Sensitivity Analysis and Shadow Prices

Definition

Sensitivity analysis asks how much a linear program's coefficients — profit per unit, or the amount of a resource available — can change before the optimal solution itself would change. Two numbers do the work: the range of optimality for each objective coefficient (how far it can move while the current decision stays best) and the dual value, "the change in the objective function value per unit increase in a constraint's right-hand side" — what economists elsewhere call a shadow price, the marginal worth of one more unit of a scarce resource.

In the Book

Chapter 3 reopens the Par, Inc. golf-bag problem from Chapter 2 to show why a single optimal answer isn't the end of the analysis. It asks what happens if the profit contribution for the standard bag falls from $10 to $8.50, and shows that as long as an objective coefficient stays within its computed range (for the deluxe bag, the book works through whether that range is a comfortable $6.67–$14.29 or a fragile $8.90–$9.25), the same 540-standard/252- deluxe production plan remains optimal — so management doesn't need to resolve the whole model every time an estimate shifts. It then turns to the right-hand sides: because the optimal solution uses all 630 available hours of cutting-and-dyeing time, the book computes the dual value of that constraint ($4.375) — the added profit from one more hour — and cautions that this per-unit value predicts the effect of resource changes only within a computed range, and only for constraints that are actually binding at the optimum.

Why It Matters

A single optimal solution invites false confidence in inputs that were only estimates. Sensitivity analysis lets a decision-maker separate the numbers worth double-checking (those near the edge of their range of optimality) from the ones that don't matter much, and it converts a scarce resource from an abstract constraint into a priced object — telling you exactly how much it would be worth to acquire more of it, which is the basis for any real build-versus-buy-more-capacity decision.