THE TRANSPORTATION AND ASSIGNMENT PROBLEMS

By -
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.)

Comments

Post a Comment

Popular posts from this blog

NETWORK OPTIMIZATION MODELS:THE MINIMUM SPANNING TREE PROBLEM

By -
Image
THE MINIMUM SPANNING TREE PROBLEM The minimum spanning tree problem bears some similarities to the main version of the shortest-path problem presented in the preceding section. In both cases, an undi r ecte d and connected network is being considered, where the given information includes some mea- sure of the positive length (distance, cost, time, etc.) associated with each link. Both problems also involve choosing a set of links that have the shortest total length among all sets of links that satisfy a certain property. For the shortest-path problem, this property is that the chosen links must provide a path between the origin and the destination. For the minimum spanning tree problem, the required property is that the chosen links must provide a path between ea c h pair of nodes. The minimum spanning tree problem can be summarized as follows: 1. You are given the nodes of a network but not the links. Instead, you are given the po- tential links and the positive length for each
Read more

DUALITY THEORY:THE ESSENCE OF DUALITY THEORY

By -
Image
THE ESSENCE OF DUALITY THEORY Given our standard form for the primal problem at the left (perhaps after conversion from another form), its dual problem has the form shown to the right. Thus, with the primal problem in maximizatio n form, the dual problem is in minimization form instead. Furthermore, the dual problem uses exactly the same pa r amete r s as the primal problem, but in different locations, as summarized below. 1. The coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem. 2. The right-hand sides of the functional constraints in the primal problem are the coef- ficients in the objective function of the dual problem. 3. The coefficients of a variable in the functional constraints of the primal problem are the coefficients in a functional constraint of the dual problem. To highlight the comparison, now look at these same two problems in matrix notation (as introduced at the beginn
Read more

DECISION SUPPORT SYSTEMS:DIALOG GENERATION AND MANAGEMENT SYSTEMS

By -
Image
DIALOG GENERATION AND MANAGEMENT SYSTEMS In our efforts in this chapter, we envision a basic DSS structure of the form shown in Figure 12. This figure also shows many of the operational functions of the database management system (DBMS) and the model base management system (MBMS). The primary purpose of the dialog generation and management system (DGMS) is to enhance the propensity and ability of the system user to use and benefit from the DSS. There are doubtless few users of a DSS who use it because of necessity. Most uses of a DSS are optional, and the decision maker must have some motivation and desire to use a DSS or it will likely remain unused. DSS use can occur in a variety of ways. In all uses of a DGMS, it is the DGMS that the user interacts with. In an early seminal text, Bennett (1983) posed three questions to indicate this centrality: 1. What presentations is the user able to see at the DSS display terminal? 2. What must the user know about what is seen at the display
Read more