Question

3. A company manufactures two types of products: A and B. Three resources R1, R2, and R3 are required to make these products. a. Each unit of product A requires 8 units of R1 and 4 units of R2. b. Each unit of product B requires 2 units of R1, 5 units of R2, and 3 units of R3. The availabilities of resources R1, R2, and R3 are 500, 400, and 180, respectively. The manufacturer also makes a profit of $20 and $50 for products A and B, respectively. Formulate a linear programming problem whose solution will tell the manufacturer how to maximize the total profit. • What are the decision variables? • What is the objective? • What is the objective function? • What are the constraints?

Answer 1

Answer:

(i)The units of production of Product A and B (x and y) are the decision variables

(ii)Objective is to maximize profit

(iii)Objective function, Max P=20x+50y

(iv)The Contraints are:

8x+2y$\leq$500....(i)

2x+5y$\leq$400.......(ii)

3y$\leq$180

x>0, y>0

Explanation:

Let the number of Product A=x

Let the number of Product B=y

Each unit of product A requires 8 units of R1 and 4 units of R2.

Each unit of product B requires 2 units of R1, 5 units of R2, and 3 units of R3

The availabilities of resources R1, R2, and R3 are 500, 400, and 180

Since R1 $\leq$500,

Product A requires 8 units of R1 per production unit

Product B requires 2 units of R1 per production unit

Total Unit of R1 possible is given by the inequality: 8x+2y$\leq$500....(i)

Since R2 $\leq$400,

Product A requires 2 units of R2 per production unit

Product B requires 5 units of R2 per production unit

Total Unit of R2 possible is given by the inequality: 2x+5y$\leq$400....(ii)

Since R3 $\leq$180,

Product B requires 3 units of R2 per production unit

Total Unit of R2 possible is given by the inequality: 3y$\leq$180....(iii)

Since the manufacturer also makes a profit of $20 and $50 for products A and B, our objective is to maximize profit

Therefore: Objective function, Max P=20x+50y ......(iv)