Length : around 25 minutes
You are now going to write the dual programme of the following base model which you wrote in the GAMS introduction lesson.
« A given farm can grow two types of crop : wheat and maize. The area to be cultivated must be determined for each crop, knowing that :
The objective of the farmer is to maximize his income.
Wheat yields 450€ per hectare, maize 1000€.
Wheat requires 25 hours of labour per hectare and per year, and maize 50 hours.
The farm area is 50 ha and the farmer can work 2000 hours a year. »
The primal linear programme is written as follows :
Maximizing Z = 450×1 + 1000×2
with x1 + x2 ≤ 50
25×1 + 50×2 ≤ 2000
x1, x2 positive
x1 : wheat primal area
x2 : maize primal area
Optimal solution, x1*=0; x2*=40; Z=40000
We determined in the unit that the dual programme is written as follows :
Minimizing Z=50y1 + 2000y2
with y1 + 25y2 ≥ 450
y1 + 50 y2 ≥ 1000
y1, y2 positive
y1 : dual value associated to the land constraint
y2 : dual value associated to the labour constraint
Write this dual programme and check that the value of the objective function is the same as for the primal programme, and find the relationships between the solutions of the primal programme and that of the dual one.
Solution :
Download the solution model here : dual.zip.