Consider the following parametric linear programming problem, where the parameter  must be nonnegative:

Maximize Z() (5  2)x1  (2  )x2  (3  )x3, subject to 4x1  x2  2x3  5  5

3x1  x2  2x3 10  10

and x1  0, x2  0, x3  0.

Let x4 be the surplus variable for the first functional constraint, and let x5 and x6 be the artificial variables for the respective functional constraints. After we apply the simplex method with the Big M method and with  0, the final simplex tableau is

(a) Use the fundamental insight (Sec. 5.3) to revise this tableau to reflect the inclusion of the parameter  in the original model.

Show the complete tableau needed to apply the feasibility test and the optimality test for any value of . Express the corresponding basic solution (and Z) as a function of .

(b) Determine the range of nonnegative values of  over which this basic solution is feasible.

(c) Determine the range of nonnegative values of  over which this basic solution is both feasible and optimal. Determine the best choice of  over this range.