Or S.

# Linear programming word problem, formulate the model

A farmer has 200 acres of land and wants to cultivate tomatoes and pumpkins. The yield of tomatoes is 5 tons per acre, and the yield of pumpkins is 3 tons per acre. The tomatoes can be sold at a profit of 50 pounds per ton, while the pumpkins at a profit of 100 pounds per ton. Both seeds will need fertilizers and the ratio for each growing seed has a limit regarding the available fertilizer. The farmer uses 2 types of fertilizer, A and B, which are mixed in the right proportion for each seed. The mix for tomatoes 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 tomatoes needs 0.4 tons of fertilizer and each acre of pumpkins needs 0.5 tons of 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 enhancing 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 tomatoes mix. However, the farmer estimates that this will cause a decrease of 10% in tomatoes 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 there will be an additional cost of 45 pounds.

Design this problem as a linear programming model in order to maximize the profit.