THE TRANSPORTATION AND ASSIGNMENT PROBLEMS

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Chapter 3 emphasized the wide applicability of linear programming. We continue to broaden our horizons in this chapter by discussing two particularly important (and related) types of linear programming problems. One type, called the received this name because many of its applications involve determining how to optimally transport goods. However, some of its important applications (e.g., production scheduling) actually have nothing to do with transportation.

The second type, called the assignment problem, involves such applications as as- signing people to tasks. Although its applications appear to be quite different from those for the transportation problem, we shall see that the assignment problem can be viewed as a special type of transportation problem.

The next chapter will introduce additional special types of linear programming prob- lems involving networks, including the minimum cost flow problem (Sec. 10.6). There we shall see that both the transportation and assignment problems actually are special cases of the minimum cost flow problem. We introduce the network representation of the trans- portation and assignment problems in this chapter.

Applications of the transportation and assignment problems tend to require a very large number of constraints and variables, so a straightforward computer application of the simplex method may require an exorbitant computational effort. Fortunately, a key characteristic of these problems is that most of the aij coefficients in the constraints are zeros, and the relatively few nonzero coefficients appear in a distinctive pattern. As a re- sult, it has been possible to develop special streamlined algorithms that achieve dramatic computational savings by exploiting this special structure of the problem. Therefore, it is important to become sufficiently familiar with these special types of problems that you can recognize them when they arise and apply the proper computational procedure.

To describe special structures, we shall introduce the table (matrix) of constraint coeffi- cients shown in Table 9.1, where aij is the coefficient of the jth variable in the ith functional constraint. Later, portions of the table containing only coefficients equal to zero will be in- dicated by leaving them blank, whereas blocks containing nonzero coefficients will be shaded.

After presenting a prototype example for the transportation problem, we describe the special structure in its model and give additional examples of its applications. Section 9.2 presents the transportation simplex method, a special streamlined version of the simplex

Introduction to Operations Research-0102

method for efficiently solving transportation problems. (You will see in Sec. 10.7 that this algorithm is related to the network simplex method, another streamlined version of the simplex method for efficiently solving any minimum cost flow problem, including both transportation and assignment problems.) Section 9.3 focuses on the assignment problem. Section 9.4 then presents a specialized algorithm, called the Hungarian algorithm, for solving only assignment problems very efficiently.

The book’s website also provides a supplement to this chapter. It is a complete case study (including the analysis) that illustrates how a corporate decision regarding where to locate a new facility (an oil refinery in this case) may require solving many transportation problems. (One of the cases for this chapter asks you to continue the analysis for an extension of this case study.)

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