Linear and combinatorial programming:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Wiley
1976
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 567 S. zahlr. graph. Darst. |
| ISBN: | 0471573701 |
Staff View
MARC
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| 100 | 1 | |a Murty, Katta G. |d 1936- |e Verfasser |0 (DE-588)1060328593 |4 aut | |
| 245 | 1 | 0 | |a Linear and combinatorial programming |c Katta G. Murty |
| 264 | 1 | |a New York |b Wiley |c 1976 | |
| 300 | |a XXIII, 567 S. |b zahlr. graph. Darst. | ||
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Record in the Search Index
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|---|---|
| adam_text | contents
NOTATION
1 FORMULATION OF LINEAR PROGRAMS 1
1.1 Formulation of linear programs 1
1.2 Solving linear programs in two variables 19
2 THE SIMPLEX METHOD 35
2.1 Introduction 35
2.2 Transforming the problem into the standard form 36
2.3 Canonical tableau 40
2.4 Phase I with a full artificial basis 44
2.5 The simplex algorithm 46
2.6 Outline of the simplex method for a general linear program 52
2.7 The Big M Method 65
3 THE GEOMETRY OF THE SIMPLEX METHOD 75
3.1 Euclidean vector spaces: Definitions and geometrical concepts 75
3.2 Matrices 84
3.3 Linear independence of a set of vectors and simultaneous linear
equations 86
3.4 Basic feasible solutions and extreme points of convex polyhedra 97
3.5 The usefulness of basic feasible solutions in linear programming 102
3.6 The edge path traced by the simplex algorithm 109
3.7 Homogeneous solutions, unboundedness, resolution theorems 114
3.8 The main geometrical argument in the simplex algorithm 124
3.9 The dimension of a convex polyhedron. 124
3.10 Alternate optimum feasible solutions and faces of convex polyhedra 125
3.11 Pivot matrices 128
3.12 Canonical tableaus in matrix notation 130
3.13 Why is an algorithm required to solve linear programs? 131
3.14 Phase I of the simplex method using one artificial variable 136
xiv / Contents
4 DUALITY IN LINEAR PROGRAMMING 147
4.1 Introduction 147
4.2 An example of a dual problem 147
4.3 Dual variables are the prices of the items 149
4.4 How to write the dual of a general linear program 150
4.5 Duality theory for linear programming 155
4.6 Other interpretations and applications of duality 169
5 REVISED SIMPLEX METHOD 183
5.1 Introduction 183
5.2 Revised simplex algorithm with the explicit form of inverse when
Phase II can begin directly 183
5.3 Revised simplex method using Phase I and Phase II 187
5.4 Revised simplex method using the product form of the inverse 192
5.5 Advantages of the product form implementation over the explicit
form implementation 197
6 THE DUAL SIMPLEX METHOD 201
6.1 Introduction 201
6.2 Dual simplex algorithm when a dual feasible basis is known initially 201
6.3 Disadvantages and advantages of the dual simplex algorithm 208
6.4 Dual simplex method when a dual feasible basis is not known
at the start 208
6.5 Comparison of the primal and the dual simplex methods 215
7 PARAMETRIC LINEAR PROGRAMS 219
7.1 Introduction 219
7.2 Analysis of the parametric cost problem given an optimum basis
for some X 221
7.3 To find an optimum basis for the parametric cost problem 224
7.4 Summary of the results on the parametric cost problem 224
7.5 Analysis of the parametric right hand side problem given an optimum
basis for some X 225
7.6 To find an initial optimum basis for the parametric right hand side
problem, given a feasible basis for some X 226
7.7 To find a feasible basis for the parametric right hand side problem 226
7.8 Summary of the results on the parametric right hand side problem 227
7.9 Convex and concave function on the real line 227
8 SENSITIVITY ANALYSIS 243
8.1 Introduction 243
8.2 Introducing a new activity 244
8.3 Introducing an additional inequality constraint 245
8.4 Introducing an additional equality constraint 248
8.5 Cost ranging of a nonbasic cost coefficient 249 |
8.6 Cost ranging of a basic cost coefficient 250 1
Contents / xv
8.7 Right hand side ranging 251
8.8 Changes in the input output coefficients in a nonbasic column vector 252
8.9 Change in a basic input output coefficient 253
8.10 Practical applications of sensitivity analysis 253
9 DEGENERACY IN LINEAR PROGRAMMING 259
9.1 Geometry of degeneracy 259
9.2 Resolution of cycling under degeneracy 263
9.3 Summary of the simplex algorithm using the lexico minimum
ratio rule 267
9.4 Do computer codes use the lexico minimum ratio rule? 268
10 BOUNDED VARIABLE LINEAR PROGRAMS 271
10.1 Introduction 271
10.2 Bounded variable problems 271
10.3 Simplex method using working bases 274
11 PRIMAL ALGORITHM FOR THE TRANSPORTATION
PROBLEM 289
11.1 The balanced transportation problem: Theory 289
11.2 Revised primal simplex algorithm for the balanced transportation
problem 299
11.3 The unimodualarity property 307
11.4 Labeling methods in the primal transportation algorithm 309
11.5 Sensitivity analysis in the transportation problem 321
11.6 Transportation problems with inequality constraints 325
11.7 Bounded variable transportation problems 329
1 2 NETWORK ALGORITHMS 341
12.1 Introduction: Networks 341
12.2 Notation 342
12.3 Single commodity maximum flow problems 348
12.4 The primal dual approaches for the assignment and
transportation problems 360
12.5 Single commodity minimum cost flow problems 373
12.6 Shortest route problems 382
12.7 Minimum spanning tree problems 390
12.8 Multicommodity flow problems 391
12.9 Other network algorithms 392
13 FORMULATION OF INTEGER AND COMBINATORIAL
PROGRAMMING PROBLEMS 397
13.1 Introduction 397
13.2 Formulation examples 398
xvi / Contents
14 CUTTING PLANE METHODS FOR
INTEGER PROGRAMMING 419
14.1 Introduction 419
14.2 Fractional cutting plane method for pure integer programs 426
14.3 Cutting plane methods for mixed integer programs 432
14.4 Other cutting plane methods 433
14.5 How efficient are cutting plane methods? 434
15 THE BRANCH AND BOUND APPROACH 437
15.1 Introduction 437
15.2 The lower bounding strategy 438
15.3 The branching strategy 440
15.4 The search strategy 441
15.5 Requirements for efficiency 445
15.6 The 0 1 knapsack problem 446
15.7 The traveling salesman problem 449
15.8 The general mixed integer linear program 456
15.9 The set representation problem 458
15.10 0 1 problems 467
15.11 Advantages and limitations 469
15.12 Ranking methods 471
15.13 Notes on the branch and bound approach 478
16 COMPLEMENTARITY PROBLEMS 481
16.1 Introduction 481
16.2 Geometric interpretations 482
16.3 Applications in linear programming 485
16.4 Quadratic programming 486
16.5 Two person games 493
16.6 Complementary pivot algorithm 495
16.7 Conditions under which the algorithm works 503
16.8 Remarks 508
17 NUMERICALLY STABLE FORMS OF
THE SIMPLEX METHOD 521
17.1 Review 521
17.2 The LU decomposition 522
17.3 The Cholesky factorization 528
17.4 Reinversions 533
17.5 Use in computer codes 534
18 COMPUTATIONAL EFFICIENCY 535
18.1 How efficient is the simplex algorithm? 535
I
Contents / xvii
APPENDIX 1
THE DECOMPOSITION PRINCIPLE OF LINEAR
PROGRAMMING 541
APPENDIX 2
BENDER S DECOMPOSITION FOR MIXED INTEGER
PROGRAMS 553
APPENDIX 3
CRAMER S RULE 557
SELECTED REFERENCES IN LINEAR PROGRAMMING 559
INDEX 561
|
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
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| discipline | Mathematik Wirtschaftswissenschaften |
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| id | DE-604.BV002282016 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:43:19Z |
| institution | BVB |
| isbn | 0471573701 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001499709 |
| oclc_num | 2089692 |
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| physical | XXIII, 567 S. zahlr. graph. Darst. |
| publishDate | 1976 |
| publishDateSearch | 1976 |
| publishDateSort | 1976 |
| publisher | Wiley |
| record_format | marc |
| spelling | Murty, Katta G. 1936- Verfasser (DE-588)1060328593 aut Linear and combinatorial programming Katta G. Murty New York Wiley 1976 XXIII, 567 S. zahlr. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Programmation linéaire Linear programming Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001499709&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Murty, Katta G. 1936- Linear and combinatorial programming Programmation linéaire Linear programming Lineare Optimierung (DE-588)4035816-1 gnd |
| subject_GND | (DE-588)4035816-1 |
| title | Linear and combinatorial programming |
| title_auth | Linear and combinatorial programming |
| title_exact_search | Linear and combinatorial programming |
| title_full | Linear and combinatorial programming Katta G. Murty |
| title_fullStr | Linear and combinatorial programming Katta G. Murty |
| title_full_unstemmed | Linear and combinatorial programming Katta G. Murty |
| title_short | Linear and combinatorial programming |
| title_sort | linear and combinatorial programming |
| topic | Programmation linéaire Linear programming Lineare Optimierung (DE-588)4035816-1 gnd |
| topic_facet | Programmation linéaire Linear programming Lineare Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001499709&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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