Question

1. Sony, a television company, has three major departments for the manufacture of its two models, A and B. The monthly capacities are given as follows:

Per Unit Time Requirement (hours)
Model A Model B Hours Available this Month
Department I 4.0 2.0 1,600
Department II 2.5 1.0 1,200
Department III 4.5 1.5 1,600

The marginal profit per unit from model A is Birr 400 and that of model B is Birr 100. Assuming that the company can sell any quantity of either product due to favorable market conditions, determine the optimum output for both the models, the highest possible profit for this month and the slack time in the three departments.
Required
a) Formulate the problem as LPM
b) Solve the LPM using graphical method.

Answer 1

a) The problem can be formulated as a Linear Programming Model (LPM) as follows:

Maximize Z = 400x + 100y

Subject to:

4x + 2y ≤ 1600

2.5x + y ≤ 1200

4.5x + 1.5y ≤ 1600

Where x is the number of units of model A produced, y is the number of units of model B produced, and the constraints represent the production capacities of each department.

b) The LPM can be solved using the graphical method by plotting the constraints on a graph and finding the feasible region, which is the region of the graph where all constraints are satisfied. The corner points of the feasible region are then evaluated to find the optimal solution.

After plotting the constraints and finding the feasible region, the corner points are (0, 0), (0, 800), (266.67, 533.33), (400, 200), and (355.56, 0). Evaluating the objective function at each corner point, we find that the maximum profit of Birr 133,333.33 is achieved at the point (266.67, 533.33), which represents producing 266.67 units of model A and 533.33 units of model B.

The slack time in each department can be found by subtracting the time used for production from the available time. The slack times are 533.33 hours in Department I, 466.67 hours in Department II, and 66.67 hours in Department III.