Linear Assignment Calculator
The Linear Assignment Method combines the rank order of the alternatives on each criterion into one overall ranking. It assigns every alternative to a different rank position so that the total weight of the criteria supporting those positions is as large as possible, using the Hungarian algorithm.
New here? Try it with example data
Fill every step with “Choosing a laptop” and see a result in one click.
Enter at least three alternatives
Insert criteria
| No. | Name | Type | Weight | Qualitative | Actions |
|---|---|---|---|---|---|
| 1 | Benefit | ||||
| 2 | Benefit |
Complete previous steps
Complete previous steps
How Linear Assignment works
The Linear Assignment Method turns the rank order on each criterion into one overall ranking. It gives every alternative a place in the final ranking by solving an assignment problem with the Hungarian algorithm, so that the total weight of the criteria that support each placement is as large as possible.
Step by step
- Rescale the criteria weights so they sum to 1.
- Rank the alternatives on each criterion separately (largest value first for benefit criteria, smallest value first for cost criteria). Only the order matters, not how far apart the values are.
- Build the PI matrix: the entry for an alternative and a rank position is the total weight of the criteria on which that alternative has that rank.
- Assign every alternative to a different overall rank position so that the sum of the matching PI entries is maximal (Hungarian algorithm).
- The result lists the alternatives from overall rank 1 to the last rank.
Formulas
Worked example
With the laptop data above, the rank orders per criterion are: Price (cheapest first) Brisk 15, Delta Air, Aurora 14, Cirrus Pro; Performance Cirrus Pro, Aurora 14, Delta Air, Brisk 15; Battery Delta Air, Aurora 14, Cirrus Pro, Brisk 15; Weight (lightest first) Delta Air, Aurora 14, Cirrus Pro, Brisk 15. This yields the following PI matrix.
| Alternative | Price (EUR) (cost, weight 3) | Performance score (benefit, weight 3) | Battery (hours) (benefit, weight 2) | Weight (kg) (cost, weight 2) |
|---|---|---|---|---|
| Brisk 15 | 900 | 6 | 9 | 1.8 |
| Cirrus Pro | 1500 | 9 | 10 | 1.6 |
| Aurora 14 | 1200 | 8 | 12 | 1.3 |
| Delta Air | 1050 | 7 | 15 | 1.2 |
| Alternative | Rank 1 | Rank 2 | Rank 3 | Rank 4 |
|---|---|---|---|---|
| Brisk 15 | 0.30 | 0.00 | 0.00 | 0.70 |
| Cirrus Pro | 0.30 | 0.00 | 0.40 | 0.30 |
| Aurora 14 | 0.00 | 0.70 | 0.30 | 0.00 |
| Delta Air | 0.40 | 0.30 | 0.30 | 0.00 |
The assignment with the largest total (2.2) puts Delta Air first (0.40), Aurora 14 second (0.70), Cirrus Pro third (0.40) and Brisk 15 fourth (0.70).
Use the "Load example" button at the top of this page to fill in this data and run it yourself.
Frequently asked questions
What does the PI matrix show?
For every alternative it shows how much criteria weight supports each rank position. For example, a value of 0.70 in the Rank 4 column means that criteria worth 70% of the total weight put that alternative last. You can open it under "Show more" in the result.
Does this method use the actual values or only their order?
Only the order within each criterion. The cheapest laptop gets rank 1 on price whether it is 1 euro or 500 euros cheaper than the next one, so use TOPSIS, SAW or WPM when the size of the differences matters.
Does the result include a score?
No. The result is an ordered list from first to last place; the PI matrix in the details explains how the ranking was built.
Why use an assignment algorithm for ranking?
The Hungarian algorithm finds the best one-to-one match of alternatives to rank positions, so every position is given to exactly one alternative and the combined supporting weight is as large as possible.
How many alternatives and criteria can I use?
You need 3 to 7 alternatives and 2 to 7 criteria, and every performance value must be greater than zero. Weights are relative importance: enter any positive numbers and the calculator rescales them so they sum to 1. Weights of 3, 3, 2 and 2 are the same as 30%, 30%, 20% and 20%.