MGMT20005 Chap.6 LP Sensitivity Analysis
LP Sensitivity Analysis
Week 6 interprets how a solved LP responds to local changes. Variable-cell rows in the Solver report contain final values, objective coefficients, allowable increases and decreases, and reduced costs. Constraint rows contain final left-hand-side values, slack or surplus, shadow prices, RHS values and allowable ranges.
An objective coefficient's range of optimality preserves the current decision-variable solution inside the interval, but the objective value must be recalculated. A constraint RHS range of feasibility preserves the reported shadow price, so objective change equals shadow price times RHS change within the band. Reduced cost concerns a variable currently at zero; shadow price concerns a resource or requirement.
Simultaneous changes, constraint-coefficient changes and values outside the printed ranges require the model to be solved again.
What this chapter covers
- 01
Variable and constraint sections of a Solver report
- 02
Objective-coefficient ranges of optimality
- 03
Reduced cost for zero-valued variables
- 04
Shadow prices for constraint RHS changes
- 05
RHS ranges of feasibility
- 06
One-at-a-time limits and re-solving triggers
Worked example · free
Price a coefficient change and a capacity increase
- 2The x coefficient range is [40 minus 5, 40 plus 10] = [35,50]. Coefficient 46 lies inside, so x=10 and y=20 remain optimal.
- 2Recompute the objective with the new coefficient: 46(10)+30(20)=460+600=1,060.
- 2The packaging RHS range is [80 minus 20, 80 plus 12] = [60,92]. New RHS 89 lies inside.
- 2Objective improvement is 6 times 9 = 54, so the new objective is 1,054. The report does not reveal the new decision-variable values after this RHS change.
Key terms
- Sensitivity analysis
- Post-optimality analysis of how local parameter changes affect a solved model.
- Range of optimality
- The objective-coefficient interval over which the current decision-variable solution remains optimal.
- Reduced cost
- The coefficient improvement required before a zero-valued variable can enter the optimal solution.
- Shadow price
- The change in optimal objective value from a one-unit RHS increase within the valid range.
- Range of feasibility
- The RHS interval over which the reported shadow price remains valid.
- Allowable increase
- The greatest upward change certified by the current Solver report for the relevant coefficient or RHS.
LP Sensitivity Analysis FAQ
What stays unchanged inside a range of optimality?
The current decision-variable values remain optimal when one objective coefficient changes inside its interval. The objective value does not stay fixed; substitute the new coefficient into the objective and recalculate. At a boundary there may be an alternative optimum, but the current solution remains optimal.
What stays unchanged inside a range of feasibility?
The reported shadow price remains valid for the one-at-a-time RHS change. Multiply it by the RHS change to calculate the local objective change. The decision-variable values can change, and the sensitivity report does not by itself identify those new quantities.
How are reduced cost and shadow price different?
Reduced cost is attached to a variable at zero and measures how much its objective coefficient must improve before it enters. Shadow price is attached to a constraint RHS and measures local objective benefit from one more RHS unit. One prices entry of an activity; the other values relaxation of a restriction.
When must the model be solved again?
Re-solve when a proposed objective coefficient or RHS falls outside its allowable range, when several parameters change together without an applicable combined-change rule, when a constraint coefficient changes, or when model structure changes. The printed report is local evidence and cannot certify a different basis beyond its stated limits.
Exam move
Begin each sensitivity answer with a classification. If the proposed change affects an objective coefficient, calculate the range of optimality and test the new coefficient. Inside it, keep the decision values and recalculate the objective; outside it, re-solve. If the change affects a constraint RHS, calculate the range of feasibility.
Inside it, multiply the shadow price by the RHS change and add that effect to the objective; outside it, re-solve. If a question asks about a product not currently used, inspect reduced cost and move the coefficient in the favourable direction by the required amount. Keep units beside every shadow price and preserve Solver signs.
Finally say whether the result is a certified local what-if answer or a scenario requiring a fresh optimisation run. Turn each what-if prompt into a three-column note: changed input, applicable range and certified conclusion. Do not calculate until the input has been classified as an objective coefficient, a constraint RHS or a new-variable condition. After calculating, add one sentence stating what remains unchanged.
This makes it obvious when the report supports a local answer and when a fresh solve is required.
Working through LP Sensitivity Analysis in MGMT20005? Sia is AskSia’s AI Management tutor — ask any MGMT20005 LP Sensitivity Analysis question and get a clear, step-by-step explanation grounded in how MGMT20005 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
