Scheduling in Green Supply Chain Management: A Mixed-Integer Approach (International Series in Operations Research & Management Science, 303) 3030674770, 9783030674779

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International Series in Operations Research & Management Science

Wolfgang Albrecht

Scheduling in Green Supply Chain Management A Mixed-Integer Approach

International Series in Operations Research & Management Science Volume 303

Series Editor Camille C. Price Department of Computer Science, Stephen F. Austin State University, Nacogdoches, TX, USA Associate Editor Joe Zhu Foisie Business School, Worcester Polytechnic Institute, Worcester, MA, USA Founding Editor Frederick S. Hillier Stanford University, Stanford, CA, USA

The book series International Series in Operations Research and Management Science encompasses the various areas of operations research and management science. Both theoretical and applied books are included. It describes current advances anywhere in the world that are at the cutting edge of the field. The series is aimed especially at researchers, doctoral students, and sophisticated practitioners. The series features three types of books: • Advanced expository books that extend and unify our understanding of particular areas. • Research monographs that make substantial contributions to knowledge. • Handbooks that define the new state of the art in particular areas. They will be entitled Recent Advances in (name of the area). Each handbook will be edited by a leading authority in the area who will organize a team of experts on various aspects of the topic to write individual chapters. A handbook may emphasize expository surveys or completely new advances (either research or applications) or a combination of both. The series emphasizes the following four areas: Mathematical Programming: Including linear programming, integer programming, nonlinear programming, interior point methods, game theory, network optimization models, combinatorics, equilibrium programming, complementarity theory, multiobjective optimization, dynamic programming, stochastic programming, complexity theory, etc. Applied Probability: Including queuing theory, simulation, renewal theory, Brownian motion and diffusion processes, decision analysis, Markov decision processes, reliability theory, forecasting, other stochastic processes motivated by applications, etc. Production and Operations Management: Including inventory theory, production scheduling, capacity planning, facility location, supply chain management, distribution systems, materials requirements planning, just-in-time systems, flexible manufacturing systems, design of production lines, logistical planning, strategic issues, etc. Applications of Operations Research and Management Science: Including telecommunications, health care, capital budgeting and finance, marketing, public policy, military operations research, service operations, transportation systems, etc.

More information about this series at http://www.springer.com/series/6161

Wolfgang Albrecht

Scheduling in Green Supply Chain Management A Mixed-Integer Approach

Wolfgang Albrecht Faculty of Law and Economics University of Greifswald Greifswald, Germany

ISSN 0884-8289 ISSN 2214-7934 (electronic) International Series in Operations Research & Management Science ISBN 978-3-030-67477-9 ISBN 978-3-030-67478-6 (eBook) https://doi.org/10.1007/978-3-030-67478-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Because of negative consequences of anthropogenic climate change, protecting our environment should be something that matters to all of us. An important paradigm shift is necessary for the worldwide economy. Due to increasing globalization, business operations are typically organized in complex supply chain networks, while a coordinated scheduling of production, distribution, and sales allows for realizing transient monetary benefits. Although the field of green supply chain management has been established for a few years in the literature, there is still a backlog in modeling objectives and constraints of sustainability. For this reason, this book proposes new integrated mathematical optimization models and problemtailored solution algorithms that may contribute to a reconciliation of economic and environmental issues. The initial manuscript of this book has been accepted in January 2020 by the University of Greifswald’s Faculty of Law and Economics. It is a habilitation thesis, i.e., a research monograph for postdoctoral qualification in Germany. First of all, I would like to express my deep gratitude to my academic advisor, Professor Dr. Martin Steinrücke (University of Greifswald). During my time as a postdoctoral researcher at his chair, several common articles have been published in international peer-reviewed journals in the field of supply chain management. In addition, he finally encouraged me to write this monograph. While acting as the first reviewer of my habilitation thesis, his broad expertise was the basis of many fruitful discussions. Additional external reviewers were Professor Dr. Richard Lackes (Technical University of Dortmund) and Professor Dr. Hans Corsten (Technical University of Kaiserslautern). I greatly appreciate the valuable comments of all three reviewers. Besides, I am much obliged to Professor Dr. Jan Körnert (University of Greifswald) for chairing the habilitation commission. Sincere thanks to Springer and the series editor of “International Series in Operations Research & Management Science” for accepting my manuscript for publication.

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Preface

Finally, I would like to acknowledge the support of all friends, colleagues, and students who accompanied me on my way to completing my habilitation. Warm thanks go to my family. Greifswald, Germany March 2021

Wolfgang Albrecht

About this Book

This book offers practical tools and new perspectives to researchers and professionals in the field of supply chain management. It deals with hierarchical scheduling of operations at/between sites of generic multi-stage networks taking into account aspects of sustainability. Driven by an increasing environmental awareness as well as current initiatives of legislation for reducing greenhouse gas emissions and waste, it proposes new mixed-integer programming models combining problems of procurement, production, distribution, sales, recycling, disposal, and emissions trading simultaneously in consideration of existing interdependencies. The modularized approach distinguishes between material flows of non-perishable and perishable goods and additionally captures an aligned financial planning. Discrete-time models are used for establishing closed-loop structures on the medium-term level. On the short-term level, continuous-time scheduling in 24/7 operating networks allows for coordinating decisions exactly while striving for a reconciliation of economic and environmental issues. Computational experiments conducted on state-of-the-art high-performance software and hardware reveal that instances of realistic scope cannot be solved to optimality within acceptable times. For this reason, problemtailored variants of relax-and-fix heuristics and genetic algorithms are proposed.

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Framework of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

1 1 3 5 6

2

Scheduling in Supply Chain Management . . . . . . . . . . . . . . . . . . . . 2.1 Characteristics and Management of Supply Chains . . . . . . . . . . . 2.2 Discrete-Time and Continuous-Time Supply Chain Scheduling . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

9 9 14 17

3

Green Supply Chain Management . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Origins and Drivers of Green Supply Chain Management . . . . . . 3.2 Reverse Logistics and Closed Loop Supply Chains . . . . . . . . . . . 3.2.1 Scope and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Aspects of Organization, Coordination, and Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Implementation and Modeling of Closed Loop Supply Chain Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Emission Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Scope and Market-Based Regulation . . . . . . . . . . . . . . . . 3.3.2 Implementation of Emission Management . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

21 22 24 24

.

25

. . . . .

28 32 32 35 37

Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Review of Static Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Review of Discrete-Time Approaches . . . . . . . . . . . . . . . . . . . . 4.2.1 Approaches without Elements of Green Supply Chain Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Approaches Including Elements of Green Supply Chain Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

41 42 47

.

47

.

50

4

ix

x

Contents

4.3

Review of Continuous-Time Approaches . . . . . . . . . . . . . . . . . . 4.3.1 Approaches without Elements of Green Supply Chain Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Approaches Including Elements of Green Supply Chain Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Research Gap and Contributions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

54

.

54

. . .

58 59 61

5

Discrete-Time Scheduling in Green Supply Chain Management . . . 5.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Networks with Recycling and Emission Trading . . . . . . . 5.1.2 Integration of Financial Planning . . . . . . . . . . . . . . . . . . 5.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

65 66 72 80 83 91

6

Continuous-Time Scheduling in Green Supply Chain Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Model Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Networks with Recycling of Non-perishable Goods . . . . . 6.1.2 Networks with Recycling of Perishable Goods . . . . . . . . . 6.1.3 Integration of Financial Planning . . . . . . . . . . . . . . . . . . 6.2 Heuristic Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Relax-and-Fix Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

93 94 94 115 127 136 136 140 149 159

7

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Abbreviations

CPU EU EU ETS GA GAMS GB GHz GSCM GT/s ISO MILP MIP RAM R&F SC SCM

Central processing unit European Union European Union’s Emissions Trading System Genetic algorithm General Algebraic Modeling System gigabyte gigahertz Green supply chain management gigatransfers per second International Organization for Standardization Mixed-integer linear program Mixed-integer program Random-access memory Relax and fix Supply chain Supply chain management

xi

List of Symbols

Sets and Indices A Csτ

F I K

set of liquidity periods; a 2 A ≔ {1, . . ., |A|} set of capacity profiles, selectable for a site s 2 Sσ , if the site has been set up for operations before the planning horizon (τ ¼  1) or the site is set up for operations in time τ ¼ 0, . . ., tE  1, respectively; c 2 Csτ set of quality grades; f 2 F set of generations in genetic algorithm, ι 2 I ≔ {1, . . ., |I|} set of feasible combinations of SC stages for describing potential material flows [depending on the specific model formulation] K¼{(σ ¼ 1, . . ., W + 1)  (λ ¼ σ + 1) ^ (σ ¼ W + 2)  (λ ¼ 1, . . ., W )} within the discrete-time model [Sect. 5.1] K¼{(σ ¼ 1, . . ., W)  (λ ¼ σ + 1, . . ., W + 1) ^ (σ ¼ 2, . . ., W )  (λ ¼ 1, . . ., σ  1) ^ (σ ¼ 2, . . ., W )  (λ ¼ W + 2)} within the continuous-time model for internal recycling [Sect. 6.1.1.1] K¼{(σ ¼ 1, . . ., W)  (λ ¼ σ + 1, . . ., W + 1) ^ (σ ¼ W + 1 )  (λ ¼ W + 2) ^ (σ ¼ W + 2)  (λ ¼ 1, . . ., W )} within the continuous-time model for external recycling [Sect. 6.1.1.2] K¼{(σ ¼ 1, . . ., W)  (λ ¼ σ + 1, . . ., W + 1) ^ (σ ¼ W + 1 )  (λ ¼ W + 2) ^ (σ ¼ W + 2)  (λ ¼ 1, . . ., W ) ^ (σ ¼ 2, . . ., W )  (λ ¼ 1, . . ., σ  1) ^ (σ ¼ 2, . . ., W )  (λ ¼ W + 2)} within the continuous-time model for combined recycling [Sect. 6.1.1.3] K¼{(σ ¼ 1, . . ., W )  (λ ¼ σ + 1, . . ., W + 1) ^ (σ ¼ W + 1)  (λ ¼ W + 2) ^ (σ ¼ 2, . . ., W )  (λ ¼ W + 2)}

xiii

xiv

L

NsqU O

P Sσ SG

List of Symbols

within the continuous-time model for perishable goods [Sect. 6.1.2] set of SC stages with recycling [depending on the specific model formulation] L ¼ {2, . . ., W} within the continuous-time model for internal recycling [Sect. 6.1.1.1] L ¼ {W + 2} within the discrete-time model [Sect. 5.1], the continuous-time model for external recycling [Sect. 6.1.1.2], and the continuous-time model for perishable goods [Sect. 6.1.2] L ¼ {2, . . ., W, W + 2} within the continuous-time model for combined recycling [Sect. 6.1.1.3] set of sites belonging to one potential material flow between sites s and q; NsqU ≔ U [ {s, q}; U 2 ℘ '(SW); (s, q) 2 S1  SW + 1 set of financial transactions; o 2 O [note: credits (o ¼ 1) and investments (o ¼ 2) in discrete-time scheduling, financial alternatives in continuous-time scheduling] set of human-induced greenhouse gases; p 2 P set of sites assigned to SC stage σ 2 Γ; s, q, i, j 2 Sσ set of sites selected for consideration in a genetic algorithm; Wþ1

SG ⊆ [ Sσ σ¼1

SW

W

set of all sites belonging to SC stages σ ¼ 2, . . ., W; SW ¼ [ Sσ σ¼2

T

T T+ Z Γ




 ΓN sqU , 

Λ Θ

set of points in time representing decisions on site states; t, τ 2 T ≔ {1, 0, . . ., tE  1} [note: t ¼  1 represents the initial site state] set of points in time representing decisions on operations; t, τ 2 T ≔ { 0, . . ., tE  1} set of points in time representing monetary surpluses/withdrawals; t, τ 2 T+ ≔ { 0, . . ., tE} set of event boundaries; z 2 Z; Z ≔ {1; 2} [note: z ¼ 1 for start of an event, z ¼ 2 for end of an event] set of SC stages; σ, λ 2 Γ; Γ ≔ {1, . . ., W + 2} [note: the assignment of operations to SC stages depends on the model; in general, σ ¼ 1,. . .,W are before-market stages (e.g., for production, distribution, or recycling), σ ¼ W+1 represents the market stage, and σ ¼ W+2 is an after-market stage (e.g., for recycling, disposal)] ordered set of SC stages representing the sites, which belong to a potential material flow according to NsqU; ΓN sqU ⊆ Γ set of steps in relax-and-fix algorithm, h 2 Λ ≔ {1, . . ., |Λ|} set of genes in genetic algorithm, θ 2 Θ ≔ {1, . . ., |Θ|}

List of Symbols

Υh Ω ℘'(SW)

xv

subset of binary variables to be optimized in the h-th step of a relax-and-fix algorithm set of chromosomes in genetic algorithm, ω 2 Ω ≔ {1, . . ., |Ω|} [note: ω represents the rank of the chromosome within a set that is ordered according to a descending sequence of fitness values] filtered power set of the set SW; ℘'(SW) ≔ {U|U ⊆ SW ^ | U \ Sσ|  1, σ ¼ 2, . . ., W}

Parameters Bλσ Bσ BBλ BEt BMt CAst CCsct CDst CEp

COιθ

CSst Ds Dλs Dst DCs DDs DQst E EDps

usable units of a product manufactured in SC stage λ ¼ 1, . . ., σ  1, which are required to manufacture one unit of a product in SC stage σ ¼ 2, . . ., W usable units of a product manufactured in SC stage λ ¼ σ  1, which are required to manufacture one unit of a product in SC stage σ ¼ 2, . . ., W units of a product manufactured in SC stage λ ¼ 1, . . ., W, which can be obtained from recycling one unit of a final product in SC stage σ ¼ W + 2 buying price of an emission allowance in time t 2 T maximum number of buyable emission allowances in time t 2 T availability costs for site s 2 Sσ of SC stage σ ¼ 1, . . ., W + 2 in time t 2 T [note: represents marketing costs for σ ¼ W + 1] capacity costs for site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 in time t 2 T, if capacity profile c 2 Csτ is selected shutdown costs for site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 in time t 2 T tons of carbon dioxide that are assumed to be comparable to one ton of greenhouse gas p 2 P (factor for determining the carbon dioxide equivalent) [note: factor is one for carbon dioxide itself] equals 1 if the allele of the gene θ 2 Θ belonging to a specific chromosome of the generation ι  1 is to be transferred to the same gene on the chromosome of the new generation ι 2 I during crossover procedure, or 0 otherwise [randomly generated binary parameter for the genetic algorithm] setup costs for site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 in time t 2 T demand at market site s 2 SW + 1 demand of the product manufactured in SC stage λ ¼ 1, . . ., W at market site s 2 SW + 1 demand of the final product at market site s 2 SW + 1 in time t 2 T costs of tardiness in demand satisfaction at market site s 2 SW + 1 (per day) due date for meeting the demand at market site s 2 SW + 1 returnable quantity of the final product at market site s 2 SW + 1 in time t 2 T end time of the planning horizon tons of greenhouse gas p 2 P emitted from recycling at site s 2 Sσ of SC stage σ 2 L (per product)

xvi

EDpst EFps EM EMt EPps EPpst ETpsq ETpsqt EV FAo FCs FDs FLotτ FPot FT FVs FZo io iotτ IAs M MCs MUιω

MVιωθ

MZ NFs

List of Symbols

tons of greenhouse gas p 2 P emitted from recycling at site s 2 SW + 2 in time t 2 T (per product) tons of greenhouse gas p 2 P emitted from disposal at site s 2 SW + 2 (per product) emission cap (tons of carbon dioxide equivalent) during the entire planning horizon emission cap (tons of carbon dioxide equivalent) in time period t 2 T tons of greenhouse gas p 2 P emitted from production at site s 2 Sσ of SC stage σ ¼ 1, . . ., W (per product) tons of greenhouse gas p 2 P emitted from production at site s 2 Sσ of SC stage σ ¼ 1, . . ., W in time t 2 T (per product) tons of greenhouse gas p 2 P emitted from transportation from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K (per product) tons of greenhouse gas p 2 P emitted from transportation from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K in time t 2 T (per product) maximum emission (tons of carbon dioxide equivalent) per emission allowance credit limit of financing alternative o 2 O maximum disposal capacity at site s 2 SW + 2 speed of disposal at site s 2 SW + 2 limit of a single credit (o ¼ 1) or investment (o ¼ 2), respectively, which starts in time t 2 T and ends in time τ 2 T+ limit of all credits (o ¼ 1) or investments (o ¼ 2), respectively, which start in time t 2 T overall credit limit during the planning horizon variable costs of disposal at site s 2 SW + 2 (per product) term of financing alternative o 2 O (difference between the end time and the start time of a financing alternative) credit rate of financing alternative o 2 O interest rate of a credit (o ¼ 1) or an investment (o ¼ 2), respectively, which starts in time t 2 T and ends in time τ 2 T+ equals 1 if site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 has been set up for operations before the planning horizon, and 0 otherwise a sufficiently large number fixed marketing costs at market site s 2 SW + 1 integer number in the interval [1;|Θ|] representing the index of a mutated gene on a chromosome ω 2 Ω of generation ι 2 I [randomly generated integer parameter for the genetic algorithm] binary number representing the new allele of a mutated gene θ 2 Θ on a chromosome ω 2 Ω of generation ι 2 I [randomly generated binary parameter for the genetic algorithm] minimum processing time for operating sites maximum number of transports starting from a site s 2 Sσ of SC stage σ ¼ 1, . . ., W + 2

List of Symbols

NL PCs PCsct PFs PIst PVs

PVst

Rfs es R Rλs Rst RCs RCsct RDs RFs RGfsq

RTs RVs RVst SBsq SDs SEt SMt TCsq

xvii

maximum quantity of recyclable products being disposed of maximum production capacity at site s 2 Sσ of SC stage σ ¼ 1, . . ., W [interpretable as maximum storage capacity for distribution stages] maximum production capacity at site s 2 Sσ of SC stage σ ¼ 1, . . ., W in time t 2 T, if capacity profile c 2 Csτ is selected fixed production costs at site s 2 Sσ of SC stage σ ¼ 1, . . ., W [interpretable as fixed storage costs for distribution stages] variable inventory costs at site s 2 Sσ of SC stage σ ¼ 1, . . ., W in time t 2 T (per product) variable production costs at site s 2 Sσ of SC stage σ ¼ 1, . . ., W (per product) [interpretable as variable storage costs for distribution stages] variable production costs at site s 2 Sσ of SC stage σ ¼ 1, . . ., W in time t 2 T (per product) [interpretable as variable storage costs for distribution stages] revenue of a product package of perishable goods in quality grade f 2 F according to the demand of market s 2 SW + 1 expected revenue of a perishable product recycled at site s 2 SW + 2 revenue of the product manufactured in SC stage λ at market site s 2 SW + 1 (per product) revenue of the final product at market site s 2 SW + 1 in time t 2 T (per product) maximum recycling capacity at site s 2 Sσ of SC stage σ 2 L maximum recycling capacity at site s 2 SW + 2 in time t 2 T, if capacity profile c 2 Csτ is selected speed of recycling at site s 2 Sσ of SC stage σ 2 L fixed costs of reverse operations (recycling and/or disposal) at site s 2 Sσ of relevant SC stage σ maximum distribution range (i.e., time difference between the end and the beginning of a realized material flow) between sites q 2 SW + 1 and s 2 S1 that is allowed for a product package to be sold in grade f 2 F time period at a market s 2 SW + 1 between demand satisfaction and product return variable costs of recycling at site s 2 Sσ of SC stage σ 2 L (per product) variable costs of recycling at site s 2 SW + 2 in time t 2 T (per product) maximum temporary storage time allowed for transports from site s 2 Sσ to site q 2 Sλ of SC stages (σ, λ) 2 K speed of production at site s 2 Sσ of SC stage σ ¼ 1, . . ., W selling price of an emission allowance in time t 2 T maximum number of saleable emission allowances in time t 2 T maximum transportation capacity for transports from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K

xviii

TCsqt TFsq TFsqt TSsq TVsq TVsqt TZsq W XCs XCst αfs αs βfs βs γs δs ηt κo υt

List of Symbols

maximum transportation capacity for transports from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K in time t 2 T fixed transportation costs for transports from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K fixed transportation costs for transports from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K in time t 2 T variable costs of temporary storage for transports from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K (per day) variable transportation costs for transports from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K (per product) variable transportation costs for transports from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K in time t 2 T (per product) transportation time from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K number of SC stages belonging to forward operations [equals the number of SC stages before the market stage] maximum throughput capacity for recycled products at site s 2 Sσ of SC stage σ ¼ 1, . . ., W maximum throughput capacity for recycled and stocked products at site s 2 Sσ of SC stage σ ¼ 1, . . ., W in time t 2 T share of quantities in quality grade f 2 F at site s 2 SW + 1 that needs to be processed in reverse logistics (recycling or disposal) share of quantities at site s 2 SW + 1 that needs to be processed in reverse logistics (recycling or disposal) share of quantities in quality grade f 2 F at site s 2 SW + 2 that can be recycled [note: (1  βfs) represents the share of quantities that cannot be recycled and, thus, needs to be disposed of] share of quantities at site s 2 Sσ of relevant SC stage σ that can be recycled [note: (1  βs) represents the share of quantities that cannot be recycled and, thus, needs to be disposed of] share of perishable quantities at site s 2 Sσ of SC stage σ ¼ 2, . . ., W that needs to be disposed of ratio between products supplied from SC stage λ that must be processed in reverse logistics, and products supplied from SC stage λ that can be used for manufacturing at site s 2 Sσ of SC stage σ ¼ 2, . . ., W technical parameter: η0 ¼ 0, η1 ¼ . . . ¼ ηtE 1 ¼ 1 technical parameter: κ1 ¼ 1, κ 2 ¼  1 weighting factor of surpluses realized in time t 2 T+

Variables af zoa apsa

credit amount (z ¼ 1) or repayment amount (z ¼ 2) of financing alternative o 2 O, assigned to liquidity period a 2 A costs of forward and reverse operations at site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2, assigned to liquidity period a 2 A

List of Symbols

arsa atsa b drs ebt est fizo fiotτ

frs gfs grfsq haιωθ hrιωθ hsθ inst lf zoa lpzsa

nus nuλs ocιω

orιω prs prst qa qt

xix

revenues (adjusted for marketing costs) at market site s 2 SW + 1, assigned to liquidity period a 2 A transportation costs for transports starting from site s 2 Sσ of SC stage σ ¼ 1, . . ., W + 2, assigned to liquidity period a 2 A auxiliary variable to be used as a counter recycled quantity at site s 2 Sσ of SC stage σ 2 L number of emission allowances bought in time t 2 T number of emission allowances sold in time t 2 T amount paid in after realizing a financing alternative o 2 O (z ¼ 1) or amount paid out for repaying the financing alternative (z ¼ 2) amount paid in after realizing a credit (o ¼ 1) or amount paid out after realizing an investment (o ¼ 2), which starts in time t 2 T and ends in time τ 2 T+ disposed quantity at site s 2 SW + 2 equals 1 if market s 2 SW + 1 is supplied with a product package of grade f 2 F, and 0 otherwise quantity in quality grade f 2 F at market site s 2 SW + 1 that needs to be processed in reverse logistics at site q 2 SW + 2 allele of the gene θ 2 Θ on chromosome ω 2 Ω in generation ι 2 I allele of the gene θ 2 Θ that belongs to the chromosome, which is on rank ω 2 Ω in generation ι 2 I allele of the gene θ 2 Θ that belongs to the chromosome, which is on the first rank ω ¼ 1 in the final generation ι ¼ |I| inventory at site s 2 Sσ of SC stage σ ¼ 1, . . ., W in time t 2 T, t  1 [note: initial stocks can be considered by a parameter ins0] equals 1 if financing alternative o 2 O starts (z ¼ 1) or ends (z ¼ 2) within liquidity period a 2 A, and 0 otherwise equals 1 if operation at site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 starts (z ¼ 1) or ends (z ¼ 2) within liquidity period a 2 A, or if demand at site s 2 SW + 1 is satisfied (z ¼ 1) or returned (z ¼ 2) within liquidity period a 2 A, respectively, and 0 otherwise recyclable quantity at site s 2 SW + 2, which is disposed of recyclable quantity of the product manufactured in SC stage λ at site s 2 Sσ of SC stage σ 2 L, which is disposed of fitness value of the chromosome ω 2 Ω in generation ι 2 I [note: equals the objective value of the submodel that results from the fixation of the binary variables according to the chromosome’s alleles] fitness value of the chromosome on rank ω 2 Ω in generation ι 2 I production quantity at site s 2 Sσ of SC stage σ ¼ 1, . . ., W [interpretable as storage quantity for distribution stages] production quantity at site s 2 Sσ of SC stage σ ¼ 1, . . ., W in time t 2 T [interpretable as storage quantity for distribution stages] liquidity in period a 2 A liquidity withdrawal in time t 2 T+

xx

qr λsq qrst rr λs sf zo sls snst spzs

stsq sust tpsq tpsqt uds ufs ups urs vsqU vet vot vst xsq xsqt ys

List of Symbols

quantity of the product manufactured in SC stage λ at market site s 2 SW + 1, which needs to be processed in reverse logistics at site q 2 SW + 2 quantity of the final product, which needs to be processed in reverse logistics at site s 2 SW + 2 in time t 2 T quantity of the product manufactured in SC stage λ available at site s 2 Sσ of SC stage σ ¼ 2, . . ., W, which needs to be processed in reverse logistics start time (z ¼ 1) or end time (z ¼ 2) of financing alternative o 2 O tardiness in demand satisfaction at market site s 2 SW + 1 (in days) equals 1 if site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 is shut down in time t 2 T, and 0 otherwise start time (z ¼ 1) or end time (z ¼ 2) of operations (e.g., production, storage, recycling, disposal) at site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2, or time of satisfying demand (z ¼ 1) or returning products (z ¼ 2) at market site s 2 SW + 1 temporary storage time for transports from site s 2 Sσ to site q 2 Sλ of SC stages (σ, λ) 2 K equals 1 if site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 is set up in time t 2 T, and 0 otherwise equals 1 if transportation from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K is conducted, and 0 otherwise equals 1 if transportation from site s 2 Sσ to site q 2 Sλ of the SC stages (σ, λ) 2 K is conducted in time t 2 T, and 0 otherwise auxiliary variable representing the variable recycling costs at site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 auxiliary variable representing the variable disposal costs at site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 auxiliary variable representing the fixed and variable production costs at site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 auxiliary variable representing the fixed costs of reverse operations at site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 auxiliary variable (see set NsqU) auxiliary variable representing the overall monetary consequences of environmental decisions in time t 2 T auxiliary variable representing the overall monetary consequences of decisions on operations in time t 2 T+ auxiliary variable representing the overall monetary consequences of decisions on site states in time t 2 T transportation quantity from site s 2 Sσ to site q 2 Sλ of SC stages (σ, λ) 2 K transportation quantity from site s 2 Sσ to site q 2 Sλ of SC stages (σ, λ) 2 K in time t 2 T equals 1 if production/storage at site s 2 Sσ of SC stage σ ¼ 1, . . ., W is conducted [for all continuous-time models], or if market site s 2 SW + 1 is selected [for all continuous-time models except the one for perishable goods, see Sect. 6.1.2], and 0 otherwise

List of Symbols

ysct ymst yrs Φ

xxi

equals 1 if site s 2 Sσ of SC stage σ ¼ 1, . . ., W, W + 2 is available in capacity profile c 2 Csτ in time t 2 T, and 0 otherwise equals 1 if market site s 2 SW + 1 is selected in time t 2 T, and 0 otherwise equals 1 if reverse operations (recycling and/or disposal) at site s 2 Sσ of relevant SC stage σ are conducted, and 0 otherwise objective value

List of Figures

Fig. 1.1 Fig. 2.1 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8

Length of planning horizons and implications for scheduling . . . . . . Structure of SC network in macro- and micro-perspective . . . . . . . . . General network structure in discrete-time scheduling . . . . . . . . . . . . . . Time representation in discrete-time scheduling . . . . . . . . . . . . . . . . . . . . . Coordination of operational and financial decisions in discrete-time scheduling . .. . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . .. . . Optimal network structure in discrete-time scheduling . . . . . . . . . . . . . Computation times using different modes of the CPLEX-D solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General network structure for internal recycling of non-perishable goods in continuous-time scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General network structure for external recycling of non-perishable goods in continuous-time scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General network structure for recycling of perishable goods in continuous-time scheduling . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. Liquidity periods and assignment of monetary consequences . . . . . . Basic elements and principles of the relax-and-fix algorithm . .. . .. . Basic elements and principles of the genetic algorithm . . . . . . . . . . . . . Flowchart of the genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal network structure in continuous-time scheduling . . . . . . . . . .

3 10 67 68 81 86 90 95 108 117 128 137 141 144 152

xxiii

List of Tables

Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8

Quantities of demand and return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interest rates for credits and investments . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquidity balancing in discrete-time scheduling . . . . . . . . . . . . . . . . . . . Results of the scenario analysis for discrete-time scheduling . . . . . Big M method in constraint (6.62) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Big M method in the system of constraints (6.76)–(6.91) and (6.105)–(6.109) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operational parameters . . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . Available financial alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquidity balancing in continuous-time scheduling . . . . . . . . . . . . . . . Test of different relax-and-fix strategies for two-step algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the relax-and-fix algorithms in the scenario analysis for continuous-time scheduling .. . . .. . . . .. . . .. . . .. . . . .. . . .. . . .. . . . .. . . .. . Results of the genetic algorithms in the scenario analysis for continuous-time scheduling .. . . .. . . . .. . . .. . . .. . . . .. . . .. . . .. . . . .. . . .. .

84 85 87 89 124 130 150 151 153 155 157 158

xxv

Chapter 1

Introduction

Abstract Based on existing structures of sites and capacities, new mathematical models for hierarchical scheduling of operations in generic multi-stage networks are developed in the book Scheduling in Green Supply Chain Management: A MixedInteger Approach. This chapter introduces into the problem of scheduling in green supply chain management. It contains the motivation of the book, the framework of analysis, and an outline.

1.1

Motivation

The integration of environmental concerns in supply chain management evolved into an important field of research over the past decades. The latter is typically covered by the term “green supply chain management” (Srivastava 2007; Sarkis 2012; Sarkis and Dou 2018). Besides, many other concepts with considerable overlaps exist. For instance, sustainable supply chain management (e.g., Seuring and Müller 2008; Brandenburg et al. 2014; Rajeev et al. 2017) includes a social dimension in addition to the economic and the environmental one. Reverse logistics strive for recycling products and establishing closed loop supply chains (e.g., Dyckhoff et al. 2004; Lebreton 2007; Govindan and Soleimani 2017). Green logistics (e.g., Dekker et al. 2012; Blanco and Sheffi 2017) take into account environmental aspects such as greenhouse gases, noise, and use of scarce resources for planning distribution and transportation. Other separable aspects of GSCM are green procurement and green supplier development (e.g., Blome et al. 2014) or green marketing (e.g., Dangelico and Vocalelli 2017). The diversity of literature is supported by the fact that more than 20 different definitions of GSCM exist, while the broadest perspective includes upstream, downstream, organizational, and intraorganizational efforts to link SC practices with issues of environment (Ahi and Searcy 2013; Sarkis and Dou 2018, pp. 9–10).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6_1

1

2

1 Introduction

The impact of GSCM in scientific literature was evaluated by de Oliveira et al. (2018) within a systematic bibliometric analysis of articles published in the field from 2006 to 2016. They extracted 339 papers from 5 relevant databases entailing thousands of citations. The quantity of publications per year increases constantly since 2009. It is noteworthy that most of the articles contain empirical studies (52%). Besides papers classified as literature review (11%), a considerably lower share (37%) contains mathematical models of GSCM. According to common classifications, the latter models comprise a huge variety of approaches spanning mathematical programming (e.g., linear programming (Chanchaichujit et al. 2016), mixed-integer linear programming (Memari et al. 2015; Nurjanni et al. 2017), or goal programming (Tsai and Hung 2009)), simulation (e.g., van der Vorst et al. 2009), heuristics (e.g., Yeh and Chuang 2011; Kannan et al. 2010), hybrid models combining mathematical programming and simulation (e.g., Fichtner et al. 2004), as well as analytical models (e.g., data envelopment analysis (Mirhedayatian et al. 2014; Mahdiloo et al. 2015), analytical hierarchy process (Lu et al. 2007; Shen et al. 2015), analytical network process (Dou et al. 2014; Tseng et al. 2014), or game theory (Chen and Sheu 2009; Sheu and Chen 2012)). Several empirical studies outline general benefits of implementing GSCM. The analysis of Green Jr. et al. (2012) provides first evidence that supports performancerelated needs for manufacturing organizations to implement GSCM in collaboration with customers and suppliers. According to the empirical study of Choi and Hwang (2015) considering Korean manufacturers, GSCM can improve both environmental and financial performances. The meta-analysis of Geng et al. (2017) focusing on Asian emerging economies indicates that implementing GSCM leads to a better economic, environmental, and operational performance. The application to retailers in Croatia is considered by Petljak et al. (2018). They reveal a positive correlation between green in-store activities and environmental and economic performance in food industry. Moreover, Hartmann and Vachon (2018) analyze the influence of the industry context on the relationship between environmental management and organizational performance. Fang and Zhang (2018) consider the practice-performance relationship for several moderators. According to their empirical study, both internal and external GSCM practices are positively related to the firm performance, while effects of industry type, ISO certification, export orientation, and the cultural dimension of uncertainty avoidance exist. Besides the scientific discussion, a growing acceptance of GSCM in practice can be observed. Starting from a widespread attitude of resistance, important industry leaders of the manufacturing sector changed their business strategy to supply chain transparency with regard to environmental sustainability (Doorey 2011). Driven by an increasing ecological awareness of the customers and the effects on the public image, a majority of companies attributes considerable importance to carbon dioxide emissions. Broadening the perspective of their businesses, most successful market players pursue an integrated view of their supply chains while taking the ecological sustainability of their partners into account (Straube and Doch 2011, p. 353).

1.2 Framework of Analysis

1.2

3

Framework of Analysis

This monograph focuses on issues of coordinated scheduling in GSCM. Determining optimal schedules for all sites belonging to a common network necessitates the implementation of mathematical models. In particular, mixed-integer programming models are developed for decisions on sites, operations, and financial transactions in line with environmental goals and restrictions. As far as an optimization is not possible within acceptable computation times even for small-sized problem structures, compatible heuristics are proposed in addition. In course of the following analysis, MILP for two different types of planning horizons characterized by different lengths, and thus different predictability of data, are developed (see Fig. 1.1). The models can be composed in a hierarchical manner. Acknowledging unavoidable inaccuracies arising from rough forecasts of demand, returns, revenues, and/or costs in a medium-term planning horizon, which can span several months, the application of discrete-time scheduling based on the big bucket method seems to be most rational. In this context, the monetary consequences of

planning horizons medium-term (see Chapter 5)

…

…

t =0

t =1

t = tE

potential sites maximum capacities emission limits

t =0

t =1

potential sites maximum capacities emission limits

short-term (see Chapter 6)

t =0 potential sites maximum capacities emission limits

t = 0,...,t E

discrete-time scheduling

t =1

… t = tE

t = tE continuous-time scheduling

continuous-time scheduling

…

Fig. 1.1 Length of planning horizons and implications for scheduling

continuous-time scheduling

4

1 Introduction

decisions are assigned to time points at the beginning or the end of time periods the planning horizon is divided into (e.g., Steinrücke and Albrecht 2015, 2016a, b, 2018). Resulting decisions on setups and shutdowns of given sites, the provision of maximum technical and human capacities at these sites, and network-wide environmental restrictions (as an outcome of emission trading) become requirements in each time period that is subsequently scheduled in separation. As data can be assumed to be much better predictable at the beginning of each of the multiple short-term planning horizons of only several days, continuous-time scheduling allows for detailing all relevant decisions exactly to the minute (e.g., Steinrücke 2007, 2011a, b, 2015; Albrecht and Steinrücke 2017, 2018, 2020a, b) so as to be directly executable. Modeling is based on the following general assumptions: All model formulations are deterministic. The relevant data is assumed to be predictable, while quality depends on the length of the planning horizons. The latter are clearly defined and, furthermore, must be divided into time periods of equal length in case of discrete-time scheduling. In opposite, an optional discretization into liquidity periods is only required within continuous-time scheduling, if financial planning should be integrated. Planning is based on a given strategic network design (e.g., Albrecht 2014). This includes potential sites that can neither be opened nor closed during the considered horizons. On that basis, changes of site state affecting the sites’ availability (setups, shutdowns, capacity provision) are possible on the medium-term, but not on the short-term level. Demand is fixed according to existing contracts (that contain both quantities and due dates), but does not need to be satisfied in full. On the medium-term level, demand is associated with the main outcome of the SC (finished products). On the short-term level, demand can comprise products of different maturities, while non-perishables and perishables are distinguished. Operations are detailed according to the length of the planning horizon. Bills of materials are given, but limited to relationships between adjacent SC stages on the medium-term level. Each SC stage is assigned to the production of a good in a specific maturity. Potential direct transports between sites result from the general network structure, while deliveries between sites of the same SC stage are excluded. According to long distances of transportation, problems of vehicle routing are not subject of the following analysis. Returns are to be handled mandatory by the SC while being classified according to the length of the planning horizon. On the medium-term level, returns of used products are considered at the markets. As useful life at the customers exceeds the length of the horizon, returnable quantities are given independently of the periodic demand. On the short-term level, there are rejects of defective products. These can occur at the production sites on the one hand and at the market sites on the other hand, which entails immediate internal or external recycling, respectively. On both levels, there are given shares of quantities to be returned, recycled, or disposed of, which are assumed to be predictable according to years of experience.

1.3 Outline

5

Financial planning is based on nonspeculative transactions. The latter include medium-term credits and financial investments with different duration-dependent interest rates that are used to balance liquidity so as to prevent insolvency and shortterm credits that can be used for bridging the time period between operations and sales if equity financing is not applicable.

1.3

Outline

The remainder of this monograph is structured as follows: General theories and approaches of scheduling in SCM are systematized in Chap. 2 while referring to related definitions and time representations. Chap. 3 deals with the peculiarities of GSCM, especially its origins and drivers. This allows for identifying the main aspects in this field, which can be integrated into mathematical models of mediumand short-term optimization of multi-stage networks. Firstly, capturing reverse logistics entails the incorporation of recycling processes, which may be required for reasons of waste reduction. In consequence, the formation of closed loop systems may occur. Secondly, emission management contains assessing and limiting greenhouse gas volumes caused by network operations. In line with existing frameworks for market-based regulation, implications for implementation are given. Acknowledging the aforementioned requirements, a literature review in Chap. 4 reveals the lack of suitable approaches so far and specifies the research gap. Besides general insights from static modeling, the reviewed publications are grouped to continuousand discrete-time formulations according to their applicability to problems with planning horizons of different length. Chapter 5 proposes approaches for discretetime master scheduling in GSCM, which comprises setup and shutdown decisions for making the existing network sites available for operations in time periods. Based on aggregated material flows, modeling includes end-of-use returns and the adjustment of periodic emission caps. The integration of financial planning allows for balancing liquidity by taking into account financial transactions. The model formulation is validated by a small-scale test instance in course of a numerical analysis. Subsequently, Chap. 6 provides modularized formulations for continuous-time scheduling in GSCM, which can be distinguished according to the perishability of goods. For non-perishables, three different types of recycling are considered. Internal recycling focuses on handling defective material occurring during the manufacturing process or production scrap, while external recycling takes immediate customer returns into account. Both network structures can be merged so as to obtain combined recycling. Specific requirements on modeling arise from the considerations on perishables, as their reverse logistics are usually affected by the product quality that is realized at the markets. The aforementioned approaches also allow for the integration of financial planning, which is based on a coordination of exactly scheduled operations with liquidity periods. Due to the complexity of the continuous-time formulations, problem-tailored heuristic solution methods are proposed to enhance their computability. On the one hand, benefits of problem

6

1 Introduction

decomposition can be realized by the application of relax-and-fix algorithms. On the other hand, genetic algorithms exploit evolutionary principles for improving solutions within an iterative procedure. Going beyond model validation, a subsequent numerical study includes a scenario analysis for evaluating the performance of the heuristics. Chapter 7 summarizes the main results of this monograph and provides starting points for further research.

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1 Introduction

Shen, L., Muduli, K., & Barve, A. (2015). Developing a sustainable development framework in the context of mining industries: AHP approach. Resources Policy, 46, 15–26. Sheu, J.-B., & Chen, Y. J. (2012). Impact of government financial intervention on competition among green supply chains. International Journal of Production Economics, 138(1), 201–213. Srivastava, S. K. (2007). Green supply-chain management: A state-of-the-art literature review. International Journal of Management Reviews, 9(1), 53–80. Steinrücke, M. (2007). Termin-, Kapazitäts- und Materialflussplanung bei auftragsorientierter Werkstattfertigung. Wiesbaden: Deutscher Universitäts-Verlag. Steinrücke, M. (2011a). An approach to integrate production-transportation planning and scheduling in an aluminium supply chain network. International Journal of Production Research, 49 (21), 6559–6583. Steinrücke, M. (2011b). Integrierte Produktions-, Distributions- und Terminplanung in globalen Supply Chains. Zeitschrift für betriebswirtschaftliche Forschung, 63, 19–47. Steinrücke, M. (2015). Integrated production, distribution and scheduling in the aluminium industry: A continuous-time MILP model and decomposition method. International Journal of Production Research, 53(19), 5912–5930. Steinrücke, M., & Albrecht, W. (2015). Mehrperiodige Kapazitäts- und Materialflussplanung in dynamischen Supply Chain Netzwerken mit Zulieferer- und Zielmarktauswahl. Betriebswirtschaftliche Forschung und Praxis, 67(4), 436–456. Steinrücke, M., & Albrecht, W. (2016a). Quantitative decision support for network integration of start-up companies. International Journal of Globalisation and Small Business, 8(1), 73–99. Steinrücke, M., & Albrecht, W. (2016b). A flow-to-equity approach to coordinate supply chain network planning and financial planning with annual cash outflows to an institutional investor. Business Research, 9, 297–333. Steinrücke, M., & Albrecht, W. (2018). Integrated supply chain network planning and financial planning respecting the imperfection of the capital market. Journal of Business Economics, 88 (6), 799–825. Straube, F., & Doch, S. (2011). A contribution to sustainable logistics and supply chain—Conceptual design to evaluate ecological and economical cause-effect relations in logistics planning processes. In G. Seliger, M. Khraisheh, & I. Jawahir (Eds.), Advances in sustainable manufacturing (pp. 351–362). Berlin/Heidelberg: Springer. Tsai, W.-H., & Hung, S.-J. (2009). A fuzzy goal programming approach for green supply chain optimisation under activity-based costing and performance evaluation with a value-chain structure. International Journal of Production Research, 47(18), 4991–5017. Tseng, M.-L., Lin, R.-J., Lin, Y.-H., Chen, R.-H., & Tan, K. (2014). Close-loop or open hierarchical structures in green supply chain management under uncertainty. Expert Systems with Applications, 41(7), 3250–3260. Van der Vorst, J. G. A. J., Tromp, S.-O., & van der Zee, D.-J. (2009). Simulation modelling for food supply chain redesign; integrated decision making on product quality, sustainability and logistics. International Journal of Production Research, 47(23), 6611–6631. Yeh, W.-C., & Chuang, M.-C. (2011). Using multi-objective genetic algorithm for partner selection in green supply chain problems. Expert Systems with Applications, 38(4), 4244–4253.

Chapter 2

Scheduling in Supply Chain Management

Abstract Based on existing structures of sites and capacities, new mathematical models for hierarchical scheduling of operations in generic multi-stage networks are developed in the book Scheduling in Green Supply Chain Management: A MixedInteger Approach. In this chapter, general theories and approaches of scheduling in supply chain management are systematized while referring to related definitions and time representations.

2.1

Characteristics and Management of Supply Chains

Considering the current structure of international business, it can be observed that most organizations function as part of a larger supply chain. Within the latter, both manufacturers and service providers work together in order to satisfy the given demand at the markets (Bozarth and Handfield 2013, p. 21). Typically, supply chains are networks which build a system of multiple suppliers and customers with relationships between them. Accordingly, a supply chain can be defined as “network of organizations that are involved, through upstream and downstream linkages, in the different processes and activities that produce value in the form of products and services in the hands of the ultimate consumer” (Christopher 2016, p. 13). Linking is realized by flows of products, information, and funds (Fandel et al. 2009, p. 2; Bozarth and Handfield 2013, p. 21; Chopra and Meindl 2013, p. 14). Besides the predominating perspective of an interorganizational collaboration, which describes a network organization of legally separated companies, even concepts of intraorganizational supply chains exist. The latter represent companies with interconnected resources (Stadtler 2015a, pp. 15–17). For structuring SC networks, they can be split up into different SC stages. Each of them contains sites of the same function, e.g., component or raw material suppliers, manufacturers, wholesalers, distributors, retailers, or customers (Chopra and Meindl 2013, pp. 14–15). The highest level of generalization is reached if SC networks with

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6_2

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SC stage I

SC stage II

SC stage N

Sites of SC stage I

Sites of SC stage II

information flow

forward material flow

…

site II,1

site I,2

site II,2

site I,3

site II,3

…

…

…

…

…

reverse material flow

site I,1

site I,m

financial flow

site II,n

material flows in both directions

Fig. 2.1 Structure of SC network in macro- and micro-perspective

an arbitrary number of stages and sites are modeled (e.g., Steinrücke 2011a, b, 2015; Steinrücke and Albrecht 2016, 2018; Albrecht and Steinrücke 2017, 2018, 2020a, b). On the one hand, aggregated flows between SC stages can be depicted in a macro-perspective. On the other hand, detailed flows between individual sites of the SC stages are part of a micro-perspective (see Fig. 2.1). In both cases, relationships are assumed to be potential flows, on whose realization is decided in the context of the problem’s overall optimization. Driven by growing competition in global markets and supported by continuing advances in communication technologies, holistic concepts for managing SC networks were developed. In their context, supply chain management can be defined as “a set of approaches utilized to efficiently integrate suppliers, manufacturers, warehouses, and stores, so that merchandise is produced and distributed at the right quantities, to the right locations, and at the right time, in order to minimize systemwide costs while satisfying service level requirements” (Simchi-Levi et al. 2008, p. 1). Besides cost minimization, alternative objective functions focus on the maximization of profits or of the net present value. Even nonmonetary objectives regarding customer service, risk, and flexibility can be found in the literature (Fleischmann and Koberstein 2015, p. 110). The characteristic feature of SCM is the global optimization of the entire network. Typical challenges of this task arise from different reasons. Firstly, a considerable complexity of the network exists if facilities are dispersed over a large region or even all over the globe. Secondly, different facilities of the network may have different and often conflicting objectives. Thirdly, dynamic effects need to be considered as an SC is an evolving system (e.g., due to changes in capabilities of the partners or in network relationships). Finally, there are system variations over time caused by timevarying parameters (e.g., demand, costs). Taking into account the aforementioned biases, an optimization across the network facilities and processes is required (Simchi-Levi et al. 2008, pp. 4–5). Some origins of SCM can be found in logistics, which provide a framework for seeking and creating plans for flows of products and information through a single

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business. Building upon this base, SCM seeks for linkage and coordination between different processes along the entire SC. In this context, both upstream and downstream relationships with suppliers are taken into account (Christopher 2016, pp. 2–3). Besides logistics, features of other subjects like marketing, operations research, organizational behavior, as well as purchasing were integrated in the concept of SCM. This integration follows the five principles of logistics thinking, in particular, the thinking in values and benefits, the systems thinking, the total cost thinking, the service orientation, as well as the striving for efficiency (Stadtler 2015a, p. 17). By that, SCM aims to achieve more profitable outcomes for all entities belonging to the chain. Besides, competitive advantages can result from generating enduring superiority over competitors, which may occur in terms of customer preferences. Different sources of competitive advantages can be found, for example, in the ability of the organization to differentiate itself or to operate at lower costs (Christopher 2016, p. 4). A concept for systematizing the major targets, functions, and bases is the House of SCM. Due to its structure, it subordinates relevant dimensions and tasks to the ultimate goal of maintaining competitiveness. The latter is closely related to focusing on customer service, while there are different ways of improvement (e.g., reducing costs, increasing flexibility, providing superior qualities of products and services). The main functions of SCM are depicted by the pillars of the house. On the one hand, they contain the integration of network organizations; on the other hand, they include the coordination of information, material, and financial flows. With regard to integration, the medium-term oriented selection of external partners, the effective structuring of the network organization, the interorganizational collaboration between legally independent sites, and the development of leadership concepts for aligning the strategies of the SC partners are to be considered. Relevant aspects of coordination include application of information and communication technologies for automating and facilitating transactions, consequent process orientation going along with a striving for standardization, as well as identification and connection of various planning tasks within advanced planning frameworks. Serving as the base of the house, it becomes obvious that different disciplines of business administration and economics including logistics, production, marketing, organizational theory, and operations research are to be connected by existing interdependencies within research in the field of SCM (Stadtler 2005, 2015a, pp. 5–6). Detailing the aforementioned requirement of process orientation, the supply chain operations reference model is proposed for standardization and benchmarking of processes in SC networks. It consists of different levels of aggregation, in particular, the types, categories, elements, and implementations of processes (Sürie and Reuter 2015, pp. 33–37). Doubts regarding the ability of the approach to represent structures of SC networks appropriately in practice can be found in the literature (Fandel et al. 2009, pp. 273–274). Each supply chain can be categorized by a set of attributes, which are part of the following SC typology proposed by Meyr and Stadtler (2015). In the first major group, there are functional attributes that describe an SC’s type of procurement, production, distribution, and sales. Firstly, the procurement type specifies the

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number and type of products that are obtained by external suppliers. Moreover, different sourcing types (e.g., single, double, or multiple sourcing) can be distinguished according to given contracts. The flexibility of suppliers refers to the supplied amounts that can be fixed, varied between specific bounds, or even chosen freely. Both the lead time and the reliability can be used for the evaluation of a supplier. The life cycles of the products to be procured have impact on their quality and obsolescence and, thus, are relevant for planning of reverse logistics. Secondly, SCs can be characterized by their production type. Different forms of organizing the production process (e.g., job shop problems or flow shop problems) are possible. With regard to the repetition of operations, mass production, batch production, or one-of-a-kind production can be distinguished, while the aforementioned classification influences both setup costs and times. It can be specified by taking other changeover characteristics additionally into account. Due to limited production capacities in the SC, bottlenecks can be identified. The latter do refer not only to technical equipment (e.g., machines or shelves) but also to human resources. Work time flexibility can be characterized by possibilities to adapt working time to changing demand patterns. Thirdly, there are attributes regarding the distribution type. In this context, the distribution structure refers to the number and the hierarchical arrangement of SC stages for storage and delivery. Besides direct transportation from the plants to the customers, there may be stages for central, regional, or even local warehouses. The pattern of delivery can be cyclic (i.e., goods are delivered between fixed bounds of times) or dynamic (i.e., goods are delivered if demand is occurring). With regard to the deployment of transportation means, vehicle routing (with standard routes or specific demand-dependent routes) or individual direct transportation is alternatively possible. Additional constraints for distribution may result from restrictions on the loading capacity. Fourthly, the sales type of the SC can be determined by several attributes. Different physical or electronic relations to the customer can occur. Dependent on them, there may be differences in the availability of information about future demand (i.e., contractbased data or forecasts). In this context, different forms of demand curves (e.g., static, sporadic, or seasonal demand) need to be taken into account. Again, considerations on life cycles need to be connected to expectations on product return, recycling, and disposal. The composition and decomposition of the sold products usually adhere to given bills of materials. Efforts of sales are connected to the number of product types to be offered and the degree of customization. Finally, the proportion of service operations that is linked to the sales is another relevant attribute for categorization. In a second major group, structural attributes are used to describe an SC network. Firstly, the network topography is to be analyzed. It describes the structure of forward material flows (i.e., from upstream to downstream SC stages) and reverse material flows (i.e., from downstream to upstream SC stages), which can be serial, convergent, divergent, or a mixture of all. According to the degree of globalization, regional, national, or international networks may occur. In this context, the integration of different taxes, duties, or exchange rates can be required for the model formulation. According to the location of decoupling points, the network forms engineer-to-order, manufacture-to-order, assemble-to-order, or

2.1 Characteristics and Management of Supply Chains

13

deliver-to-order can be distinguished. The identification of the system’s major constraints usually refers to given capabilities and capacities. Secondly, attributes of integration and coordination need to be specified. According to the legal position of the network sites, either interorganizational or intraorganizational SCM is applicable. Taking into account the given balance of power in interorganizational structures, a dominant member acting as a focal company in the network may evolve. The existence of the latter would allow for the application of central planning approaches. Otherwise, decentral planning approaches can be used to optimize polycentric structures (see Lee and Billington 1993; Dudek and Stadtler 2005). Possible directions of coordination are vertical (between different SC stages in a hierarchical sequence), horizontal (between sites of the same SC stage), or a mixture of both. Different types of information can be exchanged within coordination (e.g., demand, costs, capacities), possibly to a different extent (Meyr and Stadtler 2015, pp. 55–61). A systematization of important tasks arising in SCM contains the framework of advanced planning systems (Stadtler 2005). The according matrix includes different modules that result from a classification by two dimensions, in particular, the planning horizon (long, medium, and short term) and the SC processes (procurement, production, distribution, and sales). With regard to the long-term range, strategic network design covers all processes within simultaneous planning. In particular, it includes facility location planning, the design of the physical distribution structure, as well as aspects of strategic sales planning. Stepping down the hierarchy to the medium-term range, master planning needs to be considered. Based on forecasts and estimates of demand planning, it focuses on matching aggregated material flows between the network sites with given capacities, and furthermore, it includes master production scheduling. In this context, both technical and human capacities are considered simultaneously. Finally, there are several tasks that can be assigned to the short-term level of SC planning that deals with detailing decisions to a level that allows for their direct execution. The purchasing and material requirement planning is based on given bills of materials and takes into account specific conditions of procurement contracts (e.g., quantity discounts). Production planning refers to the determination of exact quantities to be produced at the network sites based on the existing production system (e.g., job shop or flow shop production). A subsequent distribution planning that is based on splitting up the production quantities in appropriate transportation lots can be considered as the starting point for determining exact material flows (that can be directed either forwardly or reversely). In the context of transportation planning, suitable means of transport are to be selected for specific paths. Scheduling in course of short-term planning allows for the exact temporal specification of operations on the level of lots, machines, or sites (Fleischmann and Meyr 2003; Meyr et al. 2015, pp. 99–101). Although structuring the main tasks of SC planning is being required for commercial software applications, a main shortcoming of advanced planning systems results from modularization and hierarchization entailing an insufficient consideration of existing interdependencies. Further problems of implementation arise from the deterministic perspective, the quality of parameters, and the user requirements (Fandel et al. 2009,

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pp. 146–147). Besides aspects of planning, the emphasis of this monograph is on discrete-time and continuous-time scheduling of supply chain networks.

2.2

Discrete-Time and Continuous-Time Supply Chain Scheduling

In general, scheduling aims at a determination of start times and durations of SC operations (i.e., production, transportation, sales) within a given planning horizon. It also includes the sequencing of processes on given resources. The planning horizon to be chosen for a specific industrial sector may vary from days to weeks, while it should at least cover the largest production time of a single order. Furthermore, the planning horizon needs to respect the availability of information about demand and customer orders and, therefore, should be in line with forecasts (Stadtler 2015b, pp. 195–196). Starting from focusing on scheduling within the relationship between two sites of the SC (Herrmann 2010, pp. 103–104), current approaches strive for optimizing multi-stage and multi-site network structures. With regard to details of modeling, scheduling needs to consider existing bottlenecks which affect the input of the production-distribution system. Structural parameters being part of related model formulations typically concern locations, parts, bills of materials, routings and associated operations instructions, production and transportation resources, specifications of suppliers, setup matrices, and timetables. Additional situation-dependent data may include initial inventories, the setup state of resources, and the set of orders to be processed within a given time period. Furthermore, lot-sizing rules, priority rules, or instructions with regard to the choice of routings may be prescribed (Stadtler 2015b, pp. 199–203). Objectives chosen for scheduling problems can be time-oriented. Typical examples are the minimization of the makespan, the sum of lateness, the maximum lateness, the sum of throughput times, or the sum of setup times. In contrast, objectives based on costs or profits are also applicable to scheduling problems. Besides revenues, the objective functions can include variable or fixed production costs, setup costs, or even penalty costs. In case of the latter, their occurrence can be linked to soft constraints, which are set with regard to the fulfilling of an order’s given due date (Stadtler 2015b, pp. 203–204). In opposite to penalties, bonus payments can be granted for deliveries that are distributed to the customers or markets before the given due date is reached (Steinrücke 2011a, b, 2015). As it is not to be expected that there is an optimal solution that fulfills different relevant objectives at the same time, multi-objective decision problems may arise (e.g., Sabri and Beamon 2000; Chen and Lee 2004; Mirzapour Al-e-hashem et al. 2011; Wang et al. 2011; Khalili-Damghani and Tajik-Khaveh 2015; Tosarkani and Amin 2018). Objectives with an environmental emphasis contain Kadziński et al. (2017). In the simplest case, optimal schedules can be given as lists of activities, which may contain the time periods (in case of discrete-time scheduling) or the start and the

2.2 Discrete-Time and Continuous-Time Supply Chain Scheduling

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end times (in case of continuous-time scheduling) of a process that have been assigned to a resource. By the use of electronic data exchange, these lists can be transferred and analyzed within the aforementioned advanced planning systems (Fleischmann and Meyr 2003; Fleischmann et al. 2015). Another way of representation of solutions is using Gantt charts. The latter show all resources of the system in parallel over a specific horizon. Moon et al. (2008) exemplify the usage of a Gantt chart for integrated SC planning and scheduling. Although most of the model formulations are based on the ideal assumption of a deterministic data situation (i.e., all data is known with certainty), different concepts for dealing with uncertainties (e.g., due to changes in production rates or unexpected breakdowns of the resources) exist in the literature. Firstly, optimal schedules can be evaluated by means of simulation, i.e., an extensive generation and testing of alternative scenarios which can be supported by software tools. An operational approach for SC networks that combines linear programming and discrete-event simulation is proposed by Preusser et al. (2005). Secondly, incremental planning describes a two-step planning procedure. It starts when a new order needs to be considered within the planning horizon of the system, which is to be inserted into given sequences of orders on the required resources. Time gaps in the existing scheduling are exploited in order to maintain the feasibility of the previous optimal schedule. Based on this procedure, the planned due date for the new order is submitted to the customer. Subsequently, different sequences of orders are tested, while the re-optimization strives to find new sequences at reduced costs. To avoid nervousness due to short-term changes of the start and end time of processes on the resources, parts of the previous optimal schedule can be fixed so that the re-optimization cannot start before a specific “frozen” part of the planning horizon (Stadtler 2015b, pp. 204–207). Thirdly, rolling horizon procedures can be applied, which are based on solving a series of smaller subproblems. The size and the number of the subproblems depend on the specific algorithm. In general, subsequent overlapping time intervals allow for updating the schedules with changing parameters. Origins of the method can be traced back to the single-machine dynamic scheduling problem (Ovacik and Uzsoy 1994) and the dynamic scheduling of parallel identical machines (Ovacik and Uzsoy 1995). Whereas the considerations on rolling horizon planning usually assume a single planning layer, adjusted forms of this approach even focus on multiple planning layers of supply chains, which include issues of information sharing and coordination (Sahin et al. 2013). Quantitative models of SC scheduling can be classified according to the chosen representation of the time axis. Discrete-time formulations are characterized by a sequence of time periods. As there is no common definition in the literature, time periods can be of either fixed length or variable length in general. The uniform type results if all time periods are of equal length; otherwise the type can be classified as nonuniform. Common time grids exist if all resources of the SC operate on the same time scale (Sürie 2005, pp. 56–57). In many cases, big bucket models (typically based on weeks or months) are proposed (Drexl and Kimms 1997). The latter assume that an operation started within a time bucket has to be finished by the end of the same bucket. Small bucket models (typically based on hours) allow for scheduling

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processes more precisely (Maravelias and Sung 2009; Sel et al. 2015). In the same context, integrated production-distribution problems distinguish between macroperiods and microperiods, which can even be characterized by a variable length (Amorim et al. 2012). Furthermore, hierarchical systems of time periods allow for the application of the aforementioned rolling horizon procedure. The approach contains a scheduling of all macroperiods of the entire planning horizon and a detailed scheduling of the microperiods conducted at the point in time the related macroperiod begins. By that, new information about disturbances and updated SC states can be considered (Laínez-Aguirre and Puigjaner 2015, pp. 225–226). Problems in the application of models for discrete-time scheduling arise if both the length of planning horizon and the quality of input data allow for a fine resolution of time, i.e., a huge number of small time periods within the planning horizon that results in large-scale programs. Roundings or other approximations conducted for realizing the time discretization can jeopardize the optimality or even feasibility of solutions (Sürie 2005, p. 61). Inherently, discrete-time scheduling is inferior to continuous-time scheduling of operations in terms of accuracy. Continuous-time formulations of scheduling allow for determining the exact start and end times of all processes within the entire planning horizon. Pseudo-forms of time continuity can be stated if (fixed or relaxed) period boundaries for the start and end time of operations exist. Event-based formulations assume the planning horizon to be divided into a predefined number of events, while events on different resources are coupled by sequencing constraints. Major drawbacks of this method arise from the requirement of an ex ante determination of the number of relevant events. Whereas a too small number may foreclose the determination of the overall system’s optimal solution, a too big number (including a lot of unnecessary events from the ex post perspective) would have negative effects on the computability of the problem. To overcome this conflict, iterative procedures for determining the number of events in the chemical industry is proposed by Ierapetritou and Floudas (1998) and Castro et al. (2001). At first, the problem is solved for a small number of event points. After that, the number of event points is augmented and the problem is resolved. The iterative procedure stops if the objective value cannot be increased by an additional step. As a common feature of the aforementioned modeling technique, different effects on computability are expected in general. On the one hand, the number of events is supposed to be much higher than the number of time periods relevant in discrete-time formulations, but on the other hand, the given number of events results in MILP of small sizes in comparison to other forms of modeling time continuity (Sürie 2005, pp. 61 and 67). However, due to a reduction of computational efforts in course of recent developments in high-performance hardware and software, simplifications of continuous-time formulations to predefined event points become less important. In contrast, scheduling models not being based on event points were developed (e.g., Steinrücke 2011a, b, 2015). They allow for determining start and end times exclusively for the specific processes, which are to be realized according to the optimization of the overall system.

References

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Maravelias, C. T., & Sung, C. (2009). Integration of production planning and scheduling: Overview, challenges and opportunities. Computers & Chemical Engineering, 33(12), 1919–1930. Meyr, H., & Stadtler, H. (2015). Types of supply chains. In H. Stadtler, C. Kilger, & H. Meyr (Eds.), Supply chain management and advanced planning (5th ed., pp. 55–69). Berlin/Heidelberg: Springer. Meyr, H., Wagner, M., & Rohde, J. (2015). Structure of advanced planning systems. In H. Stadtler, C. Kilger, & H. Meyr (Eds.), Supply chain management and advanced planning (5th ed., pp. 99–106). Berlin/Heidelberg: Springer. Mirzapour Al-e-hashem, S. M. J., Malekly, H., & Aryanezhad, M. B. (2011). A multi-objective robust optimization model for multi-product multi-site aggregate production planning in a supply chain under uncertainty. International Journal of Production Economics, 134(1), 28–42. Moon, C., Lee, Y. H., Jeong, C. S., & Yun, Y. S. (2008). Integrated process planning and scheduling in a supply chain. Computers & Industrial Engineering, 54(4), 1048–1061. Ovacik, I. M., & Uzsoy, R. (1994). Rolling horizon algorithms for a single-machine dynamic scheduling problem with sequence-dependent setup times. International Journal of Production Research, 32(6), 1243–1263. Ovacik, I. M., & Uzsoy, R. (1995). Rolling horizon procedures for dynamic parallel machine scheduling with sequence-dependent setup times. International Journal of Production Research, 33(11), 3173–3192. Preusser, M., Almeder, C., Hartl, R. F., & Klug, M. (2005). LP modelling and simulation of supply chain networks. In H. O. Günther, D. C. Mattfeld, & L. Suhl (Eds.), Supply chain management und logistik (pp. 95–113). Heidelberg: Physica-Verlag. Sabri, E. H., & Beamon, B. M. (2000). A multi-objective approach to simultaneous strategic and operational planning in supply chain design. Omega, 28(5), 581–598. Sahin, F., Narayanan, A., & Robinson, E. P. (2013). Rolling horizon planning in supply chains: Review, implications and directions for future research. International Journal of Production Research, 51(18), 5413–5436. Sel, C., Bilgen, B., Bloemhof-Ruwaard, J. M., & van der Vorst, J. G. A. J. (2015). Multi-bucket optimization for integrated planning and scheduling in the perishable dairy supply chain. Computers & Chemical Engineering, 77, 59–73. Simchi-Levi, D., Kaminsky, P., & Simchi-Levi, E. (2008). Designing and managing the supply chain (3rd ed.). New York: McGraw-Hill/Irwin. Stadtler, H. (2005). Supply chain management and advanced planning—Basics, overview and challenges. European Journal of Operational Research, 163(3), 575–588. Stadtler, H. (2015a). Supply chain management: An overview. In H. Stadtler, C. Kilger, & H. Meyr (Eds.), Supply chain management and advanced planning (5th ed., pp. 3–28). Berlin/Heidelberg: Springer. Stadtler, H. (2015b). Production planning and scheduling. In H. Stadtler, C. Kilger, & H. Meyr (Eds.), Supply chain management and advanced planning (5th ed., pp. 195–211). Berlin/ Heidelberg: Springer. Steinrücke, M. (2011a). An approach to integrate production-transportation planning and scheduling in an aluminium supply chain network. International Journal of Production Research, 49 (21), 6559–6583. Steinrücke, M. (2011b). Integrierte Produktions-, Distributions- und Terminplanung in globalen Supply Chains. Zeitschrift für betriebswirtschaftliche Forschung, 63, 19–47. Steinrücke, M. (2015). Integrated production, distribution and scheduling in the aluminium industry: A continuous-time MILP model and decomposition method. International Journal of Production Research, 53(19), 5912–5930. Steinrücke, M., & Albrecht, W. (2016). A flow-to-equity approach to coordinate supply chain network planning and financial planning with annual cash outflows to an institutional investor. Business Research, 9, 297–333.

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Steinrücke, M., & Albrecht, W. (2018). Integrated supply chain network planning and financial planning respecting the imperfection of the capital market. Journal of Business Economics, 88 (6), 799–825. Sürie, C. (2005). Time continuity in discrete time models. Berlin/Heidelberg: Springer. Sürie, C., & Reuter, B. (2015). Supply chain analysis. In H. Stadtler, C. Kilger, & H. Meyr (Eds.), Supply chain management and advanced planning (5th ed., pp. 29–54). Berlin/Heidelberg: Springer. Tosarkani, B. M., & Amin, S. H. (2018). A possibilistic solution to configure a battery closed-loop supply chain: Multi-objective approach. Expert Systems with Applications, 92, 12–26. Wang, F., Lai, X., & Shi, N. (2011). A multi-objective optimization for green supply chain network design. Decision Support Systems, 51(2), 262–269.

Chapter 3

Green Supply Chain Management

Abstract Based on existing structures of sites and capacities, new mathematical models for hierarchical scheduling of operations in generic multi-stage networks are developed in the book Scheduling in Green Supply Chain Management: A MixedInteger Approach. This chapter deals with the peculiarities of green supply chain management, especially its origins and drivers. This allows for identifying the main aspects in this field, which can be integrated into mathematical models of mediumand short-term optimization. Basics of reverse logistics and emission management are discussed.

Extracting core elements from a number of definitions found in the literature, the following one is used throughout this monograph. Green supply chain management is defined by the integration of traditional SCM and corporate environmental management. It includes the establishment of closed loop supply chain networks with a simultaneous coordination of forward and reverse material flows. It considers the implementation of processes and instruments for aligning network-wide operations to environmental goals and legislation (Srivastava 2007; Emmett and Sood 2010, pp. 3–5; Khan et al. 2017, pp. 5–7; Sarkis and Dou 2018, pp. 9–10). Typically, five major practices are assigned to the field of GSCM: Firstly, ecodesign determines the attributes of goods manufactured in an SC. Its importance results from the fact that most environmental influence results from a product’s design, which affects the use of materials and energy. Furthermore, its characteristics regarding reuse, recycling, and recovery of material and components are set. Design alternatives and related manufacturing processes should be assessed with regard to the avoidance or reduction of hazardous substances, while both internal and external partners should be involved. Secondly, green purchasing concerns environmentally friendly ways of procuring materials or components. The assessment of SC partners comprises the implementation of electronic systems for processing orders (e-procurement), the use of long-term contracts with ecological agreements for the reduction of non-eco-friendly behavior, cooperation for common objectives,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6_3

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auditing of internal environmental management systems of the suppliers, and finally integration of environmental performance measures into existing systems of supplier assessment and evaluation. In this context, suppliers offering environmentally labeled and/or reusable, recyclable, or remanufactured goods or packaging should be preferred. Thirdly, an internal environmental management should be established throughout the entire SC. It includes the selection of an environmental management system (which entails considerations on aspects of organization and coordination of relevant issues), the reduction of resource consumption, and finally the reduction of pollutant emissions. Fourthly, customer cooperation with environmental concerns can be initiated by the SC managers, as recent studies revealed the customers’ behavior (e.g., preferences regarding characteristics of products or delivery and packaging) to be one of the most important drivers for sustainability in SC operations. Specific projects (e.g., Carbon Disclosure Project, scorecards for determining the carbon dioxide footprint) can support common objectives. Fifthly, strategies of investment recovery can be applied. They are connected to the main processes of reverse logistics in closed loop supply chains and focus on the sale of excess inventories of products, components, and materials, excess equipment, as well as scrap and used materials (Zhu and Sarkis 2004; Zhu et al. 2005, 2008; Sarkis and Dou 2018, pp. 10–11). Inherently, quantitative targets in GSCM are related to different environmental concerns. Thus, they usually strive for limiting or reducing (or in some cases even for eliminating) the usage of hazardous chemicals (e.g., mercury), energy consumption (e.g., energy or water), emissions (e.g., carbon dioxide or dust), or solid waste (Chin et al. 2015). With regard to the latter, an increasing share of product recovery within SC operations can be intended. However, as a coordination with financial issues of SCM (e.g., Steinrücke and Albrecht 2016, 2018; Albrecht and Steinrücke 2017, 2020) is typically required, approaches in the literature are nevertheless dominated by objective functions based on profit or costs. Possibilities to consider the aforementioned environmental goals in SC planning and scheduling result from their integration by additional constraints or alternative objective functions. In this context, an evaluation of different multi-objective optimization approaches for solving design problems in GSCM can be found in Kadziński et al. (2017).

3.1

Origins and Drivers of Green Supply Chain Management

Whereas the collection and melting of metallic parts according to their economic value can be traced back to the Old and Middle Ages, various industries started putting an emphasis on feeding back scrap and spoilage into their production processes in the 1980s and 1990s (Steven 2004, p. 168). Holistic concepts of designing green SCs evolved at the end of the same century (e.g., Beamon 1999). Systematic origins of reverse logistics can be traced back to investigations on waste

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and trash disposal in the context of the traditional SCM in the early 1990s. Based on a growing importance of the products’ life cycles, related research focused on specific segments, e.g., computer technology, office automation, or military systems. As a result, the disposal of failed parts and assemblies was revealed to be a major cost contributor in these industries. In the following, both modularization and open architecture interfaces entailed new opportunities for handling, as it became possible to change or upgrade specific units instead of complete products or systems. Furthermore, the development from analog hardware to digital components increased the importance of components remaining at the end of traditional SCs. Besides the considerations on reducing costs and increasing values, additional drivers of reverse logistics can be found at consumer goods markets, as customers increasingly pay attention to returns of non-defective goods (Blumberg 2005, pp. 2–3). Environmental awareness. A major reason for greening processes of SCM can be found in addressing environmental burdens, which are caused by the economy. The latter affect air, water, and land at global, regional, and local levels. At first, negative effects can be traced back to reaching limits of our ecosystem. One of the most urgent issues regards global warming and climate change, as increasing global temperatures affect human activities. Other problems to be solved on the international level are biomass appropriation, ozone shield rupture, land degradation, and biodiversity loss. Especially the decimation of species on the global level is supposed to have a negative influence on regional ecosystems. Other regional problems are deforestation and water pollution in rivers or lakes. The latter may also affect water supply to industries. Acid rain can be caused by increased manufacturing, especially in developing countries. Inherently, there is a relationship to local problems being limited to municipal areas. Hazardous materials (e.g., pesticides or herbicides) can damage waterways and agriculture. Moreover, they may have an impact on animal or human health (Dyckhoff et al. 2004, p. 1; Sarkis and Dou 2018, pp. 3–4). Customer behavior. Besides internal institutional awareness, a main driver of GSCM concerns changing preferences of customers. This implies that customers are guided not only by the price, quality, or convenience of products but also by the impact on the environment or society. In this context, the whole life cycle of products including production, consumption, and disposal is assumed to be relevant for individual decisions. Additionally, social issues (e.g., compliance with fair labor conditions, local or regional engagement, and responsibility) can be taken into account. A major concern of a growing number of customers is that food is harvested and processed healthy and in environmentally friendly manner, e.g., without using chemicals or additives while minimizing the water consumption (Lanzini 2018, pp. 15–17). In this context, empirical studies analyze the customers’ behavior depending on specific attributes of goods (e.g., homemade, locally produced/ retailed, ingredients, reusability, packaging material) and their implications for marketing (Wagner 1997, p. 129). Regulatory effects. In the international context, the Montreal Agreement, the Basel Treaty, the Kyoto Protocol, the Waste Electrical and Electronic Equipment

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Directive, the Restriction of Hazardous Substances Directive, and the Eco-design for Energy-Using Products Directive are of particular importance (Huang and Yang 2015, p. 99). Furthermore, the specific legal basis for businesses on the national level needs to be taken into account. For instance, Germany can still be considered as a country with the most progressive laws for implementing reverse flows into SCs. Relevant legislation includes the Waste Management Act from 1986, which was replaced by the Commercial and Industrial Waste Avoidance and Management Act in 1996. The latter came into force to obligate the manufacturers to be environmentally responsible after the customers’ usage of their products, which includes mandatory processes of collection, treatment, and disposal. Additional regulations were raised for packaging, used cars and batteries, electronic scrap, and information processing devices (Steven 2004, p. 168). Other drivers and pressures. Besides the aforementioned biases, the empirical study of Huang and Yang (2015) identified several other factors relevant for the implementation of GSCM. Besides legislation, governmental regulations or corporate environmental goals exist. The authors distinguish three different forms of institutional pressures that may stem from interest groups (including investors and nongovernmental organizations), competitors, and industry sectors. Furthermore, the authors analyze cognitive and emotional mechanisms affecting the posture of SC managers (Huang and Yang 2015, pp. 104–110).

3.2 3.2.1

Reverse Logistics and Closed Loop Supply Chains Scope and Definitions

Summing up a variety of characteristic features from the literature, reverse logistics “comprises all activities involved in managing, processing, reducing and disposing of hazardous or non-hazardous waste from production, packaging and use of products, including the processes of reverse distribution” (Steven 2004, p. 164). In particular, it deals with failed or broken goods, obsolete goods that still have a substantial value, unwanted or unsold goods that remained at the sites of the retailers, or goods that have been recalled by the manufacturers. These goods may include products, parts, subassemblies, as well as materials. Issues of reverse logistics affect different groups of stakeholders, e.g., executives, managers, and operational staff, but also vendors and developers of technology, infrastructure, and software systems (Blumberg 2005, pp. 1–2). As being a starting point for reuse, recycling, remanufacturing, or professional disposal, reverse logistics can be considered as an important task in the context of aligning business activities to environmental goals (Madu 2004, p. 168; Sople 2012, p. 269). A growing importance of reverse logistics could be observed in SCM within the last decades, as related developments were supposed to relax existing contradictions of needs and supplies of natural resources in the traditional flow economy (Steven 2004, p. 167).

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According to a common definition, a closed loop supply chain is characterized by a simultaneous coordination of forward and reverse logistics so as to reintegrate required resources after appropriate processing. In particular, closed loop SCM covers all activities of forward logistics, including the coordination and control of full-service logistics providers. Relevant flows may be composed of raw materials, intermediates, or finished products, which are directed from suppliers via different production and distribution stages to final customers. Another integral part of closed loop SCM is reverse logistics, which deals with picking up and delivering goods recently being part of forward SCM for reasons of further processing, recycling, or disposition but also initiates subsequent returns to forward SCM where appropriate. Furthermore, related services to receiving the returns (i.e., depot repair, processing, diagnostics, and disposal) are considered (Blumberg 2005, p. 9; Lebreton 2007, p. 7). A classification of closed loop supply chains can be based on the following criteria. Focusing on the goods to be handled within reverse logistics, different kinds (e.g., used goods, waste, packaging) can be distinguished on different levels of aggregation (e.g., products, components, materials). Another point of view results from identifying the reasons for returns initiating the reverse flows (Sect. 3.2.3.2). Furthermore, the flows can be classified by their sources, which can be either SC partners (i.e., legally dependent or independent companies) or customers (e.g., wholesalers, households, consumers). Specific features of the network structure result from the type of sites, which are harnessed for operating the reverse flows (Sect. 3.2.3.1) and their affiliation (i.e., external or internal status in the network). Further detailing is reached if different processes at the sites (e.g., recycling, remanufacturing, refurbishment) are specified. Finally, processes in closed loop SCs can be characterized by the time horizon. Whereas several returns occur during (or immediately after) traditional SC operations (e.g., return of packaging materials) and can be well estimated with regard to regularity and quantity, others are expected to be returned after a medium- or long-term period (e.g., warranty returns or end-ofuse returns) in uncertain circumstances (Fleischmann et al. 1997; Steven 2004, pp. 169–171).

3.2.2

Aspects of Organization, Coordination, and Standardization

Reverse logistics can be organized in different forms depending on the complexity of processes and the number of partners involved: Firstly, an internal processing that takes place within an individual company can be considered. In this case, scrap or other residues of manufacturing are used as input in the same process (e.g., melting broken pieces in the chocolate fabrication) or in other processes (e.g., industrial waste heat is used for heating offices). Benefits arise from reducing the organizational effort and the related costs in small- and medium-sized enterprises. Secondly, process cells can be established as an elementary external organization structure for

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managing reverse flows. These cells are implemented between legally and economically independent companies. Typically, the exchanged goods are classified as waste by the sender, but as a raw material by the accepter (e.g., sawdust that is considered as waste in a saw mill, but as an input within the production of furniture). Due to the exchange, waste with a negative economic value turns into secondary raw material with a positive one. On the one hand, both partners can take monetary advantages of the cooperation if it is possible to save costs at both the sender and the submitter. Otherwise, it is possible to negotiate compensating payments. On the other hand, it becomes possible to save the environment due to a reduction of waste and material consumption. Relationships are usually realized in repeating market transactions, while recurrent exchanges can be regulated by long-term contracts containing fixed quantities, qualities, and/or prices. Problems of cell implementation may result from volatility of markets, a lack of transparency, or difficulties in the coordination of demand and supply. As a consequence, intermediaries (e.g., secondhand dealers, reprocessing companies, exchange agencies) can be harnessed. Thirdly, processes of reverse logistics can be implemented into network structures. In many cases, companies sustain various relations in process cells while often acting simultaneously as an accepter and a sender. Furthermore, different types of goods can be considered for the exchange. As a result of the multilateral interaction of sites and existing interdependencies, the formation of closed loop SC networks results. However, different types and structures of these networks can be observed. They range in the scope of loose cooperation based on single contracts and hierarchical organization with a strategic alignment of the partners and an own corporate identity. The ideal-typical principles of network formation based on externalization and internalization of organizational functions that were introduced for traditional networks by Sydow (1992) can be applied accordingly. Strengthening the relationships within the network can be required in order to ensure the long existence of the participating companies. In general, the economic and ecological importance of integrating reverse logistics into SC planning increases with increasing network sizes (Steven 2004, pp. 172–175). For closed loop systems, the following types of coordination approaches are distinguished in the literature: Firstly, there are successive approaches which apply traditional and reverse SCM subsequently. Often, there are returns to a central location for processing and disposal (e.g., a traditional waste dealer or a similar service organization that may be organized as both a business and a governmental facility), which operates independently of the forward SC. With regard to processing returns, the approaches mostly contain an emphasis on economic disposal through land or sea dumping or recycling. Secondly, integrated approaches for coordinating forward and reverse logistics can be found in the literature. On the one hand, they can be applied to high-tech products provided by original equipment manufacturers. The latter take simultaneous responsibility for both directions of material flows and even strive for combining them. Typically, finished products are returned and recovered on behalf of the manufacturer through indirect dealer channels or by the manufacturer’s own representatives. Reverse flows may be directed to a depot and either can be put back into inventory for resupply or can go through a process of

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qualification and reconfiguration that results in reuse at manufacturing stages or in disposal. On the other hand, integrated coordination can be applied to supply chain networks specialized on standard low-tech products, which are characterized by a lower level of integration between both directions of material flows. Instead of an overall management by the original equipment manufacturer, there are mostly independent structures of plants and maintenance facilities. Usually, organizations of a number of small dealers take responsibility for reverse flows and repairing while using their own means of transportation or even third-party logistics. Whereas the aforementioned forms of integrated coordination focus on relationships between businesses, variants designed for customer-oriented closed loop SCs can be found additionally in the literature (Fleischmann et al. 1997; Debo et al. 2004; Blumberg 2005, pp. 9–12). As various dynamics and structures in coordinating flows and processes in closed loop SCs could be observed in different industrial, commercial, and military markets, a stream of recent research strived for the identification and standardization of basic management issues in the field. In this context, Blumberg (2005) proposes a multi-step framework, which covers both processes of strategic and tactical management. It starts with the establishment of a market focus. Therein, a series of markets needs to be examined and segmented with regard to appropriate characteristics (e.g., demand for green goods, preferences of packaging). Subsequently, data on existing structures need to be collected. In this context, internal capabilities, activities and dynamics, as well as market requirements should be identified, analyzed, and evaluated thoroughly. In particular, data assignable to the following issues is required: • Internal sites (e.g., sales volumes, costs, direct and reverse logistics expenses, average turnover of inventory) and external suppliers (e.g., average sales and estimated returns) • Existing distribution structures and available providers of third-party logistics (e.g., operating costs with regard to transportation and distribution) • Impact of reverse logistics (e.g., size and dimension of the problem, average occurrence of returns, and/or unsaleable products) • Return policies or commitments in general and for the considered goods type • Statistical experiences with returns to vendors Based on this information, a business strategy including details of reverse logistics and closed loop operations can be elaborated. In this context, harnessing thirdparty service providers and secondary markets is to be discussed. Taking into account benchmarks of the relevant sector, the structures of management and organization (e.g., networks of legally dependent or independent sites, central vs. decentral coordination between the sites) must be determined. The following step includes the development and solution of a model, which allows for the full coordination of sites by means of optimization (see Sect. 3.2.3). Modeling should be aligned to the main target of the organization by formulating suitable economic or environmental objectives (Blumberg 2005, pp. 1–3 and 153–156).

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Implementation and Modeling of Closed Loop Supply Chain Networks

With regard to the strategic dimension to be captured for closed loop SCM, the focus is on product selection, recovery path determination, and location planning in the following: Considering the selection of products to be recovered, it needs to be stated that most of the original equipment manufacturers outsource the development of components to their suppliers. In this context, approaches for evaluating the profitability of refurbishing and, furthermore, for identifying a core refurbishing program can be applied. In a more detailed perspective, the optimal number of retreading cycles could be determined. Moreover, capturing time aspects is required in order to catch the dynamics of remanufacturing that may span psychological, qualitative, and technological obsolescence at the same time. A comprehensive framework may be needed to identify factors that are conflicting with the products’ reuse or recycling. Moreover, the reuse potential of different materials is to be evaluated. According to the life cycle paradigm, the goods can be characterized by three key factors that include the return behavior of the customers, the technological evolution, and the market segmentation. Another important task of strategic planning of closed loop SCs concerns the determination of an appropriate recovery path. In general, decisions on the starting point of recycling depend on a network’s expectations on the development of external prices for supply, internal prices for recycling, and disposal costs. In-house recycling paths need to be compared to the outsourcing of recycling to other sites of the existing SC network or to external partners. With regard to specific details of path formation with original equipment manufacturers, a proprietary closed loop would ensure that materials are retained in an internal cycle and, thus, remain in the property of the original equipment manufacturer that is being part of a bigger SC network. Consequently, the procurement of materials from externals (that may belong to extended versions of closed loop supply chains) can be postponed until available potentials are exhausted. Further relationships to the determination of the optimal recovery path result from insights of disassembly planning, which strives for evaluating the products’ optimal assembly depth. Finally, there are basic decisions on location planning in closed loop SCs accompanied by the determination of the optimal network design (e.g., Fleischmann et al. 2004, pp. 65–76). They concern the provision of appropriate sites in the reverse channel (see Sect. 3.2.3.1). The selection between different alternatives, probably located in different regions or countries, could be affected by differences in labor costs. The latter are assumed to be the main cost driver in refurbishing due to its manual nature. Other relevant parameters are transportation costs being influenced by the efforts resulting from different types of reverse flows (see Sect. 3.2.3.2). Compared to integrated methods of network design, successive methods (i.e., models that optimize production and recovery sites of the network subsequently) are inferior in general but can be a good approximation in case that the production sites are already located in the customer sites. Further biases on location planning, which can arise

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from both environmental and legislative issues, need to be taken into account (Lebreton 2007, pp. 71–76).

3.2.3.1

Network Sites

Sites of collection. Product collection systems can be driven by a company’s field force, by channel members, or by third-party logistics companies. If the return of the products needs to be organized by the customers themselves, it is necessary to motivate them for delivering relevant goods by given proper guidelines or instructions, or even incentives. To facilitate returns, additional services of picking up the goods from the customers can be offered. In this context, recapturing means getting the returned products from their locations to collection centers. It includes processes of collection, transportation, and warehousing. Hurdles for the system can arise from common behavioral patterns (e.g., customers that do not want to part from things they own). Determinants of building or leasing appropriate locations are aspects of coverage and costs. The operations at the collection centers are typically characterized by primary processing, packaging, and transportation. In particular, it includes breaking down the products into parts, cleaning, and grouping. Non-recoverable parts can be disposed of immediately, whereas valuable goods can be distributed to network sites for purposes of recycling, remanufacturing, or reuse. Appropriate packaging of these goods might be required by the logistics providers. As an alternative to the operation of separate collection centers in small-scale SC networks or in networks of small- and medium-sized companies, retailers and sales representatives of the SC can additionally take on some of these functions (Sople 2012, p. 279; Liu 2012, pp. 483–517). Sites of recovery. The classical form of recovering materials for returning them into an SC is recycling. It covers processing by manual handling or automated systems with effects on the goods’ characteristics (e.g., size, shape, or density). If streams arriving from the collection centers are too complex, and thus unsuitable for final processing, the first step of recycling contains the reduction of the particle size in order to perform mechanical separation steps. In this context, physical property differences between the particles in the stream are used. Further physical purification or processing by chemical or metallurgical means may follow (Heiskanen 2014, p. 40). Specific processes depend on the kind of material to be recycled. Primary sources of scrap metal that can be recycled into high-quality new metal are packing materials (e.g., cans of aluminum and steel), vehicles, electrical and electronic products, as well as batteries (Grant et al. 2015, p. 166). Especially, the extraction of rare materials (e.g., gallium, germanium, indium, tellurium, tantalum, and platinum group metals) is of particular importance due to their geological scarcity (Ayres et al. 2014, p. 27). Other materials relevant for recycling industries are steel, cooper, lead, zinc, lumber, paper, plastic, glass, and textiles. Besides recycling, recovery may additionally include value-adding operations. In particular, remanufacturing is an extensive treatment of restoring used goods to like-new condition that includes processes of disassembly, cleaning, repairing, and replacing parts of the products

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before their reassembly. An attenuated variant is refurbishing, which is characterized by lower efforts of disassembly in comparison. Parts harvesting includes the recovery of selected parts from returns that are used in order to provide spare parts for fulfilling service contracts with customers. The resale option is relevant for used devices being of full functionality if an active secondary market exists for selling them. In case that reselling of the goods is not intended, an internal reuse can be considered for goods that do not require substantial processing (Ferguson and Souza 2010, pp. 4–5). While the highest priority is given to the avoidance of waste or the extension of product life cycles, also the aforementioned processes of recovery sites can be assigned to a hierarchical system. According to the latter, a treatment of returns resulting in their resale or reuse has priority to remanufacturing, which should be preferred over recycling. If no other alternative is available, disposal can be considered. Disposal with energy recovery has priority to landfill (Steven 2004, p. 165). Sites of final disposal. Due to the characteristics of specific returns, it might not be possible to keep them within the closed loop SC and, thus, the related materials or products need to be disposed of. Possible reasons include serious failures in parts or subassembly, damages during shipping or transit, the return to clear shelves, obsolescence, age, or manufacturing recalls or even customer remorse (Blumberg 2005, p. 132). On the one hand, land or sea deposits are used. Specific jurisdictional exceptions might be relevant for landfilling, as various kinds of electronic equipment are considered as hazardous waste. On the other hand, incineration is harnessed to reduce the amount of solid waste up to a share of 95 percent in general. Moreover, it is possible to recover energy from waste. Further benefits arise for cities or regions with limited areas available for landfilling. However, major drawbacks result from incinerator emissions (e.g., mercury). Although incineration is supposed to be the most proven technology for converting waste and trash into energy, alternative technologies (e.g., gasification, pyrolysis, and plasma conversion) can be taken into account (Ferguson and Souza 2010, p. 4). Regardless of the function of a network site, intertemporal relationships of site configuration can be modeled in dynamic systems on the strategic level. Emphasizing the long-term character and reliability of decisions on location openings/closings and capacity adjustments, it becomes possible to limit the scope of changes appropriately. For instance, it might be necessary to ensure that a recycling site that has been opened cannot be closed within the same planning horizon, or vice versa. Moreover, extensions or reductions of capacities at the sites (e.g., the provision or decommissioning of conveyor belts in sorting plants) can be forced to be maintained for a specific duration (Albrecht 2014, pp. 118–137).

3.2.3.2

Reverse Material Flows

According to the classification by Fleischmann (2001), five different categories of reverse material flows are distinguished. They can be allocated to different stages of the SC, and furthermore, they result in different consequences for handling the

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goods: Firstly, end-of-use returns are considered to be the most important in the literature. Related material flows that are originating from consumers or waste processors occur after individual completion of usage, although the goods do not necessarily reach the end of their technical or economic usability. Even leased products can be included within the aforementioned definition. In general, the time period between manufacturing and return is expected to be relatively long. With regard to this category, companies (i.e., original manufacturers or specialized recovering facilities) may be interested to win back valuable resources within recovery processes (e.g., reuse, remanufacturing, recycling). Probably, end-of-use returns may also occur due to environmental regulations if manufacturers are charged with the responsibility of legally compliant disposal (e.g., Closed Substance Cycle Waste Management Act in Germany). As an alternative to material recycling, the outsourcing of the aforementioned duties to third parties (which potentially bears additional financial risks) can be considered. Further motivation for end-of-use returns may arise from asset protection goals. The latter become relevant if manufacturers strive for retrieving their products in order to prevent competitors from taking advantage of them. Before disposal, sensitive components and information must be destroyed. Secondly, there are commercial returns that occur as a result of rescinding business transactions. Typically, a buyer returns a product (that is unused and not defective) to the sender against refunding. Although this form of return may be applicable to two arbitrary partners of the SC that are connected by forward material flows, it is mostly applied in the relationship between a manufacturer/ retailer and a customer due to current legislation (e.g., Distance Selling Act in Germany). As a result of the aforementioned practices, the financial risk is transferred from the buyer to the seller, which may be problematic for products with rapid obsolescence. In general, the consideration is limited to goods, whose characteristics allow for reselling them directly at the same market or even at an alternative market. As commercial returns may indicate a lack of demand and, furthermore, they entail considerable costs for the seller, an upgrade of the returned products should be taken into account before a material recycling or even disposal. Thirdly, warranty returns are reverse material flows of defective products to the original vendor, triggered by product failures or damages that occurred while usage or delivery. The definition also includes product recalls. Usually, warranty returns are made possible by legislation (e.g., material defect liability in Germany). Moreover, they can be offered voluntarily by the companies for marketing reasons. Typically, these forms of returns are processed by repair or disposal, while the latter alternative requires replacement or refunding. Fourthly, the return of production scrap or by-products may be considered in an SC. In this context, excess material that occurs at the plants during manufacturing can be reintegrated into the production process provided that both quality and economic targets are met. By that, it becomes possible to save resources and to reduce traditional forward material flows. If not needed within a specific SC, by-products can be sold or transferred to alternative companies or networks. Finally, the return of packaging should be considered. In some cases (e.g., bottles, pallets, reusable boxes), materials can be reused after cleaning without major reprocessing. Other packaging (e.g., paper or plastic bags) can be recycled. In

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general, packaging is expected to be returned quickly after sales to the original senders, to logistics service providers, or to collection centers, as it is mainly required for the distribution and the transportation of goods. Due to the growing importance of waste that is caused by packaging material, additional legal and administrative requirements for its handling exist (Fleischmann 2001, pp. 19–24). For instance, German packaging law stipulates taking back all materials of secondary packaging by manufacturers, distributors, and retailers. The system is based on a fee to be paid by the companies, which is usually forwarded to customers by a surcharge to the product price (Sople 2012, p. 275). Due to difficulties in synchronization of returning material flows with their processing, an additional focus can be put on the inventory control. The latter is required to integrate the flow of returns into the material planning at the plant sites being part of traditional SCs. Well-established inventory control methods can be applied if returns are exclusive resources in specific production processes. If returns are considered as an alternative resource at the original manufacturer otherwise, policies for external ordering of raw materials and for recovering available returns need to be combined appropriately. Therefore, inventory management requires the simultaneous control of both alternatives while acknowledging existing trade-offs between service levels and costs. In case the recycling of returns is cheaper than the production of new materials, issues of an increased uncertainty in planning reliability leading to increasing safety stocks may become relevant. Moreover, excess inventories of returns can be avoided by assessing them with costs (Fleischmann et al. 1997, pp. 7–8). In this context, a variety of stochastic models for inventory management in closed loop SCs contains Fleischmann and Minner (2004). In general, the management of returns within closed loop supply chains entails challenges, which can be differentiated according to the types of network sites being involved. For business transactions between companies, a lack of standardization may impede seamless flows. For instance, nonuniform product qualities, packaging, and returning outdated and obsolete items influence utilization and expenses. A detailed exchange of data is required for processing. For transactions with customers, the unpredictability of returns is considered to be one of the most important challenges in reverse logistics. Due to a lack of reliable information, a high number of transactions may be required for processing returns within a customer-focused strategy of SCM (Sople 2012, p. 268).

3.3 3.3.1

Emission Management Scope and Market-Based Regulation

The emissions being most relevant for global environmental reasons are greenhouse gases. As acting like the glass in a greenhouse, they serve to trap heat energy. Although being part of a cycle through the global biochemical system, an exceeded volume of greenhouse gases that results from human activity causes trapping too

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much heat and overheating the atmosphere. Different types of anthropogenic greenhouse gases exist, while most of them occur naturally. In particular, carbon dioxide does result not only from burning fossil fuels, which is relevant for different industries and processes (e.g., power and heat generation, civil aviation, as well as all energy-intensive industry sectors including refineries, steelworks, etc.), but also from the decomposition of vegetation in farming. The latter is also relevant for the emission of methane that results from an anaerobic decomposition of organic matter. Nitrous oxide is released from fertilizers. In the chemical industry, it results from the production of nitric, adipic, and glyoxylic acids as well as glyoxal. Because of their strong absorbance and long durability, the most powerful greenhouse gases that have several practical uses in manufacturing are halocarbons including fluorocarbons, methyl halides, carbon tetrachlorides, carbon tetrafluorides, and halons. For instance, perfluorocarbons are of particular importance in the aluminum production (Casper 2010a, pp. 18–21; European Commission 2016). Striving for a reduction of the human-induced greenhouse effect, considerations are typically focused on carbon dioxide, while quantities of other relevant greenhouse gases are converted into equivalents by the use of emission weighting factors (Chaabane et al. 2012). For regulating greenhouse gas emissions, the cap-and-trade system is currently the most important environmental policy tool. Caps are defined as mandatory emission limits set by legislation, which usually refer to an economy-wide carbon market equivalent. The limits are split up into single emission allowances. Each of the latter gives the holder the right to emit a specific amount of carbon dioxide (e.g., 1 ton) or an equivalent amount of other environmentally relevant greenhouse gases. Each allowance can only be used once, while companies usually have to surrender allowances for every ton that was emitted during a previous period. While a specific number of these allowances can be allocated for free, the trading procedure is used for a market-based allocation. The default trading method is auctioning. This enables adjusting the emissions actually caused by a company with the number of held allowances by means of buying and selling. In order to ensure the functionality of the overall system, heavy fines are imposed for companies that do not hold enough certificates. Surplus allowances can be transferred to a following period. In general, cap-and-trade systems ensure market flexibility, while companies should strive for exploring the most efficient and innovative ways of meeting mandatory limits. In this context, voluntary actions to reduce emissions can be rewarded by subsidies, tax credits, or other governmental incentives. In opposite to the aforementioned allowance-based markets, there are project-based markets. The latter enable companies to invest in emission reductions in developing countries as an alternative to possible reductions in their own countries that would be more expensive (Casper 2010a, p. 92, b, pp. 55–59; European Commission 2016). The combination of capand-trade policies with rate-based emission policies that fix the average emission intensity (tradable performance standards) is analyzed by Fischer (2003). Other types and variants of emission trading systems are discussed by Nentjes (2016). Problems of implementation are ranging from market manipulation (Hintermann 2016) to financial crimes (Nield and Pereira 2016).

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The first large-scale carbon dioxide emission trading program based on the capand-trade system was the European Union’s Emissions Trading System (EU ETS), which began operating at the beginning of the year 2005. Origins can be found in the global negotiations on the Kyoto Protocol in 1997 that were accompanied by multiple sources of resistance. Independently from the aforementioned motivation, the EU ETS was embedded in European law. By that, the EU ETS became an important instrument of European climate change policy as it covers around 45% of the greenhouse gas emissions in the EU member states. Furthermore, it ensures that their obligations under the Kyoto Protocol are met. The implementation of the EU ETS can be described by the following phases: The first trading period (2005–2007) was characterized as trial phase for establishing the world’s biggest carbon market. However, as the number of allowances turned to be excessive, the price fell to zero until the end of this period. The second trading period (2008–2012) was affected by three additional countries joining the system. Although the number of allowances was reduced by 6.5% for this period, the surpluses of unused allowances continued because of an economic downturn. At the beginning of the year 2012, the aviation sector was additionally covered by the system. The third trading period (2013–2020) was characterized by a progressive shift toward auctioning of allowances instead of their cost-free allocation. For this purpose, the emission cap for all countries belonging to the EU was reduced by 1.74% per year. Furthermore, the share of free allowances allocated to the companies by the government has been decreased to 30% of the overall volume for the manufacturing industries and to 82% for the aviation sector. Initial fines for exceeding the company-specific cap were amounted to 100 Euros per ton of carbon dioxide (or an equivalent quantity of other greenhouse gases) in 2013, while this penalty rose annually in line with the European consumer price index. Further revisions of the system are intended for the following fourth trading period (2021–2030). They include a further decline of the number of allowances, a better targeted and more dynamic allocation of free allowances, as well as the implementation of innovation and modernization funds that should support meeting the challenges of a transition to a low-carbon economy. According to the ambitious targets of the EU ETS, greenhouse gas emissions are supposed to be reduced by at least 40% until 2030 compared to the level of 1990 (Ellerman and Buchner 2007; European Commission 2016). Going beyond that, other real-life applications of emission trading systems can be found in the Swiss Emissions Trading Scheme, the Australian Carbon Pricing Mechanism, the New Zealand Emissions Trading Scheme, as well as China’s emission trading systems. The US Sulfur Dioxide Cap-and-Trade-Program, the Regional Greenhouse Gas Initiative, the Western Climate Initiative, and the Midwestern Greenhouse Gas Reduction Accord are further examples from North America (Weishaar 2014). Facing the diversity of national and supranational legislation and emission trading systems, both strengthening and harmonization are required to enhance the global environmental impact (Chaabane et al. 2012).

3.3 Emission Management

3.3.2

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Implementation of Emission Management

In the following, implications of the aforementioned regulations and trading systems for companies and SC networks should be analyzed. For managing emissions appropriately, systems of accounting and reporting need to be established. Although business practices still vary considerably in terms of emission reporting, there are emerging elements of standardization. A common tool for cooperate measuring, accounting, and reporting is the Greenhouse Gas Protocol (WRI and WBCSD 2004). It covers six greenhouse gases being part of the Kyoto Protocol. With regard to relevant boundaries of accounting, it distinguishes three different scopes. The first scope covers direct emissions from greenhouse gas sources that are owned or controlled by the company. The second scope refers to indirect emissions that do not physically occur within these boundaries, but are controlled by the company (e.g., emissions caused by electricity, heat, cooling, or steam consumption). The third scope deals with emissions from sources that are neither owned nor controlled by the company. It includes the extraction of externally purchased materials or fuels, external activities of transportation, or other outsourced processes such as waste disposal. The estimation of overall emissions generally follows five steps that include the identification of relevant emission sources, the selection of a calculation approach, the collection of activity data and the choice of emission factors, the application of calculation tools, and the rollup of data to the corporate level. However, practical difficulties in the aforementioned procedure, especially in collecting data, arise from the absence of internationally agreed standards and methodologies for analysis (Green 2010; OECD 2010; Boukherroub et al. 2017). Broadening the focus from companies to entire supply chains, specific guidelines for allocating shared emissions exist. According to them, the sum of the emissions allocated to the outputs of a system should equal exactly the overall emission from the system. Effects of over-allocation, which would result from double counting of emissions, can be avoided by the implementation of network-wide trading schemes (Boukherroub et al. 2017). Based on accounting the relevant greenhouse gas emissions and reporting the environmental information, reduction plans can be established. In line with industrial requirements (e.g., Gielen and Moriguchi 2002; Li et al. 2011), emission targets should be based on different types of greenhouse gases. Both the stringency and the timeframe of these targets should adhere to the expected medium-term participation in external emission trading systems and, furthermore, the long-term environmental strategy. In general, three main types of emission reduction targets are distinguished in literature. Firstly, there are absolute emission targets, which impose reduction levels not being dependent on the network performance. Although those targets are clearly communicable to all stakeholders, difficulties in achieving the goals may arise if the network expands or activities grow. Secondly, intensity targets can be considered. The latter allow for an increase of total emissions in line with an organic growth on the one hand or future network acquisitions on the other hand. Although these targets seem to be useful for measuring efficiency, they do not systematically

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lead to the desired reductions in the absolute emission volume. Finally, carbon neutrality targets exist. In this context, the networks strive for reaching zero net emissions by combining internal actions and external offsets. A main advantage of this kind of target formulation can be seen in the flexibility of strategy choices, but— in extreme cases—it may not lead to any internal reduction. Inherently, the specific formulation of targets can vary considerably with regard to their concrete nature, scope, and methodology. A catalogue for setting emission targets may contain the following steps: In line with existing management guidelines, decisions on the target type and boundaries need to be taken. The target base year, the completion date, as well as the target commitment period should be clearly defined. Furthermore, the use of offsets or credits can be taken into account. Based on a fixation of the target level, measures for tracking and monitoring can be planned (Jotzo and Pezzey 2007; OECD 2010; Bailey 2017). Comparing available information with the defined targets creates the basis for taking actions of management and coordination. Initially, general strategies for emission reduction can be considered individually at each site belonging to the network. They range from improving energy efficiency, reducing waste generation, implementing low-carbon technologies, reorganizing logistics, and changing product characteristics to using renewable energies and less carbon-intensive inputs. Sites unable to meet their commitments on emission reduction because of unexpected circumstances can take advantage of carbon offsets. Sector-specific mixes of actions can be arranged. In contrast to uncoordinated measures of single companies, management and coordination of emissions should be established throughout the entire supply chain network for reasons of efficiency and effectivity. In practice, decentralized approaches are still dominating. Starting from the perspective of individual sites being involved, mitigation opportunities can be negotiated with adjacent partners. With regard to upstream suppliers, the negotiation content may include the substitution of materials. Regarding downstream purchasers, possibilities of reuse and recycling can be relevant. Considering bidirectional relationships, reduction of packaging as well as optimization of transports can be of particular importance. Related coordination approaches assume that the definition of sitespecific objectives and standards has impact on business partners and finally engages them to reduce their own emissions. These approaches may contain the establishment of emission criteria in procurement decisions, technical and administrative collaborations with suppliers, or the involvement of partners in business reorganization. However, the aforementioned coordination processes can be costly and protracted due to efforts on information exchange and monitoring within multiple bilateral negotiations (OECD 2010). Further perspectives for the implementation of emission management and coordination can be seen in the development of holistic approaches applicable to entire supply chain networks consisting of several stages and sites. Due to the existence of bargaining power or specific legal frameworks, the prerequisites for centralized optimization are met in many cases. By that, it becomes possible to coordinate emission reduction with the network-wide planning and scheduling of operations so as to find an appropriate trade-off between economic and environmental objectives in line with relevant legislation.

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Gielen, D., & Moriguchi, Y. (2002). CO2 in the iron and steel industry: An analysis of Japanese emission reduction potentials. Energy Policy, 30(10), 849–863. Grant, D. B., Trautrims, A., & Wong, C. Y. (2015). Sustainable logistics and supply chain management. London et al.: KoganPage. Green, J. F. (2010). Private standards in the climate regime: The Greenhouse gas protocol. Business and Politics, 12(3), 1–37. Heiskanen, K. (2014). Theory and tools of physical separation/recycling. In E. Worrell & M. A. Reuter (Eds.), Handbook of recycling (pp. 39–61). Amsterdam et al.: Elsevier. Hintermann, B. (2016). Emissions trading and market manipulation. In S. E. Weishaar (Ed.), Research handbook on emission trading (pp. 89–110). Cheltenham/Northampton: Edward Elgar. Huang, Y.-C., & Yang, M.-L. (2015). The effect of institutional pressures and top managers’ posture on green supply chain management. In V. Kachitvichyanukul, K. Sethanan, & P. Golinska-Dawson (Eds.), Toward sustainable operations of supply chain and logistics systems (pp. 99–121). Heidelberg et al.: Springer. Jotzo, F., & Pezzey, J. C. V. (2007). Optimal intensity targets for greenhouse gas emissions trading under uncertainty. Environmental and Resource Economics, 38(2), 259–284. Kadziński, M., Tervonen, T., Tomczyk, M., & Dekker, R. (2017). Evaluation of multi-objective optimization approaches for solving green supply chain design problems. Omega, 68, 168–184. Khan, M., Hussain, M., & Ajmal, M. M. (2017). Green supply chain management for sustainable business practice. Hershey: Business Science Reference. Lanzini, P. (2018). Responsible citizens and sustainable consumer behavior. Oxon: Routledge. Lebreton, B. (2007). Strategic Closed-Loop Supply Chain Management. Berlin/Heidelberg: Springer. Li, L., Tan, Z., Wang, J., Xu, J., Cai, C., & Hou, Y. (2011). Energy conservation and emission reduction policies for the electric power industry in China. Energy Policy, 39(6), 3669–3679. Liu, J. J. (2012). Supply chain management and transport logistics. Oxon: Routledge. Madu, C. N. (2004). Competing on quality and environment. Fairfield: Chi Publishers. Nentjes, A. (2016). Emission targets and variants of emissions trading. In S. E. Weishaar (Ed.), Research handbook on emission trading (pp. 27–49). Edward Elgar: Cheltenham/Northampton. Nield, K., & Pereira, R. (2016). Financial crimes in the European carbon markets. In S. E. Weishaar (Ed.), Research handbook on emission trading (pp. 195–231). Cheltenham/Northampton: Edward Elgar. OECD [Organisation for Economic Co-operation and Development]. (2010). Transition to a low-carbon economy: Public goals and corporate practices. OECD Publishing. https://doi.org/ 10.1787/9789264090231-en. Sarkis, J., & Dou, Y. (2018). Green supply chain management. New York: Routledge. Sople, V. V. (2012). Supply chain management. Delhi et al.: Pearson. Srivastava, S. K. (2007). Green supply-chain management: A state-of-the-art literature review. International Journal of Management Reviews, 9(1), 53–80. Steinrücke, M., & Albrecht, W. (2016b). A flow-to-equity approach to coordinate supply chain network planning and financial planning with annual cash outflows to an institutional investor. Business Research, 9, 297–333. Steinrücke, M., & Albrecht, W. (2018). Integrated supply chain network planning and financial planning respecting the imperfection of the capital market. Journal of Business Economics, 88 (6), 799–825. Steven, M. (2004). Networks in reverse logistics. In H. Dyckhoff, R. Lackes, & J. Reese (Eds.), Supply chain management and reverse logistics (pp. 163–180). Berlin/Heidelberg: Springer. Sydow, J. (1992). Strategische Netzwerke. Wiesbaden: Gabler. Wagner, S. A. (1997). Understanding green customer behaviour. London/New York: Routledge. Weishaar, S. E. (2014). Emissions trading design. Cheltenham/Northampton: Edward Elgar.

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Chapter 4

Literature Review

Abstract Based on existing structures of sites and capacities, new mathematical models for hierarchical scheduling of operations in generic multi-stage networks are developed in the book Scheduling in Green Supply Chain Management: A MixedInteger Approach. The literature review in this chapter reveals the lack of suitable approaches in green supply chain management so far and specifies the research gap. Besides general insights from static modeling, the reviewed publications are grouped to continuous- and discrete-time formulations according to their applicability to problems with planning horizons of different length. In the field of SCM, an overwhelming number of publications exist. As being relevant for the development of mathematical models for optimizing network systems, which typically include binary decisions on entities and continuous decisions on operations, the following consideration is limited to mixed-integer programs. The latter may be either linear or non-linear (Cottle and Thapa 2017). The structure of the following section is determined by the focus on scheduling. Although static formulations (Sect. 4.1) are limited to a single time period, and thus do not allow for a temporal differentiation of relevant decisions at all, they may exclusively contain specific features of modeling with regard to GSCM. Discrete-time approaches (Sect. 4.2) are suitable for a rough scheduling within long-term or medium-term planning horizons, i.e., optimal decisions are assigned to time periods. Continuous-time approaches (Sect. 4.3) allow for scheduling exactly to the minute. As a result of the literature review, the existing research gaps (Sect. 4.4) to be treated in the following parts of this monograph are identified.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6_4

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4.1

4 Literature Review

Review of Static Approaches

Origins of GSCM can be found in dealing with product returns in course of static reverse logistics. In this context, Fleischmann (2001) proposes a generic recovery network model. On the one hand, it includes forward flows from plants via warehouses to markets. On the other hand, there are reverse flows from the markets via disassembly centers to the plants. The static formulation establishes a link between two different types of markets, i.e., the reuse markets that are supplied with recovered products in the first case and the disposer markets that are assumed to be sources of the used products to be collected by the remanufacturer. If necessary, additional material flows can be directed from the disassembly centers to disposal facilities. The formulation is based on binary decisions on the opening of potential facilities, as well as on continuous variables representing material flows between the network sites. Furthermore, unsatisfied fractions of customer demand and uncollected fractions of customer returns are determined in course of the optimization problem. The objective of the MILP is to minimize the sum of investment and operational costs. Another basic approach for the design of reverse distribution networks was introduced by Jayaraman et al. (2003). The static MILP covers exactly three stages for origination, collection, and refurbishing facility sites and, thus, neglects forward material flows. It allows for handling products that have been recalled, are to be recycled or disposed of, or are hazardous. In course of the minimization of the total network costs, it needs to be decided on the opening of potential collection and refurbishing sites, while both of the latter are characterized by given maximum capacities. In this context, there are parameters specifying both the minimum and the maximum number of these sites to be opened. Material flows are modeled by a continuous variable that represents the fraction of returnable product units at the origination, which need to be directed through the network on a specific path, i.e., either directly or indirectly (via collection sites) to the refurbishing facilities. An uncapacitated facility location problem combining forward and reverse flows in multi-stage logistics systems propose Lu and Bostel (2007). While reverse flows of return products are possible from customer zones via intermediate centers to remanufacturing centers, forward flows can occur between producers and customer zones. Each of the aforementioned network stages may contain different sites. The problem is based on given costs for setup and operations, as well as given percentages of ingoing material flows at intermediate and remanufacturing centers, which need to be disposed of due to insufficient product quality. Recycling of used products is assumed to supersede the manufacturing of new items at the producers. It needs to be decided on the location and setup of the network facilities, as well as on fractions of demand or return quantities that are to be directed on specific material flow paths. The objective is to minimize the overall costs of the network. The static facility location model by Erkut et al. (2008) is designed for municipal solid waste management and, furthermore, applied to a real-life case in North Greece. Relevant facilities building up the considered network are waste producers, transfer stations, material recovery facilities, incinerators, as well as landfills. The facilities of the

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recycling network can be operated in different types representing alternative technologies that are available for processing different materials. Whereas decisions of facility location and their typology are indicated by binary variables, continuous ones are used for determining the quantity of reverse material flows within the network. The MILP contains five alternative objectives, which include the minimization of the greenhouse effect, the minimization of the final disposal to the landfill, the maximization of energy recovery, the maximization of material recovery, as well as the minimization of the sum of installation, transportation, and treatment costs. Wang et al. (2011) model the problem of designing a green supply chain network. Although their original mixed-integer formulation is non-linear, a linearized version is additionally provided by the authors. The network is composed of three different types of sites, i.e., suppliers, facilities, and customers. The static model formulation is limited to forward material flows of products, which can satisfy the customer demand according to given forecasting. During the strategic horizon of the static approach, it needs to be decided on the opening of the available facilities and the transportation of products between them. Furthermore, it is possible to choose between different environment protection levels for each facility. In this context, a higher level corresponds to higher investments in facility equipment or technology but to lower carbon dioxide emissions. The multi-objective formulation measures the total costs of network design, on the one hand, and the total carbon dioxide emissions caused by the supply chain, on the other hand, and moreover allows for the analysis of trade-offs between both objectives. Kannan et al. (2012) propose an MILP for reverse logistics systems design focusing on the carbon dioxide footprint. The general network structure includes stages for customers, as well as collection/ inspection, recovery, and disposal centers, but does not consider the formation of closed loops. Driving force of operations is a given quantity of returned products at the customer centers. There is an average fraction of products that need to be disposed of. Furthermore, the number of collection/inspection centers to be opened during the single-period planning horizon is limited. Besides site capacities as well as fixed and variable costs of operations, the distances between the potential network sites are given. The latter are used to calculate the carbon dioxide emissions resulting from realized transports, which need to be merged with fixed emissions that result from operating the collection/inspection centers. Further decision variables refer to facility opening and material flows. The objective is to minimize total costs. Besides fixed and variable costs of operation, the objective function takes the costs of carbon dioxide emissions into account. For this purpose, the difference between actual emissions and the given emission cap is multiplied with the costs of carbon credits per emitted ton. A two-phase approach for closed loop supply chain configuration and supplier selection, which is limited to the analysis of a single planning period, is formulated by Amin and Zhang (2012). The first phase uses a fuzzy method for evaluating suppliers based on qualitative criteria and determining specific weights representing the suppliers’ importance. These weights are considered within a multiobjective MILP formulation in the second phase, which combines both strategic and tactical network decisions. The modeled multi-stage network includes a flow of returned products to disassembly sites, whose output can be directed to disposal

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(waste) or refurbishing sites (reusable parts). The resulting inventory of as new parts can be supplemented by new parts from external suppliers and, finally, used by a manufacturer to initiate forward material flows. There are binary decisions on the selection of suppliers and on the setup of refurbishing and disassembly sites, as well as continuous decisions on the number of parts and products to be processed within the network. The first objective function maximizes the total network profit, the second one minimizes defect rates, and the third one maximizes the importance of external suppliers. For solving the multi-objective formulation, the compromise programming method is applied. Ramezani et al. (2013) present a multi-objective model for forward/reverse logistics design. The scenario-based MILP includes multiple products and multi-level capacities. The considered closed loop network contains five stages. With regard to forward flows, new products are manufactured in plants from raw material that can be obtained from suppliers. The products are shipped via distribution centers to customers in order to meet the given demand. Considering the reverse flows, there are products returned from the customers that need to be transferred to collection centers for testing and inspection. Afterward, recoverable and disposable products are separated and directed to plants and disposal centers, respectively. Modeling includes hybrid processing facilities, which allow for establishing distribution and collection processes at the same site. Continuous decision variables specify the transportation quantities, while binary ones are related to the opening of facilities and the single sourcing of customers. The first objective is to maximize the total profit, the second one is to maximize the customer service level of weighted forward and reverse flows, and the third one is to minimize the weighted total number of defects in raw material obtained from suppliers. For solving the problem, the authors propose to optimize the first objective function while considering the other two objective functions as constraints with allowable bounds. Hatefi and Jolai (2014) optimize a logistics network with simultaneous forward and reverse flows while taking uncertain parameters and facility disruptions into account for their static MILP. The approach is limited to a single period and a single product. It considers production and distribution centers in the forward flow, as well as collection, recovery, and disposal centers in the reverse flow. For reasons of cost-saving and pollution reduction, hybrid facilities for production/recovery and distribution/ collection are introduced. According to a given percentage (average disposal fraction), the returned products are separated into scrap to be shipped to the disposal centers and recoverable items to be processed in the hybrid production/recovery facility. For meeting the demand at the customer zones, both new and recovered products can be supplied by the network, while it is possible to satisfy the demand fully or partially. There are continuous decision variables representing the shipments between the network sites. Furthermore, the non-satisfied demand of each customer is quantified, as it entails penalty cost. Binary decision variables are indicating the opening of network sites. The objective of the problem is to minimize the overall costs. The MILP by Zhang et al. (2014) with multiple objective functions is developed for sustainable supply chain optimization. The general network structure includes three stages (raw material suppliers, production plants, and customer locations). The problem covers tactical planning within a single time period,

4.1 Review of Static Approaches

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which is assumed to be 1 year. Although the model formulation includes parameters for processing and transportation times in order to determine the total lead time of the overall system, it does not allow for continuous-time scheduling. Modeling does not capture an exact coordination between the processes at/between the sites, and consequently, there are no continuous decision variables for start and end time of network operations. Instead, there are decisions on quantities of raw materials to be purchased, products to be allocated to plants and units, quantities of products to be manufactured in each unit, as well as material flows from suppliers to plants, between plants, and from plants to customers. By the use of three different objective functions, it is possible to minimize the total costs, the total greenhouse gas emissions, and the total lead time. Al Dhaheri et al. (2014) propose a mixed-integer non-linear program for sustainable supply chain management. It strives for integrating environmental footprint strategies for efficient network design within a twofold approach. The structure is based on both forward and reverse material flows that occur between a given number of SC stages. The latter are arranged in a fixed closed loop of manufacturers, warehouses, end users, collection centers, and recycling centers. Additional disposal centers can only be used for rejects from the collection centers. The static formulation assumes a tactical-operational planning horizon and aims for minimizing the overall network costs, which are composed of costs of closed loop supply chain planning (opening and operating the facilities, distributing the products) as well as costs of carbon emission management. The latter allows for calculating the volume of carbon dioxide emissions that result from locating and operating facilities, as well as from transportation. Deviations of the total emission volume from the given cap in combination with constant emission trading prices are used for calculating the emission costs. Chen et al. (2015) propose a closed loop supply chain model, which classifies product returns according to the quality. The MILP covers SC stages for suppliers, manufacturers, warehouses, retailers, customer regions, collection points, recycling companies, as well as waste disposal centers. Despite the latter, each of the stages may comprise different network sites. Besides direct deliveries between the sites, a vehicle routing problem is incorporated. In particular, it is assumed that collection points can arrange a round trip to fetch used products from different customers. Within the recycling centers, products with good quality are transported back to the manufacturers, while products with poor quality are disassembled in order to extract reusable raw materials. Remaining parts are sent to disposal. The static approach contains decisions on the quantities of forward and reverse material flows between the sites as well as decisions on the opening of network facilities. The overall objective is to maximize the total profit, which is composed of revenues from selling products to the customers as well as different types of costs from facility opening, operations, and transportation. The multiobjective approach for optimizing a closed loop supply chain network by Ghayebloo et al. (2015) focuses on supplier selection and disassembly of products. The MILP is based on five SC stages, which can be assigned to the oppositely directed material flows. In forward SCM, different suppliers deliver required parts to assembly centers. Besides traditional manufacturing, principles of “design for disassembly” can be applied. The latter entail additional costs and, furthermore, determine the

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greenness level of the products, which are subsequently delivered to the customer centers. Reverse SCM considers the return of the products to disassembly or recycling centers. Only a specific proportion of ingoing products at a disassembly center can be reintegrated into forward SCM, while the remaining proportion is directed to the recycling center. Relevant decisions concern process quantities, facility location, and the selection of the greenness level for assembly lines. The alternative objective functions refer to maximizing total profit or total greenness. For handling these conflicting objectives, the weighted sum method and the ε-constraint method are proposed. Kheirkhah and Rezaei (2016) propose a model that deals with locating cross-docking centers in a reverse logistics network, which consists of given customer zones and recovery centers. The cross-docking centers allow for collecting used products in a single location and distributing them to different types of recovery centers (e.g., remanufacturing, repairing, or recycling facilities). Consequently, site decisions are limited to a single SC stage. Additional decisions on material flows specify the transportation quantities between the network sites. The objective function of the MILP is to minimize the overall costs of facility location and transportation. The mathematical model of a closed loop supply chain by Jindal and Sangwan (2017) combines issues of management and environment within a multiobjective fuzzy approach. The network structure is based on five SC stages of forward SCM (external suppliers, manufacturers, distributors, retailers, customers) and five stages of reverse SCM (collection, disassembly, recycling, disposal, and refurbishing centers). According to given maximum proportions, which represent different grades of the returned products’ usability, material flows between the different sites of the network can occur. It is assumed that new products and remanufactured products can be sold at the same market and price. The first objective function strives for maximizing the total profit, which includes revenues from sales and recycling as well as different types of cost. The second objective function focuses on the environmental impact in terms of carbon footprints, as it determines the overall carbon dioxide emissions that occur from transportation. Nurjanni et al. (2017) propose a static multi-objective optimization approach for green supply chain design, which strives for determining trade-offs between environmental and financial issues. The closed loop supply chain network contains exactly four SC stages, whose sites are connected by potential material flows. The forward path can be directed from a factory via a warehouse to a customer in order to meet the given demand with new or remanufactured products. The reverse path spans the collection of returnable products from the customer, as well as the delivery to a distribution center for recycling and subsequently to a factory for remanufacturing. The MILP is limited to a single product, which can occur in three different types of product form (new product, product to be disposed of, product to be dismantled). Relevant decisions concern the opening of facilities and the material flows. The first objective minimizes the overall costs, while the second one minimizes the total emission of carbon dioxide that results from SC operations. Three different scalarization approaches are discussed for handling the multi-objective problem. Tombuş et al. (2017) model a closed loop supply chain problem with forward and reverse flows. Their static MILP contains SC stages for plants, distribution and return centers, as well as customer

4.2 Review of Discrete-Time Approaches

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zones. Calculating return quantities is based on a recovery ratio, i.e., a given fraction of returns that can be remanufactured after inspection. The objective function is to minimize the total costs of transportation and opening facilities. The authors confirm the benefits of simultaneous optimization of forward and reverse material flows, and furthermore, they analyze effects of co-locating distribution and return centers within the network. The static formulation of a mixed-integer non-linear programming model by Liao (2018) is based on a structure, which includes 12 specific types of network stages connected by potential material flows. The reverse logistics problem focuses on modularized products. According to given percentages of maximum recovery, returned products can be repaired or dismantled into modules. The latter can subsequently be used in different ways, e.g., refurbishing for selling in spare part markets, recycling, or remanufacturing. For purposes of remanufacturing, the quantity of available modules can be supplemented by new ones. Binary variables are used for indicating the opening of network facilities and for the assignment of customers to regional collection centers. Continuous variables represent the quantities of reverse flows (products, modules) between different types of facilities, as well as remanufacturing and inventory quantities. The objective is to maximize the overall network profit. The MILP for optimizing closed loop supply chain networks with combined forward and reverse material flows by Shahparvari et al. (2018) is also based on modularized products, which can be distinguished by different quality levels depending on the extent of using recycled products during the manufacturing process. The material flows pass a predefined number of SC stages in a fixed sequence. The latter spans suppliers, manufacturers, distribution/ collection centers, and customers in forward SCM. Flows of reverse SCM are directed from customers via distribution/collection centers to disposal or recycling centers. Starting from the latter, reusable modules are sent back to the manufacturers. Binary variables refer to decisions of location and allocation, while continuous variables are used for determining process quantities. The objective function is to maximize the total profit of the overall system, which is composed of qualitydependent sales of products, on the one hand, and different types of cost, on the other hand. Although all static model formulations of this section cannot be applied to discrete-time or continuous-time problems, the review provides basic insights in modeling of interdependencies relevant for GSCM.

4.2 4.2.1

Review of Discrete-Time Approaches Approaches without Elements of Green Supply Chain Management

By now, an overwhelming variety of discrete-time SC planning and scheduling models dealing with the integration of production, distribution, and transportation

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exists in literature. Over the past few years, the tendency of modeling actual interdependencies between relevant decisions more faithful to detail can be observed. Initial approaches in the treated field (Erengüç et al. 1999) were developed without reference to issues of environment or sustainability. For instance, a capacitated approach for integrated production-distribution planning in supply chains is proposed by Lee and Kim (2002). It covers multiple periods, products, and SC stages and strives for minimizing the overall costs of production, distribution, inventory holding, and shortage. Driving force of processing is the periodic product demand of the retailers. The planning horizon is divided into time periods of fixed length, which limit the duration of operations. Tempelmeier (2002) combines short-term supplier selection and order sizing under dynamic demand conditions. The formulation includes time-varying data, which result from period-based quantity discount structures of specific discount levels. By that, rising or falling prices of quantities purchased from suppliers can be considered. Park (2005) proposes a mixed-integer approach for integrating production and distribution planning in supply chains. According to the objective function, the total net profit realizable over the entire planning horizon is to be maximized. Whereas core business demand needs to be satisfied, stock-out is tolerated for forecasted demand. The compliance of operations with available production capacities is ensured for every time period. The MILP by Guillén-Gosálbez et al. (2006) applies discrete-time scheduling within a rolling planning horizon. The authors consider simultaneous operational and financial planning in multi-product, multi-echelon distribution networks with multipurpose batch plants. Cash balancing (including marketable securities and short-term financing sources) is ensured for each time period. The discrete-time model by Laínez et al. (2007) strives for maximizing the corporate value at the end of the planning horizon and combines operations and financing with strategic decisions on facility opening and capacity increment. Periodic liquidity control uses a specific cash function. Sawik (2009) proposes a mixed-integer model for coordinated discrete-time supply chain scheduling. The customer-driven SC network with three stages includes material supply, manufacturing, and product assembly. Each order to be processed must be completed in one time period. The MILP by Hahn and Kuhn (2011) is developed for a make-to-stock supply chain. It combines the physical domain (procurement, production, distribution at sites of different network stages) with a financial domain (one-period borrowings and investments). For each time period with a length of 1 month, all relevant cash flows are merged. However, detailed decisions on quantities and financial positions cannot be obtained. Bhatnagar et al. (2011) propose an approach for coordinated planning and scheduling in global supply chains with different supply modes. Whereas a shipment planning problem includes weekly decisions on transportation quantities and inventories, the scheduling problem (based on a dynamic version of the newsboy problem) determines order quantities on a daily basis. The aforementioned problems are to be connected in an iterative sequence. Steinrücke and Jahr (2012) formulate a tactical MILP that integrates decisions on production, distribution, and transportation. There are single SC stages for suppliers, final producers, and distribution centers. Different capacities, transportation modes, and product types are considered. Furthermore,

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customer-oriented single sourcing is implemented. The discrete-time approach is based on the big bucket method so as to ensure the stability of optimal plans by providing sufficient time buffers. Relvas et al. (2013) model integrated scheduling and inventory management in supply chains. Due to their focus on the petroleum industry, the authors strive for determining a discrete-time schedule for a pipeline, which is used for pumping different oil products from a refinery to a distribution center. Inventory management at the distribution centers is based on daily volume balances. Two MILP formulations with different assumptions regarding the batch size are proposed. Rodriguez et al. (2014) combine both strategic and tactical perspectives in a multi-period planning horizon. On the one hand, there are longterm decisions on installation, expansion, and elimination of multi-product warehouses and factories. On the other hand, medium-term decisions include inventory levels at distribution centers and customer plants, as well as connections between network sites. The mixed-integer non-linear model can be reformulated as MILP by piecewise linearization. An approach for medium-term planning of capacities and material flows in an SC network is proposed by Steinrücke and Albrecht (2015). The MILP allows for supplier and market selection at the edges of the supply chain. The network consists of four subsequent stages for suppliers, plants, warehouses, and markets. Both plants and warehouses can be operated in multiple capacity levels. Within the multi-period planning horizon, a central coordinator needs to decide on discrete-time contracting with the available network partners. Steinrücke and Albrecht (2016a) develop a multi-period MILP for network design and extension of supply chains. It combines configurational decisions with aggregated decisions on production, storage, inventory, distribution, transportation, and sales. The profitmaximizing model formulation allows for assessing the integration of multiple sites of new network partners into existing structures. Moreover, it is possible to reformulate the approach in order to determine threshold levels for supply, production, or distribution quantities provided by the sites to be integrated. Steinrücke and Albrecht (2016b) combine supply chain network planning and financial planning. The assumed network structure may comprise an arbitrarily definable number of production and distribution stages, while each site of an SC stage can supply each site of a subsequent adjacent SC stage. Transports of products in different maturities, as well as inventory holding, are included. For financial planning, credits with given duration and maximum extent are available. Credit rates are calculated according to the Capital Asset Pricing Model. The objective function of the MILP maximizes the present value of equity while determining periodic cash outflows to an institutional investor. Steinrücke and Albrecht (2018) propose an MILP for integrated supply chain network planning and financial planning respecting the imperfection of the capital market. Consequently, the objective function does not use an external discounting rate, but maximizes the weighted payouts to an investor. For liquidity balancing, both credits and financial investments can be taken by the network managers. The formulation is based on a general network structure that allows for defining an arbitrary number of SC stages, which can be assigned to procurement, production, or distribution processes. Each of the latter stages can supply a subsequent one, which can be either adjacent or nonadjacent.

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Due to the high level of abstraction, the aforementioned structure allows for appropriate flexibility in coordinating material flows, as it includes all paths of forward SCM. However, modeling neglects reverse flows and emission trading completely. Regardless of the complexity of the assumed network structure, the following paragraph reviews discrete-time approaches containing features of GSCM.

4.2.2

Approaches Including Elements of Green Supply Chain Management

Kim et al. (2006) consider a planning model for a reverse logistics system, which can be part of a supply chain. Their multi-period approach considers relationships between single sites, while forward flows are excluded from consideration. The manufacturer uses a part inventory for production that is replenished by an external supplier, a refurbishing site, and a remanufacturing subcontractor. Whereas the refurbishing site obtains reusable parts via a disassembly site from the collection site, the latter directly supplies the remanufacturing subcontractor with products returned from the customer. Internal waste is shipped to a disposal site. Perioddependent binary decisions on setting up the sites for specific parts or products are combined with continuous decisions on processed or outsourced quantities. The objective function strives for maximizing the cost-saving from remanufacturing. Gomes Salema et al. (2010) propose a generic modeling framework for simultaneous design and planning of supply chains with reverse flows. The MILP is based on time discretization of a strategic planning horizon, while two interconnected time scales are considered. The latter refer to two main levels of decisions. Macroperiods (e.g., years) are relevant for the design level, which is related to the structure of the closed loop supply chain and the selection of network sites. Furthermore, it is decided on the satisfaction of demand and return. In opposite, microperiods (e.g., months) allow for detailing the aforementioned planning, i.e., it is determined how the aforementioned satisfaction can be attained. In this context, production levels, storage levels and quantities of material flows are relevant. The authors study a network that consists of SC stages for plants, warehouses, end customers, and disassembly centers, while potential material flows need to respect the predetermined sequence. Important assumptions refer to product recovery. There is no distinction between new and remanufactured products. Returned quantities are a fraction of product supply in forward SCM. There are non-recoverable products that need to be disposed of. The objective function of the model formulation exclusively covers the minimization of total supply chain costs. A model for multi-period reverse logistics network design is proposed by Alumur et al. (2012). It covers the determination of optimal sites and capacities for different types of facilities (collection centers, inspection centers, remanufacturing facilities, recycling plants). The MILP includes modular capacities and their extensions at the facilities, reverse bills of materials, minimum throughput

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constraints, variable costs of operation, as well as finite demand at a secondary market. Due to time discretization, all decisions to be taken on the network design (e.g., operating facilities, adding capacity modules) and on material flows (holding inventory, shipping products, and components) need to be assigned to the beginning or the end of the time periods. The objective is to maximize the overall profit of the network, which includes revenues from recycling centers, external remanufacturing plants, and the secondary market, as well as costs from facilities, capacities, operations, transportation, inventory, and purchasing. Instead of modeling the disposal of products not being suitable for recycling, the authors consider the latter as recyclable products with a negative profit. The MILP by Cardoso et al. (2013) allows for discrete-time designing and planning of supply chain networks. It considers processes of production, distribution, and reverse logistics simultaneously. The network can be described by a fixed sequence of four SC stages. Plants can be used for production and disassembly, warehouses for assembly and storage, and retailers for storage and distribution. Forward flows from each of these stages to the final market stage are possible. The latter is also starting point for reverse flows, which can be directed to sites of each previous SC stage. Additional reverse flows can occur from plants, warehouses, or retailers to disposal facilities. The consideration is limited to three types of products (raw materials, intermediate products, final products). The main decisions cover capacity provision, capital investments, inventories, as well as material flows. The objective of the model formulation is to maximize the expected net present value that results from discounting the overall cash flows, which are realized in different time periods of the planning horizon. Alshamsi and Diabat (2015) develop an MILP for optimizing the configuration of a general reverse logistics network, which considers a reverse flow passing four subsequent stages: Firstly, products are sorted in a collection center. Secondly, an inspection center is used to identify reusable parts. Thirdly, remanufacturing plants create new products based on the incoming parts. A replacement by external suppliers is alternatively possible. Damaged products can be sent from all sites of the first three SC stages to recycling/disposal facilities. Finally, the remanufactured products can be sold at secondary markets. Relevant decisions concern the optimal selection of sites and capacities. With regard to transportation, it needs to be decided whether an in-house fleet or an outsourcing option is chosen. The objective is to maximize the profit while taking investment on facilities or vehicles into account. Decisions on the size of initial investment at the beginning of the planning horizon and the part of revenue to be invested in subsequent time periods are introduced for limiting the available projects. An integrated approach for production-distribution planning in green supply chains is established by Memari et al. (2015). The formulation is based on a three-stage network structure of forward logistics with manufacturers, distributors, and dealers and, thus, neglects any issue of reverse logistics. Although exact processing times required for the manufacturing of products at the facilities are given in the discrete-time formulation, their usage is limited to ensure that the overall production time within a specific time period cannot exceed a given limit, which is consequently determined by the period length. Decision variables for exact start and end times of operations do not exist, and moreover, a period-overlapping

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coordination of production is not possible. Instead, the MILP contains variables for determining integer product flows between the network sites within the time periods. While the first objective function minimizes the total costs of the supply chain, the second one covers aspects of green supply chain management. In particular, transportation amounts for both echelons are multiplied with the expected productdependent carbon dioxide emissions. Finally, the latter are summed up in order to determine the overall emission volume for the entire planning horizon. Fahimnia et al. (2015) propose a mixed-integer non-linear model for green supply chain planning. The multi-site network structure consists of subsequent stages for manufacturers, warehouses, and end users, which are exclusively connected by material flows of forward SCM. The relevant demand is deterministic and needs to be satisfied sooner or later within a time period of the planning horizon. Backordering, i.e., shifting the demand satisfaction from one period to a following one, is possible but entails a penalty cost. Capacity limitations at the network facilities are defined for regular and overtime production. A first objective function represents the overall costs of manufacturing, transport, inventory holding, and backlog. For analyzing trade-offs, further objective functions can be used to minimize the generated carbon pollution, the energy consumption, as well as the waste generation. Soleimani et al. (2016) model a multi-period and multi-product closed loop supply chain network. Their MILP combines decision levels of network design and planning. The forward SC contains stages for suppliers, manufacturers, warehouses, distributors, and customers, while the reverse SC spans disassembly centers, redistributors, disposal centers, and second customers. While only new products can be sold to first customers, the returned products are sold (while respecting their different quality, demand, and price) after processing to second customers. Operations in reverse logistics are based on specific ratios for return, recycling, remanufacturing, repairing, and disposal. According to existing uncertainties, the approach takes different scenarios into account. Relevant decisions on the network design concern the selection of potential locations. Moreover, period-specific planning covers decisions on the establishment of transportation links, as well as on the quantity of material flows and residual inventories. The objective is to maximize the total profit of the system, which includes sales to the first and second customers, on the one hand, and the sum of costs for locations, materials, manufacturing, non-utilized capacities, delivery shortage, purchasing, disassembly, recycling, remanufacturing, repairing, disposal, transportation, and inventory holding, on the other hand. Entezaminia et al. (2016) propose an approach for multi-objective aggregate production planning in green supply chain management. The discrete-time model with a planning horizon of several months allows for making a trade-off between costs and environmental criteria. The general structure of the considered supply chain network distinguishes SC stages for suppliers, plants, customer zones, as well as collection and recycling centers, which are connected by potential forward or reverse material flows. Major decisions concern the opening of sites, the allocation of products and production methods to sites, as well as the determination of process quantities. The first objective function minimizes the total costs of the SC network, while the second objective function strives for maximizing the total score of products in terms of

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environmental criteria, which is based on product weights obtained from an analytical hierarchy process, on the one hand, and from realized quantities of first-class and second-class products, on the other hand. These conflicting objectives are handled by the LP metrics method. Specific constraints limit the amount of production waste as well as the greenhouse gas emissions from production and transportation to a predetermined level. The mixed-integer non-linear program by Asrawi et al. (2017) optimizes a green supply chain network at the tactical and operational management levels while splitting up the planning horizon into different time periods of equal length. The aggregate production planning takes multiple sites, products, drivers, and types of vehicles into account. The assumed network can be described by exactly three SC stages, i.e., suppliers, distribution centers, and retailers. Each site of one stage can be connected to each site of the subsequent stage by forward material flows, while transports between nonadjacent stages are excluded. Relevant decisions refer to quantities of production, transportation, and inventory. Further variables refer to the optimization of workforce. On the one hand, drivers can be selected according to their environmental behavior. On the other hand, measures for reaching the optimal number of workers are taken. A first model variant contains a single objective function and minimizes the total costs of labor, inventory, transportation, and purchasing, whereas limits for greenhouse gas emissions are considered within a constraint. A second model variant minimizes a bi-objective function that combines both total costs and greenhouse gas emissions by a multi-criteria decisionmaking method. Qiu et al. (2018) propose an MILP for production routing problems with reverse logistics and remanufacturing. The relevant network can be described by a graph that includes several customers, manufacturing depots and remanufacturing depots. Multiple homogenous vehicles at the depots can serve the customers characterized by specific pickup and delivery requests over a finite set of time periods. Remanufacturing a product is assumed to be less costly than manufacturing a new one, while returned units cannot be fully remanufactured due to an assumed rate. Period-specific decisions on the determination of manufacturing and remanufacturing quantities at the facilities, pickup and delivery quantities at the customers, and transportation routes between the customers and facilities are subordinated to the objective of minimizing the total costs of production, inventory, and routing during the entire planning horizon. The approach for green supply chain planning by Gao et al. (2018) focuses on additional customer issues. In particular, four model variants result from selecting between the consideration/nonconsideration of customers’ transportation cost and a strict carbon dioxide cap. The latter is assumed to be unalterable while being integrated in specific constraints, which restrict the overall emission that results from shipments and inventory holding. The mixed-integer non-linear models are limited to forward SCM with exactly three stages of suppliers, distribution centers, and markets. The firm produces only one kind of product, whose demand needs to be satisfied. Penalty costs occur for backordering. Continuous decision variables specify the quantities shipped on the network arcs, as well as inventory and backordering quantities at the end of the time periods. Binary decision variables are used to indicate the opening of a distribution center during a time period. Furthermore, there are integer decision variables

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determining the number of stores to be placed in the region of a market area. By optionally taking into account the customers’ transportation costs within the minimization of overall network costs, the approach strives for attracting customers by making purchases more convenient. Steinrücke and Jahr (2018) develop an MILP for the simultaneous optimization of forward and reverse distribution processes with multiple types of reuse within an industrial tool supply chain. The discrete-time approach is based on the big bucket method so as to cover multiple time periods within a medium-term planning horizon between 6 and 12 months. The general network structure spans exactly four SC stages for suppliers, plants, distribution/ preparation centers, and customers. Whereas the forward flow is organized so as to pass all the aforementioned SC stages subsequently in order to meet the given customer demand in full, the reverse flow is based on the delivered amounts from the previous time period and takes the way back through the aforementioned SC stages. In this context, the preparation centers allow for assorting the used products to four categories (recycling, remanufacturing, scrap disposal, and sales on secondary markets), which require different processes. The proportions of used products to be assigned to these processes are deterministic, so it is not possible to increase or decrease the quantities to be reintegrated in the closed loop supply chain network while taking trade-offs between economic and environmental criteria into account. The objective is to minimize the total supply chain costs of production, storage, transportation, and disposal, which need to be adjusted for sales at secondary markets. In summary, the literature review in this section reveals that the aforementioned models for discrete-time planning and scheduling in GSCM contain limitations with regard to the general network structure. In particular, potential material flows are predefined according to a fixed number of SC stages or echelons, which are additionally accompanied by a fixed and unalterable assignment of specific processes in most of the models. An adjustment of these model formulations so as to capture an arbitrarily definable number of SC stages, on the one hand, and resulting potential material flows that can flexibly skip one or more of these SC stages depending on the assignment of processes to the SC stages, on the other hand, would require essential restructuring.

4.3 4.3.1

Review of Continuous-Time Approaches Approaches without Elements of Green Supply Chain Management

Firstly, continuous-time models were developed for intraorganizational production networks without considering any issue of GSCM. For instance, the approach by Mockus and Reklaitis (1999) strives for profit maximization in a state-task network. For specific plants, it combines batch tasks (with given processing times) and

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continuous tasks (with given processing rates) that need to be scheduled and sequenced. The mixed-integer non-linear program allows for determining the timing of operations for the units (which are capable of performing one or more tasks), while processing time for each task is assumed to be known independently of the batch size. Further decisions refer to the corresponding material flow through the network. The implementation of the problem is realized by modeling a number of intervals with unequal duration. General benefits of continuous-time approaches are revealed by Floudas and Lin (2004) for process industries in the chemical sector. The authors distinguish between different processing sequences. Besides sequential processes, there are general network-represented processes that additionally allow for merging and splitting of batches and, thus, require an explicit focus on mass balancing. The resulting “global event-based models” contain continuous variables to determine the exact timings of events and binary variables that are used to assign state changes of the system (e.g., the start or end of a process) to the events. Often, problems of linearization and estimation (number of events or time points) arise in this context. Steinrücke (2007) introduces an integrated multi-modular decision model for job shop scheduling, which covers multiple jobs, and multiple production stages with several machines in each of them. Convergent, divergent, serial, or mixed structures of production can be handled. Planning includes quantity decisions on splitting up the jobs into production and transportation lots, as well as decisions on sequencing. Simultaneously, continuous-time scheduling allows for determining start and end times of all production and transportation activities exactly to the minute. There are machines for both consecutive production and simultaneous production of jobs (e.g., in processing industries). Various model variants result, as both sequence-dependent and sequence-independent setup cost and setup times can be set in any combination. Furthermore, renewable and nonrenewable resources are distinguished. For optimizing the mixed-integer linear formulations, different time-based and money-based objective functions are proposed. In contrast, Mohammadi et al. (2012) develop a slot-based approach for job shop scheduling, i.e., the authors postulate asynchronous time slots for each machine the products can be assigned to. The length of these slots and start and end times of operations that are assigned to these slots need to be determined with respect to sequence-dependent preparation times. The approach considers multiple objectives (minimizing preparation times or total tardiness) that do not take into account costs or profits due to the focus on sequencing and scheduling. Mokhtari et al. (2012) describe a flow shop model. The mixed-integer non-linear formulation includes outsourcing decisions. Besides given processing times of the manufacturer, parameters of subcontractors are considered. There are a given number of orders, while in-house capacities are limited. According to the objective of simultaneously minimizing the costs of outsourcing (processing and transportation) and costs of mean weighted flow time, operations (required to fulfill orders) need to be assigned to appropriate resources (i.e., inside or outside machines). It is possible to determine the exact start times of operations. Flexible job shop scheduling in the make-to-order industry is modeled by Gomes et al. (2013). As there are jobs that can visit machine groups more than once, the formulation includes recirculation. The MILP contains a time-based objective

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function, which weights the sum of order earliness, order tardiness, and intermediate storage times. It is necessary to determine the start time of components on machine groups for different stages that represent multiple visits. Additionally, the start times of order assembly need to be determined, as the aforementioned components need to be assembled to form an order. A model for short-term scheduling of a make-andpack plant is proposed by Baumann and Trautmann (2013). The facility performs production processes (operations on parallel, nonidentical groups of machines), storage processes (storage tanks with limited capacities), as well as packing processes (nonidentical parallel packing lines with continuous material flow). While respecting a given due date for the packed products at the end of the planning horizon, the objective is to minimize the makespan of the overall production schedule. Continuous start times of all required tasks need to be determined. An MILP for combined lot sizing and scheduling based on the block planning principle is developed by Günther (2014). Several product types are integrated into a product family that is scheduled block-wise in a predetermined sequence. Within given boundaries, both the start time and the duration of the blocks are to be determined. The time-based objective function requires the minimization of makespan, while the given demand needs to be satisfied in full. Different variants of deterministic and stochastic models for scheduling single and parallel machines, as well as flow shops and job shops, are systematized by Pinedo (2016). Although structural elements of modeling intraorganizational networks can be adopted to an interorganizational perspective in general (i.e., shops can be interpreted as SC stages, machines can be interpreted as sites), modified decision situations with different emphases on distribution and sales become relevant for network-wide coordination of sites and forward material flows in supply chains. Secondly, the following approaches extend the perspective of continuous-time scheduling within single production facilities, as they additionally take distribution problems based on vehicle routing into account. An MILP for production and distribution scheduling is proposed by Ullrich (2013), which strives for minimizing total tardiness. The author distinguishes between two subproblems. On the one hand, jobs are scheduled on identical parallel machines. In this context, continuous completion times of the jobs need to be determined based on given job-dependent processing times and due dates. On the other hand, a vehicle routing problem is included for organizing distribution. In line with the realizable start times of the tours as well as earliest and latest delivery dates that form the bounds of delivery time windows, continuous delivery times of the jobs can be determined in course of the optimal solution. Acknowledging the need of integrated scheduling of production and distribution due to reductions in inventory levels and short lead times, Chang et al. (2013) propose a mixed-integer non-linear program for integrated two-level scheduling. According to the objective function, the weighted sum of total weighted job completion time and total distribution costs is minimized. The production level is based on an identical parallel machine problem, which allows for determining the continuous completion time of the jobs. A vehicle routing problem is used for modeling the distribution level that includes decisions on the continuous delivery times of the jobs. The mixed-integer non-linear program by Low et al. (2013) allows

4.3 Review of Continuous-Time Approaches

57

for an integrated scheduling, which coordinates the processing of orders in a distribution center and the delivery to retailers. The objective function is to minimize the total completion time, while it is necessary to satisfy the given demand of the retailers in full within specific time windows. Relevant decisions comprise the composition of delivery tours of vehicles starting at distribution centers. Due to the continuous-time representation, there are continuous decision variables representing the makespan and the departure time at the retailers. As an interim summary, it becomes obvious that the distribution problems based on vehicle routing are inappropriate for long-distance transports around the globe. Furthermore, considerations are limited to the coordination of distribution with exactly one production facility. Thirdly, only few continuous-time MILP for coordinated planning and scheduling in complex supply chain networks exist. The modeling technique applied in all following approaches is based on Steinrücke (2007), i.e., shops and machines from the intraorganizational perspective can be interpreted as SC stages and sites, respectively, within interorganizational structures. The problem class of continuous-time scheduling in multi-stage SC networks was introduced by Steinrücke (2011a). The considered networks may consist of an arbitrary number of stages, whereas each stage can comprise several sites performing the same or at least homogenous processes. Therein, each site of an SC stage can supply each site of an adjacent SC stage with a homogenous product. Coordination comprises processes of production, distribution, and transportation at/between all considered network sites, while relevant quantities are detailed to production or transportation lots. As being necessary for linking worldwide activities by intercontinental transports, decisions on material flows as well as on production processes can be taken exactly to the minute (or even to the second) within a planning horizon of several months. A generalized model for simultaneous planning and scheduling with regard to possible temporary storages before and/or after site production proposes Steinrücke (2011b, 2015), which links decisions on process quantities and times. The multi-stage production-shipping and distribution-scheduling problem with multiple sites enables that either stock-free material flows within the entire network or site-specific storage times could be prescribed. Both of these models contain given due dates for delivery, while continuous-time scheduling is driven by minimizing the sum of costs adjusted for bonus payments. The latter are granted by the customer for early deliveries. The network structure is characterized by a fixed sequence of SC stages that has to be passed through and cannot be skipped by material flows and, thus, does not allow for selling intermediate products to the final customer. Alternatively, the MILP by Albrecht and Steinrücke (2017) includes financial and sales planning. While it becomes possible to determine the optimal time points of sale at different markets, on the one hand, the related monetary consequences can be captured within systemwide liquidity planning, on the other hand. Liquidity balancing is implemented to prevent insolvency of the entire network. For this purpose, all cash inflows and outflows are assigned to liquidity periods. The overall objective is to maximize the network-wide profit obtainable in the final time period. Although liquidity planning is based on the principle of time discretization, continuous-time scheduling of all

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relevant processes is applied. The latter allows for the determination of exact start and end times of all operations and financial transactions. Whereas the aforementioned study is based on non-perishable products and the assumption that sales prices are constant all over the multi-day planning horizon, Albrecht and Steinrücke (2018) focus on production and transportation in multi-stage distribution networks dealing with perishable products. The latter products are typically affected by a rapid decline in prices, which is assumed to be the driving force for continuous-time scheduling in this case. According to the general network structure, direct deliveries from a supplier to a market or indirect deliveries via warehouses of at least one distribution stage are possible. The distribution stages are arranged in a hierarchical sequence. For modeling the discrete price decline of products, quality grades are introduced. In this context, the profit-maximizing model connects delivery times that are spent on the distribution from suppliers via warehouses to markets with quality grades representing different prices. The MILP by Albrecht and Steinrücke (2020a) assumes a continuous decline in sales prices at the markets, which can be applicable to photovoltaic products. While determining the optimal sales times exactly, a strict adherence to them must be ensured for all preceding operations of production, storage, and transportation at/between different sites of the multi-stage supply chain network. In each SC stage with plants, multiple products of the same maturity can be manufactured in contrast to the aforementioned approaches, which are limited to one stage-specific product. As a consequence, the formulation initially necessitates subsequent distribution stages with warehouses that can simultaneously store and distribute products of different maturities. Albrecht and Steinrücke (2020b) propose an approach for assessing site integration into existing supply chains. In this context, continuous-time scheduling of network-wide operations allows for using time-based assessment criteria such as the minimum production time to be provided by the integrated sites. Although all models systematized in this paragraph contain approaches suitable for continuous-time scheduling of multi-stage supply chain networks, none of them contains any elements of GSCM such as recycling, disposal, or emission trading.

4.3.2

Approaches Including Elements of Green Supply Chain Management

Very few continuous-time approaches dealing with problems of GSCM exist in the literature. For instance, Kassem and Chen (2013) study the combination of forward and reverse material flows. The paper does not consider a multi-stage SC network, but a vehicle routing problem with a single depot and multiple customers. The latter require a specific quantity of new products, on the one hand, and return a specific quantity of used products, on the other hand. The customers are available during given time windows, which need to be considered for scheduling the pickup and delivery trips of the capacitated vehicles. Production or recycling processes at the

4.4 Research Gap and Contributions

59

depot, which is the common starting point and final destination of each trip, are excluded from consideration. Instead, there are decision variables specifying the load of the vehicles after leaving the depot or customer, the realized transports between the nodes, as well as the continuous start times of transports. The objective of the planning model is to minimize the total transportation cost. Due to the lack of mixed-integer optimization models for continuous-time scheduling of SC networks, a heuristic approach by Tanimizu and Amano (2016) is described in the following. It is developed for integrated production and transportation scheduling in multiobjective green supply chain network design. However, it is limited to the analysis of the relationship between a single client and supplier stage. The negotiation-based algorithm is structured as follows: The client generates a product order and sends it to the suppliers. Taking into account the new order, the suppliers strive for improving their production and transportation schedules. On this basis, they generate offers for the client. The latter can modify his order by relaxing his requirements. The negotiation process is repeated until the final acceptance or rejection of an order. Modeling takes into account continuous completion and delivery times, while penalties for delays are imposed. The integration of environmental features is limited to the consideration of carbon dioxide emissions calculated from the shipments by specific transportation vehicles. As a result, it becomes obvious that both continuous-time GSCM approaches above are not applicable to supply chain networks, which require the coordination of an arbitrarily definable number of SC stages and sites.

4.4

Research Gap and Contributions

The previous section reveals that there is no mixed-integer approach for hierarchical discrete-time and continuous-time scheduling in green supply chain management combining aspects of closed loop logistics and emission trading for multi-stage supply chain networks in the existing literature. With regard to the models to be developed in course of this monograph, the main contributions assignable to the following five categories can be stated: • Structure of GSCM networks. The proposed model formulations are not based on a fixed sequence of sites that needs to be passed through by potential material flows, but allow for arbitrarily defining the number of SC stages for production, distribution, and/or recycling. Both forward and reverse flows are not limited to sites of adjacent stages, but may also occur between nonadjacent subsequent or previous SC stages, respectively. Material flows can skip SC stages, if necessary. Moreover, bidirectional product transformation is implemented. It covers both the new products’ composition and the returned products’ decomposition from/into materials of different stage-specific maturities. Due to generic modeling, flexibility for implementing the approaches to various forms of SC networks with a specific need for different types of recovery facilities results.

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• Modularization of recycling problems. Based on a novel systematization according to the length of the planning horizon, the perishability of goods, and the resulting options of establishing recycling processes, modularized model components are developed. For short-term scheduling of non-perishables, they refer to constraints enabling internal recycling of rejects from the production/ distribution system and external recycling of returns from the markets. In this context, a combination of both is proposed for the first time. If perishables are considered alternatively, the external recycling quantities depend on the quality realized at the markets and, thus, the exact time that has been spent on the material flow path. Furthermore, new trade-offs are considered in the context of disposal. Unrecyclable quantities can be increased by recyclable ones for economic reasons to a limitable extent. • Integration of GSCM contents. To the best of knowledge, the developed formulations are the first ones, which integrate closed loop logistics and emission trading so as to capture temporal interdependencies between relevant decisions. Furthermore, the synchronized forward and reverse logistics are coordinated with an emission trading, which is not based on an unalterable emission cap valid for the entire planning horizon, but on market principles. The latter respect a possible scarcity of emission allowances by distinguishing between different buying and selling prices while capturing different types of relevant greenhouse gases by determining carbon dioxide equivalents. • Time representation in GSCM. The developed approach allows for relating GSCM decisions assignable to planning horizons of different length (which need to be distinguished according to different quality of input data) by a hierarchical framework. In contrast to the existing literature, the latter does not contain connected time periods with different levels of discretization (e.g., macroperiods and microperiods) that would represent an inappropriate form of continuation, but links decision variables and parameters of separate models instead. The latter are based on either discrete-time scheduling or continuoustime scheduling of operations according to the characteristics of the relevant GSCM problems. To the best of knowledge, the proposed short-term optimization model is the first one, which does apply continuous-time scheduling not only to production, distribution, transportation, and sales but also to recycling and disposal in a multi-stage closed loop supply chain network. By applying the modeling technique originating in Steinrücke (2007, 2015), it is possible to determine the start and end times of network-wide operations exactly within the entire planning horizon. As a main advantage of respecting time continuity in GSCM, modeling allows for capturing transient economic opportunities while respecting principles of both environmental regulations and ecological awareness. • Problem-tailored solution algorithms. Due to the integration of different problems relevant for GSCM, complex model formulations characterized by various interdependencies arise. Especially with regard to continuous-time modeling, the computability is limited to small instances used for validation. In order to obtain satisfying solutions for problems of realistic scope, relax-and-fix decomposition

References

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algorithms as well as genetic algorithms were revealed to be suitable. Differing from existing standard procedures, promising adjustments of the heuristics with regard to capturing variables, and specifying iterations are applied. Moreover, it becomes possible to avoid computational interfaces due to the structure of the algorithms, because all of the latter are implementable into the same commercial software that is used for the optimizations.

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Chapter 5

Discrete-Time Scheduling in Green Supply Chain Management

Abstract Based on existing structures of sites and capacities, new mathematical models for hierarchical scheduling of operations in generic multi-stage networks are developed in the book Scheduling in Green Supply Chain Management: A MixedInteger Approach. This chapter proposes models for discrete-time master scheduling in green supply chain management, which comprises setup and shutdown decisions for making the existing network sites available for operations in time periods. Based on aggregated material flows, modeling includes end-of-use returns and the adjustment of periodic emission caps.

Besides planning problems of network design, which are covering a long-term horizon of several years for enabling decisions on opening and closing of network sites (e.g., Albrecht 2014; Steinrücke and Albrecht 2016, 2018), discrete-time modeling is applicable to master scheduling. According to a general definition, the latter determines the quantity of final products to be processed in time periods while taking into account the coordination of different functional areas along the supply chain. Furthermore, sufficient capacities need to be provided for the operations at the existing network sites (Venkataraman and Pinto 2018, p. 643). Due to input data that does not need to meet the requirements of highest accuracy within a medium-term planning horizon (limited predictability of data), master schedules are typically based on aggregated material flows. For enabling the latter, setup and shutdown decisions need to be taken so as to make existing sites and capacities available in the time periods they are required for production, distribution, transportation, sales, recycling, or disposal. Detailing the schedules is left to continuous-time approaches (e.g., Albrecht and Steinrücke 2017, 2018, 2020a, b; see also Chap. 6). Mediumterm emission management includes coordinated adjustments of periodic standard limits by buying or selling emission allowances. The optimization of the network should be focused on maximizing weighted monetary surpluses during the entire planning horizon. If financial management is integrated (see Sect. 5.1.2), both credits and financial investments can be considered in order to balance the monetary consequences of SC decisions so as to obtain a

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6_5

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predefined structure of weighted withdrawals while preventing the insolvency of the overall system. With regard to recycling and the formation of closed loop systems, the following multi-period model formulations include returns of non-perishable final products at the end of their usage. The returns can be initiated for different reasons (Wong et al. 2016, p. 61). On the one hand, taking back the waste by the manufacturers can be mandatory due to environmental regulations. On the other hand, manufacturers can be interested in recycling for economic reasons. In both cases, end-of-use returns are expected to occur several months or even years after the sales of the products. Consequently, their handling is out of the range of a short-term planning horizon in continuous-time scheduling. Instead, quantities of product returns expected to be available in different time periods can be included in discrete-time scheduling so as to optimize their reintegration into forward material flows of GSCM.

5.1

Model Formulation

Handling end-of-use returns is organized in a supply chain network with external facilities of recycling. The latter are in charge of initiating transports of picking up the available quantities of waste from the markets. Due to the long-range life cycle, the current returns are separated from the current sales at the same sites. In general, it is assumed that only markets that are selected by the SC for sales are relevant for taking back goods. According to the specific products and the relevant legal framework, mandatory or voluntary recycling can be modeled. In both cases, the returns are handled at sites of the final network stage. There, processing (e.g., cleaning, disassembly) aims at the recovery of products of all maturities that can be used instead of new ones within the production stages. A closed loop system results after the recovered goods are redistributed and reintegrated into the production process. However, recycling can be limited to a specific share of ingoing returns according to the quality of supplies. The general structure of the SC network is described as follows (see Fig. 5.1): a. W production stages (σ ¼ 1, . . ., W ). Each of these SC stages is assigned to the production of goods in one specific maturity (e.g., raw materials in the first SC stage, intermediate products in the second SC stage, final products in the third SC stage). b. One market stage (σ ¼ W + 1). This stage contains markets, whose demand of the final product can be satisfied by sites of the preceding SC stage. c. One recycling stage (σ ¼ W + 2). Sites of the last SC stage are available for recycling of product returns obtained from markets of the preceding SC stage. The network sites can be connected by transports. Potential material flows are defined between sites of the following SC stages σ and λ (see Fig. 5.1):

5.1 Model Formulation

67

time period t production stages σ=1

time period t+1

1,m

1,m 1,1

1,1

2,n

2,n σ=2

2,1

2,1

…

… …

…

…

W,o

W,o W,1

W,1

σ=W

market stage σ=W+1 W+1,1

…

W+2,q

W+2,1

W+1,p

…

…

W+1,p

…

W+1,1

W+2,q

recycling stage σ=W+2

W+2,1

Legend: forward flows of saleable goods

σ,s

network site s of SC stage σ

σ,s

market site s of SC stage σ

reverse flows of returned goods reverse flows of recycled goods

transfer from one time period to the following one (inventory)

Fig. 5.1 General network structure in discrete-time scheduling

68

5 Discrete-Time Scheduling in Green Supply Chain Management

planning horizon

before planning horizon first time period

second time period

…

last time period points

t = −1 t = 0

t =1

t=2

…

t = tE − 1

t = t E in time

Fig. 5.2 Time representation in discrete-time scheduling

(a) Material flows of forward logistics. Each site of an SC stage with production (σ ¼ 1, . . ., W) can supply each site of an adjacent stage (λ ¼ σ + 1) with stagespecific products. (b) Material flows of reverse logistics. The following two types can be distinguished: • Flows of returned goods. Material flows are directed from market sites (σ ¼ W + 1) to sites of the final SC stage (λ ¼ W + 2). • Flows of recycled goods. Material flows are directed from sites of the final SC stage (σ ¼ W + 2) to the SC stage of the products’ original manufacturing (λ ¼ 1, . . ., W ). The medium-term planning horizon is divided into time periods of equal length and needs to be chosen so that each of these periods can be considered as an individual planning horizon itself within subsequent continuous-time scheduling. For respecting the chronology of events while merging the monetary consequences of SC decisions linked to the time periods, the latter are separated by the time points t ¼ 0, . . ., tE (see Fig. 5.2). The initial site state available before the beginning of the planning horizon is denoted by t ¼  1. For reasons of coordination, different groups of decisions and their monetary consequences are assigned to the aforementioned points in time. The sets of time points relevant for decisions on site states and operations, as well as for determining surpluses, are introduced: – Site state. Besides the initial state describing the sites’ availability before the beginning of the planning horizon, decisions on the site state can be taken in time points denoting the beginning of each time period the planning horizon consists of. Consequently, the set of time points representing site states is T ≔ {1, 0, . . ., tE  1}. The monetary consequences of these decisions are assigned to the same points in time. – Operations. Based on the current site state, decisions on operations during a time period (their exact scheduling is not required within the discrete-time consideration) are to be taken at the beginning of the same time period, so relevant time points for the decisions on operations can be described by the set T ≔ {0, . . ., tE  1}. The monetary consequences of operations realized during a specific time period are assigned to the time point that denotes the beginning of the following time period for reasons of chronology.

5.1 Model Formulation

69

– Surpluses. Every specified point in time of the planning horizon is used for merging the assigned monetary consequences for determining surpluses (that may even be negative in case that there is no integration of financial planning; see Sect. 5.1.2). The set of relevant time points for determining the surpluses from merging the monetary consequences of decisions on site state and operations is T+ ≔ {0, . . ., tE}. Considerations on the site state concern the availability of the network sites in the time periods of the planning horizon as well as an appropriate provision of capacity. In particular, site availability requires setting up the sites for operations, e.g., by preparing the machines or instructing the workforce. If sites have been set up before the planning horizon (which can be considered by the initial site state in t ¼  1), an initial availability is guaranteed. Otherwise, setups can be initiated at the beginning of each time period. In both cases, the site is available for operations (which entails periodic costs, e.g., energy cost for the machines’ operational readiness) during the following time period(s) until the site is shut down or until the planning horizon ends. Analogously to setups, shutdown decisions can be taken at the beginning of the time periods. As a consequence, a site is no longer available for operations until the planning horizon ends or another setup is conducted. For all available sites, the capacity to be provided during the time periods needs to be determined. Within a medium-term planning horizon, it is only possible to consider capacity adjustments that do not require essential restructuring (e.g., arranging periods of overtime for the existing workforce). These adjustments can be modeled by capacity profiles, which limit sequences of the maximum capacity available at the sites in subsequent time periods to the feasible ones. In general, increasing, constant, decreasing, or even arbitrarily structured sequences can be modeled. Decisions on the selection of a capacity profile are required at the point in time a site is initially available (i.e., at the beginning of the planning horizon or at the beginning of the time period the site is set up). Inherently, all operations at the network sites (depending on the SC stage the site belongs to) are restricted to the availability of the site and the provision of capacity during the relevant time period. The sites being available at the end of the planning horizon are not shut down as their continuing availability can be considered in the initial site state of the upcoming horizon. Operations include aspects of processing at/between available network sites. Forward SCM considers the manufacturing of stage-specific products, which is driven by a given demand of finished products at the market sites. In this context, it is assumed that the product maturity increases from stage to stage. Whereas primary goods are obtained in the first SC stage (e.g., raw materials obtained from mining), secondary goods are produced beginning with the second SC stage. In context of straightforward modeling of discrete-time master scheduling, the quantity of goods to be delivered from the previous SC stage for manufacturing one unit of the good belonging to the current SC stage is determined by given bills of materials. Assuming that each site of an SC stage can supply each site of a following adjacent SC stage, a predefined number of potential material flows result. Additional flows

70

5 Discrete-Time Scheduling in Green Supply Chain Management

belong to reverse SCM, which is initiated by returnable goods that are expected to be available at the market sites during the time periods the planning horizon consists of. Modeling does not include a connection between quantities of return and demand, as the medium-term consideration is limited to end-of-use returns that occur after an unpredictable period of use by customers. Acknowledging that the collection can be voluntary or mandatory according to specific environmental regulation for markets that are delivered by the SC with goods in forward SCM, returnable quantities are transported to external sites of the final SC stage. Each of the recycling sites is assumed to be able to recover products that stem from each of the markets. Reusable goods can be obtained at the sites after the recovery treatment as specified by given coefficients. The formation of closed loops is forced by the fact that recycled products need to be reintegrated into forward SCM for environmental reasons and, thus, they are redelivered to the sites of their original manufacturing. For all transports between network sites in forward and reverse logistics, different capacitated vehicles can be used. In order to maintain flexibility at the production sites, which may be required to balance overflows of recycled goods in specific time periods, inventory management is included. The latter allows for transferring stagespecific kinds of goods from one time period to the following while taking into account given costs of inventory holding. Even initial stocks being available at the beginning of the planning horizon can be considered. The model formulation is based on the following sets and indices: Firstly, there are index sets for SC stages, sites, and capacities. It becomes obvious from Fig. 5.1 that the overall number of SC stages depends on the number of production stages W. Forward flows are initiated from the production stages, whereas reverse flows have their origin in sites of the market or recycling stage. Each SC stage may comprise an arbitrary number of sites. The set of capacity profiles describes available alternatives for sequences of maximum capacities at a site depending on the time point of the profile’s assignment (before the planning horizon or at the beginning of each time period the planning horizon consists of). Secondly, discrete-time modeling requires the definition of a set that includes the time points relevant for different types of decisions (site state, operations, surpluses). Finally, a set of relevant anthropogenic greenhouse gases needs to be defined for emission management. The following parameters need to be set according to the specific problem: Costs and revenues. Production and recycling sites entail fixed costs for their availability in each period they are operated by the SC. With regard to their site state, there are period-dependent setup and shutdown costs. Fixed capacity costs occur according to the selection of a capacity profile. Furthermore, there are period-dependent costs of operations. With regard to the sites, variable costs per processed unit need to be considered for production, recycling, and inventory holding. With regard to transports between the sites, there are fixed and variable

5.1 Model Formulation

71

costs of transportation. The sales of each finished product at a market result in revenues. Capacities. The maximum production or recycling capacity at a site in a time period that results from the selection of a capacity profile restricts the manufactured or recovered product quantity, respectively. Different types of technical or human resources can be taken into account. Maximum throughput capacities are considered regardless of the selected capacity profile. Furthermore, there are maximum transportation capacities that limit the quantity of goods to be delivered between network sites of two different SC stages in both forward and reverse logistics according to given features of available vehicles (e.g., volume, area). Coefficients. At first, there are coefficients specifying the input-output relation between the products of adjacent production stages, which can be taken from relevant bills of materials. Another coefficient describes the possible decomposition of a final product returned from the markets into goods originally manufactured in the production stages of the same SC. With regard to processing at the external sites of the final SC stage, there is a coefficient for the share of ingoing products that are recyclable due to their expected quality. Environmental parameters. First of all, GSCM requires the specification of the occurrence of emissions during different types of processing. Production, recycling, and transportation are assumed to entail a specific amount of different greenhouse gas emissions per product unit. They are made comparable by conversion into a carbon dioxide equivalent. The emission cap (see Sect. 3.3.1) results from free allocation of allowances by the government. It represents the standard emission tolerance for each time period the planning horizon consists of. Internal emission management allows for adjusting this bound by trading emission allowances that are characterized by specific volumes as well as buying/ selling prices. According to given market conditions, the maximum number of tradable allowances can be fixed. Other parameters. For each market, both the demand and the returnable quantity of the final product are given for each time period. In order to specify the initial site state, there is a binary parameter indicating whether the site has already been set up before the planning horizon. The monetary surpluses to be calculated in different time points can be weighted according to the network managers’ preferences. Additional technical parameters are defined for purposes of modeling [see constraints (5.7) and (5.9)]. The following decisions need to be taken in course of the optimization of the overall system: (a) Decisions with regard to the sites operated by the SC • Usage of potential network sites for production, storage, and recycling – Setup or shutdown at the beginning of time periods – Availability and capacity provision during time periods • Processed quantities of stage-specific products during time periods

72

5 Discrete-Time Scheduling in Green Supply Chain Management

• Inventory holding of stage-specific products between time periods (b) Decisions with regard to transportation between the sites • Realization of potential material flows between the network sites during time periods • Transportation quantities of stage-specific products during time periods (c) Decisions with regard to the market sites • Selection of potential markets for demand satisfaction during time periods • Quantities delivered to the markets (scope of demand satisfaction) during time periods (d) Environmental decisions • Number of bought or sold emission allowances at the beginning of time periods

5.1.1

Networks with Recycling and Emission Trading

The mixed-integer linear program for discrete-time scheduling in networks with recycling and emission trading can be formulated as follows:

5.1.1.1

Objective Function Max Φ X Φ¼ υt  ðvot  vst  vet Þ

ð5:1Þ

t2T þ

vo0 ¼ vstE ¼ vetE ¼ 0

ð5:2Þ

According to the objective function (5.1), the sum of weighted surpluses obtained during the multi-period planning horizon is to be maximized. For its determination, the monetary consequences resulting from three main groups of decisions are merged in the points in time they are assigned to. Site state decisions and environmental decisions are possible at the beginning of each time period, and the resulting payments are assigned to the same time points t ¼ 0, . . ., tE  1. Operational decisions are also relevant for these points in time. However, as affecting the SC operations in the following time period, their monetary consequences are assigned to the beginning of the next period t ¼ 1, . . ., tE. Acknowledging the exceptions implemented by constraint (5.2), the residuals are determined for each point in time t ¼ 0, . . ., tE. The period-specific weighting factor can be chosen according to the time preference of the network managers. Even the usage of discounting factors based on external opportunities would be possible.

5.1 Model Formulation

vst ¼

W þ2 X

X

σ¼1

s2Sσ

73

CSst  sust þ CDst  snst þ

t X X τ¼0

! CC sct  ysct ; t 2 T

ð5:3Þ

c2C sτ

σ6¼Wþ1

The monetary consequences resulting from decisions on the state of the sites belonging to the production or recycling stage (i.e., each SC stage except the market stage) are determined by Eq. (5.3) at the beginning of each time period (i.e., in time points t ¼ 0, . . ., tE  1): Firstly, there are fixed setup costs that occur if the site is made available. Secondly, shutdown costs need to be considered if the site’s availability ends at the beginning of a specific time period. Thirdly, capacity costs are taken into account. They occur at the beginning of each time period a site is available during the planning horizon and are determined by the selected capacity profile. The latter does depend not only on the specific network site but also on the point in time it is assigned to the site. In general, all time points up to the current one (τ ¼ 0, . . ., t) are relevant for the assignment of a capacity profile. vet ¼ ebt  BE t  est  SE t ; t 2 T

ð5:4Þ

The balance of payments resulting from environmental decisions on emission trading is obtained by Eq. (5.4). At the time points t ¼ 0, . . ., tE  1, it is possible to affect the given cap that is valid for all sites of the entire SC network during the following time period by buying or selling emission allowances. Period-specific buying and selling prices that take into account the current market conditions can be used as parameters. By integrating non-linear functions, it would even be possible to integrate prices that consider the scarceness of allowances (i.e., increasing prices at an increasing number of allowances to be bought by the SC or decreasing prices at an increasing number of allowances to be sold by the SC). vot ¼

X X

Rs,t1  xqs,t1

q2SW s2SWþ1



W X X σ¼1



X

X

PV s,t1  pr s,t1 þ PI s,t1  ins,t1 

s2Sσ

RV s,t1  qr s,t1

s2SWþ2

X X TF sq,t1  tpsq,t1 þ TV sq,t1  xsq,t1

ðσ, λÞ2K s2Sσ q2Sλ



W þ2 X σ¼1

t XX X s2Sσ τ¼0

c2C sτ

CAs,t1  ysc,t1 

X

CAs,t1  yms,t1 ; t 2 T þ , t  1

s2SWþ1

σ6¼Wþ1

ð5:5Þ

74

5 Discrete-Time Scheduling in Green Supply Chain Management

The monetary consequences of operations are described by (5.5). The specific equations need to be determined for the time points t ¼ 1, . . ., tE. However, as the payments result from the operations during the previous time period, the equations refer to the decisions that are assigned to the previous points in time t  1. In particular, there are different types of decisions relevant for payments: Firstly, the revenues from selling the final product are to be calculated. They depend on the quantities delivered to the sites of the market stage. Secondly, there are processing costs at the sites of the production stages. They include variable costs depending on the manufactured quantities, on the one hand, and the quantities hold in inventory, on the other hand. Thirdly, for the sites of the final SC stage, variable costs of recycling are added. The latter occur for each unit of the recyclable quantity that is taken back from the market sites. Fourthly, transportation costs need to be taken into account. Each realization of potential material flow paths between two sites of different SC stages by means of a transport entails both fixed and variable costs. Fifthly, the costs for maintaining the availability of network sites of production and recycling stages are calculated. For this purpose, a certain amount of fixed costs is charged to a time period if a site is currently available in a capacity profile (that has been assigned to the site at the beginning of the current or a previous time period). In case of a site’s unavailability during a period, no costs are added. Finally, the selection of a market for both sales and returns entails fixed costs.

5.1.1.2

Constraints of Site State

snst þ

X X

ysct  1; 8σ ¼ 1, . . . , W, W þ 2; s 2 Sσ ; t 2 T

ð5:6Þ

τ2T  c2C sτ τt

Constraint (5.6) ensures that the state of a site belonging to a production or recycling stage is unambiguously defined within a time period. Maximally one capacity profile can be assigned to an available site. It becomes obvious from the constraint that this assignment is possible in each point in time up to the current one (τ  t). Furthermore, the availability of a site in a time period is excluded if this site has been shut down at the beginning of the current time period. In this case, even the assignment of the capacity profile ends. sust 

P P τ2T  c2Csτ

ysct  ηt 

P P τ2T  c2C sτ

ysc,t1  ð1  ηt Þ  IAs ;

8σ ¼ 1, . . . , W, W þ 2; s 2 Sσ ; t 2 T

ð5:7Þ

Constraint (5.7) defines prerequisites for the setup of a production or recycling site at the beginning of a time period. At the beginning of the planning horizon (t ¼ 0), the technical parameter η0 ¼ 0 is valid. Consequently, the setup is forced if the site has not been available before the planning horizon (i.e., is not part of the

5.1 Model Formulation

75

initial site state), but is operated in a specific capacity profile during the first time period. Analogously, the setup of the site is forced at the beginning of each following time period (η1 ¼ . . . ¼ ηtE 1 ¼ 1 is valid) if the site has not been available during the previous time period but is available during the current time period. The availability can be determined regardless of the capacity profile assigned to the site. In case the aforementioned conditions are not met, the constraint becomes obsolete, and the variable on the left side of the constraint needs to be restricted by the remaining system. sust 

X

ysct ; 8σ ¼ 1, . . . , W, W þ 2; s 2 Sσ ; t 2 T

ð5:8Þ

c2C st

Due to (5.8), the setup of a production or recycling site at the beginning of a time period requires that the site is available for operations during the following time period. Again, this availability can be determined regardless of the capacity profile. If the site is not available during a time period otherwise, its setup at the beginning of the time period is excluded. snst  ηt 

X X τ2T 

ysc,t1 þ ð1  ηt Þ  IAs ; 8σ ¼ 1, . . . , W, W þ 2; s 2 Sσ ; t 2 T

c2C sτ

τt1

ð5:9Þ

According to constraint (5.9), shutdown decisions of sites belonging to the production stages or the recycling stage are enabled. Due to η0 ¼ 0, shutdowns are possible in the time point t ¼ 0 for sites that have been available before the planning horizon starts and, thus, are part of the initial site state. Considering the case of η1 ¼ . . . ¼ ηtE 1 ¼ 1, sites can also be shut down at the beginning of each following time period (i.e., in the time points t ¼ 1, . . ., tE  1) if the site has been available (regardless of its capacity profile) during the previous time period. The constraint becomes redundant if previous availability is not applicable. ysc,t1  snst þ ysct ; 8σ ¼ 1, . . . , W, W þ 2; s 2 Sσ ; c 2 Csτ ; τ 2 T  , τ  t  1; t 2 T, t  1

ð5:10Þ

Constraint (5.10) refers to the continuation of capacity profiles once they have been assigned to a site of a production or recycling stage. In particular, the availability of a site in a specific profile (that has been assigned to the site in τ 2 T, τ  t  1) requires that this site is available in the same profile during the following time period if it has not been shut down before. Once it is shut down, it is no longer available until it is set up for operations again.

76

5 Discrete-Time Scheduling in Green Supply Chain Management

IAs  sns0 þ

X

ysc0 ; 8σ ¼ 1, . . . , W, W þ 2; s 2 Sσ

ð5:11Þ

c2C s,1

Constraint (5.11) forces an appropriate consideration of the initial site state at the beginning of the first time period the planning horizon consists of. In particular, if the production or recycling site has been set up before the planning horizon (IAs ¼ 1), there are two options for decisions in t ¼ 0. On the one hand, the site can be shut down immediately if it is no longer needed for processing in the SC network. On the other hand, its availability can be continued. In this case, the selection of a capacity profile from the set Cs,  1 is required. In case the site is not part of the initial site state (IAs ¼ 0), the constraint becomes redundant.

5.1.1.3

Constraints of Forward Flows X

xqst ¼ Bσ  pr st ; 8 σ ¼ 2, . . . , W; s 2 Sσ ; t 2 T

ð5:12Þ

q2Sσ1

Constraint (5.12) considers the production processes at sites of the SC stages σ ¼ 2, . . ., W. It becomes obvious that the realized production quantity determines the quantities that need to be supplied to the production site from sites of the previous SC stage according to given bills of materials. The latter are considered by a parameter for stage-specific input-output relations. inst ¼ ins,t1 þ prs,t1 þ

X q2SWþ2

xqs,t1 

X

; 8 σ ¼ 1, . . . , W; s 2 Sσ ; t 2 T, t  1

q2Sσþ1

ð5:13Þ

For discrete-time scheduling on the medium-term level, inventory holding is implemented. It enables transferring product items from one time period to the following. According to (5.13), the inventory quantity available at a site in a time point t ¼ 1, . . ., tE  1 is composed of the inventory being available at the beginning of the previous period, adjusted by ingoing and outgoing quantities of the same product during the current period. On the one hand, the inventory is increased by produced quantities and, furthermore, by recycled quantities that are directed back to the SC stage of their origin after the treatment within the final SC stage. On the other hand, a decrease of the inventory by deliveries to sites of the subsequent SC stage (which can be another production stage or even the market stage) is to be taken into account. An initial stock of inventory being available at the beginning of the planning horizon can be considered by the parameter ins0.

5.1 Model Formulation

X

77

xsqt  pr st þ inst þ

q2Sσþ1

X

xqst ; 8 σ ¼ 1, . . . , W; s 2 Sσ ; t 2 T

ð5:14Þ

q2SWþ2

Constraint (5.14) restricts the overall quantity that is delivered from a site of a production stage to sites of a subsequent SC stage within a specific time period. It cannot exceed the sum of current production and inventory quantities, which can be increased by the quantities of recycled products obtainable from different sites of the final SC stage. X

xqst  Dst  ymst ; 8s 2 SWþ1 ; t 2 T

ð5:15Þ

q2SW

If a market is selected for demand satisfaction according to constraint (5.15), it can be supplied with finished products manufactured at different sites of the preceding SC stage. However, the given demand does not need to be satisfied in full. Delivery shortages, which cannot be balanced by backordering in a following time period according to the assumptions, may occur due to economic or operational reasons.

5.1.1.4

Constraints of Reverse Flows X

xsqt ¼ DQst  ymst ; 8 s 2 SWþ1 ; t 2 T

ð5:16Þ

q2SWþ2

The selection of a market entails the duty of handling the entire quantity of endof-use returns available at the site during the time period. By constraint (5.16), the network is assumed to be forced (e.g., by environmental legislation) to distribute the amounts to different sites of the final SC stage for further processing. By slight modifications, voluntary returns could be modeled instead. X

xqst ¼ qr st ; 8 s 2 SWþ2 ; t 2 T

ð5:17Þ

xsqt ; 8 s 2 SWþ2 ; λ ¼ 1, . . . , W; t 2 T

ð5:18Þ

q2SWþ1

βs  qr st  BBλ ¼

X

q2Sλ

The overall quantity which needs to be handled at a site of the final recycling stage within a time period is determined by (5.17). It can be composed of various amounts of the final product that are directed to the site from different markets during the same time period. According to (5.18), the available quantity is adjusted by a given factor that determines the share of ingoing amounts, which is expected to be unrecyclable due to insufficient quality. The remaining share is assumed to be decomposable into different products originally manufactured in the production

78

5 Discrete-Time Scheduling in Green Supply Chain Management

stages of the SC network. A given coefficient on the left-hand side of the equation specifies the relevant stage-specific quantities of products that can be obtained after the recycling of one unit of the return. According to the right-hand side, the material flows directed back to the production stages can be distributed to different sites. At the destinations, recycled quantities can replace the production of new items.

5.1.1.5

Capacity Constraints

βs  qr st 

t X X τ¼0

pr st 

t X X τ¼0

inst þ

X q2SWþ2

RC sct  ysct ; 8 s 2 SWþ2 ; t 2 T

ð5:19Þ

c2Csτ

PC sct  ysct ; 8 σ ¼ 1, . . . , W; s 2 Sσ ; t 2 T

ð5:20Þ

c2Csτ

xqst  XC st 

t X X τ¼0

ysct ; 8 σ ¼ 1, . . . , W; s 2 Sσ ; t 2 T ð5:21Þ

c2C sτ

xsqt  TCsqt  tpsqt ; 8 s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K; t 2 T

ð5:22Þ

Constraint (5.19) restricts the recycling at sites of the final SC stage during specific time periods to the maximum capacities that result from the selection of a capacity profile. The latter can be assigned to a site at the beginning of the planning horizon in case that the site has been available before the horizon (i.e., it is part of the initial site state) or at the beginning of the time period a site is set up for operations during the horizon. Consequently, all assignments up to the current time period (τ ¼ 0, . . ., t) need to be taken into account on the right-hand side of the constraint, while a unique assignment of a profile to a site within a time period is ensured by (5.6). Analogously, constraint (5.20) refers to the sites of the production stages. Here, the manufactured quantity of a stage-specific product during a time period is restricted to the maximum production capacity that results from the profile. Respecting the maximum throughput capacity at sites of the production stages according to (5.21) ensures that the sum of the recycled quantities obtained from different sites of the final SC stage increased by the inventory quantities does not exceed a given upper bound. The latter is assumed to be valid for available sites regardless of the selected capacity profile that exclusively affects production. Restrictions on transports are considered within (5.22), which covers all possible relationships between two different sites of the SC network in general. These relationships can be described by potential material flows of both forward and reverse logistics. Transportation capacity during a time period is restricted to the maximum capacity of the vehicle to be used on the path if the material flow is realized. The latter is indicated by the binary variable on the right-hand side of the constraint.

5.1 Model Formulation

5.1.1.6

79

Constraints of Emission Trading 0

W P P

P

1

pr st  EPpst þ qr st  EDpst C B B σ¼1 s2Sσ C s2SWþ2 CE p B C P P P @ A p2P þ xsqt  ET psqt

X

ð5:23Þ

ðσ, λÞ2K s2Sσ q2Sλ

 EM t þ ebt  EV  est  EV; 8t 2 T ebt  BM t ; 8t 2 T

ð5:24Þ

est  SM t ; 8t 2 T

ð5:25Þ

Additionally, there are constraints for regulating emission trading within the discrete-time planning horizon. In particular, (5.23) accounts all emissions of different greenhouse gases that are caused by the SC network during a specific time period on the left-hand side: Firstly, there are emissions from production, which occur per unit of the stage-specific products that are manufactured at sites of the production stages. Secondly, handling each unit of end-of-use returns at sites of the recycling stage causes additional emissions. Thirdly, further emissions need to be taken into account for each unit transported between two different sites of the SC network in terms of forward or reverse logistics. In order to calculate the overall emission volume, the total volumes of different greenhouse gases need to be converted into a carbon dioxide equivalent by the use of gas-specific weighting factors. Inherently, this factor needs to be one for carbon dioxide itself. The overall amount of emissions is limited by the given emission cap, which is also measured in carbon dioxide equivalents. In accordance with a system of economic incentives, the SC is assumed to participate in emission trading, which allows to sell indivisible emission allowances in times of a lower deviation from the standard limit and to buy these allowances in times of a transgression. In line with given market conditions, the adjustment of the emission cap can be restricted. According to (5.24) and (5.25), the maximum number of allowances to be bought or sold, respectively, can be limited.

5.1.1.7

Constraints of Domain Definition

The model is completed by the following non-negativity constraints (5.26)–(5.28), binary constraints (5.29)–(5.32), and integrality constraint (5.33). pr st , inst  0; 8σ ¼ 1, . . . , W; s 2 Sσ ; t 2 T

ð5:26Þ

xsqt  0; 8 s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K; t 2 T

ð5:27Þ

80

5 Discrete-Time Scheduling in Green Supply Chain Management

qr st  0; 8 s 2 SWþ2 ; t 2 T sust , snst 2 f0, 1g; 8 σ ¼ 1, . . . , W, W þ 2; s 2 Sσ ; t 2 T

ð5:29Þ

tpsqt 2 f0, 1g; 8 s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K; t 2 T

ð5:30Þ

ysct 2 f0, 1g; 8σ ¼ 1, . . . , W, W þ 2; s 2 Sσ ; c 2 Csτ ; τ 2 T  , τ  t; t 2 T

5.1.2

ð5:28Þ

ð5:31Þ

ymst 2 f0, 1g; 8s 2 SWþ1 ; t 2 T

ð5:32Þ

ebt , est 2 f0, 1, 2, . . .g; 8 t 2 T

ð5:33Þ

Integration of Financial Planning

Going beyond the usage of weighting factors in constraint (5.1), the network managers can be interested in adjusting the structure of withdrawals. The latter is especially relevant for SC networks with a lack of equity that cannot tolerate negative surpluses during specific time periods of the planning horizon. In this context, liquidity balancing can be applied to prevent insolvency of the overall system. The following model captures financial transactions with specific durationdependent interest rates, on the one hand, and market-based limits, on the other hand. By that, a discrete-time concept of liquidity planning is implemented for optimizing the structure of an SC network within a medium-term planning horizon while taking the monetary consequences of recycling and emission trading into account. The integration requires the coordination of various payments arising from both the operational and the financial level with periodic withdrawals (see Fig. 5.3). For coordination, different types of payments need to be merged. Regarding the operational level, there are payments resulting from adjustments of site state (setups, shutdowns, provision of capacity) and emission trading (buying or selling allowances) at the beginning of each time period. Based on the realized settings, operations (production, storage, sales, recycling, transportation) are possible at/between available network sites during the following time period. As these events cannot be scheduled exactly to the minute due to uncertainties arising from the length of the assumed medium-term planning horizon, the payments of operations and site availability are assigned to the point in time representing the beginning of the following time period. While it is not possible to connect different payments of the operational level directly, exactly two payments are linked in financial transactions. The latter are assumed to be based on given interest rates while avoiding speculation. On the one hand, there are financial investments. A specific amount can be invested and is repaid in full at maturity plus interest, as typical for a zero bond. On the other hand, there are loans available. Receiving a credit entails the full redemption of the amount plus interest at maturity, which is also referred to as final date redemption. All financial transactions may be limited due to market conditions. For merging all

5.1 Model Formulation operational decisions

t = –1

initial site state

t =0

set-up, shut-down, capacity adjustment

81 financial decisions

start of credits, investments

withdrawal

start and/or end of credits, investments

withdrawal

production, storage, sales, recycling, transportation set-up, shut-down, capacity adjustment

t =1

production, storage, sales, recycling, transportation

…

…

…

…

set-up, shut-down, capacity adjustment

t = tE – 1

start and/or end of credits, investments

withdrawal

end of credits, investments

withdrawal

production, storage, sales, recycling, transportation

t = tE

time

Fig. 5.3 Coordination of operational and financial decisions in discrete-time scheduling

payments of both the operational and the financial level, the time points t ¼ 0, . . ., tE are relevant. In each of the latter, liquidity needs to be balanced while satisfying the network managers’ periodic claims of withdrawals. Respecting the financial balance within each and every time period prevents insolvency of the entire network. The model proposed for discrete-time scheduling (see Sect. 5.1.1) is to be extended by index sets that distinguish the two main types of financial transactions (i.e., credits, investments). In consequence, new parameters and variables need to be introduced. Financial parameters. Dependent on the transaction type (credit or investment) and duration (from the beginning of one time period to the beginning of a subsequent time period or even the end of the medium-term planning horizon), interest rates are given. A limit for each single financial transaction can be set. Moreover, it is possible to limit the overall amount of all financial transactions starting from the beginning of the same time period. The decisions on SC operations (see Sect. 5.1) can be complemented as follows: (e) Decisions with regard to liquidity planning – Realization of available credits and investments – Extent of realized credits and investments (amount of financial transactions) – Assignment of realized credits and investments to time periods

82

5 Discrete-Time Scheduling in Green Supply Chain Management

After replacing the previous objective function for maximizing the weighted overall surpluses (5.1) by (5.34), the financial constraints (5.35)–(5.39) can be added to mixed-integer linear programs for discrete-time scheduling in networks with recycling and emission trading. Max Φ X Φ¼ υt  qt

ð5:34Þ

t2T þ

In the context of liquidity planning, the objective function (5.34) changes into the maximization of weighted withdrawals. The latter can be taken from the residual liquidity in each time point t ¼ 0, . . ., tE. The weighting factors can be chosen according to the preferences of the network managers. By slight modifications of the objective function, it is possible to obtain consistent withdrawals or a single withdrawal at the end of the planning horizon. The distribution of withdrawals among network partners can be part of further research. vot  vst  vet þ

XX o2O τ2T þ τ>t

f iotτ  κo 

XX

ð1 þ ioτt Þ  f ioτt  κ o  qt ; t 2 T þ

o2O τ2T þ τ t) can be taken. If there are liquidity surpluses not being used for the withdrawals to the network managers on the other hand, financial investments can be taken analogously. In this context, a technical parameter is introduced to obtain the required algebraic signs for payments of credits (κ1 ¼ 1) and investments (κ2 ¼  1). Based on the realized settings, SC operations can take place during the following time period, while the resulting surpluses are assigned to time point t ¼ 1. They are merged with the new balance of payments from site state and environmental decisions, before the withdrawal qt results as a residual. This is the same at all following time points t ¼ 2, . . ., tE. Until the end of the planning horizon (i.e., time point tE), all credit and investment transactions need to be completed. fiotτ  FLotτ ; 8 o 2 O; t 2 T; τ 2 T þ , t < τ

ð5:36Þ

5.2 Numerical Analysis

83

X

fiotτ  FPot ; 8o 2 O; t 2 T

ð5:37Þ

τ2T þ τ>t

Limits for financial transactions can be justified by market conditions. According to constraint (5.36), the amount of a single financial transaction can be restricted to an upper bound. The latter can be set separately for credits and investments and even for transactions with different durations. Due to (5.37), it is possible to set a limit for all financial transactions of a kind that are simultaneously starting from the same point in time (e.g., credit lines). Non-negativity constraints (5.38) and (5.39) must be added.

5.2

qt  0; 8 t 2 T þ

ð5:38Þ

fiotτ  0; 8o 2 O; t 2 T; τ 2 T þ , t < τ

ð5:39Þ

Numerical Analysis

The model for discrete-time scheduling in networks with recycling and emission trading (Sect. 5.1.1) taking into account financial planning (Sect. 5.1.2) is validated by the following illustrative small-scale example of an SC network: The medium-term planning horizon spans three time periods. Consequently, the time points t ¼ 0, 1, 2, 3 are considered. The initial site state is represented by t ¼  1. The network is assumed to consist of four subsequent SC stages (W ¼ 2). Both the first and the second SC stage σ ¼ 1, 2 include production sites. The third SC stage σ ¼ 3 is composed of market sites. The final SC stage σ ¼ 4 contains recycling sites. There are four potential sites in each SC stage. Each production and recycling site can be operated in two capacity profiles that are distinguished by different capacity sequences and costs: • Both capacity profiles provide a maximum capacity that allows for producing 20 units or recycling 10 units in the first time period the profile is assigned to a site. • The first capacity profile provides the same maximum capacity in each following time period until the site’s shutdown or the end of the planning horizon, whereas the second one entails a reduction of the maximum capacity according to a given pattern. The sites 1,1; 1,2; 2,3; 4,2; and 4,4 have been set up before the planning horizon, which is considered in the initial site state. A capacity profile needs to be assigned to these sites at the beginning of the first time period.

84 Table 5.1 Quantities of demand and return

5 Discrete-Time Scheduling in Green Supply Chain Management t¼0 Site 3,1 Site 3,2 Site 3,3 Site 3,4

10, 22

t¼1 20, 0 10, 20 15, 30

t¼2 10, 0 10, 10

Two products of different maturity are considered. They are assumed to be arbitrarily divisible. In particular, all sites of the first SC stage can manufacture intermediates, whereas all sites of the second SC stage are able to manufacture final products that can be used to satisfy the demand at the markets. According to a simple bill of materials, exactly two units of the intermediate product are required for one unit of the final product. Regarding the latter, the following quantities of demand and return are given (see Table 5.1). Additional parameters for revenues, fixed and variable costs of operations, and capacities were set to specific values or generated randomly with respect to given bounds. In the context of GSCM, the following specifications need to be taken into account. They refer to the recycling of end-of-use returns and the implementation of emission trading: – The end-of-use returns available at the market sites in the time periods need to be taken back completely by the SC in case that the market is selected for demand satisfaction. It is expected that only 50% of the quantities returned to sites of the final SC stage can be recycled there due to quality reasons. – After recycling 1 unit of a returned final product, exactly 0.5 units of the intermediate product (belonging to the first SC stage) and 0.5 units of the final product (belonging to the second SC stage) are obtained. These products are immediately directed to the SC stages of their origin in order to replace the manufacturing of new units. – The consideration on emissions is limited to a single greenhouse gas, which is carbon dioxide. Inherently, the factor for converting its volume into carbon dioxide equivalents is 1. Manufacturing a unit of an intermediate or a final product in one of the first two SC stages causes 1 ton of emission. During the recycling of 1 product unit, 0.8 tons are emitted. For the transportation of a product unit between two different network sites, the emission of 0.5 tons is assumed. The emission cap in each time period of the medium-term planning horizon allows for emitting an overall volume of 100 tons. – In the context of emission trading, allowances can be bought or sold. By that, an emission volume of 100 tons is added to or subtracted from the SC’s emission cap during a time period, respectively. The buying price of an allowance ($ 500) exceeds its selling price ($ 400). A maximum number of two allowances can be bought or sold during a time period of the medium-term planning horizon. Financial planning ensures that the residual liquidity cannot fall below zero before and after each time period while taking into account the withdrawals. The

5.2 Numerical Analysis Table 5.2 Interest rates for credits and investments

85 From . . . to . . . t¼0 t¼1 t¼2

t¼1 5%, 4%

t¼2 10%, 9% 5%, 4%

t¼3 15%, 14% 10%, 9% 5%, 4%

latter are assumed to be of equal weights, as there are no time preferences of the network managers. For liquidity balancing, the following financial transactions are available (see Table 5.2). According to assumed credit lines, a single credit cannot exceed $ 5000, while there is an overall limit of $ 8000 for all credits taken by the SC during the planning horizon. There are no restrictions on the maximum volume of financial investments. For computing the illustrative example, a cluster of the University of Greifswald was used. It consists of 78 machines in the compute partition with 2 Intel Xenon E5-2623 v3 Quad-Core-CPUs, 3.00 GHz, 8 GT/s, and 64 GB RAM in each of them. Exactly 16 threads are available per machine. The model formulation was implemented in GAMS 24.9.1 containing the solver CPLEX-D 12.7.1. The latter allows for Distributed MIP, i.e., computing mixed-integer linear problems in a distributed way by utilizing more than one machine of the cluster. In this context, models can run on a master that is connected to multiple workers. In general, different modes of optimization can be selected before the start of computations by specific settings. Within the Concurrent Mode, each worker of the cluster applies different searches to the same problem. In opposite, the Distributed Mode advises each worker to compute one specific part of a common search tree and to communicate its findings to the master, while the latter coordinates the workers. A combination of both modes results in the Combined Mode. After conducting the Concurrent Mode for a specific period of time, the master selects the worker with the best performance so far and uses its search tree for the optimization in the subsequent Distributed Mode. Besides, the default settings of the solver were used with the following exceptions. The termination tolerance was adjusted so as to avoid that the solver stops before an optimality gap of zero is reached. The latter is defined as the relative difference between the best objective value of the relaxed subproblems not being discarded so far (upper bound of a maximization problem) and the best integer solution that is feasible but often suboptimal (lower bound of a maximization problem). Furthermore, appropriate tolerance parameters of solution feasibility were set (GAMS 2018). For each of the following computations, up to two machines of the cluster were harnessed simultaneously. The same optimal objective value of $ 8642 was obtained after all 3 optimization runs. As expected, the illustrative example is swiftly optimizable by using the cluster. The exact computation times are 0:00:56 [h:mm:ss] at the Concurrent Mode, 0:01:10 [h:mm:ss] at the Distributed Mode, and 0:00:50 [h:mm:ss] at the Combined Mode. The solution corresponds to the following optimal network structure (see Fig. 5.4) and the optimal composition of cash flows (see Table 5.3) during the medium-term planning horizon of three time periods. At the beginning of the planning horizon (time point t ¼ 0), the five sites that have been set up before (according to the initial site state) are available for operations

86

5 Discrete-Time Scheduling in Green Supply Chain Management

time t=0

0.5

site 2

-

x

0

7.8

site 3

12.5

0

0 2

1

-

16.6

0

site 4

SC stage σ=4 (recycling)

0.5

0

20

site 1

SC stage σ=3 (sales and return)

SC stage σ=2 (production)

SC stage σ=1 (production)

0

-

0

10 5

5 time t=1

site 2

x

15

20

0

5

x

0

12.5

site 3

10

x

0

20

2.5

5

5 0

20

site 1

-

0

0

site 4

5 time t=2

2.5

site 2

-

0

0

5

-

17.5

0

site 4

0

x

0

0

site 3

5

2.5 0

20

site 1

10 2.5

0

x

0

Legend production/recycling sites (quantity inside the box)

market sites

material flows (quantity next to the arrow)

unavailable site

-

unselected site

transportation

site available in first capacity profile

x

selected site

inventory

site available in second capacity profile

Fig. 5.4 Optimal network structure in discrete-time scheduling

during the following time period. Except site 4,2 that needs to operate in the second capacity profile, the first capacity profile is assigned to all available sites. Consequently, up to 20 units can be produced at sites 1,1; 1,2; and 2,3 during this time period. The capacity provision, which will remain constant for site 4,4 and decrease

5.2 Numerical Analysis

87

Table 5.3 Liquidity balancing in discrete-time scheduling

t¼0 t¼1 t¼2 t¼3

Financial transactions Credit Credit Investment 10% 5% 4% 100 4089 110 4293 4497 4677

Site state 100 1115 2450

Emissions

Operations

Withdrawals

500

2474 11,350 3965

8642

for site 4,2, allows for the recycling of 10 units at both sites initially. Based on the aforementioned settings, the site 1,1 manufactures 20 units of the intermediate product. The latter quantity is increased by 5.5 units of the same intermediate product (which are redirected to site 1,1 after the recycling treatment in sites 4,2 and 4,4) and subsequently delivered to the production site 2,3 of the following SC stage. There, it is merged with the supply of 7.8 units of the intermediate obtained from site 1,2 in order to manufacture 16.6 units of the final product. The latter quantity at this site is affected by ingoing amounts from recycling (0.5 units from site 4.2 and 5 units from site 4.4) as well as by inventory holding (12.5 units are transferred to the following time period). The remaining quantity of 9.6 units of the final product is transported to market 3,2, whose demand of 10 units is satisfied only in a part. The market selection requires mandatory handling of 22 units of the finished product, which are assumed to be available after an unspecified period of usage. Due to the optimal solution, the return quantity is distributed to the two available recycling sites of the final SC stage. According to the given share, only half of the ingoing quantity of 2 units at site 4,2 can be recycled to 0.5 units of the intermediate and 0.5 units of the finished product. Both quantities are immediately redirected to the SC stages of their original manufacturing. Recycling at site 4,4 takes place analogously. At the beginning of the second time period (time point t ¼ 1), site 4,1 needs to be additionally set up for operations. In adherence with the capacity profile, it allows for recycling 10 more units of the finished product during the period. All available production sites 1,1; 1,2; and 2,3 exhaust the production capacity provided, so their manufactured quantity can only be increased by deliveries of recycled quantities from the sites of the final SC stage (12.5 units of the intermediate product and the same quantity of the finished product) and by inventory holding (12.5 units of the finished product transferred from t ¼ 0). 62.5% of the quantity manufactured at site 1.1 are kept in stock in order to be available at the beginning of the following time period. As a result, it is possible to satisfy the given demand at the markets 3,1; 3,2; and 3,3 in full. Consequently, the SC is forced to take back the available return quantities from the same markets. Due to existing capacity limits, the latter quantities need to be distributed to three sites of the final SC stage. Analogously to the first time period, these sites are able to recycle 50% of the ingoing amounts of finished products into products (0.5 units of the intermediate and 0.5 units of the finished product from each recyclable unit of the returned finished product) that can replace the manufacturing of new items in both the first and the second SC stage. Considering the beginning of the final time period (time point

88

5 Discrete-Time Scheduling in Green Supply Chain Management

t ¼ 2), it becomes obvious that the site availability is reduced drastically. In particular, it is possible to shut down the sites 1,2; 4,1; and 4,4. The production sites 1,1 and 2,3 are still operating in the first capacity profile, so up to 20 units can be manufactured at them during the following time period. The only recycling site remaining is 4,2, whose maximum capacity provided during the following time period decreases to 5 units due to the selection of the second capacity profile. Taking into account available quantities from inventory and recycling, production site 1,1 is able to provide the whole quantity of intermediates that is required at site 2,3. The latter supplies (after receiving an additional quantity of recycled products from the final SC stage) both the markets 3,2 and 3,4 in full. Final products that need to be taken back from market 3,4 are transported to recycling site 4,2 whose maximum capacity is exhausted by the ingoing quantity (adjusted by the share of end-of-use returns that cannot be recycled). With regard to emission management, the SC is able to comply with the emission cap of 100 tons in the first and the third time period. Within the second time period, the emissions from production, recycling, and transportation exceed this given limit, so the purchase of one additional emission allowance providing 100 additional tons is required. The monetary consequences resulting from the aforementioned operational and environmental decisions are merged within liquidity balancing. In t ¼ 0, site state decisions comprise the assignment of the capacity profiles. According to the selected provision, overall capacity costs of $ 100 arise. As this amount cannot be financed by operation surpluses at the beginning of the planning horizon, it is borrowed for two time periods (credit with an interest rate of 10% for the whole credit period). Considering time point t ¼ 1, it is necessary to take another credit of $ 4089. According to its term, it needs to be repaid after one time period at an interest rate of 5%. By that credit, three different payments need to be financed: Firstly, the setup of the recycling site 4,1 and the capacity provision require an amount of $ 1115. Secondly, the SC is forced to buy the additional emission allowance at the price of $ 500. Thirdly, there are negative surpluses from operations during the previous time period in the amount of $ 2474 that need to be compensated. The beginning of the last time period is time point t ¼ 2. As it was possible to obtain a surplus of $ 11,350 from operations during the previous time period, both credits can be repaid including interest. Furthermore, capacity provision and required adjustments of the site state (shutdown of three sites) are completely financed. The residual of $ 4497 is invested at the capital market for one time period at an interest rate of 4%. Time point t ¼ 3 marks the end of the medium-term planning horizon. Merging the repayment of the financial investment ($ 4677) with the surpluses from operations that could be obtained during the previous time period ($ 3965), a residual of $ 8642 is realized. According to the optimal solution of the problem, the latter equals the only withdrawal. As liquidity is balanced completely in all time points, insolvency of the SC network is prevented. Going beyond the illustrative example presented before, the computability of the problem of discrete-time scheduling in networks with recycling and emission trading is evaluated in a scenario analysis. It includes 30 test instances assignable to 5 groups of scenarios with an identical number of network sites in each of the 4 SC stages. The

5.2 Numerical Analysis

89

site number is varied between four (which is used for the illustrative example) and eight. Each scenario is based on different parameters that are randomly generated from a uniform distribution within given intervals. Three independent optimization runs on the high-performance cluster, which are characterized by different modes of the CPLEX-D solver (Concurrent, Distributed, and Combined Mode), were conducted for each instance. After completion, both the objective values and the computation times were recorded (see Table 5.4). As a result, it becomes obvious Table 5.4 Results of the scenario analysis for discrete-time scheduling

Scen. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Avg.

Sites per stage 4

5

6

7

8

Optimization Concurrent Mode Comp. Obj. time [hh: value mm:ss] [$] 00:00:45 11,502 00:00:52 22,535 00:04:27 12,351 00:01:01 17,474 00:02:28 27,664 00:03:09 5893 00:04:04 22,811 00:05:13 10,978 00:04:43 22,990 00:04:15 19,524 00:04:51 17,678 00:09:45 33,752 00:18:55 15,264 00:11:34 29,216 00:47:30 26,715 00:07:28 16,349 00:08:29 17,447 00:23:10 20,766 00:15:04 22,881 00:46:04 36,950 00:33:30 26,986 00:16:14 27,281 01:33:15 37,038 00:08:22 23,818 00:29:02 8333 02:03:30 9491 01:37:42 23,916 01:23:20 41,850 00:20:59 34,031 00:27:20 25,853 00:25:14

Distributed Mode Comp. Comp. time time [hh: difference mm:ss] (%) 00:01:40 122 00:01:37 87 00:20:13 354 00:01:55 89 00:01:20 46 00:07:41 144 00:15:26 280 00:20:51 300 00:13:09 179 00:34:57 722 00:24:26 404 00:42:32 336 02:00:16 536 00:52:10 351 03:19:49 321 00:35:42 378 00:55:03 549 01:10:53 206 00:38:41 157 10:41:34 1293 02:07:59 282 01:06:22 309 06:10:51 298 01:05:32 683 03:15:03 572 07:33:11 267 13:33:57 733 14:54:50 974 04:02:48 1057 08:05:14 1675 02:50:31 454

Combined Mode Comp. Comp. time time [hh: difference mm:ss] (%) 00:00:44 2 00:01:02 19 00:22:20 402 00:00:59 3 00:01:40 32 00:13:47 338 00:19:49 387 00:21:05 304 00:21:58 366 00:21:55 416 00:19:52 310 00:41:17 323 01:18:21 314 00:41:59 263 03:53:50 392 00:20:07 169 00:34:45 310 01:36:35 317 00:32:40 117 07:02:32 817 01:29:24 167 00:48:35 199 09:16:07 496 00:48:04 475 01:57:44 306 05:44:40 179 07:32:12 363 13:02:12 839 03:22:13 864 14:02:52 2984 02:34:23 413

90

5 Discrete-Time Scheduling in Green Supply Chain Management

average computation times [hh:mm:ss]

10:00:00 8:00:00

6:00:00 Concurrent Mode Distributed Mode

4:00:00

Combined Mode

2:00:00 0:00:00 4

5

6

7

8

number of sites per SC stage

Fig. 5.5 Computation times using different modes of the CPLEX-D solver

that all 90 computations resulted in the proven optimum within 15 h. Inherently, the same optimal objective values were generated for each instance, while the scenario analysis reveals considerable differences in the advantageousness of the three solver modes. In summary, the Concurrent Mode is most appropriate for solving the problem, as it allows for optimizing each given instance within little more than 2 h. The average computation time over all 30 scenarios is 0:25:14 [h:mm:ss]. As considered to be acceptable for scheduling a medium-term planning horizon, these computational efforts usually do not require the application of a heuristic (see Sect. 6.2). Although the performance of the Distributed Mode and/or the Combined Mode is superior in single instances (scenario nos. 1, 4, 5), the latter are characterized by negligible computation times of less than 3 min, which marginalize the occurring absolute time differences. Acknowledging average computation times of 2:50:31 [h: mm:ss] and 2:34:23 [h:mm:ss], respectively, it becomes obvious that both modes are clearly inferior for solving the problem. The latter conclusion is supported by the fact that both of these overall averages are more than 400 percent higher than the one of the Concurrent Mode. Finally, the average computation times are analyzed separately for each of the five groups of scenarios that include instances with the same number of sites in each SC stage. Figure 5.5 confirms the previous results, while the trend approximated by specific power functions (dashed lines) allows for estimating the computation times at an increasing number of sites. Based on the results of the scenario analysis, it can be supposed that both the Distributed and the Combined Mode failed due to an inefficient coordination of workers with regard to the problem structure. In consequence, the Concurrent Mode is exclusively applied to all further computations (see also Chap. 6).

References

91

References Albrecht, W. (2014). Integrierte Netzwerk- und Liquiditätsplanung von Supply Chains. Wiesbaden: Springer Gabler. Albrecht, W., & Steinrücke, M. (2017). Continuous-time production, distribution and financial planning with periodic liquidity balancing. Journal of Scheduling, 20(3), 219–237. Albrecht, W., & Steinrücke, M. (2018). Coordinating continuous-time distribution and sales planning of perishable goods with quality grades. International Journal of Production Research, 56(7), 2646–2665. Albrecht, W., & Steinrücke, M. (2020a). Continuous-time scheduling of production, distribution and sales in photovoltaic supply chains with declining prices. Flexible Services and Manufacturing, 32(3), 629–667. Albrecht, W., & Steinrücke, M. (2020b). Assessing site integration into semi-continuous production, distribution and liquidity planning of supply chain networks. EURO Journal on Transportation and Logistics, 9, No. 100002. GAMS. (2018). GAMS Documentation, GAMS Development Corporation, Washington. Retrieved from https://www.gams.com/latest/docs/. Steinrücke, M., & Albrecht, W. (2016). A flow-to-equity approach to coordinate supply chain network planning and financial planning with annual cash outflows to an institutional investor. Business Research, 9, 297–333. Steinrücke, M., & Albrecht, W. (2018). Integrated supply chain network planning and financial planning respecting the imperfection of the capital market. Journal of Business Economics, 88 (6), 799–825. Venkataraman, R. R., & Pinto, J. K. (2018). Operations Management. Thousand Oaks et al.: Sage. Wong, C. W. Y., Lai, K., Lun, Y. H. V., & Cheng, T. C. E. (2016). Environmental Management. Heidelberg et al.: Springer.

Chapter 6

Continuous-Time Scheduling in Green Supply Chain Management

Abstract Based on existing structures of sites and capacities, new mathematical models for hierarchical scheduling of operations in generic multi-stage networks are developed in the book Scheduling in Green Supply Chain Management: A MixedInteger Approach. This chapter provides modularized formulations for continuoustime scheduling in green supply chain management. For non-perishables, three different types of recycling (internal, external, and combined recycling) are considered. Reverse logistics of perishables are affected by the product quality that is realized at the markets. Problem-tailored heuristic solution methods (relax-and-fix algorithms and genetic algorithms) are proposed.

Continuous-time scheduling is applicable to short-term planning horizons of several days. It allows to determine the start and end times of all operations (including both the processes in forward and reverse SCM) exactly to the minute. Furthermore, aspects of financial planning can be included, as an appropriate funding of all operations needs to be guaranteed at their start time. Due to the length of the planning horizon and the requirements on the accuracy of input data, the following problems of GSCM are based on continuous-time scheduling of multi-stage SC networks (Steinrücke 2011a, 2011b, 2015; Albrecht and Steinrücke 2017, 2018, 2020a, 2020b). In course of this section, the processing of perishable and non-perishable products is considered. For non-perishable goods (Sect. 6.1.1), both internal and external forms of recycling can be applied to obtain products being completely equivalent to the ones produced in the SC stages of their origin. Typically, this requires products with time-stable qualities, which are solely differentiated by their stage-specific maturity at the markets. In contrast, perishable goods (Sect. 6.1.2) are characterized by a quality that decreases in time. In this context, quality grades can be used for product differentiation at the markets.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6_6

93

94

6

6.1

Continuous-Time Scheduling in Green Supply Chain Management

Model Formulations

6.1.1

Networks with Recycling of Non-perishable Goods

6.1.1.1

Internal Recycling

Internal recycling in continuous-time scheduling of production-distribution networks captures unusable material within the production process (e.g., defective internal supplies or other rejects or production scrap). Modeling is based on the assumption that unusable material can be recycled immediately to a certain share at the sites of occurrence, before it is returned to the SC stages of its original manufacturing. As far as (partial) reintegration into production is possible, a closed loop system (see Sect. 3.2) results. The usage of external sites is limited to processing nonrecyclable parts of the material flows, which are directed to landfills for disposal. Due to the internal focus, customer returns are not included in this model. The general structure of the SC network is described as follows (see Fig. 6.1): (a) One production stage (σ ¼ 1). The first stage of the SC is exclusively assigned to the production of primary goods (e.g., raw materials). These goods are not composed of other goods obtained from preceding SC stages, and hence, rejects from ingoing material flows (that would trigger internal recycling) do not need to be considered. (b) W  1 production/recycling stages (σ ¼ 2, . . ., W ). The aforementioned SC stages can be used for production of refined stage-specific goods (e.g., intermediates, final products). As sites of these SC stages require ingoing material flows from preceding SC stages, rejects can occur. The latter are handled at the sites within internal recycling. (c) One market stage (σ ¼ W + 1). This stage contains markets, whose productspecific demand can be satisfied by sites of preceding SC stages. (d) One disposal stage (σ ¼ W + 2). Sites of the last SC stage are available for disposals. The network sites can be connected by transports. For internal recycling, potential material flows are defined between sites of the following SC stages σ and λ (see Fig. 6.1): (a) Material flows of forward logistics. Each site of an SC stage with production (σ ¼ 1, . . ., W ) can supply each site of a subsequent, but not necessarily adjacent, production or market stage (λ ¼ σ + 1, . . ., W + 1) with stage-specific products. (b) Material flows of reverse logistics. The following two types can be distinguished: • Flows of recycled goods. Material flows are directed from sites of SC stages with internal recycling (σ ¼ 2, . . ., W) to preceding production stages (λ ¼ 1, . . ., σ  1).

6.1 Model Formulations

95

production stage

1,m

σ=1

1,1

production and recycling stages

2,n

σ=2

2,1

…

…

…

W,o W,1

σ=W

market stage

W+1,1

…

σ=W+1

W+1,p

disposal stage

…

W+2,1

σ=W+2

W+2,q

Legend: forward flows of saleable goods

σ,s

network site s of SC stage σ

σ,s

market site s of SC stage σ

forward flows of saleable goods and reverse flows of recycled goods reverse flows of disposed goods

Fig. 6.1 General network structure for internal recycling of non-perishable goods in continuoustime scheduling

96

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Continuous-Time Scheduling in Green Supply Chain Management

• Flows of disposed goods. Material flows are directed from sites of SC stages with internal recycling (σ ¼ 2, . . ., W) to the SC stage of final disposal (λ ¼ W + 2). The network sites can manufacture different products, while each of them is permanently linked to a specific SC stage. With regard to forward SCM, associated sites can be used to produce quantities of stage-specific products, which can be distributed to sites of a subsequent (but not necessarily adjacent) production stage, in order to be processed further (optionally in combination with goods from other SC stages), or to sites of the market stage, in order to meet the given demand. In reverse SCM, both the (partly) recovery of products of preceding SC stages in the course of recycling and the final processing of disposals are considered at the associated sites. Regardless of their function, each of the SC stages contains a certain number of sites, which are assumed to be potentially available for operations once during the shortterm planning horizon. Therein, decisions on setups, shutdowns, or capacity adjustments (see Sect. 5) are excluded from the consideration. The sites are connected by potential material flows, while it needs to be decided on their realization in the course of the network’s overall optimization. For enabling the highest flexibility of coordination, the model captures both stock-free material flows, which adhere to given transportation times between the network sites, and material flows, which include additional temporary storage times (Steinrücke 2015). A capacitated vehicle is available on each path of the network. Due to the sites spreading around the globe, direct transports are considered instead of vehicle routing. Production processes are based on given bills of materials. As describing a special case of production, even distribution stages can be added by setting the production coefficients accordingly. Recycling processes can be distinguished according to the stage-specific product. If recycling is possible, its output needs to be redirected to the SC stage that manufactures new items of the same product. It is assumed that recycled products cannot be distinguished from new ones after the recovery treatment. In order to prevent deliveries of recycled products to the sites of their origin, which cannot be processed there, it is possible to increase the overall quantity of products that are delivered to landfills for reasons of disposal. However, the scope of this increase can be limited in accordance with internal or external regulations. Driving force of operations within a production-distribution network is the given demand of customers at markets, which does not need to be satisfied in full. The deterministic demand results from orders, based on short-term contracts. The latter do contain not only the accepted quantities of products potentially manufactured in different stages of the SC but also due dates for the delivery. Whereas a delivery by the SC partners before reaching the due date is possible without any consequences, tardiness is punished by contractual penalties, which need to be integrated in the model formulation for continuous-time scheduling. The latter penalties are assumed to be calculated exactly. The short-term planning horizon consists of several days, while it needs to be ensured that all operations are finished before its end. The model formulation is based on the following sets and indices:

6.1 Model Formulations

97

Index sets need to be defined for SC stages, sites, and events. The overall number of SC stages (ordered in a sequence according to Fig. 6.1) depends on the number of production stages W. Due to the dependency of sets, an arbitrary number of sites can be assigned to each SC stage. Except the market and the disposal stage, all sites can be considered for both forward and reverse flows in systems for internal processing of returns. The set of event boundaries specifies both the start and the end of operations by a separate index. Finally, another index is used to distinguish different types of greenhouse gas emissions. The following parameters need to be set according to the specific problem: Costs and revenues. Fixed costs arise for operating network sites (i.e., production, recycling, or disposal sites) and for transportation. Variable costs depending on the processed quantities need to be considered in the same fields of operations. Due to the time orientation of the model formulation, further variable costs occur for each day the due date of demand satisfaction is exceeded at a market and, moreover, for exceeding the given transportation times between two different sites of the network by additional times of temporary storage. Revenues can be earned at the markets (in case of their selection, which entails additional fixed costs) depending on the stage-specific type of product. Capacities. Given capacities restrict the quantity that can be processed at a site of the network. They can be chosen according to the technical features of devices or according to the availability of human resources. In particular, there are production, recycling, and disposal capacities at the sites. Furthermore, throughput capacities exist. The latter restrict the amount of recycled goods to be reintegrated into forward logistics. Finally, transportation capacities need to be respected in each relationship between two different network sites. Time parameters. The transportation times required for shipments between two sites of the SC network are given for both forward and reverse logistics. They can be increased by additional times of temporary storage with respect to given upper bounds. Furthermore, a minimum processing time can be prescribed in order to ensure the economic operation of the sites. All processes need to be finished before the end time of the planning horizon. Speed parameters. As being relevant for determining the processing times at the sites, speed parameters need to be introduced. They specify the quantity of units that can be produced, recycled, or disposed of in one unit of time. Coefficients. Within forward logistics, coefficients specifying the input-output relations between products of subsequent (but not necessarily adjacent) SC stages in the production process need to be considered. They can be taken from given bills of materials. Regarding the internal handling of rejects, a first coefficient represents the ratio between defective and usable supplies. Based on the resulting quantity of goods to be processed in reverse logistics at a network site, a second coefficient specifies the share to be recycled. Consequently, the remaining proportion needs to be disposed of. Environmental parameters. Handling stage-specific products within the SC network can entail different greenhouse gas emissions per processed product unit, which need to be converted into carbon dioxide equivalents. Based on the latter, a

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given emission cap limits the overall emissions caused by the system during the planning horizon. Furthermore, the overall disposal of recyclable products can be limited to an upper bound. Other parameters. There is given demand for products at the markets, which can be obtained from different production stages of the SC. Each market prescribes a due date for the delivery. A sufficiently large number needs to be chosen in the context of the Big M method. The following decisions need to be taken within the optimization of the overall system: (a) Decisions with regard to the sites operated by the SC. • Usage of potential network sites for production, storage, recycling, and/or disposal (Note: A subsequent site usage for production/storage and recycling is possible.) • Exact start and end times of operations. • Processed quantities of stage-specific products. (b) Decisions with regard to transportation between the sites. • • • •

Realization of potential material flows between the network sites. Exact start and end times of transports of stage-specific products. Transportation quantities of stage-specific products. Temporary storages used to increase the given transportation times.

(c) Decisions with regard to the market sites. • Selection of potential markets for demand satisfaction. • Quantities delivered to the markets (scope of demand satisfaction). • Time of sales, tardiness of demand satisfaction with regard to given due dates. (d) Environmental decisions. • Transfer of recyclable quantities of stage-specific products to disposal. The mixed-integer linear program for continuous-time scheduling in networks with internal recycling of non-perishable goods can be formulated as follows:

6.1 Model Formulations

99

Objective Function Max Φ P Φ¼

 DC s  sls  MC s  ys þ

s2SWþ1



W P P λ¼1 q2Sλ

! Rλs

 xqs

W P P

PF s  ys þ PV s  pr s P PP  TF sq  tpsq þ TSsq  st sq þ TV sq  xsq ðσ, λÞ2K s2Sσ q2Sλ



ð6:1Þ

σ¼1 s2Sσ

P

P

σ2L[fWþ2gs2Sσ

RF s  yr s 

PP σ2Ls2Sσ

RV s  dr s 

P

FV s  f r s

s2SWþ2

Objective (6.1) is to maximize the overall profit of the entire SC network during the short-term planning horizon. Monetary consequences occurring at different sites and stages of the network need to be taken into account. Firstly, they concern the customers at the market stage. Revenues result from the production quantities directed to the markets from sites of different preceding SC stages that represent different stage-specific kinds of products. The (partial) selection of a market for demand satisfaction entails fixed marketing costs. Furthermore, in case that it is not possible to meet this demand before/at the given due date, contractual penalties arise for tardiness. They are calculated exactly, based on a daily rate of cost. Secondly, production costs occurring at each production stage of the network need to be considered. The usage of sites entails fixed costs, whereas the variable costs depend on the exact quantity that is manufactured. Thirdly, different kinds of costs can occur in the context of transportation, which is realized between two different network sites. If a transport is conducted on a partial path of the network, fixed costs need to be subtracted from the objective value. Additional variable costs result from assessing the exact transport quantities. Furthermore, transportation costs are increased if it is not possible to realize just-in-time deliveries with stock-free material flows. Temporary storage times possibly required for reaching an appropriate coordination between the network sites entail additional costs, which are assumed to be independent of storage quantities. If the latter would additionally be integrated into cost calculation in order to capture capital tie-up, non-linearity would arise. Finally, costs of recycling and disposal need to be integrated into the calculation of the network’s overall profit. In case of internal processing of rejects, recycling costs can occur at production sites of the SC stages L ¼ {2, . . ., W}, whereas disposal costs are relevant for operations in the final SC stage W + 2. Fixed costs occur if the site is used for recycling or disposal, respectively, whereas the calculation of variable costs depends on the exact quantities to be processed. Due to the aforementioned assumptions with regard to internal handling, it is not possible that recycling and disposal are conducted at the same site.

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Continuous-Time Scheduling in Green Supply Chain Management

Constraints of Forward and Reverse Flows X

xqs  Dλs  ys ; 8s 2 SWþ1 ; λ ¼ 1, . . . , W

ð6:2Þ

q2Sλ

Each market of the SC network is characterized by a product-specific demand, while these products are to be obtained from the SC stages they are manufactured. The total amount of delivery can be composed of partial deliveries that may stem from different sites of the same SC stage. Due to constraint (6.2), deliveries can only be directed to a market if the latter is selected for demand satisfaction [which entails additional marketing costs; see (6.1)]. While an overfulfillment is strictly excluded, delivery shortages are possible. X

xqs ¼ ð1 þ δs Þ  Bλσ  pr s ; 8s 2 Sσ ; σ ¼ 2, . . . , W; λ ¼ 1, . . . , σ  1

ð6:3Þ

q2Sλ

rr λs ¼ δs  Bλσ  pr s ; 8s 2 Sσ ; σ ¼ 2, . . . , W; λ ¼ 1, . . . , σ  1

ð6:4Þ

Within all sites of the production stages σ ¼ 2, . . ., W, the supply of goods from preceding (but not necessarily adjacent) stages is to be calculated on the basis of given bills of materials in constraint (6.3). For this reason, the production quantity is to be multiplied with a coefficient representing the relevant input-output relation. The delivery of a required stage-specific product may stem from different sites that are capable of its manufacturing. According to the assumptions, the latter sites must belong to the same SC stage. As the supplies are expected to be defective in a certain ratio to the usable items (which entails their rejection from further operations in the forward SC in their current state), the quantity of ordered materials is increased accordingly, while the rejects are to be processed further within reverse logistics. By constraint (6.4), the exact quantity that needs to be recycled or disposed of internally by the SC network is determined. It is defined as the residual quantity of a stagespecific product, which could not be used for manufacturing at a production site. βs  rr λs ¼ nuλs þ

X

xsq ; 8s 2 Sσ ; σ ¼ 2, . . . , W; λ ¼ 1, . . . , σ  1

ð6:5Þ

q2Sλ σ 1 X λ¼1

ð1  βs Þ  rr λs þ nuλs ¼

X

xsq ; 8s 2 Sσ ; σ ¼ 2, . . . , W

ð6:6Þ

q2SWþ2

A given percentage of the rejects at the production sites can be recycled. The quantity of the stage-specific products results at the left-hand side of (6.5). According to the right-hand side of the constraint, there are two alternative ways of further processing. On the one hand, recycling can be conducted. Afterward, the recycled products are delivered to their SC stage, i.e., to the SC stage that

6.1 Model Formulations

101

manufactures new products of the same kind. In this context, quantities of the same stage-specific product can be split up into deliveries to different sites. On the other hand, it is possible to reduce the recycling quantity, e.g., for economic reasons. This reduction is realized by adding a transfer variable on the right-hand side of the constraint. The transfer variable occurs also on the left-hand side of the constraint (6.6), which refers to the determination of the disposal quantities. The latter do not need to be separated by different stage-specific products, as there is no difference in handling during the further processing. Consequently, the overall quantity of non-recyclable products (the contrary share of rejects at production site) is to be increased by the recyclable products that should be disposed of instead. As the disposal stage of the SC network may comprise different alternative sites (that can be distinguished by cost or capacity parameters), it is possible to apportion the deliveries. σ 1 X W X X λ¼1 σ¼2 s2Sσ

nuλs  NL

ð6:7Þ

The disposal of recyclable products can be required for several reasons. In particular, there may be circumstances that do not allow for a recycling of defective products due to a lack of recycling capacities at the outgoing sites or a lack of throughput capacity for recycled products at the ingoing sites. Other arguments can be seen in the context of the short-term planning horizon, as reintegrating recycled products (and thus the formation of closed loops) may extent the feasible time period for demand satisfaction at the markets. Finally, the objective of profit maximization can be contrary to recovery, especially if it is not possible to realize the recycling processes including the related reverse flows for overall costs that are less than the costs of manufacturing new products. However, as economic reasons cannot be assumed to dominate environmental concerns from each perspective (e.g., due to internal or external regulations), the quantity of recyclable products that is disposed of can be limited to an upper bound within constraint (6.7). W þ1 X

W X X X xsq ¼ pr s þ xqs ; 8s 2 Sσ ; σ ¼ 1, . . . , W

λ¼σþ1 q2Sλ

ð6:8Þ

λ¼σþ1 q2Sλ

It is guaranteed that all reverse material flows that are directed to production sites of the SC stages σ ¼ 1, . . ., W are completely integrated into the closed loop, as the reuse of recycled goods for further processing or demand satisfaction is forced by constraint (6.8). The reverse flows are added to the production quantity at these sites and, thus, they can replace the manufacturing of new goods. The forward flows starting from the production sites can be split up to sites of different subsequent production stages or the market stage.

102

6 σ 1 X

Continuous-Time Scheduling in Green Supply Chain Management

βs  rr λs  nuλs ¼ dr s ; 8s 2 Sσ ; σ ¼ 2, . . . , W

ð6:9Þ

λ¼1 W X X xqs ¼ fr s ; 8s 2 SWþ2

ð6:10Þ

λ¼2 q2Sλ

The recycled quantity at the network sites of the SC stages σ ¼ 2, . . ., W is determined by constraint (6.9). It is calculated by summing up all rejects that need to be processed within reverse logistics. It can include different stage-specific products, i.e., goods that can be returned to different SC stages λ ¼ 1, . . ., σ  1 after recycling. The aforementioned sum needs to be adjusted for a network site’s recyclable quantity that is disposed of. In case of internal handling, the disposal of goods can only occur at sites of the final SC stage σ ¼ W + 2. Due to constraint (6.10), the disposed quantities are composed of different material flows that may stem from sites of different production/recycling stages λ ¼ 2, . . ., W.

Capacity Constraints pr s  PC s  ys ; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:11Þ

W X X xqs  XC s  ys ; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:12Þ

λ¼σþ1 q2Sλ

dr s  RC s  yr s ; 8s 2 Sσ ; σ 2 L

ð6:13Þ

fr s  FC s  yr s ; 8s 2 SWþ2

ð6:14Þ

xsq  TC sq  tpsq ; 8s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K

ð6:15Þ

Constraint (6.11) limits the production quantity at a site of the SC stages σ ¼ 1, . . ., W to the maximum production capacity at this site if the latter is in use of the network. Otherwise, the production quantity is forced to zero. By slight modifications, different types of capacities (e.g., technical and personal capacities) could be distinguished. Constraint (6.12) refers to the throughput capacity of recycled goods at the production sites belonging to the SC stages σ ¼ 1, . . ., W. The upper bound of throughput is set for used sites, while any throughput needs to be excluded for sites that are not in use. Although the actual throughput can be composed of reverse material flows that may stem from production sites of all subsequent production stages λ ¼ σ + 1, . . ., W, it must consist of the same stage-specific product. Additionally, maximum capacities at sites of the reverse SC need to be respected. Due to the network structure for internal processing of rejects, two different types of SC stages can be considered separately. According to (6.13), the maximum recycling capacity is the upper bound of recycling quantities at production sites of the SC

6.1 Model Formulations

103

stages σ ¼ 2, . . ., W if these sites are used for recycling. According to (6.14), the maximum disposal capacity limits the quantity to be disposed of at landfills of the SC stage σ ¼ W + 2. Disposal sites that are not selected by the network cannot receive material flows from any other SC stages. Finally, there are restrictions on the maximum quantity to be shipped between the network sites in both forward and reverse logistics. Constraint (6.15) limits the transportation quantity if a transport is conducted and becomes redundant otherwise. These limits can be set according to the features of the specific vehicles and may refer to the loading area or volume in particular. Slight modifications would allow for the simultaneous consideration of different types of transportation capacity.

Constraints of Continuous-Time Scheduling pr s ¼ sp2s ; 8s 2 S1 SDs

ð6:16Þ

pr s dr þ s ¼ sp2s ; 8s 2 Sσ ; σ ¼ 2, . . . , W SDs RDs

ð6:17Þ

fr s ¼ sp2s ; 8s 2 SWþ2 FDs

ð6:18Þ

sp1s þ sp1s þ

sp1s þ

Start and end times of operations at the same site of the SC network are connected by the time period that needs to be spent for processing. The following constraints adhere to the assumptions on the general network structure for internal recycling: According to constraint (6.16), sites of the first SC stage σ ¼ 1 are considered exclusively for production. Thus, the exact time period of processing results from the quotient of the production quantity and the production speed. Due to constraint (6.17), the difference between the end time and the start time at production sites of the following SC stages σ ¼ 2, . . ., W may include both production and recycling times. The latter need to be added if recycling occurs due to rejecting a specific share of ingoing quantities. Based on the calculation of recycling quantities according to constraint (6.9), the time required for recycling processes results from taking the sitespecific recycling speed into account. Finally, the processing times at the disposal sites are defined by constraint (6.18). In this context, the times required for operations at the landfills are calculated by the quantities to be processed at these sites and the site-specific speed of disposal. By slight modification of the constraint, a fixed time period (e.g., setup times for preparing the machines for manufacturing or recycling) could be added to the time period that passes between the start and the end time of a site’s operation. sp2s  sp1s  MZ  ys ; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:19Þ

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Continuous-Time Scheduling in Green Supply Chain Management

sp2s  sp1s  MZ  yr s ; 8s 2 Sσ ; σ 2 L [ fW þ 2g

ð6:20Þ

According to constraints (6.19) and (6.20), a minimum processing time can be prescribed at sites of all SC stages except the market stage. For this purpose, the difference between the end time and the start time of the same site is calculated at the left-hand side of the constraint, while it is forced that this difference cannot fall below a given lower bound at the right-hand side. This bound can be set in accordance with requirements of quality management. It prevents a site’s availability for negligible processing times.   sp2s þ TZ sq þ st sq  sp1q þ M  1  tpsq ; 8s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K   sp2s þ TZ sq þ st sq  sp1q  M  1  tpsq ; 8s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K st sq  SBsq ; 8s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K

ð6:21Þ ð6:22Þ ð6:23Þ

Continuous-time scheduling also requires that the operation end time of one site is connected to the operation start time of another site if there is a material flow between them. This connection is realized by constraints (6.21) and (6.22). Potential material flows result from the general network structure depicted in Fig. 6.1. In a first case, a potential material flow is realized. This is indicated by the transportation binary variable within the optimal solution of the overall problem. In particular, the terms +M  (1  tpsq) and M  (1  tpsq), respectively, can be omitted at the righthand side of both constraints (6.21) and (6.22) in case of tpsq ¼ 1. In combination of both constraints, the equation sp2s þ TZ sq þ st sq ¼ sp1q is valid then. It becomes obvious that there is a fixed connection between the end time of the network site the material flow starts, and the start time of the network site the material flow ends. This start time can be interpreted as the time of demand satisfaction in case the material flow is directed to a market site. The transportation time is specified by the parameters for both forward and reverse logistics and can be extended by times of temporary storage. The latter can be limited to an extent maximally acceptable for the logistics company by constraint (6.23). If the upper bound of the temporary storage time is set to zero for all possible transports between two sites, stock-free material flows can be forced for the entire network. In a second case, it should be considered that the potential material flow is not realized. Obviously, the resulting value at the right-hand side of the less-than-or-equal constraint (6.21) is out of the range of the planning horizon due to the addition of the big number M and, thus, the constraint is always satisfied. Analogously, the value at the right-hand side of the greater-than-or-equal constraint (6.22) falls below zero due to the subtraction of M. Summarizing both of the aforementioned consequences, both constraints (6.21) and (6.22) become redundant due to the Big M method in the case of the non-realization of potential material flows.

6.1 Model Formulations

105

sp1s  DDs þ sls ; 8s 2 SWþ1

ð6:24Þ

Constraint (6.24) establishes a connection between the due date at a market site and the actual time of demand satisfaction. The latter results from the overall coordination of the network according to the optimal solution of the problem. Whereas a delivery before the due date is possible due to the less-than-or-equal sign, a delivery after the planned due date can be realized after the consideration of the delivery tardiness at the right-hand side of the constraint. In order to limit the occurrence of tardiness, the latter is punished by additional costs within objective function (6.1). By introducing an additional constraint, an upper limit for the time of tardiness at a market, or for the overall time of tardiness in the SC system, can be set. By slight modifications, the constraint and the objective function can be adjusted for considering bonus payments for deliveries before the given due date. sp1s  E; 8s 2 SWþ1

ð6:25Þ

 E; 8s 2 SWþ2

ð6:26Þ

sp2s

Further restrictions ensure that the short-term planning horizon cannot be exceeded by the operations to be scheduled within the SC network. Firstly, it needs to be guaranteed by constraint (6.25) that the demand is satisfied before the end of the planning horizon. In case that a market is not selected for deliveries, there are no material flows directed to it. Then, the time point sp1s is not restricted by constraints (6.21) and (6.22), and as a result, the constraint becomes redundant. Taking into account the final operation of reverse logistics, constraint (6.26) can ensure that processing of disposed goods is finished in time.

Constraints of Emission Control 0

W P P

PP

P

1

B σ¼1 s2S pr s  EPps þ σ2Ls2S dr s  EDps þ s2S fr s  EF ps C σ σ Wþ2 C CE p  B P PP @ A þ x  ET p2P sq psq

X

ðσ, λÞ2K s2Sσ q2Sλ

 EM

ð6:27Þ

Specific emission caps, which result after the adjustment by emission trading in medium-term scheduling (see Sect. 5.1.1.6), are taken into account. Firstly, greenhouse gas emissions of each unit produced at sites of the SC stages σ ¼ 1, . . ., W are calculated. Secondly, emissions of each recycled and disposed unit need to be added. In accordance with the general network structure for internal processing of rejects (see Fig. 6.1), recycling can occur at sites of the SC stages σ ¼ 2, . . ., W, and disposal takes place at sites of the SC stage σ ¼ W + 2. Thirdly, emissions from transportation

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Continuous-Time Scheduling in Green Supply Chain Management

between network sites of different SC stages need to be considered for both forward and reverse flows of materials. Emitted volumes are calculated based on realized transportation quantities. The overall volumes of specific greenhouse gas emissions are converted into their carbon dioxide equivalent. Thus, it becomes possible to make them comparable and, finally, to sum them up to the overall SC emission volume during the short-term planning horizon. The latter volume cannot exceed the given emission cap.

Constraints of Domain Definition The model formulation is completed by the following non-negativity constraints (6.28)–(6.35) and binary constraints (6.36)–(6.38):

spzs

6.1.1.2

pr s  0; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:28Þ

dr s  0; 8s 2 Sσ ; σ 2 L

ð6:29Þ

fr s  0; 8s 2 SWþ2

ð6:30Þ

sls  0; 8s 2 SWþ1

ð6:31Þ

 0; 8s 2 Sσ ; σ ¼ 1, . . . , W þ 2; z 2 Z

ð6:32Þ

xsq , stsq  0; 8s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K

ð6:33Þ

rr λs  0; 8s 2 Sσ ; σ ¼ 2, . . . , W; λ ¼ 1, . . . , σ  1

ð6:34Þ

nuλs  0; 8s 2 Sσ ; σ ¼ 2, . . . , W; λ ¼ 1, . . . , σ  1

ð6:35Þ

tpsq 2 f0, 1g; 8s 2 Sσ ; q 2 Sλ ; ðσ, λÞ 2 K

ð6:36Þ

ys 2 f0, 1g; 8s 2 Sσ ; σ ¼ 1, . . . , W þ 1

ð6:37Þ

yr s 2 f0, 1g; 8s 2 Sσ ; σ 2 L [ fW þ 2g

ð6:38Þ

External Recycling

Continuous-time scheduling of production-distribution networks is not limited to the internal processing of rejects, as even some kinds of products delivered to the final customers are suitable for triggering immediate returns. In particular, defective deliveries or packaging material can be expected to be returned after a short time period that passes after demand satisfaction by the SC network. These returns are assumed to be processed by external sites, which check the ingoing material flows and redirect them after a recycling treatment to the production SC stages they are assigned to. In order to ensure the reintegration into the production process within the resulting closed loop system, the externals organize the delivery of appropriate

6.1 Model Formulations

107

quantities of recycled materials to the preceding SC stages. Quantities that are not recycled for technical or economic reasons are disposed of. The following general network structure is given (see Fig. 6.2): (a) W production stages (σ ¼ 1, . . .W ). The SC stages are used exclusively for the production of different stage-specific products (e.g., raw materials, intermediates, final products). (b) One market stage (σ ¼ W + 1). This stage contains markets, whose productspecific demand can be satisfied by sites of preceding SC stages. Going beyond consumption, the markets can be considered as starting point of reverse logistics. (c) One recycling/disposal stage (σ ¼ W + 2). Sites of this SC stage can be used for external recycling and disposal of goods returned from the markets. Potential material flows can occur between sites of the SC stages σ and λ (see Fig. 6.2): (a) Material flows of forward logistics. Each site of an SC stage with production (σ ¼ 1, . . ., W ) can supply each site of a subsequent, but not necessarily adjacent, production or market stage (λ ¼ σ + 1, . . ., W + 1) with stage-specific products. (b) Material flows of reverse logistics. • Flows of returned goods. Material flows are directed from market sites (σ ¼ W + 1) to sites of the SC stage of external recycling (λ ¼ W + 2). • Flows of recycled goods. Material flows are directed from sites of the SC stage with external recycling (σ ¼ W + 2) to the production stages (λ ¼ 1, . . ., W). • Material flows of disposed goods cannot occur in course of external recycling, as disposal and recycling are processed at the same network site. For external recycling, it is assumed that there are no rejects during the production process. Hence, the supplied quantities are equal to the required quantities (according to given bills of materials) taking into account the ingoing returns of recycled products from the external sites. Instead, returns are expected from the customers at the market stage. In accordance with the requirements of short-term continuous-time scheduling, the return of defective deliveries as well as packaging material can be relevant for this problem, as the time period elapsed between demand satisfaction and product return usually does not exceed several days. The aforementioned period is assumed to be exactly determinable if there are standardized procedures for handling returns (e.g., fixed terms for picking up packaging material by third-party logistics providers or an immediate initialization of return shipments of defective goods at the markets). In contrast, longer time periods of months or even years (which are exceeding the short-term planning horizon) would usually be expected in case of warranty returns or end-of-use returns and, thus, the latter cannot be considered in the following approach. A specific share of the quantities returned from the markets to the external sites can be recycled and redistributed to the SC stages of their original manufacturing. The contrary share is disposed of at the same external network sites, while it is even possible to transfer recyclable products to disposal. The remaining characteristics of processes in forward and reverse logistics

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production stages

1,m

σ=1

1,1

2,n σ=2

2,1

…

…

…

W,o W,1

σ=W

market stage

W+1,1

…

σ=W+1

W+1,p

recycling and disposal stage

…

W+2,1

σ=W+2

W+2,q

Legend: forward flows of saleable goods

σ,s

network site s of SC stage σ

σ,s

market site s of SC stage σ

reverse flows of returned goods reverse flows of recycled goods

Fig. 6.2 General network structure for external recycling of non-perishable goods in continuoustime scheduling

6.1 Model Formulations

109

(including production, recycling, and demand satisfaction) and the features of the planning horizon are equal to those applicable to internal recycling systems (see Sect. 6.1.1.1). Consequently, the model formulation is based on the same sets, indices, parameters, and variables taking into account the following exceptions: An additional time parameter specifying the aforementioned delay period for returns at the markets is introduced. An additional variable is required to determine quantities of stagespecific products to be shipped from a market to an external site for reasons of handling in reverse logistics. With regard to the description of relevant decisions, it needs to be mentioned that a subsequent usage of a site for different processes (in particular, recycling and disposal) is subject to external sites exclusively. The mixed-integer linear program for continuous-time scheduling in networks with external recycling of non-perishable goods can be formulated as follows: For maximizing the overall profit in production-distribution networks for external handling of returned goods, objective function (6.1) can be applied accordingly. Further parts that can be taken from the model of internal recycling (Sect. 6.1.1.1) are the constraints for demand satisfaction [(6.2)], for respecting production and transportation capacities [(6.11) and (6.15)] as well as recycling and disposal capacities [(6.13) and (6.14)], for respecting the minimum operation times at sites [(6.19) and (6.20)], for linking end and start times of sites to be connected by a material flow [(6.21) and (6.22)], for limiting the time of temporary storage [(6.23)], for coordinating deliveries to markets with given due dates of demand satisfaction [(6.24)], for respecting the end of the planning horizon [(6.25) and (6.26)], and for limiting the environmental impact of SC operations with regard to greenhouse gas emissions [(6.27)]. The non-negativity [(6.28)–(6.33)] and binary constraints [(6.36)–(6.38)] are also applicable. Furthermore, the following constraints need to be added to the model formulation: X xqs ¼ Bλσ  pr s ; 8s 2 Sσ ; σ ¼ 2, . . . W; λ ¼ 1, . . . , σ  1

ð6:39Þ

q2Sλ

As external recycling focuses on returns originating from the customers at the markets, there are no rejects to be processed internally at the production sites of the SC stages σ ¼ 2, . . ., W. Consequently, the required supply of goods at these sites can be calculated according to (6.39) by multiplying the production quantity (to be determined within the overall optimization of the system) with the input-output coefficient resulting from the given bills of materials. The whole supplied quantity of stage-specific goods (i.e., goods that are manufactured in one of the preceding SC stages λ ¼ 1, . . ., σ  1) is assumed to be suitable for production. X q2SWþ2

qr λsq ¼ αs 

X xqs ; 8s 2 SWþ1 ; λ ¼ 1, . . . , W q2Sλ

ð6:40Þ

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The driving force of recycling can be found at sites of the market stage σ ¼ W + 1. There, products obtained from different preceding production stages satisfy the given demand. At each of the markets, a specific share of ingoing products needs to be processed further in reverse logistics due to (6.40). According to the left-hand side of the constraint, the product quantity of a stage-specific type that is to be returned from a market can be split up to sites of the SC stage σ ¼ W + 2. The latter is composed of external partners specialized on recycling and disposal. The consideration is limited to specific types of returns, as an immediate handling is a prerequisite for applying continuous-time scheduling. By slight modifications of the formulation, even product-specific shares of returned quantities can be introduced. W X

qr λsq ¼ xsq ; 8s 2 SWþ1 ; q 2 SWþ2

ð6:41Þ

λ¼1

Constraint (6.41) defines the material flows between sites of the market stage σ ¼ W + 1 and the recycling/disposal stage σ ¼ W + 2. This definition is necessary due to the integration of the flows of returned goods into scheduling as well as transportation capacity planning [see constraints (6.15), (6.21), and (6.22)]. In the aforementioned constraints, the flows are handled regardless of the stage-specific product, so the return quantities at the left-hand side of (6.41) can be summed up accordingly. X

βs  qr λqs ¼ nuλs þ

q2SWþ1 W X λ¼1

X xsq ; 8s 2 SWþ2 ; λ ¼ 1, . . . , W q2Sλ

" nuλs

þ

X

ð1  βs Þ 

ð6:42Þ

# qr λqs

¼ fr s ; 8s 2 SWþ2

ð6:43Þ

q2SWþ1

A site-specific coefficient allows for determining the share of the quantities returned from different market sites of the SC stage σ ¼ W + 1, which can be recycled. However, it becomes obvious from constraint (6.42) that it is not necessary to process the whole recyclable quantity in each case, as the possibility to transfer recyclable quantities to disposal exists. After the recycling treatment, the recovered quantities of stage-specific products can be returned to the SC stages, which are able to manufacture new products of the same kind. In the context of reverse logistics, the recycling quantity can be split up to different sites of the same SC stage. Subsequently, the external sites of the final SC stage σ ¼ W + 2 can be used for disposal. Therefore, the contrary share of recyclable returns from different market sites is determined. It is increased by the aforementioned transfers of recyclable products that should be disposed of for economic reasons. As there are no differences in treatment at the landfills, all stage-specific types of products can be summed up to obtain the overall disposal quantity at an external site of the SC stage σ ¼ W + 2 within constraint (6.43).

6.1 Model Formulations

111 W X X λ¼1 s2SWþ2

nuλs  NL

ð6:44Þ

The system-wide overall quantity of recyclable products that is disposed of at the landfills can be limited to an upper bound in constraint (6.44). In the context of external handling of returns, it needs to be considered that transferred amounts can only occur at sites of the final SC stage, whereas the site-specific types of products that are returned from the markets may stem from all production stages λ ¼ 1, . . ., W. W þ1 X

X

X

xsq ¼ pr s þ

λ¼σþ1 q2Sλ

xqs ; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:45Þ

q2SWþ2

The characteristics of a closed loop system are also applicable to network structures with external recycling. The latter becomes obvious from constraint (6.45), as the manufactured quantity of a stage-specific product at sites of the SC stages σ ¼ 1, . . ., W can be supplemented or even (fully or partially) replaced by the recycled quantities obtained from different sites of the final SC stage σ ¼ W + 2. The sum of ingoing and produced quantities at a production site needs to be (re-) distributed to sites of subsequent (but not necessarily adjacent) SC stages that can be other production stages or even the market stage. W X

" nuλs

λ¼1

þ

X

# βs 

qr λqs

¼ dr s ; 8s 2 SWþ2

ð6:46Þ

q2SWþ1

By constraint (6.46), the recycling quantity at each external site of the SC stage σ ¼ W + 2 is exactly determined. For this purpose, the recyclable quantity calculated at the left-hand side of constraint (6.42) is adjusted for the recyclable quantity of the same stage-specific product that is transferred to disposal at the same site. As characteristic parameters of recycling are assumed to be independent of the product type (e.g., variable costs of recycling, maximum recycling capacity), the resulting quantity can be summed up as depicted. X

xqs  XC s  ys ; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:47Þ

q2SWþ2

Specific requirements for limiting the throughput capacity of recycled products at sites of the production stages σ ¼ 1, . . ., W are modeled in constraint (6.47). In particular, the returned material flows can only stem from the external sites of the SC stage σ ¼ W + 2. Moreover, an upper bound for the sum of recycling and disposal quantities according to the given maximum capacity is only relevant for sites of this SC stage.

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Continuous-Time Scheduling in Green Supply Chain Management

sp1s þ

pr s ¼ sp2s ; 8s 2 Sσ ; σ ¼ 1, . . . , W SDs

ð6:48Þ

sp1s þ RT s ¼ sp2s ; 8s 2 SWþ1

ð6:49Þ

sp1s þ

fr dr s þ s ¼ sp2s ; 8s 2 SWþ2 RDs FDs

ð6:50Þ

The connection of start and end times of processes at the same network site needs to be adjusted to the particularities of external recycling. From constraint (6.48), it becomes obvious that the SC stages σ ¼ 1, . . ., W are considered for production exclusively. The processing time solely depends on the production quantity and speed. Constraint (6.49) needs to be added to the model formulation in order to capture the time period that elapses between the time of sales at a market and the point in time that is expected for returning (parts of) the sales deliveries from the same market. Inherently, the aforementioned time period needs to be in line with the short-term planning horizon. As recycling and disposal processes can be conducted subsequently at the external sites, the overall processing time at these sites is composed of both components according to constraint (6.50). Finally, non-negativity constraints (6.51) and (6.52) need to be added

6.1.1.3

qr λsq  0; 8s 2 SWþ1 ; q 2 SWþ2 ; λ ¼ 1, . . . , W

ð6:51Þ

nuλs  0; 8s 2 SWþ2 ; λ ¼ 1, . . . , W

ð6:52Þ

Combined Internal and External Recycling

In the following model for combined recycling, both the internal recycling of rejects during the production process (see Sect. 6.1.1.1) and the external recycling of defective goods occurring after delivery at the markets (see Sect. 6.1.1.2) are assumed to be applicable. The latter processes can be integrated from a formal perspective, as they are initiated at different SC stages of the network. Moreover, a connection is reasonable if the same kinds of raw materials or intermediate products are recovered in both internal and external recycling. As a consequence, recovered quantities available for manufacturing at the production sites are composed of different sources. The latter are considered to be equivalent and, moreover, even equivalent to the manufacturing of new products. Furthermore, available disposal sites may benefit from a better workload, as they can receive material flows from all SC stages except the market stage. As a result of all the aforementioned features of the network, the analysis of the resulting closed loop systems allows for the determination of trade-offs between internal and external recycling in course of the overall optimization. The general SC network is structured as follows (combination of Fig. 6.1 and Fig. 6.2):

6.1 Model Formulations

113

(a) One production stage (σ ¼ 1). See internal recycling in Sect. 6.1.1.1. (b) W  1 production/recycling stages (σ ¼ 2, . . ., W). See internal recycling in Sect. 6.1.1.1. (c) One market stage (σ ¼ W + 1). See external recycling in Sect. 6.1.1.2. (d) One recycling/disposal stage (σ ¼ W + 2). See external recycling in Sect. 6.1.1.2. Potential material flows can occur between sites of the SC stages σ and λ: (a) Material flows of forward logistics. Each site of an SC stage with production (σ ¼ 1, . . ., W ) can supply each site of a subsequent, but not necessarily adjacent, production or market stage (λ ¼ σ + 1, . . ., W + 1) with stage-specific products. (b) Material flows of reverse logistics. The following types can be distinguished: • Flows of returned goods. Material flows are directed from market sites (σ ¼ W + 1) to sites of the SC stage of external recycling (λ ¼ W + 2). • Flows of recycled goods. Material flows are directed from sites of the SC stage of external recycling (σ ¼ W + 2) to the production stages (λ ¼ 1, . . ., W), on the one hand, and from sites of SC stages with internal recycling (σ ¼ 2, . . ., W ) to preceding production stages (λ ¼ 1, . . ., σ  1), on the other hand. • Flows of disposed goods. Material flows are directed from sites of SC stages with internal recycling (σ ¼ 2, . . ., W) to the SC stage of final disposal (λ ¼ W + 2). Recall that material flows of disposed goods cannot start from the first SC stage. Due to the limitation of the latter to the production of primary goods, rejects from deliveries to this SC stage cannot occur. The model formulation is based on the sets, indices, parameters, and variables introduced in Sects. 6.1.1.1 and 6.1.1.2. Merging the decisions relevant for internal and external recycling appropriately, the mixed-integer linear program for continuous-time scheduling in networks with combined recycling of non-perishable goods can be formulated as follows: Several components being part of previous model formulations can be taken on accordingly. The latter applies to the profit-maximizing objective function [(6.1)], as well as the constraints for demand satisfaction [(6.2)], for determining quantities of internal [(6.3)–(6.6) and (6.9)] and external recycling [(6.40)–(6.42) and (6.46)], for respecting the capacities [(6.11), (6.13), (6.14), and (6.15)], for determining the processing times at the sites [(6.16), (6.17), (6.49), and (6.50)], for respecting the minimum operation times at the sites [(6.19) and (6.20)], for the coordination of sites and material flows in continuous-time scheduling [(6.21)–(6.26)], and for limiting the overall impact of emissions caused by SC operations [(6.27)]. Finally, the previous non-negativity [(6.28)–(6.35) and (6.51)–(6.52)] and binary constraints [(6.36)–(6.38)] need to be added. The remaining constraints are to be modeled as follows:

114

6 W þ1 X

X

xsq ¼ pr s þ

λ¼σþ1 q2Sλ

Continuous-Time Scheduling in Green Supply Chain Management W X X X xqs þ xqs ; 8s 2 Sσ ; σ ¼ 1, . . . , W λ¼σþ1 q2Sλ

ð6:53Þ

q2SWþ2

For combined recycling, it is assumed that recycled goods being returned to the production stages σ ¼ 1, . . ., W of their original manufacturing (for supplementing or replacing new products) can be supplied from two different sources. Due to (6.53), there are material flows from sites of the subsequent production stages λ ¼ σ + 1, . . ., W (resulting from internal recycling) and flows from sites of the external recycling stage λ ¼ W + 2. As it is guaranteed that both the ingoing and the produced quantities at a site of the same SC stage are composed of the same kind of product, the material flows can be merged accordingly at the right-hand side of the constraint. Considering the left-hand side additionally, it becomes obvious that the overall quantity of goods available at a specific site can be split up into material flows directed to sites of subsequent production stages or even the market stage. W þ2 X

X

λ¼σþ1

q2Sλ

xqs  XC s  ys ; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:54Þ

λ6¼Wþ1

Constraint (6.54) restricts the sum of ingoing material flows from internal and external recycling at a site of the production stage σ ¼ 1, . . ., W to a maximum throughput capacity. W X X λ¼2 q2Sλ

xqs þ

W X λ¼1

" nuλs þ

X

# ð1  βs Þ  qr λqs ¼ fr s ; 8s 2 SWþ2

ð6:55Þ

q2SWþ1

The composition of disposal quantities at external sites of the SC stage σ ¼ W + 2 becomes obvious from constraint (6.55). Firstly, there are disposed quantities of rejects from the production sites. If these rejects could not be recovered in course of internal recycling (that has been taken place at the production sites), they are directed (regardless of their stage-specific product type) to an external disposal site. Secondly, there are disposed quantities of returns from the markets. If these returns could not be recovered in course of external recycling (that has been taken place at the same external sites), they are collected (regardless of their stage-specific product type) at the external disposal site. Both of the aforementioned disposal quantities are merged and commonly processed for their final storage at landfills. σ 1 X W X X λ¼1 σ¼2 s2Sσ

nuλs þ

W X X λ¼1 s2SWþ2

nuλs  NL

ð6:56Þ

6.1 Model Formulations

115

Again, it is possible to transfer recyclable quantities to disposal. Constraint (6.56) takes into account that the disposal of recyclable quantities (that may cover different stage-specific products) is part of both the internal and the external recycling treatment. The sum of different transfers is restricted to a maximum overall quantity in order to ensure the formation of closed loops. It is possible that the available limit is solely used by one of the two recycling types, or alternatively by a combination of both. By slight adjustments of modeling, it is possible to impose separate limits for internal and external recycling.

6.1.2

Networks with Recycling of Perishable Goods

The quality of perishable goods is predominantly associated with freshness that is decreasing within the shelf life of products, while the latter is affected by chemical deterioration and physical instability (Kong and Singh 2011). As most perishables to be sold at the markets meet the criteria of primary products (e.g., fruits and vegetables), the organization of the related SC network needs to focus on the optimization of the distribution structure instead of the production system. Different alternative channels of distribution (e.g., direct deliveries on the one hand, usage of hierarchically arranged SC stages of warehouses on the other hand) need to be assessed in order to reduce lead times and maximize the overall profit of the network. For coordination, a high degree of flexibility regarding the time points of production and processing is required, which can be obtained by machines or mechanically assisted devices that are available around the clock. Furthermore, the usage of modern just-in-time transportation systems providing suitable transport conditions for short- and long-distance shipments is necessary. Upstream to retail distribution, existing scopes of action regarding intermediate storage (in order to bundle and split up deliveries or to delay sales until acceptable prices can be realized at the markets) should be exploited (Shewfelt et al. 2014). Specific challenges arise for production-distribution systems of perishable goods with regard to GSCM. Typically, a recycling of the fresh produce after its quality decrease so as to obtain products of their original quality level is not possible. Due to the aforementioned processes of deterioration, a recycling treatment cannot result in recovered goods being equal (or even comparable) to the goods that are available for sales after their initial production process. Hence, the formation of closed loop SC networks (as described for the recycling of non-perishable products; see Sect. 6.1.1) is not possible in this case. Nevertheless, recycling processes resulting in secondary products (e.g., jam from fruits) can be considered to reduce the waste from nonmarketable fresh produce. Besides environmental aspects, additional revenues (that increase the main revenues obtainable by in-time selling of perishables) can be generated by the network partners. In common with external recycling (see Sect. 6.1.1.2), the recycling processes are initiated by returns at the markets. Although the quality decrease of the fresh produce occurs continuously, discrete changes in the empirical perception of product features can be assumed. For

116

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Continuous-Time Scheduling in Green Supply Chain Management

standardizing the marketability of perishable products and facilitating the related business transactions, quality grades were widely introduced by trade organizations and authorities (USDA 2018). These grades are related to product characteristics (e.g., color, smell, consistency) that are assumed to change after a certain time period has passed. Consequently, the time period that has been spent from production of the perishables until their final sales at the markets needs to be taken into account for short-term continuous-time network planning. Furthermore, additional effects of the realized quality grades on the prices (and thus the revenues) as well as the share of goods to be handled in reverse logistics are relevant for modeling. The general structure of the SC network is described as follows (see Fig. 6.3): (a) One production stage (σ ¼ 1). The first SC stage of the network is reserved for production of perishable goods (e.g., by harvesting). (b) W  1 distribution stages (σ ¼ 2, . . .W ). These SC stages contain a hierarchically structured distribution system of warehouses. During the delivery of perishables to the warehouses, rejects that need to be directed to the disposal sites occur. (c) One market stage (σ ¼ W + 1). This stage contains markets, whose demand for a product package of the perishable is to be satisfied by sites of preceding SC stages. (d) One recycling/disposal stage (σ ¼ W + 2). Sites of the last SC stage can be used for the recycling and the disposal of perishable goods. Potential material flows are defined between sites of the SC stages σ and λ (see Fig. 6.3): (a) Material flows of forward logistics. Each site of a production or a distribution stage (σ ¼ 1, . . .W ) can supply each site of a subsequent, but not necessarily adjacent distribution or market stage (λ ¼ σ + 1, . . ., W + 1). (b) Material flows of reverse logistics. The following two types can be distinguished: • Flows of returned goods. Material flows are directed from market sites (σ ¼ W + 1) to sites of the last SC stage (λ ¼ W + 2) for recycling and/or disposal. • Flows of disposed goods. Material flows are directed from sites of distribution stages (σ ¼ 2, . . ., W) to the last SC stage (λ ¼ W + 2) for disposal. The considered perishable is produced at sites of the first SC stage (e.g., greenhouses or farms). After leaving this stage, production is finished as the product reaches marketability according to the given demand of the customers, which are assumed to be organized as wholesalers. The latter do not order single items, but packages containing the requested quantity of the primary product. Even goods with similar properties with regard to shelf life, such as product variants, could be taken into account. The demand needs to be satisfied at once (i.e., for marketing reasons it is not allowed to split up the product packages), while the product quantities can be composed of different deliveries arriving at the same point in time. Feasible time

6.1 Model Formulations

117

production stage

1,m

σ=1

1,1

distribution stages 2,n σ=2

2,1

…

…

…

W,o W,1

σ=W

market stage

W+1,1

…

σ=W+1

W+1,p

recycling and disposal stage

…

W+2,1

σ=W+2

W+2,q

Legend: forward flows of saleable goods

σ,s

network site s of SC stage σ

σ,s

market site s of SC stage σ

reverse flows of returned goods reverse flows of disposed goods

Fig. 6.3 General network structure for recycling of perishable goods in continuous-time scheduling

windows for supplying the product packages can be prescribed by the customers according to their sales planning. Both direct deliveries (from a producer to a market) and indirect deliveries (via warehouses of at least one distribution stage) are possible.

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Continuous-Time Scheduling in Green Supply Chain Management

Due to the assumed hierarchical distribution structure (e.g., central, regional, and local warehouses that need to be passed in that sequence in case they are used), transportation can only be directed via warehouses belonging to different SC stages, whereas exchanges between sites of the same SC stage are prohibited. All sites of the distribution stages (being continuously available to the SC within the planning horizon) can be used for merging and splitting of transportation quantities that occur on different partial paths of material flows. For enabling a better coordinated scheduling of the flows between the sites, temporary storage can be used to extend the given transportation times. According to the general assumption for all models in continuous-time scheduling, each site is only available (and thus schedulable) once during the short-term planning horizon, i.e., exactly one start time and one end time can be assigned to common operations of a network site. Consequently, goods must arrive at the same point in time, and furthermore, they need to be processed and stored as a whole before they are redelivered from the warehouse simultaneously. In this context, it needs to be ensured that the sites’ maximum throughput capacity is respected so that the total material flow does not exceed this upper bound. For reasons of appropriate processing (e.g., incoming inspection, warehousing), minimum storage times can be prescribed for warehouses. All markets are characterized by a given demand of the perishable good that is provided by the SC network. This demand is to be satisfied in full, while revenues are only determined by the quality grades realized for the product packages. The grades are classified by a discrete set of alternatives (e.g., high, medium, or low quality), while the realization of an alternative depends on the time period elapsed between picking up the goods from the production site and selling them at a market. In particular, given maximum delivery times must be met for reaching a specific quality grade. Inherently, the sales price of a product package decreases with decreasing quality grades. Exactly one quality grade is to be assigned to a product package. Decayed perishables not being marketable anymore after reaching their expiry date can be taken into account if the poorest quality grade is associated with a sales price of zero, and furthermore, a hundred percent share of quantities is to be disposed of at the landfills. As the deliveries to a market can be composed of different material flows, the quality grade is determined by the maximum of total delivery times, which are spent on material flows starting from any production site that is used to supply a specific market. The aforementioned assumption is justified if a price differentiation according to product qualities is not intended at the markets for at least one of the following practicality reasons: (a) The wholesalers at the markets strive for a homogenous offering to avoid inadvertent or even fraudulent mixing of products by the customers or the sales staff if the goods are sold loosely. The latter is typical for many types of perishables (e.g., fruits). (b) The wholesalers may be interested in avoiding a differentiated product offer, as the latter would draw their customers’ attention to the fact that they also sell

6.1 Model Formulations

119

goods that do not meet the highest-quality standard. The latter is usually applicable to discounters. Due to the aforementioned assumptions on the assignment of quality grades, the system-wide optimization of the SC network will strive for keeping the product quality at a single market (and thus the delivery times of relevant material flows to this market) as homogenous as possible, because parts of the delivery that would be good for a higher product quality could be considered for selling at another market at a higher price if they were separated. The overall schedule of forward and reverse logistics is optimized by a central coordinator (e.g., the fresh producer to be considered as a focal company due to its bargaining power), so as to maximize its overall profit realizable at the given demand at the markets. Moreover, the analysis reveals trade-offs between providing freshness at the markets (which influences the realization of quality grades and, thus, the sales prices for the product packages) and organizing the delivery of perishable goods to the markets (which entails the composition of material flow paths between producers and markets possibly directed via different warehouses and influences the costs of distribution). Inherently, a continuous-time consideration allows for the best possible coordination of 24/7 operations. The multi-day planning horizon should align with the demand period of the customers and is supposed to be repeated (taking into account possible changes of data) consecutively over time. The model formulation is based on the following sets and indices: The main index sets are defined for SC stages, sites, events, and quality grades. The overall number of SC stages depends on the number of distribution stages W  1 that are arranged subsequently between the production and the market stage. The final SC stage is reserved for recycling and disposal. The sets of sites and event boundaries are defined as in Sect. 6.1. A set of quality grades is introduced for taking discrete product deterioration into account. As the optimization of material flow paths should be based on the method of power sets, the following definitions are additionally required: W

SW

Set of all warehouse sites; SW≔ [ Sσ

℘0(SW)

Filtered power set of the set SW; ℘0(SW) ≔ {U| U ⊆ SW^| U \ Sσ |  1, σ ¼ 2, . . ., W} Set of sites belonging to one potential material flow between sites s and q; NsqU ≔ U [ {s, q}; U 2 ℘0(SW); (s, q) 2 S1  SW + 1

NsqU

σ¼2

The following parameters need to be determined for each specific problem: Costs and revenues. Comparable to SC networks for handling non-perishable goods, the model formulation contains fixed and variable costs of production, storage, and recycling/disposal. Due to the assumed network structure (see Fig. 6.3), the aforementioned costs are unambiguously assigned to specific SC stages. Further fixed and variable costs need to be considered for transportation. Peculiarities arise in the context of revenues, as it is not possible to reintegrate recycled quantities into the forward logistics of the perishable good. Besides

120

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Continuous-Time Scheduling in Green Supply Chain Management

revenues of realized sales of primary products (perishables) at the markets, there are additional revenues expected for the sale of secondary products (non-perishables or other perishables with a shelf life exceeding the assumed planning horizon). Consequently, only the revenues of the first group depend on the realization of the quality grade. Capacities. Production capacities are relevant for the first SC stage, whereas storage capacities limit the throughput quantities at the sites of the following distribution stages. Recycling and disposal capacities need to be considered at sites of the last SC stage. Transportation capacity is to be respected for all potential paths of material flows. Time parameters. For both transports in forward and reverse logistics, fixed transportation times are given. Temporary storage times can be limited to an upper bound. Minimum processing times can be prescribed for sites of production, distribution, and recycling/disposal stages. There are maximum distribution ranges for an ordered product package of the perishable, which need to be respected for realizing the potential quality grades. These ranges refer to the maximum time differences between the sale at the market and the end of production at a site that is used to supply this market directly or indirectly. As exactly one quality grade must be assigned to each market, the distribution range of the poorest grade should cover the maximum possible delivery time that could be taken by a material flow path. Another parameter specifies the delay of returns at the markets. The end time of the planning horizon is given. Speed parameters. The production speed is relevant for determining processing times at the first SC stage. The recycling and disposal speed is to be considered within the last SC stage. Coefficients. A first coefficient specifies the share of quantities delivered to a market, which needs to be processed in reverse logistics at sites of the final SC stage. A second coefficient determines the share of the returned quantity that needs to be disposed of. Both of the aforementioned coefficients depend on the realized quality level of the perishable. Finally, a last coefficient specifies the expected share of ingoing quantities at sites of the distribution stages that need to be disposed of immediately due to processing damages. Environmental parameters. The environmental parameters used for determining and limiting emissions can be taken from the model for non-perishables (see Sect. 6.1). The overall disposal of recyclable perishables can also be limited. Other parameters. For each market, the demand for the perishable product obtainable from each of the preceding SC stages is given. A sufficiently large number M needs to be chosen. The decisions with regard to the operations at the sites and the transportation between the sites can be taken from the model for handling non-perishables (see Sect. 6.1) while taking into account the following exceptions: Firstly, the fixed assignment of products to SC stages (in terms of “stage-specific products”) can be revoked, as the production of the perishable good is finished in the first SC stage. Secondly, the assumptions on external recycling with regard to a subsequent site usage for different processes (see Sect. 6.1.1.2) need to be applied accordingly.

6.1 Model Formulations

121

Decisions relevant for each market site do include not only the time of sales but also the assignment of a quality grade to the ordered product package. The mixed-integer linear program for continuous-time scheduling in networks with recycling of perishable goods is based on a method using power sets. Several constraints can be taken from previous model formulations as they are basic elements of continuous-time scheduling in forward and reverse logistics. In particular, this applies to constraints for respecting capacities [(6.11), (6.13), (6.14), and (6.15)], for determining the required production time at sites of the first SC stage [(6.16)], for setting minimum processing times [(6.19) and (6.20)], for implementing continuous-time scheduling [(6.21) and (6.22)], for limiting times of temporary storage [(6.23)], for considering the end time of the planning horizon [(6.25) and (6.26)], for limiting the overall emissions caused by the SC network [(6.27)], for capturing the delay at the market sites [(6.49)], and for determining processing times of recycling and disposal at the sites of the last SC stage [(6.50)]. Furthermore, the non-negativity constraints [(6.28)–(6.30), (6.32), and (6.33)] and the binary constraints [(6.36) and (6.38)] can be applied accordingly.

6.1.2.1

Objective Function Max Φ XX X es Φ¼ gfs  Rfs þ dr s  R s2SWþ1 f 2F



s2SWþ2

W X X PF s  ys þ PV s  pr s σ¼1 s2Sσ

X XX  TF sq  tpsq þ TSsq  st sq þ TV sq  xsq

ð6:57Þ

ðσ, λÞ2K s2Sσ q2Sλ



X

RF s  yr s þ RV s  dr s þ FV s  fr s

s2SWþ2

Objective function (6.57) is used to calculate the overall profit of the SC network. Firstly, it contains revenues that are composed of two different types. On the one hand, there are revenues for selling the ordered product packages of the perishable good at the markets. As the demand needs to be satisfied in full [see constraint (6.58)], these revenues are not affected by the quantity delivered to the markets, but by the realized quality grade instead. In this context, it is possible to define revenues of zero (or even negative revenues interpretable as penalty costs) for completely decayed products in the poorest quality level. On the other hand, a second component includes expected revenues from secondary products resulting after the recycling treatment of the perishable. As the recycled products are typically characterized by non-perishability (see models in Sect. 6.1) or by perishability with a shelf

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life that exceeds the assumed short-term planning horizon, related material flows to markets are excluded from consideration. Secondly, the objective function accounts for fixed and variable costs of production that occur at sites of the first SC stage. Thirdly, fixed and variable costs of transportation need to be considered. Variable costs do refer not only to the quantities delivered on the realized material flow paths but also to the extension of given transportation times by additional times of temporary storage. The potential material flow paths contain transports from a site of an SC stage to another site of a subsequent (but not necessarily adjacent) SC stage. Besides flows of forward logistics, these paths include reverse flows directed from market sites to external recycling/disposal sites. Fourthly, fixed and variable costs of recycling and disposal need to be subtracted from the overall profit. The latter occur at external sites of the last SC stage.

6.1.2.2

Constraints of Forward Flows W X X

xqs ¼ Ds ; 8s 2 SWþ1

ð6:58Þ

λ¼1 q2Sλ

The demand of the markets being part of the SC stage σ ¼ W + 1 needs to be satisfied in full according to constraint (6.58). Due to the consideration of product packages whose supply is contractually guaranteed, delivery shortages must not occur. Each market can order a different package containing different quantities of the perishable good. The requested quantities can be obtained from the sites of the goods production (SC stage σ ¼ 1) or from sites of any distribution stage (SC stages σ ¼ 2, . . ., W ), while the overall quantity can be split up into different material flows that are simultaneously directed to the same market. W þ1 X

X

xsq ¼ pr s ; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:59Þ

λ¼σþ1 q2Sλ

From (6.59), the distribution of product quantities available at network sites becomes obvious. These quantities (determined by the variable at the right-hand side of the constraint) result from production at sites of the SC stage σ ¼ 1 or from storage at sites of the SC stages σ ¼ 2, . . ., W. The production or storage quantities can be split up into different material flows according to the left-hand side of the constraint. The flows can be directed to sites of subsequent distribution stages λ ¼ σ + 1, . . ., W or even to sites of the market stage λ ¼ W + 1.

6.1 Model Formulations

123

σ 1 X X

xqs ¼ ð1 þ γ s Þ  pr s ; 8s 2 Sσ ; σ ¼ 2, . . . , W

ð6:60Þ

λ¼1 q2Sλ

Rejects of the perishable good caused by processing damages can occur throughout the whole distribution system. In this context, a specific share of ingoing material flows at a warehouse of the SC stages σ ¼ 2, . . ., W is expected to be unusable for redistribution and, thus, needs to be disposed of at the landfills. Acknowledging the occurrence of the aforementioned rejects, the quantities delivered to sites of the distribution stages σ ¼ 2, . . ., W need to be increased so as to obtain the required storage quantities after subtracting the expected disposal. According to (6.60), the aforementioned increase is to be taken into account for determining the overall warehouse supply that is possibly being composed of material flows starting from sites of different preceding SC stages λ ¼ 1, . . ., σ  1.

6.1.2.3

Constraints of Quality Control X

2þ j U j 

tpij ¼ vsqU ; 8ðs, qÞ 2 S1  SWþ1 ; U 2 ℘0 ðSWÞ

ði, jÞ 2 N sqU i 2 Sσ , j 2 Sλ , σ  λ σ, λ 2 ðΓ N sqU , Þ ð6:61Þ sp1q



sp2s

 RG f sq þ ðvsqU  g f q Þ  M; 8ðs, qÞ 2 S1  SWþ1 ; U 2 ℘0 ðSWÞ; f 2 F X gfq ¼ 1; 8q 2 SWþ1

ð6:62Þ ð6:63Þ

f 2F

The assignment of a quality grade to a product package delivered to a specific market depends on the delivery time elapsed during the complete material flow. This time starts when the production is finished at site s 2 S1 and ends with the sales at market q 2 SW + 1. With regard to the set of potential producers supplying a market directly or even indirectly, different delivery times sp1q  sp2s result. However, only the delivery times connected to realized material flows become relevant for meeting the maximum distribution range RGfsq, while the latter is associated with a specific quality grade. Realized material flows between two sites s 2 S1 and q 2 SW + 1 are part of the potential material flows. The latter can be obtained if (besides direct transportation) all possibilities of indirect transportation are taken into account. Indirect transportation is associated with the usage of one or more warehouses along the material flow. In this context, SW is defined as the set of all warehouse sites. Exactly 2jSWj thinkable combinations U of warehouses result from the power

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Table 6.1 Big M method in constraint (6.62) Assignment of a specific quality grade (gfq ¼ 1)

Potential material flow is realized (vsqU ¼ 1) Potential material flow is not realized (vsqU > 1)

Non-assignment of a specific quality grade (gfq ¼ 0) M is added once sp1q  sp2s  RGfsq ) redundancy of constraint (6.62) M is added (vsqU  gfq) times (but at least once) ) redundancy of constraint (6.62)

set ℘(SW) ≔ {U| U ⊆ SW}. As it is not possible to harness more than one warehouse of a specific SC stage, the modified power set ℘0(SW) must be filtered to ℘0(SW) ≔ {U| U ⊆ SW^| U \ Sσ | 1, σ ¼ 2, . . ., W}. If the remaining combinations of warehouses U are complemented by the starting and ending points of the material flow (i.e., the producer s 2 S1 and the market q 2 SW + 1), the new set NsqU results. This set contains all sites belonging to a potential material flow and can be described by NsqU ≔ U [ {s, q}; U 2 ℘0(SW); (s, q) 2 S1  SW + 1. For keeping the model formulation linear, all transportation binary variables tpij of one potential material flow are chained by addition in order to define realized material flows. The variables are built up from the tuples (i, j) 2 NsqU, which represent the sites i 2 Sσ and j 2 Sλ belonging to the SC stages σ, λ 2 ðΓ N sqU , Þ. By σ ≺ λ, it is ensured that only tuples of sites belonging to consecutive SC stages are selected. According to constraint (6.61), vsqU ¼ 1 if a potential material flow is realized (i.e., all sections of a material flow path are filled with a transport), and vsqU > 1 otherwise (i.e., at least one section of a material flow path is not filled with a transport). Due to constraint (6.62), the specific delivery time sp1q  sp2s between a producer s 2 S1 and a market q 2 SW + 1 needs to be within a maximum distribution range RGfsq that is guaranteeing a specific quality grade f 2 F if both of the following two conditions are met: (a) The considered material flow is realized, i.e., the perishable is distributed between the sites (s, q) 2 S1  SW + 1 using the warehouses U 2 ℘0(SW). Then, vsqU ¼ 1 is valid [constraint (6.61)]. As each market needs to be satisfied in full [constraint (6.58)], at least one potential material flow must be realized for each market. (b) A specific quality grade is assigned to a market. The latter is forced by constraint (6.63). The system strives for selling a product package at the highest price [see objective function (6.57)]. As more than one quality grade cannot be assigned to a market, the best possible quality grade is selected from the set of possible alternatives. Formally, sp1q  sp2s  RGfsq results from fulfilling both of the aforementioned conditions (see Table 6.1). Otherwise, constraint (6.62) becomes redundant, as the big parameter M (that needs to be chosen so as to exceed the length of the planning horizon T ) is added at least once at the right-hand side of the constraint. Then, the

6.1 Model Formulations

125

difference on the left-hand side representing the feasible delivery time is unrestricted, as its upper bound exceeds the length of the entire planning horizon. If more than one material flow to a market is realized (e.g., as different producers are harnessed or a combination of direct and indirect deliveries is optimal), the one with the highest delivery time determines the selection of the quality grade, because gfq needs to be chosen so as to satisfy constraint (6.62) for each realized material flow associated with this market. Different means of transportation could easily be implemented by slight modifications.

6.1.2.4

Constraints of Reverse Flows X

xsq ¼ γ s  pr s ; 8s 2 Sσ ; σ ¼ 2, . . . , W

ð6:64Þ

q2SWþ2

Due to constraint (6.64), the disposal quantity resulting from rejects of the perishable good throughout the distribution system [see (6.60)] is immediately directed to the last SC stage, while it can be split up to different sites. αfs  Ds  gfs ¼

X

gr fsq ; 8s 2 SWþ1 ; f 2 F

ð6:65Þ

q2SWþ2

X gr fsq ¼ xsq ; 8s 2 SWþ1 ; q 2 SWþ2

ð6:66Þ

f 2F

By constraint (6.65), the quantities returned from the market sites of the SC stage σ ¼ W + 1 can be determined. The share of the completely satisfied demand that needs to be processed in reverse logistics (including both recycling and disposal) depends on the quality grade that is realized at the market. Inherently, decreasing quality grades are accompanied by an increase of the return share. As reverse processing entails costs and, furthermore, the revenues of secondary products resulting after the recycling treatment are usually expected to be lower than the revenues of the primary perishable, the system will strive for keeping the return share as low as possible. According to the right-hand side of the constraint, it is possible to split up the returned quantities to different sites of the last SC stage. The transportation quantities directed from the market stage to the recycling/disposal stage are required for the calculation of transportation costs in objective function [(6.57)], as well as for capacity planning [(6.15)] and scheduling [(6.21) and (6.22)]. For their determination, the returned quantities on the relevant network paths need to be summed up by all possible quality grades in (6.66), while only one of the latter is realized due to (6.63).

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Continuous-Time Scheduling in Green Supply Chain Management

X X βfs  gr fqs ¼ nus þ dr s ; 8s 2 SWþ2

ð6:67Þ

q2SWþ1 f 2F

nus þ

X X

W X X  1  βfs :gr fqs þ xqs ¼ fr s ; 8s 2 SWþ2

q2SWþ1 f 2F

ð6:68Þ

λ¼2 q2Sλ

The returned quantities arriving from different sites of the market stage σ ¼ W + 1 at a recycling/disposal site of the last SC stage σ ¼ W + 2 are separated according to a share depending on the quality level realized at the market. In this context, the recyclable quantity is determined in (6.67). It can be reduced by an economically motivated disposal, which increases the quantities that need to be disposed of for reasons of an insufficient product quality. The latter amounts are determined in the second term at the left-hand side of constraint (6.68) and result from applying the contrary share to return quantities that arrive at the recycling/disposal site. Moreover, the disposal quantities representing rejects at the warehouse sites [constraints (6.64)] need to be added additionally. The resulting overall quantity to be disposed of at the landfills is processed according to (6.14) and (6.50). X

nus  NL

ð6:69Þ

s2SWþ2

Finally, constraint (6.69) needs to be part of the model formulation in order to capture further environmental aspects of GSCM. Similar to constraints (6.7), (6.44), and (6.56), it limits the quantity of recyclable products that is disposed of.

6.1.2.5

Constraints of Domain Definition

The binary constraints (6.70) and (6.71) as well as the non-negativity constraints (6.72) and (6.73) must be added in order to complete the model formulation. ys 2 f0, 1g; 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:70Þ

gfs 2 f0, 1g; 8f 2 F; s 2 SWþ1

ð6:71Þ

gr fsq  0; 8f 2 F; s 2 SWþ1 ; q 2 SWþ2

ð6:72Þ

nus  0; 8s 2 SWþ2

ð6:73Þ

6.1 Model Formulations

6.1.3

127

Integration of Financial Planning

Continuous-time scheduling in networks with recycling of non-perishable or perishable goods (see Sects. 6.1 and 6.1.2) requires an exact coordination of operations. Beyond that, an appropriate financial management should accompany the scheduling in order to avoid temporary lacks of liquidity. As an enduring provision of sufficient liquidity entails an organization’s ability to make payments as they fall due, it should be managed within short-term intervals, e.g., on a day-to-day basis (Moir 1997, pp. 1–5). For this purpose, all monetary consequences need to be balanced within liquidity-relevant intervals while taking a monetary objective (e.g., profit maximization at the end of the planning horizon) into account. Modeling the problem described above requires an integration of short-term SC planning and financial planning. It combines the continuous-time scheduling of operations and financial transactions with discrete liquidity periods. The following consideration focuses on SC networks of sites with an insufficient capital base (e.g., small- and medium-sized enterprises) and a resulting limited access to capital markets. Facing a lack of liquidity that does not allow for starting the operations without further ado, pre-financing is required in order to initialize operations such as production, recycling, or transportation. Assuming that pre-financing cannot be taken on by the customers at the market stage due to the competitive situation, the network partners are reliant on short-term credits, often characterized by high interest rates and daily interest calculation. In expectation of the revenues that are generated at the end of the multi-day planning horizon at the latest, even a series of follow-up financing can be required, whereas unbalanced payments during the same interval can be compensated by bank overdrafts. In order to run smoothly, it is necessary that the full financing amount is available at the start of each operation. Due to the resulting obligations, the SC companies are forced to cooperate in terms of financial planning. Network-wide liquidity management is based on risk-free financing alternatives. The latter are assumed to be standardized arrangements for bridging operations and sales, such as short-term bank loans. They are permanently available to the SC during the planning horizon. The alternatives can be distinguished by different credit periods and credit rates. The credit amount (to be chosen freely with respect to given limits of creditworthiness) is assumed to be immediately available at the point of time the contract enters into force. The repayment amount including interest is due and payable immediately after the term. Analogously to SC operations, continuoustime scheduling of financial transactions allows for the determination of exact times of payments and repayments. As such an accuracy is not required for liquidity control, the planning horizon is additionally subdivided into a specific number of liquidity periods of equal length (see Fig. 6.4). The latter are used to balance different monetary consequences that stem from both financial and operational planning. The related cash flows are assigned to these periods completely at the start times of production, storage, recycling, disposal, and transportation processes

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6

monetary consequences

credit payment

Continuous-Time Scheduling in Green Supply Chain Management

costs of transportation

costs of operations

financial level

credit repayment

revenues

financing alternative

transport

operational level

production, storage, recycling sales

points in time liquidity periods

a =1

3. E A

2. E A

E A

0

a =2

a =3

… …

time

( A – 1) . E

E

A

a= A

Fig. 6.4 Liquidity periods and assignment of monetary consequences

and at the times of sale, as well as at the start and end times of realized financing alternatives. In general, all of the proposed models for continuous-time scheduling in GSCM (i.e., internal, external, or combined recycling of non-perishable goods (see Sects. 6.1.1.1, 6.1.1.2, and 6.1.1.3), and recycling of perishable goods (see Sect. 6.1.2)) can be extended by integrating the following elements. With regard to financial planning, two new sets of indices need to be introduced: Firstly, there is an index representing the discrete liquidity periods. As being relevant for the model formulation in the following, it should be mentioned that the cardinality of the set is equal to the last index being element of this set. Secondly, an index for the financing alternatives is added. The latter can be interpreted as potential credits that can be taken on by the network managers to a limited extent within the bounds of the short-term planning horizon. Financial parameters. Detailed conditions of financing are to be taken into account, as these are crucial for network-wide liquidity balancing. There are given credit terms (fixed time difference between the end and the start of potential financing alternatives in case of their usage) that cannot be changed. Whereas a prolongation is excluded, a follow-up financing can be realized by taking on another credit immediately at the end of the preceding one. Furthermore, fixed credit rates are assumed to be offered regardless of the capital structure of the SC network. According to the short-term planning horizon, they are given in percent per time unit, while the length of the units depends on the time scale that is used for continuous-time scheduling (e.g., days). Credit limits according to the creditworthiness need to be respected for single transactions as well as for the overall credit amount. As a result of the integration, the following financial decisions complement the previous operational SC decisions (see Sect. 6.1.1.1, taking into account the adjustments for the models described in Sects. 6.1.1.2, 6.1.1.3 and 6.1.2):

6.1 Model Formulations

129

(e) Decisions with regard to liquidity planning. • • • •

Realization of available financing alternatives. Extent of realized financing alternatives (credit amount). Start and end times of realized financing alternatives. Assignment of the monetary consequences resulting from the realization of financing alternatives (credit payments and repayments) to liquidity periods. • Assignment of the monetary consequences resulting from SC operations (costs and revenues) to liquidity periods.

Intuitively, the integration of liquidity planning into continuous-time scheduling of SC networks results in a mixed-integer non-linear model formulation. The latter could arise from the multiplication of the terms of the monetary consequences [e.g., revenues from sales and costs of operations according to (6.1) or (6.57), payments and repayments to be additionally considered for financing alternatives] with binary variables indicating the required event-period-assignment. As the terms for determining the monetary consequences inherently consist of variables themselves (e.g., binary variables indicating the usage of sites, continuous variables for production, recycling or transportation quantities, amounts for determining the extent of realized financing alternatives), non-linearity occurs. Due to expected problems of computability, a linearized model formulation is applied. It is based on introducing non-negative auxiliary variables that represent the realized monetary consequences of both operational and financial events within liquidity periods. A specific system of constraints needs to be established in order to fix the aforementioned auxiliaries to the relevant cash flows in case of an event-period-assignment. According to the considered types of events, there are four different groups of auxiliary variables (apsa for operations at network sites, arsa for sales at markets, atsa for transports starting from network sites, af zoa for financial transactions). After being summed up for the relevant intervals, the resulting period liquidity (represented by the non-negative variable qa in order to avoid temporary lacks of liquidity) can be calculated. The following constraints are to be integrated into mixed-integer linear programs for continuous-time scheduling in networks with recycling of non-perishable goods: Max Φ W þ2 X W þ2 X X X X X  ar sa  apsa  at sa þ af 1oa  af 2oa  qa ; 8a 2 A s2SWþ1

σ¼1

σ¼1 s2Sσ

s2Sσ

o2O

σ6¼Wþ1

 qa ¼

ð6:74Þ 0

8a ¼ 1, . . . , j A j 1

Φ

8a ¼j A j

ð6:75Þ

Objective function (6.74) maximizes the liquidity surplus of the SC network. According to constraint (6.75), the network managers prefer one withdrawal at the

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Table 6.2 Big M method in the system of constraints (6.76)–(6.91) and (6.105)–(6.109)

Constraints (6.76), (6.79), (6.86), (6.89) Constraints (6.77), (6.80), (6.87), (6.90) Constraints (6.78), (6.81), (6.88), (6.91) Constraints (6.105), (6.106), (6.108), (6.109) Combining the constraints

Assignment of event to liquidity Non-assignment of event to period liquidity period Setting upper bound for auxiliary variable: Event-related liquidity consequences Setting lower bound for auxiliary variable: Event-related liquidity consequences Constraints become redundant due to big M

Constraints become redundant due to big M

Fixing the auxiliary variable to eventrelated liquidity consequences

Fixing the auxiliary variable to value of zero

Setting upper bound for auxiliary variable: Value of zero Setting lower bound for auxiliary variable: Value of zero

end of the short-term planning horizon, as the objective variable Φ is assigned to the last liquidity period a ¼ j Aj. For all of the preceding liquidity periods a ¼ 1, . . ., j A j  1, appropriate balancing needs to be guaranteed so as to ensure that the residual liquidity at the end of each period does not fall below zero [see non-negativity constraint (6.110)]. The latter is required to avoid cash flow problems. The aforementioned residual liquidity is composed of different components being connected to different events. The first term refers to the sales at the market sites, the second one to the operations (production, storage, recycling, disposal) at the remaining network sites as well as to the transports possibly starting from each of the available network sites, and the last one to the realization of financial transactions. All three terms can be merged, as they consist of monetary consequences being assigned to the same liquidity period. Inherently, liquidity balancing will strive for zeroing the surpluses of the periods a ¼ 1, . . ., j A j  1 in order to reach the highest possible surplus in the last period a ¼ j Aj. By slight modifications of (6.75), other distributions of the withdrawals (e.g., a uniform distribution or any other arbitrary structures) can be realized. In this context, the consideration of an additional cash management in (6.74) could be useful. It can easily be implemented by adding a continuous variable, which allows for transferring liquidity surpluses from one period to the following, while an interest rate for short-term financial investments can be taken into account. By introducing a system of interrelated constraints based on the Big M method, the auxiliary variables being part of constraint (6.74) are fixed unambiguously (see Table 6.2). The result of combining the constraints in both cases (assignment and non-assignment) is a fixation of the auxiliary variable. According to the constraints being most restrictive, the lower and the upper bound on the fixed value are equal. In the following, the aforementioned system of constraints needs to be specified for

6.1 Model Formulations

131

different types of auxiliary variables and, moreover, for different compositions of event-related liquidity consequences: ar sa  MC s  ys þ

W X X

Rλs  xqs ; 8s 2 SWþ1 ; a 2 A

ð6:76Þ

λ¼1 q2Sλ

ar sa  MC s  ys þ

W X X λ¼1 q2Sλ

!

Rλs  xqs

   M  1  lp1sa ; 8s 2 SWþ1 ; a 2 A ð6:77Þ

ar sa  M  lp1sa ; 8s 2 SWþ1 ; a 2 A

ð6:78Þ

First of all, the auxiliary variable arsa depends on the assignment of the event “sales at a market” to a liquidity period. The upper bound for this value defined in constraint (6.76) is characterized by the monetary consequences of sales. On the one hand, there are costs of marketing, which need to be considered, if the market is selected by the network. On the other hand, there are revenues from stage-specific non-perishable products, which can be delivered from all of the preceding SC stages to the market. The sum of cash flows resulting from the optimization of the overall system at the right-hand side of the constraint can be considered as the common maximum value of a market’s auxiliary variables regardless of the liquidity period. Aspects of the potential assignment of an event to a liquidity period are taken into account in the following constraints. Considering (6.77), the auxiliary variable arsa needs to be greater than or equal to the same aforementioned sum of cash flows if the sales event is assigned to a specific liquidity period. The latter is indicated by lp1sa ¼ 1, while the term on the right-hand side of the constraint containing the big number M is to be omitted in this case. Combining the less-than-or-equal and greater-than-or-equal constraints (6.76) and (6.77) in case of the event-periodassignment, the auxiliary variable is fixed to the sum of cash flows. However, if the binary indicates a non-assignment of an event to a liquidity period by lp1sa ¼ 0 in the opposite case, the constraint (6.77) becomes redundant due to the subtraction of the big number M from its right-hand side. Then, it is necessary to fix the auxiliary variable to the value of zero in order to avoid infeasible degrees of freedom with regard to the calculation of revenues in the objective function, especially as there is an implicit maximization of the variable according to (6.74). This fixation is realized by constraint (6.78) in connection with the non-negativity constraint (6.105). In contrast, constraint (6.78) is redundant for liquidity periods being connected to an event, due to the occurrence of M at the right-hand side of the inequality (see Table 6.2). apsa  ups þ ur s þ ud s þ uf s ; 8s 2 Sσ ; σ ¼ 1, . . . , W, W þ 2; a 2 A

ð6:79Þ

132

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Continuous-Time Scheduling in Green Supply Chain Management

apsa  ups þ ur s þ ud s þ uf s  M  ð1  lp1sa Þ; 8s 2 Sσ ; σ ¼ 1, . . . , W, W þ 2; a 2 A apsa  M  lp1sa ; 8s 2 Sσ ; σ ¼ 1, . . . , W, W þ 2; a 2 A  PF s  ys þ PV s  pr s 8s 2 Sσ ; σ ¼ 1, . . . , W ups ¼ 0 8s 2 SWþ2  RF s  yr s 8s 2 Sσ ; σ 2 L [ fW þ 2g ur s ¼ 0 8s 2 Sσ ; σ 2 Γ∖ðL [ fW þ 1, W þ 2gÞ  RV s  dr s 8s 2 Sσ ; σ 2 L uds ¼ 0 8s 2 Sσ ; σ 2 Γ∖ðL [ fW þ 1gÞ  FV s  fr s 8s 2 SWþ2 uf s ¼ 0 8s 2 Sσ ; σ ¼ 1, . . . , W

ð6:80Þ ð6:81Þ ð6:82Þ ð6:83Þ ð6:84Þ ð6:85Þ

Based on the method explained in Table 6.2, constraints (6.79)–(6.81) and (6.106) refer to the start of operations at network sites of the SC stages σ ¼ 1, . . ., W and σ ¼ W + 2. As the monetary consequences are defined for different domains, it becomes necessary to introduce additional auxiliary variables ups, urs, uds, and ufs in constraints (6.82)–(6.85) and (6.107) in order to obtain variables, which are defined for sites of each SC stage σ ¼ 1, . . ., W, W + 2. Then, the auxiliary variable apsa is fixed to the sum of production  and recycling costs in case  of theevent’s assignment to a liquidity period lp1sa ¼ 1 and to zero otherwise lp1sa ¼ 0 . at sa 

X X TF sq  tpsq þ TSsq  st sq þ TV sq  xsq ; 8s 2 Sσ ; σ ðσ  , λÞ2K q2Sλ σ  ¼σ

¼ 1, . . . , W þ 2; a 2 A ð6:86Þ P P at sa  ð TF sq  tpsq þ TSsq  st sq þ TV sq  xsq Þ  M  ð1  lp2sa Þ;  q2S λ ðσ , λÞ 2 K σ ¼ σ 8s 2 Sσ ; σ ¼ 1, . . . , W þ 2; a 2 A ð6:87Þ at sa  M 

lp2sa ; 8s

2 Sσ ; σ ¼ 1, . . . , W þ 2; a 2 A

ð6:88Þ

Within constraints (6.86)–(6.88) and (6.108), the start of transports from sites of all SC stages σ ¼ 1, . . ., W + 2 is considered. For each SC stage σ selected in the universal quantifier, the SC stages λ, these transports can potentially be directed to

6.1 Model Formulations

133

result from the set of feasible relationships (σ, λ) 2 K. The latter is valid for the specific type of continuous-time models. The monetary consequences are composed of variable costs being dependent on the transportation quantities and on times of temporary storage, as well as of fixed costs for the realization of material flow paths. According to Table 6.2, the auxiliary variable atsa is fixed to the sum of the costs in case of the event’s assignment to a liquidity period aforementioned   lp2sa ¼ 1 and to zero otherwise lp2sa ¼ 0 . af zoa  fizo ; 8z 2 Z; o 2 O; a 2 A af zoa



f izo

 M  ð1 

lf zoa Þ; 8z

2 Z; o 2 O; a 2 A

af zoa  M  lf zoa ; 8z 2 Z; o 2 O; a 2 A

ð6:89Þ ð6:90Þ ð6:91Þ

Constraints (6.89)–(6.91) and (6.109) deal with the usage of financing alternatives by the network managers. Regarding the monetary consequences, the credit amount (z ¼ 1) and the repayment amount (z ¼ 2) need to be distinguished. Based on the Big M method explained in Table 6.2, the auxiliary af zoa is fixed to one of the aforementioned amounts in caseof the event’s assignment to a liquidity period  z   lf oa ¼ 1 and to zero otherwise lf zoa ¼ 0 . f i1o  ð1 þ io ÞFZ o ¼ f i2o ; 8o 2 O

ð6:92Þ

sf 1o þ FZ o ¼ sf 2o ; 8o 2 O

ð6:93Þ

Both payments belonging to a specific financial transaction are linked by constraint (6.92). If an available financing alternative is used by the network managers, the credit amount including interest must be repaid in full at the end of the term. Interest calculation is based on the same units of time being relevant for continuoustime scheduling (e.g., days). According to constraint (6.93), the start and end times of a financial transaction are connected by its given term. Although being part of the model formulation, the constraints become redundant for alternatives that are not realized within the short-term planning horizon. The latter are characterized by a credit (and repayment) amount of zero.     E E  M  1  lpzsa  spzs < a  þ M  1  lpzsa ; jAj jAj 8z 2 Z; s 2 Sσ ; σ ¼ 1, . . . , W þ 2; a 2 A     E E ða  1Þ   M  1  lf zoa  sf zo < a  þ M  1  lf zoa ; 8z jAj jAj ð a  1Þ 

2 Z; o 2 O; a 2 A

ð6:94Þ

ð6:95Þ

It needs to be ensured that all events leading to monetary consequences are assigned to exactly one of the liquidity periods the planning horizon is divided

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Continuous-Time Scheduling in Green Supply Chain Management

into. As this horizon is characterized by an overall length of E, the length of each liquidity period a ¼ 1, . . ., j Aj is E/ j Aj. Accordingly, the start and end times of the resulting for the h liquidity i periods can be defined as h  h intervals  h ðjAj1ÞE E E 2E 3E 0; jAj ; jAj , 2E ; ; E . The lower and upper bounds of , jAj jAj jAj , . . . , jAj these intervals are part of constraints (6.94) and (6.95), which restrict the time points of events. These events must be scheduled continuously on the one hand while being assigned to a discrete liquidity period on the other hand. In particular, constraint (6.94) refers to the start time of production, storage, recycling, or disposal at a network site of the SC stage σ ¼ 1, . . ., W or σ ¼ W + 2 (event z ¼ 1), the time of sales at a market site of SC stage σ ¼ W + 1 (event z ¼ 1), and the time of starting a transport from a network site of the SC stage σ ¼ 1, . . ., W + 2 (event z ¼ 2). The transportation start time is equal to the end time of the operation at the same site. Due to omitting both of the Big M terms, all of the aforementioned times are restricted to thezbounds  of the liquidity period if the operational event  is assigned  to this period lpsa ¼ 1 . In the opposite case of non-assignment lpzsa ¼ 0 , the constraint becomes redundant, as the point in time to be considered is unbounded. Formally, this results from the subtraction (addition) of the big number M from (to) the regular interval bound at the left-hand (right-hand) side of the constraint. Parameter M needs to be chosen so as to exceed the length of the entire planning horizon. Analogously, constraint (6.95) can be applied to financial alternatives. It refers to the start time (event z ¼ 1) and the end time (event z ¼ 2) of a credit o 2 O. These times are restricted to the relevant bounds   of the interval in case of the event’s assignment to the liquidity period lf zoa ¼ 1 the constraint becomes redundant due to the  z , whereas  Big M method otherwise lf oa ¼ 0 . In order to ensure that the time point E is part of the planning horizon [0, E] and, thus, also part of its last liquidity period, both of the less-than signs must be replaced by less-than-or-equal signs in (6.94) and (6.95) for a ¼ j Aj. In general, both constraints guarantee a unique event-period-assignment, as it is not possible for an event’s time point to be within the bounds of more than one of the nonoverlapping intervals. ys 

X lp1sa ; 8s 2 Sσ ; σ ¼ 1, . . . , W þ 1

ð6:96Þ

a2A

yr s 

X

lp1sa ; 8s 2 Sσ ; σ 2 L [ fW þ 2g

ð6:97Þ

a2A

An event-period-assignment is required if an available network site is in use during the planning horizon. Due to constraint (6.96), production, storage, or sales necessitate the assignment of the event’s start time to a specific liquidity period. The latter also applies to recycling or disposal according to constraint (6.97). The same binary variables are summed up at the right-hand side of both constraints (6.96) and (6.97), as the monetary consequences of forward and reverse logistics belong to the same type of event according to (6.79) and (6.80).

6.1 Model Formulations

X X

135

tpsq  NF s 

ðσ, λÞ2K q2Sλ

X lp2sa ; 8s 2 Sσ ; σ ¼ 1, . . . , W þ 2

ð6:98Þ

a2A

σ¼σ

Further event-period-assignments are necessary for transports realized between the sites of the SC network. According to the feasible relationships of SC stages defined for the specific continuous-time model formulation, transportation may include both material flows of forward logistics (new or recycled products directed to the markets) as well as reverse logistics (returned or disposed products directed to external sites, recycled products redirected to production sites). Due to (6.98), a transport starting from a site of the SC stage σ ¼ 1, . . ., W + 2 entails the assignment of a transportation event to a liquidity period, while the number of transports starting from the same network site can be limited for administrative reasons. f i1o  FAo 

X

lf zoa ; 8z 2 Z; o 2 O

ð6:99Þ

a2A

X

fi1o  FT

ð6:100Þ

o2O

Due to constraint (6.99) being applicable to financial transactions, the assignment of monetary consequences to liquidity periods needs to be forced if a credit is taken. This is indicated by a positive credit amount occurring at the left-hand side of the constraint. Each credit entails two liquidity-relevant consequences, i.e., the first one at its beginning (payment, event z ¼ 1) and the second one at its end (repayment, event z ¼ 2). Consequently, there are always two different events, which need to be assigned separately to liquidity periods. At the same time, the constraint limits the credit amount (which needs to be determined in course of the optimization of the overall system) to an upper bound, which can be considered as limit for the specific transaction. Furthermore, the overall credit amount that is taken by the network managers during the short-term planning horizon can be restricted according to constraint (6.100) with regard to creditworthiness. Binary and non-negativity constraints must be added for all additional variables (including the auxiliaries required for the linearization of the problem) according to (6.101)–(6.110). lf zoa 2 f0, 1g; 8z 2 Z; o 2 O; a 2 A lpzsa 2 f0, 1g; 8z 2 Z; s 2 Sσ ; σ ¼ 1, . . . , W þ 2; a 2 A

ð6:101Þ ð6:102Þ

 0; 8z 2 Z; o 2 O

ð6:103Þ

sf zo  0; 8z 2 Z; o 2 O

ð6:104Þ

fizo

136

6

Continuous-Time Scheduling in Green Supply Chain Management

ar sa  0; 8s 2 SWþ1 ; a 2 A

ð6:105Þ

apsa  0; 8s 2 Sσ ; σ ¼ 1, . . . , W, W þ 2; a 2 A

ð6:106Þ

ups , ur s , uds , uf s  0; 8s 2 Sσ ; σ ¼ 1, . . . , W, W þ 2

ð6:107Þ

at sa  0; 8s 2 Sσ ; σ ¼ 1, . . . , W þ 2; a 2 A

ð6:108Þ

af zoa  0; 8z 2 Z; o 2 O; a 2 A

ð6:109Þ

qa  0; 8a 2 A

ð6:110Þ

Constraints for integrating liquidity planning into mixed-integer linear programs for continuous-time scheduling in networks with recycling of perishable goods can be derived from objective function (6.57) analogously if the relevant revenues and costs are considered for calculating the auxiliary variables in (6.76)–(6.88).

6.2

Heuristic Solution Methods

Due to challenges of complexity that arise from both the simultaneous coordination of forward and reverse flows in GSCM, and additional requirements of continuoustime scheduling of operations and financial transactions, further considerations on the solution method are required as the usefulness of commercial standard solvers is usually limited to small-scale test instances. To facilitate the computation of problems of realistic scope, two alternative heuristics applicable to the continuous-time model formulations are proposed in the following.

6.2.1

Relax-and-Fix Algorithm

Relax-and-fix algorithms are applicable to integer and mixed-integer problems, as they prescribe specific rules for handling binary variables. The heuristics are based on a decomposition of the original model into subproblems that can be optimized subsequently by the use of commercial standard solvers. Briefly summarized, these subproblems are a result of partitioning the overall set of binary variables into disjoint subsets. As only selected subsets are considered for optimization, while the binary variables of other subsets are subject to relaxation or fixation, lower computation times can be expected for solving the subproblems in comparison to solving the original optimization model. Ideally, the algorithm passes through a predefined number of steps, and ends, if all previously relaxed subsets have been replaced by fixation or optimization. By that, it is ensured that a feasible solution obtained after the last step of the algorithm cannot be infeasible to the original problem.

6.2 Heuristic Solution Methods

subset

1

137

subset

2

subset

subset

3

|Λ|

relaxation

step h = 2

fixation

optimization

relaxation

…

relaxation

step h = 3

fixation

fixation

optimization

…

relaxation

…

optimization

step h = |Λ|

fixation

fixation

fixation

…

…

…

relaxation

…

relaxation

…

optimization

…

step h = 1

Fig. 6.5 Basic elements and principles of the relax-and-fix algorithm

Roots of the relax-and-fix algorithm can be found in Dillenberger et al. (1994), who define it as a decomposition heuristic that does not consider the integrality of all binary variables at once, but successively. In general, the approach is based on partially relaxed submodels to be optimized in several subsequent steps. Major tasks of designing a problem-tailored algorithm include the decomposition of the set of binary variables into subsets and the establishment of rules for fixation (Federgruen et al. 2007; Ferreira et al. 2009). Further considerations on solution quality, criteria of termination, and the occurrence of infeasibilities can be found in the literature (Akartunalı and Miller 2009; Mohammadi et al. 2010; Wu et al. 2012). Originally, Dillenberger et al. (1994) revealed the advantageousness of R&F algorithms in the context of large-scale mixed-integer optimization models that could not be solved quickly and/or optimally by conventional branch-and-bound methods. A general applicability to continuous-time scheduling in process industries was confirmed by Kelly and Mann (2004). Beyond that, Steinrücke (2011a, 2011b, 2015) successfully applied R&F heuristics to continuous-time planning and scheduling in supply chain networks for the first time. The implementation of an R&F algorithm into a continuous-time model of GSCM requires the definition of additional sets and indices. The fundamental principle of an R&F algorithm based on different handling of subsets in subsequent steps is depicted in Fig. 6.5. Steps. Each basic step h 2 Λ ≔ {1, . . ., | Λ| } of the R&F algorithm represents the optimization of a submodel that results from the decomposition of the original problem. If the heuristic can be aggregated so as to optimize exactly one subset of binary variables within a submodel, the number of basic steps to be passed through is equal to the number of subsets. Subsequent to the basic steps, additional backtracking steps can be added to the algorithm if no feasible solution is obtained after the final basic step.

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Continuous-Time Scheduling in Green Supply Chain Management

Subsets. The set of binary variables to be considered by the heuristic needs to be decomposed into disjoint subsets. Different methods of decomposition exist in general: • Functions. Binary variables representing the same process, task, or purpose are grouped. In many cases, process-oriented decomposition is applied. By that, it becomes possible to distinguish processes of forward and reverse logistics in GSCM. • SC stages. Different binary variables belonging to the same SC stage are pooled. With regard to their consideration within the R&F algorithm, upstream (from markets to suppliers) and downstream (from suppliers to markets) methods can be distinguished. • Time periods. Binary variables representing decisions within the same time period are grouped to a subset. This method is not applicable to models of continuous-time, but of discrete-time scheduling (see Sect. 5). • Others. In general, all further indices the binary variables are dependent on can also be used for decomposition. Typical examples are products or transportation modes. Regardless of the selected decomposition method in the R&F algorithm, its general principle can be described as follows: In the h-th basic step, the subset Υ h is assumed to be  optimized,  while all subsets to be optimized in other basic steps h0 6¼ h are fixed Υ 1h0 20 results in orι2 ¼ 40). After initializing rank ω  ¼ 3 (orι3 ¼  M), it is assigned to chromosome ω ¼ 1 (20 >  M results in orι3 ¼ 20), but not to chromosome ω ¼ 2 (M + 1 < 20). The worst rank ω  ¼ 4 remains for the chromosome ω ¼ 2 associated with the infeasible solution (M + 1 >  M). As a consequence, all four chromosomes are unambiguously assigned to their ranks, i.e., ω ¼ 3 to ω  ¼ 1, ω ¼ 4 to ω  ¼ 2, ω ¼ 1 to ω  ¼ 3, and ω ¼ 2 to ω  ¼ 4. Ranking(ι) Remove all previous allocations of chromosomes to ranks. FOR each rank ω (starting with the best one ω  ¼ 1) of generation ι DO Initialize orιω (current best objective value of this rank) by setting it to M. Initialize hrιωθ by setting all values to zero. FOR each chromosome ω (except the ones already allocated to a better rank) DO IF objective value of chromosome ocιω is superior to the current best objective value orιω corresponding with the current rank THEN Set orιω to ocιω. FOR all genes θ DO Set hrιωθ to haιωθ END FOR Allocate chromosome ω to rank ω. [Note: If another chromosome is already allocated to the current rank, this allocation can be superseded. In this case, the previously allocated chromosome becomes unallocated again.] END IF END FOR END FOR The transformation module covers the alternative procedures of cloning, crossover, and mutation, which can be used to create chromosomes of a new generation from selected chromosomes of the previous generation. The aforementioned selection is based on the results of the ranking module. Firstly, all chromosomes up to a specific rank are cloned, i.e., the alleles of all its genes are taken over without any modifications. Formally, the transition is realized by setting the allele variables of the new generation’s chromosome (that is still unranked) to the allele variables of the previous generation’s chromosome (that is selected for cloning due to its rank). Implementing the cloning procedure in the transformation module for at least one chromosome ensures carrying on the best feasible solution found so far to the following iteration. Secondly, two chromosomes of different ranks can be selected for crossover in order to combine their alleles for building up at least one chromosome of the new generation from them. A randomly generated binary parameter is

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Continuous-Time Scheduling in Green Supply Chain Management

used to determine whether the new allele is taken over from the first or the second previous chromosome. Accordingly, each allele variable of an unranked new chromosome is set to the value of exactly one of the two allele variables of the previous chromosomes on the ranks selected for crossover. Thirdly, single chromosomes of specific ranks can be considered for mutation. All alleles of its genes are taken over, but for a specific randomly selected gene, the allele is replaced by a randomly generated binary number. Formally, the basic principle of exchanging variables introduced for the cloning procedure can be applied, whereas the allele variable of the gene selected for mutation according to MUιω needs to be set to MVιωθ. The following example considers the transformation of the two best-ranked chromosomes consisting of six genes with the alleles hrι  1, 1, θ ¼ 1, . . ., 6 ¼ (0, 1, 0, 1, 1, 1) and hrι  1, 2, θ ¼ 1, . . ., 6 ¼ (0, 0, 1, 1, 0, 1) to the following generation. In case of cloning, the alleles of the best chromosome can be identically transferred to the alleles of the first chromosome of the new generation that are represented by the binary variables haι, 1, θ ¼ 1, . . ., 6 ¼ (0, 1, 0, 1, 1, 1). Within crossover, the alleles of the genes θ ¼ 4, 5, 6 from the best chromosome ω ¼ 1 and the alleles of the genes θ ¼ 1, 2, 3 from the second-best chromosome ω ¼ 2 can be combined to the second chromosome of the new generation according to haι, 2, θ ¼ 1, . . ., 6 ¼ (0, 0, 1, 1, 1, 1). If the second allele of the best chromosome should be replaced by the random binary number zero for creating the third chromosome of the new generation by mutation, the vector haι, 3, θ ¼ 1, . . ., 6 ¼ (0, 0, 0, 1, 1, 1) results. Transformation(ι) FOR each chromosome ω of the new generation ι that results from cloning of one chromosome of the previous generation ι  1 allocated to the rank ω DO FOR each gene θ DO Set haιωθ to hrι  1, ωθ END FOR END FOR Set COιθ to random binary numbers [0;1] for determining the crossing scheme. FOR each chromosome ω of the new generation ι that results from crossover of two chromosomes of the previous generation ι  1 allocated to the ranks ω and ω0 DO FOR each gene θ DO Set haιωθ to hr ι1,ωθ  COιθ þ hr ι1,ω0 θ  ð1  COιθ Þ END FOR END FOR Set MUιω to random integer numbers from the interval [1; | Θ| ]. Set MVιωθ to random binary numbers. FOR each chromosome ω of the new generation ι that results from mutation of one chromosome of the previous generation ι  1 allocated to the rank ω DO FOR each gene θ 6¼ MUιω DO Set haιωθ to hrι  1, ωθ END FOR FOR each gene θ ¼ MUιω DO Set haιωθ to MVιωθ END FOR END FOR

6.3 Numerical Analysis

149

After the GA passed through the predefined number of iterations and the generation ι ¼ j Ij is reached, the final outcome is generated by the solution module. In general, the formal procedure of exchanging the variables’ values is comparable to the implementation module. However, it is no longer necessary to take the entire population of chromosomes into account. Instead, the alleles of the best chromosome’s genes are transferred to auxiliary variables. A final submodel is built up from fixing the binary variables of the original SC network problem to the auxiliaries. Its optimization reveals the result of the GA. Solution(ι=|I|) FOR each gene θ DO Set hsθ to hrι1θ END FOR Set counter b to zero (b¼0). FOR each site s 2 SG DO Increase counter b by one (b+1). Set θ ¼ b. Fix binary ys to the allele hsθ. END FOR Solve the resulting submodel after the aforementioned fixation of binaries. As a main advantage, feasible solutions obtained in any step are also feasible to the overall problem, while their transfer (including the possibility of improvement from step to step) is guaranteed by the structure of the algorithm.

6.3

Numerical Analysis

An illustrative small-scale example is used for model validation. After composition from the relevant modules described before, the approach for continuous-time scheduling in supply chain networks with combined recycling of non-perishable goods (Sect. 6.1.1.3) taking into account financial planning (Sect. 6.1.3) is analyzed in the following: The network consists of four subsequent SC stages (W ¼ 2). The first stage σ ¼ 1 is exclusively reserved for production. The second stage σ ¼ 2 includes operations of production and internal recycling. The third stage σ ¼ 3 is the market stage. The last stage σ ¼ 4 is assigned to operations of external recycling and disposal. Except the market stage containing two sites with given demand, each of these SC stages comprises four potential sites. The SC network produces two different products characterized by a different maturity. Intermediate products are manufactured at sites of the first stage, whereas final products are manufactured at sites of the second SC stage. The products are assumed to be arbitrarily divisible. Exactly two units of the intermediate product are required in order to produce one unit of the final product. Both products are

150

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Continuous-Time Scheduling in Green Supply Chain Management

Table 6.3 Operational parameters Parameter Transportation time between sites of SC stages σ, λ Production speed at sites of SC stages σ ¼ 1, 2 Recycling speed at sites of SC stages σ ¼ 2, 4 Disposal speed at sites of SC stage σ ¼ 4 Time period between sales and return at markets of SC stage σ ¼ 3 Minimum processing time at sites of SC stages σ ¼ 1, 2, 4 Maximum temporary storage time between sites of SC stages σ, λ

Value 0.5 10 10 10 0.3 0.3 5

Dimension Days Units/day Units/day Units/day Days Days Days

marketable. There is a demand for 5 units of the intermediate at market 3,1 and for 10 units of the final product at market 3,2. According to the model’s assumptions, the demand does not need to be satisfied in full. Parameters relevant for continuous-time scheduling of SC operations are given in Table 6.3. Additional parameters for revenues, fixed and variable costs of operations, and capacities were set to specific values or generated randomly with respect to given bounds. With regard to GSCM, there are requirements for the recycling of rejects and returns on the one hand, and limitations of greenhouse gas emissions occurring from the network operations on the other hand. The following specifications are assumed: • Firstly, there are rejects of intermediate products delivered from the first SC stage to the second one. It is assumed that the quantity ordered by sites of the second SC stage (as required to realize the production quantity) needs to be increased by 50% due to defective items, while the resulting overflow is to be handled within internal recycling. • Secondly, there are returns of final products from the markets. It is expected that 50% of the goods do not meet the requirements and, thus, are delivered to sites of the final SC stage for reasons of external recycling. • Within both external and internal recycling, there is a share of 10% of the treated quantity that can be recovered so as to obtain recycled products equivalent to new ones. Consequently, 90% of the aforementioned quantity needs to be disposed of. • In order to force the formation of a closed loop SC, the overall recyclable product quantity that is disposed of instead is limited to 0.5 units within the planning horizon. • Emission management is limited to carbon dioxide. The production of each unit in the first or second SC stage causes 1 ton of emission. If a unit of a product is recycled or disposed of, 0.8 tons are emitted. The transportation of a product unit between two different network sites entails 0.5 tons. For the entire short-term planning horizon, the SC is allowed to emit a maximum of 100 tons. The short-term planning horizon spans 7 days. It is divided into two liquidity periods of equal length. Within financial planning, three potential credits are available (Table 6.4). The overall credit limit for the entire planning horizon is $ 2000.

6.3 Numerical Analysis Table 6.4 Available financial alternatives

151

Credit o ¼ 1 Credit o ¼ 2 Credit o ¼ 3

Credit rate 1% per day 2% per day 3% per day

Credit term 2 days 3 days 4 days

By using high-performance hardware and software (see Sect. 5.2), the optimal solution of $ 3330 was found after 5:09:06 [h:mm:ss]. It corresponds to the optimal network structure (see Fig. 6.8) and the optimal composition of cash flows (see Table 6.5). Only few of the network sites potentially available for operations are used during the short-term planning horizon. In particular, two sites of the first SC stage (sites 1,1 and 1,2), one site of the second SC stage (site 2,3), and two sites of the final SC stage (sites 4,2 and 4,4) are harnessed. Both markets (sites 3,1 and 3,2) are partially delivered with the goods demanded. Production starts at time point 1.564 with the manufacturing of 16.36 units of intermediates at site 1,1. According to the given production speed, the operations are finished at time point 3.200. As just-in-time operations are assumed, the latter point in time can also be considered as starting time of a transport, which is directed to a production site of the subsequent SC stage in order to transform the delivery into final products. Both of the aforementioned operations (production and transportation) are assigned to the first liquidity period, as their starting times belong to the first half of the 7-day planning horizon. After merging the corresponding cash flows, it becomes obvious that an amount of $ 495.46 is required to initiate these SC operations. As there is no liquidity assumed to be available, appropriate financing is necessary. For this purpose, the available credit o ¼ 1 is taken by the network managers during the second day (time point 1.500) to the extent of exactly $ 495.46. As a consequence, the liquidity of the first period is balanced completely (balance of zero). Within the second liquidity period, the manufacturing of 5.46 units of the final product at site 2,3 starts after the transport from site 1,1 arrives (taking into account a transportation time of half a day) at the time point 3.700. According to the relevant input-output relation, approximately 10.91 units are required. The latter quantity is increased by 50% to an order of 16.36 units according to long-term experiences with rejects. The defective 5.46 units of intermediates are to be handled in internal recycling at the same site. However, only 10% of them can be recovered completely during this treatment so they can be used equivalently to new products. These 0.54 units are sent back to the SC stage of the items’ production, in particular, to site 1,2. As the latter product quantity needs to be reintegrated into forward flows directed to the customers, a closed loop supply chain results. The remaining share of 90% needs to be disposed of and, thus, it is sent to the disposal site 4,4 belonging to the last SC stage. As the overall processing time at site 2,3 comprises the production time (0.546 days) and the time for internal recycling (0.055 days), the operations end in the time point 4.300. The latter can be considered as the starting point for all

1.564

2

liquidity period a=1

1,1 P:16.36

transportation quantity [time of temporary storage]

3

3.200

16.36

4

2,3 P:5.46 R:0.54 0.54

4.300

4.91

6

3,1

1,2 P:4

5

5.700 6.000

3,2

5.700 6.000

4.800 5.200

5.46 [0.9]

4.54

5.291

4,4 D:4.91

4.800

end time

2.27

4,2 D:5

7 time

6.500 7.000

recycling/ disposal site σ=4

start time

2.73

transportation quantity [time of temporary storage]

liquidity period a=2

market site σ=3

sales time return time transportation quantity [time of temporary storage]

3.700

end time

production/ recycling site σ=2

start time

Fig. 6.8 Optimal network structure in continuous-time scheduling

1

end time

production site σ=1

start time

6

0

P: produced quantity R: recycled quantity D: disposed quantity

Legend

152 Continuous-Time Scheduling in Green Supply Chain Management

6.3 Numerical Analysis

153

Table 6.5 Liquidity balancing in continuous-time scheduling

Cash flow

Site(s)

Revenues

3,1 3,2 1,1 1,2 2,3 2,3 4,2 4,4 1,1 to 2,3 2,3 to 1,2 2,3 to 3,2a 2,3 to 4,4 1,2 to 3,1 3,1 to 4,2 3,2 to 4,2

Production

Recycling Disposal Transportation

Financing o ¼ 1 Balance a

Liquidity periods a¼1 a¼2 2172.50 2627.50 363.64 240.00 254.55 10.55 15.00 14.91 131.82 52.73 104.28 74.55 72.73 61.37 63.64 495.46 505.41 0.00 3330.31

Including cost of temporary storage

outgoing transports, including a transport to a market. In particular, 5.46 units of the final product are delivered to market 3,2 to satisfy more than a half of its demand. As the optimal sales time is 5.700, the standard transportation time of 0.5 days needs to be increased by 0.9 days of temporary storage. The second location producing intermediates in the first SC stage is site 1,2. Manufacturing starts in time point 4.800 and results in exactly 4 units in time point 5.200. The production quantity is increased by the ingoing amount of 0.54 recycled units, so as to obtain an overall quantity of 4.54 units. This amount is directed to market 3,1 by an additional transport. Respecting the transportation time of 0.5 days, the time of sales is also 5.700. Approximately 90.8% of the market’s demand are satisfied by this supply. However, there are returns expected from both of the markets according to the assumptions. In particular, half of the ingoing amount is redirected to a recycling facility of the final SC stage after 0.3 days have been passed. As site 4,2 is selected for external recycling according to the optimal solution vector of the overall problem, two transports (2.73 units from market 3,2, and 2.27 units from market 3,1) are arranged to it accordingly. Taking into account the standard transportation time, both transports are expected to arrive simultaneously in time point 6.500. The amount to be handled in reverse logistics (resulting as sum of the ingoing) comprises 5 units of the final product. According to the given share, 0.5 units of it (10%) can be recycled. However, the optimization of the overall system advises the network managers to make use of the possibility of transferring recyclable products to disposal. As a result, not only 4.5 units (90%) of the final product are disposed of at this site, but the complete 5 units being available. It is

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noteworthy that the maximum amount of recyclable products actually transferred to disposal is exhausted by that. Taking into account the aforementioned quantity as well as the given speed of disposal, the operations end after 0.5 days at the end of the planning horizon (time point 7.000). The second location of the last SC stage being harnessed is site 4,4. It is not used for external recycling, as it is only supplied with disposal quantities that remain after internal recycling in the second SC stage and, thus, cannot be recovered. A quantity of 4.91 units of the intermediate product arrives in time point 4.800. Its disposal requires 0.491 days due to the given speed parameter, so the operations end in time point 5.291 here. All operations at the sites 1,2; 2,3; 4,2; and 4,4 as well as the transports departing from them can be assigned to the second liquidity period, as their starting times are in the second half of the 7-day planning horizon. The same applies to the sales time at both of the markets. All related revenues and costs are merged. Besides overall revenues of $4800 for both products, the operations entail costs of $964.27. However, the residual amount of $3835.73 needs to be decreased by the repayment of the credit. The latter is to be assigned to the second liquidity period due to the given term of 2 days. The final balance at the end of the planning horizon $ 3330.31 is equal to the optimal objective value of the overall problem and can be interpreted as the (maximum) withdrawal by the network managers. Cash flow problems are prevented, as the liquidity cannot fall below zero in both periods relevant for its accounting. The initialization of the operations in the first liquidity period entailing production and transportation costs is financed by taking on the short-term credit, which is repaid at the beginning of the second liquidity period. Within this time interval, the revenues from demand satisfaction at the markets are realized, which do compensate not only this repayment but also all further costs as well as the final withdrawal. This compensation even comprises monetary consequences of events scheduled before the time of sales. Assuming the existence of financial scopes of the network within the same liquidity period (tolerated overdrafts), there is no need for taking on additional financing for this reason. In summary, the solution reveals an appropriate coordination of continuously scheduled operations and financial alternatives with the requirements of periodic liquidity balancing. Acknowledging a computation time of more than 5 hours that results for the small-scale example, the proposed heuristics (see Sect. 6.2) are implemented in order to improve the computability of the MILP while striving for at least satisfying solutions. Firstly, the two-step R&F heuristic is applied. The starting point is a functional decomposition that groups the overall set of binary variables according to five different types (ys, yrs, tpsq, lpzsa , lf zoa ). Then, 30 strategies can be built up so as to cover all possible combinations of assigning these variable groups to exactly 2 disjoint subsets (see Sect. 6.2.1). These strategies are tested for the illustrative example presented before (see Table 6.6). The algorithm is terminated after a computation

6.3 Numerical Analysis

155

Table 6.6 Test of different relax-and-fix strategies for two-step algorithms Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Determination of binary variable ys yrs tpsq lpzsa lf zoa I II II II II II I II II II II II I II II II II II I II II II II II I I I II II II I II I II II I II II I II I II II II I II I I II II II I II I II II I II II I II II I I II II II I II I II II II I I I I I II II I I II II I I II II I I II II I I I II I I I II II I I II I II I II I I I II I II I I II I I II I I II I II II I I I I I II I I I I I II I I I I I II I I I I I II

Results of computation Comp. time [hh:mm:ss] 00:01:24 00:00:02 02:00:00a 00:00:26 00:16:44 00:00:48 02:00:00a 00:00:14 00:00:53 02:00:00a 00:00:13 00:00:57 02:00:00a 02:00:00a 02:00:00a 00:01:58 00:00:36 02:00:00a 02:00:00a 02:00:00a 02:00:00a 02:00:00a 02:00:00a 02:00:00a 00:00:53 02:00:00a 02:00:00a 00:01:48 00:02:29 00:16:40

Obj. value [$] 3330 ++ ++ 0 3278 3220 ++ ++ ++ ++ ++ 3204 ++ ++ ++ 2633 3166 ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ 2490 3278

I: Optimization in step h ¼ 1, fixation in step h ¼ 2 II: Relaxation in step h ¼ 1, optimization in step h ¼ 2 a Computation terminated after 2 hours ++ no feasible solution found

time of 2 hours (which is assumed to be maximally acceptable for the heuristic) without any backtracking in case of infeasibility. The best performing strategies can be identified by evaluating both the computation time and the heuristic’s solution after finishing the second step of the algorithm. Requirements for a further consideration of the strategy are feasibility after the last basic step on the one hand and extraordinary high reductions in computation times (>99%) in combination with low reductions in the objective value (99.9% 00:00:46 3346 99.9% 00:00:39 3303 99.5% 00:00:48 3247 98.9% 00:00:47 3426 99.8% 00:00:43 3321 99.8% 00:00:39 3311 99.9% 00:00:40 3329 99.8% 00:00:42 3421 99.9% 00:01:15 3411 99.4% 00:00:41 3378 99.8% 00:00:46 3313 99.9% 00:00:46 99.6%

Table 6.7 Results of the relax-and-fix algorithms in the scenario analysis for continuous-time scheduling

Obj. value difference 1.0% 1.1% 0.8% 4.3% 4.8% 0.5% 8.7% 0.5% 0.4% 2.9% 3.5% 5.0% 0.7% 4.6% 3.4% 3.1% 0.5% 1.2% 1.5% 3.8% 2.6%

6.3 Numerical Analysis 157

Gap 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 23.9% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

Computation terminated after 24 hours

Optimization Comp. time [hh: Obj. mm:ss] value [$] 03:29:30 3428 00:31:45 3437 07:05:44 3448 03:48:36 3440 12:32:00 3438 05:51:13 3454 13:57:43 3438 03:35:17 3440 24:00:00a 3444 09:09:55 3448 02:06:46 3425 01:09:36 3417 05:44:46 3450 07:17:28 3483 20:16:36 3428 04:28:39 3437 19:36:13 3439 03:39:49 3452 06:34:14 3428 13:58:33 3445 08:26:43

Genetic algorithm with two generations Comp. time [hh: Obj. Comp. mm:ss] value [$] time difference 00:01:33 3421 99.3% 00:03:28 3437 89.1% 00:01:49 3427 99.6% 00:01:57 3385 99.1% 00:02:59 3430 99.6% 00:28:59 3445 91.7% 00:01:31 3408 99.8% 00:01:16 3434 99.4% 00:01:09 3432 99.9% 00:15:16 3445 97.2% 00:03:09 3425 97.5% 00:03:38 3404 94.8% 00:31:47 3424 90.8% 00:04:00 3421 99.1% 00:02:25 3402 99.8% 00:01:58 3433 99.3% 00:02:10 3437 99.8% 00:01:36 3439 99.3% 00:04:59 3371 98.7% 00:02:03 3443 99.8% 00:05:53 97.7% Obj. value difference 0.2% 0.0% 0.6% 1.6% 0.2% 0.3% 0.9% 0.2% 0.4% 0.1% 0.0% 0.4% 0.8% 1.8% 0.8% 0.1% 0.1% 0.4% 1.7% 0.1% 0.5%

Genetic algorithm with three generations Comp. time [hh: Obj. Comp. Obj. mm:ss] value [$] time difference value difference 00:02:57 3428 98.6% 0.0% 00:04:18 3437 86.5% 0.0% 00:02:29 3427 99.4% 0.6% 00:03:21 3385 98.5% 1.6% 00:04:32 3430 99.4% 0.2% 00:29:37 3454 91.6% 0.0% 00:02:30 3408 99.7% 0.9% 00:02:07 3434 99.0% 0.2% 00:02:11 3432 99.8% 0.4% 00:16:25 3445 97.0% 0.1% 00:05:36 3425 95.6% 0.0% 00:13:42 3404 80.3% 0.4% 00:32:25 3424 90.6% 0.8% 00:05:50 3424 98.7% 1.7% 00:03:27 3402 99.7% 0.8% 00:03:39 3433 98.6% 0.1% 00:03:03 3437 99.7% 0.1% 00:02:26 3439 98.9% 0.4% 00:05:58 3371 98.5% 1.7% 00:03:16 3443 99.6% 0.1% 00:07:29 96.5% 0.5%

6

a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Avg.

Scen.

Table 6.8 Results of the genetic algorithms in the scenario analysis for continuous-time scheduling

158 Continuous-Time Scheduling in Green Supply Chain Management

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optimization run (0.5%). The average reduction of computation time, which is affected by backtracking procedures that were required for 5 instances, is still acceptable (95%). Strategy no. 6 dominates regarding an extraordinary average computation time reduction (99.6%). Backtracking was required for none of the instances. However, the solution quality is considerably lower, i.e., the average relative deviation of objective values is higher (2.6%), while the optimal solution could not be found by the heuristic in any instance. Considering the two variants of the genetic algorithm, it becomes obvious that the aforementioned trade-offs are less important. The GA with two generations allows for a considerable average reduction of computation times (97.7%). At the same time, the average relative deviation of the objective values (0.5%) is typical for a high solution quality. The optimal solution could be found for two instances. In contrast, increasing computation times are to be expected if an additional iteration is added to the algorithm. Consequently, the average relative computation time reduction decreases (96.5%) for the GA with three generations. Although it is possible to find the exact optimal solution for two additional instances, there are only minor improvements in the average solution quality that remains almost constant (0.5%). A solution quality corresponding with an average relative deviation of the heuristics’ solutions from the ones obtained after the optimization runs of approximately 0.5% is supposed to be sufficient for practical purposes. Although the optimal amount to be withdrawn by the network managers after the second liquidity period of the 7-day planning horizon may decrease slightly, this is usually considered to be negligible due to remaining uncertainties in data. Acknowledging the relevance of rapid planning and scheduling in GSCM problems within a short-term planning horizon, the GA with two generations can be considered as preferable. It results in the highest average computation time reduction while meeting the aforementioned requirements for the average solution quality in course of this scenario analysis.

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Chapter 7

Summary and Conclusions

Abstract Based on existing structures of sites and capacities, new mathematical models for hierarchical scheduling of operations in generic multi-stage networks are developed in the book Scheduling in Green Supply Chain Management: A MixedInteger Approach. This chapter summarizes the main results of the book and provides starting points for further research in green supply chain management.

During the past few years, the importance of environmental awareness evolved from a desirable goal to a compelling necessity for global development. According to their recent report, the members of the Intergovernmental Panel on Climate Change urge for rapid, far-reaching, and unprecedented changes in all aspects of society in order to limit global warming to 1.5  C above pre-industrial level. Consequently, official regulatory procedures are expected to be introduced or strengthened in the near future. In context of reducing greenhouse gas emissions, the European Union’s Emissions Trading Scheme currently turns into the phase of auctioning allowances instead of their cost-free allocation. Further initiatives of legislation deal with waste reduction and reuse of resources. As companies are increasingly organized in complex networks due to progressing globalization, a common approach for implementing the aforementioned requirements into business practices can be found in GSCM. Based on existing structures of sites and capacities, this monograph deals with scheduling on both the medium- and the short-term level, which allows for realizing transitory market opportunities while striving for a reconciliation of economic and environmental issues. Considering the existing literature in the aforementioned field, an extensive review of recent publications reveals that aspects of scheduling are predominantly neglected so far. To the best of knowledge, an optimization model for continuoustime scheduling in GSCM integrating aspects of both closed loop logistics and emission control for multi-stage networks does not exist. Moreover, this approach is implemented into a hierarchical framework, which allows for coordinating decisions to be taken in medium- and short-term planning horizons according to existing dependencies. Other contributions refer to details of modeling. In line with the proposed systematization of different problem classes according to the quality of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6_7

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available input data as well as the characteristics of products and processes relevant for GSCM, modularized programming formulations capturing all reasonable sequences of SC stages as well as material flow paths are developed. Furthermore, modeling explicitly considers relationships between operational, financial, and environmental problems so as to reveal existing trade-offs. Discrete-time modeling is applicable to master scheduling, which determines the quantity of final products to be processed while taking into account different functional areas along the supply chain. As input data of the assumed mediumterm planning horizon cannot be supposed to meet the requirements of highest accuracy, a division into time periods of equal length seems to be rational. Nevertheless, the chronology of events is respected, as decisions on site usage, operations, and surpluses as well as related monetary consequences are unambiguously assigned to time points at the beginning or the end of the time periods. In particular, the approach includes setup and shutdown decisions for making given sites and capacities available for operations of production, transportation, sales, recycling, and/or disposal. Stage-to-stage material flows of non-perishable products are considered on an aggregated level, while inventory holding from one time period to the following is possible. Acknowledging long-range life cycles of the relevant goods, quantities of sales and returns are unconnected and given separately in this case. Mandatory or voluntary recycling taking place at external sites of the final SC stage can be modeled. As the recovered goods of different maturities are to be reintegrated in the SC stages of their original manufacturing, the formation of closed loop systems results. Furthermore, the management of emission trading is relevant for the medium-term level of scheduling. Based on a given cap, which represents the standard volume of carbon dioxide equivalents that is allowed to be emitted by the entire network, decisions of buying or selling of emission allowances can be taken in each time period. The resulting adjusted caps are considered as given limits in subsequent short-term approaches. For continuous-time scheduling in GSCM, different model variants are distinguished with regard to product constitution and recycling type. First of all, non-perishable goods are considered. Internal recycling occurs in the context of manufacturing and refers to materials that can be immediately processed and returned. It is applicable to rejects of defective deliveries in business-to-business transactions or to scrap. Recycling is performed at the supplied production sites, which are also starting point of subsequent reverse flows of recovered goods. In contrast, external recycling captures materials that are expected to be returned after a short time period, which passes after demand satisfaction by the SC network at the markets. These returns may comprise defective deliveries or packaging material. Their processing takes place at external sites of the final SC stage, which are exclusively provided for operations of recycling and disposal, and represent the starting point of reverse flows in this case. In general, the latter are based on retransformation according to given reverse bills of materials and are directed back to the SC stages of the materials’ origin as far as a reintegration of the recovered products is possible for quality reasons. Consequently, a closed loop system results. As recycled quantities supersede the manufacturing of new items, resources required

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165

for the original production process can be conserved. The remaining part of the returns, which needs to be discarded due to quality reasons at the processing sites, is delivered to or remains at the sites of the final SC stage for reasons of final disposal in case of internal or external recycling, respectively. The models even allow for transferring recyclable quantities to disposal for economic reasons in course of the optimization of the overall system. However, this transfer can be limited to a specific amount, which can be set in line with legislation and/or corporate goals. The analysis of trade-offs between internal and external recycling is possible if combined recycling is applied. The latter is characterized by recovered quantities available at the production sites, which are composed of different sources, but can be merged for reasons of equivalency. A second group of products to be considered within continuous-time scheduling in GSCM are perishables. As the quality of the latter is typically affected by chemical deterioration and physical instability, traditional network structures are not applicable. Instead of subsequent production stages that allow for increasing the product maturity, modeling focuses on a multi-stage distribution network that is organized in a hierarchical manner. The latter allows for bundling and/or splitting up deliveries so as to better coordinate sales at the markets. Both the sales prices and the return quantities depend on the quality of the products delivered to the markets and, thus, the time spent on the material flow paths. As discrete quality grades are commonly used for the standardization of business transactions, a coordination with the continuous-time scheduling of operations is required. The latter does refer not only to processes of forward logistics but also to recycling and disposal. Inherently, the treatment cannot result in recovering products equivalent to their previous constitution due to an irreversible deterioration. Thus, the formation of closed loop systems cannot occur in this case. Instead, recycling processes resulting in secondary products are implemented for reducing waste from nonmarketable fresh produce. Besides issues of reverse processing, emission control is part of the models for continuous-time scheduling in GSCM. All operations need to be aligned to a given cap, which limits the overall emissions during the entire planning horizon. As the latter represents one time period in medium-term scheduling, the limit is obtained from a previous optimization. With regard to different emission types harmful to the climate, all greenhouse gases resulting from production, transportation, and recycling can be transformed into carbon dioxide equivalents. Another module for extending the aforementioned approaches is financial planning. In the context of medium-term scheduling, it can be integrated so as to adjust the resulting structure of periodic withdrawals by the network managers. A set of alternative credits and investments with duration-dependent interest rates can be used to overcome financial imbalances, which may result from the need of legally required recycling or from buying additional allowances in the context of emission trading, and could even entail the insolvency of the entire network in the worst case. With regard to short-term scheduling, additional financial planning allows for bridging temporary lacks of liquidity that cannot be covered by bank overdrafts. Balancing necessitates merging relevant monetary consequences within discrete

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7 Summary and Conclusions

liquidity periods, while an appropriate assignment of exactly scheduled operations to these periods needs to be ensured. As a result of the proposed modeling technique, it becomes possible to realize momentary benefits from exact coordination while respecting environmental requirements. Going beyond the insights from the small-scale examples used for model validation, all proposed approaches can be implemented to different industrial sectors facing problems of supply chain network optimization between the poles of economic rationality and environmental awareness. Due to the generic structure of modeling, networks with an arbitrary number of subsequent SC stages as well as sites within these SC stages can be analyzed. Moreover, an individual selection and composition of required modules is possible. Formally, all approaches are mixedinteger programming models, which can be handled by different commercial solvers available for research and practice. Especially with regard to the complexity in continuous-time scheduling, computational experiments conducted on highperformance software and hardware revealed that problem instances of realistic scope could not be solved to optimality within acceptable times. For this reason, problem-tailored variants of relax-and-fix heuristics and genetic algorithms are finally developed. Further research in the treated field may include long-term considerations on the selection of potential sites, products, and even financial alternatives according to environmental criteria. With regard to the medium- and short-term scheduling that was picked out as the central theme of this monograph, the focus of modeling could be broadened from the environmental perspective of green supply chain management to a sustainability perspective that additionally covers social aspects.

Index

A Acceptance, 2 Advanced Planning Systems, 13 Applications, 34 Assumptions, 4

Demand, 4 Discrete-time approaches, 47–54 Discrete-time formulations, 15 Discrete-time scheduling, 3, 59–60 Disposal, 30

B By-products, 31

E Eco-design, 21 Emission control, 105–106 Emissions trading, 79 Emission targets, 35 End-of-use returns, 31 Environmental awareness, 23 Environmental parameters, 71, 97, 120 European Union’s Emissions Trading System (EU ETS), 34 Example, 83, 149 External recycling, 107

C Capacities, 71, 97, 120 Cap-and-trade system, 33 Chromosomes, 140 Closed loop supply chain, 25 Coefficients, 71, 97, 120 Collection, 29 Combined recycling, 112 Commercial returns, 31 Continuous-time approaches, 54–59 Continuous-time formulations, 16 Continuous-time scheduling, 4, 93–159 Cooperate measuring, 35 Coordination, 11, 26, 36 Costs and revenues, 70, 97, 119 Crossover, 142 Customer behavior, 23 Customer cooperation, 22

D Decisions, 71, 81, 98, 120, 128 Decomposition, 138

F Financial parameters, 81, 128 Financial planning, 5, 80–83, 127–136 Functional attributes, 11

G Gantt-charts, 15 Generations, 141 Genes, 140 Genetic algorithms, 140 Global optimization, 10 Greenhouse gases, 32

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 W. Albrecht, Scheduling in Green Supply Chain Management, International Series in Operations Research & Management Science 303, https://doi.org/10.1007/978-3-030-67478-6

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Index

Greenhouse Gas Protocol, 35 Green logistics, 1 Green purchasing, 21 Green Supply Chain Management, 1, 21, 50–54, 58–59

O Objectives, 14 Operations, 4, 68 Origins, 22 Other parameters, 71, 98, 120

H House of SCM, 11

P Packaging, 31 Parts harvesting, 30 Perishable goods, 115 Planning, 4 Practices, 21 Process cells, 25 Production scrap, 31 Products, 28 Pseudocode, 144

I Incremental planning, 15 Integration, 11, 60 Internal environmental management, 22 Internal processing, 25 Internal recycling, 94 Internal reuse, 30 Inventory control, 32 Investment recovery, 22

Q Quality grades, 116, 118, 123 Quantitative targets, 22

K Kyoto Protocol, 35

L Liquidity, 80, 127 Lists of activities, 14 Location planning, 28 Logistics, 10

M Management, 32, 36 Master scheduling, 65 Medium-term planning horizon, 3, 65 Mixed-integer linear programs, 72, 82, 98, 109, 113, 121, 129, 136 Modularization, 60 Mutation, 142

N Networks, 9, 26, 66, 94, 107, 116 Non-perishable final products, 66

R Recovery, 21 Recovery path, 28 Recycling, 24 Regulatory effects, 23 Relax-and-fix algorithms, 136 Remanufacturing, 29 Resale, 30 Returns, 4 Reverse logistics, 1, 24 Reverse material flows, 30 Rolling horizon procedures, 15

S Scenario analysis, 88, 156 Scheduling, 14 SC stages, 9 Short-term planning horizons, 4, 93 Simulation, 15 Sites, 29–30 Site state, 68

Index Solution algorithms, 60 Speed parameters, 97, 120 Standardization, 27 Static approaches, 42–47 Strategic network design, 4 Structural attributes, 12 Structure, 59 Supply chain, 9 Supply Chain Management, 1, 10 Surpluses, 68 Sustainable supply chain management, 1

169 T Time parameters, 97, 120 Time points, 68 Time representation, 60

W Warranty returns, 31