Discuss if a firm engaged in producing two models A and B performs three operations: painting, assembly, and testing. The relevant data are as follows: Hours required for each unit... Discuss if a firm engaged in producing two models A and B performs three operations: painting, assembly, and testing. The relevant data are as follows: Hours required for each unit: Model A: Assembly 1.0, Painting 0.2, Testing 0.0; Model B: Assembly 1.5, Painting 0.2, Testing 0.1. Total hours available: assembly 600, painting 100, testing 30. Determine weekly production schedule to maximize profit. Formulate the LPP model for the problem. Read more

Steps to Solve

1. Define variables for production units

Let:

• $x_1$ = number of units produced for Model A
• $x_2$ = number of units produced for Model B

2. Determine the objective function

The profit from each model is calculated as follows:

• Profit from Model A: $50 \cdot x_1$
• Profit from Model B: $80 \cdot x_2$

Thus, the objective function to maximize profit is:

$$ Z = 50x_1 + 80x_2 $$

3. Establish constraints based on labor hours

From the problem, the constraints based on the hours available for each activity are:

• Assembly hours constraint: $$ 1.0x_1 + 1.5x_2 \leq 600 $$

• Painting hours constraint: $$ 0.2x_1 + 0.2x_2 \leq 100 $$

• Testing hours constraint: $$ 0.0x_1 + 0.1x_2 \leq 30 $$

4. Include non-negativity constraints

The number of units produced cannot be negative:

$$ x_1 \geq 0 $$ $$ x_2 \geq 0 $$

5. Compile the complete LPP model

The complete linear programming problem (LPP) can be summarized as follows:

Maximize: $$ Z = 50x_1 + 80x_2 $$

Subject to:

• $$ 1.0x_1 + 1.5x_2 \leq 600 $$
• $$ 0.2x_1 + 0.2x_2 \leq 100 $$
• $$ 0.1x_2 \leq 30 $$
• $$ x_1 \geq 0 $$
• $$ x_2 \geq 0 $$