Consider the following problem.

Maximize Z 10x1  4x2, subject to 3x1  x2 30 2x1  x2 25 and x1  0, x2  0.

Let x3 and x4 denote the slack variables for the respective functional constraints. After we apply the simplex method, the final simplex tableau is Now suppose that both of the following changes are made simultaneously in the original model:

1. The first constraint is changed to 4x1  x2 40.

2. Parametric programming is introduced to change the objective function to the alternative choices of Z() (10  2)x1  (4  )x2, where any nonnegative value of  can be chosen.

(a) Construct the resulting revised final tableau (as a function of

), and then convert this tableau to proper form from Gaussian elimination. Use this tableau to identify the new optimal solution that applies for either  0 or sufficiently small values of .

(b) What is the upper bound on  before this optimal solution would become nonoptimal?

(c) Over the range of  from zero to this upper bound, which choice of  gives the largest value of the objective function?