Old hens can be bought at rs. 2 each and young ones at rs. 5 each. The old hens lay 3 eggs per week and young ones lay 5 eggs per week, each egg being worth 30 paisa. A hen costs rs. 1 per week to be fed there are only rs. 80 available to spend on purchasing the hens and it is not possible to house more than 20 hens at a time. Formulate the lpp and solve it by the graphical method to find how many of each kind of hens should be bought in order to have a maximum profit per week.
Answers
The answer is: 16 young hens and no old hen
Suppose one buys x old hens and y young hens.
Since, the old hens lay 3 eggs per week and young ones lay 5 eggs per week, the total number of eggs one has per week is 3x +5y.
Consequently, each egg being worth 30 paisa, the total income per week is
0.3(3x+4y)
Also, the expenses for feeding (x +y) hens at the rate of 1 Rs per hen per week,
= Rs (x+y)
Thus, total profit earned Z, in Rs per week is given as,
Z = 0.3(3x+5y) – (x+y)
Z = 0.5y – 0.1x
Since, old hens can be bought at Rs. 2 each and young ones at Rs. 5 each and one has 80 Rs to spend for hens,
Therefore,
2x + 5y has to be less then or equal to 80.
2x + 5y ≤ 80
Since one cannot house more than 20 hens,
Therefore,
x + y has to be less than or equal to 20.
x + y ≤ 20
Where both x and y has to be positive.
x ≥ 0 and y ≥ 0
Thus the LPP formulated for given problem is,
Maximise,
Z =0.5y – 0.1x
Subject to constraints,
2x + 5y ≤ 80 and x + y ≤ 20
And the non negative constraint,
x ≥ 0 and y ≥ 0.
By solving this problem using graphical method we get,
Z is maximum for x = 0 and y = 16 and the maximum value of Z is
Z = 0.5 * 16 + 0
Z = 8
In order to have maximum profit, one should buy 16 young hens and no old hen to get profit of Rs 8 per week.
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