Mathematical Aspects. Scheduling and Applications


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1 Mathematical Aspects of Scheduling and Applications by R. BELLMAN University of Southern California Los Angeles, USA A. O. ESOGBUE Georgia Institute of Technology Atlanta, USA and I. NABESHIMA University of ElectroCommunications Tokyo, Japan TECHNISCHE HOCHSCHULE DARMSTADT Fachbereich 1 Gesamt b ib I iothek Be t f i eb s wi r 1 schof t s I eh r e Invemo r Nr. : AbstoilNr. : Saciigebiete:... < A PERGAMON PRESS OXFORD NEW YORK TORONTO SYDNEY PARIS FRANKFURT
2 1. Network Flow, Shortest Path and Control Problems Introduction The Routing Problem Dynamic Programming Approach Upper and Lower Bounds Existence and Uniqueness Optimal Policy Approximation in Policy Space Computational Feasibility Storage of Algorithms Alternate Approaches "Travelingsalesman" Problem Reducing Dimensionality Via Bounding Strategies Stochastic Travelingsalesman Problem Applications to Control Theory Stratification Routing and Control Processes Computational Procedure Feasibility Perturbation Technique Generalized Routing Discussion 22 Bibliography and Comments Applications to Artificial Intelligence and Games Introduction An Operational Point of View Description of a Particular Problem Imbedding A Fundamental Equation Geometrie Ideas Conversion of a Decision Process into an Equation The Concept of a Solution Determination of ehe Solution Determination of a Number Decisionmaking by Computers Enumeration Writinga Program Storage Dimensionality Structure Heuristics Discussion Application to Game Playing Mathematical Abstractions Intelligent Machines The Winepouring Problem 36 ix
3 x Formulation as a Multistage Decision Process Cannibals and Missionaries Formulation as a Multistage Decision Process Chinese Fifteen Puzzle The Puzzle Again Feasibility Doable Positions Associated Questions The Original Puzzle Lewis Carroll's Game of Doublets Chess and Checkers Solving Puzzles by Computer Discussion 44 Bibliography and Comments Scheduling Problems and Combinatorial Programming Introduction Classification of Scheduling Problems Sequencing Problem Project Scheduling Problem Assemblyline Balancing Problem Mutual Relations Summary of Combinatorial Programming as a Solution Technique State Transformation Process Branch and Bound Method (BAB Method) Relative Error of Approximate Value and Heuristic Algorithm for Deciding a Reliable Approximate Solution (Extended BAB Method) Backtrack Programming and Lexicographical Search Method 58 Bibliography and Comments The Nature of the Sequencing Problem Introduction Assumption and Objective Functions Objective Function (Measure of Performance) Objective Functions of the other Type Classification of Sequences and Nonnumerical Judgement Judgement for Determination Generation of Feasible Sequences Potentially Optimal Sequence and Nonoptimal Sequence Determination of Potentially Optimal Sequences and Nonoptimal Sequences Classification and Generation of Schedules Semiactive Schedule and Inadmissible Schedule Start and Completion Times of Each Operation Active Schedule and Nonactive Schedule aoptimal Schedule and Nonaoptimal Schedule Generation of Active Schedules 82 Bibliography and Comments Sequencing Involving Capacity Expansion Introduction A Simple Expansion Sequencing Problem the Onedimensional Version Why Dynamic Programming? Conventional Dynamic Program (DPI) for the Onedimensional Sequencing Problem 89
4 xi 5.5. The Imbedded States Space Dynamic Program (DP2) for the Onedimensional Sequencing Problem Discussion Formulation An Illustrative Example Reducing M&E Imbedded State Space for Largecapacity Expansion Problems Discussion Multidimensional Sequencing Problem Theory A Graphical Illustration Computational Experience on Realworld Problems Variations and Extensions Miscellaneous Exercises 104 Bibliography and Comments Sequencing Problems with Nonserial Structures Introduction Serial Multistage Sequencing Processes Nonserial Multistage Sequencing Processes CPMCost Problem: the Basic Model CPMCost Problems with Serial Structures CPMCost Problems with Nonserial Phase Structure Nonserial Networks: Project Cost Minimization Approach [PCM] Project Time Minimization Approach [PTM] A Dynamic Programming Model of the PTM Problem Example 1. The Pseudostage Concept and CPMCost Problem Example 2. A CPMCost Problem with Many Paths Departing from Junction Example 3. A Complex Nonserial System as in Fig Discussion 127 Bibliography and Comments Analytical Results for the Flowshop Scheduling Problem Introduction Characteristics of Schedules Calculation of Makespan Determination of Machine Idle Time and Waiting Time of Job Flow Time T k (i,) Recurrence Relation for Flow Time 7\(i,) Expressions of Makespan Calculation of Machine Idle Time Flow Network Expression in Critical Path Method Critical Path Length between Two Nodes and Makespan Twomachine MinMakespan Problem Solution by Using Machine Idle Time Solution by Dynamic Programming Solution by using Flow Time T 2 (i) Working Rules General Working Rule Restricted Working Rule 158 Bibliography and Comments Flowshop Scheduling Problems: Analytical Results II and Extensions Introduction 162
5 xii 8.2. Threemachine MinMakespan Problem Limited Solution under Special Processing Times Solution by Using Machine Idle Time Formulation by Flow Times T 2 (i), T 3 (i) Flow Times T 2 (i), T 3 (i) and Makespan Recurrence Relation Between Tß q ), Tß q ) (q = 1, 2,..., n) Dynamic Programmingtype Formulation Values of T 3 (i), Tßj), and T 3 (Jl T 3 (ji) Proofby Using Flow Times Case(a) Case(b) Limited Solution under Other Special Processing Times Sufficient Conditions for Two Adjacent Jobs to be in a Definite Order in the Optimal Sequence Sufficient Inequalities not Including T 2 (l) and T 3 (l) Sufficient Inequalities that Satisfy the Transitive Property Twomachine MinMakespan Problem where Time Lags Exist On the Generalizations to m Machine Permutation Scheduling 185 Bibliography and Comments Flowshop Scheduling Problems: General Solutions Introduction m Machine MinMakespan Problem (mä 3, no Passing is Allowed) Concrete Procedures in BAB Algorithms BAB Algorithms for Flow Shops Standard Lower Bound Efficiency of BAB Algorithm Relations Between Machine Order, Processing Times, and Optimal Sequence Revised Lower Bound and Composite Lower Bound Backtrack Programming and Lexicographical Search Method m Machine MinMean Completion Time Problem A Lemma and Analytical Results A BAB Algorithm m Machine MinPenalty Cost by Tardiness Problem Backtrack Programming and Lexicographical Search Method A BAB Algorithm and its Lower Bound 230 Bibliography and Comments Flowshop Scheduling Problems: Unified Algorithms and Approximate Solutions Introduction Unified Multistage Combinatorial Algorithms for Setuptime Imbedded Processing Times A Formulation by the State Transformation Process Simple Function q{f, P(S k )) for Each Objective Function F Relation Between MinMakespan Problem and MinOther Objective Problems Unified Multistage Combinatorial Algorithm Application to Parallel Scheduling Numerical Examples of MinMean Intermediate Waitingtime Problem and Related Parallel Scheduling Unified Multistage Combinatorial Algorithm where Explicit Setup Times Exist 244
6 xiii Setup Times A Formulation by the State Transformation Process Approximate Algorithms for MinMakespan Problem Application of a Specified Function to Construct an Approximate sequence 251 Bibliography and Comments Flowshop Scheduling under Sequence Dependence Conditions Sequence Dependent Setup Times Problem Definition Optimality of Permutation Schedules Dynamic Programming Formulations for ID and DI Forward DP Recursion for Problem Type ID Computational Aspects Backward DP Recursion for Problem Type DI Implications of Theorem Computational Aspects Discussion of the Dynamic Programming Approach Optimal Schedules for Problem Types DU, IDI and HD Dominance Conditions for Problem Types DU, IDI, and HD Backward Dynamic Programming Formulations Reducing Dimensionality Selection of Value for c, Computational Requirements Extensions 277 Bibliography and Comments The Jobshop Scheduling Problem Introduction The n x 2 MinMakespan Problem Graphical Solution of the 2 x m MinMakespan Problem Graphical Representation of the Schedule Construction of the Path Approaches by Integer Linear Programming Standard Formulation by ILP BAB Algorithms by Following Active Schedule Generation Procedure BAB Algorithm by Active Schedule Generation Procedure BAB Algorithm I Lower Bound in BAB Algorithm I Graph C o = (X, Z) Expressing Orderings Disjunctive Graph Representation by Limited Machine Availability BAB Algorithm on Disjunctive Graphs by following an Active Schedule Generation Procedure BAB Algorithm by Resolving the Pairs of Disjunctive Ares on a Disjunctive Graph Representation by Graph G o = (X, Z) Conflict Set Disjunctive Graph Complete Selection of Disjunctive Ares and Schedules Formulation of Two Types BAB Algorithm by Partial Selection of Disjunctive Ares Generalization of the Sequencing Problem: The Multiproject Scheduling Problem with Limited Resources Solution Based on Disjunctive Graphs Problem Statement Conflict Set 308
7 xiv Disjunctive Graphs Based on Conflict Sets of Activities Formulation by Disjunctive Graph BAB and EXTBAB Algorithms by Partial Selection of Disjunctive Ares BAB Algorithm Justification of Proposed BAB Algorithm and Remarks Numerical Examples by BAB Algorithm and Related EXTBAB Algorithm 315 Bibliography and Comments 318 Author Index 323 Subject Index 327
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