THE TRANSPORTATION AND ASSIGNMENT PROBLEMS:CONCLUSIONS

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CONCLUSIONS

The linear programming model encompasses a wide variety of specific types of problems. The general simplex method is a powerful algorithm that can solve surprisingly large versions of any of these problems. However, some of these problem types have such simple formulations that they can be solved much more efficiently by streamlined algorithms that exploit their special structure. These streamlined algorithms can cut down tremendously on the computer time required for large problems, and they sometimes make it computa- tionally feasible to solve huge problems. This is particularly true for the two types of linear programming problems studied in this chapter, namely, the transportation problem and the assignment problem. Both types have a number of common applications, so it is important to recognize them when they arise and to use the best available algorithms. These special-purpose algorithms are included in some linear programming software packages.

We shall reexamine the special structure of the transportation and assignment prob- lems in Sec. 10.6. There we shall see that these problems are special cases of an impor- tant class of linear programming problems known as the minimum cost flow problem. This problem has the interpretation of minimizing the cost for the flow of goods through a net- work. A streamlined version of the simplex method called the network simplex method (described in Sec. 10.7) is widely used for solving this type of problem, including its various special cases.

A supplementary chapter (Chap. 23) on the book’s website describes various additional special types of linear programming problems. One of these, called the transship- ment problem, is a generalization of the transportation problem which allows shipments from any source to any destination to first go through intermediate transfer points. Since the transshipment problem also is a special case of the minimum cost flow problem, we will describe it further in Sec. 10.6.

Much research continues to be devoted to developing streamlined algorithms for special types of linear programming problems, including some not discussed here. At the same time, there is widespread interest in applying linear programming to optimize the operation of complicated large-scale systems. The resulting formulations usually have spe- cial structures that can be exploited. Being able to recognize and exploit special structures is an important factor in the successful application of linear programming.

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