Question

Gotham City must determine how to allocate ambulances during the next year. It costs $6,000 per year to run an ambulance. Each ambulance must be assigned to one of two districts. Let x = number of ambulances in district 1 y = number of ambulances in district 2 The average time in minutes for an ambulance to respond to a call is

District 1: 40 – 3x

District 2: 50 – 4y

Gotham City has three goals (listed in order of priority)

1. At most $120,000 per year should be spent on ambulance service

2. Average response time in district 1 should be at most 5 minutes

3. Average response time in district 2 should be at most 5 minutes Remember that the number of ambulances must be integer.

a. Use preemptive goal programming to determine how many ambulances to assign to each district.

b. If each minute in extra average response time is worth $4,000 in district 1 and $2,000 in district 2, determine how many ambulances to assign to each district (assuming there are no other priorities among the goals).

Please solve in excel

Answer 1

(a)

Step-1: Satisfy the first goal

Objective function: Minimize Z1 = o1
Subject to,
6x + 6y + u1 - o1 = 120
40 - 3x + u2 - o2 = 5
50 - 4y + u3 - o3 = 5
x, y = positive integers, u1, o1, u2, o2, u3, o3 >= 0

Result

	x	y	u1	o1	u2	o2	u3	o3
	12	8	0	0	1	0	0	13
Z1				1					0

Step-2: Satisfy the second goal, keeping the achievement of the first goal in step-1 intact.

Objective function: Minimize Z2 = o2
Subject to,
6x + 6y + u1 - o1 = 120
40 - 3x + u2 - o2 = 5
50 - 4y + u3 - o3 = 5

o1 = 0
x, y = positive integers, u1, o1, u2, o2, u3, o3 >= 0

Result

	x	y	u1	o1	u2	o2	u3	o3
	12	8	0	0	1	0	0	13
Z2						1			0

Step-3: Satisfy the third goal, keeping the achievement of the first and second goal in step-1 and 2 intact.

Objective function: Minimize Z3 = o3
Subject to,
6x + 6y + u1 - o1 = 120
40 - 3x + u2 - o2 = 5
50 - 4y + u3 - o3 = 5

o1=0
o2=0
x, y = positive integers, u1, o1, u2, o2, u3, o3 >= 0

	x	y	u1	o1	u2	o2	u3	o3
	12	8	0	0	1	0	0	13
Z3								1	13

So, when we consider these goals as preemptive, the best solution is x=12, y=8 with goal 1 and 2 achieved and goal 3 not achieved.

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(b)

For non-preemptive priority goals, we can assign weighs considering the worth of extra average response time i.e. say 4 for goal 1, 2 for goal 2 and 1 for goal 3

Objective function: Minimize Z = 4o1 + 2o2 + 1o3
Subject to,
6x + 6y + u1 - o1 = 120
40 - 3x + u2 - o2 = 5
50 - 4y + u3 - o3 = 5
x, y = positive integers, u1, o1, u2, o2, u3, o3 >= 0

	x	y	u1	o1	u2	o2	u3	o3
	11	9	0	0	0	2	0	9
Z				4		2		1	13