Produktion und Logistik Herausgegeben von C. Bierwirth, Halle, Deutschland B. Fleischmann, Augsburg, Deutschland M. Fleischmann, Mannheim, Deutschland M. Grunow, München, Deutschland H.-O. Günther, Berlin, Deutschland S. Helber, Hannover, Deutschland K. Inderfurth, Magdeburg, Deutschland H. Kopfer, Bremen, Deutschland H. Meyr, Stuttgart, Deutschland K. Schimmelpfeng, Stuttgart, Deutschland Th. S. Spengler, Braunschweig, Deutschland H. Stadtler, Hamburg, Deutschland H. Tempelmeier, Köln, Deutschland G. Wäscher, Magdeburg, Deutschland

Diese Reihe dient der Veröff entlichung neuer Forschungsergebnisse auf den Gebieten der Produktion und Logistik. Aufgenommen werden vor allem herausragende quantitativ orientierte Dissertationen und Habilitationsschriften. Die Publikationen vermitteln innovative Beiträge zur Lösung praktischer Anwendungsprobleme der Produktion und Logistik unter Einsatz quantitativer Methoden und moderner Informationstechnologie.

Herausgegeben von Professor Dr. Christian Bierwirth Universität Halle

Professor Dr. Herbert Kopfer Universität Bremen

Professor Dr. Bernhard Fleischmann Universität Augsburg

Professor Dr. Herbert Meyr Universität Hohenheim

Professor Dr. Moritz Fleischmann Universität Mannheim

Professor Dr. Katja Schimmelpfeng Universität Hohenheim

Professor Dr. Martin Grunow Technische Universität München

Professor Dr. Thomas S. Spengler Technische Universität Braunschweig

Professor Dr. Hans-Otto Günther Technische Universität Berlin

Professor Dr. Hartmut Stadtler Universität Hamburg

Professor Dr. Stefan Helber Universität Hannover

Professor Dr. Horst Tempelmeier Universität Köln

Professor Dr. Karl Inderfurth Universität Magdeburg

Professor Dr. Gerhard Wäscher Universität Magdeburg

Kontakt Professor Dr. Thomas S. Spengler Technische Universität Braunschweig Institut für Automobilwirtschaft und Industrielle Produktion Mühlenpfordtstraße 23 38106 Braunschweig

Jens Kuhpfahl

Job Shop Scheduling with Consideration of Due Dates Potentials of Local Search Based Solution Techniques Foreword by Prof. Dr. Christian Bierwirth

Jens Kuhpfahl Halle, Germany Dissertation University of Halle (Saale), 2015

Produktion und Logistik ISBN 978-3-658-10291-3 ISBN 978-3-658-10292-0 (eBook) DOI 10.1007/978-3-658-10292-0 Library of Congress Control Number: 2015941508 Springer Gabler © Springer Fachmedien Wiesbaden 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Gabler is a brand of Springer Fachmedien Wiesbaden Springer Fachmedien Wiesbaden is part of Springer Science+Business Media (www.springer.com)

Foreword A bulk of research in job shop scheduling is dealing with problems where finding a solution schedule with minimum makespan is desired. Recently, also tardiness objectives received attention because on-time order fulfillment is of increasing importance particularly in pull-oriented supply chain systems. Keeping job due dates is a prerequisite for avoiding stock outs and for serving customers within promised delivery times. In case of highly utilized machine capacities, however, not every jobs might be producible on time. In such situations the minimization of the total tardiness of the jobs turns out as an appropriate objective for production scheduling. The book of Jens Kuhpfahl investigates scheduling methods, more precisely local search based heuristics, for the job shop problem with total weighted tardiness (JSPTWT) as objective. The operation research literature treating this problem class dates back over fifteen to twenty years. In this relatively short period an area of intense research has developed which is systematically reviewed and refined by the author. Based on the well-known disjunctive graph model for the classical minimum makespan problem, he analyses existing neighborhood search operators for the JSPTWT and derives new definitions to modify solution schedules in so far unknown ways. Feasibility guarantees and connectivity properties are proved analytically for these neighborhoods and their performance is compared empirically. To reduce computation times, a fast method for assessing the schedule quality, called Head Updating, is proposed. These and further components are thoroughly integrated into the metaheuristic framework of a Greedy Randomized Adaptive Search Procedure (GRASP) delivering a new state-of-the-art method for the JSPTWT. Although research in deterministic machine scheduling is highly developed, the book succeeds in closing existing gaps and contributing new ideas to the design of scheduling heuristics. I wish it a great success. Christian Bierwirth

Acknowlegdement I would like to express my gratitude to the people who supported and helped me achieving this dissertation. Special thanks goes to: - Prof. Dr. Christian Bierwirth - my PhD supervisor - for his continuous support and guidance, as well as the chance for me to discover the fascinating world of scheduling. - Prof. Dr. Dirk C. Mattfeld for the enlightening discussions at several conferences and the permission to use the code of its genetic algorithm. - My colleagues from the chair of Production and Logistics for their support and the pleasant working atmosphere. - My family for their love, the constant encouragement and the financial support particularly at the beginning of my studies. - Isabell for her love, warmth, support, motivation and vitality.

Jens Kuhpfahl

Contents List of Figures

xiii

List of Tables

xv

List of Notations

xix

List of Abbreviations

xxiii

1 Introduction

1

1.1

Aims and Contributions of the Thesis . . . . . . . . . . . . . . . . . .

4

1.2

Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3

Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2 Job Shop Scheduling - Formulation and Modeling

9

2.1

Problem Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Classification into the Scheduling Theory . . . . . . . . . . . . . . . . 11

2.3

Mathematical Model and Complexity . . . . . . . . . . . . . . . . . . 13

2.4

The Disjunctive Graph Model . . . . . . . . . . . . . . . . . . . . . . 15

2.5

The Concept of the Critical Tree . . . . . . . . . . . . . . . . . . . . 17

2.6

Exemplification on the instance ft06 (f = 1.3) . . . . . . . . . . . . . 19

3 Literature Review

9

25

3.1

Exact Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2

Dispatching Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3

Shifting Bottleneck Heuristic . . . . . . . . . . . . . . . . . . . . . . . 26

3.4

Local Search based Algorithms and Techniques

3.5

Other Heuristic Approaches . . . . . . . . . . . . . . . . . . . . . . . 27

3.6

Hybrid Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

. . . . . . . . . . . . 27

Contents

x

4 Neighborhood Definitions for the JSPTWT

31

4.1

The Basic Concept of Neighborhood Search . . . . . . . . . . . . . . 31

4.2

Existing Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3

New Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4

4.5

4.6

Characteristics of the proposed Neighborhoods . . . . . . . . . . . . . 41 4.4.1

Feasibility Property . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4.2

Connectivity Property . . . . . . . . . . . . . . . . . . . . . . 45

4.4.3

Estimate of the Size of the Neighborhoods . . . . . . . . . . . 48

Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5.1

Test Suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5.2

Local Search with a Single Neighborhood Operator . . . . . . 52

4.5.3

Local Search with Pairs of Neighborhood Operators . . . . . . 57

4.5.4

Local Search with all Neighborhood Operators . . . . . . . . . 60

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Neighbor Evaluation Procedures in Local Search based Algorithms for solving the JSPTWT

65

5.1

Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2

Lower Bound Procedure for the CET Neighborhood . . . . . . . . . . 70

5.3

Lower Bound Procedure for the SCEI Neighborhood . . . . . . . . . . 72

5.4

A new approach: Heads Updating . . . . . . . . . . . . . . . . . . . . 74

5.5

Performance Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Solving the JSPTWT - a new Solution Procedure 6.1

6.2

81

Metaheuristic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.1.1

Basic Concept of Metaheuristics . . . . . . . . . . . . . . . . . 82

6.1.2

Some Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . 84

6.1.3

The Fitness Landscape: a brief Side Trip . . . . . . . . . . . . 87

Algorithmic Concept for a new Solution Procedure . . . . . . . . . . 95 6.2.1

Motivation and Overview of the Algorithmic Concept . . . . . 96

6.2.2

Construction Algorithm . . . . . . . . . . . . . . . . . . . . . 99

6.2.3

Improvement Algorithm . . . . . . . . . . . . . . . . . . . . . 102

6.2.4

Adaptive Components . . . . . . . . . . . . . . . . . . . . . . 104

6.2.5

Configuration and Parameter Values . . . . . . . . . . . . . . 109

Contents

xi

7 Computational Study 7.1

7.2

7.3

Benchmark Instances of the JSPTWT

119 . . . . . . . . . . . . . . . . . 119

7.1.1

Modification of JSP Instances . . . . . . . . . . . . . . . . . . 119

7.1.2

Standard Benchmark Set of Singer and Pinedo . . . . . . . . . 120

7.1.3

Lawrence’s Instances . . . . . . . . . . . . . . . . . . . . . . . 131

Other Objective Functions . . . . . . . . . . . . . . . . . . . . . . . . 136 7.2.1

JSP with minimizing the Total Flow Time . . . . . . . . . . . 137

7.2.2

JSP with minimizing the Number of Tardy Jobs . . . . . . . . 141

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8 Conclusion

147

Bibliography

151

A Applying Neighborhood Operators: the example ft06 (f = 1.3)

167

Appendices B Overview of considered Dispatching Rules

171

C Computational Results from the Literature

175

D Analysis of the EGRASP result for the problem instance orb08 (f = 1.6) E Computational Results for the JSPTWU

181 187

List of Figures 2.1

Disjunctive Graph Model for minimum makespan JSP. . . . . . . . . 16

2.2

Disjunctive Graph Model for JSPTWT. . . . . . . . . . . . . . . . . . 17

2.3

Example of a critical tree. . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4

Gantt-Chart of the optimal schedule with makespan objective. . . . . 20

2.5

Gantt-Chart of the optimal schedule with TWT objective. . . . . . . 20

2.6

Directed Graph G of the ft06 (f = 1.3) schedule based on FCFS rule. 21

2.7

Gantt-Chart of the ft06 (f = 1.3) schedule based on the FCFS rule. . 22

2.8

CT (0, F ) of the ft06 (f = 1.3) schedule based on FCFS rule. . . . . . 22

4.1

CET move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2

CET+2MT move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3

CE3P move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4

SCEI move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5

CSR move.

4.6

ECET move.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.7

ICT move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.8

CE4P move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.9

DOCEI move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.10 DICEI move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.11 BCEI+2MT move. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.12 Course of a potential cycle produced by CET+2MT (I). . . . . . . . . 44 4.13 Course of a potential cycle produced by CET+2MT (II). . . . . . . . 45 4.14 Counterexample for Connectivity Property. . . . . . . . . . . . . . . . 47 4.15 Ranks according to the eight instance classes in Exp. 1 and Exp. 2. . 54 4.16 Correlations in Experiments 1 and 2. . . . . . . . . . . . . . . . . . . 55 4.17 Grouping of operator performance in Experiments 1 and 2. . . . . . . 57

List of Figures

xiv

4.18 Correlations in Experiments 5 and 6. . . . . . . . . . . . . . . . . . . 62 5.1

Predecessors and successors of an operation v in G . . . . . . . . . . . 68

5.2

Example to the Heads Updating procedure. . . . . . . . . . . . . . . 76

5.3

Neighbor evaluation process in the steepest descent algorithm. . . . . 79

6.1

Classification of solution procedures. . . . . . . . . . . . . . . . . . . 83

6.2

Morphology of metaheuristics. . . . . . . . . . . . . . . . . . . . . . . 84

6.3

Fictive example of a fitness landscape. . . . . . . . . . . . . . . . . . 89

6.4

Flowchart of the solution procedure EGRASP. . . . . . . . . . . . . . 97

6.5

Construction scheme for inserting the next unscheduled operation. . . 101

6.6

Demonstration of the LOPA strategy. . . . . . . . . . . . . . . . . . . 103

6.7

General concept of path relinking. . . . . . . . . . . . . . . . . . . . . 107

6.8

Process of changing the operation sequence in Path Relinking Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

List of Tables 2.1

Results of CPLEX 12.2 for several problem instances. . . . . . . . . . 14

2.2

Data to the example 3 x 3 instance. . . . . . . . . . . . . . . . . . . . 15

2.3

Data of the instance ft06 (f = 1.3). . . . . . . . . . . . . . . . . . . . 20

2.4

Job orders of the FCFS solution. . . . . . . . . . . . . . . . . . . . . 22

2.5

Overview of the critical blocks for the FCFS solution of ft06 (f = 1.3). 23

3.1

Literature overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1

List of existing neighborhood operators.

4.2

List of new neighborhood operators. . . . . . . . . . . . . . . . . . . . 37

4.3

Classification of neighborhoods based on perturbation schemes. . . . . 40

4.4

Data to the example 4 x 2 instance. . . . . . . . . . . . . . . . . . . . 46

4.5

Classification of neighborhoods based on feasibility and connectivity. . 48

4.6

Computational results using a single neighborhood operator. . . . . . 53

4.7

Computational results using pairs of neighborhood operators. . . . . . 59

4.8

Computational results using all neighborhood operators. . . . . . . . 61

4.9

Compact results of neighborhood operator. . . . . . . . . . . . . . . . 63

5.1

Computation times in Experiment 7. . . . . . . . . . . . . . . . . . . 80

6.1

Results of the diversity measurements. . . . . . . . . . . . . . . . . . 94

6.2

Average improving steps of the steepest descent algorithms. . . . . . . 95

6.3

List of tested dispatching rules. . . . . . . . . . . . . . . . . . . . . . 111

6.4

Average ranks of the dispatching rules. . . . . . . . . . . . . . . . . . 112

6.5

Hierarchy of the dispatching rules. . . . . . . . . . . . . . . . . . . . . 113

6.6

Stabilitity of the selected dispatching rules. . . . . . . . . . . . . . . . 115

. . . . . . . . . . . . . . . . 33

List of Tables

xvi

7.1

Computational results of EGRASP for the benchmark set of Singer

7.2

Computational results of EGRASP for the benchmark set of Singer

7.3

Computational results of EGRASP for the benchmark set of Singer

and Pinedo (f = 1.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 and Pinedo (f = 1.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 and Pinedo (f = 1.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.4

Comparison of computational results of EGRASP with best perform-

7.5

Computational results of ZSW (2013), EMD (2008) and EGRASP for

7.6

Computational results of ZSW (2013), EMD (2008) and EGRASP for

ing solution procedures, solving the instances of Singer & Pinedo. . . 125 the benchmark set of Singer and Pinedo (f = 1.3). . . . . . . . . . . . 127 the benchmark set of Singer and Pinedo (f = 1.5). . . . . . . . . . . . 128 7.7

Computational results of ZSW (2013), EMD (2008) and EGRASP for the benchmark set of Singer and Pinedo (f = 1.6). . . . . . . . . . . . 129

7.8

Termination limits for the computations to Lawrence’s instances. . . . 132

7.9

Computational results of EMD (2008), KB (2011), GGVV (2012) and EGRASP for Lawrence’s instances (f = 1.3). . . . . . . . . . . . . . . 133

7.10 Computational results of EMD (2008), KB (2011), GGVV (2012) and EGRASP for Lawrence’s instances (f = 1.5). . . . . . . . . . . . . . . 134 7.11 Computational results of EMD (2008), KB (2011), GGVV (2012) and EGRASP for Lawrence’s instances (f = 1.6). . . . . . . . . . . . . . . 135 7.12 Time limits for the computations of JSPTFT instances. . . . . . . . . 139 7.13 Computational results of GVSV (2010) and EGRASP for the benchmark set of González et al. . . . . . . . . . . . . . . . . . . . . . . . . 140 7.14 Computational results of CF (2008), MB (2004) and EGRASP for the benchmark set of Chiang and Fu. . . . . . . . . . . . . . . . . . . 144 A.1 Overview of the critical blocks for the FCFS solution of ft06 (f = 1.3).167 C.1 Comparison of the computational results for the benchmark set of Singer and Pinedo (f = 1.3). . . . . . . . . . . . . . . . . . . . . . . . 177 C.2 Comparison of the computational results for the benchmark set of Singer and Pinedo (f = 1.5). . . . . . . . . . . . . . . . . . . . . . . . 178 C.3 Comparison of the computational results for the benchmark set of Singer and Pinedo (f = 1.6). . . . . . . . . . . . . . . . . . . . . . . . 179

List of Tables

xvii

D.1 Job orders in the central global optimum. . . . . . . . . . . . . . . . . 182 D.2 Job orders in the best found solution of EGRASP. . . . . . . . . . . . 182 D.3 Data to the instance orb08 (f = 1.6). . . . . . . . . . . . . . . . . . . 185 E.1 Computational results of EGRASP for the benchmark set of Singer and Pinedo with TWU objective (f = 1.3). . . . . . . . . . . . . . . . 188 E.2 Computational results of EGRASP for the benchmark set of Singer and Pinedo with TWU objective (f = 1.5). . . . . . . . . . . . . . . . 189 E.3 Computational results of EGRASP for the benchmark set of Singer and Pinedo with TWU objective (f = 1.6). . . . . . . . . . . . . . . . 190

List of Notations n

-

number of jobs

J

-

set of jobs, J = {1, 2, . . . n}

m

-

number of machines

M

-

set of machines, M = {1, 2, . . . m}

oij

-

the i-th operation of job j

μij

-

associated machine of operation oij

pij

-

processing time of operation oij

rj

-

release time of job j

wj

-

weight of job j

dj

-

due date of job j

cj

-

completion time of job j

tj

-

tardiness of job j, tj = max{0, cj − dj }

uj

-

tardiness indicator of job j, uj = 1, if tj > 0; uj = 0, else start time of operation oij

sij

-

cmax

-

makespan of a schedule, cmax = max{cj | j = 1, . . . , n}

f

-

due date factor

G

-

(Disjunctive) Graph G = (N, A, E)

(i/j)

-

node representing operation oij

0,1

-

dummy nodes

N

-

set of nodes in graph G

A

-

set of consecutive arcs in graph G

E

-

set of disjunctive arcs in graph G

v→w

-

directed arc leading from node v to node w

Bj

-

completion node of job j

Fj

-

finishing node of job j

LP(u, v)

-

longest path from node u to node v

List of Notations

xx CT (0, F )

-

critical tree with root 0 and set of leaves F

L(.)

-

length of an object

# LP

-

number of longest paths

x

-

solution

X

-

solution set

c

-

cost function

NB

-

neighborhood

P(u, v)

-

path from node u to node v

C

-

cycle in a graph

σl

-

length of critical block l, i. e. the number of critical arcs in the block

τ

-

total number of critical blocks

τ1

-

number of critical blocks with length σl = 1

τ2

-

number of critical blocks with length σl = 2

τ3

-

number of critical blocks with length σl ≥ 3

ZN B

-

maximum number of neighboring schedules in N B

xi

-

sequence of solutions xi = (x1i , x2i , . . . )

Gap(j)

-

average gap of neighborhood operator j

# Eval

-

total number of schedule evaluations

# LOpt

-

total number of computed local optima

# Imp

-

average number of improving steps

Δ Gap

-

Gap difference according to absolute and relative

h(v)

-

head of node/operation v, i. e. h(v) = L(LP(0, v))

q j (v)

-

tail(j) of node/operation v, i. e. q j (v) = L(LP(v, Fj ))

P J(v)

-

predecessor of operation v in the job sequence

performance quality

(on the corresponding machine) SJ(v)

-

successor of operation v in the job sequence

P M (v)

-

predecessor of operation v in the machine sequence

SM (v)

-

successor of operation v in the machine sequence

(on the corresponding machine) of the corresponding job of the corresponding job l(v)

-

level value of operation v

h (v)

-

estimation value of the head of operation v

List of Notations

xxi

qvj

-

estimation value of tail(j) of operation v

LB

-

lower bound value

c¯max

-

exact makespan (of neighboring schedule)

TWT ¯ h(v)

-

exact total weighted tardiness value (of neighboring schedule)

-

exact value of the head of operation v (in neighboring schedule)

q¯j (v)

-

exact value of tail(j) of operation v (in neighboring schedule)

nxm

-

size of a problem instance based on n and m

n:m

-

ratio of jobs to machines

P

-

pool of solutions

AHD

-

average hamming distance

E

-

entropy

ωmjk

-

number of bits with value 1

TB

-

time bound

α

-

look-ahead parameter of time bound

β

-

scaling factor

ILP

-

number of iterations in learning phase

IAP

-

number of iterations in Amplifying Procedure

IP R

-

number of iterations in Path Relinking procedure

I

-

total number of iterations

g(i)

-

criticality of block i

Rk,i,ILP

-

k-th best rule of the learning phase, solving problem instance i

δk,i,ILP δ¯I

-

recurrence indicator of rule Rk,i,ILP

-

average recurrence of rule selection using ILP iterations

δ¯IbvLP

-

average recurrence of rule selection, based on the criterion of

with the setting of ILP iterations

LP

producing the best objective function value (bv), using ILP iterations δ¯IavLP

-

average recurrence of rule selection, based on the criterion of producing the best average objective function value (av), using ILP iterations

TotalGap -

total gap

# BKS

number of best known solutions computed

-

# ops

-

number of operations

fj

-

flow time of job j, fj = cj − rj

List of Notations

xxii

tinst

-

Tinst

-

time limit of the computations on instance inst summarized time limit of the computations on instance inst

Z(oij )

-

priority value of operation oij

t

-

time

SO

-

set of schedulable operations

List of Abbreviations ACO

- Ant Colony Optimization

ATC

- Apparent Tardiness Cost rule

BCEI+2MT

-

Backwards Critical End Insert + 2-Machine Transpose neighborhood

BFS

- Best Found Solution

BKS

- Best Known Solution

BS

- Benchmark Set

CE3P

- Critical End 3-Permutation neighborhood

CE4P

- Critical End 4-Permutation neighborhood

CET

- Critical End Transpose neighborhood

CET+2MT

-

COVERT

- Cost Over Time rule

CSR

- Critical Sequence Reverse neighborhood

CT

- Critical Transpose neighborhood

DD+PT+WT

-

DICEI

- Double Inwards Critical End Insert neighborhood

Critical End Transpose + 2-Machine Transpose neighborhood

Combination of Due Date, Processing Time and Waiting Time rule

DOCEI

- Double Outwards Critical End Insert neighborhood

ECET

- Extended Critical End Transpose neighborhood

ECT

- Earliest Completion Time rule

EDD

- Earliest Due Date rule

EGRASP

- Extended GRASP

FCFS

- First Come First Served rule

FDC

- Fitness-Distance-Correlation

FPTAS

- Fully Polynomial-Time Approximation Scheme

GA

- Genetic Algorithm

List of Abbreviations

xxiv

GRASP

- Greedy Randomized Adaptive Search Procedure

ICT

- Iterative Critical Transpose neighborhood

ILS

- Iterated Local Search

JSP

- Job Shop scheduling Problem

JSPTFT

- JSP with Total Flow Time objective

JSPTWT

- JSP with Total Weighted Tardiness objective

JSPTWU

- JSP with weighted number of tardy jobs objective

LOPA

- Longest Path sorting strategy

LPT

- Longest Processing Time rule

MDD

- Modified Due Date rule

ODD

- Operational Due Date rule

PPC

- Production Planning and Control

PTAS

- Polynomial-Time Approximation Scheme

PT+PW

- Combination of Processing Time and Waiting Time rule

PT+PW+ODD

-

Combination of Processing Time, Waiting Time and Operational Due Date rule

PT+WINQ+SL -

Combination of Processing Time, Work In Next Queue and Slack rule

RCL

- Restricted Candidate List

SCEI

- Single Critical End Insert neighborhood

SL

- Slack rule

SPT

- Shortest Processing Time rule

SPT+S/RPT

-

S/OPN

- Slack per number of operations rule

TS

- Tabu Search

TFT

- Total Flow Time

TWT

- Total Weighted Tardiness

TWU

- Total Weighted number of tardy jobs

VNS

- Variable Neighborhood Search

WCR

- Weighted Critical Ratio rule

WEDD

- Weighted Earliest Due Date rule

WI

- Weight rule

WINQ

- Work In Next Queue rule

Combination of Shortest Processing Time and Slack per Remaining Processing Time rule

List of Abbreviations

xxv

WMDD

- Weighted Modified Due Date rule

WRA

- Weighted Remaining Allowance rule

WSL+WSPT

-

Combination of Weighted Slack and

- Weighted Shortest Processing Time rule WSPT

- Weighted Shortest Processing Time rule