Question 4 (15 points): The municipality of a new city must decide about the location of some police stations in the city such that different districts (A, B, C, D, E, F, G, H in the following map) can be covered by the opened stations. In the following map, the 7 potential locations for the police stations are shown as circles labeled from 1 to 7. Location Fixed cost ($) Districts that can be covered from the related location 4 5 6 7 A, B, C B,C C. DE GET GF D, F, H G, H 600000 450000 150000 350000 690000 470000 240000 Part a) Write a mathematical programming model to help the authorities decide where to open police stations such that all districts are covered by at least one police station with the minimum cost. (9 points) Part b) Since so many crimes have recently happened in district C, the authorities have decides to increase the coverage of this area. How would you change the proposed model in part a to ensued that district C is covered by at least 2 police stations? (3 points) Part c) The authorities have realized that locations 4 and 5 are too close to each other and therefore both of them cannot be simultaneously selected to open police stations. How would you change the proposed model in part a to consider this extra information? (3 points) Part d) Ignoring parts a, b, and c consider the following case: The authorities realized that they can spend at most 1700000 $ for the opening cost of police stations. Therefore, because of this limited budget, they may not be able to cover all districts. Considering this information, formulate a mathematical programming model such that the total number of covered districts is maximized. (5 points) Question 4 (15 points): The municipality of a new city must decide about the location of some police stations in the city such that different districts (A, B, C, D, E, F, G, H in the following map) can be covered by the opened stations. In the following map, the 7 potential locations for the police stations are shown as circles labeled from 1 to 7. Location Fixed cost ($) Districts that can be covered from the related location 4 5 6 7 A, B, C B,C C. DE GET GF D, F, H G, H 600000 450000 150000 350000 690000 470000 240000 Part a) Write a mathematical programming model to help the authorities decide where to open police stations such that all districts are covered by at least one police station with the minimum cost. (9 points) Part b) Since so many crimes have recently happened in district C, the authorities have decides to increase the coverage of this area. How would you change the proposed model in part a to ensued that district C is covered by at least 2 police stations? (3 points) Part c) The authorities have realized that locations 4 and 5 are too close to each other and therefore both of them cannot be simultaneously selected to open police stations. How would you change the proposed model in part a to consider this extra information? (3 points) Part d) Ignoring parts a, b, and c consider the following case: The authorities realized that they can spend at most 1700000 $ for the opening cost of police stations. Therefore, because of this limited budget, they may not be able to cover all districts. Considering this information, formulate a mathematical programming model such that the total number of covered districts is maximized. (5 points)
