Consider the following parametric linear programming problem.

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

x1  2x2  3x3 6  2

and x1  0, x2  0, x3  0, where  can be assigned any positive or negative values. Let x4 and x5 be the slack variables for the respective functional constraints.

After we apply the simplex method with  0, the final simplex tableau is

(a) Express the BF solution (and Z) given in this tableau as a function of . Determine the lower and upper bounds on  before this optimal solution would become infeasible. Then determine the best choice of  between these bounds.

(b) Given that  is between the bounds found in part (a), determine the allowable range to stay optimal for c1 (the coefficient of x1 in the objective function).