A company manufactures x units of Product A and y units of Product B, on two machines, I and II. It has been determined that the company will realize a profit of $4/unit of Product A and a profit of $3/unit of Product B. To manufacture a unit of Product A requires 6 min on Machine I and 5 min on Machine II. To manufacture a unit of Product B requires 9 min on Machine I and 4 min on Machine II. There are 5 hr of machine time available on Machine I and 3 hr of machine time available on Machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?
x,y
what is the optimal profit?
impatient much? Can't wait even one minute?
To solve this problem, we need to use linear programming to find the optimal values of x and y that maximize the company's profit. Linear programming involves setting up a system of inequalities and finding the point that satisfies all the constraints while maximizing (or minimizing) the objective function.
Let's define our variables:
x = number of units of Product A
y = number of units of Product B
Objective Function:
We want to maximize the profit, which is given by:
Profit = 4x + 3y
Constraints:
1. Machine I time constraint: 6x + 9y ≤ 300 minutes (5 hours = 300 minutes)
2. Machine II time constraint: 5x + 4y ≤ 180 minutes (3 hours = 180 minutes)
3. Non-negativity constraint: x ≥ 0, y ≥ 0
Now we can solve this linear programming problem to find the optimal solution.
One way to solve this is by graphing the feasible region and finding the corner point(s) that maximize the objective function. However, for more complicated problems, we can use software like Excel or linear programming solvers.
Using a linear programming solver, we can set up the problem and find the optimal solution:
- Objective function: Maximize 4x + 3y
- Constraints:
6x + 9y ≤ 300
5x + 4y ≤ 180
x ≥ 0
y ≥ 0
Solving this problem, we find that the optimal solution is:
x = 20 units of Product A
y = 10 units of Product B
To determine the optimal profit, substitute the values of x and y into the objective function:
Profit = 4(20) + 3(10) = 80 + 30 = $110
Therefore, the optimal profit for the company is $110.