by Russell C. Walker 



This text is a soft cover, spiral bound, custom published version of the hard cover text of the same name. It or its hardcover predecessor have been used in the Business Administration program at Carnegie Mellon for more than 10 years. It is the sole text for Models and Methods for Optimization and one of the texts for Multivariate Analysis.
There have been several improvements in the custom version which is now in its third edition:

The text can be ordered by contacting the author: rw1k AT andrew DOT cmu DOT edu or from Amazon. 
The table of contents and chapter summaries: 
Chapter 1: Introduction to the Problems
Section 1.1 Introduction Section 1.2 Types of problems to be considered Section 1.3 Sample problems Section 1.4 Graphical solution of linear programs Section 1.5 Summary and objectives 
The first chapter provides a survey of problem types to be considered to indicate the possible applications.
In some cases the contribution of the solution to an organization is indicated to emphasize the relevance of these skills. The sample problems in the third section suggest the linear structure involved in most models we consider and issues associated with model formulation. The concluding section presents a graphical approach to solving two variable linear programs. 
Chapter 2: Vectors and Matrices
Section 2.1 Introduction Section 2.2 Vectors Section 2.3 The span of a set of vectors Section 2.4 Matrices Section 2.5 Linear independence Section 2.6 Systems of equations Section 2.7 The inverse of a matrix Section 2.8 Summary and objectives  This chapter develops the matrix algebra needed
to treat linear problems.
Section 2.5 is important since there is a linearly independent set of vectors corresponding to each basic solution in the simplex algorithm. The discussion of linear independence includes some principles of basic mathematical reasoning which should be understood by any student who has studied mathematics. These ideas are then used in proving propositions concerning linear independence. The section on systems of equations is important because the row operations used there are the same as those needed later in the simplex algorithm. Matrix inverses are discussed but are needed only in the Exercises of Section 3.3 and in Sections 4.4 and 4.5. 
Chapter 3 Linear Programming
Section 3.1 Introduction Section 3.2 Slack variables Section 3.3 The simplex algorithm Section 3.4 Basic feasible solutions and extreme points Section 3.5 Formulation examples Section 3.6 General constraints and variables Section 3.9 Summary and objectives  The central topic in the text is linear programming. Sections 3.2 and 3.3 develop the simplex algorithm. In Section 3.4 we establish that the simplex algorithm is correct. Section 3.5 discusses the formulation of problems and is of particular importance to those most motivated by applications. Section 3.6 extends the simplex algorithm to problems with nonstandard constraints or unsigned variables. 
Chapter 4 Duality and Post Optimal Analysis Section 4.1 Introduction Section 4.2 The dual and minimizing problems Section 4.3 Sensitivity analysis Section 4.4 The matrix setting for the simplex algorithm Section 4.5 Adding a variable Section 4.6 Summary and objectives  Section 4.2 discusses the solution of minimization problems by using the associated dual maximization problem. The power of linear programming as a managerial tool is shown in an example which helps to motivate the discussion of sensitivity analysis in Section 4.3. Section 4.4 takes a closer look at the linear algebra involved in the simplex algorithm, and in Section 4.5 that linear algebra is used to show that a variable can be added to a solved linear program without having to resolve it. 
Chapter 5 Network Models
Section 5.1 Introduction Section 5.2 The transportation problem Section 5.3 The critical path method Section 5.4 Shortest path models Section 5.5 Minimal spanning trees Section 5.6 The maximum flow problem Section 5.7 Summary and objectives  Chapter 5 treats six network problems: the transportation problem, the transhipment problem, the critical path method, the shortest path problem, minimal spanning trees, and maximum flow. Sample models in LINGO and LINDO are provided. The discussions of shortest paths and minimal spanning trees require some introduction to graph theory. The effectiveness and correctness of an algorithm are also introduced and considered for minimal spanning tree algorithms. 
Chapter 6 Unconstrained Extrema
Section 6.1 Introduction Section 6.2 Locating extrema Section 6.3 The economic lot size model and convexity Section 6.4 Location of extrema in two variables Section 6.5 Least squares approximation Section 6.6 The nvariable case Section 6.7 Numerical search Section 6.8 Summary and objectives  In this chapter we discuss classical optimization techniques. Some knowledge of differential calculus is required. Convexity is discussed in connection with the economic order quantity problem and inventory management. A section is devoted to the application of least squares curve fitting. There is a discussion of the theory underlying optimization and also an introduction to the use of Maple to solve optimization problems. Numerical search techniques are also introduced. 
Chapter 7 Constrained Extrema
Section 7.1 Introduction Section 7.2 Two variable problems Section 7.3 More variables; more constraints Section 7.4 Problems having inequality constraints Section 7.5 The convex programming problem Section 7.6 Linear programming revisited Section 7.7 Summary and objectives  This chapter extends the discussion begun in the previous one to problems in which the solution is subject to constraints. The key theorem is the KarushKuhnTucker theorem for solving convex problems. The main applications presented are the minimization of the cost of a cardboard box, the maximization of utility, the minimization of the cost of equipment replacement, and choosing an investment portfolio to achieve an acceptable return at minimum risk. The chapter concludes with a look back at linear programming as a special case of convex programming. 
Chapter 8 Integer Programming
Section 8.1 Introduction Section 8.2 The knapsack problem Section 8.3 The dual simplex algorithm Section 8.4 Adding a constraint Section 8.5 Branch and bound for integer programs: Section 8.6 Basic integer programming models Section 8.7 The traveling salesman problem Section 8.8 Summary and objectives  Branchandbound algorithms are central to this topic. The knapsack problem is considered first to introduce
the branchandbound method. The dual simplex algorithm is discussed next, and then employed to reoptimize a solved problem after a constraint has been added. This then forms the basis of a branchandbound approach to solving integer programs. A variety of integer programming models is then discussed, and the chapter concludes with a branchandbound approach to the traveling salesman problem. 
Chapter 9 Introduction to Dynamic Programming
Section 9.1 Introduction to recursion Section 9.2 The longest path Section 9.3 A fixed cost transportation problem Section 9.4 More examples Section 9.5 Summary and objectives  Problems for which a solution can be obtained through a succession of independent decisions can often be solved by dynamic programming. Dynamic programming requires an introduction to recursion. This leads to a brief excursion through the Towers of Hanoi, Fibonacci numbers, and the binomial expansion. Applications discussed are the longest path problem, which is similar to the determination of earliest times in the CPM model, the fixed cost transportation problem, and the cargo loading problem. We also return to the traveling salesman problem and investigate the computational challenge of that problem. 
Chapter 10 Case Studies
Section 10.1 Tweaking Widget's production Section 10.2 A furniture sales opportunity Section 10.3 Building storage lockers Section 10.4 The McIntire farm Section 10.5 Cylinders for beverages Section 10.6 Books by the holidays Section 10.7 Into a blind trust Section 10.8 Max's taxes Section 10.9 A supply network  Chapter 10 presents several more openended problems suitable for longer assignments
and group projects. Solutions to the cases and suggestions for their class
use are available for instructors in the solutions manual from the author.
Methods required for their solution are linear programming, integer programming, critical path management, and nonlinear optimization. 
Appendix A Brief Introductions to LINDO and LINGO
Section A.1 LINDO Section A.2 LINGO  The LINDO program is extremely useful in solving
linear programs including those with integer constraints. Examples of its
use are presented along with the use of the basic commands.
LINGO is a related package allowing the solution of nonlinear problems. As a modelling language, LINGO is particularly useful for its ability to efficiently express problems with repetitive constraints. 
Appendix B A Brief Introduction to Maple
Section B.1 The basics Section B,2 Using packages  The symbolic computation software Maple can be very useful, particularly in solving classical optimization problems such as are presented in Chapters 5 and 6 as well as in doing matrix computations, curve fitting, solving linear programs, and solving network models. 
Appendix C Using Excel Solver
Section C.1 A basic example Section C.2 Two network examples Section C.3 Two nonlinear examples
 The Excel spreadsheet includes the Solver addin which is extremely useful in linear optimization problems. A brief introduction to Solver with several examples of its use is provided. 
Appendix D Selected Answers and Hints  The answers to all odd problems are provided here. Solutions to the even numbered problems are available from the author in a pdf file. 
References p. 585  
Index p. 588 