Question

A firm manufactures two products A and B on which the profits earned per unit are ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M₁ and M₂. Product A requires one minute of processing time on M₁ and two minutes on M₂, While B requires one minute on M₁ and one minute on M₂. Machine M₁ is available for not more than 7 hrs 30 minutes while M₂ is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.

Answer 1

(i) Variables: Let x₁ represents the product A and x₂ represents the product B.

(ii) Objective function:

Profit earned from Product A = 3x₁

Profit earned from Product B = 4x₂

Let Z = 3x₁ + 4x₂

Since the profit is to be maximized, we have maximize Z = 3x₁ + 4x₂

(iii) Constraints:

	M1	M2
Requirement for A	1 min	2 min
Requirement for B	1 min	1 min

M₁ is available for 7 hrs 30 min = 7 × 60 + 30 = 450 min

M₂ is available for 10 hrs = 10 × 60 = 600 min

∴ x₁ + x₂ ≤ 450 .....[for M₁]

2x₁ + x₂ ≤ 600 ......[for M₂]

(iv) Non-negative restrictions:

Since the number of products of type A and B cannot be negative, x₁, x₂ ≥ 0.

Hence, the mathematical formulation of the LLP is maximize

Z = 3x₁ + 4x₂

Subject to the constraints

x₁ + x₂ ≤ 450

2x₁ + x₂ ≤ 600

x₁, x₂ ≥ 0