1. 1 Let us consider that local authorities want to locate fire brigades at some places from the set 1, 2, 3 and 4 so that a distance from the worst located dwelling place from set {1, 2, …, 10} to fire brigade location be at most 25 km. The number of fire brigades should be minimised. All dwelling places must be covered. Transportation network with distances is on the picture below. Use the shortest distances matrix and calculate incidental matrix from shortest distances matrix in Mosel. 12 34 5 6 7 8 9 10 22 26 10 12 10 16 26 10 1210 12 14 10 12 10 20 customers i i’ D j a ij =1 a i’j =0

2. 2 Mathematic model of the Maximum distance problem

3. 3 Solve the previous problem but now a distance from the worst located dwelling place from set {1, 2, …, 10} to fire brigade location be at most 20 km. Can we obtain feasible solution now? Use getprobstat function. procedure print_status ! To find out the status of the solution declarations status:array({XPRS_OPT, XPRS_UNF, XPRS_INF, XPRS_UNB}) of string end-declarations status:=['Optimum found', 'Unfinished', 'Infeasible', 'Unbounded'] writeln(status(getprobstat)) end-procedure OR if (getprobstat != XPRS_OPT ) then writeln(“Optimal solution not found”)

4. 4 Let us consider that local authorities want to locate 1 ambulance vehicle at one place from the set 1, 2, 3 and 4 so that the size of the part of population from set {1, 2, …, 10}, which is out of the time limit T max =30, should be minimized. Solve using the allocation approach. Population b j in nodes {1, 2, …, 10}: 100, 150, 100, 200, 150, 100, 100, 200, 150, 100 Transportation network with travel times is on the picture below. Use the shortest distances matrix. 12 34 5 6 7 8 9 10 22 26 10 12 10 16 26 10 1210 12 14 10 12 10 20 We define c ij = b j, if d ij >T max and c ij = 0 otherwise.

5. 5 d ij 12345678910 101224382210 262234 2120 26341020202622 32412014484822 2010 43826140626236 222412 Distance Matrix (travel time)

6. 6

7. 7 Let us consider that local authorities want to locate 1 ambulance vehicle at one place from the set 1, 2, 3 and 4 so that the size of the part of population from set {1, 2, …, 10}, which is out of the time limit T max =30, should be minimized. Solve using the covering approach. Population b j in nodes {1, 2, …, 10}: 100, 150, 100, 200, 150, 100, 100, 200, 150, 100 Transportation network with travel times is on the picture below. Use the shortest distances matrix. 12 34 5 6 7 8 9 10 22 26 10 12 10 16 26 10 1210 12 14 10 12 10 20 customers i i’ T max j a ij =1 a i’j =0 x j =0 The x j is allowed to be zero only if no ambulance is located in the radius T max from j.

8. 8