Question

Grave City is considering the relocation of several police substations to obtain better enforcement in high-crime areas. The locations under consideration together with the areas that can be covered from these locations are given in the following table:

Potential Locations for Substations	Areas Covered
A	1, 3, 5
B	2, 4, 5
C	3, 4, 6
D	4, 5, 6
E	1, 5, 6, 7
F	1, 5, 7
G	1, 2, 5, 7

(a)	Formulate an integer programming model that could be used to find the minimum number of locations necessary to provide coverage to all areas. If your answer is zero enter “0” and if the constant is "1" it must be entered in the box.
	Enter values below in place of xA,xB,xC and so on and so forth.
	Min xA + xB + xC + xD + xE + xF + xG s.t. xA + xB + xC + xD + xE + xF + xG - Select your answer - ≤ ≥ = (?) (area 1 covered) xA + xB + xC + xD + xE + xF + xG - Select your answer - ≤ ≥ = (?) (area 2 covered) xA + xB + xC + xD + xE + xF + xG - Select your answer - ≤ ≥ = (?) (area 3 covered) xA + xB + xC + xD + xE + xF + xG - Select your answer - ≤ ≥ = (?) (area 4 covered) xA + xB + xC + xD + xE + xF + xG - Select your answer - ≤ ≥ = (?) (area 5 covered) xA + xB + xC + xD + xE + xF + xG - Select your answer - ≤ ≥ = (?) (area 6 covered) xA + xB + xC + xD + xE + xF + xG - Select your answer - ≤ ≥ = (?) (area 7 covered)
(b)	Solve the problem in part (a). Choose the correct optimal solution.
	If your answer is zero enter “0” and if the constant is "1" it must be entered in the box. Enter values in below boxes.
	Locations for Substations Select ? A B C D E F G

Answer 1

a.

Min   xA   +   xB   +   xC   +   xD   +   xE   +   xF   +   xG          
s.t.                                                              
1xA   +   0xB   +   0xC   +   0xD   +   1xE   +   1xF   +   1xG   >=1       (?) (area 1 covered)
0xA   +   1xB   +   0xC   +   0xD   +   0xE   +   0xF   +   1xG   >=1       (?) (area 2 covered)
1xA   +   0xB   +   1xC   +   0xD   +   0xE   +   0xF   +   0xG   >=1       (?) (area 3 covered)
0xA   +   1xB   +   1xC   +   1xD   +   0xE   +   0xF   +   0xG   >=1       (?) (area 4 covered)
1xA   +   1xB   +   0xC   +   1xD   +   1xE   +   1xF   +   1xG   >=1       (?) (area 5 covered)
0xA   +   0xB   +   1xC   +   1xD   +   1xE   +   0xF   +   0xG   >=1       (?) (area 6 covered)
0xA   +   0xB   +   0xC   +   0xD   +   1xE   +   1xF   +   1xG   >=1       (?) (area 7 covered)

b.

Solver model

Solver formula

Solver window

Sovin in solver we get, location C and G should be chosen

Locations for	Select ?
Substations
A	0
B	0
C	1
D	0
E	0
F	0
G	1