A firm manufactures two types of products A and B and sells them at a profit of Rs 2 on type A and Rs 3 on type B. Each product is processed on two machines M1 and M2. Type A requires one minute of processing time on M1 and two minutes of M2; type B requires one minute on M1 and one minute on M2. The machine M1 is available for not more than 6 hours 40 minutes while machine M2 is available for 10 hours during any working day. Formulate the problem as a LPP.

Matching exercise

Let required production of product A be x units and product B be y units.
Given, profit on one unit of product A and B are Rs 2 and Rs 3 respectively, so profits on x units of product A and y units of product B will be Rs 2x and Rs 3y respectively.
Let total profit be Z, so Z = 2x + 3y
Given, production of one unit of product A and B require 1 and 1 minute on machine M1 respectively, so production of x units of product A and y units of product B require x minutes and y minutes on machine M1 but total time available on machine M1 is 600 minutes, so
x + y ≤ 400 (First constraint)
Given, production of one unit of product A and B require 2 minutes and 1 minutes on machine M2 respectively. So, production of x units of product A and y units of product B require 2x minutes and y minutes respectively on machine M2, but machine M2 is available for 600 minutes, so
2x + y ≤ 600 (Second constraint)

Hence, the mathematical formulation of LPP is:
Find x and y which
maximize Z = 2x + 3y
Subject to constraints,
x + y ≤ 400
2x + y ≤ 600
and, x, y ≥ 0 [Since production of the product cannot be less than zero]