Seers Ltd., a firm that supplies crystal balls to fortune tellers, has expanded its client base in recent years. Because of this expansion, it now needs to establish a set of warehouses from which it...

Seers Ltd., a firm that supplies crystal balls to fortune tellers, has expanded its client base in recent years. Because of this expansion, it now needs to establish a set of warehouses from which it will resupply its customers. Since demand for crystal balls is unpredictable (you never know when a crystal ball will become clouded) and time sensitive (when a crystal ball fails, you need a new one NOW) and since Seers is not the only firm in this business, Seers feels a need to provide timely deliveries to its customers. In particular, Seers would like to guarantee all of its customers’ deliveries within 48 h and to maximize the number of customers that are served within 24 h. Seers estimates that any customer within 250 miles of the nearest warehouse can be served within 1 day, while those within 500 miles can be served within 2 days.

(a) Formulate the problem of locating the minimum number of warehouses needed to satisfy all demands within 500 miles and (from among the solutions that serve all demands within 500 miles) maximizing the number of demands within 250 miles of the nearest warehouse as an integer linear programming problem.

(b) Using the first demand set in the 49-node data set (SORTCAP.GRT) as a proxy for demand and assuming that facilities can only be located at one of these 49 cities, solve the problem using the SITATION program.

Note that you may need to modify the distance matrix in some way to solve this problem using the SITATION software. Also, note that you may need to modify the default Lagrangian options rather significantly to obtain a solution in which the lower and upper bounds are adequately close to each other.