Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per…
2023
Food X contains 6 units of Vitamin D per gram and 7 units of Vitamin E per gram and cost is Rs 12 per gram.Food Y contains 8 units of Vitamin D per gram and 12 units of Vitamin E per gram and cost is Rs 20 per gram. The daily minimum requirements of Vitamin D and E are 100 units and 120 units respectively.
Suppose \(x\) is quantity (in gram) of food X, \(y\) is quantity (in gram) of food Y.
Answering the following question based on the above paragraph given.
The dual of the formulated LPP is :
- A.
Max
\(Z = 100u + 120v\)s.t.
\(6u + 7v \leq 12 \\ 8u + 12v \leq 20 \\ \ \ \ \ \ \ \ \ \ \ \ u,v \geq 0\) - B.
Max
\(Z = 12u+ 20u\)s.t.
\(6u + 7v \leq 100 \\ 8u + 12v \leq 120 \\ \ \ \ \ \ \ \ \ \ \ \ u,v \geq 0\) - C.
Max
\(Z = 100u + 120v\)s.t.
\(6u + 7u \leq 12 \\ 8u + 7v \leq 20 \\ \ \ \ \ \ \ \ \ \ \ \ u,v \ \text{are unrestricted}\) - D.
Max
\(Z = 100u + 120u\)s.t.
\(6u + 7v \geq 12 \\ 8u + 12v \geq 20 \\ \ \ \ \ \ \ \ \ \ \ \ u,v \geq 0\)
Attempted by 7 students.
Show answer & explanation
Correct answer: A
Dual of the LPP:
Start from the primal (minimization) formulation:
Primal (minimize): Minimize cost 12x + 20y
Subject to: 6x + 8y ≥ 100 (Vitamin D requirement) and 7x + 12y ≥ 120 (Vitamin E requirement), with x, y ≥ 0
Form the dual by introducing dual variables u and v for the two constraints (u for Vitamin D, v for Vitamin E). Because the primal constraints are ≥ type and primal is minimization, the dual variables are nonnegative.
Dual objective:
Maximize 100u + 120v
Dual constraints (one for each primal variable):
For x: 6u + 7v ≤ 12 (coefficients from the two constraints compared to cost coefficient of x)
For y: 8u + 12v ≤ 20 (coefficients from the two constraints compared to cost coefficient of y)
Non-negativity:
u ≥ 0, v ≥ 0
Therefore the correct dual is: Maximize 100u + 120v subject to 6u + 7v ≤ 12, 8u + 12v ≤ 20, with u, v ≥ 0.