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A company manufactures two products, A and B, on two machines, I and II. It has been determined that the company will realize a profit of $9/unit of Product A and a profit of $10/unit of Product B. To manufacture 1 unit of Product A requires 6 min on Machine I and 5 min on Machine II. To manufacture 1 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?
To maximize the company's profit, we need to find the number of units of each product that should be produced in each shift. We can approach this problem using linear programming.
Let's define the decision variables:
Let x be the number of units of Product A to be produced in each shift.
Let y be the number of units of Product B to be produced in each shift.
Now, let's write the objective function, which represents the company's profit:
Profit = 9x + 10y
Next, we need to establish the constraints based on the available machine time:
Machine I constraint: 6x + 9y ≤ 5 (since there are 5 hours of machine time available on Machine I)
Machine II constraint: 5x + 4y ≤ 3 (since there are 3 hours of machine time available on Machine II)
Additionally, we have the non-negativity constraints:
x ≥ 0
y ≥ 0
Now, we can graph the feasible region defined by the constraints and find the corner points.
Using a graphing calculator or software, graph the inequalities:
6x + 9y ≤ 5
5x + 4y ≤ 3
x ≥ 0
y ≥ 0
The feasible region should be a polygon bounded by lines. The corners of this polygon represent the possible combinations of x and y that satisfy the constraints.
Next, calculate the objective function (profit) at each corner point:
Corner 1: (x1, y1) = (0, 0) → Profit = 9(0) + 10(0) = 0
Corner 2: (x2, y2) = (0, 5/3) → Profit = 9(0) + 10(5/3) ≈ $16.67
Corner 3: (x3, y3) = (5/6, 0) → Profit = 9(5/6) + 10(0) ≈ $7.50
Corner 4: (x4, y4) = (5/11, 10/11) → Profit = 9(5/11) + 10(10/11) ≈ $15.91
The corner point that maximizes the profit is (x4, y4) ≈ (5/11, 10/11). Therefore, the company should produce approximately 5/11 units of Product A and 10/11 units of Product B in each shift to maximize the profit.