ETF5248 Chap.3 Duality and Sensitivity Analysis
Duality and Sensitivity Analysis
Duality gives each primal constraint an economic price and each primal variable a corresponding dual restriction.
Weak duality provides a bound, strong duality equates the optimal primal and dual values under standard feasibility conditions, and complementary slackness links positive activity with tight opposing rows.
Sensitivity analysis uses that structure to ask what a resource or coefficient change is worth without blindly resolving every nearby case.
A shadow price is local, not permanent, and an allowable range protects the basis rather than promising that all decision values remain unchanged. This chapter turns a solver report into evidence, separates slack from value, and frames recommendations with their valid ranges and units.
The dual is most useful when its numbers retain their units.
If a primal constraint measures labour hours and the objective measures dollars, the paired dual variable is measured in dollars per labour hour. Multiplying that price by the available hours contributes dollars to the dual objective.
Complementary slackness then supplies a diagnostic pattern: unused capacity normally pairs with a zero dual value, while positive primal activity makes its paired dual restriction tight, subject to degeneracy. Sensitivity output answers one-change questions around the current optimum, not arbitrary redesigns. An allowable increase or decrease states where the current marginal story remains valid under the report's conditions.
Outside that interval, a different constraint may bind and the model should be resolved. A strong recommendation therefore names the resource, quotes the marginal benefit with its unit, limits the claim to the stated range, and compares that benefit with the cost of obtaining the resource.
What this chapter covers
- 01
Primal rows and dual variables
- 02
Primal variables and dual constraints
- 03
Weak and strong duality
- 04
Complementary slackness
- 05
Shadow prices and resource values
- 06
Allowable ranges and local interpretation
Worked example · free
Use a shadow price inside its valid range
- 1Eight hours are within the allowable increase of 15 hours, so the reported shadow price is valid locally.
- 1Estimated improvement is 8 times $24 = $192.
- 1The estimated new objective is $9,600 + $192 = $9,792.
- 1Twenty hours exceed the allowable increase, so the active basis may change and the model must be resolved for a reliable value.
Key terms
- Primal
- The original optimisation model whose decisions and constraints are being analysed.
- Dual
- A related model that assigns variables to primal constraints and bounds the primal objective.
- Shadow price
- The local objective change produced by one unit of a right-hand-side resource.
- Complementary slackness
- The relationship linking slack in one problem with zero value in the paired variable of the other.
- Allowable range
- An interval over which a sensitivity interpretation retains its stated basis conditions.
Duality and Sensitivity Analysis FAQ
Does a binding constraint always have a positive shadow price?
No. Binding means the row has zero slack at the current solution, while the shadow price measures the local value of relaxing it. Degeneracy or redundancy can make a binding row worth zero. Report both facts rather than treating tightness and economic value as synonyms.
What does strong duality tell me?
When the relevant linear programs have finite optima, the optimal primal and dual objective values are equal. This supplies a certificate: a feasible primal solution and feasible dual solution with the same objective value prove that neither can be improved.
What stays constant within an allowable range?
The current basis remains optimal under the stated one-at-a-time change, which preserves the associated marginal interpretation. Variable values and the objective can still move. The report should not be read as saying the entire solution is numerically unchanged throughout the interval.
How should sensitivity become a recommendation?
Name the resource or coefficient, state the marginal value with units, specify the valid change range, compare the benefit with the purchase or policy cost, and mention what must be resolved beyond the range. This turns software output into a conditional business case.
Exam move
Write the primal and dual side by side, using arrows from every primal row to its dual variable and from every primal variable to its dual constraint. Check sign and direction conventions systematically rather than memorising an isolated table. For each solver sensitivity line, translate the output into a complete sentence with the resource name, currency per unit and allowable change.
Practise rejecting overclaims, especially extending one shadow price beyond its range or assuming slack alone proves a resource has no strategic importance in every scenario.
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