The shadow prices can be used to determine the effect of a new variable (like a new product in a production linear program). Suppose that, in formulation (1.1), a new variable w has coefficient 4 in the first constraint and 3 in the second. What objective coefficient must it have to be considered for adding to the basis?
If we look at making w positive, then this is equivalent to decreasing the right hand side of the first constraint by 4w and the right hand side of the second constraint by 3w in the original formulation. We obtain the same effect by making and . The overall effect of this is to decrease the objective by . The objective value must be sufficient to offset this, so the objective coefficient must be more than 10 (exactly 10 would lead to an alternative optimal solution with no change in objective).
(a) From the final tableau, we read that is basic and are nonbasic. So 100 units of should be produced and none of , and . The resuting profit is $ 600 and that is the maximum possible, given the constraints.
(b) The reduced cost for is 2 (found in Row 0 of the final tableau). Thus, the effect on profit of producing units of is . If 20 units of have been produced by mistake, then the profit will be lower than the maximum stated in (a).
(c) Let be the profit margin on . The reduced cost remains nonnegative in the final tableau if . That is . Therefore, as long as the profit margin on is less than 4.5, the optimal basis remains unchanged.
(d) Let be the profit margin on . Since is basic, we need to restore a correct basis. This is done by adding times Row 1 to Row 0. This effects the reduced costs of the nonbasic variables, namely , , and . All these reduced costs must be nonnegative. This implies:
Combining all these inequalities, we get . So, as long as the profit margin on is 6 or greater, the optimal basis remains unchanged.
(e) The marginal value of increasing capacity in Workshop 1 is .
(f) Let be the capacity of Workshop 1. The resulting RHS in the final tableau will be:
in Row 1, and
in Row 2.
The optimal basis remains unchanged as long as these two quantities are nonnegative. Namely, . So, the optimal basis remains unchanged as long as the capacity of Workshop 1 is in the range 0 to 800.
(g) The effect on the optimum profit of producing units of would be . If the profit margin on is sufficient to offset this, then should be produced. That is, we should produce if its profit margin is at least 3.