A farmer has 200 acres of land and wants to cultivate potato

A farmer has 200 acres of land and wants to cultivate potatoes or pumpkins or a combination of both. He has discovered that there is sufficient demand for these products and does not consider other alternatives. The maximum yield of potatoes is five tons per acre, and if pumpkins will grow only three tons per acre will be produced. The potatoes can be sold at a profit of 50 pounds per ton, while the pumpkins at a profit of 105 pounds per ton. There is a defined demand for both species. A maximum of 750 tons of potatoes and of 300 tons of pumpkins should be produced per year in order to be placed freely in the market. Both seeds will need fertilizers and the ratio for each growing seed has a limit regarding the available fertilizer. The farmer uses two types of fertilizer, A and B, which are mixed in the right proportion for each seed. He believes that the mix for potatoes should be composed of 40% of fertilizer A and 60% of fertilizer B. The mix for the pumpkins should consist of 55% of fertilizer A and 45% of fertilizer B. Each acre of potatoes needs 0.4 tons of fertilizer and each acre of pumpkins needs 0.5 tons of fertilizer. There is a limit to the amount of available fertilizer. The farmer can buy up to 30 tons of fertilizer A and 100 tons of fertilizer B. Fertilizer A is of better quality. The farmer can improve the quality of B by adding enhaneing ingredients. If he does so, the improved tons of B can be used as partial or total supplement for 40% of A which is required in the potatoes mix. However, the farmer estimates that this will cause a decrease of 10% in yield. Its use is not possible on the pumpkin mix because the result would be disastrous. For every ton of fertilizer B that will be improved in this way 0.1 tons of additional components are required, with an additional cost of 45 pounds. 1) Design (without solving) this problem as a linear programming model in order to maximize the profit. 2) Give arguments for how to strengthen this plan, assuming that the optimal solution has already been calculated.

Solution

Here the farmer have 200 acres and he cultivate the two type of products one potatoes ao pumpkins or combination of here only he cultivate only potatoes

He will take profit in 200 acres 50,00,000 pounds by the potatoes

And the another is pumpkins when he cultivate pumpkins

63,00,000 pounds profit.

When both species 1,38,00,000

Here he taken two various types of fertilizers one potato takes in type fertilizers A is 40% B is 60%

And for pumpkins 55% A 45% B

The total tones of potato fertilizers he take 80 tones

For pumpkins 100 tones.

In this the better way for farmer the fertilizer A is and he spent amount is to be low .