Consider the following problem.

Maximize Z  8x1  24x2, subject to x1  2x2 10 2x1  x2 10 and x1 0, x2 0.

Suppose that Z represents profit and that it is possible to modify the objective function somewhat by an appropriate shifting of key personnel between the two activities. In particular, suppose that the unit profit of activity 1 can be increased above 8 (to a maximum of 18) at the expense of decreasing the unit profit of activity 2 below 24 by twice the amount. Thus, Z can actually be represented as Z(

)  (8 

)x1  (24  2

)x2, where is also a decision variable such that 0 10.

(a) Solve the original form of this problem graphically. Then extend this graphical procedure to solve the parametric extension of the problem; i.e., find the optimal solution and the optimal value of Z(

) as a function of

, for 0 10.

I

(b) Find an optimal solution for the original form of the problem by the simplex method. Then use parametric linear programming to find an optimal solution and the optimal value of Z(

) as a function of

, for 0 10. Plot Z(

).

(c) Determine the optimal value of

. Then indicate how this optimal value could have been identified directly by solving only two ordinary linear programming problems. (Hint: A convex function achieves its maximum at an endpoint.)