Sensitivity Analysis: Methods, Types, and Applications

Sensitivity analysis asks: how much can the data of a linear programming problem change before the optimal solution changes? Coefficients, resource limits and costs are estimates, so we need to know how robust the answer is.

This matters because real data is never exact. A manager wants to know: "If the price of steel rises 10%, is my production plan still optimal?" Sensitivity analysis answers such questions without solving the whole problem again.

Key Concepts

• Optimal basis: the set of basic variables in the optimal solution; sensitivity ranges keep it unchanged.
• Range of optimality: the interval a cost coefficient cⱼ can vary while the same corner stays optimal.
• Range of feasibility: the interval a right-hand-side value bᵢ can vary while the same corner stays feasible.
• Shadow price (dual value): the change in the optimum per unit change in a resource, valid only inside the feasibility range.
• 100% rule: for simultaneous changes, sum the fractions of each allowable change; the solution stays optimal if the total ≤ 100%.

Formulas

• Shadow price of resource i: yᵢ = the dual variable value at optimum; new optimum ≈ old optimum + yᵢ·(Δbᵢ), within the feasibility range.
• Feasibility range: values of bᵢ for which B−¹b ≥ 0 keeps the current basis feasible.
• Optimality range: values of cⱼ for which Zⱼ − Cⱼ ≤ 0 (maximisation) keeps the current basis optimal.

Assumptions: only the LP data changes; the structure (constraints, variables) stays the same. Ranges apply to one change at a time, or combined via the 100% rule.

Worked Example

Recall: maximise Z = 3x + 5y, x + 2y ≤ 10, x + y ≤ 6, optimum (2, 4) with Z = 26 and shadow prices u = 2, v = 1.

1. If the first resource increases from 10 to 12 (Δ = +2): new Z ≈ 26 + 2·2 = 30.
2. Verify by re-solving: x + 2y = 12 and x + y = 6 → y = 6, x = 0 → Z = 3·0 + 5·6 = 30. ✓
3. If instead it increases to 20: the prediction 26 + 2·10 = 46 would fail, because beyond the feasibility range the basis changes. Re-solving x + 2y = 20, x + y = 6 gives y = 14, x = −8 — infeasible, so a different corner binds and Z = 5·10 = 50 at (0, 10).

Shadow prices are exact only inside their valid range — here the small change worked perfectly, the large one did not.

Common Misconceptions

• "Shadow prices work for any change." They are valid only within the feasibility range.
• "Sensitivity ranges let me change everything at once." Individual ranges assume one change at a time; use the 100% rule for combinations.
• "If data changes, the old plan is useless." Small changes usually keep the same basis; only the numbers shift.

Real-World Applications

• Procurement teams decide how much extra raw material is worth buying using shadow prices.
• Pricing analysts see how far a product's margin can drop before the product mix changes.
• Capacity planners find which machine is the true bottleneck worth expanding.

Sensitivity analysis turns a single optimal answer into a map of safe operating ranges. It tells decision-makers which numbers their plan depends on, and which can drift without consequence — exactly what management needs from optimisation.