a. Consider a hierarchical facility location model. There are three customer sites, i.e., N = {1, 2, 3} and 3 potential facility locations, i.e., M = {4, 5, 6}. Locations 4 and 6 are candidates for a type 1 facility, whereas locations 5 and 6 are candidates for a type 2 facility, i.e., M1 = {4, 6} and M2 = {5, 6}. We would like to open one facility for each type, i.e., p1 = p2 = 1. The number of customers at sites 1, 2, 3 for service types 1 and 2 are 500 (site 1, type 1), 2000 (site 1, type 2), 400 (site 2, type 1), 200 (site 2, type 2), 750 (site 3, type 1) and 1000 (site 3, type 2) respectively. We would like to decide which sites of potential facilities to open considering minimization of total distance between customers and facility sites. Below is the distance matrix between customers and potential facilities. Formulate this problem in open form.

potential facility locations

customers 4 5 6

1 5 1 4

2 2 3 6

3 3 7 2

b. Consider the model discussed in case study CARE. Why would constraints (6) and (7) be always binding at an optimal solution?

c. Consider the distance matrix used in the lecture hour regarding nearest allocation constraints and remember Alternative III nearest allocation (closest assignment) constraints we have discussed. Write these constraints for each customer and potential facility location (in open form).

please answer a, b and c