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  • STOCHASTIC RESOURCE-CONSTRAINEDPROJECT SCHEDULING

    vorgelegt vonDipl.-Math. techn. Frederik Stork

    aus Munchen

    Vom Fachbereich Mathematikder Technischen Universitat Berlin

    zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften

    genehmigte Dissertation

    Berichter: Prof. Dr. Rolf H. Mohring,Technische Universitat Berlin

    Berichter: Prof. Dr. Peter Brucker,Universitat Osnabruck

    Tag der wissenschaftlichen Aussprache: 09. April 2001

    Berlin 2001D 83

  • PREFACE

    After having received my diploma from the Technische Universitat Berlin in 1996,Rolf Mohring, the supervisor of my diploma thesis, offered me a research positionin his group. At that time I was employed at a Berlin software company thehead of which, Gert Scheschonk, strongly encouraged me to accept the offer. Iaccepted and in 1997 I began to work within a research initiative funded by theDeutsche Forschungsgemeinschaft DFG. The members engaged in this initiativebelong to five research groups in Germany which are located at universities inBonn, Karlsruhe, Kiel, Osnabruck, and Berlin. In Berlin, the scope of the projectwas to develop algorithms and theory for stochastic resource-constrained projectscheduling problems which is the main topic of this thesis.

    I am thankful to Rolf Mohring for his support, his encouragement, and the su-pervision of my thesis. In particular, I greatly benefited from his guidance duringmy work on AND/OR precedence constraints and scheduling policies.

    My special thanks go to my colleagues Martin Skutella and Marc Uetz. Martingreatly helped to establish, generalize, and improve many of my original consider-ations on AND/OR precedence constraints which finally led to the results presentedin Chapters 2 and 3. The continuous fruitful discussion with Marc led to new in-sights in the field of deterministic resource-constrained project scheduling. Theresults presented in Chapter 4 on different representations of resource constraintsare one example of this productive collaboration.

    I am also very grateful to my colleagues Andreas Schulz and Matthias Muller-Hannemann. I gained a lot from Andreas expertise and his co-authorship in pa-pers on deterministic project scheduling (which are not part of this thesis). Myformer roommate Matthias was always willing to interrupt his work in order todiscuss the questions I raised.

    I would also like to mention the fruitful collaboration with the other membersof the DFG research initiative on resource-constrained project scheduling. In par-ticular, I thank Peter Brucker for the willingness to serve as a member of my thesiscommittee.

    Some parts of this thesis rely on software implementations that would not havereached the current quality without the support of Ewgenij Gawrilow. I thankhim for introducing me to the concept of generic programming; he had a greatshare in establishing the basis of our programming environment, a collection offundamental scheduling algorithms and data structures.

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  • iv

    Finally, I am grateful to Marc Uetz, Martin Skutella, Andreas Schulz, MarcPfetsch, Michael Naatz, Ekkehard Kohler, and Andreas Fest for their carefulproof-reading of different parts of the manuscript.

    It has been a great pleasure to share both research and leisure activities with thecolleagues at the Technical University in the groups of Rolf Mohring and GunterZiegler. It is hard to imagine a better working environment.

    Berlin, February 2001 Frederik Stork

  • CONTENTS

    Introduction 1

    1 Project Scheduling 71.1 Deterministic Resource-Constrained Project Scheduling . . . . . . 71.2 Stochastic Project Networks (PERT-Networks) . . . . . . . . . . 101.3 Stochastic Resource-Constrained Project Scheduling . . . . . . . 12

    2 AND/OR Precedence Constraints: Structural Issues 172.1 Motivation and Related Work . . . . . . . . . . . . . . . . . . . . 172.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Detecting Implicit AND/OR Precedence Constraints . . . . . . . . 23

    2.4.1 Problem Definition and Related Work . . . . . . . . . . . 232.4.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . 25

    2.5 Minimal Representation of AND/OR Precedence Constraints . . . 272.6 An NP-Complete Generalization . . . . . . . . . . . . . . . . . . 30

    3 AND/OR Precedence Constraints: Earliest Job Start Times 333.1 Problem Definition and Related Work . . . . . . . . . . . . . . . 333.2 Arbitrary Arc Weights . . . . . . . . . . . . . . . . . . . . . . . 35

    3.2.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 A Simple Pseudo-Polynomial Time Algorithm . . . . . . 373.2.3 A Game-Theoretic Application . . . . . . . . . . . . . . . 38

    3.3 Polynomial Algorithms . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Positive Arc Weights . . . . . . . . . . . . . . . . . . . . 413.3.2 Non-Negative Arc Weights . . . . . . . . . . . . . . . . . 42

    3.4 The Linear Time-Cost Tradeoff Problem . . . . . . . . . . . . . . 47

    4 Representation of Resource Constraints in Project Scheduling 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Threshold and Forbidden Set Representations . . . . . . . . . . . 53

    4.2.1 Relations to Threshold (Hyper-)Graphs . . . . . . . . . . 534.2.2 From Thresholds to Minimal Forbidden Sets . . . . . . . 54

    v

  • vi Contents

    4.2.3 Related Topics . . . . . . . . . . . . . . . . . . . . . . . 554.3 Computing Minimal Forbidden Sets . . . . . . . . . . . . . . . . 56

    4.3.1 Counting Minimal Forbidden Sets . . . . . . . . . . . . . 564.3.2 Description of the Algorithm . . . . . . . . . . . . . . . . 574.3.3 Analysis of the Algorithm . . . . . . . . . . . . . . . . . 584.3.4 Implementation and Fast Reduction Tests . . . . . . . . . 594.3.5 Compact Representation of Forbidden Sets . . . . . . . . 61

    4.4 Computational Evaluation . . . . . . . . . . . . . . . . . . . . . 624.4.1 Setup and Benchmark Instances . . . . . . . . . . . . . . 624.4.2 Computational Results . . . . . . . . . . . . . . . . . . . 64

    4.5 Further Remarks and Examples . . . . . . . . . . . . . . . . . . . 68

    5 Robust Scheduling Policies 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 General Scheduling Policies . . . . . . . . . . . . . . . . . . . . 735.3 Earliest Start Policies . . . . . . . . . . . . . . . . . . . . . . . . 775.4 Preselective Policies . . . . . . . . . . . . . . . . . . . . . . . . 79

    5.4.1 Definition and Characteristics . . . . . . . . . . . . . . . 795.4.2 Domination . . . . . . . . . . . . . . . . . . . . . . . . . 82

    5.5 Linear Preselective Policies . . . . . . . . . . . . . . . . . . . . . 845.5.1 Definition and Characteristics . . . . . . . . . . . . . . . 845.5.2 Domination . . . . . . . . . . . . . . . . . . . . . . . . . 865.5.3 Acyclic Preselective Policies . . . . . . . . . . . . . . . . 87

    5.6 Job-Based Priority Policies . . . . . . . . . . . . . . . . . . . . . 895.6.1 Definition and Characteristics . . . . . . . . . . . . . . . 895.6.2 Domination . . . . . . . . . . . . . . . . . . . . . . . . . 90

    5.7 Relationship between Optimum Values . . . . . . . . . . . . . . . 91

    6 Branch-and-Bound Algorithms 956.1 Introduction and Related Work . . . . . . . . . . . . . . . . . . . 956.2 Branch-and-Bound and Random Processing Times . . . . . . . . 986.3 Dominance Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    6.3.1 Earliest Start Policies . . . . . . . . . . . . . . . . . . . . 1046.3.2 Preselective Policies . . . . . . . . . . . . . . . . . . . . 1046.3.3 Linear Preselective Policies via Forbidden Sets . . . . . . 1066.3.4 Linear Preselective Policies via the Precedence-Tree . . . 1076.3.5 Job-Based Priority Policies . . . . . . . . . . . . . . . . . 108

    6.4 Improving the Performance . . . . . . . . . . . . . . . . . . . . . 1106.4.1 Initial Upper Bound . . . . . . . . . . . . . . . . . . . . 1106.4.2 The Critical Path Lower Bound and Jensens Inequality . . 1106.4.3 Single Machine Scheduling Relaxations . . . . . . . . . . 112

  • Contents vii

    6.4.4 Sorting the Minimal Forbidden Sets . . . . . . . . . . . . 1136.4.5 Flexible Search Strategy . . . . . . . . . . . . . . . . . . 114

    6.5 Computational Study . . . . . . . . . . . . . . . . . . . . . . . . 1146.5.1 Computational Setup . . . . . . . . . . . . . . . . . . . . 1146.5.2 The Test Sets . . . . . . . . . . . . . . . . . . . . . . . . 1156.5.3 Comparison of the Procedures . . . . . . . . . . . . . . . 1166.5.4 Impact of Additional Ingredients . . . . . . . . . . . . . . 1216.5.5 Application to other Instances . . . . . . . . . . . . . . . 127

    Concluding Remarks 131

    List of Algorithms 134

    Bibliography 135

    Symbol Index 147

    Index 149

    Zusammenfassung 151

    Curriculum Vitae 153

  • INTRODUCTION

    Motivation. Scheduling theory is an important and dynamic subject within com-binatorial optimization and has attracted numerous researchers. Scheduling isconcerned with the planning of activities over time subject to various side con-straints with the intention to minimize some objective function. Activities areseparate pieces of work and are commonly referred to as jobs. In this thesis weconsider a fairly general scheduling model that has numerous applications andcontains many other models as a special case. Let us sketch the characteristics ofthe model. First, precedence constraints have to be respected, that is, certain jobsmust be completed before others can be executed. During its execution, each jobrequires capacity of

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