The 100% optimality Rule. A rule similar to the 100% feasibility rule outlined in problem 12, set 3.6c, can also be developed for testing the effect of simultaneously changing all cj to cj + dj , j = 1, 2,..., n, on the optimality of the current solution. Suppose that uj ≤ dj ≤ vj is the optimality range obtained as a result of changing each cj to cj + dj one at a time, using the procedure in Section 3.6.3. In this case, uj ≤ (vj ≥ 0), because it represents the maximum allowable decrease (increase) in cj that will keep the current solution optimal. For the cases where uj ≤ dj ≤ vj, define rj equal to dj/vj if dj is positive and dj is negative. By definition, 0 ≤ rj ≤ 1. The 100% rule says that a sufficient (but not necessary) condition for the current solution to remain optimal is that r1 + r2 + ... + rn ≤ 1. If the condition is not satisfied, the current solution may or not remain optimal. The rule does not apply if dj falls outside the specified ranges.
Demonstrate that the 100% Optimality rule is too weak to be consistently reliable decision-making tool by applying it to the following cases:
(a) Parts (ii) and (iii) of Problem 1.
(b) Parts (b) of Problem 7.