Consider the problem

Maximize z = x1 + x2

Subject to

2x1 + x2 ≤ 6

X1 + 2x2 ≤ 6

X1 + x2 ≥ 0

(a) Show that the optimal basic solution includes both x₁ and x₂ and that the feasibility ranges for the two constraints, considered one at a time, are – 3 ≤ D₁ ≤ 6 and – 3 ≤ D₂ ≤ 6.

(b) Suppose that the two resources are increased simultaneously by ∆ > 0 each. First, show that the basic solution remains feasible for all ∆ > 0. Next, show that the 100% rule will confirm feasibility only if the increase is in the range 0 < ∆ ≤ 3 units. Otherwise, the rule fails for 3 < ∆ ≤ 6 and does not apply for ∆ > 6.