WPM Calculator

The Weighted Product Model (WPM) is a mathematical approach used to rank and select alternatives based on multiple criteria

New here? Try it with example data

Fill every step with “Choosing a laptop” and see a result in one click.

Step 1: Alternatives

Enter at least three alternatives

Alternatives

Step 2: Criteria

Insert criteria

No.	Name	Type	Weight	Qualitative	Actions
1		Benefit
2		Benefit

Step 3: Performance Matrix

Complete previous steps

How WPM works

The Weighted Product Model (WPM) is similar to SAW but multiplies instead of adding. It compares two alternatives at a time: the ratio of their normalized values on each criterion is raised to the power of that criterion's weight, and the ratios are multiplied together.

Step by step

Rescale the criteria weights so they sum to 1.
Normalize each criterion column: divide by the column maximum for benefit criteria, and divide the column minimum by the value for cost criteria.
For every ordered pair of alternatives (A, B), divide A's normalized value by B's on each criterion and raise the ratio to the criterion weight.
Multiply those weighted ratios to get the score S(A, B) for the pair.
If S(A, B) is greater than 1, A is preferred to B; if it is below 1, B is preferred.
The result lists every pair with a ">" or "<" sign, and the details panel shows the full score matrix.

Formulas

Normalization (benefit and cost criteria)

\[r_{ij} = \frac{x_{ij}}{\max_i x_{ij}} \ \text{(benefit)}, \qquad r_{ij} = \frac{\min_i x_{ij}}{x_{ij}} \ \text{(cost)}\]

Pairwise score of A over B

\[S(A,B) = \prod_{j} \left( \frac{r_{Aj}}{r_{Bj}} \right)^{w_j}\]

Worked example

With the laptop data above, every pair gets a score S(first, second). A value below 1 means the second alternative is preferred.

Example data: choosing a laptop
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

WPM pairwise scores
Pair (first vs second)	S(first, second)	Calculator output
Brisk 15 vs Cirrus Pro	0.987	Brisk 15<Cirrus Pro
Brisk 15 vs Aurora 14	0.885	Brisk 15<Aurora 14
Brisk 15 vs Delta Air	0.833	Brisk 15<Delta Air
Cirrus Pro vs Aurora 14	0.896	Cirrus Pro<Aurora 14
Cirrus Pro vs Delta Air	0.843	Cirrus Pro<Delta Air
Aurora 14 vs Delta Air	0.941	Aurora 14<Delta Air

Delta Air is preferred to every other laptop, Aurora 14 to Cirrus Pro and Brisk 15, and Cirrus Pro to Brisk 15, which gives the order Delta Air, Aurora 14, Cirrus Pro, Brisk 15. The closest pair is Brisk 15 against Cirrus Pro (0.987, so Cirrus Pro is only about 1.3% better overall).

Use the "Load example" button at the top of this page to fill in this data and run it yourself.

Frequently asked questions

How do I read the WPM result?

Each line compares two alternatives. "A>B" means the product of weighted ratios is above 1 and A is preferred; "A<B" means B is preferred. Read all lines together to get the overall order.

Why does WPM use a product instead of a sum?

Multiplication means that a very low value on one criterion drags the whole comparison down more than it would in a weighted sum, so weak spots are harder to compensate for.

Can I enter zero or negative values?

No. WPM divides values by each other, so every performance value must be greater than zero.

Do I have to normalize my data first?

No. The calculator normalizes every criterion column itself, so you can enter raw values such as prices, scores and hours.

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%.