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Based on earlier work by a variety of authors in the 1930s and 1940s, the simplex method for solving linear programming problems was developed in 1947 by the mathematician George B. Dantzig. Helped by the computer revolution, it has been described by some as the overwhelmingly most significant mathematical development of the last century. Owing to the simplex method, linear programming (or linear optimization, as some would have it) is pervasive in modern society for the planning and control of activities that are constrained by the availability of resources such as manpower, raw materials, budgets, and time.
The purpose of this book is to describe the field of linear programming. While we aim to be reasonably complete in our treatment, we have given emphasis to the modeling aspects of the field. Accordingly, a number of applications are provided, where we guide the reader through the interactive process of mathematically modeling a particular practical situation, analyzing the consequences of the model formulated, and then revising the model in light of the results from the analysis.
Closely related to the issue of building models based on specific applications is the art of reformulating problems. Some of these models may at first appear not to be amenable to a linear representation, and we devote an entire chapter to this topic. A properly balanced treatment of linear programming will necessarily require a full discussion of both duality and postoptimality, and we dedicate one chapter to each of these two topics. As far as solution methods are concerned, we cover the simplex method as well as interior point techniques. During the last two decades, the latter have become serious challengers to the simplex method for solving large scale practical problems