ETF2480 Chap.4 Duality and Sensitivity Analysis
Duality and Sensitivity Analysis
Duality pairs an LP with another optimisation problem that exchanges the roles of rows and columns. Primal constraints create dual variables, primal variables create dual constraints, and the coefficient matrix is transposed. Feasible dual solutions bound a primal maximisation value, while strong duality connects optimal objective values.
Complementary slackness links unused resources with zero marginal value and active decisions with tight dual rows. Sensitivity analysis then asks how the optimum responds to changes in right-hand sides or objective coefficients. A shadow price is a local slope within a validity range, stated with units and conditions, not a permanent market value.
What this chapter covers
- 01
Primal and dual correspondence
- 02
Weak and strong duality
- 03
Resource bounds and dual values
- 04
Complementary slackness
- 05
Shadow-price units
- 06
Allowable ranges and basis changes
Worked example · free
Value a capacity increase inside its range
- +1Eight additional hours lie inside the allowable increase of twelve.
- +2The local gross objective increase is 8 times $9 = $72.
- +2Net gain after the $48 purchase cost is $24 if cost is not already included in the LP objective.
- +1Recommend the purchase conditionally and re-solve for larger changes or altered data.
Key terms
- Dual problem
- The optimisation problem paired with the primal through transposed coefficients, exchanged data and sign rules.
- Weak duality
- The rule that feasible primal and dual objective values bound one another in the appropriate direction.
- Strong duality
- The equality of optimal primal and dual objective values when the linear programs have suitable optimal solutions.
- Complementary slackness
- Conditions linking positive variables to tight paired constraints and slack constraints to zero paired variables.
- Shadow price
- The local change in optimal objective per unit change in a right-hand side within a valid interval.
Duality and Sensitivity Analysis FAQ
What does the dual represent?
It assigns values to primal constraints and creates a bound on the primal objective. In a resource model, those values can describe local marginal worth, but their exact interpretation depends on objective units, row units and the current optimal basis.
How does weak duality check an answer?
For the standard max-primal and min-dual pairing, a feasible primal value cannot exceed a feasible dual value. If it does, at least one stated solution is infeasible, one objective calculation is wrong, or the dual was formed with incorrect signs.
When can I use a shadow price?
Use it for a change to the named right-hand side that stays inside the reported allowable range, while holding the remaining model data fixed. Beyond that interval, binding constraints and marginal values can change, so solve the modified model again.
What must a sensitivity recommendation state?
Name the changed resource, its units, the proposed right-hand-side movement and the supported interval. Separate the gross objective effect from acquisition cost, then re-solve whenever the change exceeds the reported range.
Exam move
Build the dual mechanically before interpreting it. Create a correspondence table showing each primal row, its dual variable and sign; then each primal column, its dual constraint and direction. Verify dimensions and use a feasible pair to test weak duality. Next, practise complementary slackness by identifying slack primal rows and positive primal variables.
For sensitivity questions, write the unit of the shadow price, check the allowable interval, multiply only the valid change, and compare the marginal value with the true cost of adjustment. Separate row sensitivity from reduced-cost information about columns. End every answer with a boundary sentence explaining when the current estimate stops being justified.
This is the difference between reading a solver report and converting it into a responsible management decision. Maintain a four-column dual construction sheet: primal feature, dual feature, sign or direction rule, and economic unit. Check the transpose by counting primal rows against dual variables and primal columns against dual constraints before doing arithmetic.
Create one feasible primal-dual pair and calculate both objectives to rehearse the weak-duality bound. Then use an optimal pair to mark which slack rows require zero prices and which positive activities require tight dual restrictions. In sensitivity drills, distinguish a proposed right-hand-side change from an objective-coefficient change before consulting the report.
Calculate the largest justified marginal estimate when an offer crosses an allowable endpoint, and label the remainder as requiring re-optimisation. Compare marginal benefit with an actual quoted acquisition cost, not with the resource's accounting label. Finally, change two inputs simultaneously and explain why separate one-at-a-time ranges may no longer certify the combined prediction.
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