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Preface xiii
Acknowledgements xv
1 Introduction 1
1.1 Modelling using Linear Programming 1
1.2 Solving linear programmes 3
1.2.1 The graphical solution and the importanceof visual displays 3
1.2.2 The Simplex Method and its main variants 5
1.2.3 Computer software packages 6
1.2.4 Complementary information and SensitivityAnalysis 6
1.3 Linear Programming: the approach par excellence for
understanding modelling 7
1.3.1 The variants of Linear Programming 7
1.3.2 LP's related topics 9
1.4 The approach of the book 11
Part I Linear Programming and Sensitivity Analysis 13
The Geometric Approach 15
2.1 The founding concepts of Linear Programming 16
2.2 The Maximization Form 21
Application # 1: An advertising campaign [Aurel 2D] 21
2.2.1 The mathematical formulation 22
2.2.2 The graphical solution: the solution space and
the optimal solution 22
2,2.3 Interpreting the slack and surplus variables:
used and unused resources 29
2.2,4 Shadow prices: value of an extra unit of
a given resource 29
Application # 2: Computer games [2D] 30
21.3 The Minimization Form 32
Application # 3: A portfolio selection [2D] 32
Chapter 2 Exercises and applications 34
viii CONTENTS
3 The Simplex Method
3.1 The Maximization Form
3.1.1 The Standard Form
3.1.2 The simplex algorithm (using tableaux)
3 1,3 Shadow prices and reduced costs
3.1,4 The algorithm (using Matrix Algebra)3 1,5 Introduction of artificial variables
3.1.6 The remarkable features of the simplex algorithm
3.2 The Minimization Form
3.3 The Revised Simplex Method
3.3.1 The revised simplex algorithm3.3.2 Using artificial variables: adjusting the algorithm
Chapter 3 Exercises and applications
4 Understanding Special Cases and Mixed Function Problems 77
4.1 Identifying special cases: graphical and simplex approaches 77
4.1.1 Alternative optimal solutions 77
4.1.2 Unboundedness ^
4.1.3 Infeasibility versus point solution 81
4.1.4 Degenerate solutions 83
4.1.5 Special types of constraints 84
4.2 The mixed function problem 87
Chapter 4 Exercises and applications 90
5 Duality 95
5.1 Theorems of duality and relationships 95
5.1.1 The theorems of duality 95
5.1.2 Primal/dual relationships 98
5.1.3 Formulating duals using the general primal formats 1035.1.4 Primal/dual interrelationships: the Complementary
Slackness Theorem 106
5.2 The Dual Simplex Method 107
5.3 Particular cases 1095.3.1 Unrestricted-in-sign variables (free variables) 109
5.3.2 Revisiting the special cases: study of thebehaviour of their duals 115
Chapter 5 Exercises and applications 118
6 Sensitivity Analysis6A A visual approach to Sensitivity Analysis
6.1 The Maximization Form
6.1.1 The range of optimality: separate andsimultaneous changes
6.1.2 The range of feasibility6.1.3 Notes on a few specific cases
6.2 The Minimization Form
6B Sensitivity Analysis under the Simplex Method,using Matrix Algebra6.3 The Maximization Form
6.3.1 Ranges of optimality: simple changes
39
39
40
42
46
47
50
57
58
62
64
69
73
123124
124
124
127
130
137
137
137
CONTENTS ix
6.3.2 Optimality ranges: simultaneous changes and
restoring optimality 139
6.3.3 Simple and simultaneous ranges of feasibility 1416.3.4 Restoring feasibility 144
6.3.5 The 100% Rule: optimality and feasibility tests 1466.4 Introduction of a new variable or of a new constraint 149
6.5 Note on the Minimization Form [The Portfolio
3D modified] 151
6.6 Embedded modifications 152
6C Revisiting mixed function problems 156
6.7 Discussion on optimality ranges: simplex and
graphical approaches 156
Chapter 6 Exercises and applications 160
7 Understanding Computer Outputs and LP Applications 171
7A Highlighting Outputs 171
7.1 Using software packages to solve LP problems 172
7.1.1 Lindo: How to take advantage of its
features and facilities 172
7.1.2 The Management Scientist 174
7.1.3 Solving LPs with Excel 174
7.2 Study of outputs with respect to Chapters 3 and 6:
the Simplex Method and Sensitivity Analysis 177
7.3 Commented outputs with respect to Chapters 4 and 5:
special cases and duality 183
7B The Various Fields of Application 190
7.4 Production and make-or-buy 191
7.5 Purchase plans 196
7.6 Finance 198
7.7 Advertising 203
7.8 Staff scheduling 205
7.9 Blending and nutrition 209
7.10 Efficiency problems 216
Chapter 7 Applications 219
Part II Variants and Related Topics 227
8 The Variants of Linear Programmes 229
8.1 Integer Programming 230
8.1.1 Pure and Mixed Integer Programming:the graphical insight 230
8.1.2 Binary Integer Linear Programming 233
8.1.3 Formulating logical constraints 240
8.2 Game Theory 243
8.2.1 Strictly determined games 244
8.2.2 Non-strictly determined games and
solution approach by LP 248
X CONTENTS
8.3 The Transportation Problem
8.3.1 The balanced problem: solution approach
through simplex multipliers8.3.2 The unbalanced problem8.3.3 Comment on the LP formulation of unbalanced
problems and on Sensitivity Analysis8.3.4 Special Transportation Problems: LP formulation
and solution approach8.3.5 Maximization problems
8.4 The Assignment Model
8.4.1 The solution approach: Konig's Algorithm8.4.2 The maximization problem (example also
displaying an unbalance)8.4.3 Note on the LP formulation and on
Sensitivity Analysis
Chapter 8 Exercises and applications
9 Related Topics: Graphs and Networks 301
9.1 The main building concepts of Graph Theory 301
9.1.1 Definitions and examples 302
9.1.2 From the graph to the matrix: adjacency and
incidence matrices 305
9.1.3 Directed graphs 306
9.2 Flow networks 308
9.2.1 The LP formulation and solution 308
9.2.2 Solving the capacitated network graphically 314
9.2.3 The Max-Flow Min-Cut Theorem (Ford-Fulkerson) 319
9.2.4 Transshipments 321
9.3 The shortest path 3259.3.1 The LP formulation and solution 326
9.3.2 The graphical solution (Dijkstra's Algorithm) 3299.3.3 Floyd's Algorithm 331
9.4 The Minimal Spanning Tree 3389.4.1 The graphical solution 3399.4.2 The limits of the LP formulation 344
Chapter 9 Exercises and applications 347
25b
256265
268
269
272
274
274
279
283
288
Part III Mathematical Corner and Note on Nonlinear
Programming 355
10 Mathematical Corner 35710.1 Coping with infeasibility 357
10.1.1 Graphical insights 35310.1.2 Discussion on changes ^61
10.2 Flow networks30g
10.2.1 Highlighting the cut on outputs 36610.2.2 The cut revisited by duality 357
CONTENTS Xi
10.3 The Shortest Route Algorithm: discussion on
Sensitivity Analysis 37010.4 The Minimal Spanning Tree 374
10.4.1 Minimal Spanning Trees and hierarchical
clustering schemes 37410.4.2 The LP formulation of Minimal Spanning
Trees: a heuristic approach 379Note on Sensitivity Analysis 382
Chapter 10 Exercises and applications 383
11 Note on Nonlinear Programming 38511.1 Quadratic Programming: definition 38611.2 Illustrations and graphical displays: solution method
using Lagrange multipliers 38711.3 Formulating the quadratic programme 39211.4 Comment on shadow prices and 'RHS ranges' 396
Chapter 11 Exercises 400
Basic Review Chapter 401
R.l Basic Matrix Algebra 401R.l.l Vectors: definitions, addition, subtraction and
multiplication 401
R.l.2 Matrices: types, addition and subtraction,
multiplication and inverses 402
R.1.3 Finding the inverse of m x m matrices usingthe Gauss-Jordan method 406
- Using inverses to solve systems of
simultaneous equations 408
R.2 Derivatives and local extrema 408
R.2.1 Brief review of derivatives in the
single-variable case 408Comments on limits, continuity and differentiability 410
R.2.2 Partial derivatives 411
Answers to Selected Problems and Applications 417
Study Applications 435
Index 436
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