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Beyond the Printout
Quantitative Analysis · Chapter 9: Linear and Integer ProgrammingSection 9.5 · lesson 44 of 5510 min · free with a free account
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In this lesson
Turning linear programming solver output into managerial decisions.
- Identify the key elements of a typical LP solver report
- Interpret shadow prices, reduced costs, slack and surplus
- Apply sensitivity ranges to test a solution's stability
- Decide on capacity expansion from solver output alone
Transcript
Everything said in the video, section by section (1,607 words). Read along, or search it.
Linear and Integer Programming · Section 9.5
Beyond the Printout
A linear programming solver hands you an answer in seconds. The hard part is no longer the algebra. It is the reading. This section is about the report the software prints after it solves. It tells you which resources are scarce, and which are sitting idle. It puts a price on every bottleneck in the operation. And it tells you how far things can drift before the plan collapses. By the end, you will read a solver printout the way an accountant reads accounts. You will interpret shadow prices, reduced costs, slack and surplus. And you will use them to decide whether buying more capacity is worth it.
From Algebra To Action
The Solver Does the Algebra. You Do the Reading.
In practice, almost nobody solves a linear program by hand. Firms lean on software instead. Excel Solver, the PuLP library in Python, and the big engines, C P L E X and Gurobi. They return optimal solutions in seconds, even for models with thousands of variables. Excel Solver is built in, so most managers already own it. PuLP is open source, and free to embed in a Python workflow. The enterprise engines are built for models with many thousands of rows. The mathematics disappears behind a button. What appears instead is a report, and a report has to be read. A solver report says which resources are scarce, and what each constraint is worth. Ignoring all that is like buying a smartwatch and only checking the time. Almost all of the value sits beneath the surface.
Anatomy Of A Report
Six Outputs, Six Questions
Layouts differ between tools, but the content does not. Every solver report carries the same six categories of information. First, the optimal objective value. That is the headline number. Second, the decision variable values, which are the plan itself. Third, reduced costs, for the options the model left out. Fourth, slack and surplus, which reveal what is genuinely scarce. Fifth, shadow prices, which put a money value on that scarcity. And sixth, sensitivity ranges, which say how long the answer holds. Get into the habit of asking what question each output answers. That habit is the whole foundation of post optimal analysis.
The Headline
Objective Value and Decision Variables
Start with the two outputs that everybody reads first. The optimal objective value is the single headline number. It is the best profit, or the lowest cost, available under every constraint. In a four product model it might read: objective value, twenty-four thousand eight hundred dollars. Treat it as the benchmark against which every alternative scenario is judged. The decision variables are the plan itself. What to produce, what to allocate, what to assign. They turn the solver's mathematics into instructions a factory can follow. And now the part that most people skip entirely. A variable that comes back as zero is still an answer. It tells you what the model has deliberately chosen not to do.
Reduced Costs
What Would Have to Change
A variable sitting at zero is called non basic. It sits outside the plan, and the solver attaches a number to it: the reduced cost. It says how much the objective coefficient must improve, before including the variable pays. Take Premium, the third product, with a reduced cost of minus three dollars. Its profit margin has to rise by at least three dollars a unit. Only then does Premium deserve a place in the production plan. Read that number as an activation threshold, the point where a dormant option wakes up. It is a precise instruction for the marketing team, not a vague hint. Raise the price by three dollars, or strip three dollars out of unit cost. Variables already in the plan show a reduced cost of zero. They are already earning their place, so nothing needs to change.
Slack And Surplus
Which Resources Are Actually Scarce
Now turn from the variables to the constraints, where the resource story is told. Slack and surplus answer one question. Is this resource actually scarce? For a less than or equal to constraint, the value reported is slack, the unused amount of that resource. Positive slack means the constraint is non binding, with capacity left over. For a greater than or equal to constraint, the value is surplus. Surplus is how far you have overshot the minimum requirement. Positive surplus means you are carrying a cushion above what was demanded. An equality constraint always shows zero, because it must be met exactly. Zero slack is the tell-tale sign of a binding constraint. The resource is fully consumed, and it is actively limiting profit. Binding constraints are exactly where shadow prices come into play.
Shadow Prices
The Marginal Value of One More Unit
A shadow price is sometimes called a dual value. It is the marginal value of relaxing a constraint by one unit. Put plainly: give me one more hour, and profit rises by this much. That is why it behaves like the internal price of scarcity. Now here is the condition that catches students out. A shadow price is meaningful only for binding constraints. If a constraint still has slack, its shadow price is zero. That zero is not a rounding artefact. It is the economics. You already hold spare capacity, so one more unit buys you nothing. Binding constraints carry a non zero price. Slack constraints carry none. So read the slack column first, and only then read the shadow price.
Sensitivity Ranges
How Far Before the Answer Breaks
Every number a solver reports comes with an expiry date. That is what the sensitivity ranges are telling you. A range says how far a value can move without altering the optimal basis. The basis is simply which variables are in the solution. Two kinds of range get reported, and they answer different questions. Coefficient ranging asks how far a profit margin can move, before the plan changes. Right hand side ranging asks how far a resource limit can move, before its shadow price lapses. Inside the range, the shadow prices and reduced costs you were given remain valid. Outside it, they are simply wrong, and the model has to be re-solved. This is a validity boundary, not a forecast of what will actually happen. Treat it as the fine print on every recommendation a solver makes.
Worked Printout
Reading the Variable Cells
Let us read a real printout, the way a plant manager would. A manufacturer makes four products. Standard, Deluxe, Premium and Custom. Three machines constrain production, and Custom also faces a demand ceiling. Standard comes out at one hundred and twenty units, Deluxe at sixty. Custom comes out at twenty units, and Premium at zero. Premium is the row worth studying, because it is the product left out. Its reduced cost is minus three dollars, on an objective coefficient of ninety-five. So the margin must reach ninety-eight dollars before Premium enters the plan. One caution about the two range columns on that row. For a product left out of a maximisation plan, the three dollar gap is the allowable increase. This printout files it under allowable decrease, which is the wrong way round. When the columns and the reduced cost disagree, trust the reduced cost.
Worked Printout
Reading the Constraint Cells
Now the constraint cells, where the money questions live. Machine A has forty hours available, and the plan uses all forty. Slack is zero, so Machine A is binding. Its shadow price is eight dollars an hour. One more hour of Machine A is worth eight dollars of profit. Machine B tells the opposite story: sixty hours available, forty-five used, fifteen of slack. Its shadow price is zero, and buying more B time would be wasted money. Machine C is binding too, worth five dollars an hour, and the Custom cap three dollars. Now a warning. This report claims a profit of twenty-four thousand eight hundred dollars. But the plan it prints earns only twelve thousand. That is one hundred and twenty at fifty, sixty at eighty, and twenty at sixty. Always reconcile the headline against the variable values and their coefficients.
Acting On The Output
Shadow Prices as Willingness to Pay
Here is how a shadow price turns into a decision rule. It is the most you should ever pay for one more unit. Machine A time is worth eight dollars an hour, so an offer below that adds profit. An offer above eight dollars destroys value, so turn it down. At exactly eight dollars you are indifferent. That is break even. But a price on its own is not yet a decision. You also need to know how many units that price covers. For Machine A the eight dollar price holds from thirty-two to forty-eight hours. So four extra hours add four times eight, which is thirty-two dollars. Ask for twenty extra hours and you leave the range entirely. The eight dollar figure lapses, and the model has to be re-solved.
Decision Case
Should We Add a Second Shift on Machine A?
Now put all of it to work on a real proposal. The plant manager wants a second shift on Machine A. It costs two hundred dollars a week, and supplies thirty extra hours of capacity. Step one. Identify the binding resource, the one running at zero slack. Machine A has zero slack and a shadow price of eight dollars an hour. Step two. Check the sensitivity range on that resource. The allowable increase on Machine A is only eight hours. So the eight dollar price covers just eight of the thirty hours offered. Step three. Compute the gain you can actually guarantee. Eight hours at eight dollars is sixty-four dollars a week. Against a cost of two hundred, that is minus one hundred and thirty-six a week. So on current information, reject the proposal. But do not stop there. The other twenty-two hours lie outside the range, and might still add value. Re-run the model with Machine A set to seventy hours, then compare the profit. Accept the second shift only if profit improves by more than two hundred dollars. And in general, pay at or below the shadow price, and stay inside the range.
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