Consider again the CHB, Inc. problem described in Problem 12. Suppose only a limited number of PPBs can be placed. CHB would like to place this limited number of PPBs in counties so that the allowable branches can reach the maximum possible population. The file CHBPop contains the county adjacency matrix described in Problem 12 as well as the population of each county.
a. Assume that only a fixed number of PPBs, denoted by k, can be established. Formulate a linear binary integer program that will tell CHB, Inc. where to locate the fixed number of PPBs in order to maximize the population reached. (Hint: Review the Ohio Trust formulation in Section 13.4. Introduce a binary variable yi such that yi 51 if county i, that is, if county i can be reached by a PBB (because there is a PBB in county i or in an adjacent county to county i), and yi 5 0 otherwise.
b. Suppose that two PPBs can be established. Where should they be located to maximize the population served?
c. Solve your model from part (a) for an allowable number of PPBs ranging from 1 to 10. In other words, solve the model 10 times, k set to 1, 2, . . . , 10. Record the population reached for each value of k. Graph the results by plotting the population reached versus the number of PPBs allowed. Based on their cost calculations, CHB considers an additional PPB to be fiscally prudent only if it increases the population reached by at least 500,000 people. Based on this graph, how many PPBs do you recommend to implement?
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