Q8 of 118 Page 30

A manufacturer can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.


The available capacities of these operations in hours for the given time period are: grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs 20 for each unit of A and Rs 30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.


Let required production of product A and B be x and y respectively


Given, profits 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 are given by 2x and 3y respectively. Let Z be total profit, so


Z = 2x + 3y


Given, production of 1 unit of product A and B require 1 hour and 2 hours of grinding respectively, so, production of x units of product A and y units of product B require x hours and 2y hours of grinding respectively but the maximum time available for grinding is 3 hours, so


x + 2y ≤ 30 (First constraint)


Given, production of 1 unit of product A and B require 3 hours and 1 hours of turning respectively, so x units of product A and y units of product B require 3x hours and y hours of turning respectively but total time available for turning is 60 hours, so


3x + y ≤ 60 (Second constraint)


Given, production of 1 unit of product A and B require 6 hour and 3 hours of assembling respectively, so production of x units of product A and y units of product B require 6x hours and 3y hours of assembling respectively but total time available for assembling is 200 hours, so


6x + 3y ≤ 200 (Third constraint)


Given, production of 1 unit of product A and B require 5 hours and 4 hours of testing respectively, so production of x units of product A and y units of product B require 5x hours and 4y hours of testing respectively but total time available for testing is 200 hours, so


5x + 4y ≤ 200 (Fourth constraint)


Hence, mathematical formulation of LPP is,


Find x and y which


maximize Z = 2x + 3y


Subject to constraints,


x + 2y ≤ 30


3x + y ≤ 60


6x + 3y ≤ 200


5x + 4y ≤ 200


and, x, y ≥ 0 [Since production cannot be negative]


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Vitamins A and B are found in two different foods F1 and F2. One unit of food F1 contains 2 units of vitamin A and 3 units of vitamin B. One unit of food F2 contains 4 units of vitamin A and 2 units of vitamin B. One unit of food F1 and F2 cost ₹ 50 and 25 respectively. The minimum daily requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that anything in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F1 and F2 at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.

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An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man - days on each truck but only 2 man - days on each automobile. Shop B which performs finishing operations, must work 3 man - days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man - days per week available while shop B has 135 man - days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.