The Ball Company manufactures three types of lamps, labeled A,B,and C. Each lamp is processed in two departments, I and II. Total available work-hours per day for departments I and II are 400 and 600, respectively. No additional labor is available. Time requirements and profit per unit for each lamp type are as follows:
A. B. C.
Work-hours in I 2 3 1
Work-hours in II 4 2 3
Profit per Unit $5 $4 $3
The company has assigned you as the accounting member of its profit planning committee to determine the numbers of types of A, B, and C lamps that it should produce in order to maximize its total profit from the sale of lamps.
To determine the optimal quantities of lamps that the Ball Company should produce to maximize its total profit, we can use linear programming. Linear programming is a mathematical technique to optimize a linear objective function subject to linear constraints.
Let's denote the number of lamps of type A, B, and C produced as xA, xB, and xC, respectively.
Objective function: The objective is to maximize the total profit. Since the profit per unit for each lamp type is given, the total profit can be calculated as:
Total Profit = Profit per unit of A * Number of A lamps + Profit per unit of B * Number of B lamps + Profit per unit of C * Number of C lamps
Maximize: 5 * xA + 4 * xB + 3 * xC
Constraints:
1. Work-hours in department I: The total work-hours used in department I should not exceed the available work-hours of 400.
2 * xA + 3 * xB + xC <= 400
2. Work-hours in department II: The total work-hours used in department II should not exceed the available work-hours of 600.
4 * xA + 2 * xB + 3 * xC <= 600
3. Non-negativity constraints: The number of lamps of each type should be non-negative.
xA >= 0, xB >= 0, xC >= 0
By solving this linear programming problem, we can find the optimal values of xA, xB, and xC that maximize the total profit.
You can use software tools like Excel Solver, MATLAB, or Python libraries like SciPy's linprog function to solve this linear programming problem and find the optimal solution.