Refer to the transportation alternatives problem of Example 5.2. The bridge at Cay Road is actually part of the road between Augen and Burger. Therefore it is not reasonable for the bridge to have fewer lanes than the road itself. This means that if projects 2 or 4 are carried out, either projects 6 or 7 must also be carried out. Formulate a zero-one programming problem that includes this additional requirement. Solve the problem.

Data from Example 5.2 

Suppose that the goals and specific projects shown in Table 5.2 are being considered by the County Transportation Authority.

There are three independent goals and a total of 10 projects. Table 5.2 shows the cost and the net present value (after the cost has been deducted) for each of the projects. The total available budget is \($5 million.\) To formulate this problem we introduce a zero—one variable for each project. (However, for simplicity we index these variables consecutively from 1 through 10, rather than using the double indexing procedure of the general formulation presented earlier.) The problem formulation can be expressed as

This problem and its solution are clearly displayed by a spreadsheet, as illustrated in Figure 5.2. The solution is that projects 2, 5, and 10 should be selected, for a cost of \($4,500,000\) and a total present value of \($8,000,000.\)

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TABLE 5.2 TRANSPORTATION ALTERNATIVES Cost NPV ($1,000) ($1,000) Road between Augen and Burger 1 Concrete, 2 lanes 2,000 4,000 2 Concrete, 4 lanes 3 Asphalt, 2 lanes 4 Asphalt, 4 lanes Bridge at Cay Road 3,000 5,000 1,500 3,000 2,200 4,300 5 Repair existing 500 1,000 6 Add lane 1,500 1,500 7 New structure 2,500 2,500 Traffic Control in Downsberg 8 Traffic lights 100 300 9 Turn lanes 600 1,000 10 Underpass 1,000 2,000 At most one project can be selected for each major objective.