Testing the Boundaries, Explained with Examples

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Testing the Boundaries

Quantitative Analysis · Chapter 10: Sensitivity AnalysisSection 10.3 · lesson 47 of 5511 min · free with a free account

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In this lesson

What sensitivity analysis reveals about LP optimal solutions.

  • Tell a robust optimal solution from a fragile one
  • Trace an optimum with parametric programming
  • Read a shadow price and construct the dual
  • Price the cost of overriding the model

Transcript

Everything said in the video, section by section (1,658 words). Read along, or search it.

Sensitivity Analysis & Advanced LP · Section 10.3

Testing the Boundaries

A linear program always ends with a confident looking answer. Produce this much, of these products, for exactly this much profit. But that answer rests entirely on the numbers you fed into it. Profit forecasts, capacity estimates, engineering rates. Every one of them is an estimate. Sensitivity analysis is how you find out whether the answer survives those estimates. In this section we separate robust solutions from fragile ones. We trace a whole optimum path using parametric programming. We read a shadow price, build the dual problem, and apply complementary slackness. And we put an exact price on overruling the model for strategic reasons.

From Numbers To Decisions

The Optimum Is Only as Good as Its Data

An optimal solution is only as trustworthy as the data that produced it. Profit coefficients are forecasts. Resource availabilities are projections. Consumption rates are engineering estimates. Every single parameter in the model carries some uncertainty. Sensitivity analysis studies how the optimum responds when those parameters move. Done well, it turns a one shot calculation into a decision support tool. It answers three questions the optimal solution alone simply cannot. First. How confident should I be in this recommendation? Second. Which resource, if I could acquire more of it, would be most valuable? Third. What does it cost to override the model, and is that worth paying? Notice that all three are managerial questions, not arithmetic ones.

Robustness Spectrum

Robust Solutions and Fragile Ones

The quantity that decides all this is called the range of optimality. It is the interval a profit coefficient can move over without changing the optimal basis. And the basis is simply the set of products the model chose to make. A robust solution has wide ranges and stable shadow prices. Unit profit can drop ten per cent, or rise fifteen per cent, and the decision holds. Implement a robust solution with confidence. A fragile solution is the exact opposite. A two per cent shift in one coefficient can overturn the entire product mix. Either invest in better data, or pre-specify a plan for each alternative basis. The worst response is to treat a fragile solution as though it were robust. A rule of thumb. Ranges narrower than plus or minus ten per cent mean fragile. Ranges wider than plus or minus twenty five per cent are safe to act on.

A Continuous View

Parametric Programming Traces the Whole Path

A standard sensitivity report hands you one interval around the current value. Parametric programming is greedier. It traces the whole trajectory. You choose one parameter, a profit coefficient or a right hand side. Everything else is held fixed while that one moves. Then you start from the current basis and re-solve as the parameter changes. At each value you record the optimal objective and the optimal basis. Wherever the basis changes, you have found a breakpoint. Between two consecutive breakpoints the optimum is a straight line. Join those linear pieces and you have the parametric profile. A sensitivity report tells you whether the basis will change. The parametric profile tells you where. That difference is the whole point.

Reading The Profile

Tracking the Profit Coefficient from 0 to 80

Here is that idea running on a real feasible region. Three vertices compete for the optimum. Call them B, C and D. Vertex B makes twenty units of the first product and none of the second. Vertex C makes sixteen and eight, and vertex D makes none and twenty four. Now let the profit on the first product climb from zero to eighty rupees. Below thirty, the optimum sits at D, making only the second product. Between thirty and sixty it sits at C, the current plan, making both. Above sixty it jumps to B, and the second product is dropped completely. At sixty, C and B must earn the same, which forces the second profit to thirty. So vertex D earns twenty four times thirty, or seven hundred and twenty rupees. Now the payoff. Suppose a price change could push the coefficient to fifty. Fifty sits inside thirty to sixty, so nothing changes at all. Sit still. But if it could fall to twenty five, the basis flips. Prepare the other plan now.

Shadow Prices

The Internal Value of a Resource

The shadow price of a constraint has a very precise definition. It is the change in the optimal objective, when you add one unit of that resource. Everything else must be held fixed. That condition is not decorative. Algebraically it is a partial derivative of the optimum with respect to that resource. Economically it is the firm's own internal value for one more unit. Notice what it is not. It is not the market price of the resource. It is what the resource is worth inside this particular production plan. And it only holds across the range of feasibility. Push the right hand side far enough and the basis changes, and the price falls. A shadow price of zero is informative too. That resource is already in surplus.

Buy More Capacity?

Shadow Price Against Market Price

This makes the buy more capacity decision almost mechanical. If the market price sits below the shadow price, buy. Each unit acquired earns you exactly the difference between the two. If the market price sits above the shadow price, do not buy. The resource is worth less to you than the supplier is charging for it. Take a bakery. Its shadow price on flour is twelve rupees a kilogram. The wholesale market is asking nine rupees a kilogram. So every extra kilogram bought generates three rupees of incremental profit. Keep buying until the shadow price falls, or until the market price rises. Now run it backwards. A trucking firm values diesel at eighteen rupees a litre. The pump charges twenty two. Every litre bought destroys four rupees of value. The answer is no, and the model has told you precisely why.

The Dual Problem

Resource Valuation as Its Own LP

Every linear program has a partner program called its dual. The primal asks how to produce. The dual asks how to value resources. One condition first. The dual we build assumes the primal is in standard maximisation form. Each primal constraint, which is a resource limit, becomes a dual variable. That variable is the value the model places on that resource. Each primal variable, a production quantity, becomes a dual constraint. That constraint says the resources one unit consumes must cover its contribution. The two problems are joined by a result called strong duality. At optimality the two objective values are exactly equal. And the optimal dual variables are precisely the shadow prices of the primal. Picture an investor offering to buy your entire stock of resources outright. The dual finds the smallest payment that would make you stop producing. Strong duality says that payment equals the profit you gave up. Nothing is lost.

Worked Example

Writing a Dual, and Checking It

Let us actually build one, and then check it. The primal maximises five times the first variable plus four times the second. Two resource constraints limit it, and both variables must be non negative. The dual turns that inside out. Two constraints become two variables. The right hand sides, eight and twelve, become the dual objective coefficients. The objective coefficients, five and four, become the dual right hand sides. And every inequality reverses direction, because we are now minimising. Solve the primal. You get two of the first product and three of the second. That gives an optimal objective value of twenty two. Solve the dual. You get one half, and one and a half. Multiply those out and the dual objective is also twenty two. Read the duals as prices. The second resource is worth three times the first.

Complementary Slackness

The Diagnostic Link

Complementary slackness is the algebraic glue between primal and dual. It states two facts, and they are two faces of the same coin. First. If a constraint has positive slack, its dual variable must be zero. The resource is not fully used, so one more unit of it is worth nothing. Second. If a dual variable is positive, its constraint must be binding. A resource with real value is always a resource you have run out of. Here is that diagnostic on a real solution. Machine hours show zero slack and a shadow price of twenty rupees. Labour hours show four hours of slack and a shadow price of zero. That is perfectly consistent, and it names the bottleneck immediately. Machines are the constraint. Expanding labour would not add a single rupee. And that stays true however expensive the labour was to buy.

Case Study

The 'All Four Products' Debate

Now the case study that makes all of this worth learning. A consumer goods firm makes four products from shared machine time and labour. The model recommends only the first two products, for eight hundred and eighty rupees a day. The other two are simply unprofitable in the current mix. Marketing objects, and their argument is not a silly one. Shelf presence, brand breadth, retailer relationships. They want all four products. The operations manager asks the one question that can settle this. How much profit will that strategic preference actually cost? Because the excluded products sit outside the basis, their reduced costs answer it. The third product carries a reduced cost of thirty rupees per unit. The fourth carries forty five. Force a unit in, and the objective falls by that much.

Pricing The Argument

What Shelf Presence Actually Costs

Do nothing, and the firm earns eight hundred and eighty rupees a day. Force in one unit of the third product, and profit falls to eight hundred and fifty. Force in one unit of each excluded product, and it falls to eight hundred and five. That is a penalty of seventy five rupees a day. Now the disagreement is answerable. Is shelf presence worth seventy five rupees a day? If it is not, decline it politely, with evidence rather than opinion. One warning before you extrapolate. Reduced costs are local rates. They are exact for the first unit, and after that they can only get worse. Ten units of each therefore costs at least seven hundred and fifty rupees a day. So treat that as a floor on the damage, not an estimate of it. Sensitivity analysis does not replace strategic judgement. It prices it. That is what turns a turf battle into a measurable trade off.

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FAQ

Questions students ask us

About Testing the Boundaries

What is the difference between a robust and a fragile LP solution?
A robust solution has wide ranges of optimality and stable shadow prices, so profit coefficients can move substantially, such as 10% down or 15% up, without changing the optimal basis. A fragile solution can flip to a different product mix after a small shift, such as 2% in one coefficient. A rule of thumb treats ranges narrower than about plus or minus 10% as fragile and wider than 25% as safe.
What is parametric programming in linear programming?
Parametric programming varies one parameter, such as a profit coefficient or a right-hand side, across a wide interval while holding everything else fixed, recording the optimal value and basis along the way. Points where the basis changes are breakpoints, and between them the optimal value changes linearly. For example, as one product's profit rises from 0 to 80, the optimum may jump to a new corner at 30 and again at 60.
What is the dual of a linear program?
Every linear program has a partner problem called the dual. For a maximization primal, each resource constraint becomes a dual variable that values the resource, each production variable becomes a dual constraint, and the dual minimizes total resource value. Strong duality says the optimal objective values are equal, and the optimal dual variables are the primal's shadow prices. Complementary slackness adds that a constraint with slack has a dual value of zero.
How can reduced costs show the cost of overriding an LP recommendation?
If management forces an excluded product into the plan, its reduced cost gives the profit lost per unit. In one case, a model recommends two products for 880 rupees a day, and the two excluded products have reduced costs of 30 and 45 rupees. Forcing in one unit of each lowers profit to 805, a penalty of 75 rupees a day. Reduced costs are exact for the first unit, and further units cost at least as much.

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