parallel. Note that {tn, t2n, t3n, . . .}is a subset of {tl, t2, t3, . . .}. The expected total cost consists of the setup cost KS, the expected quality control cost E(QC), the expected inventory holding cost E(HC), and the expected preventive maintenance cost E(PM). Ben-Daya and Makhdoum (1998) derived expressions for both the expected quality control cost per cycle E(QC) and the expected cycle length E(T). The expected production cycle length includes (1) the expected time for inspection intervals when the process is in control, (2) the expected time for detecting the presence of an assignable cause, (3) the preventive maintenance time, and (4) the repair time. The expected quality control cost includes the expected cost of operating while in control with no alarm, the expected cost of false alarm, the expected cost of operating while out of control with no alarm, the repair cost, and the cost of sampling minus the salvage value for working equipment of age t. If PM is not carried out, it is assumed that the time for quality inspection is negligible. However, if PM activities are performed, then it is assumed that the expected time to perform a PM is proportional to the PM level and is given by:
 Z 1 ¼ Z max C PM =Cm 1 PM

ð7:1Þ

Theorem 7.1 In Policy 7.1, E(T ) is given by (Ben-Daya and Makhdoum 1998): EðT Þ ¼Z 2 þ "

m X j¼1

L j ð 1  pi Þ þ

m1 X j¼1

Z 1 j1

j1 Y

ð 1  pi Þ þ β

i¼1

m1
X   l j  Z 1i1 βij1 þ lm βmj1

#

m1 X j¼1

pj

j1 Y

ð 1  pi Þ

i¼1

i¼jþ1

ð7:2Þ where:

554

7 Maintenance

 Z1 j ¼

Z1 0

j ¼ n, 2n, 3n, . . . otherwise and for j ¼ m

ð7:3Þ

The quality control cost E(QC) is given by: ( E ðQCÞ ¼ðK n þ Cn nÞ 1 þ

j m2 Y X

ð1  pi Þ þ β

m2 X

j¼1 i¼1

pj

j¼1

j Y

ð1  pi Þ

mj2 X

i¼1

) β

i

i¼0

Q j1 Z j1 m m Y X X p j i¼0 ð1  p i Þ y j
 tf ðt Þdt þ ðCout  Cin Þ y jp j ð1  p i Þ þ ðCin  Cout Þ w j1 j¼1 F ðyi Þ  F w j1 j¼1 i¼1 " # j1 j1 m m m X Y X Y Y Lj ð1  pi Þ þ Ca pj ð1  pi Þ þ ð1  pi Þ þ Cin j¼1

þ Cout β

j¼1

i¼1

" m1 X

pj

j¼1

j1 Y

ð1  pi Þ

m X

i¼1

i¼1

#

Li βij1 þ αCf

i¼jþ1

i¼1

j m1 Y X

ð1  p i Þ 

j¼1 i¼1

m Y

ð1  pi ÞLðtm Þ

i¼1

ð7:4Þ The proof of this theorem is given in Appendix A of Ben-Daya and Makhdoum (1998). The total expected inventory holding cost, E(HC), is defined as: Z

T

EðHCÞ ¼ h

I ðt Þdt

ð7:5Þ

0

The integral in (7.5) is determined by computing the expected area E(A) under the function I(t). Hence: Z E ðHCÞ ¼ h

T

I ðt Þdt ¼ hE ðAÞ

ð7:6Þ

0

In order to compute E(A), the expression of the inventory levels at times tj + Z1j is required and is given by the following lemma: Lemma 7.1 Let Ij be the inventory level at time tj + Z1j, j ¼ 1, 2, . . ., m  1, Im be the inventory level at time tm, and then: Ij ¼ I

j1

þ ðP  DÞL j  DZ 1 j

j ¼ 1, 2, . . . , m

ð7:7Þ

where Z1j is given by (7.3), I0 ¼ 0, and if Ij < 0 it is set equal to zero.The proof of this lemma is clear from Fig. 7.2. The expected area E(A) under the function I(t) is given by the following theorem:

7.2 No Shortage

555

.

I(t)

..

Fig. 7.2 Inventory levels, where tPMj is the time at which jth PM is performed (Ben-Daya and Makhdoum 1998)

-D IPM3

P-D

Z1

Z1

Lj

t1 . . . tPM3

t

. . . tPM3 . . . tm

Production Cycle Inventory Cycle

Theorem 7.2 Let: 8 >  > < 2I

 Lj I þ j1 þ ðP  DÞL j 2 Uj ¼ >  Lj  > : 2I j1 þ ðP  DÞL j þ I 2 ¼ 1, 2, . . . , m

þ ðP  DÞL j 2D

j1

2 if I j ¼ 0

 Z1 j j1 þ ðP  DÞL j þ I j 2

for j

if I j > 0 ð7:8Þ

where Z1j is given by (7.3). Let B j ¼ I 2j =2D, j ¼ 1, 2, . . ., m. Then E(A) is given by: E ðA Þ ¼

m X j¼1

þβ

Uj

j1 Y

ð 1  pi Þ þ ð 1  β Þ

i¼1 m1 X j¼1

pj

j1 Y i¼1

" ð 1  pi Þ

m1 X

B jp j

j¼1 m X

i¼jþ1

β

ij1

j1 Y

ð 1  pi Þ

i¼1

Ui þ β

mj

# Bm þ Bm

m 1 Y

ð 1  pi Þ

i¼1

ð7:9Þ The proof of this theorem is given in Appendix B of Ben-Daya and Makhdoum (1998). In many preventive maintenance models, the system is assumed to be as good as new after each preventive maintenance action. However, a more realistic situation is one in which the failure pattern of a preventively maintained system changes. One way to model this is to assume that, after PM, the failure rate of the system is somewhere between as good as new and as bad as old. This concept is called imperfect maintenance and was introduced by many authors (Nakagawa 1980; Pham and Wang 1996). It can be assumed that the failure rate of the equipment is decreased after each PM. This amounts to a reduction in the age of the equipment. In this problem, Ben-Daya and Makhdoum (1998) assumed that the reduction in the

556

7 Maintenance

age of the equipment is a function of the cost of preventive maintenance CPM. Let yk(wk) denote the effective age of the equipment right before (right after) the kth PM. Let: γ k ¼ ηk1

C PM Cm PM

ð7:10Þ

where 0 < η  1. Linear and nonlinear relationships between age reduction and PM cost are considered. In the linear case: wk ¼ ð1  γ k Þyk

ð7:11Þ

and in the nonlinear case:
 wk ¼ 1  γ εk yk ,

0  > < 2I

 Lj I þ j1 þ ðP  DÞL j 2 Uj ¼ >    > : 2I j1 þ ðP  DÞL j L j þ I 2 ¼ 1, 2, . . . , m

j1

þ ðP  DÞL j 2D

2

 Z1 j j1 þ ðP  DÞL j þ I j 2

if I j ¼ 0

for j

if I j > 0 ð7:20Þ

Note that Z1j is given by (7.18) (Ben-Daya and Makhdoum 1998). The expected maintenance cost during a complete cycle is the same as (7.15) except that C PM j is redefined as follows:

558

7 Maintenance

 CPM j ¼

C PM


 if r t j  r max

0

otherwise and for j ¼ m

ð7:21Þ

The expected number of PMs nPM during the production cycle is given by (7.17). Note that the age of the equipment at time tj right before PM is given by (7.14) and right after the PM is given by: (
wj ¼

 1  γj yj

yj


 if r t j  r max otherwise

ð7:22Þ

Policy 7.3 The process is inspected at times tl, t2, . . ., to assess the state of the process. PM activities are performed only at those intervals at which two consecutive values of sample means fall in the warning zone. At those intervals, the process is shut down, and both quality control inspections and PM tasks are carried out in parallel. Under the assumptions described before and the requirements of Policy 7.3, the expected cycle length E(T) is the same as (7.1) except that Z1j is replaced by p2w Z 1j , where pw is the probability that a sample mean falls in the warning zone (Ben-Daya and Makhdoum 1998). The quality control cost E(QC) per cycle is exactly the same as (7.4). The total expected inventory holding cost, E(HC), is defined by (7.5), and the inventory levels are given by the following lemma. Lemma 7.3 Let Ij be the inventory level at time tj + Z1j, j ¼ 1, 2, . . ., m  1, Im is the inventory level at time tm, and then:
   I j ¼ 1  p2w I j1 þ ðP  DÞL j  p2w I ¼ 1, 2, . . . , m  1

j1

 þ ðP  DÞL j  DZ 1 ,

I m ¼ I m1 þ ðP  DÞlm

j ð7:23Þ ð7:24Þ

where I0 ¼ 0 and if Ij < 0 is equal to zero. The proof of this lemma is clear from Fig. 7.2. The expected area at a given inspection interval can be computed as (the probability of performing PM)  (inventory area if PM is performed) + (the probability of not performing PM)  (inventory area if PM is not performed):

7.2 No Shortage

559

8  2 !  Lj > I j1 þ ðP  DÞL j > 2 > pw 2I j1 þ ðP  DÞL j þ > > 2 2D > > > > >   > >
  Lj > > for I j ¼ 0, j ¼ 1, 2, . . . , m  1 > 2I j1 þ ðP  DÞL j þ 1  p2w > > 2 > <    Lj   Z1 Uj¼ > > 2I j1 þ ðP  DÞL j 2 þ I j1 þ ðP  DÞL j  I j 2 > > > >   >
>   Lj > 2 > > þ ð P  D ÞL 2I þ 1  p j1 j > for I j > 0, j ¼ 1, 2, . . . , m  1 w > 2 > > > >   > Lj : 2I j1 þ ðP  DÞL j for j ¼ m 2

ð7:25Þ The expected maintenance cost during a complete cycle is given by: EðPMÞ ¼ CPM nPM

ð7:26Þ

where nPM is the expected number of PMs. Since no PM is performed at time tm and the probability of performing PM at time tj, j ¼ 1, 2, . . ., m  1 is p2w , then nPM is given by: nPM ¼ ðm  1Þp2w

ð7:27Þ

Again the expected age of the equipment at a given inspection interval can be computed as (the probability of performing PM)  (age of the equipment if PM is performed) + (the probability of not performing PM) (age of the equipment if no PM is performed). So, the age of the equipment at time tj right before PM is given by (7.14) and right after the PM is given by:  w j ¼ p2w 1  η

7.2.1.1

j1


 CPM y j þ 1  p2w y j m CPM

ð7:28Þ

Integrated Model and Solution Method

The expected total cost per unit time is given by (Ben-Daya and Makhdoum 1998): ETC ¼

K S þ EðHCÞ þ E ðQCÞ þ E ðPMÞ E ðT Þ

The expression of E(T ) is given by:

ð7:29Þ

560

7 Maintenance

E ðT Þ ¼

P ðE ðT Þ  nPM Z 1 Þ D

ð7:30Þ

Next, they discussed the problem of solving the above integrated production, quality, and maintenance models to obtain the optimal values for the decision variables. Also, they discussed the way by which the frequency of sampling should be regulated and the optimization procedure used to determine the optimal design parameter values and the optimal preventive maintenance effort (Ben-Daya and Makhdoum 1998). The problem is to determine simultaneously the optimal production run time and hence the optimal EPQ, the optimal preventive maintenance level, and the optimal design parameters of the x-control chart, namely, Ll, L2, . . ., Lm, the sample size n, and the control limit coefficient k. In addition, the optimal values of 1, rmax, and warning limit coefficient for Policies 7.1–7.3, respectively, will be also determined simultaneously with the abovementioned decision variables. For a Markovian shock model, a uniform sampling scheme provides a constant integrated hazard over each interval. Banerjee and Rahim (1988) extended this fact to non-Markovian shock models by choosing the length of sampling intervals such that the integrated hazard over each interval is the same for all intervals, that is (Ben-Daya and Makhdoum 1998): Z

t

jþ1

Z r ðt Þdt ¼

tj

t1

r ðt Þdt

ð7:31Þ

0

Since the failure rate is reduced at the end of each interval because of PM activities, condition (7.31) becomes: Z

yj

w

Z

l1

r ðt Þdt ¼

r ðt Þdt,

j ¼ 2, . . . , m

ð7:32Þ

0

j1

If the time that the process remains in the in-control state follows a Weibull distribution, that is, its probability density function is given by: f ðt Þ ¼ λvt v1 eλt , v

t > 0, v  1, λ > 0

ð7:33Þ

Then using (7.32), the length of the sampling intervals Lj, j ¼ 2, . . ., n can be determined recursively as follows: Lj ¼




w

v j1

þ Lv1

1=v

w

j1 ,

j ¼ 2, . . . , m

ð7:34Þ

The problem is to determine the values of the decision variables m, n, k, L1, and CPM, which define the PM level, that minimize the expected total cost ETC. Recall that the age of the equipment after a PM is reduced proportional to the PM cost CPM.

7.2 No Shortage

561

The cost function is minimized using the pattern search technique of Hooke and Jeeves (1961). Example 7.1 Ben-Daya and Makhdoum (1998) presented an example with parameter values Z max ¼ 0.1, Z2 ¼ 1.0, Kn ¼ $2.0, Cn ¼ $0.5, Cf ¼ $500, Ca ¼ $1100, 1 Cin ¼ $50, Cout ¼ $950, δ ¼ 0.5 (δ is the amount of shift in the mean when the process is out of control, measured in standard deviations), L(tm) ¼ 100etm, D ¼ 1400 units, P ¼ 1500 units, h ¼ $0.1, KS ¼ $20, η j1 ¼ 0.99, λ ¼ 0.05, and ʋ ¼ 2. They used a computer program coded in Fortran implementing the Hooke and Jeeves (1961) procedure and run on a 486 PC to obtain the results presented below. The results of Policy 7.1 with L set equal to 1 obtained for different PM levels corresponding to C m PM ¼ $300 for both cases are summarized in Tables 7.3 and 7.4. These results illustrate clearly the trade-offs between PM levels and quality control costs. The increase in PM level yields reductions in quality control costs. With no PM, the quality control cost is $400.1. With a PM level of $100, the quality control cost is reduced to $276.1 and $235.9 for both Case I and Case II, respectively. The optimum PM level when C m PM ¼ $300 is obtained when CPM ¼ $300 (linear case), leading to a quality cost of $183.9 and an overall cost of $367.6 much less than without PM ($429.7). One might also notice that at low values of CPM, more reductions in both quality and overall costs are obtained for Case II than that for Case I. This is because for the same cost of PM, Case II yields much more reduction in the age of the equipment than Case I (Ben-Daya and Makhdoum 1998). PM activities also affect the economic production quantity (EPQ). As a matter of fact, it is noticeable from Tables 7.2 and 7.3 that PM does affect the production cycle (tm) since for higher PM levels we have longer production cycles. This is because when PMs are performed during the production cycle, quality control costs are reduced and hence longer procedure cycle will be still feasible. Some different numerical examples are provided by Ben-Daya and Makhdoum (1998) for other policies.

7.2.2

Imperfect Preventive Maintenance

Ben-Daya (2002) developed an integrated model for the joint determination of economic production quantity and preventive maintenance (PM) level for an imperfect process having a general deterioration distribution with increasing hazard rate. The effect of PM activities on the deterioration pattern of the process is modeled using the imperfect maintenance concept. It is assumed that after each PM, the age of the system is reduced proportional to the PM level. Consider a production process producing a single item. A production cycle begins with a new system which is assumed to be in an in-control state, producing items of acceptable quality. However, after a period of time in production, the process may shift to an out-of-control state. The elapsed time for the process to be in the in-control

CPM 0 100 200 300

m 26 37 38 29

N 49 56 64 70

k 2.51 2.51 2.49 2.47

L1 1.88 1.54 1.54 1.56

α 0.012 0.0121 0.0128 0.0136

1β 0.8387 0.8909 0.9344 0.9569 tm 4.3 7.3 10.3 13.4

QC 400.1 294.7 238.6 199.9

Table 7.3 Case I: Effect of PM level for linear improvement (Q* ¼ EPQ) (Ben-Daya and Makhdoum 1998) HC 25.4 24.7 17 13.3

PM 0 107.2 152.3 166.8

Q* 6502 10,539 14,717 18,987

ETC 429.7 429.3 409.7 381.6

562 7 Maintenance

CPM 0 100 200 300

m 26 24 22 29

n 49 53 64 70

k 2.51 2.53 2.49 2.47

L1 1.88 1.28 1.48 1.56

α 0.012 0.0114 0.0129 0.0136

1β 0.8387 0.8667 0.9349 0.9569 tm 4.3 9.5 11.2 13.4

QC 400.1 235.9 218.2 199.9

Table 7.4 Case II: Effect of nonlinear PM level for nonlinear improvement (Ben-Daya and Makhdoum 1998) HC 25.4 31.6 23.5 13.3

PM 0 103.4 138.2 166.8

Q* 6502 13,719 16,065 18,987

ETC 429.7 373 381.6 381.6

7.2 No Shortage 563

564

7 Maintenance

Table 7.5 Specific notations (Ben-Daya 2002) α τi tj Ni pj ETC

Percentage of non-conforming units produced when the process is in the out-of-control state (%) Detection delay during interval i (time) Pj Time of the jth PM, t j ¼ i¼1 L j (time) Number of non-conforming items produced (ti  1, ti) (item) The conditional probability that the process shifts to the out-of-control state during the time interval (tj  1, tj) given that the process was in control at time tj  1 Expected total cost (%)

Fig. 7.3 Inventory cycle; the jth PM is performed at time tj (Ben-Daya 2002)

I(t)

P-D

Lj

t1

t2

-D

. . . tj

. . . tm

t

Production Run Inventory Cycle

state, before the shift occurs, is a random variable assumed to follow a general distribution with increasing hazard rate. The process is inspected at times t1, t2, . . ., tm to assess its state, and at the same time, PM activities are carried out. The production cycle ends either when the system is out of control or after m inspection intervals whichever occurs first. The number m is a decision variable. The process is then restored to the in-control state and to the as good as new condition by maintenance and/or replacement. The usual assumptions of the classical EPQ model apply here. In particular, the demand is constant and continuous, and it must be met (Ben-Daya 2002). The specific notations which are used for this problem are presented in Table 7.5. The total expected cost per cycle consists of the setup cost, inventory holding cost, PM cost, inspection cost, cost of producing non-conforming items, and restoration cost. Before deriving these costs, let us determine the expected production cycle length (Ben-Daya 2002). The expected inventory cycle length is given by (see Fig. 7.3): E ðCTÞ ¼

P E ðT Þ D

ð7:35Þ

where E(T ) is the expected production run length which is given by the following lemma:

7.2 No Shortage Table 7.6 Associated probabilities (Ben-Daya 2002)

565 State In control Out of control

Probability 1 – p1 p1

Expected residual cost E(T1) 0

Lemma 7.4 The expected production cycle is given by (Ben-Daya 2002): E ðT Þ ¼

m X

Lj

j¼1

j1 Y

ð 1  pi Þ

ð7:36Þ

i¼1

Proof Let E(Tj) be the expected residual time in the cycle beyond tj given that the process was in control at time tj, E(T0) ¼ E(T). Then:

 F y j  F w j1
 pj ¼ F w j1

ð7:37Þ

In order to find the expression of E(T), consider the possible states of the process at the end of the first interval (at time t1 ¼ l1). For each possible state, the expected residual time in the cycle and the associated probabilities are presented in Table 7.6. Consequently, E(T) ¼ L1 + (1  p1)E(T1). Similarly, for j ¼ 1, 2, . . ., m  2, one has E(Tj) ¼ Lj + 1 + (1  pj + 1)E(Tj + 1). Note that E(Tm1) ¼ Lm; therefore (Ben-Daya 2002): E ðT Þ ¼

m X j¼1

Lj

j1 Y

ð 1  pi Þ

ð7:38Þ

i¼1

In this model, the setup cost is KS, and the inventory holding cost is hE (T )2(P  D)P/2D. As mentioned earlier, Ben-Daya (2002) used the concept of imperfect maintenance. After each PM, the age of the system is somewhere between as good as new and as bad as old depending on the level of PM activities. The reduction in the age of the equipment is a function of the cost of preventive maintenance. Let yk(wk) denote the effective age of the equipment right before (after) the kth PM. Similar to previous model of Ben-Daya and Makhdoum (1998), Ben-Daya (2002) used Eqs. (7.10), (7.11), (7.13), and (7.14) including γ k ¼ ηk1 CCPM m , wk ¼ (1  γ k)yk, y1 ¼ l1, and yj ¼ wj  1 + lj, j ¼ 2, . . ., m. PM

Now, let us turn to the problem of determining the expected preventive maintenance cost. Since the inspection is error-free and after each inspection PM activities are carried out, the expected PM cost per cycle, E(PM), is given by the following lemma:

566

7 Maintenance

Lemma 7.5 The expected PM cost per cycle is given by (Ben-Daya 2002): E ðPMÞ ¼ C PM

j m1 Y X

ð1  p i Þ

ð7:39Þ

j¼1 i¼1

It is assumed that no PM is carried out at the end of the last interval. The proof of this lemma is similar to that of Lemma 7.4. The expected number of inspections is equal to the number of PMs in addition to one inspection at the end of the cycle. Hence, inspection cost is E(IC) ¼ CI(nPM + 1). The inspection cost is separated from PM so that alternative PM policies can be considered. Here, a PM is carried out with each inspection. An alternative policy would be to inspect the system at l1, l1 + l2, . . . but perform PM only at a subset of these time epochs. The reader is referred to Ben-Daya and Duffuaa (1995) for alternative PM policies (Ben-Daya 2002). The expected number of non-conforming items during the jth interval is given by (Ben-Daya 2002):




Z

yj

E Nj ¼ w

αðyi  t ÞPf c ðt Þdt

j1





¼ αP y j F c y j  F c w

 j1

Z

yj

 w

! tf c ðt Þdt

ð7:40Þ

j1

where:
 f c ¼ f =F w j1
 F c ¼ F=F w j1

ð7:41Þ ð7:42Þ

The total expected number of non-conforming items per production run is given by the following lemma: Lemma 7.6 E ðN Þ ¼

m X

j1
Y p jE N j ð1  pi Þ

j¼1

i¼1

ð7:43Þ

Proof Let E(Bj) be the expected residual number of non-conforming items produced in the cycle beyond tj given that the process was in control at time tj, E(N ) ¼ E(Bo) (Ben-Daya 2002). In order to find the expression of E(N ), consider the possible states of the process at the end of the first interval (at time t1 ¼ l1). For each possible state, the expected residual time in the cycle and the associated probabilities are presented in Table 7.7.

7.2 No Shortage

567

Table 7.7 Associated probabilities (Ben-Daya 2002) State In control Out of control

Probability 1 – p1 p1

Expected residual number of non-conforming items E(B1) E(N1)

Consequently: E ðN Þ ¼ p1 EðN 1 Þ þ ð1  p1 ÞE ðB1 Þ

ð7:44Þ

Similarly, for j ¼ 1, 2, . . ., m – 2, one has
 E Bj ¼ p

jþ1 E




N

 jþ1


þ 1p

jþ1


E B



ð7:45Þ

jþ1

Note that E(Bm1) ¼ pmE(Nm). Therefore: E ðN Þ ¼

m X

j1
Y p jE N j ð1  pi Þ

j¼1

i¼1

ð7:46Þ

The total cost of producing non-conforming items per unit time is given by: E ðDCÞ ¼ αC S P (

m X

pj

j¼1

j1 Y

ð 1  pi Þ

i¼1





 y j Fc y j  Fc w

 j1

Z

)

yj

 w

tf c ðt Þdt

ð7:47Þ

j1

Since the restoration cost depends on the detection delay, the restoration cost during the jth interval is given by: Z E ðRCÞ ¼ w




yj

τðyi  t Þ f c ðt Þdt

j1

¼ CR0 þ CR1 y j

 



Fc y j  Fc w

 j1

Z  C R1 w

yj

! tf c ðt Þdt

ð7:48Þ

j1

because the restoration cost changes linearly with the detection delay as below:

 C R y j  t ¼ C R0 þ C R1 y j  t

ð7:49Þ

where CR0 and CR1 are some constants. The restoration cost per cycle is given by the following lemma:

568

7 Maintenance

Lemma 7.7 The expected restoration cost per cycle is given by (Ben-Daya 2002): EðRCÞ ¼

m X

pj

j¼1

"



j1 Y


1  pj



i¼1






CR0 þ CR1 y j F c y j  F c w

 j1

Z  CR1 w

yj

# tf c ðt Þdt

ð7:50Þ

j1

The proof of this lemma is similar to that of Lemma 7.6. Let E(QC) ¼ E(DC) + E(RC) denote quality-related costs. To solve the problem, recall that the expected total cost per unit time is given by (Ben-Daya 2002): ETC ¼

K S þ E ðICÞ þ E ðHCÞ þ EðQCÞ þ EðPMÞ E ðCTÞ

ð7:51Þ

where KS, E(IC), E(HC), E(QC), E(PM), and E(CT) are the setup cost, inspection cost, inventory holding cost, quality-related costs (cost of non-conforming items and restoration cost), PM cost, and the expected inventory cycle length, respectively. Next, Ben-Daya (2002) discussed the problem of solving the above integrated model to obtain the optimal values for the decision variables. Also, they would have discussed the way by which the inspection frequency should be regulated and the optimization procedure used to determine the optimal solution. The problem is to determine simultaneously the optimal lengths of the inspection intervals, namely, l1, l2, . . ., lm; the optimal PM level, CPM; and the number of inspections, m. For a Markovian shock model, a uniform inspection scheme provides a constant integrated hazard over each interval. Banerjee and Rahim (1988) extended this fact to non-Markovian shock models by choosing the length of inspection intervals such that the integrated hazard over each interval is the same for all intervals, as presented by Ben-Daya and Makhdoum (1998) in Eqs. (7.31) and (7.32), and Eqs. (7.33) and (7.44) are used too. The problem is to determine the values of the decision variables m, L1, and CPM, which define the PM level, that minimize the expected total cost ETC. Recall that the age of the equipment after a PM is reduced in proportion to the PM cost CPM. The cost function is minimized using the pattern search technique of Hooke and Jeeves (1961). However, due to the characteristics of the cost function, some modifications to the standard method have to be made to account for the inherent integrality constraint on the number of inspections. The optimal value of m  2 is determined by the inequalities ETC(m  1)  ETC(m) and ETC(m)  ETC(m + 1). Example 7.2 Ben-Daya (2002) presented numerical examples to illustrate important aspects of the developed integrated model. The process shift mechanism is assumed to follow a Weibull distribution. The Weibull scale and shape parameters are λ ¼ 5 and v ¼ 2.5, respectively. The following data are used for other parameters: D ¼ 500 units, P ¼ 1000 units, h ¼ $0.5, KS ¼ $150, CS ¼ $20, CI ¼ $10, CR0 ¼

7.2 No Shortage Table 7.8 Effect of PM on quality and total cost (Ben-Daya 2002)

569 α 0.2

0.4

CPM 0 0.5 1 0 0.5 1

M 9 10 5 8 10 5

Q 387 692 747 350 685 713

E(QC) 33.22 22.41 19.68 36.66 24.74 21.55

ETC 315.36 265.09 255.8 346.81 276.26 265.09

$10, CR1 ¼ 0.15, and ƞ ¼ 0.99. The relationship between PM cost, CPM, and the improvement in the age of the system will be used to investigate the effect of PM level on both quality control-related costs and total expected costs. The results ¼ 20 are summarized in obtained for different PM levels corresponding to C PM 0 Table 7.8, where Q is the lot size. These results illustrate clearly the trade-offs between PM levels and qualityrelated costs. The increase in PM level yields reductions in quality control costs. With no PM, the quality-related costs amount to $33.22 and $36.66 for α ¼ 0.2 and 0.4, respectively. With a PM level of 50%, the quality-related costs are reduced to $22.41 ($24.74). The optimum PM level when C m PM ¼ 20 is obtained when CPM ¼ $20, leading to quality-related costs of $19.68 (21.55) and an overall cost of $255.80 ($265.09) much less than without PM ($315.36 ($346.81)).

7.2.3

Imperfect Maintenance and Imperfect Process

Sheu and Chen (2004) developed an integrated model for the joint determination of both economic production quantity and level of preventive maintenance (PM) for an imperfect production process. This process has a general deterioration distribution with increasing hazard rate. The effect of PM activities on the deterioration pattern of the process is modeled using the imperfect maintenance concept. The production operation system is considered to produce a single item. A production cycle begins with a production system assumed to be in an in-control state: that is, the system produces items of acceptable quality. However, after a period of time in production, the process may shift to an out-of-control state. The elapsed time for the process in the in-control state, before the shift occurs, is a random variable that is assumed to follow a general distribution with increasing hazard rate. The process is inspected at times t1, t2, . . ., tm the state of the production system whether it is kept in the in-control state or not. At the same time,PM activities are carried out, but the production system has to stop. The production cycle will be stopped either when the system is transferred to the type II out-of-control state or after the mth inspection, whichever occurs first. The process is then restored to the in-control state and to the as good-as-new condition by a complete repair or replacement if necessary (Sheu and Chen 2004).

570

7 Maintenance

Table 7.9 Specific notations (Ben-Daya 2002) θ γk α1 α2 Cmr ðIÞ

Nj

ðIIÞ

Nj

nmr nPM

Probability of a type II out-of-control state when the system is out of control Imperfectness coefficient at the kth PM Non-conforming rate with type I out-of-control state Non-conforming rate with type II out-of-control state Cost of minimal repair by type I out-of-control state ($) Produced non-conforming items within (tj  1, tj) due to the type I out-of-control state (item) Produced non-conforming items within (tj  1, tj) due to the type II out-of-control state (item) The expected number of minimal repairs per production cycle (integer) The expected number of PMs per production cycle (integer)

Fig. 7.4 Inventory cycle; the jth PM is performed at time tj (Sheu and Chen 2004)

I(t)

P-D

Lj

t1

t2

-D

. . . tj

t

. . . tm

Production Run Inventory Cycle

The specific notations which are used for this problem is presented in Table 7.9. The total expected cost per cycle can be presented as below: TC ¼setup cost þ holding cost þ PM cost þ inspection cost þ quality‐related cost ðnonconforming items cost þ restoration costÞ ð7:52Þ Before deriving these costs, let us determine the expected production cycle length. The expected inventory cycle length is given by (see Fig. 7.4) (Sheu and Chen 2004): E ðCTÞ ¼

P E ðT Þ D

ð7:53Þ

where E(T ) is the expected production run length which is given by the following lemma (Sheu and Chen 2004):

7.2 No Shortage

571

Lemma 7.8 The expected production cycle is given by (Sheu and Chen 2004): E ðTÞ ¼

m X

Lj

j¼1

j1 Y

ð1  θpi Þ

ð7:54Þ

i¼1

Proof The related proof can be performed similar to Lemma 7.4. In this model, the setup cost is KS, and the inventory holding cost is hE (T )2(P  D)P/2D similar to the previous presented models. For preventive maintenance cost and inspection cost, Sheu and Chen (2004) followed Ben-Daya and Makhdoum (1998) and Ben-Daya (2002) and used γ k ¼ ηk1 CCPM m , wk ¼ (1  γ k)yk, PM

y1 ¼ l1, and yj ¼ wj  1 + lj, j ¼ 2, . . ., m in their model. Then to calculate the expected PM cost per cycle, E(PM), they presented Lemma 7.9. Lemma 7.9 The expected PM cost per cycle is given by (Sheu and Chen 2004): E ðPMÞ ¼ C PM

j m1 Y X

ð1  pi Þ þ ð1  θÞCmr

j¼1 i¼1

m1 X j¼1

pj

j1 Y

ð1  θpi Þ

ð7:55Þ

i¼1

It is assumed that no PM is carried out at the end of the last interval. The proof of this lemma is similar to that of Lemmas 7.4 and 7.8. Remark 7.1 nPM can be explained as the expected number of PMs per production cycle as below: nPM ¼

j m1 Y X

ð1  θpi Þ

ð7:56Þ

j¼1 i¼1

Remark 7.2 nmr can be explained as the expected number of minimal repairs per production cycle. nmr ¼ ð1  θÞ

m1 X j¼1

pj

j1 Y

ð1  θpi Þ

ð7:57Þ

i¼1

Moreover, the expected number of inspections is equal to the number of PMs in addition to one inspection at the end of the cycle. Hence, E(IC) ¼ CI(nPM + 1) (Sheu and Chen 2004). In order to calculate the production cost of non-conforming items, Lemma 7.10 is presented by Sheu and Chen (2004).

572

7 Maintenance

Lemma 7.10



Z ðIÞ E Nj ¼ w

   ð1  θÞf ðt Þ F ðt Þ θ α1 P y j  t 
1θ dt F w j1 j1


yj

ð7:58Þ

Proof For more detailed description, see Sheu and Chen (2004). Lemma 7.11

E

ðIIÞ Nj



Z

   θf ðt Þ F ðt Þ θ1 αII P y j  t 
θ dt F w j1 j1

yj

¼ w




ð7:59Þ

Proof For more detailed description, see Sheu and Chen (2004). The expected number

of non-conforming items during the jth interval is given by ð IÞ ðIIÞ E Nj and E N j . The total expected number of non-conforming items per production run is given by the following lemma (Sheu and Chen 2004): Lemma 7.12 E ðN Þ ¼

m h X

j1



i Y ð IÞ ðIIÞ ð1  θÞE N j þ θE N j ð1  θpi Þ pj

j¼1

ð7:60Þ

i¼1

where:  θ
 ð1  θÞf ðt Þ F ðt Þ αI P y j  t 
1θ dt w j1 F w j1  

Z yj
 θf ðt Þ F ðt Þ θ1 ðIIÞ αII P y j  t 
E Nj ¼ θ dt w j1 F w j1



Z ðIÞ E Nj ¼

yj

ð7:61Þ

ð7:62Þ

Proof For more detailed description, see Sheu and Chen (2004). The total cost of producing non-conforming items per unit time is given by Sheu and Chen (2004): E ðDCÞ ¼ C S E ðN Þ ¼ CS

j1 m h



i Y X ðIÞ ðIIÞ ð1  θÞE N j þ θE N j ð1  θpi Þ pj j¼1

ð7:63Þ

i¼1

To derive the restoration cost per production cycle, the following lemma is presented by Sheu and Chen (2004). Lemma 7.13 The expected restoration cost per production cycle is given by Sheu and Chen (2004):

7.2 No Shortage

E ðRCÞ ¼

m X

573

θp j

j¼1

0

j1 Y

ð1  θpi Þ

i¼1

2   @ CR0 þ CR1 y j 41 



 !θ 3  θ1 1 Z yj F yj θf ðt Þ F ðt Þ
 5  r1 t 
θ Adt F w j1 w j1 F w j1 ð7:64Þ

where CR0 and CR1 are similar to what is presented in the work of Ben-Daya (2002). Proof For more detailed description, see Sheu and Chen (2004). Remark 7.3 The quality-related cost E(QC) ¼ E(DC) + E(RC) (Sheu and Chen 2004). Remark 7.4 There are several special cases in the integrated model. For example, when m ¼ θ ¼ 1, this is the classical economic production quantity model. Another case was considered by Ben-Daya (2002) when θ ¼ 1 (Sheu and Chen 2004). The expected total cost is composed of the setup cost, inspection cost, inventory holding cost, quality-related costs (i.e., cost of non-conforming items, restoration cost), and PM cost. For a renewal reward process (Ross 1996), one has the expected total cost per expected cycle length as follows (Sheu and Chen 2004): ETC ¼

K S þ E ðICÞ þ E ðHCÞ þ EðQCÞ þ EðPMÞ E ðCTÞ

ð7:65Þ

The problem is to determine simultaneously the optimal lengths of the inspection intervals, namely, L1, L2, . . ., Lm; the optimal PM level, CPM; and the number of inspections, m (Sheu and Chen 2004). For a Markovian shock model, Sheu and Chen (2004) used Eqs. (7.31)–(7.34) of Ben-Daya and Makhdoum (1998). As mentioned before, the problem is to determine the values of the decision variables m, L1, and CPM, which define the PM level, that minimize the expected total cost ETC. Recall that the age of the equipment after a PM is reduced in proportion to the PM cost CPM. The cost function is minimized using the pattern search technique of Hooke and Jeeves (1961). However, due to the characteristics of the cost function, some modifications to the standard method have to be made to account for the inherent integrality constraint on the number of inspections. The optimal value of m  2 is determined by the inequalities ETC(m  1)  ETC(m) and ETC(m)  ETC(m + 1). Therefore, the optimal value m* and L1 can be obtained by the following procedure if the level of PM is determined (Sheu and Chen 2004). Step 1. Estimate m0, the maximum number of inspections undertaken during each production run, either from historical records or from the condition of production. Step 2. For m ¼ 1, one can search an optimal value l1 subject to the expected total cost ETC1.

574

7 Maintenance

Step 3. One has the expected total cost from ETC2 to ETCm0 by repeating Step 2 for m ¼ 2, 3, . . ., m0. Step 4. The optimal values l1 and m satisfy the following condition:
  ETC L1 , m ; C PM ¼ Min ETC j , j ¼ 1, . . . , m0 Example 7.3 Sheu and Chen (2004) presented numerical examples to illustrate important aspects of the developed integrated model. The process shift mechanism is assumed to follow a Weibull distribution similar to the previous cases with parameters λ ¼ 5 and v ¼ 2.5. They considered θ ¼ 0.1, KS ¼ $150, D ¼ 500 units, P ¼ 1000 units, h ¼ $0.5, CS ¼ $20, CI ¼ $10, CR0 ¼ $10, CR1 ¼ $0.15, ƞ ¼ 0.99, αI ¼ 0.2, αII ¼ 0.4, C m PM ¼ 20, and Cmr ¼ 10 (Sheu and
Chen 2004). The resultsfor no PM under the setup cost KS ¼ $150 is L1 , m , Q , EðTCÞ, C PM ¼ ð0:308, 3, 472:42, 300:90, 0Þ, and the optimal value of PM obtained is 20 leading
cost  to the expected total cost amount L1 , m , Q , EðTCÞ, C PM ¼ ð0:261, 3, 768:03, 262:63, 20Þ.

7.2.4

Aggregate Production and Maintenance Planning

Aghezzaf et al. (2007) developed an aggregate production and maintenance planning problem in which a set of items must be produced in lots on a capacitated production system throughout a specified finite planning horizon. They assumed that the production system is subject to random failures and that any maintenance action carried out on the system, in a period, reduces the system’s available production capacity during that period. The objective is to find an integrated lot-sizing and preventive maintenance strategy of the system that satisfies the demand for all items over the entire horizon without backlogging and which minimizes the expected sum of production and maintenance costs. At the aggregate planning level, the only work they were aware of is that of Weinstein and Chung (1999). They proposed a three-part model to evaluate an organization’s maintenance policy. In their approach, an aggregate production plan is first generated, then a master production schedule is developed to minimize the weighted deviations from the goals specified at the aggregate level, and finally work center-loading requirements are used to simulate equipment failures during the planning horizon. We have used several experiments to test the significance of various factors for maintenance policy selection. These factors include the category of maintenance activity, maintenance activity frequency, failure significance, maintenance activity cost, and aggregate production policy. The fundamental difference between Weinstein–Chung’s approach and ours lies in the fact that our model takes, explicitly, into consideration the reliability parameters of the system at the early stage of the planning process, that is, when the aggregate plan is to be developed. As

7.2 No Shortage

575

Table 7.10 Specific notations (Ben-Daya 2002) H ¼ Nτ N τ Dit Cmax Lp Lr ρi Kit Cit hit Cr Qit Iit yit T

Planning horizon (time) Number of periods (integer) Each period fixed length (time) Demand rate of ith item should be satisfied in each period t (item/period) Production system nominal capacity (item) Capacity units consumed for each planned preventive (item) Capacity units consumed for each unplanned maintenance (item) The processing time for each unit of product i (time) Fixed cost of producing item i in period t ($/setup) Variable cost of producing one unit of item i in period t ($/item) Variable cost of holding one unit of item i by the end of period t ($/unit/unit of time) Cost to carry out a corrective maintenance action (CPM < Cr) ($) Quantity of item i produced in period t (item) Inventory of item i at the end of period t (item) Binary variable (yit equals to 1 if item i is produced in period t and 0 otherwise) Preventive maintenance cycle (time)

a consequence, the chances that their model performs better at the simulation phase are higher (Aghezzaf et al. 2007). The specific notations which are used for this problem is presented in Table 7.10. They are given a planning horizon including N periods of fixed length and a set of products P to be produced during this planning horizon. They assumed that the production system has a known nominal capacity and that each maintenance action consumes a certain percentage of this capacity. Thus, they assumed that each planned preventive and unplanned maintenance action consumes, respectively, Lp ¼ aCmax and Lr ¼ bCmax capacity units (with 0  a  b  1). Note that the assumption a  b may be justified by the fact that more capacity resources may be consumed in case of a random failure since some offline activities for repair must in this case be accomplished online. Finally, it is assumed that the failure probability density function f(t) and the cumulative distribution function F(t) of the production system are known. They let r(t) be the failure rate of a system at time t. It is well known that r(t) is given by Aghezzaf et al. (2007): r ðt Þ ¼

f ðt Þ 1  F ðt Þ

ð7:66Þ

The maintenance policy suggests to replace the production system at predetermined instances T ¼ kτ, 2 kτ, 3kτ, . . . and to carry out a minimal repair whenever an unplanned failure occurs. All maintenance actions are supposed to be perfectly performed (Aghezzaf et al. 2007):

576

Minimize

7 Maintenance

  Z T XX Nτ ðK it yit þ Cit Qit þ hit I it Þ þ C PM þ Cr r ðt Þdt ð7:67Þ T 0 t2H i2P s:t: Qit þ I it1  I it ¼ Dit ! X Qit  Dis yit

for t 2 H, i 2 P for t 2 H, i 2 P

ð7:68Þ ð7:69Þ

s2H, st

X

ρi Qit  C ðt Þ for t 2 H

ð7:70Þ

i2P

Qit , I it , T  0;

yit 2 f0, 1g

for t 2 H, i 2 P

ð7:71Þ

where the function C(t) defines the available capacity in each period t. This capacity C(t) is given by (Aghezzaf et al. 2007): Z C ðt Þ ¼ C max  Lp  Lr

τ

r ðu þ ðt  1ÞτÞdu

ð7:72Þ

0

if the preventive maintenance takes place in period t. For other period t, the capacity C(t) is given by: Z C ðt Þ ¼ Cmax  Lr

τ

r ðu þ ðt  1ÞτÞdu

ð7:73Þ

0

To solve the above mathematical programming problem (PPM), the length of the planning horizon H is assumed, and the length of the preventive maintenance cycle T is given in multiples of the basic planning period duration τ (i.e., H ¼ Nτ and T ¼ kτ). Let nI ¼ [N/k] if the ratio N/k is integer and nI ¼ t[N/k] + 1 otherwise (where [N/k] is the highest integer smaller or equal than N/k). The maintenance and planning model (PPM) can now be rewritten as follows (Aghezzaf et al. 2007):

Minimize

Z ðk Þ ¼

nI X n¼1

0 @C PM þ

nk X t¼ðn1Þkþ1, tN

Z

τ

Cr

r ðu þ ðt  ðn  1Þk  1ÞτÞ

0

þ

X

! ðK it yit þ C it Qit þ hit I it Þ

i2P

ð7:74Þ s:t: : Constraints Eqs: (7.72) and (7.73), and :

7.2 No Shortage

577

Table 7.11 Products’ periodic demand (Aghezzaf et al. 2007)

X i2P

( ρI Qit  Cðt Þ ¼

Period 1 2 3 4

D1t 2 3 2 3

D2t 3 2 3 2

Period 5 6 7 8

D1t 2 3 2 3

D2t 3 2 3 2

Rτ C max  LP  Lr 0 r ðu þ ðt  1ÞτÞ if t ¼ ðn  1Þk þ 1 Rτ C max  Lr 0 r ðu þ ðt  1ÞτÞ if t ¼ ðn  1Þk þ 2  t  nk ð7:75Þ

The decision variables remain, for each product i and each period t, Qit, Iit, and yit together with the variable k which defines the optimal length of the preventive maintenance cycle T (T ¼ kτ). To determine the optimal values of production plan and the length T of the maintenance cycle, the following procedure has been used (Aghezzaf et al. 2007). Step 1. Based on the value of k, determine nI and the corresponding maintenance cost function terms, and then determine the available capacities Ck(t) in period t. Step 2. Solve the resulting pure production planning problem (PPMr for fixed values of k) using any selected algorithm. In this case, the production planning problem using the mixed integer solver of CPLEX is solved. Step 3. Compare the resulting values Z(k), and select for the optimal preventive maintenance period size of the value k* such that Z(k*) ¼ mink{Z(k)}. The production plan associated with Z(k*) is selected as the final production plan (Aghezzaf et al. 2007). Example 7.4 Aghezzaf et al. (2007) presented an example in which planning horizon composed of 8 production periods, each with an available maximal capacity of Cmax ¼ 15. Two products are to be produced in lots so that the demands are satisfied. They assumed hit ¼ 2, Cit ¼ 5, and hit ¼ 2 for i ¼ 1, 2. Also Table 7.11 shows the setup, production, and holding costs for each product and the periodic demands of each product, respectively. Table 7.12 shows the optimal plan for the two products without taking into account the capacity lost in maintenance (assuming that the system will not fail and does not require any preventive maintenance). The total cost for the optimal production plan is equal to 417 (Aghezzaf et al. 2007). Now, one considers the preventive maintenance model with minimal repair at failure as the selected maintenance strategy of the production system with the following parameters: the cost of a preventive maintenance action is set to CPM ¼ $28, and the cost of minimal repair action at failure is given by Cr ¼ $35. Aghezzaf et al. (2007) assumed that the system lifetime is distributed according to Gamma distribution with the parameters G(α ¼ 2, λ ¼ 1).

578

7 Maintenance

Table 7.12 Optimal production plan with maximum capacity (Aghezzaf et al. 2007)

Table 7.13 Specific notations (Salameh and Ghattas 2001)

7.3 7.3.1

Qit Products i 1 2 5 10 0 0 7 0 0 0 0 10 8 0 0 0 0 0

Periods t 1 2 3 4 5 6 7 8

Q T t P D h B(t)

Iit 1 3 0 5 2 0 5 3 0

yit 2 7 5 2 0 7 5 2 0

1 1 0 1 0 0 1 0 0

2 1 0 0 0 1 0 0 0

Buffer stock level (item) Running time of the production unit per cycle (time) Preventive maintenance time per cycle (time) Buffer replenishment rate (units/unit time) (item/year) Consumption rate from the buffer during t (units/year) Buffer holding cost ($/unit/unit time) Shortage per preventive maintenance cycle t (item)

Backordering Preventive Maintenance

Preventive maintenance, an essential element of the just-in-time structure, induced the idea of this problem. Performing regular preventive maintenance results in a shutdown of the production unit for a period of time to enhance the condition of the production unit to an acceptable level. During such interruption, a just-in-time buffer is needed so that normal operations will not be interrupted. The optimum just-in-time buffer level is determined by trading off the holding cost per unit per unit of time and the shortage cost per unit of time such that their sum is minimum (Salameh and Ghattas 2001). The specific notations which are used for this problem is presented in Table 7.13. The main goal of Salameh and Ghattas (2001) was to determine the optimum justin-time buffer level to withstand the regular preventive maintenance interruption. The following assumptions are applied throughout this problem: • The just-in-time buffer is not subject to deterioration or obsolescence. • The regular preventive maintenance guarantees that the probability of a breakdown of the production unit during T is approximately zero. • Before the beginning of any normal preventive maintenance, the just-in-time buffer is Q. • T is large enough compared with t, so that during any time period T, buffer replenishment starts from a zero level.

7.3 Backordering

579

Q D Q/D t Prev. Maint.

D

P Q/P T Running Time

Q/D Time

t Preventive Maintenance

Fig. 7.5 Buffer level behavior (Salameh and Ghattas 2001)

• Unused buffer inventory during t is depleted to zero during the next cycle T. The behavior of the system is depicted in Fig. 7.5. Defining I(t) to be the average inventory level during the period (t + T ), then:

I ðt Þ ¼

Q2 Q2 þ 2D 2P

h

1 tþT

i

ð7:76Þ

Therefore, the expected average inventory per cycle of length (t + T ) and hence per unit of time will be: Z 1 h i

2 f ðt Þ P þ D Q2 1 Q Q2 dt ¼ E þ E ½ I ðt Þ ¼ PD tþT 2 2D 2P 0 t þ T

ð7:77Þ

where:

Z 1 f ðt Þ 1 dt ¼ tþT 0 tþT

E

Therefore, the average buffer carrying cost will be: hQ2 ðD þ PÞ 1

E 2PD tþT If the buffer supply time Q/D is less than the preventive maintenance time, then the stock out time will be t  Q/P; otherwise, the stock out time will be zero, i.e.:

580

7 Maintenance

stock out time ¼

8 >

D t

> :t  Q D

Therefore, the shortage per preventive maintenance cycle can be expressed as:

Bðt Þ ¼

8 >

D t



> : D t  Q ¼ Dt  Q D

ð7:78Þ

The expected shortage per preventive maintenance cycle t is: Z E½Bðt Þ ¼ D



Q f ðt Þdt t D Q=D 1

ð7:79Þ

The expected cycle length is: E ðt þ T Þ ¼ T þ E ðt Þ

ð7:80Þ

Therefore, the expected number of cycles per unit time is 1/T + E(t), and the expected number of units short per unit time will be: E ½Bðt Þ D ¼ E ðt þ T Þ T þ E ðt Þ

Z

1

t

S=D

Q f ðt Þdt D

ð7:81Þ

Hence, the storage cost per unit time is: Cb

D T þ E ðt Þ

Z

1 Q=D



t

Q f ðt Þdt D

Let TCU(S) be the expected total cost per unit time. TCU(S) can be expressed as: TCUðQÞ ¼

hQ2 ðP þ DÞ 1

Cb D E þ 2PD T þt T þ E ðt Þ

Z

1



t

Q=D

Q f ðt Þdt D

ð7:82Þ

Differentiating Eq. (7.82) with respect to the buffer stock level Q yields to:

dTCUðQÞ hQ 1 Cb D ¼ ðP þ DÞE  dQ PD T þt T þ E ðt Þ

Z

1

Q=D



t

Q f ðt Þdt D

ð7:83Þ

7.3 Backordering

581





d2 TCUðQÞ hQ 1 Cb D Q ð P þ D ÞE þ f ¼ 2 PD T þ t D D ð T þ E ð t Þ Þ dQ

ð7:84Þ

d2TCU(Q)/dQ2  0 for all Q  0. Hence, TCU(Q) admits a unique minimum at Q ¼ Q*, where:  dTCUðQÞ ¼0 dQ Q¼Q or: Z

1

Q=D

f ðt Þdt ¼

hQðP þ DÞðT þ E ðt ÞÞ 1

E C b PD T þt

ð7:85Þ

Example 7.5 To illustrate the use of the model developed in previous section, consider a situation where T ¼ 30 days, P ¼ 15,000 units/year, D ¼ 30,000 units/ year, h ¼ $28.00/unit/year, and Cb ¼ $1.00/unit. Preventive maintenance time t is uniformly distributed over the interval [a, b], with a ¼ 1 day and b ¼ 3 days. Therefore:  f ðt Þ ¼

1=2 0

1t3 otherwise

The total cost function in Eq. (7.82) reduces to:   hð P þ D Þ T þb 2 2C b D TCUðSÞ ¼ Q þ ln T þa ð2T þ a þ bÞðb  aÞ 2PDðb  aÞ  2  b b Q2   Qþ 2 2 D 2D

ð7:86Þ

The optimal value of Q that minimizes the above equation is Q* ¼ 170.44 units, with TCU(Q*) ¼ $671.31. For Q ¼ 0 (no just-in-time buffer), the expected total annual cost is TCU(0) ¼ $2109.38. The minimum just-in-time buffer level that reduces the probability of stock out during preventive maintenance to zero can be obtained by setting Eq. (7.79) equal to zero and solving for S. In this example, this minimum value of Q is 250 units. The corresponding expected annual cost is $984.70. The above results are summarized in Table 7.14.

582

7 Maintenance

Table 7.14 Results of example No buffer Optimal buffer level Minimum buffer such that the probability of stock out is zero

7.4 7.4.1

Q 0 170.44 250

TCU(Q) $2109.38 $671.31 $984.70

Partial Backordering Preventive Maintenance

Taleizadeh (2018) developed a multi-product single-machine economic production quantity model with preventive maintenance and scraped and rework process. Shortages are permitted and a fraction of them is backlogged. Capacity and service level are limitations of the production system. It is assumed that preventive maintenance can be performed when the inventory level is positive or negative. Indeed, two different scenarios are modeled, and according to the comparisons between their costs, a new scenario according to the best time of preventive maintenance is investigated and modeled. The aim of this problem is to determine the best time for preventive maintenance, production, and backordered quantities of each item and common cycle length, such that the expected total cost is minimized. The more details of this study can be found in Sect. 5.5.5 of Chap. 5.

7.5

Conclusion

In this chapter, six inventory models with maintenance are presented. These models consider the traditional production problem in which preventive maintenance is considered. All models are categorized in three sections about the type of shortages including no shortage, backordering, or partial backordering. Preventive maintenance, imperfect preventive maintenance, imperfect maintenance and production, aggregate production, and maintenance planning are the main topics investigated and differ models in this chapter.

References Aghezzaf, E. H., Jamali, M. A., & Ait-Kadi, D. (2007). An integrated production and preventive maintenance planning model. European Journal of Operational Research, 181(2), 679–685. Ben-Daya, M. (2002). The economic production lot-sizing problem with imperfect production processes and imperfect maintenance. International Journal of Production Economics, 76(3), 257–264. Banerjee, P. K., & Rahim, M. A. (1988). Economic design of control charts under Weibull shock models. Technometrics, 30(4), 407–414.

References

583

Ben-Daya, M., & Duffuaa, S. O. (1995). Maintenance and quality: the missing link. Journal of Quality in Maintenance Engineering, 1(1), 20–26. Ben Daya, M., & Makhdoom, M. (1998). Integrated production and quality model under various preventive maintenance policies. Journal of the Operational Research Society, 49, 840–853. Hooke, R., & Jeeves, T. A. (1961). “Direct Search'” solution of numerical and statistical problems. Journal of the ACM (JACM), 8(2), 212–229. Nakagawa, T. (1980). A summary of imperfect preventive maintenance policies with minimal repair. RAIRO-Operations Research, 14(3), 249–255. Pham, H., & Wang, H. (1996). Imperfect maintenance. European Journal of Operational Research, 94(3), 425–438. Ross, S. M. (2013). Applied probability models with optimization applications. Chelmsford: Courier Corporation. Ross, S. M. (1996). Stochastic processes (2nd ed.). New York: Wiley. Salameh, M. K., & Ghattas, R. E. (2001). Optimal just-in-time buffer inventory for regular preventive maintenance. International Journal of Production Economics, 74, 157–161. Sheu, S. H., & Chen, J. A. (2004). Optimal lot-sizing problem with imperfect maintenance and imperfect production. International Journal of Systems Sciences, 35, 69–77. Taleizadeh, A. A. (2018). A constrained integrated imperfect manufacturing-inventory system with preventive maintenance and partial backordering. Annals of Operations Research, 261, 303– 337. Weinstein, L., & Chung, C. H. (1999). Integrating maintenance and production decisions in a hierarchical production planning environment. Computers & Operations Research, 26(10–11), 1059–1074.

Index

A Aggregate planning, 574 Aggregate production, 574, 575, 582 Algebraic manipulation, 121 Annual rework cost, 452 Annual total inventory, 387 Arithmetic–geometric mean inequality method, 186–188 Auction, 385–390 Average profit, 116

B Backorder case, 33 Backorder cost (BC), 94, 98, 101, 145 Backordering and rework (see Rework and backordering) backorder costs, 261 cost function, 267 defective products, 268 EPQ model, 261 imperfect quality and inspection, 70–74 imperfect-quality items, 109–112 inventory average, 263 learning in inspection, 101–109 mathematical expression, 263, 267 multiple quality characteristic screening, 74–82 notations, 70, 71 partial (see Partial backordering) partial derivatives, 266 preventive maintenance, 578–580 rejection, 82–87 second partial derivatives, 265 solution, 270

© Springer Nature Switzerland AG 2021 A. A. Taleizadeh, Imperfect Inventory Systems, https://doi.org/10.1007/978-3-030-56974-7

total cost function, 264 Backordering cost (BC), 90, 101, 102, 109 and lost sale costs, 113 Backordering level, 274 Backordering, multi-product single-machine EPQ model defective items annual production cost, 402 annual shortage cost, 405 data, 408, 409 joint production policy, 403–405 normal distribution, 410 objective function, 402, 405, 407 optimal production quantity, 401 production cost per unit, 402 production–inventory cycle, 403 production rate, 401, 407 scrapped, 401, 404 shortages, 401 total production and setup time, 406 uniform distribution, 409 interruption, manufacturing process (see EPQ inventory model with interruption, manufacturing process) multidefective types (see Multidefective types) repair failure, 443, 444, 446–448 rework proces, 433–441 Backordering period, 118 Backorders, 30, 34, 36 Batches imperfect-quality, 13 maximum purchasing price, 38 multiple, 13 received, 37

585

586 Batches (cont.) rejection of defective supply batches, 16–18 Behavior of inventory level, 93 Beta-binomial distribution, 51 Binomial distribution, 49, 51 Buy and repair options, 56–61 Buyer makes decisions, 50 Buyer’s purchasing price, 500 Buyer’s selling price, 37, 39–42, 500

C Capacity constraint model, 321, 384, 439, 447, 454, 466, 475, 488 Classical EOQ model, 244 Commodity, 62 Commodity flow, 62, 68, 69 Complete backordering, 124 Computer numerical control (CNC), 492 Continuous delivery, EPQ models with scrap cyclic inventory cost, 158 expected annual cost, 158 fully backordered arithmetic–geometric mean inequality method, 186–189 designer window treatments manufacturer, 199, 200, 202 random breakdown, 196, 197, 200, 204, 207, 208 random defective rate, 189–191, 193–199 HGA vs. GAMS software, 168 imperfect production process, 155 imperfect-quality items, 156 manufactured product, 159 MINLP problem, 164, 166 multi-product and multi-machine, 160–164 on-hand inventory, 158 optimal production quantity, 159 probability density function, 158 production downtime, 157 production rate, perfect-quality items, 156 production uptime, 157 proposed HGA, 165, 168 solution procedures, 157 with partial backordered shortage, 223–230 Convex nonlinear programming problem (CNLPP), 389, 469 Corrective maintenance, 575 Cost-efficiency, 125 Cost equation, 292 Cost function, 63 Cumulative number, 28 Cutting plane method, 380, 381 Cycle duration, 100

Index Cycle time, 93, 96 Cyclic backordering cost, 482 Cyclic inventory-related cost, 84

D Defective and defective delivery occurrences, 82 and deterioration costs, 20 fraction, 74 and good items, 18, 24 HC, 25, 27, 78 inventory level, 19, 57, 60, 76, 77 learning curve, 101 non-inspected items, 53 number and portion, 75 percentage, 7, 70, 71 produced items, 17 random proportion, 10 supply batches, 82–87 uninspected items, 50 Defective items, 311, 313, 383, 435, 443, 501 cost expression, 257 on-hand inventory, 255, 256 production, 255, 258 reworked, 259 Defective products, 370, 385, 392 decision variables, 268 inventory level, 269 notation and analysis, 271 optimal backordering level, 276 optimal total cost, 273 production–inventory model, 269 production time, 272 total cycle time, 271 Demand rate, 1 Deterioration, 8, 10, 18, 19, 23 Different values, 107 Discrete delivery, EPQ models with scrap machine breakdown buyer’s cost, 177, 178, 180, 181 buyer’s inventory level, 173 cost of transportation, 173 expected production–inventory cost per cycle, 180 on-hand inventories, perfect and defective vendor items, 173 optimal run time, 185 random variable, 173 stochastic breakdown and multiple shipments, 173 total cost function, 185 vendor’s cost, 170, 173–176, 179 with and without breakdowns, 175, 181, 182, 184

Index machine failures, 185 multi-delivery algebraic approach, 170 finished items, 166 integer number, 172 optimal number of shipments, 172 optimal replenishment lot size, 172 random scrap rate, 172 randomness of scrap rate, 169 supplier’s inventory holding during delivery time, 168 total holding costs, 169 total production–inventory–delivery costs per cycle, 166 produced items, 166 vendor–buyer integrated EPQ model with scrap, 166 Discrete shipments, 153, 154 Disposal cost, 18 Distribution function, 50, 87, 294

E Economic frequency, 373 Economic lot size model, 1 Economic manufacturing quantity (EMQ), 312 Economic operating policy, 375 Economic order quantity (EOQ) model, 2, 4, 50, 147 assumptions, 498 buy and repair options, 56–61 cost function, 62 formulation, 7 holding and ordering costs, 153 inspection shifts from buyer to supplier, 37–42 inventory models, 153 learning in inspection, 506 maintenance, 16–18 partial backordering, 506–510 quality levels with backordering, 502–505 reworkable items, 153 robust production–inventory model, 153 sampling inspection plans, 43–50 scrapped items, 153, 155 traditional models, 497 with no return without shortage, 502 Economic policy, 374 Economic production quantity (EPQ), 92, 99, 324, 561 breakdown, 1 economic lot size model, 1 inventory models, 2, 3 manufacturing system, 2 multi-product single machine, 3, 4

587 production model, 367 rework process, 3 single machine, 367 single-product single-stage manufacturing system, 367 traditional model, 1 End-of-cycle backorders, 84 End-of-cycle inventory, 82 Entropy cost, 64–65, 68, 69 Entropy EOQ with screening, 65–70 without screening, 61–65 EOQ imperfect system, 506 EOQ model with return defective quality levels and partial backordering, 525–530, 533, 534 Hessian matrix, 519 holding costs per cycle, 517 holding costs, defective items, 520 imperfect inspection process, 522–525 inspection and sampling, 515–517 NBBARY algorithm, 522 purchase cost per cycle, 520 EPQ inventory model with interruption, manufacturing process annual joint production policy, 429 backlogging situation, 430 backorder-filling stage, 421 data, 432, 433 decision variables, 419, 431 defective products, 419 holding cost, 423–426 shortage cost, 424 imperfect products, 419, 422 inventory level, perfect-quality items, 421 inventory system cost, 421 no shortages, 421, 422, 424, 426 objective function, 431 on-hand inventory defective items, 426 scrapped items, 427 optimal solution, 431 production capacity, 429 production cycle length, 422 random defective rate, 421 reasons, 421 rework process, 425 shortage cost, 423 shortages, 419, 422, 426 total cost function per cycle, 428 total defective products, 427 total objective cost function, 427 total scrap products, 427 EPQ inventory models, 2, 3 EPQ inventory system, 329

588 EPQ model, 235, 248 categories, 236 classical, 237 EOQ, 237, 238 GP, 238 notations, 236, 237 Porteus’s model, 238 EPQ model with return continuous quality characteristic without shortage, 537–542 inspections errors without shortage, 542–546 sensitivity analysis, 542 EPQ model without return quality assurance without shortage, 535, 536 quality screening and rework without shortage, 536 Equipment condition, 549 Equivalent number, 28, 29 Error type I, 500, 501, 515, 516, 522 Error type II, 500, 515–517, 522 Expectation value, 124 Expected cost, 82 Expected cycle time, 36 Expected profit per cycle, 37 Expected total cost (ETC), 305 Expected total profit (ETP), 240 Expected total profit per time unit, 38 Extended cutting plane method, 379, 380

F Failure rate, 555, 557, 560, 575 Fill rate, 118, 123 Finite planning horizon model, 106 Finite production model, 307 First-order condition, 123 First-order derivatives, 85, 106 Fixed backorder cost, 261 Fixed cost, 87, 88 Fixed lifetime, 8 Fixed setup, 89, 94, 97 Fixed transportation cost, 9, 56, 57, 75, 87, 88, 126, 128

G General and specific data, 324 Geometric programming (GP) approach, 238 Geometric random variable, 84 Geometric series, 45 Goodwill cost, 10, 18, 23–27, 72, 77, 81

Index H Harmony search algorithm (HS), 379 Heat flow, 62 Hessian matrix, 54, 91, 92, 95, 96, 111, 112, 117, 311, 316, 322, 330, 407, 431, 447, 456, 469, 491 Holding cost (HC), 10, 11, 13, 15, 18, 20, 23–28, 30, 31, 35, 36, 51, 53, 57, 58, 63, 77, 78, 87, 88, 90, 94, 98, 101, 102, 114, 127–129, 131, 134, 137, 140 order quantities, 27 with learning effects, 27 without learning effects, 27 Hypergeometric distribution, 51

I Imperfect EOQ model categories, 7, 8 deterioration, 8 HC, 10 imperfect quality, 10–15 inventory control purposes, 9 JIT manufacturing environment, 10 lifetime constraints, 8 notations, 7, 9 obsolescence, 8 perishability, 8 probability distribution, 10 random proportion, 10 real manufacturing environment, 10 vendor–buyer inventory policy, 10 Imperfect inventory system, 56 Imperfect item sales, 242 differentiating and equating, 243 inspection process, 239 inventory level per cycle, 240, 241 production period, 243 Imperfect maintenance expected PM cost per cycle, 571 expected production cycle length, 570 inspection intervals, 573 integrated model, 573 inventory holding cost, 571 non-conforming items, 572 pattern search technique, 573 PM activities, 569 process shift mechanism, 574 and production process, 569 renewal reward process, 573 restoration cost per production cycle, 572 specific notations, 570

Index Imperfect preventive maintenance (PM) associated probabilities, 565, 567 classical EPQ model, 564 expected cost, 564, 565 expected inventory cycle length, 564 expected restoration cost per cycle, 568 inspection cost, 566 inventory holding cost, 565 non-conforming items, 566 process shift mechanism, 568 production cycle, 561, 564 quality control costs, 569 restoration cost, 567 specific notations, 564 Imperfect production processes, 223 Imperfect production system, 236 Imperfect products, 409 partial backordering cycle time, 128, 136 decision variables, 140 demand, 127 demand and inspection rates, 128 expected total profit, 140 fraction of cycle time, 134 HC, 128, 129, 131, 134, 137 identified and studied, 127 optimal cycle time, 128 optimal policy, 140 optimal values, decision variables, 135 process failure, 126 profit function, 137 repair cost, 134 repair process, 128 SC, 129, 132, 134, 137 screening period, 127 total cost, 127 TP, 129, 132, 135 uniform distribution, 140 variable cost, 127 Imperfect quality, 7, 10, 18, 25, 56 expected profit, 11, 12 fixed cost, 13 fixed ordering cost, 11 and inspection, 70–74 inventory level, 11 optimal value, 14 perfect and imperfect inventory levels, 13, 15 perfect-quality item, 11 probability, 82 screening process, 11 Imperfect rework process, 324 defective items, 251

589 EPQ model, 249 notations, 250, 312 optimal production, 253 perfect-quality items, 314 practical production process, 249 production uptime, 250 scrap items, 252 solution procedures, 249 Imperfect-quality EMQ model, 313 Imperfect-quality items, 69, 235, 239, 246, 252, 308 In control, 553, 565–567 Inspection with backordering, 506 buyer’s responsibility and supplier pays, 507 economic production model, 536 errors, 542, 543 imperfect inspection process, 522 and rework processes, 503 and sampling, 515, 516 Inspection cost, 47, 166, 193, 223 Inspection cost per time unit, 90 Inspection errors, 501, 517, 522, 523, 525, 537, 542, 543 Inspection process, 239 Inspection/screening period, 60 Instantaneous replenishment acceptance sampling plan, 50 AOQL, 55, 56 average inventory per cycle, 52 beta-binomial distribution, 51 binomial distribution, 49 Hessian matrix, 54 integer nonlinear program, 54 inventory behavior, 52 inventory-related costs, 51, 52 notations, 50, 51 parameters, 55 probability of acceptance, 55 quality-related cost, 53 renewal–reward theorem, 52 solution method, 55 varying, 55, 56 Integer nonlinear program, 54 Integrated model, 303 Integrated procurement–production–inventory system, 208–216 Inventories control systems, 1 EPQ inventory models, 2 inventory order quantity models, 2 managing inventories, 1

590 Inventories (cont.) variables, 4 Inventory average, 287 Inventory control problem, 331, 450 Inventory cost analysis, 247 Inventory cost per cycle, 252 Inventory cycle, 88, 97, 302 Inventory cycle length, 46 Inventory level, 129, 132, 134, 136, 142, 143 behavior, 11, 51, 52, 70, 71, 88, 89, 93, 97, 101, 103, 109, 134 cycle time, 128 defective items, 76, 77 differential equation, 114 dynamics, 51 FTD, 131 graphic representation, 119 maximum, 97, 111 over time, 24 perfect and imperfect, 15 problem on hand, 24 repair option, 57, 60 repaired products, 87, 127 replenishment cycle, 75, 76 starting and ending, 82 variation, 19, 104 zero, 11, 30, 93 Inventory level, EPQ model, 376 Inventory management, 76 Inventory model, 358, 500 Inventory order quantity models, 2 Inventory-related costs, 51, 52 Investment cost, 508, 512, 513

J Joint production policy, 383 Joint production system, 321 Just-in-time (JIT) manufacturing environment, 10

L Lagrangian function, 39–41 Lagrangian relaxation method, 394, 397 Learning curve, 101 Learning effects EOQ model, 26 HC, 23–28 in inspection with backorders, 34 in inspection with lost sales, 31 notations, 29 transfer of learning, 28–37

Index Linear backorder cost, 261 Lost profit, 387 Lost sale cost, 30, 31, 34, 62, 113, 125, 128, 129, 227, 453 Lower specification level (LSL), 537

M Machine repair, 154, 173, 179, 200 Maintenance, 4, 16–18 aggregate production and maintenance planning, 574 imperfect maintenance and process (see Imperfect maintenance) imperfect preventive maintenance, 561 preventive (see Preventive maintenance (PM)) production and quality, 549 production planning models, 549 Maintenance and planning model (PPM), 576 Maintenance planning, 549, 574, 575, 582 Maintenance policy, 575 Management systems, 113 Manufacturing processes, 4 defective products, 497 Manufacturing system, 357 Marketing, 1 Markovian shock model, 568 Material cost, 64 Materials, 1, 2 Mathematica 6.0, 106 Mathematical equation, 287, 291 Mathematical expressions, 98 Mathematical model, 382, 386 Mathematical modeling and analysis, 235 MATLAB, 397 Maximum limit/minimum limit, 49 Maximum purchasing price, 42 Mean of yield, 508, 512 Mixed integer nonlinear programming (MINLP), 379 Multidefective types annual backorder cost, 413 annual constant production rate, 409 annual holding cost, 413 annual production cost, 413 backorder level, 415 budget, 414 capacity, single machine, 414 cycle length, 412 EPQ model, 415 imperfect single-machine production system, 416

Index inequalities, 411, 415 inventory and shortages, product, 411, 412 inventory model, 411 objective function, 415 optimal cycle length, 415 parameter types, 409 perfect and imperfect-quality products, 409 rework production rate, 410 rework rates, 409 reworks, 413 scrapped items, 411 service level, 414 shortage quantity, 417 space, 413 total shortage quantity/safety factor, 415 total warehouse construction cost, 414 Multi-delivery, 166, 168 See also Discrete delivery, EPQ models with scrap Multi-delivery policy costs per cycle, 333 defective items, 332, 334 on-hand inventory, 332 notations, 335 optimal production, 335 partial rework defective items, 339 EPQ model, 337 non-conforming items, 339 production quantity, 339 proposed model, 337 and quality assurance, 332 rework, 332 Multi-delivery single machine algebraic derivation, 344 defective rate, 343 non-conforming items, 340 notations, 342 on-hand perfect-quality inventory, 341 Multi-item production system, 392, 393 Multi-product and multi-machine, 160–164 Multi-product case, 382 Multi-product inventory control problem, 231 Multi-product model, 468 Multi-product single-machine EPQ model backordering (see Backordering, multiproduct single-machine EPQ model) capacity, 368 categories, 367, 368 classification, 367 decision variables, 370 defective products, 370

591 features, reviewed studies, 372 mathematical model, 371 non-conforming items, 370 no shortage auction, 385–391 discrete deliveries, 375, 377, 379, 380 rework, 380, 382, 383, 385 scrapped, 391, 392, 394–397, 399, 400 simple model, 371, 373–375 optimal cycle length, 370 optimal production quantity, 370 partial backordering shortage (see Partial backordering shortage, multiproduct single-machine EPQ model) preventive maintenance, 371 production capacity limitation, 370, 371 production quantities, products, 368 random defective production rate, 371 service level, 370, 371 shortages, 370 space, 368 total annual cost, 371 total budget, 370 trade credit policy, 371 Multi-product single-machine production system, 223–230 Multi-product two-machine fabrication costs, 349 notations, 345 on-hand inventory level, 346 parameters, 351 quality screening, 345 Multi-products single manufacturing system, 3

N Negative inventory, 113 Non-conforming item, 552, 564, 566, 567, 570–572 Non-defective items, 310, 380, 441 Nonlinear programming model (NLPP), 70 Non-Markov shock model, 305 Non-shortage cycle, 481

O Objective function, 384, 385, 389, 391, 467 Obsolescence, 8 On-hand inventory, 246 Optimal cost, 64 Optimal decision variables, 79 Optimal lot size, 501 Optimal order quantity, 50

592 Optimal PM level, 568, 573, 577, 578 Optimal policy, 61, 123, 125 Optimal production, 248, 249, 312, 351, 560 Optimal replenishment policy, 500 Optimal solution, 501 Optimal values, 244, 309 Optimization software, 165 Order quantity, 500 Ordering cost, 11, 15, 51, 53, 102 Ordering cycle, 11, 13, 37, 120, 121 Out of control, 550, 552, 553, 564, 565, 567

P Partial backordering, 506–510, 582 economic manufacturing model, 318 EMQ model, 318 EOQ model imperfect products (see Imperfect products) imperfect-quality products, 112–118 replacement, imperfect products, 142–147 screening, 118–126 joint production systems, 321 modeling procedure, 318 non-defective and defective items, 318 partial differentiations, 323 policy production, 320 production and demand rates, 318 production capacity limitation, 323 production cost, 319 single product problem, 319 Partial backordering shortage, multi-product single-machine EPQ model preventive maintenance, 479–481, 483–492 repair failure, 460 rework, 449–452, 454–457, 459, 470, 471, 473–475, 477 scrapped, 461–467, 469, 470 Partial derivatives, 447 Particle swarm optimization (PSO), 379 Penalty cost shortage, 78 Penalty cost purchasing cost, 102 Perfect-quality item, 70, 245, 250, 275 Perishability, 8, 9 Pharmaceutical companies, 190 Physical thermodynamic system, 61 Planning horizon, 574–577 Positive inventory, 294 Practical production systems, 300

Index Preventive maintenance (PM), 371, 479–481, 483–492 age reduction and PM cost, 556 computer program coded in Fortran, 561 cost functions, 553 expected maintenance cost, 556, 557, 559 expected production cycle length, 553 expected shortage, 580 imperfect maintenance, 555 imperfect process, 561 integrated model and solution method, 559, 560 inventory levels, 554, 555 joint production, 551 just-in-time structure, 578 maintenance, 551 minimal repair, 577 optimal preventive maintenance level, 560 optimum just-in-time buffer level, 578 out of control, 550 partial backordering, 582 PM effect, 551 PM level for linear improvement, 562 production cycle, 550, 552 quality control cost, 554, 558 quality control inspections and PM tasks, 557 quality inspection activities, 552 quality model, 551 specific notations, 550, 552, 578 system changes, 555 total expected inventory holding cost, 557 Price–demand relationship, 39–41 Probability, 82 Probability density function, 43, 47, 49, 50, 70, 80, 81, 109, 112, 117, 315, 339 Probability distribution, 10, 239, 260 Probability function, 23 Product handling, 9 transportation, 9 Product cycle, 449 Product deterioration, 8 Product flows, 2 Production cost, 94, 97 Production cycle length, 325, 450 Production–inventory cycle, 225, 403 Production inventory model, 295 Production–inventory situation, 238 Production operation system, 569 Production period, 93 Production planning, 549

Index Production process, 239 Production rate, 2, 401 Production system, 61, 304 Products categorization, 7 Profit function, 100, 357 Proposed manufacturing problem, 357 Purchase cost, 118

Q QCU, 53 Quality continuous quality characteristic, 537–540 defective levels and partial backordering, 525–527 economic order quantity model, 500 EPQ models, 501, 537 imperfect-quality items, 500 inventory models, 500 investments, 498 quality assurance without shortage, 535, 536 quality screening and rework without shortage, 536 with backordering, 502–505 Quality-related cost, 53 Quality screening defective items, 255 notations, 255 production and screening, 255 production quantity model, 254

R Random defective rates, 344 backorders, 283 beta distribution, 289 cost equation, 284, 292 cyclic backordering, 283 mathematical equation, 282 maximum inventory level, 280 production process, 307 production time, 280 uniform distribution, 277 Random variable, 71, 82, 85, 92, 99 Real manufacturing environment, 23 Rejection defective supply batches, 82–87 Renewal reward theorem, 52, 72, 79, 85, 115, 247, 252 Repair, 552, 575, 577

593 Repair cost, 87, 134, 552, 553 Repair failure, 371, 443, 444, 446–448, 460, 461 defective items, 325 material, 324 modeling procedure, 325 notations, 328 production capacity, 324 random defective rate, 328 rework process, 325 traditional EPQ model, 324 Replenishment cycle, 76, 103, 104 Replenishment policy, 107 Restoration cost, 564, 567, 568, 572, 573 Return policy, 501 Rework, 3 Rework and backordering avoiding interruptions, 87 BC, 90 distribution function, 87 fixed cost, 87, 88 HC, 87, 90 Hessian matrix, 91, 92 inventory cycle, 88 production cycle, 87 production period, 87, 88 random variable, 87 repair cost, 87 total cost, 87 total profit, 91 variable cost, 87 Rework policy, 296 and preventive maintenance inventory control systems, 300 inventory system, 300 linear and nonlinear relationships, 304 non-conforming items, 301 production cycle, 300, 301, 304 profit function, 306 quantity, 303 EPQ model, 244, 298 inventory cost per cycle, 246 parameter values, 300 perfect-quality items, 245 production rate, 245 regular production process, 245 setup cost, 299 Rework process, 313, 501, 503, 538, 542–545 Rework process and scraps backordering, 294 EPQ model, 293 optimal solutions, 296

594 Rework process and scraps (cont.) production cycle time, 294 total cost function, 295 Reworkable items, 357, 500, 502, 503, 538, 545

S Salameh and Jaber’s model, 66 Sales revenue per time unit, 89 Sample sizes, 500, 515, 516, 522 Sampling, 515–517 Sampling inspection plans buyer draws, 43 decisions, 43 economic order quantity, 43 notations, 43, 44 pp1, 44, 46, 47 p0