Consider the Wyndor Glass Co. problem described in Sec.

3.1. Suppose that, in addition to considering the introduction of two new products, management now is considering changing the production rate of a certain old product that is still profitable. Refer to Table 3.1. The number of production hours per week used per unit production rate of this old product is 1, 4, and 3 for Plants 1, 2, and 3, respectively. Therefore, if we let  denote the change

(positive or negative) in the production rate of this old product, the right-hand sides of the three functional constraints in Sec. 3.1 become 4  , 12  4, and 18  3, respectively. Thus, choosing a negative value of  would free additional capacity for producing more of the two new products, whereas a positive value would have the opposite effect.

(a) Use a parametric linear programming formulation to determine the effect of different choices of  on the optimal solution for the product mix of the two new products given in the final tableau of Table 4.8. In particular, use the fundamental insight of Sec. 5.3 to obtain expressions for Z and the basic variables x3, x2, and x1 in terms of , assuming that  is sufficiently close to zero that this “final” basic solution still is feasible and thus optimal for the given value of .

(b) Now consider the broader question of the choice of  along with the product mix for the two new products. What is the breakeven unit profit for the old product (in comparison with the two new products) below which its production rate should be decreased (  0) in favor of the new products and above which its production rate should be increased ( 0)?

(c) If the unit profit is above this breakeven point, how much can the old product’s production rate be increased before the final BF solution would become infeasible?

(d) If the unit profit is below this breakeven point, how much can the old product’s production rate be decreased (assuming its previous rate was larger than this decrease) before the final BF solution would become infeasible?