Foundations of integer programming:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
North-Holland
1989
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [705] - 743 |
| Physical Description: | XIX, 755 S. graph. Darst. |
| ISBN: | 0444012311 |
Staff View
MARC
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| 100 | 1 | |a Salkin, Harvey Marshall |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Foundations of integer programming |c Harvey M. Salkin ; Kamlesh Mathur |
| 264 | 1 | |a New York u.a. |b North-Holland |c 1989 | |
| 300 | |a XIX, 755 S. |b graph. Darst. | ||
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| 650 | 4 | |a Integer programming | |
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Record in the Search Index
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| adam_text | Contents
Preface xvii
1 Introduction to Integer Programming 1
1.1 Linear Programs with Integer Variables 1
1.2 Uses and Applications 3
1.2.1 Formulations That Allow Integer Variables .... 3
1.2.2 Classical Applications and Case Studies 8
Problems 20
2 Review of Linear Programming 27
2.1 The Linear Programming Problem 27
2.2 Graphical Solution and Geometric Concepts 28
2.3 The Simplex Algorithm 31
2.3.1 Definitions 31
2.3.2 Fundamental Theorems 34
2.3.3 The Simplex Algorithm 35
2.3.4 The Two Phase Simplex Method 42
2.4 The Revised Simplex Algorithm 47
2.5 Duality in Linear Programming 53
2.6 The Dual Simplex Algorithm 57
2.7 The Beale Tableau 62
Problems 67
3 Using Linear Programming to Solve Integer Programs 70
3.1 Graphical Solutions to Mixed Integer or Programs .... 70
3.2 Solving an Integer Programming Problem as a Linear
Program 73
3.2.1 Unimodularity 74
3.3 Obtaining Integer Programming Solutions by Rounding
Linear Programming Solutions 76
vii
viii Contents
3.4 An Overview of Approaches for Solving Integer (or Mixed
Integer) Problems 78
3.4.1 Cutting Plane Techniques (Chapters 4, 5, 6, 7) . . 79
3.4.2 Enumerative Methods (Chapters 8, 9) 79
3.4.3 Partitioning Algorithms (Chapter 10) 79
3.4.4 Group Theoretic Algorithms (Chapter 11) .... 79
Problems 80
4 Dual Fractional Integer Programming 84
4.1 The Basic Approach 84
4.2 Notation: The Beale Tableau 87
4.3 The Form of the (Gomory) Cut 89
4.4 Illustrations 90
4.5 The Derivation of the Cut 99
4.5.1 Congruence 99
4.5.2 Derivation 100
4.6 Some Properties of Added Inequalities 102
4.7 Algorithm Strategies 116
4.7.1 Number of Possible Inequalities 116
4.7.2 Choosing the Source Row 116
4.7.3 Dropping Inequalities 117
4.8 A Geometric Derivation 118
4.9 Finiteness 121
Problems 123
Appendix A: Convergence using the Dantzig Cut .... 132
Appendix B: A Variation of the Basic Approach: The
Accelerated Euclidean Algorithm 135
Appendix C: Geometrically Derived Cuts 142
5 Dual Fractional Mixed Integer Programming 150
5.1 The Basic Approach 150
5.2 The Form of the Cut 151
5.3 An Illustration 152
5.4 The Derivation of the Cut 155
5.5 Applying the Mixed Cut to the Integer Program 159
5.6 Finiteness 163
Problems 166
Contents ix
6 Dual All Integer Integer Programming 170
6.1 The Basic Approach 170
6.2 The Form of the Cut 172
6.2.1 The Rules for Finding A 173
6.3 Illustrations 173
6.4 Derivations: The Form of the Cut, the Pivot Column,
and the A Selection Rules 179
6.4.1 The Form of the Cut 179
6.4.2 Choosing the Pivot Column 181
6.4.3 A Selection Rules 182
6.5 Some Properties of the Added Inequalities 184
6.5.1 The Relative Strength of the Added Inequalities . 184
6.5.2 A Second Derivation of the Fractional Cut, and
Relation to the All Integer Cut 186
6.6 Algorithm Strategies 187
6.6.1 Choosing the Source Row 187
6.6.2 Dropping Inequalities 188
6.7 Finiteness 188
6.7.1 If Only the Cost Row Is All Integer 189
Problems 190
7 Primal All Integer Integer Programming 197
7.1 Introduction 197
7.2 The Tableau, the Rudimentary Primal Algorithm . . . .198
7.2.1 The Rudimentary Algorithm 199
7.2.2 The Rudimentary Primary Algorithm 200
7.3 A Convergent Algorithm: The Simplified Algorithm (SPA)204
7.3.1 Modifications 1 and 2: Introducing a Reference
Row and Selecting the Pivot Column 205
7.3.2 Modification 3: Acceptable Source Row Selection
Rules 206
7.3.3 The Simplified Primal Algorithm (SPA) 211
7.4 A Second Reference Equation: Using the Dual Variables 213
7.5 Illustrations (SPA) 216
7.6 Optimality without Dual Feasibility 226
7.7 Convergence 228
7.7.1 Finiteness under Rule 1 231
7.7.2 Finiteness under Rule 2 232
7.7.3 Finiteness under Rule 3 233
x Contents
Problems 234
Appendix A: Cycling in Rudimentary Promal Algorithm 240
8 Branch and Bound Enumeration 245
8.1 Introduction , 245
8.2 The Problem, Notation, and the Basic Result 247
8.2.1 The Problem, Notation 247
8.2.2 The Basic Result and Geometric Interpretations . 248
8.3 The Enumeration Tree, Algorithm Formulation, An Ex¬
ample 255
8.3.1 The Tree, Algorithm Formulation 255
8.3.2 An Example 257
8.4 The Basic Approach, A Second Example 261
8.4.1 The Basic Approach 261
8.4.2 A Second Example 263
8.5 A Variation of the Basic Approach 264
8.6 Specialization for the Zero One Problem 267
8.7 Node Selection, Branching Rules, and Penalties 269
8.7.1 Node Selection 269
8.7.2 Branching Rules and Penalties 272
Problems 277
Appendix A: Computational Details of Examples 8.3 and
8.4 287
9 Search Enumeration 298
9.1 Introduction 298
9.2 The Basic Approach, the Tree 299
9.2.1 The Basic Approach 300
9.2.2 The Tree 300
9.3 The Point Algorithm: Implicit Enumeration Criteria . . 304
9.3.1 Ceiling Tests 305
9.3.2 Infeasibility Test 306
9.3.3 Cancellation Zero Test 306
9.3.4 Cancellation One Test 307
9.3.5 Linear Programming 308
9.3.6 Post Optimization, Penalties 309
9.3.7 Surrogate Constraints 310
9.4 The Point Algorithm: Branching Strategies 314
9.4.1 Preferred Sets 314
Contents xi
9.4.2 The Balas Test 317
9.4.3 Other Branching Rules 318
9.5 The Generalized Origin, Restarts 318
9.6 Search Enumeration in 0 — 1 Mixed Integer
Programming 322
Problems 325
Appendix A: A Sample Search Algorithm and Its Imple¬
mentation 339
A.I The Basic Approach; The Point Algorithm .... 340
A.2 The Bookkeeping Scheme 341
Appendix B: Computational Experience 343
B.I Program Description 343
10 Partitioning in Mixed Integer Programming 346
10.1 Introduction 346
10.2 Posing the Mixed Integer Program as an Integer
Program 347
10.3 The Partitioning Algorithm 352
10.4 Properties of the Partitioning Algorithm 356
10.5 Finiteness 360
Problems 363
Appendix A: Application of the Partitioning Algorithm
to the Uncapacitated Plant Location Problem 370
A.I Problem [Pl] s Equivalent Integer Program [I] 371
A.2 Solving the Linear Program [DL] 372
A.3 Summary of the Partitioning Algorithm 374
A.4 An Illustrative Example 375
11 Group Theory in Integer Programming 380
11.1 Introduction 380
11.2 The Group Minimization Problem 383
11.2.1 The Group G{a) 387
11.2.2 Solving the GMP 393
11.3 Solving the Group Minimization Problem 394
11.3.1 Dynamic Programming Algorithms 395
11.3.2 A Sufficient Condition for xfl 0 405
11.3.3 An Enumeration Algorithm 410
11.4 The Group Minimization Problem Viewed as a Network
Problem 415
xii Contents
11.4.1 Formulation 415
11.4.2 Solving the Integer Program by Solving the Net¬
work Problem 420
11.5 An Equivalent Group Minimization Problem 422
11.6 The Isomorphic Factor Group G(A)/G(B) 430
11.6.1 Definitions, Results 430
11.6.2 The Isomorphic Groups G(a) and G(X) 434
11.6.3 The Subgroup Decomposition of G(A)/G(B) . . 439
11.6.4 The Order of G(a) and G{ ) 442
11.7 The Geometry 443
11.7.1 The Corner Polyhedron (x Space) 443
11.7.2 The Corner Polyhedron (xN Space) 446
11.7.3 Relating the Corner Polyhedra 448
11.7.4 The Master (Comer) Polyhedron 453
11.7.5 Generating Valid Inequalities from the Faces of
Master Polyhedra 461
Problems 467
Appendix A: A Shortest Route Algorithm 485
A.I Specialization of the Shortest Path Algorithm for the
Group Minimization Problem 491
Appendix B: Diagonalizing the Basis Smith s Normal
Form 492
Appendix C: Computational Experience 499
C.I Computer Programmed Group Minimization Algorithms 499
C.I.I A Dynamic Programming Algorithm 499
C.1.2 Branch and Bound Algorithm 501
C.1.3 Shortest Route Algorithms 502
C.2 Reducing The Order Of The Group 503
C.2.1 Scaling 503
C.2.2 Relaxation 505
C.2.3 Decomposition 506
Appendix D: Implementation of a Group Theoretic Al¬
gorithm 508
D.I Linear Programming Module 508
D.2 Construction of FGMP 508
D.3 Solution of the FGMP 509
D.4 Branch and Bound Algorithm 509
Contents xiii
D.4.1 Node Selection Rule 509
D.4.2 Implicit Branching Rule 509
D.4.3 Node Omission Rule 510
D.4.4 Combining the Implicit Branching Rule and the
Node Omission Rule 510
12 The Knapsack Problem 513
12.1 Introduction 513
12.2 Applications and Uses of Knapsack and Related
Problems 516
12.2.1 Capital Budgeting 516
12.2.2 The Cutting Stock Problem 519
12.2.3 Loading Problems 524
12.2.4 Change Making Problem 525
12.2.5 Other Uses 525
12.3 Reducing Integer Programs to Knapsack Problems: Ag¬
gregating Constraints 526
12.3.1 An Aggregation Process 527
12.3.2 An Improved Aggregation Process 533
12.4 Algorithms 538
12.4.1 Dynamic Programming Techniques 538
12.4.2 A Periodic Property 546
12.4.3 Branch and Bound Algorithms: General Knap¬
sack Problem 554
12.4.4 Lagrangian Multiplier Methods 559
12.4.5 Network Approaches 564
12.5 Branch and Bound Algorithms: 0 1 Knapsack Problem . 567
12.5.1 The Linear Programming Solution 567
12.5.2 The Upper Bound Solution 568
12.5.3 The Lower Bound Solution 569
12.5.4 Reduction Algorithm 569
12.5.5 The Branch and Bound Algorithm 571
Problems 578
13 The Set Covering Problem, the Set Partitioning Prob¬
lem 590
13.1 Introduction 590
13.2 Set Covering and Networks 593
13.2.1 The Node Covering Problem 593
xiv Contents
13.2.2 The Matching Problem 595
13.2.3 Disconnecting Paths 597
13.2.4 The Maximum Flow Problem 599
13.3 Applications 600
13.3.1 Airline Crew Scheduling 600
13.3.2 Truck Scheduling 602
13.3.3 Political Restricting 602
13.3.4 Information Retrieval 603
13.4 Relevant Results 603
13.5 Algorithms 614
13.5.1 A Search Algorithm for the Set Partitioning Prob¬
lem 615
13.5.2 A Search Algorithm for the Set Covering Problem 618
Problems 625
Appendix A: Computational Experience 633
A.I Cutting Plane Algorithms 633
A.2 Enumerative Algorithms 635
A.3 Summary 638
14 The Fixed Charge Problems: The Plant Location Prob¬
lem and Fixed Charge Transportation Problem 639
14.1 Introduction 639
14.1.1 The Plant Location Problem 639
14.1.2 The Fixed Charge Transportation Problem .... 641
14.2 Algorithms for the Plant Location Problem 643
14.2.1 A Branch and Bound Algorithm for the Fixed
Charge Problem 644
14.2.2 A Branch and Bound Algorithm for the Plant
Location Problem 646
14.3 Branch and Bound Algorithm for the Fixed Charge Trans¬
portation Problem 652
Problems 656
Appendix A: Computational Experience 664
A.I The General Fixed Charge Problem 664
A.2 The Fixed Charge (or Capacitated Plant Location) Prob¬
lem 666
A.3 The (Uncapacitated) Plant Location Problem 666
A.4 Summary 667
Contents ;cv
15 The Traveling Salesman Problem 669
15.1 Introduction 669
15.2 Mathematical Formulation 670
15.3 Algorithms 673
15.3.1 Bounding Rules 675
15.3.2 Branching Rules 677
15.4 Approximate Algorithms 682
15.4.1 Tour Construction Procedures 682
15.4.2 Tour Improvement Algorithms 689
Problems 692
Appendix A: The Assignment Algorithm 695
Appendix B: Some Counterexamples to Heuristic Algo¬
rithms for the Traveling Salesman Problem 700
B.I Cheapest Insertion Algorithm (nonsymmetric case) . . . 700
B.2 Cheapest Insertion Algorithm (symmetric case) 701
B.3 Convex Hull Heuristic 702
B.4 Two Edge Interchange (2 opt) Algorithm 703
Bibliography 705
Author Index 745
Subject Index 751
|
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| indexdate | 2024-07-09T16:13:18Z |
| institution | BVB |
| isbn | 0444012311 |
| language | English |
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| physical | XIX, 755 S. graph. Darst. |
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| spelling | Salkin, Harvey Marshall Verfasser aut Foundations of integer programming Harvey M. Salkin ; Kamlesh Mathur New York u.a. North-Holland 1989 XIX, 755 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. [705] - 743 Discrete programmering gtt Programmation en nombres entiers Integer programming Ganzzahlige Optimierung (DE-588)4155950-2 gnd rswk-swf Ganzzahlige Optimierung (DE-588)4155950-2 s DE-604 Marthur, Kamlesh Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002763871&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Salkin, Harvey Marshall Marthur, Kamlesh Foundations of integer programming Discrete programmering gtt Programmation en nombres entiers Integer programming Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| subject_GND | (DE-588)4155950-2 |
| title | Foundations of integer programming |
| title_auth | Foundations of integer programming |
| title_exact_search | Foundations of integer programming |
| title_full | Foundations of integer programming Harvey M. Salkin ; Kamlesh Mathur |
| title_fullStr | Foundations of integer programming Harvey M. Salkin ; Kamlesh Mathur |
| title_full_unstemmed | Foundations of integer programming Harvey M. Salkin ; Kamlesh Mathur |
| title_short | Foundations of integer programming |
| title_sort | foundations of integer programming |
| topic | Discrete programmering gtt Programmation en nombres entiers Integer programming Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| topic_facet | Discrete programmering Programmation en nombres entiers Integer programming Ganzzahlige Optimierung |
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