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When the Numbers Move
Quantitative Analysis · Chapter 10: Sensitivity AnalysisSection 10.2 · lesson 46 of 5511 min · free with a free account
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In this lesson
How changing coefficients, constraints and parameters affect the optimal solution.
- Compute and interpret objective coefficient ranges
- Read RHS ranges for binding and non-binding constraints
- Screen a new variable or a new constraint without re-solving
- Turn a sensitivity report into managerial recommendations
Transcript
Everything said in the video, section by section (1,659 words). Read along, or search it.
Sensitivity Analysis · Section 10.2
When the Numbers Move
You have solved the linear programme, and the optimal plan is on the table. Then someone asks a question the solution cannot answer on its own. What if the profit on our best product drops by two dollars? What if the supplier delivers less protein than promised? Sensitivity analysis is the discipline of answering those questions properly. We will work through four things in this section. How far a profit or cost coefficient can drift before the plan changes. The range over which a shadow price stays valid. Quick screening tests for a new product, or a new restriction. And finally, how to turn the whole report into a memo a director can act on.
Why It Matters
Three Sets of Numbers, None of Them Certain
Every linear programme rests on three sets of numbers. First, the objective coefficients: the profits or costs attached to each variable. Second, the technological coefficients: how much of each resource a unit of output consumes. Third, the right-hand sides: how much of each resource you actually have. Textbook problems hand you all three as certainties, and real firms do not. Prices fluctuate, supplier promises change, and demand forecasts drift. So the useful question is not what is optimal today. It is how far each number can move before today's answer stops being optimal. That question earns its keep in three ways. It shows which inputs deserve the most careful estimation. It shows which constraints are the real bottlenecks, worth spending money on. And it tells you when a change is big enough that you must re-solve.
Coefficient Ranging
How Far a Profit Figure Can Drift
Start with the profit and cost figures in the objective function. For each variable, the report gives a lower and an upper bound on its coefficient. Inside that interval, the current optimal corner stays optimal. The objective value moves, because you earn a different margin per unit. But the production quantities do not move at all. That is the useful part. The plan is stable even when the profit is not. Push the coefficient past either bound and a different corner becomes best. At that point the report is spent, and you must re-solve the model. One condition is easy to skim past, and it matters enormously. The range assumes that coefficient moves on its own. Geometrically, the objective is a family of parallel lines. Their slope is minus c one over c two, so changing a coefficient rotates them. The allowable range is simply a range of slopes.
Reduced Cost
The Variables Sitting at Zero
Some variables sit at zero in the optimal solution. The model has looked at them and declined to make them. These are the non-basic variables. For each one, the report answers a single question. How much better would its coefficient have to be, before the model would use it? That number is the reduced cost. In a maximisation problem, the profit must rise by that amount. In a minimisation problem, the cost must fall by it instead. Now compare the reduced cost with what you could genuinely achieve. If the improvement you can win is smaller, the variable stays at zero. So a large reduced cost is a quiet instruction to stop rescuing that product.
RHS Ranging
What a Shadow Price Is Worth, and For How Long
Now move to the right-hand side, the amount of each resource you hold. Loosening or tightening a resource shifts its constraint line in parallel. It slides the line, rather than rotating it. A binding constraint is one used right up to capacity, with zero slack. Its shadow price is the marginal value of one extra unit of that resource. The report gives the interval of right-hand-side values where the shadow price holds. Step outside it, and some other constraint becomes the new bottleneck. The shadow price then changes, and it can fall all the way to zero. This is the mistake managers make most often with a sensitivity report. They multiply a shadow price by a big purchase, and budget for a gain nobody promised.
Slack And Kinks
Non-Binding Constraints Are Priced at Zero
A non-binding constraint has slack at the optimum. You have more of that resource than the plan actually uses. Its shadow price is zero, and two consequences follow. Buying more of it adds nothing to profit, however much you buy. And cutting it back costs nothing, right up to the point where the slack runs out. Now picture the optimal objective value plotted against a single right-hand side. Inside the allowable range you get a straight line, and its slope is the shadow price. At each end of the range the line kinks to a new slope. That kink is the entire reason ranges get reported at all. A shadow price is a local price, not a permanent one.
Two What-Ifs
Adding a Variable, Adding a Constraint
Two questions arrive constantly once a model has been solved. Should we add a new product? And what happens if a new rule is imposed? Both have a screening test that avoids re-solving anything. For a new variable, compute its reduced cost from the current shadow prices. Take its objective coefficient, then subtract the resources it would consume. Price each of those resources at the shadow price the model already gave you. For a maximisation problem, include the new product only if that figure is positive. For a minimisation problem, it enters when the figure is negative. If neither condition is met, nothing changes, and no re-solve is needed. For a new constraint, simply ask whether the current optimum already satisfies it. If it does, that same solution is still feasible and still optimal. If it does not, your solution is infeasible and the model must be re-solved. And remember, adding a constraint can never improve a maximisation objective.
Worked Example
A 3-Product Blending Plan
A specialty-foods firm blends three products, A, B and C. They share three resources: protein, oils and processing hours. The solved plan makes forty units of A, sixty of B, and twenty of C. Protein and processing hours are binding. Oils are not. Oils finish with twelve litres of slack. The source report prints an optimal profit of three thousand eight hundred and forty dollars. Check that figure before you trust anything built on top of it. Forty units at twenty four dollars is nine hundred and sixty. Sixty at thirty two dollars is one thousand nine hundred and twenty. Twenty at eighteen dollars is three hundred and sixty. Those add to three thousand two hundred and forty, not three thousand eight hundred and forty. The printed total does not reconcile, so we use the recomputed one.
Coefficient Ranges
Where the Profit Figures Can Go
Here are the objective coefficient ranges for the three products. Product A earns twenty four dollars a unit. It can fall by four or rise by six, giving a range of twenty to thirty. Product B earns thirty two, with a range of twenty seven to forty. Product C earns eighteen, and its range starts at fifteen with no upper limit. Read the allowable decrease column first, because that is where the risk lives. B can lose only five dollars a unit before the mix changes. On a margin of thirty two, that is a cushion under sixteen per cent. And B is the largest volume in the plan, at sixty units. Product C is the opposite kind of news. Make it more profitable by any amount, and the current mix stays optimal. Only a fall below fifteen would force a re-plan there.
Shadow Prices
What Each Resource Is Actually Worth
Protein has four hundred kilograms, at a shadow price of three dollars twenty. That price holds between three hundred and sixty, and four hundred and sixty. So the sixty kilograms of headroom is worth one hundred and ninety two dollars. Processing has one hundred and twenty hours, priced at eight dollars fifty an hour. Its range runs from one hundred and five, up to one hundred and thirty five. Fifteen extra hours, at eight fifty each, is one hundred and twenty seven fifty. Oils sit at two hundred litres with a shadow price of zero, so more oil buys nothing. Now test an offer of twenty extra processing hours, at five dollars an hour. Only the first fifteen are covered by that eight fifty shadow price. Each of those earns a margin of three dollars fifty, so you gain fifty two fifty. Beyond one hundred and thirty five hours, the shadow price is simply unknown. So take the fifteen, and re-solve before committing to the rest.
Managerial Memo
Three Findings a Director Can Act On
A sensitivity report is not a deliverable. A memo is. First, processing hours are the most valuable bottleneck in the model. Each extra hour adds eight dollars fifty, the highest shadow price we have. That premium holds for any increment up to fifteen hours. So an extra shift, or outsourcing peripheral work, converts directly into profit. The recommendation is a quick cost-benefit study on extending the second shift. Second, oils are not the constraint anyone thought they were. Zero shadow price, twelve litres of slack, and room to fall to one hundred and eighty eight. Money tied up in oil inventory would work harder in processing capacity. Third, product B's margin is the most strategically exposed number in the plan. A five dollar fall is enough to change the mix, and B carries the largest volume. So tighten B's cost forecasting to a monthly cycle, and lock in supplier pricing.
Chapter Summary
What to Carry Forward
Objective coefficient ranging tells you how far a profit or cost can drift. Inside the range the corner holds. Outside it, the basis switches and you re-solve. For variables at zero, the reduced cost is the improvement needed to bring them in. Right-hand-side ranging tells you how long a shadow price stays valid. Keep the two straight in your head. Shadow prices belong to constraints. Reduced costs belong to variables. Non-binding constraints price at zero, so loosening them buys you nothing. To screen a new variable, compute its reduced cost from the current shadow prices. To screen a new constraint, test whether today's solution already satisfies it. One last check. Suppose product A's profit falls to nineteen dollars. That is a drop of five, and the allowable decrease was only four. One dollar past the edge, so the plan is no longer guaranteed. Re-solve it.
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