Consider the following linear programming problem P and its dual D:

Primal (P): Dual (D): Maximize Z = 5x₁ + 7x₂ Minimize W = 16y₁ + 19y₂ + 8y₃ subject to x₁ 16 subject to y₁ + 2y₂ + y₃ 5 2x₁ + 3x₂ 19 3y₂ + y₃ 7 x₁ + x₂ 8 and x₁ 0, x₂ 0 and y₁ 0, y₂ 0, y₃ 0

Let x₃, x₄, and x₅ denote the slack variables of the respective functional primal constraints and y₄ and y₅ denote the surplus variables of the respective dual constraints. Answer the following questions:

(a) (5 points) Using the weak duality show that W* 30.

(b) (20 points) The graphically obtained optimal solution of problem P is x₁* = 5 and x₂* = 3.

Complete the primal (augmented) optimal solution.

Find its complementary dual optimal solution using the complementary slackness property.

What are the shadow prices of the resources of problem P?

What are the reduced costs of x₁ and x₂?

What are the values of Z* and W*?

(c) (5 points) If the price of resource #2 is 2.50 per unit, should additional units of that resource be purchased, if available, in order to increase profit? Why?