Marl*«fmnm Berhy

Pmfttotfwvb *« imprint «if

#w%*< 1<I><' «* <

1j »»r* « »«l> *rn

>1 <i » >*f *

*(*jt»t

M< u>«« n «' »»•*§Wong> ij»«ta» Cif * «

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

Supporting resources

Visit www.pearsoned.co.uk/derhy to find valuable online resources

For instructors

• Instructor's manual

• PowerPoint slides

For more information please contact your local Pearson Education sales representative or

visit www.pearsoned.co.uk/derhy