A company makes tennis and squash rackets. Each tennis racket requires two units of aluminum and one unit of nylon. Each squash racket requires 1.8 units of aluminum and 0.8 units of nylon. The company has 1000 units of aluminum and 800 units of nylon available. The company is not able to manufacture more than 550 rackets in total but must manufacture at least twice as many tennis rackets as squash rackets. The profit on each tennis racket and squash racket is $7 and $9 respectively.
Let and represent the number of each type of racket made in order to maximize profit.
Formulate the above information as a Linear Programming Problem by determining
i. the objective function
ii. the linear constraints
The objective function in this case is to calculate the profit.
s=number of squash rackets (made and) sold.
t=number of tennis rackets (made and) sold.
From "The profit on each tennis racket and squash racket is $7 and $9 respectively"
we can say that the objective function, P(s,t)=7t+9s
I will show how to compose the two constraints concerning aluminium and nylon.
"Each tennis racket requires two units of aluminum and one unit of nylon. Each squash racket requires 1.8 units of aluminum and 0.8 units of nylon. The company has 1000 units of aluminum and 800 units of nylon available. "
This translates to:
2t+1.8s ≤ 1000,
2t+0.8s ≤ 800
I will leave it to you to complete the other constraints
To formulate the given information as a Linear Programming problem, we need to identify the objective function and the linear constraints.
i. The Objective Function:
The objective is to maximize the profit.
Let x be the number of tennis rackets and y be the number of squash rackets.
The profit on each tennis racket is $7, and the profit on each squash racket is $9.
Therefore, the objective function is:
Profit = 7x + 9y
ii. The Linear Constraints:
1. Each tennis racket requires two units of aluminum, and each squash racket requires 1.8 units of aluminum.
The company has 1000 units of aluminum available.
So, the aluminum constraint is:
2x + 1.8y ≤ 1000
2. Each tennis racket requires one unit of nylon, and each squash racket requires 0.8 units of nylon.
The company has 800 units of nylon available.
So, the nylon constraint is:
x + 0.8y ≤ 800
3. The company can't manufacture more than 550 rackets in total.
The total number of rackets constraint is:
x + y ≤ 550
4. The company must manufacture at least twice as many tennis rackets as squash rackets.
The ratio constraint is:
x ≥ 2y
Now, we have formulated the given information as a Linear Programming problem with the objective function and the following linear constraints:
Objective Function:
Maximize Profit = 7x + 9y
Linear Constraints:
2x + 1.8y ≤ 1000
x + 0.8y ≤ 800
x + y ≤ 550
x ≥ 2y