Network Optimization Models

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Networks arise in numerous settings and in a variety of guises. Transportation, electrical, and communication networks pervade our daily lives. Network representations also are widely used for problems in such diverse areas as production, distribution, project planning, facilities location, resource management, supply chain management and financial planning— to name just a few examples. In fact, a network representation provides such a powerful visual and conceptual aid for portraying the relationships between the components of systemsthat it is used in virtually every field of scientific, social, and economic endeavor.

One of the most exciting developments in operations research (OR) in recent decades, has been the unusually rapid advance in both the methodology and application of network optimization models. A number of algorithmic breakthroughs have had a major impact, as have ideas from computer science concerning data structures and efficient data manipulation. Consequently, algorithms and software now are available and are being used to solve huge problems on a routine basis that would have been completely intractable three decades ago.

Many network optimization models actually are special types of linear programming problems. For example, both the transportation problem and the assignment problem dis- cussed in the preceding chapter fall into this category because of their network representations presented in Figs. 9.3 and 9.5.

One of the linear programming examples presented in Sec. 3.4 also is a network optimization problem. This is the Distribution Unlimited Co. problem of how to distribute its goods through the distribution network shown in Fig. 3.13. This special type of linear programming problem, called the minimum cost flow problem, is presented in Sec. 10.6. We shall return to this specific example in that section and then solve it with network methodology in the following section.

In this one chapter we only scratch the surface of the current state of the art of net- work methodology. However, we shall introduce you to five important kinds of network problems and some basic ideas of how to solve them (without delving into issues of data structures that are so vital to successful large-scale implementations). Each of the first three problem types—the shortest-path problem, the minimum spanning tree problem, and the maximum flow problem—has a very specific structure that arises frequently in applications.

The fourth type—the minimum cost flow problem—provides a unified approach to many other applications because of its far more general structure. In fact, this structure is so general that it includes as special cases both the shortest-path problem and the maxi- mum flow problem as well as the transportation problem and the assignment problem from

Chap. 9. Because the minimum cost flow problem is a special type of linear programming problem, it can be solved extremely efficiently by a streamlined version of the simplex method called the network simplex method. (We shall not discuss even more general network problems that are more difficult to solve.)

The fifth kind of network problem considered here involves determining the most economical way to conduct a project so that it can be completed by its deadline. A technique called the CPM method of time-cost trade-offs is used to formulate a network model of the project and the time-cost trade-offs for its activities. Either marginal cost analysis or linear programming then is used to solve for the optimal project plan.

The first section introduces a prototype example that will be used subsequently to illustrate the approach to the first three of the problem types mentioned above. Section 10.2 presents some basic terminology for networks. The next four sections deal with the first four problem types in turn, and Sec. 10.7 then is devoted to the network simplex method. Section 10.8 presents the CPM method of time-cost trade-offs for project management. (Chapter 22 on the website also uses network models to deal with a variety of project management problems.)

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