Consider the following problem:

Maximize Z = 2 x1 - x2 + x3

Subject to

3 x1 2 x2 + 2 x3 15 (resource 1)

x1 + x2 + x3 3 (resource 2)

x1 - x2 + x3 4 (resource 3)

and

x1 0, x2 0, x3 0

Change the problem as follows, changes are in bold:

Maximize Z = 2 x1 - x2 + 4 x3

Subject to

3 x1 2 x2 + 3 x3 15 (resource 1)

x1 + x2 + 2 x3 3 (resource 2)

x1 - x2 + x3 4 (resource 3)

and

x1 0, x2 0, x3 0

Use duality theory directly to determine whether the original optimal solution is still optimal. Hint: first solve the dual problem and check the constraint that corresponds to the coefficients of x3 in the primal problem.