The Rappaport Communications Company provides cellular telephone services in several Midwestern states. The company is planning to expand its customer base by offering cellular service in northeastern Ohio to the cities of Cleveland, Akron, Canton, and Youngstown. The company will install the hardware necessary to service customers in each city on preexisting communications towers in each city. The X-Y coordinates of the preexisting towers are Cleveland (5.45). Akron (12, 21), Canton (17,5). and Youngstown (52.21). The location of the preexisting towers are not under the control of Rappaport. However, the company needs to construct a new communications tower somewhere between these cities to handle intercity calls. This tower will also allow cellular calls to be routed onto the satellite system for worldwide calling service. The tower that the company is planning to build can cover areas within a 40- mile radius. Thus, the tower must be located within 40 miles of the preexisting towers in each of these cities. Rappaport wants to determine the coordinates of the new communications tower that minimizes the sum of the distances between the new tower and each the four existing towers.
a) Formulate the NLP problem,
b) What is the coordinate of the optimal location and the distance from each of the four existing towers?
c) If the objective is to minimize the maximum distance between the new tower and any of the four existing towers, what will be the coordinate of the optimal location and the distance from each of the four existing towers?