GOR ■ Publications Managing Editor Kolisch, Rainer Editors Burkard, Rainer E Fleischmann, Bernhard Inderfurth, Karl Möhring, Rolf H Voss, Stefan Titles in the Series H.-O Günther and P v Beek (Eds.) Advanced Planning and Scheduling Solutions in Process Industry VI, 426 pages 2003 ISBN 3-540-00222-7 J Schönberger Operational Freight Carrier Planning IX, 164 pages 2005 ISBN 3-540-25318-1 Christoph Schwindt Resource Allocation in Project Management With 13 Figures and 12 Tables 123 Professor Dr Christoph Schwindt Institut für Wirtschaftswissenschaft TU Clausthal Julius-Albert-Straße 38678 Clausthal-Zellerfeld E-mail: christoph.schwindt@tu-clausthal.de Library of Congress Control Number: 2005926096 ISBN 3-540-25410-2 Springer Berlin Heidelberg New York This work is subject to copyright.All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag.Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, 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 Cover design: Erich Kirchner Production: Helmut Petri Printing: Strauss Offsetdruck SPIN 11409960 Printed on acid-free paper – 42/3153 – Die Seele jeder Ordnung ist ein grofier Papierkorb Kurt Tucholsky, Schnipsel Preface This monograph grew out of my research in the ficld of resourcc-constraincd project scheduling conducted from 1995 to 2004 during my work as teaching assistant and assistant professor at the Institute for Economic Theory and Operations Research of thc Univcrsity of Karlsruhe The aim of the book is to givc an introduction to quantitative concepts and methods for resource allocation in project managcmcnt with an cmphasis on an ordcr-theoretic framework allowing for a unifying treatment of various problem types In order to make the work accessible for general readers, the basic concepts nccded arc rcviewed in introductory scctions of the book Many pcople have contributed to the outcome of this research First and foremost, I would like to express my deep appreciation to my supervisor Professor Klaus Ncumann, who introduced me to the field and the community of project schcduling I have greatly benefited from his comprehensive scientific knowlcdgc and expertise, his continuous encouragement, and his support During all these years, his departmcnt has bcen a stimulating and attractive place for doing research and teaching in Operations Rcsearch Moreover, I would like to thank my formcr collcagucs for many fruitful discussions on various research topics and their continuing interest in my work A major part of my rcsearch has been done in collaboration with the colleagues of the Karlsruhe project scheduling group, Birger Franck, CordUlrich Fundeling, Karsten Gentner, Steffen Hagmaycr, Dr Thomas Hartung, Dr Roland Heilmann, Christoph Mellentien, Dr Hartwig Nubel, Dr Thomas Selle, PD Dr Norbert Trautmann, and Professor Jiirgcn Zimmcrmann Our work has been greatly influenccd by the activities of a research unit on project scheduling funded by the Deutsclle Forschungsgemcinschaft and involving colleagucs from the universities of Bcrlin (Profcssor Rolf Mijhring), Bonn (Professor Erwin Pcsch), Karlsruhe (Professor Klaus Ncumann), Kicl (Professor Andreas Drexl), and Osnabriick (Professor Peter Brucker) Numerous joint workshops on project scheduling and the "cooperative-competitive" spirit in this group havc been a great incentive to work even harder viii Preface Finally, I grateful acknowledge the help of several peoplc in preparing the manuscript of this monograph: Klaus Neumann for many valuable comments on different versions of the manuscript, Gerhard Grill for carefully proofreading and improving the English wording of the manuscript, Frederik Stork for pointing me to state-of-the-art contributions in convex programming, and Jiirgcn Zimmermann for making experimental results on rcsourcc levelling problems availablc to me Of course thc faults and dcficicncics rcmaining are entircly my own Clausthal-Zellerfeld, February 2005 Christoph Schwindt Contents Introduction 1 Models and Basic Concepts 1.1 Tcrnporal Constraints 1.1.1 Time-Feasible Schedules 1.1.2 Project Networks 1.1.3 Temporal Scheduling Computations 11 1.2 Renewablc-Resource Constraints 16 1.2.1 Rcsourcc-Fcasiblc Schedules 16 1.2.2 Forbidden Sets and Delaying Alternatives 19 1.2.3 Breaking up Forbidden Sets 21 1.2.4 Consistency Tests 23 1.3 Cumulative-Resource Constraints 28 1.3.1 Resource-Feasible Schedules 30 1.3.2 Forbidden Sets and Delaying Alternativcs 32 1.3.3 Brcaking up Forbidden Sets 35 1.3.4 Consistency Tests 36 Relations Schedules and Objective Functions 39 2.1 Resource Constraints and Feasible Relations 39 2.1.1 Renewablc-Resource Constraints 40 2.1.2 Cumulative-Resource Constraints 46 2.2 A Classification of Schcdulcs 52 2.2.1 Global and Local Extreme Points of the Feasible Region 52 2.2.2 Vcrtices of Relation Polytopes 53 2.3 Objectivc Functions 55 2.3.1 Regular and Convexifiable Objective Functions 56 2.3.2 Locally Regular and Locally Concave Objective Functions 60 2.3.3 Preorder-Decreasing Objective Functions 64 x Contents Relaxation-Based A l g o r i t h m s 65 3.1 Regular Objcctivc Functions 66 3.1.1 Enumeration Scheme 66 3.1.2 Solving the Relaxations 69 3.1.3 Branch-and-Bound 72 3.1.4 Additional Notes and References 76 3.2 Corwexifiablc Objective Functions 82 3.2.1 Enumeration Scheme 83 3.2.2 Solving thc Rclaxations: Thc Primal Approach 85 3.2.3 Solving the Relaxations: The Dual Approach 94 3.2.4 Branch-and-Bound 97 3.2.5 Additional Notes and References 99 C o n s t r u c t i v e A l g o r i t h m s 107 4.1 Schedule-Generation Schcme 109 4.2 Local Scarch 115 4.3 Additional Notcs and Rcfererlccs 118 Supplements 123 5.1 Break Calendars 124 5.2 Scquencc-Dcpcndent Changeover Times 128 Alternative Execution Modes for Activities 131 5.4 Continuous Cumulative Resources 135 Applications 141 6.1 Make-to-Order Production Scheduling 142 6.2 Small-Batch Production Planning in Manufacturing Industries 147 6.3 Production Scheduling in thc Proccss Industrics 149 6.4 Evaluation of Investment Projects 155 6.5 Coping with Uncertainty 160 Conclusions 165 References 167 List of S y m b o l s 181 I n d e x 185 178 References 209 Sourd F, Rogerie J (2005) Continuous filling and emptying of storage systerns in constraint-based scheduling European Journal of Operational Research 165:510-524 210 Sprecher A, Drexl A (1998) Multi-mode resource-constrained project scheduling by a simple, 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networks ZOR Methods and Models of Operational Research 36:423-438 232 Zheng S, Sun F (1999) Some simultaneous iterations for finding all zeros of a polynonlial with high order convergence Applied Mathematics and Computation 99:233-240 233 Zimmermann J (2001a) Ablauforientiertes Projektmanagement: Modelle, Verfahren und Anwendungcn Gabler, Wiesbaden 234 Zimmermann J (2001b) Personal communication 235 Zimmermann J, Schwindt C (2002) Parametric optimization for the evaluation of investnlent projects In: The Eighth International Workshop on Project Management and Scheduling (PMS 2002), Valencia, pp 378-382 236 Zoutendijk G (1960) Methods of Feasible Directions Elsevier, Amsterdam 237 Zwick U, Paterson M (1996) The complexity of mean payoff games on graphs Theoretical Computer Science 158:343-359 List of Symbols Equal by definition End of proof Smallest integer greater than or equal to x Maximum of and x Empty set Half open interval {x E R I a x < b} Cardinality of finite set X Ball of radius E around S in I W " + ~ Line segment in Rnf Set of all positive integers Landau's symbol Power set of set X Set of all real numbers Set of all n-tuples of real numbers Set of all nonnegative real numbers X is proper subset of Y X is subset of Y Difference of sets X and Y Intersection of sets X and Y Union of sets X and Y Set of all integers Set of all nonnegative integers Projects and project activities 6, bv d?x ,:Y' d Activity calendar for activity i Time lag calendar for arc (i,j ) Maximum time lag between the starts of activities i and j Minimum tinie lag between the starts of activities i and j Prescribed maximum project duration 182 List of Symbols Execution mode for activity i Set of alternative execution modes for activity i Duration (processing time) of activity i Duration of activity i in execution mode rn, Sequence-dependent changeover time from activity i to activity j on resource k Set of all activities Set of all real activities Set of all fictitious activities (events) Set of all real activities using renewable resource k Set of all events depleting cumulative resource k Set of all events replenishing curnulativc resource k Directed graphs and networks G = (V, E ) G = (V, E, 6) (4 j ) m = lEl 7L = IVI N Pred (i) S~rcc(i) + v Distance matrix for project network N Length of a longest directed path (distance) from node i to node j in project network N Weight of arc (i,j) Weight of arc (i,j) for mode combination (m,, m j ) Arc set of project network N Directed graph with node set V and arc set E Weighted directed graph (network) with node set V, arc set E, and vector of arc weights Arc with initial node i and terrnirlal node j Number of arcs in project network N Number of nodes in project network N Project network Set of all direct predecessors of node i E V Set of all direct successors of node i E V Node set of project network N Resources Forbidden set of activities Set of all minimal forbidden sets Set of all minimal k-surplus sets Set of all miriimal k-shortage sets Requirement of activity i for resource k Requirement of activity i for resource k in execution mode m, Loading profile for resource k given schedule S Capacity of renewable resource k or availability of nonrenewable resource k Safety stock of cumulative resource k Storage capacity of cumulative resource k Set of all (discrete) cumulative resources Set of all corltinuous cumulative resources Set of all nonrenewable resources Set of all renewable resources List of Symbols Objective functions Continuous interest rate Cash flow associated with the start of activity i Per unit cost for resource Ic Left-hand Si-derivative of f at S Right-hand &-derivative of f at S Objective function to be minimized Continuation of objective function f C1-diffe~mor~hism Derivative of f at S Left-hand derivative of f at S Right-hand derivative of f at S Directional derivative of f at S in direction z Weight of activity i Earliness and tardiness costs for activity i per unit time Weight of free floats for activity i Relations and preorders Covering relation of strict order 6' Distance matrix for relation network N ( p ) Distance from node i to node j in relation network N ( p ) Weight of arc ( i ,j ) in relation network N ( p ) Precedence graph of strict order 6' Set of all C_-minimal feasible relations Minimal point of ordered set (M,