For the given linear prgramming problem, write down the objective function and the constraints.
Kris is trying to make his business more efficient by having a system of both large and small vans. He can spend no more than $100,000 for both type of vans and no more than $600 per month for maintenance. Kris can purchase a small van for $15,000 and maintain it for $100 per month. He can purchase a large van for $25,000 and maintain it for $70 per month. Each large van carries 14 passengers, and each small van carries 7 passengers. Kris is interested in knowing how many of each van he should purchase so that he can serve the maximum number of passengers.
"large and small vans."
L=# of large, L≥0
S=# of small, S≥0
"He can spend no more than $100,000 for both type of vans"
"Kris can purchase a small van for $15,000 and maintain it for $100 per month...a large van for $25,000 and maintain it for $70 per month"
25000L+15000S≤100000
"no more than $600 per month for maintenance"
70L+100S≤600
"large van carries 14 passengers, and each small van carries 7 passengers"
Utility=objective function=
P(S,L)=7S+14L
So the above are the constraints and the objective function.
Note:
The formulation of the constraints and objective functions is a good exercise, especially when the parameters can change with time.
However, there are times that you don't need a screwdriver to do repairs. Same with linear programming:
Here:
Capacity of large van = 2*capacity of small
Cost of large van < 2*cost of small
cost of maintaining large van < 2*cost of maintaining small van
Number of large vans he can buy with budget=100000/25000 = exactly 4 (for 56 passengers)
cost of monthly maintenance = 4*70=280 < 600
So what would be your choice even without the screwdriver?
To find the objective function and the constraints for the given linear programming problem, we need to analyze the given information and translate it into mathematical equations.
Let's define the variables:
x = number of small vans to purchase
y = number of large vans to purchase
Objective function:
We want to maximize the number of passengers that can be served. Since each small van carries 7 passengers and each large van carries 14 passengers, the objective function can be written as:
Maximize: 7x + 14y
Constraints:
1. Cost constraint: Kris can spend no more than $100,000 for both types of vans. The cost of each small van is $15,000, and the cost of each large van is $25,000. So the cost constraint is:
15,000x + 25,000y ≤ 100,000
2. Maintenance constraint: Kris can spend no more than $600 per month for maintenance. The monthly maintenance cost of each small van is $100, and the maintenance cost of each large van is $70. So the maintenance constraint is:
100x + 70y ≤ 600
3. Non-negativity constraint: The number of vans cannot be negative, so we have:
x ≥ 0
y ≥ 0
So, the objective function and the constraints for this linear programming problem can be summarized as follows:
Objective function:
Maximize: 7x + 14y
Constraints:
15,000x + 25,000y ≤ 100,000
100x + 70y ≤ 600
x ≥ 0
y ≥ 0