Linear and integer optimization: theory and practice
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2015
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| Edition: | 3. ed. |
| Series: | Advances in applied mathematics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIX, 656 S. graph. Darst. |
| ISBN: | 1498710166 9781498710169 |
Staff View
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Record in the Search Index
| _version_ | 1804174931813990400 |
|---|---|
| adam_text | List of Figures xvii
List of Tables xxiii
Preface xxv
i Basic concepts of linear optimization I
i.i The company Dovetail.............................................................. 2
1.1.1 Formulating Model Dovetail ............................................... 2
1.1.2 The graphical solution method ............................................ 3
1*2 Definition of an LO-model ........................................................ 5
1.2.1 The standard form of an LO-model.......................................... 5
1.2.2 Slack variables and binding constraints .................................. 9
1.2.3 Types of optimal solutions and feasible regions ......................... 10
1.3 Alternatives of the standard LO-model ....................................... 12
14 Solving LO-models using a computer package....................................... 14
1.5 Linearizing nonlinear functions.................................................. 16
1.5.1 Ratio constraints........................................................ 16
1.5.2 Objective functions with absolute value terms............................ 17
1.5.3 Convex piecewise linear functions........................................ 19
1.6 Examples of linear optimization models........................................... 21
1.6.1 The diet problem......................................................... 22
1.6.2 Estimation by regression................................................. 24
1.6.3 Team formation in sports................................................. 26
1.64 Data envelopment analysis................................................ 29
1.6.5 Portfolio selection; profit versus risk.................................. 34
1.7 Building and implementing mathematical models.................................... 39
1.8 Exercises........................................................................ 42
I Linear optimization theory: basic techniques
2 Geometry and algebra of feasible regions 51
2.x The geometry of feasible regions................................. 51
2.1.1 Hyperplanes and halfspaces............................................... 52
2.1.2 Vertices and extreme directions of the feasible region..................... 55
2.1.3 Faces of the feasible region............................................... 58
2.1.4 The optima] vertex theorem................................................. 60
2.1.5 Polyhedra and polytopes.................................................... 62
2.2 Algebra of feasible regions; feasible basic solutions............................. 63
2.2.1 Notation; row and column indices • • ■ .................................... 63
2.2.2 Feasible basic solutions................................................... 64
2.2.3 Relationship between feasible basic solutions and vertices................. 68
2.2.4 Degeneracy and feasible basic solutions ................................... 75
2.2.5 Adjacency.................................................................. 77
2.3 Exercises......................................................................... 79
3 Dantzig’s simplex algorithm 83
3.1 From vertex to vertex to an optimal solution...................................... 83
3.2 LO-model reformulation............................................................ 89
3.3 The simplex algorithm............................................................. 91
3.3.1 Initialization; finding an initial feasible basic solution................. 92
3.3.2 The iteration step; exchanging variables................................... 92
3.3.3 Formulation of the simplex algorithm....................................... 95
3.4 Simplex tableaus.................................................................. 96
3.4.1 The initial simplex tableau................................................ 99
3.4.2 Pivoting using simplex tableaus............................................ 99
3.4.3 Why the identity matrix of a simplex tableau is row-permuted...............104
3.5 Discussion of the simplex algorithm...............................................106
3.5.1 Simplex adjacency graphs ................................................. 107
3.5.2 Optimality test and degeneracy .............................................no
3.5.3 Unboundedness .............................................................112
3.5.4 Pivot rules................................................................114
3.5.5 Stalling and cycling.......................................................115
3.5.6 Anti-cycling procedures: the perturbation method...........................118
3.5.7 Anti-cycling procedures: Bland’s rule .....................................120
3.5.8 Correctness of the simplex algorithm.......................................121
3.6 Initialization....................................................................122
3.6.1 The big-M procedure........................................................123
3.6.2 The two-phase procedure....................................................127
3.7 Uniqueness and multiple optimal solutions ......................................131
3.8 Models with equality constraints..................................................136
3.9 The revised simplex algorithm.....................................................139
3.9.1 Formulating the algorithm .................................................139
3.9.2 The product form of the inverse ...........................................141
3.9.3 Applying the revised simplex algorithm.....................................142
3.10 Exercises
144
4 Duality, feasibility, and optimality 151
4.1 The companies Dovetail and Salmonnose............................................152
4.1.1 Formulating the dual model...............................................152
4.1.2 Economic interpretation..................................................153
4.2 Duality and optimality...........................................................154
4.2.1 Dualizing the standard LO-model..........................................154
4.2.2 Dualizing nonstandard LO-models..........................................155
4.2.3 Optimality and optimal dual solutions....................................159
4.3 Complementary slackness relations................................................162
4.3.1 Complementary dual variables.............................................162
4.3.2 Complementary slackness..................................................164
4.3.3 Determining the optimality of a given solution...........................167
4.3.4 Strong complementary slackness...........................................168
4.4 Infeasibility and unboundedness; Farkas’ lemma...................................171
4.5 Primal and dual feasible basic solutions.........................................175
4.6 Duality and the simplex algorithm................................................179
4.6.1 Dual solution corresponding to the simplex tableau.......................180
4.6.2 The simplex algorithm from the dual perspective..........................182
4.7 The dual simplex algorithm ......................................................185
4.7.1 Formulating the dual simplex algorithm...................................187
4.7.2 Reoptimizing an LO-model after adding a new constraint ..................188
4.8 Exercises........................................................................190
5 Sensitivity analysis 195
5.1 Sensitivity of model parameters..................................................195
5.2 Perturbing objective coefficients................................................197
5.2.1 Perturbing the objective coefficient of a basic variable.................197
5.2.2 Perturbing the objective coefficient of a nonbasic variable..............202
5.2.3 Determining tolerance intervals from an optimal simplex tableau..........203
5.3 Perturbing right hand side values (nondegenerate case) ..........................205
5.3.1 Perturbation of nonbinding constraints...................................206
5.3.2 Perturbation of technology constraints...................................207
5.3.3 Perturbation of nonnegativity constraints ...............................213
5.4 Piecewise linearity of perturbation functions....................................216
5.5 Perturbation of the technology matrix............................................219
5.6 Sensitivity analysis for the degenerate case.....................................221
5.6.1 Duality between multiple and degenerate optimal solutions................222
5.6.2 Left and right shadow prices.............................................225
5.6.3 Degeneracy, binding constraints, and redundancy..........................231
5-7 Shadow prices and redundancy of equality constraints............................ 234
5-8 Exercises
237
6 Large-scale linear optimization 247
6.1 The interior path...............................................................248
6.1.1 The Karush-Kuhn-Tucker conditions for LO-models.........................248
6.1.2 The interior path and the logarithmic barrier function .................249
6.1.3 Monotonicity and duality................................................254
6.2 Formulation of the interior path algorithm • ...................................255
6.2.1 The interior path as a guiding hand.....................................255
6.2.2 Projections on null space and row space.................................256
6.2.3 Dikin’s affine scaling procedure...............*......................258
6.2.4 Determining the search direction .......................................259
6.3 Convergence to the interior path; maintaining feasibility ......................263
6.3.1 The convergence measure..............*................................ 263
6.3.2 Maintaining feasibility; the interior path algorithm ...................265
6.4 Termination and initialization .............*.................................267
6.4.1 Termination and the duality gap.........................................267
6.4.2 Initialization..........................................................268
6.5 Exercises.......................................................................273
7 Integer linear optimization 279
7.1 Introduction....................................................................279
7.1.1 The company Dairy Corp..................................................280
7.1.2 ILO-models...........................*..................................281
7.1.3 MILO-models.............................................................281
7.1.4 A round-off procedure • ■ ...........*..................................282
7.2 The branch-and-bound algorithm..................................................283
7.2.1 The branch-and-bound tree...............................................283
7.2.2 The general form of the branch-and-bound algorithm .....................286
7.2.3 The knapsack problem....................................................289
7.2.4 A machine scheduling problem ...........................................293
7.2.5 The traveling salesman problem; the quick and dirty method............* 298
7.2.6 The branch-and-bound algorithm as a heuristic...........................299
7.3 Linearizing logical forms with binary variables.................................301
7.3.1 The binary variables theorem ...........................................301
7.3.2 Introduction to the theory of logical variables.........................303
7.3.3 Logical forms represented by binary variables...........................305
7.3.4 A decentralization problem............................................ 310
7.4 Gomory’s cutting-plane algorithm................................................312
7.4.1 The cutting-plane algorithm...................................... • • 313
7.4.2 The cutting-plane algorithm for MILO-models.............................317
7.5 Exercises.......................................................................319
8 Linear network models 333
8.1 LO-models with integer solutions; total unimodularity...........................333
8,1.1 Total unimodularity and integer vertices............................... 334
8.1.2 Total unimoduiarity and ILO-models......................................336
8.1.3 Total unimoduiarity and incidence matrices..............................338
8.2 ILO-models with totally unimodular matrices....................................345
8.2.1 The transportation problem..............................................345
8.2.2 The balanced transportation problem ................................... 347
8.2.3 The assignment problem..................................................349
8.2.4 The minimum cost flow problem...........................................350
8.2.5 The maximum flow problem; the max-flow min-cut theorem .................353
8.2.6 Project scheduling; critical paths in networks..........................361
8.3 The network simplex algorithm..................................................367
8.3.1 The transshipment problem...............................................367
8.3.2 Network basis matrices and feasible tree solutions......................369
8.3.3 Node potential vector; reduced costs; test for optimality...............372
8.3.4 Determining an improved feasible tree solution..........................374
8.3.5 The network simplex algorithm...........................................375
8.3.6 Relationship between tree solutions and feasible basic solutions........378
8.3.7 Initialization; the network big-M and two-phase procedures .............380
8.3.8 Termination, degeneracy, and cycling....................................381
8.4 Exercises......................................................................385
9 Computational complexity 397
9.1 Introduction to computational complexity.......................................397
9.2 Computational aspects of Dantzig’s simplex algorithm...........................400
9.3 The interior path algorithm has polynomial running time........................403
9.4 Computational aspects of the branch-and-bound algorithm........................404
9.5 Exercises............................................*.......................407
II Linear optimization practice: advanced techniques
10 Designing a reservoir for irrigation 413
10.1 The parameters and the input data..............................................413
10.2 Maximizing the irrigation area.................................................417
10.3 Changing the input parameters of the model.....................................419
xo.4 GMPL model code................................................................420
ro.5 Exercises......................................................................421
11 Classifying documents by language 423
11.1 Machine learning...............................................................423
11-2 Classifying documents using separating hyperplanes.............................425
11-3 LO-model for finding separating hyperplanes....................................428
11.4 Validation of a classifier.....................................................431
11.5 Robustness of separating hyperplanes; separation width..........................432
11.6 Models that maximize the separation width.......................................434
11.6.1 Minimizing the 1-norm of the weight vector..............................435
11.6.2 Minimizing the oo-norm of the weight vector.............................435
11.6.3 Comparing the two models.............................................. 436
11.7 GMPL model code.............-..................................................438
11.8 Exercises.....................................*................................440
12 Production planning: a single product case 441
12.1 Model description...............................................................441
12.2 Regular working hours...........................................................444
12.3 Overtime........................................................................447
12.4 Allowing overtime and idle time.................................................449
12.5 Sensitivity analysis ...........................................................451
12.6 GMPL model code.................................................................454
12.6.1 Model (PP2).............................................................454
12.6.2 Model (PP3).............................................................454
12.7 Exercises.......................................................................455
13 Production of coffee machines 459
13.1 Problem setting.................................................................459
13.2 An LO-model that minimizes backlogs.............................................460
13.3 Old and recent backlogs.........................................................462
13.4 Full week productions...........................................................466
13.5 Sensitivity analysis ...........................................................467
13.5.1 Changing the number of conveyor belts...................................467
13.5.2 Perturbing backlog-weight coefficients..................................468
13.5.3 Changing the weights of the ‘old’ and the ‘new’ backlogs................469
13.6 GMPL model code.................................................................469
13.7 Exercises.......................................................................471
14 Conflicting objectives: producing versus importing 473
14.1 Problem description and input data..............................................473
14.2 Modeling two conflicting objectives; Pareto optimal points......................474
14.3 Goal optimization for conflicting objectives....................................477
14.4 Soft and hard constraints.......................................................481
14.5 Sensitivity analysis ...........................................................483
14.5.1 Changing the penalties..................................................483
14.5.2 Changing the goals .....................................................483
14.6 Alternative solution techniques.................................................484
14.6.1 Lexicographic goal optimization...........................................485
14.6.2 Fuzzy linear optimization ................................................488
14.7 A comparison of the solutions.....................................................489
14.8 GMPL model code...................................................................490
14.9 Exercises.........................................................................491
15 Coalition formation and profit distribution 493
15.1 The farmers cooperation problem...................................................493
15.2 Game theory; linear production games..............................................495
15.3 How to distribute the total profit among the farmers?.............................499
15.4 Profit distribution for arbitrary numbers of farmers .............................502
15.4.1 The profit distribution...................................................503
15.4.2 Deriving conditions such that a given point is an Owen point............505
15.5 Sensitivity analysis .............................................................506
15.6 Exercises.........................................................................509
16 Minimizing trimloss when cutting cardboard 513
16.1 Formulating the problem...........................................................513
16.2 Gilmore-Gomory’s solution algorithm...............................................516
16.3 Calculating an optimal solution...................................................519
16.4 Exercises.........................................................................521
17 OfF-shore helicopter routing 523
17.1 Problem description...............................................................523
17.2 Vehicle routing problems..........................................................524
17.3 Problem formulation...............................................................526
17.4 ILO formulation...................................................................529
17.5 Column generation.................................................................530
17.5.1 Traveling salesman and knapsack problems................................533
17.5.2 Generating‘clever’subsets of platforms....................................534
17.6 Dual values as price indicators for crew exchanges................................536
17.7 A round-off procedure for determining an integer solution.........................538
17.8 Computational experiments.........................................................540
17.9 Sensitivity analysis .............................................................542
17.10 Exercises........................................................................544
18 The catering service problem 547
18.1 Formulation of the problem........................................................547
18.2 The transshipment problem formulation.............................................549
18.3 Applying the network simplex algorithm.........................................552
184 Sensitivity analysis ..........................................................555
184.1 Changing the inventory of the napkin supplier.........................556
184.2 Changing the demand.....................................................556
184.3 Changing the price of napkins and the laundering costs................557
18.5 GMPL model code................................................................558
i
18.6 Exercises.................................................................... 560
Appendices
A Mathematical proofs 563
A. I Direct proof...................................................................564
A.2 Proof by contradiction.........................................................565
A. 3 Mathematical induction ........................................................565
B Linear algebra 567
B. i Vectors....................................................................... 567
13.2 Matrices ......................................................................570
B.3 Matrix partitions..............................................................571
B.4 Elementary row/column operations; Gaussian elimination.........................572
B.5 Solving sets of linear equalities..............................................576
B.6 The inverse of a matrix........................................................578
B.7 The determinant of a matrix....................................................579
B. 8 Linear and affine spaces...................................................... 582
C Graph theory 585
C. i Undirected graphs.............................................*..............585
C.2 Connectivity and subgraphs.................................................. ‘586
C.3 Trees and spanning trees.......................................................588
C.4 Eulerian tours, Hamiltonian cycles, and Hamiltonian paths......................589
C.5 Matchings and coverings........................................................591
C.6 Directed graphs................................................................592
C,7 Adjacency matrices and incidence matrices .......................................594
C. 8 Network optimization models ...................................................595
D Convexity 597
D. i Sets, continuous functions, Weierstrass’ theorem...............................597
D.2 Convexity, polyhedra, polytopes, and cones.....................................600
D.3 Faces, extreme points, and adjacency........................................602
D. 4 Convex and concave functions................................................608
E Nonlinear optimization 611
E. i Basics.....................................................................611
E.2 Nonlinear optimization; local and global minimizers.........................614
E.3 Equality constraints; Lagrange multiplier method ..........................615
E. 4 Models with equality and inequality constraints; Karush-Kuhn-Tucker conditions • 620
£.5 Karush-Kuhn-Tucker conditions for linear optimization.......................624
F Writing LO-models in GNU MathProg (GMPL) 627
F. 1 Model Dovetail..............................................................627
F.2 Separating model and data...................................................629
F.3 Built-in operators and functions............................................631
£4 Generating all subsets of a set.............................................633
List of Symbols 635
Bibliography 639
Author index 645
Subject index
647
|
| any_adam_object | 1 |
| author | Sierksma, Gerard 1945- Zwols, Yori |
| author_GND | (DE-588)1028551304 |
| author_facet | Sierksma, Gerard 1945- Zwols, Yori |
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| bvnumber | BV042724078 |
| classification_rvk | QH 421 SK 870 SK 890 |
| ctrlnum | (OCoLC)915140391 (DE-599)GBV825605172 |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 3. ed. |
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| id | DE-604.BV042724078 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:08:14Z |
| institution | BVB |
| isbn | 1498710166 9781498710169 |
| language | English |
| lccn | 2015288125 |
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| oclc_num | 915140391 |
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| physical | XXIX, 656 S. graph. Darst. |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Advances in applied mathematics |
| spelling | Sierksma, Gerard 1945- Verfasser (DE-588)1028551304 aut Linear and integer optimization theory and practice Gerard Sierksma ; Yori Zwols 3. ed. Boca Raton [u.a.] CRC Press 2015 XXIX, 656 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advances in applied mathematics Lineare Optimierung (DE-588)4035816-1 gnd rswk-swf Ganzzahlige Optimierung (DE-588)4155950-2 gnd rswk-swf Lineare Optimierung (DE-588)4035816-1 s Ganzzahlige Optimierung (DE-588)4155950-2 s DE-604 Zwols, Yori Verfasser aut Digitalisierung UB Augsburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028155226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sierksma, Gerard 1945- Zwols, Yori Linear and integer optimization theory and practice Lineare Optimierung (DE-588)4035816-1 gnd Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| subject_GND | (DE-588)4035816-1 (DE-588)4155950-2 |
| title | Linear and integer optimization theory and practice |
| title_auth | Linear and integer optimization theory and practice |
| title_exact_search | Linear and integer optimization theory and practice |
| title_full | Linear and integer optimization theory and practice Gerard Sierksma ; Yori Zwols |
| title_fullStr | Linear and integer optimization theory and practice Gerard Sierksma ; Yori Zwols |
| title_full_unstemmed | Linear and integer optimization theory and practice Gerard Sierksma ; Yori Zwols |
| title_short | Linear and integer optimization |
| title_sort | linear and integer optimization theory and practice |
| title_sub | theory and practice |
| topic | Lineare Optimierung (DE-588)4035816-1 gnd Ganzzahlige Optimierung (DE-588)4155950-2 gnd |
| topic_facet | Lineare Optimierung Ganzzahlige Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028155226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT sierksmagerard linearandintegeroptimizationtheoryandpractice AT zwolsyori linearandintegeroptimizationtheoryandpractice |