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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated, 2011.
Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
MANAGEMENT SCIENCE - THEORY AND APPLICATIONS
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INVENTORY SYSTEMS: MODELING AND RESEARCH METHODS
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MANAGEMENT SCIENCE - THEORY AND APPLICATIONS
INVENTORY SYSTEMS: MODELING AND RESEARCH METHODS
HUI-MING WEE
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EDITOR
Nova Science Publishers, Inc. New York Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material.
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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Wee, Hui-Ming. Inventory systems modeling and research methods / Hui-Ming Wee. p. cm. ISBN: (eBook)
1. Inventory control--Mathematical models. I. Title. TS160.W397 2010 658.7'87015118--dc22 2010025541
Published by Nova Science Publishers, Inc. © New York Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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CONTENTS Preface
vii
Acknowledgment
ix
Part I
Introduction
1
Chapter 1
Introduction
3
Chapter 2
General Concept on Inventory Modeling
11
Part II.
Methods for Independent Demand
25
Chapter 3
Independent Demand System
27
Chapter 4
Quantity Discount
39
Chapter 5
Batch-Type Production Systems
51
Chapter 6
Discrete Demand Systems: Deterministic Models
69
Independent Demand Systems: Probability Models
87
Chapter 7 Part III.
Methods for Dependent Demand
107
Chapter 8
MRP, MRP-II and ERP
109
Chapter 9
Work-in-Process Inventory, JIT and TOC
123
Chapter 10
Deteriorating Inventory Models
139
Chapter 11
Solving Inventory Problems without Derivatives
153
Index
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PREFACE
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Most organizations use, transform, distribute or sell materials of one or several kinds. The management of materials has serious implication on the financial, production, and marketing functions of an organization. As the level of complexity of a product increases, so does the importance of managing materials. To fulfill their role in providing products and services, enterprises must satisfy the needs of production, as well as services through the use of all resources such as labor, capital, equipment and materials. All these managerial activities are called materials management. This new book covers the topic of inventory systems.
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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ACKNOWLEDGMENT
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This book is partially financed by CYCU.
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
PART I
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INTRODUCTION
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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Chapter 1
1.1. INTRODUCTION Most organizations use, transform, distribute or sell materials of one or several kinds. The management of materials has serious implication on the financial, production, and marketing functions of an organization. As the level of complexity of a product increases so does the importance of managing materials.
1.2. MATERIALS MANAGEMENT
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To fulfill their role in providing products and services, enterprises must satisfy the needs of production, as well as services through the use of all resources such as labor, capital, equipment and materials. All these managerial activities are called materials management. Materials Management is the planning, execution and control of 1. material flow systems 2. inventories Material flow system is shown in figure 1.1.
Figure 1.1. Material flow system.
¾ Inventory: •
Inventories are idle or incomplete resources that possess economic value waiting for sale or use. They are the stocks on-hand at a given time. In manufacturing, they are normally classified into five categories such as supplies, raw materials, work- inprocess (WIP), final products, and deteriorating materials.
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Hui-Ming Wee
4 • • •
Unutilized labor and capital are also idle resources. Although no one really declares, it is essentially a type of inventory. Although service industries need to manage materials and deteriorated items like other industries, there is no way to store service inventory. Airline provides air travel service, but there is no way to keep stock of seats.
¾ Supplies: items (not part of a final product) needed in the normal function of an organization. Ex: Pencils, pen, paper, light bulbs….. ¾ Raw materials: items purchased from suppliers to be used in production. ¾ Work-in-process goods: partially completed final products that are shill in the production process. ¾ Final products: finished goods that are available for sale, distribution or storage. ¾ Deteriorated materials: salvaged or spoiled products that have little or no value ¾ Inventory problems: The inventory problems are common to:
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_ _ _
profit making institutions non-profit organizations military
The cost of the stock is a total investment in inventory that represents a sizable portion of the Gross National Product (GNP).
¾ Do We Need Inventory? No, we do not need inventory if the system is perfect.
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Introduction
5
¾ Different perspectives on inventory •
Inventory is critically bad (a grave) for an enterprise
1. 2. 3. 4.
No profit earned directly from inventories Produces idles and surpluses Stagnant capital Losses from discounted sales and scraps
•
Inventory is basically good (a savior) for an enterprise
1. 2. 3. 4.
Allows firm to schedule and perform more economically Prevents material shortage Reduces waiting time (production lead time) Bulk purchases with quantity discounts can reduce cost significantly.
However, due to the following factors we need to store supplies and/or raw materials and goods. _ _ _ _
time factor (uncertainty in time between the demand and the production.) disconnection factor (dependency among various parties in the production process ) uncertainty factors economic factors
Another way of classifying inventories is dependent on their functions.
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_ _ _ _ _ _
working stock (lot size stock) materials are ordered as lots to meet future requirements. safety stock (due to demand and/or lead time uncertainties) anticipation stock (inventory built up to cope with peak seasonal demand) pipeline stock (transit stock or working process) decoupling stock (inventory accumulated between dependant activities) psychic stock (retail display inventory carried to stimulate demand and act as a silent salesman)
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1.3. INVENTORY PROBLEM CLASSIFICATIONS
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Figure 1.2. Inventory Problem Classifications.
1.4. PROPERTIES OF INVENTORY ¾
Demand:
_ _
size patterns of withdrawal: at the beginning, at the end of a period or dispatch uniformly)
¾ Replenishments: _ _ _
size patterns for multi-items lead time
¾ Constraints: _ _ _ _
space capital facility equipment
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Introduction ¾ Costs _
cost patterns
1.5. INVENTORY COSTS 1. 2. 3. 4.
Purchase Cost Order/Shipment Cost Holding Cost Stockout Cost
1.6. INVENTORY MODEL
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1.6.1. Lot Sizing Problems
Figure 1.3. Lot-sizing Problems Model.
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8
Hui-Ming Wee
1.6.2. Discrete Demand Lot Sizing Problems
Figure 1.4. Discrete Demand Lot Sizing Problems Model.
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1.6.3.Multi-Item Multi-Family Dynamic Lot Sizing Problems
Figure 1.5. Multi-Family Dynamic Lot Sizing Problems. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Introduction
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Multi-Item Capacitated Lot Sizing Problems
Figure 1.6. Lot-sizing Problems Model.
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Chapter 2
GENERAL CONCEPT ON INVENTORY MODELING CASE STUDY
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A company has done a lot of improvement to computerize working processes but some troubles and bottleneck exist in the production. Material supply is unable to synchronize with the production schedule due to excessive or short supply. Excessive supplies result to high stock cost while the risk of shortages results to delivery delay, lost sales, and lost of goodwill. If you are the material manager of this company, what would you do?
2.1. COST FLOW Different method or concept of inventory flow influences the inventory cost. There are four kinds of inventory flow concepts which are commonly used in inventory control and cost accounting. 1. FIFO: First In First Out 2. LIFO: Last In First Out 3. Average Cost 4. Specific Cost A company should consider many factors in choosing the methods of inventory flow, such as type of organization, economic plan, industrial rules and tax policy. Some organizations may use one method for internal use and another method for external purposes.
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12
The accounting and tax payment are influenced by the chosen kind of inventory flow method which must be used consistently.
2.1.1.FIFO FIFO (first in first out), is an inventory costing method that based its cost flow on the chronological order of purchases. In the FIFO method, older stocks are used before the newer ones. It is recommended for perishable items such as milk and eggs. FIFO is a method used to determine the cost of goods sold in the cost accounting. The cost flow is based on the purchase price and delivery sequence of goods. It is suitable for those materials with stable price but its calculation is more complicated and less efficient.
Example 1. Table 2.1. shows the cyclical inventory record of an item. The ending inventories on 4/1 are 300 units. Use the FIFO method to find the cost of sales and ending inventory value Table 2.1.
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Date 1/1 1/31 2/28 3/31
Item Beginning inventory purchase purchase purchase Total
Quantity 200 300 400 100
purchasing price ＄1.00 1.10 1.16 1.26
1000
Total cost ＄200 330 464 126 1,120
Solution Sale quantity 200 300 200 700
Purchasing price ＄1.00 1.10 1.16
Ending inventory Purchasing on February Purchasing on March Total value
Quantity 200 100 300
Total cost ＄200 330 232 762 purchasing price ＄1.16 1.26
Total cost ＄232 126 358
¾ The cost of sales is 762. ¾ The ending inventory value is＄358 ¾ or＄1120-＄762＝＄358
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General Concept on Inventory Modeling
13
Example 2 Table 2.2 shows the perpetual inventory record. The ending inventories on 4/1 are 300 units. Use the FIFO method to find the cost of sales and ending inventory value. Table 2.2. Date
1/1 1/31
Input Quantity
Cost
Total cost
300
＄1.10
＄330
2/3
Output Quant.
200 200
2/28
400
1.16
＄1.00 1.10
100 200 100
1.26
Total cost
＄200 220
464
3/1 3/31
Cost
1.10 1.16
110 232
126
Inventory Quantity
Cost
200 200 300 100
＄1.00 1.00 1.10 1.10
Total cost ＄200 200 330 110
100 400 200
1.10 1.16 1.16
110 464 232
200 100
1.16 1.26
232 126
The ending inventory value is ＄358=(＄232+＄126) The cost of sales is ＄762=(＄200+＄220＋＄110+＄232)
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2.1.2. LIFO LIFO (last in first out), is an inventory costing method based on cost flow in a reverse chronological order. The last items stocked are sold first. It is a typical stocking method for items that have no due date associated with them. In the cost accounting and valuation method, LIFO assumes that assets produced or acquired last are used or sold first then track back forward progressively. Its calculating method is opposite to FIFO. It is suitable for those materials with varying prices.
Example 3 Table 2.3 shows the cyclical inventory record of an item. The ending inventories on 4/1 are 300 units. Use the LIFO method to solve the cost of sales and ending inventory value. Table 2.3. Date 1/1 1/31 2/28 3/31
Item Beginning inventory purchasing purchasing purchasing Total
Quantity 200 300 400 100 1000
purchasing price ＄1.00 1.10 1.16 1.26
Total cost ＄200 330 464 126 1,120
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14 (Solution) Sale quantity 100 400 200 700 Ending inventory Inventory of January Purchasing on January Total
Purchasing price ＄1.26 1.16 1.10
Total cost ＄126 464 220 ＄810 purchasing price ＄1.00 1.10
Quantity 200 100 300
Total cost ＄200 110 310
¾ The cost of sales is $ 810 ¾ The ending inventory value is $ 310
Example 4 Table 2.4 shows the perpetual inventory record. The ending inventories on 4/1 are 300 units. Use the LIFO method to find the cost of sales and ending inventory value. Table 2.4. Date
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1/1 1/31
Input Quant.
Cost＄
Total cost＄
300
1.10
330
2/3 Date
2/28
Input Quant.
Cost＄
400
1.16
Total cost＄ 464
3/1 3/31
Output Quant.
300 100 Output Quant.
300 100
1.26
126
Cost ＄
Total cost＄
1.00 1.00
330 100
Cost ＄
Total cost＄
1.16
348
Inventory Quant. Cost＄ 200 200 300 100
1.00 1.00 1.10 1.00
Inventory Quant. Cost＄ 100 400 100 100 100 100 100
1.00 1.16 1.00 1.16 1.10 1.16 1.26
Soluition ¾ The cost of sales is＄778＝(＄330+＄100+＄348) ¾ The ending inventory value is $ 342 = ($ 100+$116+$126)
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Total cost＄ 200 200 330 100
Total cost＄ 100 464 100 116 100 116 126
General Concept on Inventory Modeling
15
2.1.3. Average Cost The average cost includes three kinds of methods: 1. Simple Average The simple average cost is the total cost divided by the total purchasing quantities in a period. This method is simple and convenient. But the average price can only be calculated at the end of term, it has no relationship with the quantity bought in each batch, and it cannot represent the real price. 2. Weighted Average Weighted average is calculated taking into account the relative weights of each component of the capital structure divided by the total purchasing quantities in a period. It is more reasonable but it only can be calculated at the end of term. 3. Moving Average
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A moving average, also called running average, is a type of finite impulse response filter used to analyze a set of data points by creating a series of averages of different subsets of the full data set. It is not a single number but a set of number. Each number is the corresponding subset average of a larger set of data points. A moving average is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles. It is useful in a perpetual inventory system and more efficient with transparent information.
Example 5 Table 2.5 shows the cyclical inventory record of an item. The ending inventories on 4/1 are 300 units. Use the three kinds of average methods to find the ending inventory value and cost of sales a) Simple average b) Weighted average c) Moving Average Table 2.5. Date 1/1 1/31 2/28 3/31
Item Beginning inventory purchasing purchasing purchasing Total
Quantity 200 300 400 100 1000
purchasing price ＄1.00 1.10 1.16 1.26
Total cost ＄200 330 464 126 1,120
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16 Solution a) Simple average
= 1 + 1 . 1 + 1 . 16 + 1 . 26
= $ 1 . 13
4 ¾ Average cost ¾ The ending inventory value = 300 × 1 . 13 = $ 339
¾ The cost of sales = ¾ Weighted average
700 × 1 . 13 = $ 791
4
=
∑ P iQ i
i =1
N ¾ Weighted cost 1 . 00 ( 200 ) + 1 . 10 ( 300 ) + 1 . 16 ( 400 ) + 1 . 26 (100 ) = 1000 ¾ = $1.12
¾ The ending inventory value = 300 × 1 . 12 = $ 336
¾ The cost of sales = 700 × 1 . 12 = $ 784 b) Moving Average
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Table 2.6. Date
Quantity
Purchasing price＄
Total cost＄
1/1 1/31 2/28 3/31
200 300 400 100
1.00 1.10 1.16 1.26
200 330 464 126
Moving average cost ＄ 1.00 1.06 1.10 1.12
The moving average for the period is the last moving average. i.e.: 1.12. Therefore, ¾ The ending inventory value = 3 0 0 × 1 . 1 2 = $ 3 3 6 ¾ The cost of sales = 7 0 0 × 1 .1 2 = $ 7 8 4 Example 6 Table 2.7 shows the perpetual inventory record. Use the moving average methods to find the ending inventory value and cost of sales.
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General Concept on Inventory Modeling
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Table 2.7. Date
1/1 1/31 2/3 2/28 3/1 3/31
Input
Output
Quant.
Cost＄
Total cost＄
300
1.10
330
400
1.16
464
100
1.26
126
Quant.
Inventory Cost＄
Total cost＄
400
1.06
424
300
1.14
342
Quant.
Cost＄
200 500 100 500 200 300
1.00 1.06 1.06 1.14 1.14 1.18
Total cost＄ 200 530 106 570 228 354
Solution ¾ The ending inventory value = 300 × 1 . 18 = $ 354 ¾ The cost of sales = 424 + 342 = $ 766
2.1.4. Specific Cost The inventory value is defined according to the market price. Organization can use various kinds of the cited methods to determine the inventory value. Enterprise should consider the actual conditions, accuracy and feasibility when choosing the method of inventory flow. The following example explains the effect of choosing different cost flow methods.
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Example 7 A company produces a single product. Table 2.9 shows the records of production and sales. The other information is shown in Table 2.8. Calculate the company net income by using: a) FIFO, (b). LIFO, (c) Simple average, (d). Weighted average, and b) Special cost Table 2.8. Operation cost Start inventory Tax Inventory method Ending inventory
＄500 / year 400 units, ＄2.0 / per unit 50％ Per cycle 210 units, on November
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Table 2.9. Month
1 2 3 4 5 6 7 8 9 10 11 12 Total
Produce Quant.
Cost＄
Total cost＄
Sale Quant.
600 570 550 610 580 490 450 480 540 610 600 580 6660
2.04 2.05 2.10 2.08 2.15 2.17 2.25 2.30 2.50 2.57 2.59 2.60 ＄27.40
1,224.00 1,168.50 1,155.00 1,268.80 1,247.00 1,063.30 1,012.50 1,104.00 1,350.00 1,567.70 1,554.00 1,508.00 ＄15,222.80
500 610 650 590 600 400 470 540 570 650 670 600 6850
Cost＄ 3.00 3.00 3.00 3.00 3.20 3.20 3.20 3.20 3.50 3.50 3.50 3.50
Total cost＄ 1,500 1,830 1,950 1,770 1,920 1,280 1,504 1,728 1,995 2,275 2,345 2,100 ＄22,197
a) ending inventory ×final purchasing cost ＝210×＄2.60＝＄546 b) ending inventory ×beginning purchasing cost ＝210×＄2.00＝＄420 c) ending inventory ×cost ＝210×（＄27.40 / 12）＝＄479 d) ending inventory ×cost ＝210×（＄15,223）/ 6660＝＄480 e) ending inventory ×November cost ＝210×＄2.59＝＄544
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Solution
Sale＄ Beginning inventory＄ Produce ＄ Quantity to sale (-): ending inventory Production cost Gross income (-): operation cost Pretax income (-): income tax Net income＄
FIFO
LIFO 22,197 800
Average cost simple weighted 22,197 22,197 800 800
Special cost 22,197 800
22,197 800 15,223 16,023 546a
15,223 16,023 420b
15,223 16,023 479c
15,223 16,023 480d
15,223 16,023 544e
15,477 6,720 5,000
15,603 6,594 5,000
15,544 6,653 5,000
15,543 6,654 5,000
15,479 6,718 5,000
1,720 860 860
1,594 797 797
1,653 827 826
1,654 827 827
1,718 859 589
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General Concept on Inventory Modeling
19
2.2. REDUCTION IN COST OR MARKET VALUE The inventory value discussed in Section 2.1 is calculated based on the starting cost. However, if the item values are changing due to changing price and deterioration, those methods are not suitable. In this case, we should use the market value to calculate cost and take account of the value lost.
Example 8 A PC company purchased a lot of PCs six months ago at＄1800 per unit. However, the price dropped to＄1500 this month. The ending inventory of the company is 10 units. Find the ending value of PC in this company for cost accounting. How much does it lose? a) Ending value＝10×(1500)＝＄15,000 b) Loss＝10(1800－1500)＝＄3,000
2.3. INVENTORY CHECK The purposes of inventory check are shown as follows: a) Audit the existing stock and adjust any inconsistencies in material accounting. b) Examine the performance of inventory management and improve the inventory management system. c) Calculate the profit and loss statement.
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There are three categories of inventory check methods: 1. Periodic Physical Inventory 2. Continuous Physical Inventory 3. Combination Physical Inventory
2.3.1. Periodic Physical Inventory In the periodic inventory system, sales are recorded as they occur but the inventory is not updated immediately. All stocks are recorded periodically on a certain day or within a fairly short period. During this time, stock movements are prohibited. A physical inventory must be taken at the end of the year to determine the cost of goods sold. ¾ Advantages a) The quantities of material in process can be counted more accurately. b) The company can implement the plans of preventive maintenance during this time. c) High accuracy of inventory check and more employees are involved.
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20 ¾ Disadvantages
a) The loss of stopping production. b) The inventory check team members are not familiar with this job, thus, may not do a good job. c) In periodic physical inventory, one cannot spot the inventory loss immediately. d) One cannot update the continuous physical inventory records in real time.
2.3.2. Continuous Physical Inventory Continuous physical inventory refers to continual physical count of the stock quantity. It requires accounting records to show the amount of inventory on-hand at all times. The system maintains a separate account in the subsidiary ledger for each material in stock, and the account is updated whenever there is an input or output of materials. During this time, all the process can continue. ¾ Advantages No loss of production. High degree of control that helps management to maintain proper inventory levels. It allows stock discrepancies to be more fully investigated and corrected. Professionals maintain higher work standards and warehouse personnel knows the need to frequently count stock. e) Unauthorized changes in procedures are detected.
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a) b) c) d)
¾ Disadvantage a) Concealing and collusion can occur easily.
2.4. ABC INVENTORY CONTROL Most enterprises carry a large number of items in stock, therefore they need to control all of the stocks at a reasonable cost. It is helpful to classify these items according to their importance, such as annual dollar usage or other criteria. The ABC principle or Pareto’s law states that a small number of items usually dominate the inventory total cost, as shown in Figure 2.10. The relationship between the percentage of items and the percentage of annual dollar usage is shown in Table 2.10.
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General Concept on Inventory Modeling
21
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Figure 2.1. ABC Inventory Control.
Table 2.10. Percentage of items in ABC Inventory Control % Catalogue A B C
Quantity (%)
Value (%)
20 30 50
80 15 5
2.5. INVENTORY ORDERING SYSTEM It is important for a company to obtain the right material, in the right quantities with the right delivery time from the right source and at the right price. These factors are the goals of inventory management. The most common approaches to determine the orders time are: 1) Fixed Order Quantity system (s, Q), as shown in Fig 2.2 2) Period Order Quantity system (T, S), as shown in Fig 2.3
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Hui-Ming Wee
22
I nve nt or yl e ve l
Figure 2.2. Fixed Order Quantity system.
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I nve nt or y S
Q
Q
L
L T
Ti me T
Figure 2.3. Period Order Quantity system . Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
General Concept on Inventory Modeling
23
2.5.1. Fixed Order Quantity System Same order quantities each time or fixed quantity system (Q- system) is called (s, Q) system. It is used for items when lot sizing algorithms cannot be used, e.g.: items with limited shelf life, customer order quantities for make-to-order products, limited capacities of production equipment or process and unit packaging lot. Q-system uses continuous physical inventory and is suitable for class A items. The operation flow chart is shown as Fig 2.4.
( Re c e i v eQ) Av a i l a bl ei nv e nt o r y
Re que s to fr e qui r e me nt
Che c ki nv e nt o r yl e v e l
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NO
I nv e nt o r y r e o r de rpo i nt
YES Pur c ha s i ngo r de r Figure 2.4. Operation flow cart of Q-system.
2.5.2. Period Order Quantity System Period order quantity system (POQ) is called Economic Time Cycle or (T, S) system. The number of item required in a pre-determined number of days (time) determines the order quantity and safety stock. The POQ system was designed to avoid remnants and give lower costs for lumpy demand. Enterprises depend on their past experiences to determine the time interval. The order quantity is the difference between the warehouse capacity and the existing stock at that time.
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24
Hui-Ming Wee
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POQ-system does not need to monitor or order inventory continuously but periodically at cycle T. An organization needs a large warehouse to prevent shortage and it is suitable for class C items. The operation flow chart is shown in Fig 2.5.
Figure 2.5. Operation flow cart of POQ system.
These two kinds of inventory systems mentioned above are the most common inventory management systems. There are many other kinds of inventory management systems, such as Min-Max System, Two-Bin System, TRM cargo transfer manifest and MRP, etc.
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PART II. METHODS FOR INDEPENDENT DEMAND
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Chapter 3
INDEPENDENT DEMAND SYSTEM The types of inventory models depend on demand, lead time variation and replenishment conditions. This chapter discusses a special class of inventory problems when items are independent with each other and demand is deterministic and uniform.
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3.1. DEFINITION ¾ Independent: two items are said to be independent if one demand is not driven by the requirement of another item. Ex: automobile. ¾ Deterministic: all of the parameters and variables are known or can be calculated with any certainty. In deterministic inventory models, demand for items and the appropriate inventory costs are assumed to be known with accuracy. Also, replenishment lead time is known and constant. ¾ Uniformity: a demand is said to be uniform if the demand level does not change throughout the planning horizon. ¾ Time-phase involved: in many cases, inventory decisions are medium and short term; the time frame is usually one year.
3.2. ASSUMPTION In deterministic inventory models, the following information is assumed to be known: I.Demand forecast II.Appropriate inventory costs III.Lead time
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Hui-Ming Wee
28
Given the above information, the question is: −
What should be the order size (how many) and when should an order to be placed such that total relevant costs are minimized.
3.3. DETERMINISTIC INVENTORY MODELS We can classify deterministic inventory models as follows: (see Fig 3.1)
Fixed Order Size Systems In a fixed order size system, ¾ ¾ ¾ ¾ ¾
Demand is known and continuous. (R units / year) Lead time is known and constant. (L weeks or months) Purchase cost of an item is given. (P $ / units) Ordering cost per order is given. (C $ / units) Holding cost per unit per year is given. (U $ / units / year)
or ¾ Annual holding cost as a fraction of unit cost is given. (F %)
∴ H ＝ PF
(3.1)
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The questions are 1) How much to order? 2) When to order? Since demand is constant and known, the same number of units is always ordered. Also, the time between orders does not vary.
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Independent Demand System
29
Figure 3. Lot-sizing Problems Model.
In a fixed order size system, inventory position is continuously reviewed and whenever the inventory position reaches a predetermined point, an order for a fixed number of units is placed. In a fixed order point, therefore, there are two parameters that define how much and when, respectively.
3.3.1. They are Q and B Q is known as the order size. B is known as the reorder point.
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30
3.3.2. Economic Order Quantity
Figure 3.2. EOQ Model (Saw tooth diagram).
Total Annual Cost ＝ purchase cost ＋ order cost ＋ holding cost
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QH TC = RP + RC + Q 2
(3.2)
Figure 3.3. Total Annual Cost.
Find a value for Q such that total annual cost, TC, is minimized. Let that value be Q0.
¾
How to find B ?
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Independent Demand System
31
Reorder point is selected in such a way that during lead time L, demand is satisfied without backlog. Therefore, B ＝ demand during lead time Reorder point B ＝ RL if lead time is given in months.
(3.3)
＝ RL if lead time is given in weeks.
(3.4)
12
52
If the lead time L ＜ T there will never be more than a single order outstanding. L ＞ T there will always be more than a single order outstanding.
¾ How to find Q0 ?
dTC = H − CR = 0 2 Q2 dQ
(3.5)
Q0 = 2CR = 2CR = EOQ H PF
(3.6)
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Expected number of orders during year＝m＝
Average order interval ＝ T ＝
R = HR Q0 2C
1 = Q0 = 2C m R RH
(3.7)
(3.8)
¾ Questions:
• •
In what case there will never be an outstanding order? (L ＝ 0) In what case there will be exactly one outstanding order? (L ＝ T)
3.3.3. Assumptions of the EOQ model 1) 2) 3) 4) 5) 6) 7)
The demand rate is known and constant. The lead time is known and constant. The entire lot size is added to inventory at the same time. No stockouts are permitted. Setup/order cost and holding cost are fixed regardless of the size of the lots. No quantity discounts are given on large purchases. There is sufficient space, capacity, and capital to procure the desired quantity.
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Hui-Ming Wee
32 8) Item is a single product. ¾ Example 3.1:
Hey Song purchases 54,000 units of spare parts each year at $40.00 each. The ordering cost is $20.00 per order, and the holding cost per unit per year is $9.00. Find: 1. 2. 3. 4. 5.
The economic order quantity. Q* How many times of order are needed in a year？ Reorder point B, if lead time is given one month Reorder point B, if lead time is given two weeks. The total annual cost.
Solution 1. R
=
54,000
(units/year),
P=
$
40,
2.
2 ( 20 )( 54000 ) = 490 9 m = R = 54000 = 110 orders/year
5.
TC ( Q ) = PR + HQ * = 40 × 54000
Q* =
2 CR H
C
=
=
$
20,
H
=
$9.
units
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Q0 490 3. B = RL = 54000 (1 ) = 4500 units 12 12 4. B = RL = 54000 ( 2 ) = 2077 units 52 52
+ 9 × 490
＝ 2,164,410
3.4. BACKORDERING Backordering is an unfilled demand that will be filled later than desired. In the backordering case, the firm does not lose the sale. The consumer waits until the arrival of next order. Loyal is important. There is a special cost related with backordering because processing of backorders may require different scheme. In this model, we assume that all shortages are satisfied from the next shipment. It is also called Captive demand.
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Independent Demand System
33
3.4.1. Economic Order Quantity Model with Shortage
Inventory level
V Q
t2
t1
t3
Figure 3.4. EOQ model with shortage.
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¾ The size of the stockout ＝ Q － V units ¾ Maximum inventory level ＝ V units ¾ Inventory cycle ＝ t3
(3.9) (3.10) (3.11)
Consider period t3
¾ Average holding cost during period
t3 = V t1 H 2
¾ Average backordering cost during period
t3 = K
(3.12)
Q − V (3.13) t 2 2
where K is the backordering cost per unit per year.
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34
R
V
t1
1 year ¾ What are the values of t1 and t2? For an annual demand of R unit,
R =V 1year t1
(3.14)
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t1 = V R
t2 =
R =V 1year t1
Q −V R
(3.15)
(3.16)
2 K (Q − V ) 2 QP + C + HV + 2 2R R (3.17) Total cost in one cycle ＝
¾ ¾ We have R/Q cycles in a year. ¾ Total annual cost ＝ purchase cost ＋ order cost ＋ holding cost ＋ backorder cost 2
TC = RP + C R + HV + Q 2Q Q0 =
2 CR H
H + K K
K (Q − V ) 2 2Q
(3.18) (3.19)
Note: How to derive Q0?
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Independent Demand System
35
2 K (Q − V ) 2 TC = RP + C R + HV + Q 2Q 2Q 2 2 K ( Q − 2 QV + V 2 ) = RP + CR + HV + Q 2Q 2Q
2 KV 2 KQ CR HV = RP + + + − KV + Q 2Q 2 2Q
∂ TC = 0 − CR − HV 2 + K − 0 − KV 2 = 0 ∂Q 2 2Q 2 Q2 2Q 2
2CR + HV 2 + KV 2Q 2 2 CR + HV
2 CR + V
Q2
2
2
(H
+ KV
2
2
=
K 2
= KQ
+ K ) = KQ
2
2
2CR + V 2 (H + K ) = K
(1)
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Get V 2 K (Q − V ) 2 TC = RP + C R + HV + Q 2Q 2Q ∂ TC = 0 + 0 + 2 HV − K + 2 KV = 0 2Q 2Q ∂V HV KV + =K Q Q
V (H + K ) = KQ KQ V = H +K
(2)
Substitute (2) to (1),
KQ 2 CR + H +K Q2 = K ⎛ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ ⎠
2
(H
+K)
( KQ ) 2 = 2 CR H +K KQ 2 ( H + K ) − ( KQ ) 2 = 2 CR H +K
KQ
2
−
KQ
2
( H + K ) − ( KQ ) 2 = 2 CR
Q KH = 2 CR (H + K 2
(H
+ K
)
)
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Hui-Ming Wee
36
2 CR (H + K ) KH 2 CR H +K Q0 = H K Q2 =
When
H + K →1 K
K → ∞,
K V0 = 2CR H H +K
(3.20)
Note: How to derive V0? Substitute Q0 to (2),
V =
KQ H +K
V =
K H +K
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⎛ ⎜V ⎜ ⎝
K = H +K
2CR H 2 CR H
H +K K H +K K
⎞ ⎟ ⎟ ⎠
2
K2 2CRK V = × 2CR × H + K = K (H + K )H (H + K )2 H 2
V0 =
when
2 CRK = (H + K )H
K → 0,
2 CR H
K (H + K
)
K →0 K +H
V0 ＝ 0, no inventory What do you mean about this? Reorder point ＝ lead time demand － backorders
B = RL − (Q − V ) N
(3.21)
¾ Problem 1: Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Independent Demand System
37
The United Semi Conductor Company purchases 30,000 units of mother board each year at a cost of $40.00 per unit. The ordering cost is $10.00 per order, and the annual holding cost is estimated at 20% of the unit value. a) What is the EOQ for the USC? b) What is the total annual inventory cost for the mother board if it is ordered in economic quantities? c) What is the maximum number of mother boards in inventory at one time? d) What is the average number of units in inventory? e) What is the approximate order interval in weeks if the most economic ordering policy is followed and the USC is operated 52weeks each year?
¾ Problem 2:
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Optical Industries originally estimated its annual demand for glass as 1,000 units per year and its holding cost for glass as $1.50 annually. Optical has discovered at the end of the year that annual demand is only 700 units and holding cost per unit per year close to $2.00. The only good news is that Optical figured order cost $30.00 to be placed when they really cost just $25.00 of each. Optical now wants to know the significance of these mistakes on the last year’s decisions. a) What was the combined effect of the error on the EOQ for glass? b) What was the individual effect of each error on glass’s economic order quantity? c) What was the combined effect of the error on the minimum total variable cost of glass for last year? d) What was the individual effect of the error in demand on the minimum total variable cost for last year?
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Chapter 4
QUANTITY DISCOUNT Suppliers often offer discounts on the price of goods in exchange for larger order quantities. Suppliers can offer this discount to liquidate their inventories.
4.1. QUANTITY DISCOUNTS
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¾ Buyer may want to buy in volumes because -high volume buyouts reduce ordering cost. -of less unit cost. ¾ At the expense of -higher inventories ¾ There are two types of quantity discount models.
1) All units discount. _ purchasing larger quantities result in a lower unit price for the entire lot. 2) Incremental discount. _ lower unit price is offered to units which are purchased above a specified quantities.
4.2. ALL UNITS DISCOUNTS P0 for 1 أQ ＜ U1 price breaks P1 for U1 أQ ＜U2 P＝. Pj for Uj أQ where P0 ＞ P1 ＞ P2 ＞ ………＞ Pj and 1 ＜ U1 ＜ U2 ＜ ………＜ Uj Case of 1: price break. That is
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Hui-Ming Wee
40
P＝
P1, for 1 Q ＜ Qd P2, for Qd Q ＜Qs
Qd ¾ Procedures:
1) Starting with the lowest unit cost, calculate the EOQ at each unit cost until a valid EOQ is obtained. 2) Calculate the total annual cost for the valid EOQ and all price breaks quantities larger than the valid EOQ. 3) Pick the minimum lowest total cost.
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¾ Example 4.1:
An automobile company uses 20,000 bearings each year. The supplier of the bearing offers them at the following prices: Quantity ordered (Q) 1 ~ 799 800 ~ 1199 1200 ~ 1599 ≥ 1600
Unit price (P) $11 $10 $9 $8
The cost of an order is $50, and the holding cost is 20% of the unit value per year. (a). What order size should be placed to minimize costs with an all-units quantity discount? (b). How long will each order last? 【Sol】:
2(20,000)(50) Q * (8) = 2CR = = 1,118 H 8(0.2)
(invalid )
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Quantity Discount
Q * (9) = 1,054
41
(invalid )
Q * (10 ) = 1,000
(valid )
TC(Q*)=RP+HQ*
QH TC (Q = 1200) = RP + RC + Q 2
QH TC (Q = 1600) = RP + RC + Q 2 Each order will last=Q/R
4.3.INCREMENTAL QUANTITY DISCOUNT P0, for each of the first U1 -1 units P1, for each of the next U2 -U1 units P＝.
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Pj,for each unit in excess of Uj -1 units where P0 ＞ P1 ＞ P2 ＞ ………＞ Pj The purchase cost for a lot size of Q units is:
M i = Di + Pi Q
(4.1)
Di = ∑ (U z −1)(Pz −1 − Pz ) (Additional ordering cost)
(4.2)
i
z =1
The purchase cost per unit is:
M i Di = + Pi Q Q
(4.3)
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42
The total cost per year of a lot size of Q units is:
TC ( Q ) = purchase cos t + order cos t + holding cos t ⎛
= ⎜⎜ Pi + ⎝
= Pi R +
D i ⎞⎟ D FQ ⎛⎜ R + CR + Pi + i ⎟ ⎜ Q 2 ⎝ Q ⎠ Q
(C
⎞ ⎟ ⎟ ⎠
FD i + D i )R Pi FQ + + Q 2 2
(4.4)
(C + Di )R + Pi F = 0 dTC (Q ) =− 2 dQ Q2 i
Q0i =
2 R[C + ∑ (U Z − 1)( PZ −1 − PZ )] Z =1
(4.5)
Pi F
i
TCi (Q) =
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¾
[C + ∑ (U Z − 1)( PZ −1 − PZ ]R Z =1
Q
+ Pi R + F [ ∑ (U Z − 1)(PZ −1 − PZ ) + Pi Q] 2 Z =1 (4.6) i
Procedures:
1) Calculate Qoi for each price level. 2) Determine which Qoi `s are valid. 3) Calculate the total cost for each valid Qoi. ＊ must be in the lot size range ＊ too big or small not valid
1) Select the valid Qoi with the lowest total cost. 2) Calculate the minimum lowest total cost. ¾ Example 4.2: (From Example 4.1)
If incremental quantity discounts are offered to the automobile company mentioned in Example 4.1, what would be the optimal lot size under these conditions? Quantity ordered (Q) 1 ~ 799 800 ~ 1199 1200 ~ 1599 ≥ 1600
Unit price (P) $11 $10 $9 $8
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Quantity Discount
43
【Solution】 i
i
Pi
Ui
∑ (U Z =1
0 1 2 3
$ 11.00 $ 10.00 $ 9.00 $ 8.00
Q 00 =
Q01 =
0 799(1)=799 1199(10-9)+799=1998 1599(9-8)+ 1998=3597
i
∑ (U Z =1
Z
− 1)( PZ − 1 − PZ )]
Pi F
40 , 000 ( 50 + 799 ) = 4 ,125 (n.v.) 10 (. 20 )
=
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− 1)( PZ −1 − PZ )
2 (50 )( 20 ,000 ) = 953 ( N. valid ) (n.v.) 11(0 .20 )
2CR = PF
2 R[C +
1 800 1200 1600
Z
Q 02
40 , 000 ( 50 + 1998 ) = 6 , 746 (n.v.) 9 (. 20 )
Q 03
40 , 000 ( 50 + 3597 ) = 9 , 549 ( valid ) 8 (. 20 )
TC(Q03) ＝ $175,637.39
4.4. SPECIAL SALE PRICE A supplier may temporarily reduce the price of an item. Possible reasons: − −
Reduction in inventory competition
4.4.1. Temporary Sale Price For the time being, assume that prior to the placement of an order, a supplier is making a one-time discount on the price of a commodity.
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44 − − −
P: regular price d: reduction in price P- d: one-time price of an item
A buyer may want to take advantage of this discount. Thus may order more than usual. The order quantity prior to price decrease is:
Q0 =
2 CR PF
(4.7)
Figure 4.1. Special Sale Price.
What is the optimal special order size Q’ ?
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Consider two Cases
I. No special order is made. (TC1) II. One time special order is made. (TC’) Q’ is optimal order quantity that maximizes the difference between two total costs. Total cost of the system during Q’/ R period would be： Total cost ＝ purchase cost ＋ holding cost ＋ order cost ¾ Consider the case when a special order is placed.
TC ' = ( P − d )Q '+
F (P − d ) Q' Q' + C 2 R
(4.8)
¾ Consider the case where no special orders are placed.
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Quantity Discount
45
(4.9) Let q ＝ TC1 － TC’＝ special order cost savings
PQ 0 optimal special order size. dR + (P − d )F P−d
Q '0 =
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q0 =
C ( P − d ) Q '0 ( − 1) 2 P Q0
(4.10)
(4.11)
Suppose special discount offer expires before the replenishment time, management must decide if a special order should be placed before the regular replenishment time. Assume that during the expiration date, stock level is g and lead time is 0. Then,
Q '0 =
PQ 0 dR + −g (P − d )F P − d
q0 = C[(
(4.12)
Q'0 ) 2 −1] P (P − d ) Q
(4.13)
In this case, it is not always advantages to place the special order. The cost saving is only positive when the special order size exceeds
P P−d
tim es
Q '0 i .e . :
Q '0 >
P × Q (4.13) P − d
In the case when lead time is positive, g must be reduced by lead time demand. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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46
4.4.2. A MODIFIED EOQ MODEL WITH TEMPORARY SALE PRICE The discussion presented in this section follows the same reason of argument and assumptions as Martin. Moreover, addition conditions are added to make the formulation more robust and realistic. The added conditions are: 1. On-line instantaneous updating of EOQ values for the case when no special order is made. This is significant if discount price is large. 2. The number of replenishment is an integer as in real world.
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With the above conditions and Fig 4.2, the special order gain, g, can be modeled as
Figure 4.2. Modified inventory for a temporary sale rice.
g = TCn − TCs
(4.14)
where
TC s (P − d )Q ' +
Q ' (P − d )F +C 2R 2
(4.15)
represents the total relevant cost during that production cycle when the special order in taken; and
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Quantity Discount
47
(4.16) represents the total relevant cost during the same cycle time when the first replenishment is an updated EOQ using the reduced price and the subsequent replenishment using normal price EOQ policy. It is noted that the number of replenishment in an integer, as seen from the last term. The special symbols used in (4.16) are：
⎡o ⎤ : which represents integer value equal or greater than its argument.
⎣o ⎦ : which represents integer value equal or less than its argument. The following notation are retained or added P : regular unit price d : unit price discount
Q
: special order quantity F : annual holding cost function C : cost per order
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Q * : normal price EOQ
Q 0 : reduced price EOQ
T ' : the length of special sale cycle T 0 : economic order interval at reduced cost T * : economic order interval at normal cost From Fig 4.2 and using the classical economic order quantity policy, one has
Q* =
Q
0
2C R (P − d )F
=
T' =
2C R PF
Q' R
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48
T0 =
T
*
Q0 R
Q * R
=
⎢Q ' − Q 0 ⎥ * t = T0 + ⎢ ⎥T * Q ⎣ ⎦
q = (T ' − t ) R
Q
g*
As in Martin, closed-form expression for and cannot be derived from (4.14).
4.5. KNOWN PRICE INCREASES When a supplier plans to increase the price of an item on some future date, should the buyer order more units prior to price increase? When a buyer foresees a price decrease, should he reduce current order?
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The question is: Should a special order be placed prior to price increases or decreases? Q0 ： EOQ before price increase ＝
2 CR PF
(4.17)
Q0* ： EOQ after price increase ＝
2 CR (P + k )F
(4.18)
P ＋ k ： new price q ： inventory level before the price increase If there is B in the reorder point, then new q’ ＝ q － B
(4.19) (4.20)
Q0 ' = Q0 P + k + kR − q p PF
(4.21)
k ( FQ0* + R) −q PF
(4.22)
Or
= Q0* +
q0 = C[( Q0' >1 Q0
Q0 ' 2 ) − 1] Q0
(4.23)
(4.24)
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Quantity Discount
49
4.6. EOQ SENSITIVITY Assume that no price discount or price increase is expected.
Q0 = 2CR H
(4.25)
TVC(Q) ＝ order cost ＋ holding cost ＝
RC + QH , optimal total variable cost. Q 2
(4.26)
Q* H TVC ⎛⎜ Q* ⎞⎟ = RC + = HQ* , optimal total variable cost * 2 ⎝ ⎠ Q
(4.27)
Assume that there are errors in estimation of the parameters R, C, and H.
X X X C X R Q* X C X R X H Q = 2CR C R = Q* = H XH XH XH
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Q − Q* = Q*
XCX XH
R
,
(4.28)
− 1 = order quantity error fraction, (4.29)
where XR ＝ estimated
demand ＝ demand error factor actual demand XC ＝ estimated order cos t = order cost error factor actual order cos t XH ＝ estimated
holding cost holding cost error factor = actual holding cos t
Q = Q0
X R XC XH
(4.30)
Order quantity with parameter errors
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50
Order quantity error fraction ＝
Q − Q0 X R XC = −1 XH Q0
(4.31)
When two of the error factors are 1, (ex: XR and XC ＝ 1)
Q − Q0 = 1 −1 , XH Q0 TVC − TVC0 X H + X R X C = −1 TVC0 2 X R XC X H
(4.32)
(4.33)
¾ Problem 1:
A bicycle company has an annual demand of 12,000 units for a folding bike. The supplier sells the units for $105.00 in order quantities below 130 and for $100.00 in order quantities above 131 units. The order cost is $6.00, and the annual holding cost is 12% of unit value. a) In what quantities should the folding bike be purchased with an all-units quantity discount? b) What will be the maximum inventory level? ¾ Problem 2:
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If an incremental quantity discount schedule is in effect, what is the optimum lot size for the folding bike described in Problem 1? ¾ Problem 3:
The United Semi Conductor Company purchases 30,000 units of mother board each year at a cost of $40.00 per unit. The ordering cost is $10.00 per order, and the annual holding cost is estimated at 20% of the unit value. If a supplier is offering a special discount and temporarily is reducing the unit price of the mother board by $0.20. c) What lot size should USC order to take advantage of the discount? d) What cost savings would result from this order?
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Chapter 5
BATCH-TYPE PRODUCTION SYSTEMS In a production environment, products are usually produced in batches or lots. That is why inventory problem in production systems is often referred to lot-sizing problem. In most cases, more than one product are produced on the same equipment which has a limited capacity. This capacity often imposes a constraint on the amount to produce.
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5.1. LOT-SIZING PROBLEM
Figure 5.1. Lot-sizing problem.
Setup：A setup is required when a production facility is switched from the production of one item to the production of another item. It may include: Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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52 − −
Labor time and/or cost for setup. (Loss of production capacity) Material wastage. (Materials required for the production of one item may need to be discarded when a switch is made.) − Cost of materials for setup. (Cleaning of equipment) When the family structure is important, a major setup is incurred in addition to a minor setup during production switch.
5.2. EPQ – SINGLE ITEM EOQ assumes instantaneous replenishment (that is, the entire order enters into the inventory at one time - infinite replenishment rate). However, in production environment (EPQ model), replenishment is finite or continuous ( that is, products are being replenished during the production period). In EPQ model, inventory cycle is divided into two periods; the production period during which production takes place, and no production period when only demand occurs. In a production environment, the unit cost is usually the most important cost, and it is determined by the management. Unit cost includes：
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− − −
direct labor cost direct material cost factory burden (all other cost such as: indirect labor, indirect material, depreciation, taxes, insurance, power, maintenance …)
In economic production quantity model the decision involves the determination of the production run size (lot size). The production size that minimizes the total relevant costs is referred to as EPQ.
Figure 5.1. EPQ Model. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Batch-Type Production Systems
53
p： Production rate r： Demand rate / day
5.2.1. Production Order Quantity 1) QM ＝ Maximum inventory 2) Q M = ( p − r )t P 3) tP ＝
Q p
(5.1)
, length of production period (5.2)
4) average inventory ＝ 5) average inventory ＝
max imum inventory 2
( p − r )Q 2p
(5.3) (5.4)
6) Total annual cost ＝ production(material) cost ＋ setup cost ＋ holding cost
TC = RP + RC + Q( p − r ) H 2p Q
(5.5)
dTC = 0 dQ
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Q* =
2CRp H ( p − r)
Optimum length of production run =
B =
RL = rL N
(5.6)
Q* p
(5.7)
(5.8)
5.2.2. EPQ – Single Item ¾ Example 5.1:
A fan manufacture plans to produce 40,000 units of a special type of fan next year. The production rate is 200 fans per day, and there are 250 working days available in a year. The setup cost is $200.00 per run; the unit production cost is $15.00; holding cost is $11.50 per unit per year.
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54 a) b) c) d) e)
What is the economic production quantity? What is the total variable inventory cost for the fan on an annual basis? How many production runs should be made each year? What will be the maximum number of fans in inventory at one time? If the production lead time is 5 days, what is the reorder point?
【Solution】
a. Q* =
2CRp 2(40,000)(200)(200) = = 2637 units H ( p − r) 11.5(200 − 160)
b. Total variable cost ＝
RC + Q( p − r ) H 2p Q*
= 15.2 × 200 + 40(2637) (11.50) = $ 6076 2(200) c. # of runs / year =
R = 40,000 = 15.2 runs Q* 2637
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d. maximum inventory level=( p − r )
=
e. reorder point=
Q P
40(2637) ≅ 528 units 200
RL = 40,000(5) = 800 units 250 250
Note: There seems to be an inconsistency here! Do we have to order always since maximum inventory level ＜ reorder point? If it is so, why does the model tell us to order 15.2 times? (hints: L>t2) ¾ Exercise: Backordering- Derive the equations.
Q* =
2C R p H ( p − r)
B = RL − N
H +k k
2 CRH ( p − r ) pk ( H + k )
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Batch-Type Production Systems
55
5.3. EPQ – MULTIPLE ITEMS Consider a two-item production planning problem whose data for one item is given in the previous example. The following are the data for the second item. R ＝ 20,000 units p ＝ 200 units/day L ＝ 4 days P ＝ $ 50 per unit H ＝ $ 10 per unit per year C ＝ $ 30 per run
r = R = 50 units/day N
Q0 =
2CRp 2(30)(20,000)(200) = = 400 H ( p − r) (10)(150)
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# of runs=
20,000 = 50 times or once in 5 days. 400
Furthermore, assume that these two items share the same facility. The question is how to schedule the production of both items in that facility? The first item takes approximately 13 days to produce. However the second item must be produced once in 5 days. Therefore EPQ for each item is infeasible. The problem is feasible because it takes 200 days of production of item 1 and 50 days of production of item 2. The two item problem is difficult to solve. It is known as Economic Lot Scheduling Problem. To solve this problem the following assumption in made. − −
Assume that there is a production cycle during which all products are produced. The question then boils down to find the optimal number of production cycle and the lot size of each item. − Assume there are n items. ¾ Additional notations:
i:
index of items
i = {1,2,....., n}
Ri, Pi, Ci, pi, ri are demand rate per year, price, setup cost, production rate per day and daily demand for item i, respectively.
ti :
duration of time for the production of item i in any cycle. Assume that there are m production cycles in a year, what is the production lot size for item i in each cycle?
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56
Maximum inv＝
( p i − ri )
ti
(5.9)
( pi − ri ) ti 2
Average inv＝
(5.10)
Ri Ri = ti pi ti = p m i However, m
(5.11)
n
n
n
i =1
i =1
i =1
TC = ∑ Ri Pi + m ∑ Ci + ∑
( pi − ri )Ri H i 2mpi
(5.12)
To find the optimal production run, we take the first derivative of the total annual cost and equate it to zero.
dTC n 1 n ( pi − ri )Ri H i = ∑ Ci − =0 ∑ pi dm i=1 2m 2 i=1 n
m0 =
∑ ( p i − ri )R i H i p i
i =1
n
2 ∑ Ci
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i =1
Qi =
Ri m0
(5.13)
(production size for each product)
n
n
i =1
i =1
TC0 = ∑ Ri Pi + 2m0 ∑ Ci
Ri i =1 pi n
N≥∑
(total time in a year)
Run time for each cycle ＝
N ≥ n Qi m0 i∑ =1 pi
(5.14)
(5.15)
(5.16)
(5.17)
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Batch-Type Production Systems
57
¾ Example 5.2:
A fan manufacture produces six products on a single machine. The information available on the products is shown in the following Table. Item
1 2 3 4 5 6
a) b) c) d)
Annual Demand Ri 3,000 8,000 5,000 5,000 12,000 6,000
Qi =
Ri
m0
566 1510 943 943 2264 1132
Unit production cost Pi $8.00 $8.00 $4.00 $2.00 $5.00 $2.00
Daily production rate pi 100 200 100 250 600 100
Annual holding cost Hi $2.00 $1.80 $1.20 $0.60 $1.50 $0.50
Setup cost Ci $70 $100 $120 $80 $250 $160
What is the optimum production cycle if there are 250 working days in a year? What are the product production runs size? How long it takes to finish a complete production cycle (in days)? What is the slack time per run?
【Solution】
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(a) (1) Item
(2) pi
(3) ri
(4)
( pi − ri )Ri
(5) Hi
(4)(5)
1 2 3 4 5 6
100 200 100 250 600 100
12 32 20 20 48 24
2,640 6,720 4,000 4,600 11,040 4,560
$2.00 1.80 1.20 0.60 1.50 0.50
5,280 12,096 4,800 2,760 16,560 2,280 $43,776
∑ ( pi − ri )Ri H i pi 6
m0 =
i=1
6
2 ∑ Ci i=1
=
43776 = 5.3 2 × 780
Ci
Qi pi
$70 100 120 80 250 160 $780
5.66 7.55 9.43 3.772 3.773 11.32 41.5
runs／year
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58 (b) Ri 3,000 8,000 5,000 5,000 12,000 6,000
Product 1 2 3 4 5 6
m0 5.3 5.3 5.3 5.3 5.3 5.3
Qi 566 1,509 943 943 2,264 1,132
250 = 47.2 (c). Length of run＝ 5.3 days. 6 Q 566 1509 943 943 2264 1132 + + + + + = 41.5 ∑ pi = (d). i=1 i 100 200 100 250 600 100
Slack time/run＝47.2－41.5＝5.7 days. ¾ Example 5.3: a) Suppose the setup cost in Example 5.2 were actually 20% less than estimated. b) (a). How much difference would the error could occurs in the production run size for each product? c) (b). What difference would the error could occurs in the length of a complete production cycle?
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【Solution】 (a). 6 C = 780 − 0.2 × 780 = $624 ∑ i i =1
m0 = 43,776 = 5.9 2 × 624 Product 1 2 3 4 5 6
Ri 3,000 8,000 5,000 5,000 12,000 6,000
runs／year
m0 5.9 5.9 5.9 5.9 5.9 5.9
Qi = 508 1,356 874 874 2,034 1,017
Ri
Errors in unit
m0 566－508＝58 1509－1356＝153 943－874＝69 943－874＝69 2264－2034＝230 1132－1017＝115
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Batch-Type Production Systems
(b). Length of run＝
N 250 = = 42.4 m0 5.9
59
days.
The error resulted in the length of the run shorter by 4.8 units. Once the lot size of each product is determined, these items must be scheduled on a machine. Scheduling is the order of production. In scheduling items, machine breakdowns, operator deficiencies, production of scrap, tooling failures, or quality difficulties are not considered. This theoretical model must be modified to fit real world emergencies and contingencies. Scheduling is done on a weekly basis. In scheduling, the following information for each item must be available: • • • • •
Standard hours per item Lot size per week Demand forecast per week Current inventory position Machine capacity
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Given the above information, scheduling can be done using a heuristic procedure. One such procedure is Run Out Time (ROT). If this procedure results a feasible schedule, the problem is solved. If this procedure does not result in a feasible schedule, the lot size of each item must be modified. To modify the lot sizes, another heuristic procedure, Aggregate Run Out Time (AROT), can be used.
5.4. RUN OUT TIME (ROT) Run Out Time is the duration of time at which inventory is depleted. For any item i, ROT is computed as follows:
ROTi =
current inventory position of item i demand per period for item i
(5.18)
The decision rule is to schedule the items in ascending order of their ROT values. ¾ Example 5.4:
Gigabyte produces four different components on the same piece of PC board. Gigabyte is planning its production schedule. The expected demands and inventories on hand, production hours required per part, and standard lot sizes are shown in the following Table:
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60 Part
Stand Hrs. per part 0.02 0.04 0.04 0.05
A B C D
Lot size
demand
400 300 400 300
300 250 350 350
Inventory on hand 300 400 500 600
The production process operates 40 hours per week. a) Use the run out time method to develop a production schedule for the four parts. b) Does there appear to be a capacity problem? 【Solution】
(a) (1) Part A B C D
(2) Stand Hrs. per part 0.02 0.04 0.04 0.05
(3) Lot size
(4) demand
(5) Inventory on hand
(5)÷(4) ROT
Sequence
400 300 400 300
300 250 350 350
300 400 500 600
1.00 1.60 1.43 1.71
1 3 2 4
Check for feasibility
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(b). (1) Sequence A C B D
(2) ROT 1.00 1.43 1.60 1.71
(3) Lot size 400 400 300 300
(4) Stand Hrs. per part 0.02 0.04 0.04 0.05
Mach. Hrs. per lot 8 16 12 15
Run. Cap. in hrs. 32 16 4 -11
There is a capacity shortage of 11 hours for the given lot sizes. Therefore, we need to change the lot size of each item. Let’s answer this question in next section.
5.5. AGGREGATE RUN OUT TIME (AROT) METHOD Assume that the production ceases at the end of the week. What should be the lot size of each item such that the inventory for each item would be depleted at the same time? Let ai denote the standard hours per unit of item I and Ii denote the inventory level of item i. For simplicity n items are assumed t. To solve the problem, let m denote the number of day inventory will be depleted.
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Batch-Type Production Systems
61
AROT＝
(5.19)
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(1)
I 2 + x2 =m d2
(2)
I n + xn = m (n) dn
a1 x1 + a2 x2 + ..... + an xn = N
(n+1)
x1 = d1m − I1 x2 = d 2 m − I 2
xn = d n m − I n Substitute xi into equation (n+1) we get:
a1 (d1m − I1 ) + a2 (d 2 m − I 2 ) + ..... + an (d n m − I n ) = N (5.20) Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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62
a1d1m − a1I1 + a2 d 2 m − a2 I 2 + ..... + an d n m − an I n = N
m(a1d1 + a2 d 2 + ..... + an d n ) = N + a1I1 + a2 I 2 + ... + an I n n
∑ ai I i + N
m = i=1 n ∑ ai di i =1
(5.21)
N：total machine hours available during the planning period. n
∑ ai I i :
i =1
inventory in machine hours for all items in the family.
n
∑ ai d i :
i =1
machine hours for forecasted demand per period for all items in the family. m：AROT
¾
Example 5.5:
Using the information given in Example 5.4, develop a 40 hours capacity feasible production schedule.
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【Solution】 (1) Part
(2) Stand hours per part
A B C D
0.02 0.04 0.04 0.05
(3) Demand forecast per period 300 250 350 350
(4)=(2)×(3) Machine hour for demand 6 10 14 17.5 47.5
(5) Current Inventory position 300 400 500 600
(6)=(5)×(2) Inventory mac hine hours 6 16 20 30 72
AROT = 72 + 40 = 2.358 weeks 47.5 From (1)
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Batch-Type Production Systems
63
Lot size of A, xA：
From
I A + xA = 2.358 dA
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dA＝300 , IA＝300, xA＝ 407.4 units Similarly, Lot size of B, xB＝ 189.5 units Lot size of C, xC＝ 325.3 units Lot size of D, xD＝ 225.3 units (1) Part A B C D
(2) Stand Hrs. per part 0.02 0.04 0.04 0.05
(3) Lot size 407.4 189.5 325.3 225.3
(4)=(2)×(3) Mach. hrs 8.15 7.58 13.00 11.27 40
40－(4) Run capacity 31.85 24.27 11.27 0.00
5.6. FIXED ORDER INTERVAL SYSTEM In the economic order (production) models, we assumed that inventory is reviewed continuously. Orders are placed whenever stock level drops to reorder point. In some cases, however, inventory is reviewed periodically. In other words, there is a fixed time period, T, where inventory is reviewed, and depending on the stocking policies, an order may be placed to replenish the stock. The order size brings the present stock level up to the maximum inventory level.
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64
In this model, we determine, T, the review period or order interval and E, the maximum inventory level. ¾ Order size ＝E－current inventory position on hand＋on order－backorders Assume we do not permit stock out: ¾ m ＝ # of orders or reviews per year.
m＝
1 T
(5.22)
At each review period, an order is placed. Therefore, Total cost,
TC = Rp + mC +
RFp 2m
where mE ＝ R (max inventory E ＝
(5.23)
R = RT m
TC = RP + C + RFPT T 2
2C RFP 1 m* = = RFP 2C T*
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T* =
) (5.24) (5.25) (5.26)
In a deterministic system, a fixed order size and a fixed order interval system are same.
Q* = RT * = 2CR FP E = TR + RL = R(T + L) = Q + B
(5.27)
＝max inventory level (5.28)
5.7. ECONOMIC ORDER INTERVAL – MULTIPLE ITEMS For wholesale and retail businesses, orders are usually placed for more than one item. If all items are supplied from the same supplier, multiple items order saves ordering cost. Due to the coordination of multiple items orders, logistics and transportation cost savings may also be possible.
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Batch-Type Production Systems
65
In this model, we assume a major ordering cost, C, for the joint orders and a minor ordering cost, ci, associated with each individual item. The order size of each item, then, depends on the time interval, T, between each order for the entire group. Once T is found, the maximum inventory level for each item, Ei, can also be found. n
∑ ci n n TC = ∑ Ri Pi + C + i =1 + TF ∑ Ri Pi T T 2 i =1 i =1
(5.29)
n
dTC = − dT
C + ∑ ci T
i =1 2
+ F ∑ Ri Pi = 0 2 i=1 n
n
T *2 =
2(C + ∑ ci ) i =1
n
F ∑ Ri Pi i =1
(5.30)
= nc, if all items has the same setup cost. n
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T* =
2(C + ∑ ci ) i =1
n
F ∑ Ri Pi i =1
(5.31)
T* ：Economic order intervals per year. Ei ＝
RT i + Ri L = Ri (T + L) , N N N
(5.32)
where N is operating days in a year.
¾
Example 5.6 : A fan manufacture plans to order 8 items from the same vender, as shown in the table. The ordering cost is $10.00 per major purchase order, $0.25 per minor purchase order for each item. Carrying cost is 15% per year. a) What is the economic order interval? b) If the lead time is one month, what is the maximum inventory level for each item?
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66
n
Item
annual demand Ri
unit cost Pi
purchase cost
∑R P i =1
A B C D E F G H
175 425 70 190 810 115 90 210
1.00 0.60 4.00 5.00 0.75 2.10 3.00 2.00
i
i
175.00 255.00 280.00 950.00 607.50 241.50 270.00 420.00
【Solution】 (a) n
T* =
2(C + ∑ ci ) i =1
n
F ∑ Ri Pi
=
2(10.00 + 2.00) = 0.224 0.15(3,199)
years
i =1
or 2.7 months. Every 2 months an order would be placed for the eight items. (b)
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Ei =
Ri (T + L) 3.7 Ri = = 0.3083Ri 12 12
A 53.95 ~ 54 B 131.02 ~ 132 C 21.58 ~ 22 D 58.58 ~ 59 E 249.72 ~ 250 F 35.45 ~ 36 G 27.75 ~ 28 H 64.74 ~ 65 ¾ Problem 5.1 :
A plastics corporation uses 600 gallons/day of fuel. The holding cost is $0.003 per gallon per day. The relevant data are as follows:
Item cost Order/setup cost Replenishment rate
Purchase $14.10 $ 8.50 ∞
Manufacture $13.80 $ 101.00 2000 gallons/day
What are the minimum cost and order quantity for the solvent? Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Batch-Type Production Systems
67
¾ Problem 5.2 :
Give an annual demand of 9000 units and an economic order quantity of 650 units:
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a) What is the economic order interval for the item in weeks? b) In an economic order interval system, what is the maximum inventory level if the lead time is one week, and there are 52 operating weeks per year?
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Chapter 6
DISCRETE DEMAND SYSTEMS: DETERMINISTIC MODELS EOQ, EPQ and EOI models assume that demand occurs at constant rate. If there are T months in a planning horizon (usually planning horizon = 1 year), these models assume that demand in each month is constant or does not vary. However, there may be some instances when this assumption is seriously violated. That is, when demand in one month may vary drastically with another month. When this case occurs, we say that there is a time-varying demand.
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6.1. ASSUMPTIONS Demand with time-varying patterns can occur for both dependent and independent items. In this chapter, we study the lot sizing techniques for deterministic, time-varying demand items. Here are the assumptions: Demand is known (deterministic) and occurs at the beginning of each period. There is a finite planning horizon with several time periods. Orders (production lots) arrive at the beginning of each period. Unit cost of each item is constant. No quantity discounts. Each item is treated independently. No shortages or stock outs are allowed. Items are withdrawn from inventory at the beginning of the period. That is, demand occurs at the beginning of the period. Therefore, holding cost is applied to the ending inventory. 8) Inventory related costs (ordering and holding) and lead time are known and constant over the planning horizon. 9) Lead time is zero. This is not a restrictive assumption. If lead time is not negligible, orders are offset by the lead time. 10) Ending inventory of period T is zero. This implies that, demand in period T ＋ 1 (period right after the planning horizon) is zero.
1) 2) 3) 4) 5) 6) 7)
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70
11) Initial inventory is zero. If the initial inventory in period 1 is not zero, it is subtracted from the first period’s demand until initial inventory becomes zero. Planning inventory of lot sizes starts from the period whose beginning inventory is zero.
In all of the models discussed in this chapter, we assume that the following information is available. 1) 2) 3) 4) 5)
Demand in each period. Unit cost. Cost of holding inventory as a fraction of unit cost. Ordering (setup) cost per lot. Lead time.
Given the above information and the assumptions, we will find the lot size of each item in each period. From assumption 10, the facility seems to cease at the end of the planning horizon. However, the facility in real life runs in a continuous fashion, unless shut off decision is made by the general management. This assumption is necessary to derive the replenishment quantities (lot sizes). To eliminate the effect of this limitation, lot size (replenishment quantity) of imminent period is computed over the planning horizon. As the schedule rolls forward in time, new demand information is included and the model is executed again.
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6.2. SOLUTION PROCEDURES
Figure 6.1. Solution Procedures.
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.
6.2.1. Wagner and Whitin Algorithm Wagner and Whitin (1958) proved that the optimal solution in any period is either: 1) the beginning inventory is zero and lot-size is positive, or 2) the beginning inventory is positive and lot size is zero, or 3) The beginning inventory and lot size are both zero. This property is known as Wagner and Whitin property. All of the heuristic procedures listed above find lot size of each period that has W-W property. This implies that lot size in any period must be exactly the requirements of the proceeding several periods.
6.2.2.General Outline of W-W Algorithms
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An arc starting from any node specifies that the lot size at the beginning of that period will be equal to the sum of the demands of periods until the end of that arc. Therefore the cost of each arc can be easily computed.
Figure 6.2. Wagner and Whitin Algorithm (W-W).
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General outline of W-W algorithms 1. Find the cost of each arc on the network. Cost of any arc = cost of ordering ＋
hp
end (arc )
∑ (Qce − Qci )
i = starts ( arc )
e
Z ce = C + hp ∑ (Qce + Qci ) i =c
for 1 ≤ c ≤ e ≤
N
(6.1)
e
where
Qce = ∑ Rk
(6.2)
k =c
Demand rate in period k 2. Find the shortest path from node 0 to T. 3. Find the lot-size in each period. 4. Find the cost of each arc on the network.
6.2.3. Example 6.1
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W-W algorithms Period di
d1
d2
d3
d4
d5
d6
Demand
10
5
50
65
25
30
C=$75, h=1 Therefore
P=$1 hP=1
Find the shortest path Z11=75 Z12=75+5=80 Z13=75+2(50)+5=180 Z14=75+3(65)+2(50)+5=375 Z15=4(25)+375=475 Z16=5(30)+475=625
Z22=75 Z23=75+50=125 Z24=65(2)+125=255 Z25=25(3)+255=330 Z26=30(4)+330=450
Z33=75 Z34=75+65=140 Z35=25(2)+140=190 Z36=30(3)+190=280
Z44=75 Z45=75+25=100 Z46=30(2)+100=160
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Discrete Demand Systems: Deterministic Models Z55=75 Z56=75+30=105
e c 1 2 3 4 5 6
73
Z66=75
1 75
Zce 3 180 125 75
2 80 75
4 375 255 140 75
5 475 330 190 100 75
6 625 450 280 315 105 75
Define fe to be the minimum possible cost in periods 1 through e, given that the inventory level at the end of period e is zero.
f0 = 0 f1 = 75
f 2 = Min(Z12 + f 0 , Z 22 + f1 ) f
=
2
Min ( 80 + 0,75 + 75 ) = 80
f 3 = Min(Z13 + f 0 , Z 23 + f1 , Z 33 + f 2 )
f = Min (180,200,155 ) =155 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.
3
f
4
= M in ( Z
14
+ f ,
Z
0
24
+ f , 1
Z
34
+ f , Z 2
44
+ f ) 3
f 4 = Min(375,330,220,230) = 220 f = Min ( Z + f , 15 0 5
Z +f, 25
Z +f , Z +f ,
1
35
2
45
3
Z 55 + f 4 )
f 5 = Min(475,405,270,255,295) = 255 f = M in ( Z + f , 16 0 6
Z
26
+f , 1
Z +f , Z 36
2
46
+f , 3
Z 56 + f 4 , Z 66 + f 5 )
f 6 = Min(625,525,360,315,325,330) = 315
f 6 = Z 46 + f 3
，Last order placed in period 4,
X 46 = 65 + 25 + 30 = 120 f 3 = Z 33 + f 2
，2nd order for period 3,
X 33 = 50
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f 2 = Z12 + f 0
，1st order for period 1, 2
Period 1 Demand 10 Order quantity 15 Cum. Variable Cost 80 Total variable cost ＝ $ 315 ¾ Problem 6.1
2 5 0 80
X 12 = 10 + 5 = 15 3 50 50 155
4 65 120 285
5 25 0 315
6 30 0 $315
A product with the demand pattern as below, has a unit purchase cost, P=$25, ordering cost per order=$60, and a holding cost fraction of 4%. Derive the order quantity by WW algorithm. Period Demand
1 75
2 0
3 50
4 65
5 25
6 30
6.3. HEURISTIC PROCEDURES 6.3.1. Lot For Lot Ordering (LFL)
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In LFL, lot size in each period = demand in each period. Lot for lot ordering minimizes the inventory holding cost. Indeed, its inventory holding cost is zero. ¾ Example 6.2： Given: ordering cost, C=$300 per order Period Demand Order Quantity Var. Cost
1 120
2 50
3 0
4 360
5 70
6 0
120
50
0
360
70
0
300
300
0
300
300
0
Total cost
$1200
6.3.2. Periodic Order Quantity (POQ) Lot size in period 1 = Total demand of fixed period (Economic Order Interval, EOI) where
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Discrete Demand Systems: Deterministic Models
EOI = [
EOQ ] = 2C R RPh ,
75
(6.3)
where [X] is the smallest integer greater than or equal to X, and T
R=
(∑ Ri ) i =1
T
(6.4)
R : average demand rate per period. Example 6.3 I. Apply LFL for the following data, Given: Ordering cost, C=$100 Period Demand Order size Cost
1 10 10 100
2 3 3 100
3 30 30 100
4 100 100 100
5 7 7 100
6 15 15 100
7 80 80 100
8 50 50 100
9 15 15 100
10 0 0 0
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II. Apply POQ Unit purchase cost, P = $50 Holding cost fraction, h = 0.02 Ordering cost, C= $100 9
∑ Ri R = i=1 = 34.44 9 ⎡
EOI = ⎢ ⎢ ⎣
2C ⎤⎥ 2(100) = =3 RPh ⎥⎦ (34.44)(50)(0.02) time periods
【Solution】 Period Demand Order size Cost
1 10 43 163
2 3 0 0
3 30 0 0
4 100 122 137
5 7 0 0
6 15 0 0
7 80 145 180
8 50 0 0
9 15 0 0
10 0 0 0
100 + 63 (30×2＋3: inventory cost)
Total cost ＝ 163 ＋ 137 ＋ 180 ＝ $ 480 Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Σ 310
$480
Σ 310
$900
Hui-Ming Wee
76 Note: 63 ＝ hP × (Inventory carried))
Total carried from period 1
＝ 1[(43－10)＋1(43－13)＋(43－43)]
Total carried from period 3
Total carried from period 2
6.3.3. Silver- Meal Algorithm (SMA) Step：
Let T= 1, i = 1, L= total accumulated periods Compute ( until the end of the planning horizon)
TRC(T ) C +total holding cost to the end of T = L L
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TRC (T ) = L
T
C + hP
∑ (k k =1
− 1) R k
L
(6.5)
Compute
TRC (T + 1) = L +1
If
C + hP
T +1
∑ (k
k =1
− 1) R k
L +1
(6.6)
TRC (T + 1) TRC (T ) > L +1 L Q=
Then lot size
T
∑R k =1
k
(1st lot size)
(6.7)
Find the 2nd lot size: Set
i = i +1
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T = T + 1, L = 1 Go to step 2 TRC (T + 1) TRC (T ) ≤ L +1 L If
Then Set
T = T +1
L = L +1 Go to step 2 Example 6.4
Apply SMA to the following demand pattern. Period Demand
Step 1:
1 10
2 3
3 30
T = 1, i = 1, start
4 100
5 7
6 15
7 80
8 50
9 15
L =1
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TRC (1) 100 + 0 = = 100 1 1 Step 2:
TRC (2) 100 + 3 = = 51.5 2 2 Step 3: 100 ＜ 51.5 Therefore
T = 2 (Go to step 2)
TRC (2) 100 + 3 = = 51.5 2 2 [Step 2]:
TRC (3) 100 + [3 + 2(30)] = = 54.33 3 3 [Step 3]: TRC (3) TRC (2) = 54.33 > = 51.5 3 2 Lot size in period 1 ＝ 10 ＋ 3 ＝ 13
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i=3 T =3
start
L =1
TRC (3) = 100 1 [Step 2]: TRC (4) 100 + 100 = = 100 2 2 [Step 3]: 100 ≤ 100
TRC (4) = 100 2 [Step 2]: TRC (5) 100 + 100 + 2 × 7 = = 71.33 3 3 [Step 3]: 71.33 ≤ 100
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TRC (5) = 71.33 3 [Step 2]:
TRC (6) 100 + 100 + 2 × 7 + 3×15 = = 64.75 4 4 [Step 3]: 64.75 ≤ 71.33
TRC (6) = 64.75 4 [Step 2]: [Step 3]:
TRC (7) 100 + 100 + 2 × 7 + 3×15 + 4 × 80 = = 115.8 5 4 115 .8 > 64.75 Lot size in period 3 ＝ 30 ＋ 100 ＋ 7 ＋ 15 ＝152
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T =7 start
L =1
TRC (7) = 100 1 [Step 2]:
TRC (8) 100 + 50 = = 75 2 2 [Step 3]:
75 ≤ 100
TRC(8) = 75 2 [Step 2]: TRC(9) 100 + 50 + 2 ×15 = 60 = 3 3 [Step 3]:
60 ≤ 75
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TRC (9) = 60 3 [Step 2]:
TRC (10) 100 + 50 + 2 ×15 + 0 = 45 = 4 4 [Step 3]: Lot size in period 7 ＝ 80 ＋ 50 ＋ 15 ＝145 Period(T)
T
7 8 9 10
1 2 3 4
Dema nd 80 50 15 0
Incremental h Co. 0.00 50.00 30.00 0.00
Cum. h cost
TRC(T)
TRC/T
0 50 80 0
100 150 180 180
100 75 60 45
Lot size in period 7 ＝ 80 ＋ 50 ＋ 15 ＝145 Period Demand Lot size Cost
1 10 13 103
2 3
3 30 152 259
4 100
5 7
6 15
7 80 145 180
8 50
9 15
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10 0
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80 Total cost =＄642
¾ SMA is expected to perform badly when:
I.Sharp increase (decrease) in demand occurs. II.There are a large number of periods with zero demand (lumpy demand)
6.3.4. Part Period Algorithm (PPA or LTC)
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In PPA, a number of periods, whose requirements will be satisfied by a lot size ordered, are selected at the beginning of that period. Hence, at the beginning of the period, a lot size, which is equal to the sum of the demands of the periods, is selected. Since the requirements of n periods is satisfied with a lot size in period 1, there is an inventory holding cost associated with this plan. It is obvious that, we are usually better off if the total inventory holding cost for this plan is less than or equal to an ordering cost. This is true, because if total inventory holding cost ＞ ordering cost, then instead of producing the demand of nth period in period 1, it is better to produce the demand of nth period in period n and incur additional setup. In other words, we carry inventory until a setup is less costly then inventory holding cost. A setup is preferred when total inventory carried (aggregate part periods, APP) is greater than
C = hP Economic part-periods (EPP)
(6.8)
T APP = ∑ (k −1) Rk = C = EPP Ph k =1
(6.9)
Part period number for period k
Example 6.5 A fan has a unit purchase cost of $50.00, an ordering cost per order of $100.00 and a holding cost fraction per period of 2 %. Determine the order sizes by the incremental partperiod algorithm using the data below: period demand
1 10
2 3
3 30
4 100
5 7
6 15
7 80
8 50
9 15
10 0
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【Solution】
EPP = C = 100 = 100 Ph 50(0.02) Period
T
Demand
Accumulated Part-periods T
∑ (k − 1) R k =1
1 2 3 4 4 5 6 7 7 8 9 10
1 2 3 4 1 2 3 4 1 2 3 4
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Period Demand Lot size Cost
10 3 30 100 100 7 0 80 80 50 15 0
1 10 43 163
k
0 (13-10)=3 (43-10)+(43-13)=63 (143-10)+(143-13)+(143-43)=363 0 (107-100)=7 (122-100)+(122-107)=37 (202-100)+(202-107)+(202-122)=277 0 (80-50)=30 (145-80)+(145-130)=80 (145-80)+(145-130)+(145-145)=80
2 3
3 30
4 100 122 137
100+63
5 7
6 15
7 80 145 180
100+37
8 50
B) Q
SLu = 1 −
E ( M > B) Q + E ( M > B)
(7.32)
for complete backordering (7.33)
for lost sale case
For a normal distribution,
(7.34) Partial Expectation
E(M > B) = σE(Z )
(7.35)
Re order level B = M + Zσ
(7.36)
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¾ Problem 7.4
For SLu＝0.99, R＝1800 units/year, C＝$3.00 per order, H＝$3.00 per unit per year, See the following table, What are the order quantity and reorder point? Lead time Demand (M) 48 49 50 51 52 53 54 55 56 57 58
Probability 0.02 0.03 0.06 0.07 0.20 0.24 0.20 0.07 0.06 0.03 0.02 1.0000
P(M )
P(M > B) 0.98 0.95 0.89 0.82 0.62 0.38 0.18 0.11 0.05 0.02 0.00
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¾ Problem 7.5:
See the Problem 7.4, if the distribution is normal distribution with a mean of 53 and a standard deviation of 2 units, what should be the reorder point? ¾ Example 7.6
The demand of two bicycles is normal distribution with a mean weekly demand of 120 units. Each bicycle has a replenishment lead time of 4 weeks. Type A has a standard deviation of 30 units, while type B has a standard deviation of 40 units. If the firm’s safety stock policy is a one-week supply on all items, what is the probability of a stockout for each type? 【Solution】
Z = S = 120 = 2.00 σ 30 4 Type A:
Type B:
Z = S = 120 = 1.50 σ 40 4
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PART III. METHODS FOR DEPENDENT DEMAND
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Chapter 8
MRP, MRP-II AND ERP MRP -Materials Requirement Planning MRP-II Manufacturing Resource Planning ERP -Enterprise Resource Planning
CASE STUDY
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An automobile assembling company has more than 1,000 kinds of material and parts. The Company has been implementing MRP system since 1980, but shortages still occur. What is the problem with the material management system? Will implementing MRP plan eliminate material shortage? Why does shortage occur in this company? What is wrong with the MRP system?
8.1. MATERIALS REQUIREMENT PLANNING：MRP Materials Requirement Planning (MRP) is a computer/information based material management. It utilizes information from master production schedule (MPS), bill of materials (BOM), on-hand inventories, to initiate and allocate orders by a standard procedure. The aim of MRP is to manage the dependent items inventory (Orlicky, 1975). The primary objective of MRP system is to generate the correct inventory information and determine the right order quantities at the right time.
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The MRP process translates the MPS for finished products into a time-phased, material requirements plan.
8.1.1. The Fundamentals of MRP 1. The company must possess master production schedule (MPS). 2. There is a BOM for each final product. 3. The company needs an integrity compiler system of material and there is a unique code which can be distinguished for each item. 4. The company keeps the inventory records. 5. There are efficient information systems.
The Assumptions of MRP (1) The information system must be designed to create a database among MPS, BOM and inventory records. (2) The production or purchasing lead time are known and fix. (3) Assembling order is placed only when all the necessary parts are ready. (4) Every material in or out the warehouse should be recorded exactly. (5) All final spare parts are independent. They can be purchased or distributed individually.
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8.2. INDEPENDENT AND DEPENDENT REQUIREMENT Dr. Orlicky (1965) proposed the concept of dependent and independent demands. Capacity Requirement Planning is a tool for independent demands and MRP is a tool for dependent demands. In 1970, Dr. Olicky, Plossol and Weight, together with the American Production and Inventory Control Society, developed an integral structure of material requirement planning for MRP system. Later, APICS and IBM vigorously implement computer assisted MRP.
8.2.1. Independent Demand Independent demands are based on forecasts (e.g. demand forecast). These quantities are used in MRP to calculate material procurement and/or production quantities. These forecast data are only prediction of the real market situation. Consumers may decide their actual automobile purchase based on their economic situation. The demand for automobile and computer are two kinds of totally different markets. So, the forecasts for the automobile and computer demand can be carried out independently. Even if the company produces these two products at the same time, the demands of these two products are totally independent.
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8.2.2. Dependent Demand •
Origin of the dependent requirement
There are two reasons to consider dependent demand: (1) Using the traditional prediction method will result in too many stocks. Traditional forecast only predicts the final products (an independence demand). The information the producers has is the final product information. The company knows very little about the demand of other parts. These parts are necessary for production. To reduce stock, we must check the inventory of parts before order. (2) The development of computer and information technology Computers help to speedup data analysis capability. Orlicky extended the application to material requirement planning, and proposed the concept of dependent demand in inventory management problems. •
Dependent demand
Dependent demand is a demand that is influenced by another material. The Bill of Material (BOM) shows the relationship between components. Due to the number of parts in a product, traditional producer needs to use computer to record and calculate the needs of each component.
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8.3. THE BASIC LOGIC OF MRP MRP system consists of three basic components (input, MRP system and output). Fig 8.1 illustrates the MRP system.
Figure 8.1. MRP system. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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8.3.1. Input The input consists of basic input and assisted input. •
Basic input
− − −
MPS: Master Production Schedule BOM: Bill of Material Inventory records
¾ Other input
− −
Forecast Open order of parts MPS
• •
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• • •
The MPS translates the sales and operations plan into a schedule for final products production. The MPS sets out an aggregate plan for production. MRP translates that aggregate plan into a detailed plan. MPS is based on customer contract and market prediction. The MPS links with Materials Requirement Planning (MRP) system. MPS provides basic communication between sale and production. BOM
• • • • • • • •
BOM is a list of all of the items that make up a product or assembly. BOM has a formal structure, stage to use, quantity needed for each component. Product structure lists the composition, assembly relationship and quantity of all product spare parts. It helps to determine the quantity and time of relevant supplies needed accurately. (Fig 8.2) As show in Fig 8.2, the final products are products A (level 0th), it belongs to the independence demand. Its entire parts are the interdependent demand. Multi-layer to express more complicated product structures. As show in Fig 8.2, product A consists of one piece of B and two piece of C. The component of B consists of one piece of D and one piece of F. The component of C was derived from G directly
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Figure 8.2. BOM of product A.
Inventory Records • • •
The inventory record shows the inventory status and data requirements for each stock item. The MRP program utilizes the procedure to upgrade data at any time. The inventory record can reflect:
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(1) The state of the stock at present, including existing stock and in-transit stock. (2) Net requirement and gross requirement of each item. (3) Future order plans and quantities.
8.3.2.Benefits of MRP System MRP system is a software based production planning and inventory control system. Using MRP, enterprises are able to meet current and future customer demand at the lowest possible cost. The MRP system has the following benefits: (1) Ensure products and materials are available for production and delivery to customers. (2) Maintain the lowest possible level of inventory. (3) Plan manufacturing activities, delivery schedules and purchasing activities. It provides answers to the following questions: (4) What items are required? (5) How many are required? (6) When are they required?
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Basically, MRP answers two related decisions: timing (when to order) and quantity (how much to order). It decides on what items to purchase from outside suppliers and what items to produce internally.
8.3.3. Output ¾ The primary outputs of MRP are planned orders and special orders. These are generated to satisfy demand. Special orders are to balance supply with demand during rescheduling or order cancellation. If demand exceeds supply, MRP automatically reschedules orders and generates messages to expedite or create orders. On the other hand, if supply exceeds demand, MRP reschedules or cancels planned orders and generates messages to cancel orders. ¾ Basic report
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(1) Gross Requirement - the forecast demand for the item. (2) Planned Order Receipt - how much we will receive each period as a result of planned orders. (3) Planned Order Release - when we need to release the planned orders after taking note of the lead time required. (4) Scheduled Receipt - any scheduled receipts from previous orders that are currently being processed. (5) Projected On Hand - how much the inventory level will be if we produce as planned and the demand forecast is correct. (6) Projected Net Requirement - how much we need in each period to avoid stockout. ¾ Secondary report
(1) Performance Control Report - Used for assessing systematic operation. It can help the administrator to assess the bias of the plan, including the error of producing and stock shortage. It provides information for cost assessment. (2) Plan Report - Used for predicting the future stock. It can be used for planning purchase contract and other thing in the future. (3) Exception Report - to arouse manager's attention to the important problems in production, such as: delay, overdue order, mistakes and no demand parts.
8.4. THE IMPLEMENTATION OF MRP (1). Confirm the lead time of all materials. (including production and procurement) (2). Production control department schedules MPS. (3). Engineering department develops BOM. (4). Material department checks the status of stocks.
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(5). Calculate gross requirement of each independent and dependent item via MPS and BOM. (6). Transfer the gross requirement into the net requirement. (7). Transfer the net requirement into the purchasing plan. (8). If purchasing plan and production plan do not conform to demand, the system informs the purchasing department immediately. Immediate actions will be taken by the planner to balance supply with demand.
8.5. MANUFACTURING RESOURCE PLANNING (MRP-II) MRP II (Manufacturing Resources Planning) essentially is closed loop MRP plus manufacturing resources planning. It was developed during the 1970's by extending the original MRP features. Typically an MRP II package will include features of management reports, cost information and easy “what-if” analysis. It may also include capacity requirements planning which includes capacity restrictions in the planning process. MRP II package concentrates on planning the manufacture resources on people, machines, and storage etc. rather than limiting to the materials requirements planning. Human involvement is crucial to the success of MRP II.
8.5.1. The evolution of MRP II
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The evolution of MRP II can be divided into the following four stages: (see Fig 8.3) 1) 2) 3) 4)
Generate better purchasing method. Re-arranging the priority of orders. Form the closed-loop MRP. Accomplish Manufacturing Resources Planning.
Figure 8.3. The evolution of MRP II.
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8.5.2. Generate the Better Purchasing Method Before 1960, the idea on interdependent requirement did not exist. Enterprise used the reorder point system to make purchasing decision regardless of the type of product. It results in huge inventory and waste. When Dr. Orlicky proposed the concept of dependent requirement in 1965, he suggested that reorder point system should be used for independent demand only. The dependent demand should use MRP to generate better reorder quantity.
8.5.3. Consider Re-Arranging the Priority of Orders The dependent demand with MRP is unable to form an effective closed-loop system and feedback information due to lack of information technology. Therefore, how to use resources effectively at this stage is an important issue. In this case, enterprise should consider the priority of customers and orders in the system, schedule optimal production plan.
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8.5.4. Form The Closed-Loop of MRP MRP makes the time decision of the low level requirement and the time schedule to receive order. The main purpose is to order at the right time, for the right items and at the right quantity. The closed-loop of MRP is shown in Fig 8.4. The modified closed-loop of MRP system includes control and manufacturing data in a feedback system, as well as stock level, equipment capacity and demand of working capacity. As a result, it makes MRP system more practical. For example, Capacity requirement planning (CRP) can calculate how many employees and equipment is required to accomplish the production tasks. CRP can calculate the requirement of employees and machine time for each workstation on each time period. CRP is designed for the decision process of manpower and equipment requirement to match the production goal. The main function of CRP including long-term, medium-term and short-term plan as shown in Table 8.1. Fig 8.5 shows the CRP system flowchart, the main input of CRP process includes: 1. Notification of planed orders and replacement orders from MRP. 2. Working load information from workstation. 3. The process information from working process files. Improving capability for different processes or changing plans.
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Figure 8.4. Closed-loop of MRP.
Table 8.1. Capacity planning Tool
Time period
Plan
Solution changed
Resource requirement long-term planning (RRP)
Production Planning (PP)
Rough-Cut Capacity Planning (RCCP)
MPS
Land Equipment Process Power Employee Transfer Tool Produce/Purchase Outsourcing Employee Transfer Over time Outsourcing
medium-term
Capacity requirement short-term planning (CRP)
MRP
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Figure 8.5. CRP System Flowchart.
The main outputs of CRP system are MRP system notification of orders and load work loading reports. These outputs possess three kinds of characteristics: 1. Integrality: Including the notification of planed orders and replace-ment orders. 2. Validity: Based on the latest information priority. 3. Good fitness: What it is convenient to plan the future work.
8.5.5. Manufacturing Resources Planning MRP II (Manufacturing Resource Planning) considers resources of the whole enterprise. It cooperates flexibly with the management environment inside and outside of organization to achieve the goals of enterprises. MRP II is a kind of management information system (MIS) which utilizes the computer to imitate a real industrial environment to include all areas of management and function in its interconnected system. Then it makes the necessary decision needed by different levels to control and asses the system. The flowchart of MRP II is shown in Fig 8.5.
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8.6. ENTERPRISE RESOURCE PLANNING (ERP) With the progress in computer technology and internet network, more sophisticated system is possible. Enterprise Resource Planning (ERP) is one of the systems. For modern enterprise, there are more competitions and consumer demands. Quick response using computers and internet technologies is critical to its success.
8.6.1. The Background of ERP
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In the early period, many enterprises are production oriented, but as the era changes, they become consumers focus. In addition, the development of computer and internet technology decrease the cost of informations. Therefore, the use the computer to integrate resources and planning becomes an important issue.
Figure 8.5. Flowchart of MRP II.
MRP-II is production oriented. ERP evolved from MRP-II using e-commerce and modern technology. MRP-Ⅱsystem is structured outside the whole MIS system. ERP has Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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expanded from coordination of manufacturing processes to integrate the enterprise-wide processes and MIS system. From technological aspect, the whole ERP system improves the original MRP- system and become a more flexible tiered client-server architecture. Fig 8.6 shows the three stages of information technology application in management: MRP、MRP-II and ERP. The evolution of the whole ERP is shown as Table 8.2.
Figure 8.6. The evolution concept of ERP.
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Table 8.2. The evolution of ERP 1960’
1970’
1980’
1990’
2000’
IT
MRP
MRP -I
MRP -Ⅱ
ERP
Application Department Decision goal
Material Control
Production Control Optimize scheduling Few varieties, mass production Reduced inventory through ‘infinite’ manufacturing planning
Manufacture
Enterprise
EERP (Extended ERP) SCM system
Total decision for manufacture Variety,craft production Reduced inventory, fewer stockouts and accurate financial records through ‘infinite’ manufacturing planning and periodic transaction processing.
Optimize enterprise resources decision Variety, mass production Customer focused, dynamically balanced enterprise through asset optimization and real-time transaction processing
manufacturing oriented
manufacturing oriented
Customer oriented
Production Model Vision
Background
Materials requirement plan Few varieties, mass production Materials requirement plan
production oriented
Optimize SCM decision Mass customization Customer focused, dynamically balanced value network through asset optimization and real-time transaction processing Customer oriented
8.6.2. ERP System ERP is an accounting oriented information system to identify and plan manpower, material resources, finance, equipment, sale and customer's information. It has friendly user
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interface and comprehend relational database, computer aided software engineering tools, client/server architecture and open-system portability.
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Elements of ERP may include: (see Fig 8.7) 1. DBMS (Data Base Management System) 2. Human Resource Management system 3. Financial Management system ¾ General Ledger- Accounting records that show all the financial accounts of a business ¾ Account Payable ¾ Accountable Receivable ¾ Asset Management ¾ Cash Management 1. Decision Management system 2. Production Management system ¾ MRP ¾ Production Planning/MPS ¾ Capacity Planning ¾ Sales andandand Operation Management ¾ Production Activity Control ¾ Receiving 1. Logistic and Service Management system ¾ Sales Support andandand Sales ¾ Billing ¾ Purchasing and Shipping ¾ Warehouse and Inventory Management 2. Quality Management system
Figure 8.7. ERP System.
ERP implementation Process: (see Fig 8.8)
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Figure 8.8. ERP implementation Process.
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Finally, let’s discuss the key to successful ERP: 1. 2. 3. 4. 5. 6.
The resolution and support of top managements. Participation of top managements. Experienced special project manager and consultant. Authorization from top managements. Cooperation between departments The efforts of the special project team and good communicating coordination.
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Chapter 9
WORK-IN-PROCESS INVENTORY, JIT AND TOC In this chapter, we discuss work-in-process inventory, Just-in-Time (JIT) system and Theory of Constraints (TOC). The purpose is to extend the problems to consider logistic, financial and other functions.
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CASE STUDY In an automobile part factory, there is a constant conflict between the production manager and the administrative staff of inventory department. The production manager thinks that inventory department does not stock sufficient materials and this affects the smooth running of the production line. The inventory department wants to save holding cost by keeping inventory as low as possible. Understanding the situation, the general manager orders the inventory department to increase the inventory level. Do you think this order is logical? What is your answer? How can the problem be solved?
WORK-IN-PROCESS INVENTORY Work-in-process (WIP) inventory consists of all units currently being worked on. It can be anywhere from the raw materials to finished goods waiting for final sale.
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The value of WIP is dependent on manufacturing lead time and planned production output. Total production cost includes direct material, direct labor, factory overhead (fixed and variable) and WIP.
9.1.1. Production Processes
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Different production process will have different in-process inventory. There are three kinds of production processes: 1) a) b) c) d) e) f)
Continuous Processes and Flow Shop The flow of materials is continuous Mass production with low costs, the capacity is stable. Having standardization, few varieties. Specific equipment for each product; set-up time is long. Manufacture process cannot be changed or interrupted. Lower operation technique.
Intermittent Processes and Job Shop More product variety; equipments can be used for different products The production equipment classified according to its function. The use of space is not as good as the continuous processes, production control is more complicated. d) The manufacture process depends on different products, and the intermittent flow of raw materials. e) It has more flexible production process. f) Higher operation technique. 2) a) b) c)
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Special Project − − − − −
Large investment in a specific time. Different production location for each new product. The equipment needs to be moved to the new product. Manufacturing or the operational procedure is different each time. Highest operation technique.
Continuous Processes and Flow Shop is Make-to-stock, which are based on customer demands forecast. Intermittent Processes and Job Shop is Make-to-order, which consists of customer driven production planning, and highlights producing products in the right time at the right place. Using different production process scheduling will affect the in-process inventory directly.
9.1.2. Scheduling There are two kinds of Scheduling:
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1) Forward Scheduling It is suitable for Continuous Processes and Flow Shop production process. The starting date is the same day the first day going of operation. It begins from the first step to produce, and finish at shipment. Each operation is has schedule progress and time. This method is common in the industry where customers require fast delivery time or products with uncomplicated parts. 2) Backward Scheduling On the contrary, backward scheduling starts with shipping date and going backward to estimate starting working date of each process. This method is common in the assembling factory. Since production is run when needed, it reduces material parts and WIP inventory. The scheduling logic of MRP is usually used in Backward Scheduling. WIP inventory provides buffer for work stations. It also regulates the work stations bottleneck. But too many WIP will result in: 1) 2) 3) 4) 5)
More funds accumulated. Complicated and expensive production control system. More materials will take up more space. Higher materials carrying cost. Longer manufacturing cycle time.
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9.1.3. Lead Time Lead time is a time period to initiate any production process. Most work needs lead time period to warm up or prepare. Lead time differs in length depending on the nature of the process. If lead time is short, it may be neglected. Lead time may have different meaning for different occasion. Two kinds of lead time are purchasing and production lead time. Production Cycle Time is the period of time from raw material purchase to packed final product. Manufacturing Cycle Time is the sum of all operation time of manufacturing processes and it consists of five units as shown in Fig 9.1.
Figure 9.1. Manufacturing Cycle Time.
1) 2) 3) 4) 5)
Set up time: The preparing time of all materials, machines and work stations. Processing time: the production operation time. Waiting time: The time of material is waiting to be transfer to thenext work station. Moving time: The transfer time during the work stations. Queuing time: The time of material is waiting because of order is processed in the work station.
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Reducing the production and purchasing lead time help to improve the forecast accuracy. Improvement in forecast accuracy helps to improve customer's service level and reduce the inventory cost.
9.1.4.Time Cycle Charts Time Cycle Charts consist of manufacture operation time, assembling time and purchasing time, as shown in Fig 9.2. It can be used to calculate total production lead time. Product (1) consists of parts (2), (6) and purchasing part (11). Part (2) consists of manufacturing items (3), (4) and (5). Manufacturing items (3), (4) and (5) are produced from purchasing materials (12), (13) and (14). Part (6) consists of manufacturing items (7), (8) and purchasing part (17). Manufacturing items (7), (8) are produced from purchasing materials (15), and (16). The purchasing material (15) is the bottleneck item. The cumulative lead time is 22 weeks if there are no inventories. If the customer will purchase the finished final product after six weeks, then there is only 6 weeks lead time in the manufacture in Figure 9.2. So, manufacturing items (3), (4), (5), (7), (8) and purchasing part (11), (17) should have inventories to satisfy the customer's demand.
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Figure 9.2. Time Cycle Charts.
9.1.5. Bottleneck Work Centers In general, bottleneck work station takes place in two situations: 1) When one station work is finished but the operation of next station has not finished yet, then the work piece is unable to flow into the next work station. The work piece is blocked at this station; it is called Blocking. 2) The work piece of the preceding work station has already been done, but the operation of the work station preceding has not been done yet, then the work piece is unable to flow into this work station. It will cause the stand-by of this work station (idle) phenomenon; it is called Starving. In other words, the bottleneck work station limits production capacity. If capacity wants to be increased, the bottleneck work station capacity must be improved.
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9.1.6. Input/Output Control Table 9.1 shows the input and output record of Intermittent Processes and Job Shop. Current physical input or output plus ex- cumulative balance minus current project input or output is equal to the current cumulative balance. Ex-physical inventory plus current physical input minus physical output is equal to the current physical inventory. Table 9.1 shows that projected weekly output is 600 hours in the first four weeks, following by 540 hours in the next weeks. The project weekly input is 540 hours for all weeks. Since the first four weeks the projected weekly output is higher than the projected weekly input with a total value equal to 240, so the physical inventory must have 240 hours available. Table 9.1 Input /Output Record5 work stations Week Project input Physical input Cumulative balance Project output Physical output Cumulative balance Project inventory Physical inventory balance
12 540 540 0 600 610 +10 180 170 -10
13 540 530 -10 600 520 -70 120 180 +60
14 540 500 -50 600 600 -70 60 80 +20
15 540
16 540
17 540
18 540
19 540
20 540
21 540
600
540
540
540
540
540
540
0
0
0
0
0
0
0
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Week 14th Input cumulative balance =500+ (-10)- 540 =-50 Week 13th Output cumulative balance =520+10 - 600 =70 Week 13th Physical inventory=170+530 – 520=180
The final data shows that there is no physical inventory in week 14th. So, the capacity of the work shop can exceed the order need. Table 9.2 Input /Output Record11 work stations Week Project input Physical input Cumulative balance Project output Physical output Cumulative balance Project inventory Physical inventory balance
21 220 110 -110 220 150 -70 60 20 -40
22 220 150 -180 220 140 -150 60 30 -30
23 220 140 -260 220 160 -210 60 10 -50
24 220 130 -350 220 140 -290 60 0 -60
25 220
26 220
27 220
28 220
29 220
30 220
220
220
220
220
220
220
60
60
60
60
60
60
Table 9.2 shows that projected weekly input and output is 220 hours. But the physical output is less than 290 hours, and the physical input is less than 350 hours. In examining the reason of this problem, it is seen that the ex-work station is not being sent accurately.
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¾ There are two kinds of out of control situations:
1) Inventory exceeds upper limit: The possible reason is equipment shut down, low efficiency of the next work station. It can be improved by reducing input and increasing output.
Figure 9.3. Capacity limit.
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Fig 9.3 shows the relationship between input, output and capacity. The main function of inventory and lead time is to adjust capacity. 2) Output is lower than lower limit: The possible reason is equipment shut down, low efficiency of work station or lower input. It can be improved by increasing input and output. ¾ How to reduce work-in-process inventory:
The delivery schedule of materials close to process operation time. Release materials only when needed. Balance input and output. Start operation only when employees, materials, equipments and support services are ready. 7) To avoid low utilization rate, start with the shortest cycle job first. 8) Consider increase production time only if actual demand exceed supply. 3) 4) 5) 6)
9.1.7.Critical Ratio Technique Critical Ratio (CR) Technique primary determine the priority of the orders. We can calculate the Critical Ratio as follows:
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Re quired time Spare time = Physcial input time Operation required time Shipping date − today = Operation required time
CR =
(9.1)
The order with smaller CR represents the priority order. ¾ Example1
If the production schedule today is 34th day, Table 9.3 shows the information of each job. Use the Critical Ratio Technique to determine the order for each job. Table 9.3 Scheduling sheet Job A B C D E
Shipping date 70 39 36 64 37
Operation required time (day) 6 5 3 8 6
【Solution】
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The Critical Ratio calculation for each job, is shown in Table 9.4. Table 9.4 Critical Ratio for each job Job E C B D A
CR (37-34)/6=0.5 (36-34)/3=0.67 (39-34)/5=1.0 (64-34)/8=3.75 (70-34)/6=6
Job E and C are behind schedule. Priority is given to speed up the operation time. Job B is on schedule, Job A and D are ahead of schedule. The formula of CR can also be expressed as follows: CR =
Physcial input time Spare lead time
Physcial input time =
Holding inventory− Safety Stock Average daily required
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(9.2)
(9.3)
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9.2. JUST-IN-TIME (JIT) Due to increasing competition and changing business environment, corporations are pursuing different supply chain management strategies to fulfill a variety of customer requirements and improve profits. Under this circumstance, flexibility and adaptation are increasingly important. Enterprises should reduce lead times, shorten planning cycles, and expedite distribution to satisfy markets demand. A competitive enterprise should have the ability to adapt better and faster, such as Just in time (JIT) management system to satisfy their customers. It is called Zero Inventory Management. JIT technique reduces inventory, reduces space requirements by shortening process distances, and reduces costs by eliminating non-value-added activities and improve efficiency and quality. The fundamental concepts of the JIT system include:
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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Pull process designed Consistence high quality Small-lot production Reduce set-up time Load leveling system Standardization of parts and operations Close Cooperation with supplier Multi-skilled workers Single supplier Reduces space requirements by shortening process distances Improve machine reliability through TPM (Total Preventive Mainten-ance)
Since the Toyota Motor company has developed and applied the JIT production system successfully, the JIT manufacturing system is synonymous with Toyota Production System (TPS) or Lean Production System (LPS). LPS is a customer driven production plan, and emphasizes on production in the right time and at the right place. LPS strategies are based on actual customer demands at the end of the line, called pull production system. They are considered to be reactive-oriented and reduces inventory through JIT. JIT is a dynamic linked system based on actual usage parts. System is “linked” through the use of Kanbans. JIT system originated from replacement system in the supermarket. It is successful applied to production and operation in the manufacturing industry. When JIT system is implemented successfully, enterprise can improve in efficiency, quality, inventory, space requirements and costs. The turnover ratio can also be improved. Table 9.5 shows the goals of JIT system:
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Table 9.5 JIT Goals •
Elimination of non-value-added activities and waste
•
Zero defects (ZD)
•
Zero setup time
•
Zero residues
•
Zero stock
•
Zero handling
•
Zero waiting
•
Zero breakdowns
•
Zero lead time
•
One lot size
•
100% on time delivery service
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The basic elements of JIT production system are shown in Fig 9.4.
Total Quality Management
Elimination of Waste
Quality of Employee
Figure 9.4. Basic elements of JIT production system.
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Table 9.6 Push production vs. Pull production (JIT) system Push production Some defects are acceptable Economic batch size（more is better） Quick production system Saf ety stock Inventory regulates capacity Inventory is a kind of property Waiting if need Negotiate with suppliers Multiple suppliers Maintain machine when br eakdown s Lead time is need ed Setup time is need ed Command-oriented Specialist workers
Pull production (JIT) system Zero def ects Small batch size（less is better） Load leveling system Zero sto ck Inventory as waste Inv en tor y is a k ind of debt Zero waiting Collaborate with suppliers Single supplier Improve machine reliability through TPM Zero lead time Zero setup time Reactive-oriented Multi-skilled workers
The essence of TPS is to encourage all employees to work hard at eliminating the unnecessary stock through a pull manufacture process design. It enables an enterprise to respond quickly as well. Automation (Jidoka) and JIT are crucial aspects for efficient TPS. It shortens the manufacture throughput time. Pull production control strategies are based on actual customer demands; they are considered to be reactive-oriented. The differences between push and pull production systems are shown in Table 9.6.
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¾ Elimination of Waste:
JIT eliminates non-value-added activities and waste. The definition of wasted is any costs or activities without increasing the value of the product. Wastes include materials, equipments, defects, WIP, transportation, waiting, pipeline inventory, backing up resource, repaired time, inspection time and classification time. In general, all costs without value added are wastes. ¾ Inventory as Waste:
Overstocking is a waste; too much stock will conceal production problems, such as: 1. 2. 3. 4. 5. 6. 7. 8.
Mechanical breakdowns. Bad quality or high scrap rate. Out of standard materials. Wear and tear cutters. Worker absence or late. Delay of parts. Redundant material and equipment. Unnecessary inspection or set up.
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These problems will delay the flow of the product. To eradicate the problems, one should observe the symptom as well as the cause of the problems. ¾ The ideal goals of JIT:
1. 2. 3. 4. 5. 6. 7.
The batch must be as small as possible. Keep high and consistent quality. Capable and reliable employees. Wasteful stocks be reduce to minimum. 100% reliable machines. The production schedule must be Load leveling system. Space should be well used.
New technologies should be used to achieve these goals, such as:
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1. Shorten the set up time. 2. Training multi-skilled workers. 3. Develop the Total Quality Management (TQM) cultural, that is: all employees have responsibility of quality management. 4. Guarantees no breakdown through TPM. 5. Using group technology cells to modularize parts and improve product flow. (Cell production system) 6. Collaborate with reliable supplier. 7. Small lot/high frequency delivery strategy. Usually, the size of batch depends on the holding cost and set up cost. JIT is devoted to shorten set up time (usually smaller than 10 minutes). The small batch is more economical due to lower set up cost. With small batch production, it is easier to expose and solve problems in the process. The small batch production design can reduce lead time, capital and storage space. The quality guide of JIT is shown in Fig 9.5. Enterprises should reduce defects zero defect or close to zero. (ZD or PPM; parts per million).
9.3. THEORY OF CONSTRAINTS TOC (Theory of Constraints) is a management philosophy introduced by Israeli physicist and enterprise management advisor Dr. Eliyahu M. Goldratt. He proposed the theory in his book titled “The Goal” in 1984. The management philosophy of TOC focuses on reducing bottlenecks and improving the performance of organizations through continuous improvement. It is a simple and effective general knowledge management. TOC method is an approach to solve constraints and problems. A logic map of the problem and its roots cause (the bottleneck) are identified to improve the efficiency of the company. Bottleneck reduction is an effective way to achieve management goal.
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Work-in-Process Inventory, JIT and TOC
Figure 9.5. JIT Quality Guide.
¾ Summary of TOC
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1. 2. 3. 4. 5. 6. 7.
TOC balances production line flow. Determines the process time and WIP inventory from constraints. Maximize the non-bottleneck station. Resources must be utilized effectively. The time lost in the bottleneck is equal to the whole system time lost. The processing batch quantities can be changed by process route and overtime. Check all bottleneck conditions in decision making.
VAT PLANTS ¾ V-Plant
Figure 9.6. V-Plant. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
135
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Figure 9.7. A-Plant.
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¾ T –Plant
Figure 9.8. T-Plant.
CONTINUOUS IMPROVEMENT The basic step of continuous improvement is shown in Figure 9.8: 1. 2. 3. 4.
Identify the biggest bottleneck in the system where the output is limited. Optimize bottleneck output: make it produce near 100% outputs. Implement the optimal method for all other work stations. Improve the bottleneck output. When the bottleneck is dispelled, get back to step 1 and identify the same procedure again to avoid inertia.
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TOC is logic, systematic way to continuous improvement process. The following questions should be asked:
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1. What to Change? 2. What to Change to? 3. How to Cause the Change?
Figure 9.9. Steps of continuous improvement.
CONCLUSION To pursue higher profit, enterprise should pay more attention to inventory control. There are several topics we discussed in this chapter, such as: different production procedure, scheduling method, lead time and bottleneck. These all are related WIP inventory. Finally, we discuss the influence of JIT and TOC on the WIP inventory. Effective inventory management is a great key factor to achieve successful operation.
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Chapter 10
DETERIORATING INVENTORY MODELS Traditional Economic Lot Size (ELS) models do not consider deteriorating products in their inventory cost structures. One characteristic of deteriorating stock is the higher inventory holding costs when it is retained for a long period. Most deteriorating products are age-dependent.
CASE STUDY
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A manager from a fresh food department orders fresh milk everyday. Due to uncertain market and the short shelf life of milk, he has difficulty to decide the order quantity of the fresh milk. Too much order leads to perishable lost of inventory, but too few order results in lost sale. If you are the manager, how do you solve the difficult problem?
10.1.THE IMPORTANT OF DETERIORATING INVENTORY There are many reasons for holding inventory. They are: uncertain demand, unstable quality, management mistake and deteriorating items. Too much or too little inventory will result in financial loss. Overstock will increase holding and deteriorating costs while under stock will cause sales loss, bottleneck and unsatisfied customers. Therefore, enterprises
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should develop a good inventory management policy which will influence the competitiveness in of the company. Materials are classified into five categories: raw materials, work-in-process material, manufactured goods, assembling module and deteriorating material. Traditional inventory models do not consider deteriorating material in their inventory cost structures due to the fact that certain items have negligible deteriorating rate. But in reality, the value and quantity of most stocks will deteriorate with time; it is especially true with agro-food products and fashion goods. For example: fresh fruit, vegetables, meat, petrol, electronic good and fashion cloth will decrease in utility with time. We call these products “deteriorating items”. Deteriorating phenomenon is common; deteriorating stock can result in extra cost, especially if it is retained for a long period. Traditional Economic Lot Size (ELS) models do not consider deteriorating products in their inventory cost structures. In recent years, the studies on deteriorating stock become more important due to the flourishing fresh food supermarket, semiconductor industry and logistics industry.
10.2. CHARACTERISTIC AND CATEGORY OF DETERIORATING In general, most products will deteriorate. “Deteriorating” means products are unable to function the way they are made. The characteristics and categories of deteriorating items are explained as following.
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10.2.1 Characteristics of Deteriorating Items The deteriorating properties of inventory are classified into three kinds (Ghare and Schrader, 1963), as follows 1. Direct spoilage: For instance: vegetables, fruit and raw fresh food that spoils easily, as shown in Fig 10.1.
Figure10.1. vegetables, fruit and raw fresh food.
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Deteriorating Inventory Models
141
2. Physical depletion: For instance: Petrol, alcohol and volatile liquid, as shown in Fig 10.2.
Figure 10.2. Petrol, alcohol and volatile liquid.
3. Deterioration:
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For instance: electronic products, film and medicine that may reduce in value, lose in efficiency and changes in quality due to radiation, as shown in Fig 10.3.
Figure 10.3. Electronic products, film and medicine.
102.2. Categories of Deteriorating Items The categories of deteriorating items are usually classified by the life of product and the time value. 1. Classified by the time value (Raafat, 1991) (1). Utility constant Although the function decreasing during the holding time period, but the utility while using will not have too apparent difference, such as the liquid medicine. (2). Utility increasing
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Hui-Ming Wee
The value of products is increasing during the holding time period, such as some drinks showing in Fig 10.4. (3). Utility decreasing The value of products will reduce as time increases, such as vegetables, fruit for instance catch fresh food, etc.
Figure 10.4. Utility increasing products: some drinks etc.
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2. Classified with the life of product (Nahmias, 1978): (1). Fixed lifetime The life cycle of the products is predicable, and the life-span has nothing to do with other factors or parameters of the whole stock system. It is called time-independent deterioration. The value or the function of this product will disappear gradually within life cycle. When products exceed their life-span in retention period, it will be considered as completely perished. It has no value at all and can’t be used again such as: milk, blood stock and food etc.
Figure 10.5. Completely perished products: milk, blood stock and food etc.
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Deteriorating Inventory Models
(2). Random lifetime This kind of product has no reserve time-limit. The life-span of the products is following random probability distribution during the process of stock holding, such as: Gamma distribution, Weibull distribution, Exponential distribution etc. The products are degrading of a certain distribution rate, during the holding period. It is called time-dependent deterioration, such as: Electronic element, chemical medicines etc. Example 1: The beginning inventory of a certain product is 200, and the deteriorating rate is 20% at the end in each initial stock. So the stock will decrease periodically. Suppose the deterioration is the only factor of inventory reduction (That is: does not consider the reduction of the demand). Try calculating the final inventory amount of each period. 【Sol】
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(1). Period 1: Beginning inventory = 200 Deteriorating quantity = 200 x 0.2 = 40 Ending inventory = 200 - 40 = 160 Ending inventory of period 1= Beginning inventory of period 2= 160 (2). Period 2: Beginning inventory = 160 Deteriorating quantity = 160 x 0.2 = 32 Ending inventory = 160 - 32 = 128 Ending inventory of period 2 = Beginning inventory of period 3= 128 So as to analogize and finish the form under, then it will understand the influence of each final inventory. Period
Beginning inventory
Deteriorating
Ending inventory
1
200
40
160
2
160
32
128
3
128
25.6
102.4
4
102.4
20.48
81.92
5
81.92
16.38
65.54
6
65.536
13.11
52.426
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10.3. EOQ FOR DETERIORATING ITEMS WITH FIXED DEMAND In 1960’, the inventory models do not consider perishable products in their inventory cost structures. In the meanwhile, the traditional Economic Lot Size (ELS) model has too many restriction assumptions could not represent many real situations and reduce its utilization. Hadley and Whitin (1961) proposed an inventory model with deteriorating items during the holding period at first. Later decades, several researches had studied deteriorating inventory models. ¾ The EOQ model of deteriorating items with fixed deterioration rate:
Ghare and Schrader (1963) had studied the issues of deteriorating items. They expanded the restriction assumptions of traditional EOQ model. They assumed that the proportion of deteriorating items follow the Exponential distribution and proposed the EOQ model of deteriorating items with fixed deterioration rate:
R(t ) : The demand rate in time t Q(t ) : The stock level while in purchasing period
Q(t ) will be influence with the demand rate R(t ) Q(t )
⇐
and the deterioration rate θ , then:
dQ(t ) + θQ(t ) = − R(t ) dt (t )
(10.1)
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¾ Inventory model for deterioration
Convert and Philip (1973) had modified the deterioration EOQ model proposed by Ghare and Schrader in 1963, using two parameters Weibull distribution.
D(t ) : The demand rate in time t Q(t ) : The stock level while in purchasing period α , β : Two parameters of Weibull distribution Q(t )
⇐
dQ(t ) + αβ t β − 1Q(t ) = − D(t ) dt (t )
(10.2)
¾ Three deterioration parameters inventory model
Philip (1974) had modified the deterioration proposed by Convert and Philip in 1973, using three parameters in Weibull distribution. He discussed two difference situations of deterioration: (1) the product had deteriorated when it was received; (2) the product was deteriorating after storage for a while.
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Deteriorating Inventory Models The deterioration function
g (t ) :
g (t ) = αβ (t − γ )β −1 ,
where
(10.3)
γ : Location parameter
¾ Gamma distribution inventory model
Tadikamalla (1978) proposed an EOQ inventory model with Gamma distributed deterioration. The decision function is calculated without backorder. The deterioration function
g (t ) =
g (t ) :
f (t ) f (t ) = , t G(t ) 1 − ∫0 f (x )dx
(10.4)
where
−t
β f (t ) = t α −1 e α Γ(α )β
(10.5)
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f (t ) Probability density function of Gamma distribution.
α : Shape parameter β : Scale parameter
10.4. EPQ FOR DETERIORATING ITEMS WITH FIXED DEMAND Misra developed an EPQ model with a finite replenishment rate and considered backordering in 1975. The deterioration rate belongs to the Gamma distribution and the production model is Weibull distribution with two parameters. Mak (1982) considered backordering for unfilled orders with Exponential distribution. In the same year, Hwang and Hwang studied the effects of two inventory-issuing policies for items with Weibull deterioration. Heng et al. (1991) used the aid of computer to generate a lot size, order level inventory system with finite replenishment rate, constant demand rate, and exponential deteriorating. Wee (1992) developed the inventory model for perishable items with partial backordering,
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and then in 1993, he developed the production lot size model for deteriorating items with constant production and demand rate with partial backordering.
10.5. EOQ FOR DETERIORATING ITEMS WITH VARIED DEMAND In this section, we will discuss the issue of varied demand. For the real situation, many researches have been analyzing the effect of deterioration and variations in the demand rate with time. The early research in 1969, Silver and Meal gave a heuristic solution procedure for the inventory model with time varying demand. Then, Donaldson (1977) studied the issue of inventory replenishment policy for a linear trend in demand. The linear function of requirement in planning period (0, H),
R(t ) :
R(t ) = a + bu, where
(10.6)
a, b : Constant
This model discussed the time-dependent requirement function
()
R(t ) during the time
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period from t to ( t + Δ t ) should be R t Δt . It is a continuous function. Dave and Patel (1981) derived a lot size model for constant deteriorating items with time increasing demand. They proposed the deterioration model ( T, Si ): Si : Order quantity Ti : Time period from 0 to the end of i period, i = 1, 2, ……m m : Order times
Qi (t ) : Inventory in time t of i period
θ : Deterioration rate (constant) Assumption: (1). Each time period has same time-length. (2). Infinite replenishment rate. (3). Lead time = 0 The inventory function:
dQi (t ) + θQi (t ) = −(a + bt ) dt
(10.7)
Mitra et al. (1984) developed a simple method by modifying the EOQ model for linearly increasing and decreasing demand. Bahari-Kashani (1989) presented a heuristic model for Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
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Deteriorating Inventory Models
determining the ordering schedule of deteriorating item and linearly time demand. Chung and Ting (1994) modified the heuristic model of Bahari-Kashani (1989) to simplify and determine the replenishment schedules for deteriorating item with time proportional demand. Furthermore, Hariga and Benkherouf (1994) developed an inventory replenishment model for deteriorating items with exponential time varying demand in the same year. They concluded that the order cycle was increasing or decreasing with the increasing or decreasing demand. All of researches above do not considering backorder situation. But in practical, the backorder is unavoidable, so we are going to discuss EOQ for deteriorating items with backorder. Sachan (1984) extended the time-demand model developed by Dave and Pate (1981), considered the economic order-quantity model when backorder was allowed with fixed cycle time and time proportional demand. This model assumed elapsed from initial to ith cycle ( i = 1, 2, …, m ). Where
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ti − Ti = γ (Ti − Ti − 1), 0 ≤ γ ≤ 10 γ : Shortage rate
t
i
is the total time
(10.8)
The shortage could be replenished in next period. So did Goswami and Chaudhuri (1995), they developed a model for deteriorating items with linearly time-varying demand, finite shortage cost and equal replenishment intervals in 1991. Hariga (1995) modified the model developed by Goswami and Chaudhuri (1995) and assumed that the length of the replenishment cycles were equal for a continuous time-varying demand with shortage allow but not in the final period. Wee (1992) proposed economic production lot size model for deteriorating items with partial backordering. Abad (2000) considered the problem of determining the lot size for perishable items under finite production, exponential decay, partial backordering and lost sale.
10.6. EPQ FOR DETERIORATING ITEMS WITH VARIED DEMAND Aggarwal and Bahari-Kashani (1991) extended Hollier and Mak's (1983) model to allow flexible rates of production in each period. The model assumed fixed production interval. Both of the exponentially decreasing demand models did not allow shortages. Wee (1992) proposed an economic production lot size model for deteriorating items with partial backordering and then he provided a solution procedure that entails intermittent checking of the Hessian of the approximate cost expression in 1993. In 1995, Wee modified his model proposed in 1933 and studied the optimal production policy for perishable items with partial backordering. He concluded that the optimal production lot-size policy for perishable items has been developed for a situation when shortage is partially backordered and time is treated as a discrete variable. The total cost consists of cost due to deterioration, ordering cost, holding cost, cost due to backordering and cost due to lost sales.
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10.7. AMELIORATING INVENTORY MODEL
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The ameliorating inventory is the ameliorating items whose utility increases over the time by ameliorating activation. Generally, fast growing animals like duck, chicken, pigs, fish etc. in farm, or in pond are these types of items. When these ameliorating items are kept/ reserved at farm or pond, the weight or value or utility of the items increases due to growth. But it is also decreases due to disease, death or some other factors. Ameliorating inventory model usually grow up fast comparatively during initial stage. But the value-added speed of this kind item will slow down of later stage, so Weibull distribution is often used for being regarded as the speed of ameliorating rate. Hwang (1997) performed an EOQ model for ameliorating inventory in accordance with Weibull distribution. Then in 1999, he developed an inventory of ameliorating items of a PSO (partial selling quantity) model for selling the surplus inventory accumulated by ameliorating activation and economic production model(EPQ) considering issuing policies, FIFO and LIFO. Mondal et al., (2003) developed an instantaneous replenishment inventory model of ameliorating items for prescribed time period. The rate of amelioration in their model followed the Weibull distribution and the demand rate was assumed to be a function of selling price of an item and shortages were not allowed. Hwang (2004) studied deteriorating items to determine the minimum number of storage facilities among a discrete set of location sites and formulated this problem using the stochastic set-covering problem, which can be solved via the 0 ~ 1 programming method. Moon et al., (2005) developed model for ameliorating items with time-varying demand pattern taking into account the effects of inflation and time value of money over a finite planning horizon. So, it is important to discuss the ameliorating factor in the ameliorating inventory model.
10.8. EXAMPLE Ghare and Schrader (1963) were the first researchers to consider the effect of deterioration on inventory items. They derived an economic order quantity model (EOQ) where inventory items decay exponentially with time as shown in Fig 10.6.
Figure 10.6. Inventory items decay exponentially with time. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Deteriorating Inventory Models
149
Examle 2: Assumption of Exponential deteriorating inventory model:
R(t ) : Requirement rate Q (t ) : Inventory level θ : Deteriorating rate (constant)
Î
dQ(t ) + θQ(t ) = − R(t ) dt (t )
θ : is constant. R(t ) : is constant, it means fixed-demand. ⎡
Then
(
)
Q(t ) = ⎢ R(t )⎥ eθT − 1 ⎣
θ
⎤ ⎦
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Convert and Philip (1973) had modified the deterioration EOQ model proposed by Ghare and Schrader in 1963, using two parameters in Weibull distribution as shown in Fig 10.7.
Figure 10.7. Weibull deteriorating inventory model.
Example 3: Assumption of Weibull deteriorating inventory model:
D(t ) : The demand rate in time t Q(t ) : The stock level while in purchasing period α , β : Two parameters of Webber distribution
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Î
dQ(t ) + αβ t β − 1Q(t ) = − D(t ) dt (t )
D(t ) : is constant, it means fixed-demand. α : Shape parameter β : Scale parameter Then
0 (nβ +1) ∞ β Q(t ) = ∫0T ( D(t )eαt )dt = D(t ) ∑ α T n=0 n!(nβ + 1)
CONCLUSION
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Deteriorating products include a very large range. The common one are fruit, vegetables, fish, meat, petrol, etc. In recent years, with growing the flourishing development of the fresh supermarket, semiconductor industry and logistics industry, the management of the deteriorating inventory is being paid attention to. So one should consider the deteriorating factors in the research of the inventory model and make the model more accord with the use on the practice.
REFERENCES Abad, P. L., “Optimal lot size for a perishable good under conditions of finite production and partial backordering and lost sale,” Computers and Industrial Engineering, Vol. 38(4), (2000). pp.457-465. Aggarwal, V. and Bahari-Kashani, H., "Synchronized production policies for deteriorating items in a declining market, " Ι Ι E Transactions, Vol.23, No.2, (1991). pp.185-197. Bahari-Kashani, H., "Replenishment schedule for deteriorating items with time proportional demand," Journal of the Operational Research Society, Vol.40, (1989). pp.75-81. Chung, K. J. and Ting, P. S., "On replenishment schedule for deteriorating items with timeproportional demand," Production Planning and Control, Vol.5, No.4, (1994). pp.392396. Convert, R. P. and Philip, G. C., "An EOQ model for items with Weibull distribution deterioration", AIEE Transactions, Vol.5, No.4, (1973). pp.323-326. Dave, U. and Pate, L. K., "(T,Si) policy inventory model for deteriorating items with time proportional demand, " Journal of the Operational Research Society, Vol.32, (1981). pp. 137-142. Donaldson, W. A., "Inventory Replenishment policy for a linear trend in demand - an analytical solution," Operational Research Quarterly, Vol.28, No.3, (1977). pp.633-670.
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Deteriorating Inventory Models
151
Ghare, P. M. and Scharager, G. F., "A model for an exponentially decaying inventory', Journal of Industrial Engineering, Vol. 14, (1963). pp.23B-243. Goswami, A. and Chaudhuri, K. S., "An EOQ model for deteriorating items with shortages and a linear trend in demand, " Journal of the Operational Research Society, Vol. 42, No. 1, (1991). pp. 1105-1110. Hadley, G., and Whitin, T. M., "An Optimal Final Inventory Model ," Management Science, Vol.?, (1961). pp.179-183. Hariga, M., "An EOQ model for deteriorating items with shortages and time-varying demand," Journal of the Operational Research Society, Vol.46, No.3, (1995). pp.398404. Hariga, M. A. and Benkherouf, L., "Optimal and heuristic inventory replenishment models for deteriorating items with exponential time-varying demand," European Journal of Operational Research, Vol.7, (1994). pp.123-137. Heng, K. J., Labban, J. and Linn, J., "An order-level lot-size inventory model for deteriorating items with finite replenishment rate," Computers and Industrial Engineering, Vol.20, No.2, (1991). pp. 187-197. Hollier, R. H. and Mak, K. L., "Inventory replenishment policies for deteriorating items in a declining marker," International Journal of Production Research, Vol.39, (1983). pp.265-270. Hwang H. and Hwang, H. S., “Optimal issuing policy in production lot size system for items with Weibull deterioration,” International Journal of Production Research, Vol. 20, (1982). Pp. 87-94. Hwang, H.S., “A study on an inventory model for items with Weibull ameliorating,” Computers and Industrial Engineering, Vol. 33, (1997), pp. 701–704. Hwang, H.S., “Inventory models for both deteriorating and ameliorating items,” Computers and Industrial Engineering, Vol. 37 (1999), pp. 257–260. Hwang, H. S., “A stochastic set-covering location model for both ameliorating and deteriorating items,” Computers and Industrial Engineering, Vol. 46(2), (2004). pp. 313319. Mak, K. L., "A production lot size inventory model for deteriorating items," Computers and industrial Engineering, Vol.6, No.4, (1982). pp.309-317. Misra , R. B., "Optimum production lot size model for a system with deteriorating," International Journal of Production Research, Vol. 15, (1975). pp.495-505. Mitra, A., Cox, J. F. and Jesse, R.R., “A note on determining order quantities with a linear trend in demand,” Journal of the Operational Research Society, Vol. 35, (1984). Pp. 141144. Mondal, B., Bhunia, A. K. and Maiti, M., ”An inventory system of ameliorating items for price dependent demand rate,” Computers and Industrial Engineering, Vol. 45(3), (2003). pp. 443 – 456. Nahmias, S., "Perishable inventory theory ; a review", Operations Research, Vol.30, No.4, (1978). pp.680-708. Philip, G. C., "A generalized EOQ model for items Weibull distribution, " AIIE Transactions, Vol.6, (1974). pp.159-162. Raafat, F., "Survey of literature on continuously deteriorating inventory models". Journal of the Operational Research Society, Vol.42, No.1, (1991). pp.27-37.
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Sachan, R. S., "On (T,Si) policy inventory model for deteriorating items with time proportional demand," Journal of the Operational Research Society, Vol.35, No.1 1, (1984). pp.1013-1019. Tadikamalla, P. R., “An EOQ inventory model for items with gamma distributed deterioration”, AIIE Transactions, Vol. 10,(1978). pp. 100-103. Wee, H. M., "Perishable commodities inventory policy with partial backordering," Chung Yuan Journal, Vol.12, (1992). pp.191-198. Wee , H. M., "Economic production lot size for deteriorating items with partial backordering," Computers and Industrial Engineering, Vol.24, No. 3, (1992). pp.449458. Wee, H. M., "Economic production lot size model for deteriorating items with partial backordering," Computers and Industrial Engineering, Vol.24, No.3, (1993). pp.449-458. Wee, H. M., "Optimal production policy for perishable items with partial backordering, " Engineering Optimization, Vol. 23, (1995). pp.315-322.
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Chapter 11
SOLVING INVENTORY PROBLEMS WITHOUT DERIVATIVES In the past, most of the deteriorating inventory problems were solved by the calculus method. Recently, non-calculus method (without derivatives) has become very popular. It enables one to study inventory management problem without understanding the complicated calculus. In this chapter, we discuss how the non-calculus method applies to the deteriorating inventory problems. We start by simplifying deteriorating inventory model so the decay curves are approximately a straight line. Then, we derive the EOQ and EPQ model using noncalculus method.
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11.1. INTRODUCTION Most previous works have applied differential calculus to derive an optimal solution for economic ordering policy. Grubbstrom and Erdem (1999) elaborated the formulation of classical EOQ without differential calculus. Cardenas-Barron (2001) extended the formulation to economic production quantity with shortage model. Yang and Wee (2002) derived an optimal policy for the integrated vendor-buyer inventory system without using derivatives. Under different economic issue, Wee et al. (2003) later developed an economic ordering quantity model with temporary sale price without using derivatives. Zanoni and Grubbström (2004) used the approach of Grubbström and Erdem (1999) and considered the model by Braglia and Zavanella (2003) to develop an analytic formulation. However, the integrated multiple-stage production inventory is neglected in the inventory model development.
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11.1.1. Developing of Without Derivative Methodology
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d e t e r i o r a t i n gc u r v ea sl i n e s
Figure 11. Developing of without derivative methodology.
11.1.2. Comparative of Without Derivative Models There are many methodologies used in inventory model without derivative. The comparative of without derivative model is shown in Table 11.1.
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Solving Inventory Problems without Derivatives Table 11.1. Comparative of without derivative model Authors
Grubbström and Erdem 1999 Complete square method
Minner
Leung
Teng
Wee et al.
2007 Cost comparisons
2008 Complete square method
2009 Cost comparisons
Backorders Linear
yes yes
Utilize pair of cost Solve the cost on the convex 1 (T) EOQ、EPQ yes yes
Solve the problem by twice
Variable Solution
Change the order cost 2 variables solution 2 (Q, B) EOQ
2008 Arithmeticgeometric-meaninequality method Utilize Arithmeticgeometric-mean
Year Methodology
Characteristic
2 (U, S) EOQ yes(partial) yes (2 Slopes)
1 (Q) EOQ、EPQ yes yes
Utilize pair of cost Solve the cost on the convex 2 (r, Q) EOQ、EPQ yes yes
11.2. THE EOQ WITH BACKLOG
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Grubbström (1996) has been shown that the standard EOQ formula can be derived algebraically without derivatives. This section extends the method with backlogging. This method let the EOQ formula would be accessible for younger students in school. The EOQ with backlog (saw-tooth) development of inventory has the form:
The following notations are used:
D : demand rate C : Ordering cost per order Q : maximum inventory level B : maximum backorder level υ : backorder cost (per unit and unit time) H : holding cost (per unit and unit time) Q + B : order quantity
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The objective function (AC: average cost per time unit) may then be written:
AC =
D ⎛⎜ HQ 2 + νB 2 + C ⎞⎟ ⎟ Q + B ⎜⎝ 2D 2D ⎠
(11.1)
If we write C as:
C = H × 2CDν + ν × 2CDH 2D H (H +ν ) 2D ν (H +ν ) ,
(11.2)
With some slight algebraic development, the cost expression may be shown as:
⎡ ⎤ ⎛ 2 2CD ⎞ ν ⎛⎜ 2 2CDH⎞⎟ Q+B 2CDH D H ν ν ⎢ ⎥ ⎜ ⎟ AC= Q+ + B + − H(H+ν)⎟⎠ 2D⎜⎜ ν(H+ν) ⎟⎟ D H+ν ⎥⎥ Q+B⎢⎢2D⎜⎝ ⎝
⎣
⎠
⎦
+ 2CDHν H +ν ⎡ ⎛ ⎞ ⎛ 2CDH 2CDH ⎞⎟⎤⎥ = D ⎢⎢ H ⎜⎜ Q 2 − 2Q 2CDν + 2CDν ⎟⎟ + ν ⎜⎜ B 2 − 2B + H (H +ν ) H (H +ν ) ⎠ 2D ⎜ Q + B ⎢ 2D ⎝ ν (H +ν ) ν (H +ν ) ⎟⎟⎠⎥⎦⎥ + ⎝ ⎣
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⎡
2
⎛ ⎞ ⎛ ⎞ = D ⎢⎢ H ⎜⎜ Q − 2CDν ⎟⎟ + ν ⎜⎜ B − 2CDH ⎟⎟ ν (H +ν ) ⎠ H (H +ν ) ⎠ 2 D ⎝ Q + B ⎢ 2D ⎝ ⎣
+ 2CDHν H +ν
2CDHν H +ν
2⎤ ⎥ ⎥ ⎥ ⎦
(11.3)
The minimum is obtained when making the two quadratic non-negative terms depending on Q and on B zero, respectively:
Q* =
B* =
2CD ν H (H +ν )
2CDH ν (H +ν )
Then the EOQ and AC*:
EOQ = Q* + B* Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
(11.4)
(11.5)
Solving Inventory Problems without Derivatives
=
2CDν + 2CDH H (H +ν ) ν (H +ν )
=
2CD ⎛⎜ ν + H (H +ν ) ⎜⎝ H ν
⎞ ⎟ ⎟ ⎠
157
(11.6)
AC * = 2CDHν H +ν
(11.7)
If ν is made infinitely large, we obtain the standard results:
EOQ =
2CD H
,
(11.8)
and
AC * = 2CDH
(similarly for H).
(11.9)
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11.3. DETERIORATING INVENTORY MODEL
TRCofD represents the total EOQ deteriorating inventory cost :
(
)
TRCofD = A + hK2 eθT − θT −1 T Tθ
The EOQ model with backorders is shown in Fig 11.2
Figure 11.2. The EOQ model with backorders. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
(11.10)
Hui-Ming Wee
158
For θT (K + θ )
(11.82)
Then
TRC = nA + n v KT1 (T1 + T2 ) + n h (K + θ )T4 (T3 + T4 ) T T2 T2
= nA + Thv[hK (P − K ) + v(K + θ2 )(P − (K + θ ))] T 2nP(h + v )
(11.83)
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and
nA + Thv[hK (P − K ) + v(K + θ )(P − (K + θ ))] T 2nP(h + v )2
≥ 2 Ahv[hK (P − K ) + v(K 2+ θ )(P − (K + θ ))] P(h + v )
(11.84)
When (11.76) is an equal sign:
2 AP EOI = T* = (h + v ) hv[hK (P − K ) + v(K + θ )(P − (K + θ ))] n (11.85)
EPQ = (K )(T1 + T2 ) + (K + θ )(T3 +T 4 )
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170
⎛ ⎞ = (K )⎜⎜ h ⎟⎟ EOI + (K + θ )⎛⎜ v ⎞⎟ EOI ⎝h+v⎠ ⎝h+v⎠
[Kh + (K + θ )v]
2 AP hv[hK (P − K ) + v(K + θ )(P − (K + θ ))]
(11.86)
2 Ahv[hK (P − K ) + v(K + θ )(P − (K + θ ))] P TRC * = (h + v )
(11.87)
=
when θ
= 0 , and without deteriorating:
p EPQ = 2 AK (h + v ) P−K hv
(11.88)
(11.88) is similar to tradition EPQ model.
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11.7. INVENTORY LEVEL FOR A TEMPORARY SALE PRICE Recently, researches in EOQ and temporary price discount have been widely done (Goyal, 1996; Grubbstrom and Kingsman, 1997; Martin, 1994; Yang and Wee, 2001). Grubbstrom and Erdem (1999) elaborated the formulation of classical EOQ without differential calculus. Cardenas-Barron (2001) extended the formulation to economic production quantity with shortage model. Tersine (1994) derived the formulation of the EOQ model with temporary sale price. A supplier may temporarily cut the unit price of an item during a regular replenishment cycle. The normal response by the buyer, under the circumstances, is to order extra units during the temporary price discount period. If an extraordinary order is received, the manager must decide the optimal amount to order. Now, we extend the approach to the case with temporary price discount. When the price of the item to be purchased is temporarily reduced, an order is placed. The optimal size of special order at reduced price will be larger than those of the regular orders that are purchased at the regular price. ¾ Assumption:
• • •
Special sale order is a multiple of EOQ lot size of normal price order. When no special order is made, order size is fixed at normal price EOQ. On-line instantaneous updating of EOQ values for the case when no special order is made. This is significant if discount price is large.
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171
The number of replenishment is an integer as in real world; neither of the former authors assumes that.
¾ Modified inventory level for a temporary sale price:
Figure 11.5. Modified inventory level for a temporary sale price.
g = TC n − TC s
(special order gain) (11.89)
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where
TC s = ( p − d )
Q'
+
Q'
2
( p − d )F 2R
+C
(11.90)
total relevant cost during that production cycle when the special order is taken. 2
2 Q* pF ⎢⎢ Q ' − Q 0 ⎥⎥ Q 0 ( p − d )F TCn = ( p − d )Q + + p⎛⎜ Q ' − Q 0 ⎞⎟ + 2R ⎠ ⎝ 2R ⎢⎢ Q* ⎥⎥ 0
⎣
q (T − t q ) pF + 2
+
⎛ ⎜ ⎜ ⎜ ⎝
⎡ ⎢ ⎢ ⎢⎢
Q ' − Q 0 Q *
⎤ ⎥ ⎥ ⎥⎥
+ 1
⎞ ⎟ ⎟C ⎟ ⎠
⎦
(11.91)
total relevant cost during the same cycle time when the first replenishment is an updated EOQ using the reduced price and the subsequent replenishment using the normalprice EOQ policy.
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172
The special symbols used in (11.91) are：
⎡ο ⎤ : An integer value equal or greater than its argument ⎣ο ⎦ : An integer value equal or less than its argument The following notations are retained or added
p : Regular unit price
d : Unit price discount F : Annual percentage of holding cost C : Ordering cost per order Q ' : Special order quantity Q 0 : EOQ with reduced price Q* : EOQ with normal price
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T ' : The length of special sale cycle T 0 : The economic order interval at reduced cost T * : The economic order interval at normal cost From Figure 11.5 and using the classical economic order quantity policy, one has:
Q* = 2CR pF 2CR ( p − d )F
Q0 = T' =
Q' R
T0 =
Q0 R
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
(11.92)
(11.93)
(11.94)
(11.95)
Solving Inventory Problems without Derivatives
Q* = R
T*
173
(11.96)
⎢Q' − Q0 ⎥ ⎥T * * ⎢⎣ Q ⎥⎦
tq = T 0 + ⎢
(11.97)
q = (T ' − t q ) R
(11.98)
As in Martin, closed-form expression for
Q'
value to initialize is used to find the maximum g value.
dR + pQ* ( p − d )F ( p − d )
(11.99)
search technique using Tersine‘s From Tersine [2]
*
Q' =
Q ' and g * cannot be derived from (11.89). A
*
Compute
Q ' and
g * using (11.89) through (11.91) by Fibonacci search [2] technique.
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11.8. INTEGRATED VENDOR-BUYER INVENTORY SYSTEM In previous modeling of the integrated vendor–buyer system, the buyer’s economic order quantity and the vendor’s optimal number of deliveries are derived by setting the first derivatives to zero and solving the simultaneous equations. The Hessian matrix of second derivatives is used to prove the convexity of the objective function. This procedure can be difficult for students who lack the background of differential calculus. In this section develops algebraically the optimal policy of the integrated vendor–buyer inventory system without using differential calculus. The algebraic model is developed on the basis of the following assumptions: 1. Both the production and demand rates are constant and the production rate is greater than the demand rate. 2. The integrated system of single-vendor and single-buyer is considered. 3. The vendor and the buyer have complete knowledge of each other’s information. 4. Shortage is not allowed. The buyer’s inventory level and the vendor’s time-weighted inventory are depicted in Fig 11.6. The buyer’s and the vendor’s cycle time are Q/d and nQ/d, respectively. The buyer’s average inventory level is Q/2. The vendor’s average inventory level, Iv, is the time-weighted inventory per cycle length, and is derived as follows:
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174
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ nQ 2 + Q 2 ⎜⎜ 1 − 1 ⎟⎟ + 2Q 2 ⎜⎜ 1 − 1 ⎟⎟ + ... + (n − 1)Q 2 ⎜⎜ 1 − 1 ⎟⎟ p⎠ p⎠ p⎠ 2p ⎝d ⎝d ⎝d Iv = nQ d
⎡ nQ 2 n(n − 1)Q 2 ⎛⎜ 1 1 ⎞⎟⎤⎥ d ⎢ = + ⎜ d − p ⎟⎥ nQ ⎢⎣ 2 p 2 ⎠⎦ ⎝
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=
⎛ Q⎡ d⎞ d⎤ ⎢(n − 1)⎜⎜1 − ⎟⎟ + ⎥ p ⎠ p ⎥⎦ 2 ⎢⎣ ⎝
(11.100)
Figure 11.6. Buyer’s inventory level and vendor’s time-weighted inventory.
Referring to Grubbstrom and Erdem (1999), the integrated total cost of the vendor and the buyer is derived as follows:
TC =
dC b QH b dC v QH v ⎡ ⎛ d + d ⎞⎟ ⎤ + + + ⎢ (n − 1)⎜⎜1 − ⎥ p p ⎟⎠ ⎦⎥ 2 2 ⎣⎢ Q nQ ⎝
⎡ ⎤⎫ ⎛ ⎛ ⎞ C ⎞ Q⎧ = d ⎜⎜ Cb + v ⎟⎟ + ⎪⎨ H b + H v ⎢(n −1)⎜⎜1 − d ⎟⎟ + d ⎥ ⎪⎬ n ⎠ 2 ⎪⎩ p ⎠ p ⎦⎥ ⎪⎭ Q⎝ ⎝ ⎣⎢
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175
QB B ⎛⎜ 2 2dA ⎞⎟ ≡ dA + = Q + B ⎟⎠ Q 2 2Q ⎜⎝ 2
⎡
⎤
⎛ ⎞ = B ⎢⎢⎜⎜ Q − 2dA ⎟⎟ + 2Q 2dA ⎥⎥ B ⎠ B ⎥ 2Q ⎢⎝ ⎣
⎦
2
⎛ ⎞ = B ⎜⎜ Q − 2dA ⎟⎟ + 2dAB B ⎠ 2Q ⎝
(11.101)
where
A ≡ Cb +
Cv n
(11.102)
and
⎡
⎤
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B ≡ H b + H v ⎢(n − 1)⎜⎜1 − d ⎟⎟ + d ⎥ p ⎠ p ⎥⎦ ⎢⎣ ⎝ ⎛
⎞
(11.103)
From (11.101)－(11.103), the buyer’s economic lot size and the optimal integrated total cost are
Cv ⎞⎟ n ⎟⎠ ⎝ Q* = ⎡ ⎤ ⎛ ⎞ H b + H v ⎢(n − 1)⎜⎜1 − d ⎟⎟ + d ⎥ p ⎠ p ⎥⎦ ⎢⎣ ⎝ ⎛
2d ⎜⎜ Cb +
(11.104)
and
⎛
TC * = 2d ⎜⎜ Cb + ⎝
⎡ ⎛ Cv ⎞⎟⎧⎪ d ⎞⎟ + d ⎤ ⎫⎪ ⎜ ( ) H + H n − 1 1 − ⎥ v⎢ ⎜ n ⎟⎠⎨⎪⎩ b p ⎟⎠ p ⎥⎦ ⎬⎪⎭ ⎢⎣ ⎝ (11.105)
respectively. The results of this method supplies goods to the buyer as soon as there is enough to make up the batch-size, thus reducing the inventory cost during the production period. The minimization of the expression under the square root sign in (11.105) minimizes the square
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176
root of the expression. It can be rearranged in the form of represented by x. After rearrangement, one has ⎧
ax + b x + c , 2
where n is
⎫
⎡ A ⎞ ⎛ 2d ⎞ ⎛ d ⎞⎤ B ⎛ TC * = 2d ⎪⎨ ⎢Cb H b + H v ⎜⎜ − 1⎟⎟ + Cv H v ⎜⎜1 − ⎟⎟ ⎥ + 1 ⎜ n − 1 ⎟ + 2 A1B1 ⎪⎬ ⎜ p ⎠ ⎦⎥ n ⎝ B1 ⎟⎠ ⎝ p ⎠ ⎝ ⎪ ⎣⎢ ⎪ ⎩ ⎭
(11.106)
where
⎡
⎞⎤
A1 ≡ Cv ⎢ H b + H v ⎜⎜ 2d − 1⎟⎟⎥ > 0 ⎢⎣ ⎝ p ⎠⎥⎦
(11.107)
B1 ≡ Cb H v ⎜⎜1 − d ⎟⎟ > 0 p⎠ ⎝ q
(11.108)
⎛
and
⎛
⎞
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Since the down-stream holding cost is greater than the up-stream holding cost (higher cost for finished goods), (11.107) is positive. Equation (11.108) is also positive because it has been assumed that the production rate is greater than the demand rate. From (11.107) and (11.108), the value of n that minimizes the integrated total cost is ⎡
⎞⎤
Cv ⎢ H b − H v ⎜⎜1 − 2d ⎟⎟⎥ p ⎠⎦⎥ ⎝ ⎣⎢ n= ⎞ ⎛ Cb H v ⎜⎜1 − d ⎟⎟ p⎠ ⎝ ⎛
(11.109)
Since the value of n is positive integer, the optimal value of n is
⎢ ⎢ ⎢ n1* = ⎢ ⎢ ⎢ ⎢ ⎣
when
⎡
⎞⎤ ⎥
Cv ⎢ H b − H v ⎜⎜1 − 2d ⎟⎟⎥ ⎥⎥ p ⎠⎦⎥ ⎝ ⎣⎢ ⎥ ⎛ ⎞ ⎥ d Cb H v ⎜⎜1 − ⎟⎟ ⎥ p⎠ ⎝ ⎥
( )
⎛
⎦,
(
)
TC n1* ≥ TC n1* + 1
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(11.110)
Solving Inventory Problems without Derivatives
177
or
⎢ ⎢ ⎢ n2* = ⎢ ⎢ ⎢ ⎢⎣
⎡
⎞⎤ ⎥
Cv ⎢ H b − H v ⎜⎜1 − 2d ⎟⎟⎥ ⎥⎥ p ⎠⎥⎦ ⎢⎣ ⎝ ⎥ +1 ⎞ ⎛ ⎥ d Cb H v ⎜⎜1 − ⎟⎟ ⎥ p⎠ ⎝ ⎥ ⎛
⎦
,
when
( )
(
)
TC n 2* ≤ TC n 2* − 1 where
(11.111)
⎣x⎦ is the greatest integer ≤ x .
From (11.104), the economic lot size of the lot-for-lot condition (i.e. n=1) is
2d (Cb + Cv ) ⎛ ⎞ H b + H v ⎜⎜ d ⎟⎟ ⎝ p⎠
Q* =
(11.112)
From (11.104), when the production rate is infinite, the economic lot size is
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⎛
Q*
2d ⎜⎜ Cb +
Cv ⎞⎟ ⎟
n ⎟⎠ = H b + (n − 1)H v ⎜ ⎝
(11.113)
¾ Example 11.10:
Annual demand rate, d = 1000. Annual production rate, p = 3200. Buyer’s ordering cost, Cb = $25 per order. Vendor’s setup cost, Cv = $400 per set-up. Buyer’s holding cost, Hb = $5 per unit per year. Vendor’s holding cost, Hv = $4 per unit per year. Using (11.101), the value of n = 4.5126. Since TC deliveries is 5.
(n = 5) − TC (n = 4) = 1903.3 −1903.9 < 0
, the optimal number of
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178
Hui-Ming Wee
11.9. THREE-STAGE SUPPLY CHAIN WITH BACKORDER Multi-stage supply chain management integration provides a key to successful international business operations. This is because the integrated approach improves the global system performance and cost efficiency. The integrated production inventory models using differential calculus to solve the multi-variable problems are prevalent in operational research. This paper extends the model by Yang and Wee (2002) to a multi-stage supply chain inventory problem and simplifies the solution procedure using a simple algebraic method to solve the multi-variable problems in the supply chain. As a result, students who are unfamiliar with calculus may be able to understand the solution procedure easily and derive the optimal solution.
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11.9.1. Three-Stage Supply Chain Integration Model The behavior of the integrated production-inventory chain is shown in Fig 11.7. The supply chain integration with JIT implementation is one of the keys to successful SCM. The integrated JIT system views the distributor and the buyer as a team rather than opponents. They share the necessary information to deal with the uncertainty of demand and supply. The global optimization of the supply chain results in lesser cost than the traditionally individual system where the dominant player determines the number of deliveries. This is because the information of the upstream and the downstream is one of most important factors in the operations of an enterprise (Vidal and Goetschackx, 2000). In practice, the uncertain sub-systems result in uncertainty which influences the tactical and the operational planning of the supply chain (SC) (Hendrik Van and Hendrik, 2002). Hence, one of the critical concerns of the vendor and the distributor is how to deliver the finished goods on time. The vendor collaborates with the distributor and the buyer and implements the production planning from MPS or other advanced production scheduling (APS) system.
Figure 11.7. Three-stage supply chain integration model.
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179
Fig 11.8 depicts the time-weighted inventory (TWI) of the vendor and distributor’s inventory level. The vendor produces a production lot size of Q and delivers n times of the stock, Q/n (=Mq) to the distributor’s warehouse. The vendor’s time-weighted inventory is as follows:
Iv =
⎛ Q⎡ d⎞ d⎤ ⎢(n − 1)⎜⎜1 − ⎟⎟ + ⎥ p ⎠ p ⎦⎥ 2n ⎣⎢ ⎝
(11.114)
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Figure 11.8. The distributor’s inventory level and the vendor’s time-weighted inventory.
Fig11.9 illustrates the TWI of the distributor and the buyer’s inventory level with backorder. The distributor periodically delivers a batch size, q, to the buyer. If the stock of the buyer is depleted, the buyer’s shortages are allowed. When a maximum amount of the buyer’ shortages reaches B, an ordering quantities of q is replenished on time. The integrated system’s total cost is
TC =
⎛ dCv QH v ⎡( d ⎞ d ⎤ (M − 1)q H + ndC d + ⎢ n − 1)⎜⎜1 − ⎟⎟ + ⎥ + d p ⎠ p ⎥⎦ 2n ⎢⎣ 2 Q Q ⎝
dCb (q − B )2 H b B 2 (b + H b ) + + + q 2q 2q
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(11.115)
Hui-Ming Wee
180
Figure 11.9. The buyer’s inventory level and the distributor’s time-weighted inventory.
The integrated system’s total cost can be represented by using the method of completing square with respect to the buyer’s backlogging level. One has
TC =
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+
dCv QH v + 2n Q
ndCd ⎡ ⎛ d ⎞ d ⎤ (M − 1)q Hd + ⎢(n − 1)⎜⎜1 − ⎟⎟ + ⎥ + 2 Q p ⎠ p ⎥⎦ ⎢⎣ ⎝
(b + 2 H b ) B 2 − H 2q
(b + 2H b ) ⎡⎢B 2 − = 2q
⎢ ⎣⎢
bB +
dC q Hb + b q 2
⎤ ⎛ ⎞ Hb 2H b C ⎞⎞ d⎛ 1⎛ qB + ⎜⎜ q ⎟⎟ + ⎜⎜ Cb + ⎜⎜ Cd + v ⎟⎟ ⎟⎟⎥ (b + 2H b ) ⎝ (b + 2H b ) ⎠ q ⎝ M ⎝ n ⎠ ⎠⎥ ⎥⎦ 2
H b2 ⎫⎪ ⎡ ⎛ q⎧ d⎞ d⎤ + ⎪⎨MH v ⎢(n − 1)⎜⎜1 − ⎟⎟ + ⎥ + MH d + (H b − H d ) − ⎬ b + 2 H b ⎪⎭ 2 ⎪⎩ p ⎠ p ⎥⎦ ⎢⎣ ⎝ (11.116)
11.9.2. Notation and assumptions The following notation and assumptions are made in the development of the integrated production inventory model. ¾ Notation for the vendor and the buyer:
d
p
Hv
Demand rate. Production rate Holding cost per dollar per year for the vendor
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Solving Inventory Problems without Derivatives
Cv
n
181
The vendor’s setup cost per cycle time The number of vendor’s deliveries to the distributor per delivery cycle for
the vendor
Cd Hd M
The distributor’s ordering cost per delivery Holding cost per dollar per year for the distributor The number of distributor’s deliveries to the buyer per delivery cycle for the
distributor
q
The distributor’s delivery batch size per delivery
Hb
Holding cost per dollar per year for the buyer
b Cb
B Q
Backordering cost per dollar per year for the buyer Ordering cost per delivery for the buyer Backordering quantity for the buyer The production lot size per cycle time, where
Q = nMq
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¾ Assumptions for the integrated production inventory model:
a) Production rate is greater than demand rate. b) Production rate and demand rate are constant and are independent of production lot size. c) Shortages are allowed for the buyer. d) The system does not consider the wait-in-process items and the defective items. e) The replenishment is instantaneous and the lead time is constant. f) The players have complete information of each other. g) The number of vendor’s shipment is an integer number. h) The number of distributor’s shipment is an integer number. i) A single item is considered. j) The delivery policy is a just-in-time multiple delivery for the integrated inventory system.
11.9.3. Optimization ¾ Optimization of the delivery batch size q and the backordering level B
Our objective is to derive the optimal values of the relevant solutions and TC. Rewriting (11.115) and using the method of perfect square, one has
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182
( b + 2H b ) ⎡ ⎛⎜ ⎢B − TC = 2q
⎛ + ⎜⎜ ⎝
⎞⎤ Hb q ⎟⎥ ⎜ (b + 2 H ) ⎟⎥ b ⎠⎦ ⎝
⎢ ⎣
dW − qY q 2
⎞ ⎟ ⎟ ⎠
2
2
+ 2dWY (11.117)
where
⎛ ⎞ W = ⎜⎜ Cb + 1 Ψ1 ⎟⎟ M ⎠ ⎝
Ψ1 = ⎢Cd +
Cv ⎤ n ⎥⎦
⎧ Y = ⎪⎨MΨ2 + ⎪⎩
Hb2 ⎫⎪ (Hb − Hd ) − b + 2H ⎬ b⎪
⎡ ⎣
⎧
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(11.118)
(11.119)
⎭
⎡
⎤
(11.120)
⎫
Ψ2 = ⎪⎨H v ⎢(n −1)⎜⎜1 − d ⎟⎟ + d ⎥ + H d ⎪⎬ p ⎠ p ⎦⎥ ⎪⎩ ⎝ ⎪⎭ ⎣⎢ ⎛
⎞
(11.121)
From (11.117), setting
⎛ ⎜ ⎜ ⎝
dW − qY q 2
⎞ ⎟ ⎟ ⎠
2
=0
⎡ ⎛ ⎢B − ⎜ ⎜ ⎢ ⎝ and ⎣
The optimal delivery batch size
2
⎞⎤ Hb ⎟⎥ (b + 2 H b ) q ⎟⎠⎥⎦ = 0
.
q * is
q* = 2dW Y
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Solving Inventory Problems without Derivatives
=
⎛ Ψ⎞ 2d⎜⎜Cb + 1 ⎟⎟ M⎠ ⎝
⎡ ⎢MΨ + 2 ⎢ ⎣
and the optimal backordering level
B* =
Hb2 ⎤⎥ (Hb − Hd )− b + 2H ⎥ b ⎦
183
(11.122)
B * is
Hb (b + 2H b ) q *
(11.123)
Substituting (11.122) and (11.123) into (11.117), the minimum value of the integrated system’s total cost is
TC = 2dWY
(11.124)
q*
The optimal is a function of the vendor’s delivery number n and the distributor’s delivery number M. We use the method of perfect square to derive the optimal numbers of vendor’s deliveries n and distributor’s deliveries M. Rewrite (11.124) with respect to M, one has
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TC = 2dWY ⎤ ⎡⎛ ~ 2 dΨ1G ⎞⎟ ⎛ ⎢⎜ ~ ⎞2 ⎥ = 2⎢⎜ dMCbΨ2 − + ⎜ dΨ1Ψ2 + dCbG ⎟ ⎥ M ⎟⎟ ⎝ ⎠ ⎥ ⎢⎜⎝ ⎠ ⎦ ⎣
(11.125)
where
H b2 ⎞⎟ ⎟ ⎜ b + 2H b ⎟
⎛ ~ G = H b − H d − ⎜⎜ ⎝
⎠
(11.126)
¾ Optimization of the optimal numbers of vendor’s deliveries n and distributor’s deliveries M
In (11.125), assuming follows:
ρ=
d p , the term
dΨ1Ψ2 with respect to n can be rewritten as
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184
dΨ1Ψ2 = d ⎧
= d ⎪⎨
⎛⎜ ⎪⎩ ⎝
⎛ ⎜C ⎜ d ⎝
+
Cv ⎞⎟ (H [(n −1)(1 − ρ ) + ρ ]+ H d ) n ⎟⎠ v
Cd [H d − H v (1 − 2ρ )] + [Cv H v (1 − ρ )] ⎞⎟
2
⎠
⎛
+ ⎜⎜ nCd H v (1 − ρ ) − ⎝
Cv [H − Hv (1− 2ρ )] n d
1 2 ⎫2 ⎞ ⎪ ⎟ ⎟ ⎬ ⎠ ⎪ ⎭ (11.127) 2
From
(11.125),
and
(11.127),
setting
~⎞ ⎛ ⎜ MdC Ψ − dΨ1G ⎟ = 0 b 2 ⎜ M ⎟ ⎝ ⎠
and
2
⎛ ⎞ Cv ⎜ nC d H v (1 − ρ ) − [H d − H v (1 − 2 ρ )] ⎟⎟ = 0 ⎜ n ⎝ ⎠ , the optimal numbers of vendor’s deliveries n and distributor’s deliveries M are derived as follows
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n* =
Cv [H d + H v (2ρ −1)] Cd H v (1 − ρ )
~ Ψ1G M* = Ψ2Cb =
⎛ ⎜⎜ Cd ⎝
[
(11.128)
]
+ Cv n *⎞⎟⎟ ⋅ (H b − H d ) − H b2 (b + 2H b ) ⎠
{H v [(n *)(1 − ρ ) + 2ρ ]+ (H d − H v )}Cb
(11.129)
Because the value of n * and M * are positive integers, the corresponding conditions are:
and
(n *) ⋅ (n * −1) ≤ (n *)2 ≤ (n *) ⋅ (n * +1)
(11.130)
(M *)⋅ (M * −1) ≤ (M *)2 ≤ (M *)⋅ (M * +1)
(11.131)
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185
From (11.128) and (11.129), (11.130) and (11.131) are equivalent to
(n *) ⋅ (n * −1) ≤
Cv [H d + H v (2ρ − 1)] ≤ (n *) ⋅ (n * +1) Cd H v (1 − ρ ) (11.132)
and
[
]
⎛ C + Cv ⎞ ⋅ (H − H ) − H 2 (b + 2 H ) ⎜ d b d b b n *⎟⎠ (M *) ⋅ (M * −1) ≤ ⎝ ≤ (M *)(M * +1) {H v [(n *)(1 − ρ ) + 2ρ ] + (H d − H v )}Cb
(11.133) respectively. When the optimal conditions in (11.132) and (11.133) are satisfied, the optimal numbers of the vendor’s and the distributor’s delivery can be derived. It implies that for known n * and M * , from (11.132), (11.133), (11.122) and (11.123), the optimal distributor’s delivery batch size q * is:
⎛ ⎜ 2d ⎜ Cb ⎜ ⎜ ⎝
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q* =
(M )
* ⎛⎜ Ψ* ⎞⎟ + ⎝ 2⎠
B* =
⎛ Ψ* ⎞ ⎞ ⎜ 1 ⎟⎟ + ⎝ ⎠⎟ M * ⎟⎟ ⎠
H b2 (H b − H d )− b + 2H b
Hb (b + 2H b ) q *
(11.134)
(11.135)
where
⎡
Ψ1* = ⎢Cd + ⎣
Cv ⎤ n * ⎥⎦
and
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
(11.136)
Hui-Ming Wee
186
⎧
⎡
⎫
⎤
Ψ2* = ⎪⎨H v ⎢(n * −1)⎜⎜1 − d ⎟⎟ + d ⎥ + H d ⎪⎬ p ⎠ p ⎥⎦ ⎪⎩ ⎢⎣ ⎝ ⎪⎭ ⎛
⎞
(11.137)
the minimum value of the integrated system’s total cost from (11.125) is: ⎛ TC* = 2d ⎜ ⎜ ⎝
~ 2 Ψ1G ⎞⎟ (M *)Cb Ψ2 − + M*⎟ ⎠
( ΨΨ 1
2
~ + Cb G
)
2
(11.138)
where
H b2 ⎞⎟ ⎟ ⎜ b + 2H b ⎟
⎛ ~ G = H b − H d − ⎜⎜ ⎝
⎡
Ψ1 = ⎢Cd + ⎣
⎠,
(11.139)
Cv ⎤ n * ⎥⎦ , (11.140)
and
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⎧
⎡
⎤
⎫
Ψ2 = ⎪⎨H v ⎢(n * −1)⎜⎜1 − d ⎟⎟ + d ⎥ + H d ⎪⎬ p ⎠ p ⎥⎦ ⎪⎩ ⎢⎣ ⎝ ⎪⎭ ⎛
⎞
(11.141)
Since n* and M* must be equal and greater than one, the inequality is:
n* =
M* =
Cv [H d + H v (2ρ −1)] ≥1 Cd H v (1 − ρ ) ⎛ ⎜⎜ Cd ⎝
+ Cv
⎛ ⎞ ⎜ n *⎟⎟⎠ ⋅ ⎜⎜ H b ⎝
(11.142)
H b2 ⎞⎟ − Hd − b + 2H b ⎟⎟ ⎠
{H v [(n *)(1 − ρ ) + 2ρ ]+ (H d − H v )}Cb
≥1 (11.143)
By algebraic calculations, one has
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Solving Inventory Problems without Derivatives
187
⎞ C d + Cv Cv H d ⎟ − (⎝ 2Cv + Cd ) H v (2Cv + Cd )⎟⎠ ⎛
d ≥ p⎜⎜
(11.144)
and ⎡
p ⎢Cb (H v n * +( H d − H v )) − ( H b − H d −
⎢ d≥ ⎣
H v Cb (n * −2)
H b2 ⎛ C ⎞⎤ )⎜⎜ C d + v ⎟⎟⎥ b + 2H b ⎝ n * ⎠⎥ ⎦
= pψ (n *)
(11.145)
Since ψ (n *) is positive, the following condition must be satisfied to ensure the feasible solution of (11.145): ⎡ H b2 C ⎞⎤ ⎛ ⎢C b H v n * + ( H d − H v ) − ( H b − H d − )⎜ C + v ⎟ ⎥ ⋅ H v C b (n * − 2 ) > 0 b + 2 H b ⎜⎝ d n * ⎟⎠ ⎥⎦ ⎢⎣
(
)
[
]
(11.146) From (11.144) - (11.146), a lower bound value of the demand rate can be found.
¾
Example 11.11:
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d=1000. p=3200. Cb=$25 per order.
Cv=$400 per set-up. Hb=$5 per unit per year. Hv=$4 per unit per year.
Additional input parameters are Cd =$50 per order, b= $30 per unit and Hd=$4.2 per unit per year. Using (11.120) and (11.121), the value of n=3 and the value of M=1.The optimal number of vendor’s deliveries is 3 and optimal number of distributor’s deliveries is 1. By applying the above solution, the results of the proposed model are given in Table 11.2. Table 11.2 The results of the proposed model Proposed model No. of vendor’s delivery (n)
3
No. of distributor’s delivery (M)
1
Distributor’s delivery batch size (q)
188.8
Buyer’s backordering quantity (B)
23.6
Vendor’s production lot size (Q)
566.5
Total cost
2100.69
Total cost neglecting distributor’s cost
1888.86
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Hui-Ming Wee
188
When we consider the distributor, the relevant information of the distributor influences the decision making of the production and the dispatching planning in a supply chain inventory model. In the last row of Table 11.2, the total cost neglecting distributor’s cost of the proposed model is 1888.86. The backordering policy may reduce the system’s total cost.
11.10. COST-DIFFERENCE COMPARISONS This section presents a modified method to compute economic order quantities without derivatives by cost-difference comparisons. Extensions to allow backorders are done for the EOQ/EPQ models. In contrast to previous literatures, limiting values on a finite planning horizon are used rather than algebraic manipulations for the cost function comparisons.
11.10.1. Assumption and Analysis The assumptions are similar to the basic EOQ model. The setup cost (per order) is denoted by A, d is the demand rate (units per unit time) and h is the holding cost (per unit per unit time). Unlike the traditional EOQ model, we assume a finite planning horizon with length T. When the demand over this horizon is replenished by n equal batch sizes,
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Qn =
dT n
and
n ≥1,
it is obvious that
Qn −1 > Qn > Qn +1 .
total variable costs (i.e. cost rates) of consecutive batch numbers of are expressed as:
hQ Cu (i, T ) = Ad + i 2 Qi where
, for
The different unit-time
n − 1, n
i = n −1, n and n +1
and
n +1
(11.147)
( )
Cu (i, T ) is equivalent to Cu Qi .
In order to determine the optimal batch size for a given horizon T, we increase the value
(
)
(
)
(
)
(
)
Cu n − 1, T ≥ Cu n, T and Cu n + 1, T ≥ Cu n, T are satisfied of n until simultaneously. That is, the cost-differences of inventory cost rates between two consecutive batch numbers satisfy the following conditions.
Cu (n + 1, T ) − Cu (n, T ) ≥ 0 and
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
(11.148)
Solving Inventory Problems without Derivatives
Cu (n + 1, T ) − Cu (n, T ) ≥ 0
189
(11.149)
Thus, from (11.147), (11.148) and (11.149), for the optimal batch number n, one has:
⎛ ⎞ Ad ⎜⎜ 1 − 1 ⎟⎟ + h (Qn−1 − Qn ) ≥ 0 Q Qn ⎠ 2 ⎝ n−1
(11.150)
⎛ 1 Ad ⎜⎜ Q ⎝ n+1
(11.151)
and
⎞ h 1 − ⎟⎟ + (Qn+1 − Qn ) ≥ 0 Qn ⎠ 2
Eq. (11.150) and (11.151) can be simplified to
2 Ad ≤ Qn−1Qn h
(11.152)
2 Ad h
(11.153)
and
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Qn+1Qn ≥
If the planning horizon T and the optimal batch number n tend to infinity, then
lim n→∞
Qn−1 n = lim n→∞ =1 Qn n −1 and
lim n→∞
Qn+1 n = lim n→∞ =1 Q = Qn = Qn +1 n +1 Qn resulting in n −1
From (11.152) and (11.153), the optimal batch size converges to the economic order
2 Ad h
quantity EOQ directly.
. We use the variable batch size to express cost rate function and derive the
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Hui-Ming Wee
190
11.10.2. EOQ with Backorders In the EOQ model with backorders, we use the fill rate, r, to simplify and express the proportion of immediate filling demand in the inventory cycle. The proportion 1－r that is not
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filled from inventory is backordered. The backorder cost is υ (per unit and unit time). The cost rate function on EOQ with backorders is expressed as (see Fig. 11.10)
Figure 11.10. EOQ inventory model with backorders.
hQn r 2 vQn (1 − r ) Ad Cu (r , n, T ) = + + 2 2 Qn Q = Ad + n ⎡⎢hr 2 + v(1 − r )2 ⎤⎥ Qn 2 ⎣ ⎦
2
(11.154)
From (11.154), the cost rate function can be expressed in the form:
Cu (r , n, T ) = f1 (Qn ) + f 2 (Qn )× g (r )
(11.155)
Qn and r are independent variables for the cost rate function. The optimal solution can be obtained from the optimal Qn and r values.
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Solving Inventory Problems without Derivatives
191
¾ Derive optimal fill rate with fixed batch size
Let
rL < r < rU
, when r is the optimal fill rate, one must satisfy the conditions
Cu (rL , n, T ) − Cu (r , n, T ) ≥ 0
and
Cu (rU , n, T ) − Cu (r , n, T ) ≥ 0
as in
(11.149) must be satisfied. Then,
v ⎡⎢(1 − rL )2 − (1 − r )2 ⎤⎥ h rL2 − r 2 ⎦Q≥0 Q+ ⎣ 2 2
)
(
(11.156)
and
(
h r2 U
− r2 2
)Q +
v ⎡⎢(1 − rU )2 − (1 − r )2 ⎤⎥ ⎣
2
⎦Q≥0
(11.157)
Eq. (11.156) and (11.157) can be simplified to
rL + r v ≤ 2 h+v
(11.158)
v ≤ rU + r h+v 2
(11.159)
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and
rU tend to the optimal fill rate r, then rL = r = rU . The optimal fill rate v h + v . We have shown that the optimal fill rate does not depend on the batch
r If L and
converges to size. Using the optimal fill rate, the optimal batch size can be derived. We obtain the optimal fill rate directly by the limit of expressions. ¾ Derive optimal batch size with the optimal fill rate
Similar to the analysis in Section 11.10.1, the following equations can be obtained from (11.149):
2 ⎛ ⎞ v(1 − r )2 ⎛ − Qn ⎞⎟ ≥ 0 Ad ⎜⎜ 1 − 1 ⎟⎟ + hr (Qn−1 − Qn ) + ⎜Q ⎝ n −1 ⎠ Q Q 2 n 2 ⎝ n−1 ⎠ Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
(11.160)
Hui-Ming Wee
192
and
2 ⎛ ⎞ v(1 − r )2 ⎛ Ad ⎜⎜ 1 − 1 ⎟⎟ + hr (Qn+1 − Qn ) + − Qn ⎞⎟ ≥ 0 ⎜Q n + 1 ⎝ ⎠ 2 Q Q n 2 ⎝ n+1 ⎠
(11.161)
Eq. (11.160) and (11.161) can be simplified to
2 Ad hr 2 + v⎛⎜1 − r 2 ⎞⎟ ⎝
≤ Qn−1Qn
⎠
(11.162)
and
Qn+1Qn ≥
2 Ad
hr 2 + v(1 − r )2
(11.163)
If the horizon length T and the optimal batch number n tend to infinity, the same conditions as in Section 11.10.1 result. That is,
Qn −1 = Qn = Qn +1 . Therefore the optimal 2 Ad hr + v(1 − r )2 . 2
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batch size converges to the economic order quantity with backorders With the optimal fill rate found in the previous section, the optimal batch size can be derived.
11.10.3.EPQ with Backorders In this case, p is the production rate (units per unit time). The cost rate function on EPQ with backorders is (see Fig. 11.11)
Figure 11.11. EPQ inventory model with backorders. Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Solving Inventory Problems without Derivatives
193
hQn r 2 ⎛⎜ p − d ⎞⎟ vQn (1 − r )2 ⎛⎜ p − d ⎞⎟ Ad Cu (n, r , T ) = + + ⎜ ⎟ Qn 2 ⎜⎝ p ⎟⎠ 2 ⎝ d ⎠
Q ⎛ p − d ⎞⎟⎡ 2 = Ad + n ⎜⎜ hr + v(1 − r )2 ⎤⎥ ⎢ ⎟ Qn 2 ⎝ p ⎠⎣ ⎦ r Similar to the analysis in Section 11.10.2, if L and
(11.164)
rU tend to the optimal fill rate r, then
v rL = r = rU . The optimal fill rate converges to h + v
as in Section 11.10.2. Similar to the analysis in Section 11.10.2, if the horizon length T and optimal batch
Q
=Q =Q
n n +1 . The optimal number n tend to infinity as in Section 11.10.1, then n −1 production batch size converges to the economic production quantity with backorders
2 Ad ⎛
p ⎞ ⎟ p − d ⎟⎠ ⎝
hr 2 + v(1 − r )2 ⎜⎜
.
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In the EPQ model with backorders, our analysis applies the variable fill rate in the cost rate function.
REFERENCES [1]
[2]
[3] [4]
[5]
[6]
Braglia, M. and Zavanella, L. “An industrial strategy for stock management in Supply Chains: modelling and performance evaluation.” International Journal of Production Research, Vol. 41, (2003); pp. 3793–3808. Cardenas-Barron, L. E., “The economic production (EPQ) with shortage derived algebraically,” International Journal of Production Economics, Vol. 70, (2001), pp. 289-292. Goyal S. K., “A comment on Martin's: Note on an EOQ model with a temporary sale price.” International Journal of Production Economics, Vol. 43, (1996), pp. 283-284. Grubbström, R. W. “Material requirements planning and manufacturing resource planning.” In: M. Warner Editor, International Encyclopedia of Business and Management Routledge, London (1996). Grubbström, R. W. and Kingsman, B. G. “Ordering and Inventory Policies for Step Changes in the Unit Item Cost: A discounted Cash Flow Approach.” Research Report RR-130, Dept. of Production Economics, Linkoping Institute of Technology, (1997). Grubbström, R. W. and Erdem, A. “The EOQ with backlogging derived without derivatives,” International Journal of Production Economics, Vol. 59, (1999), pp. 529530.
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194 [7] [8] [9]
[10] [11]
[12]
[13] [14] [15] [16] [17]
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[18]
[19]
Hui-Ming Wee Harris, F. W. “What quantity to make at once.” In: Operation and Costs, The Factory Management Series A.W. Shaw Co., Chicago (1915), pp. 47–52. Hendrik, Van L. and Hendrik, V., “Robust planning: a new paradigm for demand chain planning,” Journal of Operations Management, Vol. 20, (2002), pp. 769-783. Kang S, and Kim I. T. “A study on the price and production level of the deteriorating inventory system.” International Journal Production Research, Vol. 21 (6), (1983); pp. 899- 908. Martin, G. E., “Note on an EOQ model with a temporary sale price,” International Journal of Production Economics, Vol. 37, (1994), pp, 241-243. Minner S. “A note on how to compute economic order quantities without derivatives by cost comparisons.” International Journal Production Economics, Vol. 105, (2007); pp. 293-296. Misra R. B. “Optimum production lot size model for a system with deteriorating inventory.” International Journal Production Research, Vol. 13(5), (1975); pp. 495505. Teng J. T. “A simple method to compute economic order quantities.” European Journal of Operational Research, Vol. 198(1), (2009), pp. 351- 353. Tersine R. J., Principles of Inventory and Materials Management, 4th ed., Prentice-Hall Englewood Cliffs, N.J., (1994). Vidal C. J. and Goetschalckx, M. “Modeling the effect of uncertainties on global logistics systems.” Journal of Business Logistics, Vol. 21, (2000), pp. 95-120. Wee H. M., Chung, S.L. and Yang, P. C. “The EOQ model with temporary sale price derived without derivatives.” Engineering Economist, Vol. 48, (2003), pp. 190-195. Yang P. C., and Wee, H. M. “A deteriorating inventory model with a temporary price discount: A present-value approach,” Journal of Information and Optimization Sciences, Vol. 22, (2001), pp. 579-596. Yang P. C. and Wee, H. M. “The economic lot size of the integrated vendor-buyer system derived without derivatives.” Optimal Control Applications and Methods, Vol. 23, (2002), pp. 163-169. Zanoni S. and Grubbström, R. W. “A note on an industrial strategy for stock management in supply chains: modelling and performance evaluation.” International Journal of Production Research, Vol. 37, (2004), pp. 2463-2475.
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
INDEX A accounting, 12, 19, 20, 120 adaptation, 131 age, 139 Aggregate Run out Time (AROT) Method, 60 algorithm, 74, 80, 81, 83, 85 anticipation stock, 5 Appropriate inventory costs, 27 assessment, 114 assets, 13
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B benefits, 113 bias, 114 bill of materials (BOM), 109 blood, 142 boils, 55 breakdown, 134 Bulk purchases, 5 business environment, 131 businesses, 64 buyer, 44, 48, 153, 170, 173, 174, 175, 178, 179, 180, 181, 194
C calculus, 153, 170, 173, 178 chemical, 143 Chicago, 194 chicken, 148 classical EOQ, 153, 170 classification, 133 collusion, 20 combined effect, 37 commodity, 43
communication, 112 competition, 43, 131 competitiveness, 140 complexity, vii, 3, 98 composition, 112 computer, 109, 110, 111, 118, 119, 121, 145 computer technology, 119 conflict, 123 constant rate, 69 consulting, 102 consumers, 119 coordination, 64, 120, 122 cost accounting, 11, 12, 13, 19 cost saving, 45, 50, 64 cost structures, 139, 140, 144 covering, 148, 151 CRP, 116, 117, 118 customers, 113, 116, 125, 131, 139 cycles, 15, 34, 55, 131, 147
D data analysis, 111 data set, 15 database, 110, 121 decay, 147, 148, 153 decoupling, 5 decoupling stock, 5 defects, 132, 133, 134 deficiencies, 59 Demand forecast, 27, 59 depreciation, 52 derivatives, 153, 155, 158, 173, 188, 193, 194 deteriorating materials, 3 deviation, 104, 105 discounted sales, 5 discrete variable, 147
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Index
196 distribution, 4, 89, 91, 98, 103, 105, 131, 143, 144, 145, 148, 149, 150, 151 distribution function, 98
E e-commerce, 119 economic factors, 5 Economic Time Cycle, 23 emergency, 87 employees, 19, 116, 129, 133, 134 engineering, 121 environment, 51, 52, 101, 118 equipment, vii, 3, 6, 23, 51, 52, 116, 120, 124, 125, 129, 133 evolution, 115, 120 execution, 3
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F Final products, 4 financial, vii, 3, 120, 121, 123, 139 financial records, 120 fish, 148, 150 fitness, 118 flexibility, 131 fluctuations, 15 food, 139, 140, 142 food products, 140 formula, 89, 98, 130, 155 frequency distribution, 97 funds, 125
G general knowledge, 134 GNP, 4 Gross National Product (GNP), 4 growth, 148
H Human Resource Management, 121
I ideal, 134 income, 17, 18 income tax, 18 independence, 111, 112
independent variable, 190 industries, 4 industry, 125, 131, 140, 150 inequality, 155, 165, 186 inertia, 136 inflation, 148 information technology, 111, 116, 120 institutions, 4 integration, 178 integrity, 110 interface, 121 inventory records, 20, 110 investment, 4, 125 issues, 144
L lead, 5, 6, 27, 31, 32, 36, 45, 54, 65, 67, 69, 83, 89, 90, 92, 93, 95, 96, 97, 98, 100, 101, 103, 104, 105, 110, 114, 124, 126, 129, 131, 132, 133, 134, 137, 181 Lean Production System (LPS), 131 life cycle, 142 lifetime, 142, 143 light, 4 linear function, 146, 158 liquidate, 39 logistics, 64, 140, 150, 194 LTC, 80
M management, vii, 3, 19, 20, 21, 24, 45, 52, 70, 101, 109, 111, 115, 118, 120, 131, 134, 137, 139, 150, 153, 178, 193, 194 manpower, 116, 120 manufactured goods, 140 manufacturing, 3, 101, 113, 115, 116, 120, 124, 125, 126, 131, 193 marketing, vii, 3 mass, 120 master production schedule (MPS), 109, 110 material flow systems, 3 material resources, 120 materials, vii, 3, 4, 5, 12, 13, 20, 52, 109, 113, 114, 115, 123, 124, 125, 126, 129, 133 matrix, 173 matter, iv meat, 140, 150 medicine, 141 messages, 114 methodology, 154, 158
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Index military, 4 Min-Max System, 24 modelling, 193, 194 models, iv, 27, 28, 39, 63, 69, 70, 139, 140, 144, 147, 151, 158, 162, 178, 188 Moon, 148
N New York, iv non-profit organizations, 4 normal distribution, 93, 96, 104, 105
197
researchers, 148 residues, 132 resolution, 122 resources, vii, 3, 4, 115, 116, 118, 119, 120 response, 15, 119, 170 restrictions, 115 retail, 5, 64 risk, 11, 87 root, 175 roots, 134 rules, 11
S O on-hand inventories, 109 operations, 112, 131, 178 optimization, 120, 178 overtime, 135
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P Pareto, 20 Period order quantity system (POQ), 23 permit, 64 pigs, 148 pipeline stock, 5 plastics, 66 policy, 37, 47, 105, 140, 146, 147, 150, 151, 152, 153, 172, 173, 181, 188 portability, 121 preparation, iv probability, 89, 91, 93, 98, 100, 103, 105, 143 probability distribution, 89, 98, 100, 143 producers, 111 profit, 4, 5, 19, 137 profit making institutions, 4 programming, 148 project, 122, 128 psychic stock, 5
safety, 5, 23, 87, 89, 90, 91, 92, 96, 105 safety stock, 5, 23, 87, 89, 90, 91, 92, 96, 105 school, 155 semiconductor, 140, 150 service industries, 4 services, iv, vii, 3 shelf life, 23, 139 short supply, 11 shortage, 5, 24, 33, 60, 109, 114, 147, 153, 170, 193 showing, 142 skilled workers, 131, 133, 134 software, 113, 121 solution, 71, 81, 97, 146, 147, 150, 153, 178, 187, 190 Stagnant capital, 5 standard deviation, 93, 95, 96, 104, 105 standardization, 124 state, 113 states, 20 stock movements, 19 storage, 4, 115, 134, 144, 148 structure, 15, 52, 110, 112 supplier, 40, 43, 48, 50, 64, 83, 131, 133, 134, 170 suppliers, 4, 114, 133 supply chain, 131, 178, 188, 194 support services, 129 surplus, 148 synchronize, 11
R radiation, 141 raw materials, 3, 5, 123, 124, 140 Raw materials, 4 real time, 20 reality, 140 recommendations, iv reliability, 131, 133 requirements, 5, 71, 80, 83, 110, 113, 115, 131, 193
T tax policy, 11 taxes, 52 team members, 20 techniques, 69, 97 technologies, 119, 134 technology, 119, 134 time factor, 5
Wee, Hui-Ming. Inventory Systems: Modeling and Research Methods : Modeling and Research Methods, Nova Science Publishers, Incorporated,
Index
198 time frame, 27 time periods, 69, 75 time series, 15 tooth, 30, 155 total costs, 44 total product, 126 Toyota Production System (TPS), 131 transit stock, 5, 113 transportation, 64, 133 treatment, 104 TRM cargo transfer manifest, 24 turnover, 131 Two-Bin System, 24
U
valuation, 13 variable costs, 188 variables, 27, 155 variations, 146 varieties, 120, 124 VAT, 135 vegetables, 140, 142, 150
W waste, 116, 132, 133 wholesale, 64 withdrawal, 6 work- in- process (WIP), 3 workers, 133 working stock, 5, 87 workstation, 116
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uncertainty factors, 5 uniform, 27, 104 unit cost, 28, 39, 40, 52, 66, 70, 92 United, 37, 50 Unutilized labor and capital, 4 updating, 46, 170
V
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