Linear Programming Problem

Consider the following problem.

Maximize   Z = 2x1 + 7x2 – 3x3,

subject to

x1 + 3x2 + 4x3  ≤ 30

x1 + 4x2 – x3  ≤ 10

and

x1 ≥ 0,     x2 ≥ 0,    x3 ≥ 0.

By letting x4 and x5 be the slack variables for the respective constraints, the simplex method yields the following final set of equations:

 (0)    Z      +   x2 +  x3        + 2x5 = 20

  (1)      –  x2  + 5x3  +  x4  –  x5 = 20

  (2)    x1  +   4x2    – x3       +  x5 = 10.

Now you are to conduct sensitivity analysis by independently investigating the following change in the original model. For the follwoing change, use the sensitivity analysis procedure to revise this set of equations (in tableau form) and convert it to proper form from Gaussian elimination for identifying and evaluating the current basic solution. Then test this solution for feasibility and for optimality. If either test fails, reoptimize to find a new optimal solution.

Suppose that the right-hand sides of the functional constraints are changed to

30 + (6)θ (for constraint 1)

and

10 + (-4)θ   (for constraint 2),

where θ can be assigned any positive or negative values.

Determine the lower and upper bounds of  θ  before this solution would become infeasible.