Consider again the Ohio Trust Inc. problem described in Problem 15. Suppose only a limited number of PPBs can be placed. Ohio Trust would like to place this limited number of PPBs in counties so that the allowable branches can reach the maximum possible population. The file Ohio Trust Pop contains the county adjacency matrix described in Problem 15 as well as the population of each county.

1. Assume that only a fixed number of PPBs, denoted k. can be established. Formulate a linear binary integer program that will tell Ohio Trust Inc. where to locate the fixed number of PPBs in order to maximize the population reached.

Hint:

Introduce variable yi = 1 if it is possible to establish a branch in county i, and

yi = 0 otherwise; that is, if county i is covered by a PPB, then the population can be counted as covered.

2. Suppose that two PPBs can be established. Where should they be located to maximize the population served?

Solve your model from part a for 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 of this analysis by plotting the population reached versus number of PPBs allowed. Based on their cost calculations, Ohio Trust 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, what is the number of PPBs you recommend to be implemented