a farmer has 20 hectares farm. he produces cabbage,carrot and potato. whatever he produces is fully sold in the market. he gets $ 4 per per kg for cabbage, 6 for carrot per kg and 5 for potato per kg. the average per hectare yield is 200kg of cabbage,250 kg of carrot and 150 kg of potato. to produce each of 20 kg of cabbage and carrot and 25 kg of potato, an amount of 10 birr respectively has to be used for fertilizer. labor required for each hectare to raise the vegetables is 5 man days for cabbage and potato each and 6 man days for carrot. a total of 300 man days of labor at a rate of 30 per man day are available. formulate this LPM that helps to determine the optimum number of hectares to maximize the farmer's total profit.

Question

a farmer has 20 hectares farm. he produces cabbage,carrot and potato. whatever he produces is fully sold in the market. he gets $ 4 per per kg for cabbage, 6 for carrot per kg and 5 for potato per kg. the average per hectare yield is 200kg of cabbage,250 kg of carrot and 150 kg of potato. to produce each of 20 kg of cabbage and carrot and 25 kg of potato, an amount of 10 birr respectively has to be used for fertilizer. labor required for each hectare to raise the vegetables is 5 man days for cabbage and potato each and 6 man days for carrot. a total of 300 man days of labor at a rate of 30 per man day are available. formulate this LPM that helps to determine the optimum number of hectares to maximize the farmer's total profit.

Answer 1

Well, I am not going to provide the solution for you without any thinking on your part.

You need to define the constraints and cost equations, and as they are stated pretty clearly, it should not be difficult.
Lastly, you need the Profit function, and mazimize it.
There are tools readily available for the solution after that. There are about 100 ways to do that, but one of the quickest is https://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=1e692c6f72587b2cbd3e7be018fd8960&title=Linear%20Programming%20Calculator&theme=blue
If you need more help with your thinking, ask on that. here I think you are looking for optimum number of acres allocated to each of the of the crops, not exactly what you said in the last sentence.

Answer 2

I am only confused because it's profit maximization, with no profit

Answer 3

To formulate a Linear Programming Model (LPM) to determine the optimum number of hectares to maximize the farmer's total profit, we need to define decision variables, objective function, and constraints.

Decision Variables:
Let x, y, and z represent the number of hectares of cabbage, carrot, and potato respectively.

Objective Function:
The objective is to maximize the farmer's total profit. The profit from each crop can be calculated by subtracting the cost of fertilizer per hectare from the revenue obtained by selling the crop, and then multiplying it by the yield per hectare. The objective function can be written as:

Maximize: 4x + 6y + 5z - 10(x + y + z)

Constraints:
1. The total area of the farm should not exceed 20 hectares:
x + y + z ≤ 20

2. The total amount of fertilizer used should be less than or equal to the available fertilizer:
10(x + y + z) ≤ 300

3. The labor required for each crop should not exceed the available labor:
5x + 6y + 5z ≤ 300

4. Non-negativity constraint:
x, y, z ≥ 0

Now, we can summarize the LPM as follows:

Maximize: 4x + 6y + 5z - 10(x + y + z)
Subject to:
x + y + z ≤ 20
10(x + y + z) ≤ 300
5x + 6y + 5z ≤ 300
x, y, z ≥ 0

By solving this linear programming model using appropriate software or techniques, you can determine the optimal number of hectares for each crop that maximizes the farmer's total profit.