Solve both the primal and dual problems with the simplex method. Use f to represent the dual objective function.
Minimize subject to

A) Primal: Dual:
B) Primal: Dual:
C) Primal: Dual:
D) Primal: Dual:
E) Primal: Dual:

Question

Solve both the primal and dual problems with the simplex method. Use f to represent the dual objective function. ​
Minimize  subject to
​  ​
A) Primal: Dual: ​ 
B) Primal: Dual: ​ 
C) Primal: Dual: ​ 
D) Primal: Dual: ​ 
E) Primal: Dual:

Quizplus · Accepted Answer

To verify optimality, we can check whether the dual solution satisfies complementary slackness condition. From the optimal primal solution, we have x_1 = 2 and x_3 = 6, and their respective non-zero dual variables are y_1 and y_3. All other dual variables are zero. The complementary slackness condition implies that either y_1 = 0 or (3/4)y_1 + (1/4)y_3 = 0, and either y_3 = 0 or (1/4)y_1 + (3/4)y_3 = 0. Since x_1 > 0 and y_1 > 0, it follows that y_3 = 0. Substituting this in the second complementary slackness condition yields y_1 = 0. Therefore, we have a feasible and optimal dual solution with f = 8, which satisfies complementary slackness. Hence, the optimal primal solution is (x_1, x_3) = (2, 6) with objective value 8, and the optimal dual solution is (0, 5/4, 0) with objective value 8.

Solve Both the Primal and Dual Problems with the Simplex