The grade of a fertilizer refers to the percentage of the three main nutrients: nitrogen, phosphorus, and potassium. It appears as three large numbers on the fertilizer label. For example, a fertilizer with a grade of 13-10-27 contains 13% nitrogen, 10% phosphorus and 27% potassium. For wheat cultivation, a farmer needs to apply 50 kg / ha of nitrogen, 20 kg / ha of phosphorus and between 40 and 60 kg / ha of potassium. For this, it has two fertilizers: FertiSmart: 10-20-30, at a cost of $ 1500 / kg NutraFarm: 40-30-20 at a cost of $ 2000 / kg. The farmer needs to determine the fertilizer mix that satisfies the nutritional requirements for growing 100 ha (hectares) of wheat at the lowest possible cost. For this, the following optimization model has been proposed: Min Z = 800 X1 + 2000 X2 s.a. R1) 0.2X1 + 0.4 X2 ≥ 50 R2) 0.1X1 + 0.3 X2 ≥ 20 R3) 0.3X1 + 0.2 X2 ≥ 40 R4) 0.3X1 + 0.2 X2 ≤ 60 X1, X2 ≥ 0 -Formulate the associated dual problem. -Determine the optimal solution of the dual problem using the complementary slack theorem. -Interpret the optimal solution of the dual in the context of the problem. -Write the optimal solution of the primal and dual in dictionary format. Comment on your results.
The grade of a fertilizer refers to the percentage of the three main nutrients: nitrogen, phosphorus, and potassium. It appears as three large numbers on the fertilizer label. For example, a fertilizer with a grade of 13-10-27 contains 13% nitrogen, 10% phosphorus and 27% potassium.
For wheat cultivation, a farmer needs to apply 50 kg / ha of nitrogen, 20 kg / ha of phosphorus and between 40 and 60 kg / ha of potassium. For this, it has two fertilizers:
FertiSmart: 10-20-30, at a cost of $ 1500 / kg NutraFarm: 40-30-20 at a cost of $ 2000 / kg.
The farmer needs to determine the fertilizer mix that satisfies the nutritional requirements for growing 100 ha (hectares) of wheat at the lowest possible cost. For this, the following optimization model has been proposed:
Min Z = 800 X1 + 2000 X2
s.a.
R1) 0.2X1 + 0.4 X2 ≥ 50
R2) 0.1X1 + 0.3 X2 ≥ 20
R3) 0.3X1 + 0.2 X2 ≥ 40
R4) 0.3X1 + 0.2 X2 ≤ 60
X1, X2 ≥ 0
-Formulate the associated dual problem.
-Determine the optimal solution of the dual problem using the complementary slack theorem.
-Interpret the optimal solution of the dual in the context of the problem.
-Write the optimal solution of the primal and dual in dictionary format. Comment on your results.
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