EXAMPLES ON FORMULATION OF LINEAR PROGRAMMING PROBLEM

QUESTION 1:


A company sells two different products A and B, making a profit of Rs 40 and Rs 30 per unit on them, respectively. They are produced in a common production process and are sold in two different markets. The production process has a total capacity of 30,000 man-hours. It takes 3 hours to produce a unit of A and 1 hour to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 8,000 units and that of B is 12,000 units. Subject to these limitations, products can be sold in any combination. Formulate this problem as an LP model to maximize profit.


SOLUTION:


Let x unit of A and y unit of B are produced.


Maximize z = 40x + 30y 


Subject to the constraints


3x + y <= 30000

x <= 8000

y <= 12000

x, y >= 0


QUESTION 2:

The agricultural research institute suggested the farmer to spread out at least 4800 kg of special phosphate fertilizer and not less than 7200 kg of a special nitrogen fertilizer to raise the productivity of crops in their fields. There are two sources for obtaining these mixtures A and B. Both of these are available in bags weighing 100 kg each and they cost Rs. 40 and Rs. 24 respectively. Mixture A contains phosphate and nitrogen equivalent of 20 kg and 80 kg respectively, while mixture B contains these ingredients equivalent to 50 kg each. Write this as an LPP and determine how many bags of each type the farmer should buy in order to obtain the required fertilizer at minimum.


SOLUTION:


Let x and y be the number of bags of mixture A and B.

Minimize z = 40x + 24y


Subject to the constraints


20x + 50y >= 4800

80x + 50y >= 7200

x, y >= 0


On solving equations 20x + 50y = 4800 and 80x + 50y = 7200, we get 

x = 40 and y = 80


The farmer should buy 40 bags of phosphate fertilizer and 80 bags of nitrogen fertilizer in order to obtain the required fertilizer at minimum.


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