Question

Problem 4 (24 points) Farmer Joe is planning to plant 30 hectares of rice, 40 hectares of wheat, and 30 hectares of cotton in his farm this year. He has 100 hectares of land and 125,000 m² of irrigation water to use. The first table below contains the fully irrigated water demand, the maximum yield per hectare, and the expected profit per ton for the three crops. The ratios of actual-to-maximum yield for each of these three crops under different irrigation levels are shown in the second table. Assume that Farmer Paul can select only one irrigation level for each crop. Farmer Joe needs to find an optimal cropping pattern so that he can maximize his profit. Water Demand (m/ha) Maximal Yield (tons/ha) Profit (S/ton) Rice 1,300 190 Wheat 1,200 180 Cotton 1,500 240, 100% 60% Maximum Yield by Irrigation Level 80% 92% 88% 72% Actual Yield (tons/acre) by Irrigation Level 100% 80% 60% 8 .28 7.02 10.56 9.6 5.76 5.52 9 Rice Wheat Cotton 100% 100% 100% 78% 80% 69% a. Formulate the LP model for this problem. b. Solve it with Spreadsheet modeling

Answer 1

LP Problem

Ha of Rice, Wheat and Cotton to be irrigated 100%, 80% and 60% to maximize profit

Subject to: Available land, and water

It can be shown as follows:

Maximize

190x(9R1+8.28R2+7.02R3

180x(12W1+10.56W2+9.6W3

240x(8C1+5.76C2+5.52C3

Where R1, R2, R3 is Ha of Rice irrigated at 100%, 80% and 60% respectively

Where W1, W2, W3 is Ha of Wheat irrigated at 100%, 80% and 60% respectively

Where C1, C2, C3 is Ha of Cotton irrigated at 100%, 80% and 60% respectively

Subject to

Land use

R1+R2+R3 = 30

W1+W2+W3 = 40

C1+C2+C3 = 30

Water use

1300(R1+0.8R2+0.6R3) + 1200(W1+0.8W2+0.6W3) + 1500(C1+0.8C2+0.6C3) <=125000

This is the solution we get in Excel, using Solver:

3.08 Ha of Rice at 100%, and 26.92 Ha of Rice at 80% irrigation

40 Ha of Wheat at 100% irrigation

30 Ha of Cotton at 100% irrigation

give $191,617 of profit

				Land Used				Production in tons / Ha			Profit ($/ton)
	Land used (ha)	Water (m3/ha)	Water demand @100%	Irrigation @100%	Irrigation @80%	Irrigation @60%	Subject to	Irrigation @100%	Irrigation @80%	Irrigation @60%		Profit	Water
Rice	30	1300	39000	3.08	26.92	0	30	9	8.28	7.02	190	47,617	32000	30
Wheat	40	1200	48000	40.00	-	0	40	12	10.56	9.6	180	86,400	48000	40
Cotton	30	1500	45000	30.00	-	0	30	8	5.76	5.52	240	57,600	45000	30
Total required	100		132000									1,91,617	125000
Total available	100		125000