a First assume that each ambulance station costs 150000 to b

a) First, assume that each ambulance station costs $150,000 to build. Provide the shorest ambulance time Iminutesl to each community in tabular format and the overall cost and the ambulance station that serves that city b) Second, use the cost in the table below to determine the least cost solution and provide the distancelambulance table as required in part (a) c) Finally, for the same conditions as in part (b), if the same ambulance station cannot serve communities 4 and 5 what is the least cost solution for this situation? Provide the distanceambulance table as requiredin part a Ambulance Station Cost $1 20,000 30,000 400,000 100,000 500,000 100,000 60,000 500,000 300,000 7.3. An ambulance siting problem. The nine communities on the road network shown below are all part of the same rural county. Travel times between pairs of communi- ties are shown on the arcs of the network. They have banded together to form a sin- gle ambulance service to serve all of the nine towns with the eligible sites for ambulances restricted to any of the nine communities. The county council desires to site the least number of ambulances such that all of the towns can be reached within 15 minutes. Set up the linear integer program that achieves the county\'s objective. Do not solve unless you are instructed to do so

Solution

7.3)

First determine the minimum time to travel between the cities as follows:

1

2

3

4

5

6

7

8

9

1

-

10

10

15

22.5

27.5

30

40

50

2

10

-

15

17.5

25

30

20

30

40

3

10

15

-

5

12.5

17.5

20

30

40

4

15

17.5

5

-

7.5

12.5

15

25

35

5

22.5

25

12.5

7.5

-

7.5

22.5

22.5

37.5

6

27.5

30

17.5

12.5

7.5

-

15

15

30

7

30

20

20

15

22.5

15

-

10

20

8

40

30

30

25

22.5

15

10

-

15

9

50

40

40

35

37.5

30

20

15

-

Now assign 1 to the cities pairs which are located within time distance of 15 minutes, and 0 to the cities pair more than 15 minutes time distance

1

2

3

4

5

6

7

8

9

1

1

1

1

1

0

0

0

0

0

2

1

1

1

0

0

0

0

0

0

3

1

1

1

1

1

0

0

0

0

4

1

0

1

1

1

1

1

0

0

5

0

0

1

1

1

1

0

0

0

6

0

0

0

1

1

1

1

1

0

7

0

0

0

1

0

1

1

1

0

8

0

0

0

0

0

1

1

1

1

9

0

0

0

0

0

0

0

1

1

Decision variable: Let

X_j = 1 if the city j is selected for locating ambulance site otherwise X_j = 0

Where, j = 1, 2, .. 8, 9 for the nine cities

Objective function:

The objective is to minimize the number of ambulance sites required to serve all the nine cities.

Minimize z = X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + X₇ + X₈ + X₉

Subject To:

Set of cities that can be served within 15 minutes

Sr. no.

Locating Ambulance at city

Equation

1

City 1

X₁ + X₂ + X₃ + X₄ + 0X₅ + 0X₆ + 0X₇ + 0X₈ + 0X₉ >= 1

2

City 2

X₁ + X₂ + X₃ + 0X₄ + 0X₅ + 0X₆ + 0X₇ + 0X₈ + 0X₉ >= 1

3

City 3

X₁ + X₂ + X₃ + X₄ + X₅ + 0X₆ + 0X₇ + 0X₈ + 0X₉ >= 1

4

City 4

X₁ + 0X₂ + X₃ + X₄ + X₅ + X₆ + X₇ + 0X₈ + 0X₉ >= 1

5

City 5

0X₁ + 0X₂ + X₃ + X₄ + X₅ + X₆ + 0X₇ + 0X₈ + 0X₉ >= 1

6

City 6

0X₁ +0 X₂ + 0X₃ + X₄ + X₅ + X₆ + X₇ + X₈ + 0X₉ >= 1

7

City 7

0X₁ + 0X₂ + 0X₃ + X₄ + 0X₅ + X₆ + X₇ + X₈ + 0X₉ >= 1

8

City 8

0X₁ + 0X₂ + 0X₃ + 0X₄ + 0X₅ + X₆ + X₇ + X₈ + X₉ >= 1

9

City 9

0X₁ + 0X₂ + 0X₃ + 0X₄ + 0X₅ + 0X₆ + 0X₇ + X₈ + X₉ >= 1

10

All X_j = 1 or 0

	1	2	3	4	5	6	7	8	9
1	-	10	10	15	22.5	27.5	30	40	50
2	10	-	15	17.5	25	30	20	30	40
3	10	15	-	5	12.5	17.5	20	30	40
4	15	17.5	5	-	7.5	12.5	15	25	35
5	22.5	25	12.5	7.5	-	7.5	22.5	22.5	37.5
6	27.5	30	17.5	12.5	7.5	-	15	15	30
7	30	20	20	15	22.5	15	-	10	20
8	40	30	30	25	22.5	15	10	-	15
9	50	40	40	35	37.5	30	20	15	-