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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.

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i i’

Dj

aij=1 ai’j=0

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Mathematic model of the Maximum distance problem

10,...,1}1,0{

10,...,114

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ifory

jforyatosubject

yMinimize

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iiji

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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

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if (getprobstat != XPRS_OPT) then writeln(“Optimal solution not found”)

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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 Tmax =30, should be minimized. Solve using the allocation approach. Population bj in nodes {1, 2, …, 10}: 100, 150, 100, 200, 150, 100, 100, 200, 150, 100Transportation network with travel times is on the picture below. Use the shortest distances matrix.

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We definecij = bj, if dij >Tmax and cij = 0 otherwise.

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dij 1 2 3 4 5 6 7 8 9 10

1 0 12 24 38 22 10 10 26 22 34

2 12 0 12 26 34 10 20 26 22 22

3 24 12 0 14 48 22 22 20 10 10

4 38 26 14 0 62 36 36 22 24 12

DistanceMatrix(travel time)

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Jj,Iifor},{z

Iifor},{y

py

Jj,Iiforyz

JjforztoSubject

zcMinimize

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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 Tmax =30, should be minimized. Solve using the covering approach. Population bj in nodes {1, 2, …, 10}:100, 150, 100, 200, 150, 100, 100, 200, 150, 100Transportation network with travel times is on the picture below. Use the shortest distances matrix.

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The xj is allowed to be zero only if no ambulance is located in the radius Tmax from j.

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Jjforx

Iifory

py

JjforyaxtoSubject

xbMinimize

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Iiiijj

Jjjj

}1,0{

}1,0{

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