A farmer has a 100 - acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is Rs 1 per kilogram for tomatoes, Rs 0.75 a head for lettuce and Rs 2 per kilogram for radishes. The average yield per acre is 2000 kgs for radishes, 3000 heads of lettuce and 1000 kilograms of radishes. Fertilizer is available at Rs 0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man - days for tomatoes and radishes and 6 man – days for lettuce. A total of 400 man - days of labour are available at Rs 20 per man - day. Formulate this problem as a LPP to maximize the farmer’s total profit

Question

A farmer has a 100 - acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is Rs 1 per kilogram for tomatoes, Rs 0.75 a head for lettuce and Rs 2 per kilogram for radishes. The average yield per acre is 2000 kgs for radishes, 3000 heads of lettuce and 1000 kilograms of radishes. Fertilizer is available at Rs 0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man - days for tomatoes and radishes and 6 man – days for lettuce. A total of 400 man - days of labour are available at Rs 20 per man - day. Formulate this problem as a LPP to maximize the farmer’s total profit

Answer 1

Let:

x1 = number of acres of tomatoes
x2 = number of acres of lettuce
x3 = number of acres of radishes

Objective function:
Maximize Z = 1(x1*2000) + 0.75(x2*3000) + 2(x3*1000) - 0.5(100x1) - 0.5(100x2) - 0.5(50x3) - 20(5x1 + 6x2 + 5x3)

Subject to:
x1 + x2 + x3 <= 100 (total farm size)
x1 >= 0, x2 >= 0, x3 >= 0

Solving this linear programming problem will give us the optimal number of acres of each crop the farmer should plant in order to maximize his total profit.