# Building and Solving Mathematical Programming Models in Engineering and Science

# Building and Solving Mathematical Programming Models in Engineering and Science

ISBN: 978-0-471-15043-5

Nov 2001

568 pages

## Description

Fundamental concepts of mathematical modelingModeling is one of the most effective, commonly used tools in engineering and the applied sciences. In this book, the authors deal with mathematical programming models both linear and nonlinear and across a wide range of practical applications.

Whereas other books concentrate on standard methods of analysis, the authors focus on the power of modeling methods for solving practical problems-clearly showing the connection between physical and mathematical realities-while also describing and exploring the main concepts and tools at work. This highly computational coverage includes:

* Discussion and implementation of the GAMS programming system

* Unique coverage of compatibility

* Illustrative examples that showcase the connection between model and reality

* Practical problems covering a wide range of scientific disciplines, as well as hundreds of examples and end-of-chapter exercises

* Real-world applications to probability and statistics, electrical engineering, transportation systems, and more

Building and Solving Mathematical Programming Models in Engineering and Science is practically suited for use as a professional reference for mathematicians, engineers, and applied or industrial scientists, while also tutorial and illustrative enough for advanced students in mathematics or engineering.

**Preface xiii**

I Models 1

**1 Linear Programming 3**

1.1 Introduction 3

1.2 The Transportation Problem 4

1.3 The Production Scheduling Problem 6

1.4 The Diet Problem 9

1.5 The Network Flow Problem 11

1.6 The Portfolio Problem 13

1.7 Scaffolding System 15

1.8 Electric Power Economic Dispatch 18

**2 Mixed-Integer Linear Programming 25**

2.1 Introduction 25

2.2 The 0-1 Knapsack Problem 25

2.3 Identifying Relevant Symptoms 27

2.4 The Academy Problem 29

2.5 School Timetable Problem 32

2.6 Models of Discrete Location 35

2.7 Unit Commitment of Thermal Power Units 38

3 **Nonlinear Programming** 47

3.1 Introduction 47

3.2 Some Geometrically Motivated Examples 47

3.3 Some Mechanically Motivated Examples 51

3.4 Some Electrically Motivated Examples 55

3.5 The Matrix Balancing Problem 62

3.6 The Traffic Assignment Problem 64

**II Methods 71**

**4 An Introduction to Linear Programming 73**

4.1 Introduction 73

4.2 Problem Statement and Basic Definitions 73

4.3 Linear Programming Problem in Standard Form 78

4.4 Basic Solutions 81

4.5 Sensitivities 83

4.6 Duality 84

**5 Understanding the Set of All Feasible Solutions 97**

5.1 Introduction and Motivation 97

5.2 Convex Sets 101

5.3 Linear Spaces 105

5.4 Polyhedral Convex Cones 107

5.5 Polytopes 109

5.6 Polyhedra 110

5.7 Bounded and Unbounded LPP 113

** 6 Solving the Linear Programming Problem 117**

6.1 Introduction 117

6.2 The Simplex Method 118

6.3 The Exterior Point Method 140

**7 Mixed-Integer Linear Programming 161**

7.1 Introduction 161

7.2 The Branch-Bound Method 162

7.3 The Gomory Cuts Method 172

**8 Optimality and Duality in Nonlinear Programming 183**

8.1 Introduction 183

8.2 Necessary Optimality Conditions 188

8.2.1 Differentiability 188

8.3 Optimality Conditions: Sufficiency and Convexity 207

8.4 Duality Theory 216

8.5 Practical Illustration of Duality and Separability 221

8.6 Constraint Qualifications 226

**9 Computational Methods for Nonlinear Programming 235**

9.1 Unconstrained Optimization Algorithms 236

9.2 Constrained Optimization Algorithms 254

9.2.1 Dual Methods 254

**III Software 283**

**10 The GAMS Package 285**

10.1 Introduction 285

10.2 Illustrative Example 286

10.3 Language Features 290

**11 Some Examples Using GAMS 311**

11.1 Introduction 311

11.2 Linear Programming Examples 311

11.3 Mixed-Integer LPP Examples 330

11.4 Nonlinear Programming Examples 344

**IV Applications 369**

**12 Applications 371**

12.1 Applications to Artificial Intelligence 371

12.2 Applications to CAD 378

12.3 Applications to Probability 387

12.4 Regression Models 395

12.5 Applications to Optimization Problems 401

12.6 Transportation Systems 417

12.7 Short-Term Hydrothermal Coordination 442

**13 Some Useful Modeling Tricks 451**

13.1 Introduction 451

13.2 Some General Tricks 451

13.3 Some GAMS Tricks 466

**A Compatibility and Set of All Feasible Solutions 477**

A.l The Dual Cone 478

A.2 Cone Associated with a Polyhedron 480

A.3 The Γ Procedure 483

A.4 Compatibility of Linear Systems 488

A.5 Solving Linear Systems 491

A.6 Applications to Several Examples 494

**B Notation 517**

**Bibliography 533**

**Index 541**

*Zentralblatt Math*, Vol. 1029, 2004)

"I think this textbook is worth having in the college library…” (*Interfaces*, July-August 2003)

"...can be quite valuable because of its documentation of the GAMS software product...a means to learn and utilize a sophisticated linear and nonlinear programming tool." (*Journal of Mathematical Psychology*, 2002)

"...a welcome addition to the series of publications on mathematical programming applications to engineering problems..." Note: Review features an image of wiley.com. (*IEEE Computer Applications in Power*)

"...intention is to discuss the subject from an angle different from the standard, emphasizing conditions leading to well-defined problems, compatibility and uniqueness of solutions." (*Mathematical Reviews*, 2002i)

"...a useful and welcome addition to existing books on mathematical programming…I recommend this book..." (*IIE Transactions*)

"...very well suited as a professional reference or as a text for advanced mathematics or engineering courses." (*Journal of Applied Mathematics and Stochastic Analysis*, Vol. 15, No. 4)