What it means
Many business decisions are all-or-nothing. A company either builds a new factory or it does not, and it either hires a team or it does not.
Zero-one integer programming turns such choices into a formal model, in which each project gets a variable equal to 1 if selected and 0 if rejected. The model has an objective, such as maximising total net present value (the value today of a project's future cash flows less its cost).
It also has constraints, which are limits like a capital budget, a staff limit or a rule that two projects cannot both go ahead. A computer program then searches for the combination that gives the highest objective without breaking any limit.
The method is especially useful in capital budgeting when projects cannot be split into pieces. Ranking projects by return and funding them in order can fail, because the best project may use too much of the budget and prevent a better combination.
The zero-one model checks combinations rather than relying on a simple ranking. The nuance is that the number of combinations grows very quickly.
With 30 projects there are over a billion possible combinations, so the solution needs specialist software and clear input data. The model is only as good as the estimates for costs and values, so its output should be a guide, not a verdict.
Other applications include choosing locations for warehouses, scheduling staff, selecting a portfolio of assets with fixed lot sizes and deciding which loans to buy. Spreadsheets with a solver add-in can handle small problems.
Larger problems use dedicated optimisation software. A finance team can also use the model to ask what-if questions.
For example, it can show how much the best total NPV would fall if the budget were cut by $100,000, or which project would be dropped first. That information helps leaders see what each dollar of budget is buying, which is often more useful than the single optimal answer.
In practice
Real-world examples.
Example
A retailer has 12 possible store sites and a limit of $8,000,000 of capital. A zero-one model selects the combination of sites with the highest total expected profit, while ensuring no two sites compete in the same neighbourhood. The result differs from the simple ranking by profit.
Example
A hospital chooses which equipment purchases to fund from a fixed budget. Each item is either bought or not, and some depend on each other. The model keeps the combination that improves patient capacity the most, and the finance committee uses the result to explain the choices to the board.
Example
A logistics firm decides which of 15 depots to keep open. Each depot has a fixed cost and serves certain customers. The model finds the set of depots that covers all customers at the lowest total cost.
Formula
Calculation
Maximise: Sum of (NPV of project i x x_i), where each x_i is 0 or 1
Subject to: Sum of (cost of project i x x_i) is no more than the budget
A company has a $500,000 budget and four projects. Project A costs $200,000 with an NPV of $60,000, B costs $300,000 with an NPV of $90,000, C costs $250,000 with an NPV of $70,000, and D costs $150,000 with an NPV of $40,000. The best combination within budget is A and B, with cost 200,000 + 300,000 = $500,000 and NPV 60,000 + 90,000 = $150,000. The next best, B and D at 300,000 + 150,000 = $450,000, gives 90,000 + 40,000 = $130,000.Case study
Seen in the real world.
Redfern Manufacturing is an illustrative, fictional company with a $2,000,000 capital budget and eight requests from its plants. The finance director first ranks them by profitability index, which is NPV divided by cost, and funds the best ones in order. That approach selects projects worth a total NPV of $520,000.
An analyst then builds a zero-one model in a spreadsheet solver. It finds a different combination, skipping one high-ranking project that used too much of the budget and adding two smaller ones. The total NPV rises to $585,000, which is 585,000 - 520,000 = $65,000 higher.
The illustrative lesson is that ranking is a good first pass but can leave value on the table. The finance team keeps both approaches, using the model to test the ranking and to explain the final choice to the board.
Watch out
Common mistakes.
- Using the ranking by return as the final answer, when the best combination under a budget limit may not include the top-ranked project.
- Trusting the output without checking the inputs, when poor estimates of cost and value lead to a poor result.
- Treating projects as independent, when some depend on each other or exclude each other and must be modelled that way.
Questions
People also ask.
What does zero-one mean?
Each decision variable can take only the value 0, meaning not chosen, or 1, meaning chosen.
Can I do this in a spreadsheet?
Yes, for small problems a spreadsheet solver can handle it, though larger problems need specialist optimisation software.
How is it different from linear programming?
Linear programming allows fractional values, while zero-one programming forces whole yes-or-no decisions.
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