# STOCHASTIC RESOURCE-CONSTRAINED PROJECT SCHEDULING ?· STOCHASTIC RESOURCE-CONSTRAINED PROJECT SCHEDULING…

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<ul><li><p>STOCHASTIC RESOURCE-CONSTRAINEDPROJECT SCHEDULING</p><p>vorgelegt vonDipl.-Math. techn. Frederik Stork</p><p>aus Munchen</p><p>Vom Fachbereich Mathematikder Technischen Universitat Berlin</p><p>zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften</p><p>genehmigte Dissertation</p><p>Berichter: Prof. Dr. Rolf H. Mohring,Technische Universitat Berlin</p><p>Berichter: Prof. Dr. Peter Brucker,Universitat Osnabruck</p><p>Tag der wissenschaftlichen Aussprache: 09. April 2001</p><p>Berlin 2001D 83</p></li><li><p>PREFACE</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>iii</p></li><li><p>iv</p><p>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.</p><p>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.</p><p>Berlin, February 2001 Frederik Stork</p></li><li><p>CONTENTS</p><p>Introduction 1</p><p>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</p><p>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</p><p>2.4.1 Problem Definition and Related Work . . . . . . . . . . . 232.4.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . 25</p><p>2.5 Minimal Representation of AND/OR Precedence Constraints . . . 272.6 An NP-Complete Generalization . . . . . . . . . . . . . . . . . . 30</p><p>3 AND/OR Precedence Constraints: Earliest Job Start Times 333.1 Problem Definition and Related Work . . . . . . . . . . . . . . . 333.2 Arbitrary Arc Weights . . . . . . . . . . . . . . . . . . . . . . . 35</p><p>3.2.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 A Simple Pseudo-Polynomial Time Algorithm . . . . . . 373.2.3 A Game-Theoretic Application . . . . . . . . . . . . . . . 38</p><p>3.3 Polynomial Algorithms . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Positive Arc Weights . . . . . . . . . . . . . . . . . . . . 413.3.2 Non-Negative Arc Weights . . . . . . . . . . . . . . . . . 42</p><p>3.4 The Linear Time-Cost Tradeoff Problem . . . . . . . . . . . . . . 47</p><p>4 Representation of Resource Constraints in Project Scheduling 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Threshold and Forbidden Set Representations . . . . . . . . . . . 53</p><p>4.2.1 Relations to Threshold (Hyper-)Graphs . . . . . . . . . . 534.2.2 From Thresholds to Minimal Forbidden Sets . . . . . . . 54</p><p>v</p></li><li><p>vi Contents</p><p>4.2.3 Related Topics . . . . . . . . . . . . . . . . . . . . . . . 554.3 Computing Minimal Forbidden Sets . . . . . . . . . . . . . . . . 56</p><p>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</p><p>4.4 Computational Evaluation . . . . . . . . . . . . . . . . . . . . . 624.4.1 Setup and Benchmark Instances . . . . . . . . . . . . . . 624.4.2 Computational Results . . . . . . . . . . . . . . . . . . . 64</p><p>4.5 Further Remarks and Examples . . . . . . . . . . . . . . . . . . . 68</p><p>5 Robust Scheduling Policies 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 General Scheduling Policies . . . . . . . . . . . . . . . . . . . . 735.3 Earliest Start Policies . . . . . . . . . . . . . . . . . . . . . . . . 775.4 Preselective Policies . . . . . . . . . . . . . . . . . . . . . . . . 79</p><p>5.4.1 Definition and Characteristics . . . . . . . . . . . . . . . 795.4.2 Domination . . . . . . . . . . . . . . . . . . . . . . . . . 82</p><p>5.5 Linear Preselective Policies . . . . . . . . . . . . . . . . . . . . . 845.5.1 Definition and Characteristics . . . . . . . . . . . . . . . 845.5.2 Domination . . . . . . . . . . . . . . . . . . . . . . . . . 865.5.3 Acyclic Preselective Policies . . . . . . . . . . . . . . . . 87</p><p>5.6 Job-Based Priority Policies . . . . . . . . . . . . . . . . . . . . . 895.6.1 Definition and Characteristics . . . . . . . . . . . . . . . 895.6.2 Domination . . . . . . . . . . . . . . . . . . . . . . . . . 90</p><p>5.7 Relationship between Optimum Values . . . . . . . . . . . . . . . 91</p><p>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</p><p>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</p><p>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</p></li><li><p>Contents vii</p><p>6.4.4 Sorting the Minimal Forbidden Sets . . . . . . . . . . . . 1136.4.5 Flexible Search Strategy . . . . . . . . . . . . . . . . . . 114</p><p>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</p><p>Concluding Remarks 131</p><p>List of Algorithms 134</p><p>Bibliography 135</p><p>Symbol Index 147</p><p>Index 149</p><p>Zusammenfassung 151</p><p>Curriculum Vitae 153</p></li><li><p>INTRODUCTION</p><p>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 different resources, and the resource availability is limited. Inaddition, we assume that the processing time of each job is uncertain and follows agiven probability distribution. The outlined model, which is usually referred to asstochastic resource-constrained project scheduling, integrates two different, clas-sical scheduling models both of which have been extensively studied in the past40 years. On the one hand, this is the deterministic resource-constrained projectscheduling problem where each job processing time is assumed to be fixed andknown in advance. One of the first papers which refers to this model was writtenby Wiest (1963). Since then, a vast body of literature has been established; we hereonly mention the recent publications (Brucker, Drexl, Mohring, Neumann, andPesch 1999; Weglarz 1999) for reviews of different models and algorithms. Onthe other hand, stochastic resource-constrained project scheduling generalizes so-called stochastic project networks or PERT-networks where job processing timesare assumed to be stochastic but the resource availability is unlimited. Adlakhaand Kulkarni (1989) have provided a bibliography that classifies the enormousnumber of contributions up to 1987.</p><p>Scheduling models with stochastic job processing times are important becausemany uncertain events within project execution may cause job interruptions anddelays. Weather conditions, unavailability of resources, and authorization pro-cesses are only some examples. Already Fulkerson (1962) noted that the expectedcompletion of the last job in a stochastic project network, the expected projectmakespan, is greater than or equal to the project makespan that is based on theexpected processing times of jobs.</p><p>The necessity to consider models with random job processing times is proba-bly best motivated by the following quotation taken from the final report of a re-</p><p>1</p></li><li><p>2 Introduction</p><p>cent NASA project (Henrion, Fung, Cheung, Steele, and Basevich 1996). There,space shuttle ground processing is modeled as a stochastic resource-constrainedproject scheduling problem (we give details in Chapter 6 below).</p><p>Shuttle ground processing is subject to many uncertainties and delays.These uncertainties arise from many sources, including unexpected shuttlemaintenance requirements, failure of ground test equipment, unavailabilityof resources or technical staff, manifest constraints, and delays in paper-work. (Henrion et al. 1996, Page 4)</p><p>Scheduling with policies. Due to the combination of random job processingtimes and limited resources the stochastic resource-constrained project schedul-ing problem is a stochastic dynamic optimization problem and, as such, belongsto the field of stochastic dynamic programming. Scheduling is usually done by so-called policies. A policy may be seen as a dynamic decision process that defineswhich jobs are started at certain decision times t, based on the observed past up tot. Since it is commonly believed that the class of all policies is computationallyintractable, different subclasses of policies have been considered in the literature.Mohring and Radermacher (1985) have contributed an illustrative survey. Ourwork is based upon so-called preselective policies which have been introduced byRadermacher (1981b). Let us briefly mention the basic concept of this structurallyappealing class. Preselective policies are defined via so-called minimal forbiddensets. A set F of jobs without a precedence constraint among them is called for-bidden if the total resource consumption of the jobs in F exceeds the resourceavailability. If no proper subset of a forbidden set F is forbidden, then we call Fminimal forbidden. A preselective policy defines for each minimal forbidden seta preselected job j F which is postponed until at least one job from F \{j} hasbeen completed.</p><p>Contribution. The purpose of this thesis is to provide new insights on how tosolve stochastic resource-constrained project scheduling problems. To this end,we first study the combinatorial structure of preselective policies (and appropri-ately defined subclasses thereof). Then, we develop, implement, and evaluatesolution techniques for stochastic resource-constrained project scheduling prob-lems that are based on these classes of policies. We next outline the contributionsin more detail.</p><p>The obtained results on the combinatorial structure of preselective policies relyon the concept of so-called AND/OR precedence constraints which are a general-ization of traditional precedence constraints. For a given set V of jobs, an AND/ORprecedence constraint consists of a pair (X, j) with X V and j V \ X withthe meaning that at least one job from X must have been completed before j</p></li><li><p>Introduction 3</p><p>can be executed. AND/OR precedence constraints are of relevance in its own dueto their appearance within, e. g., assembly or disassembly processes (Goldwasserand Motwani 1999). We propose a new field of application which is based onthe fact that any preselective policy can be expressed as a set of such constraints.To this end, we develop a number of basic algorithms for scheduling jobs subjectto AND/OR precedence constraints. For example, we give two different polyno-mial time algorithms to compute earliest job start times as well as a linear timealgorithm to detect transitive AND/OR precedence constraints. These algorithmslater appear as important components within procedures to comput...</p></li></ul>

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