(Allocation problem) A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be

Options

A. 20 hec for crop X and 30 hec for crop Y
B. 20 hec each for both crop X and crop Y
C. 30 hec for crop X and 20 hec for crop Y
D. 30 hec each for both crop X and crop Y 30

Answer

C

Explanation

Let x hec of land ne allocated crop X and Y hec to crop Y. Obviosly, ` x ge 0, y ge 0 `
The mathematical formulation of the problem is as follows:
Maximise Z = 10500 x + 9000 y
Subject to the constriants ,
` x +y le 50` (constraint related to land ) ...(i) ,
`20x +10y le 800` ( constraint related to use of herbicide ) ` rArr 2x +y le 80 x ge 0, y ge 0 ` (non -negative constraint ) ... (iii)
The feasible region is OABCD whose corner points
0(0, 0), A (40, 0), B (30, 20) and C (0, 50) respectively.
The values of Z at these points are as follows

Hence, the society will get the maximum profit of ₹ 495000 by allocating 30 hec for crop X and 20 hec for crop Y.