WPM (Weighted Product Model)

WPM is a ranking method for when you hold a numerical decision table and want to place the alternatives in a single order. Its output is a unit-free multiplicative score for each alternative and the ranking that follows from it; it does not produce criterion weights, taking them from outside. Miller and Starr proposed it in 1969; it traces its origin to dimensional analysis in physics, and it is one of the oldest and simplest multiplicative approaches among multi-criteria decision methods.

The idea behind WPM is to MULTIPLY criteria rather than add them. Adding measures in different units (kilograms, days, points) requires acting as though they had all been converted into the same "currency." Multiplying is different: you take each criterion's own RATIO on its own scale (its percentage relative to the best) and multiply these ratios together; the units then cancel, leaving a directly comparable, unit-free number. This is why the method draws its inspiration from dimensional analysis in physics: just as units cancel through ratios in physics, criteria cancel the same way in WPM.

This idea carries a sharp philosophical consequence: unlike additive methods such as SAW, WPM does NOT EASILY FORGIVE a weak criterion. In a sum, a small number can be lost against larger ones; in a product, a value close to zero drags the whole product down. WPM's nature is not "I partly go along with every criterion" but "you cannot be seriously weak on any criterion." While not fully eliminatory, it is less forgiving than additive methods.

The method proceeds in two steps.

First, direction-sensitive ratioing. WPM sets each cell against the best value in its own column: for a "higher is better" criterion, the value is divided by the column's largest; for a "lower is better" criterion, the column's smallest is divided by the value. This differs from TOPSIS's vector normalisation: in WPM, normalisation is itself a RATIO, not a difference or square-root operation. Every column is thereby converted into a unit-free ratio between 0 and 1.

Second, the weighted product. WPM raises each alternative's ratioed values to a power equal to the relevant criterion's weight, and multiplies these together across all criteria. A weakness on a heavily weighted criterion, thanks to this exponentiation, pulls the score down disproportionately. WPM ranks alternatives from highest to lowest on this single multiplicative score.

The formulas behind the steps and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The multiplicative score states an alternative's relative standing against the other alternatives in this particular analysis; a value such as 0.77 does not mean "77 per cent good," and it cannot be compared with a score from a different analysis, because the ratioing in each analysis is built from that analysis's own best values. Thanks to WPM's normalisation (ratioing against the column's best), scores are unaffected when criteria are rescaled by the SAME PROPORTION (from kilograms to tonnes, say). This is the source of the method's claim to be "unit-independent." But this immunity holds only for PROPORTIONAL rescaling. If a criterion's measurement ORIGIN (its zero point) is arbitrary or contested, for instance where a scoring system's zero does not represent a genuine "nothing," the result can change; the multiplication operation assumes a genuine zero point exists, that is, a ratio scale.

Thus instead of writing:

"The WPM result stays the same whatever unit the criteria are measured in"

the report should read:

"The WPM result does not change when criteria are rescaled PROPORTIONALLY (as from kilograms to tonnes); but where a criterion's measurement origin is arbitrary (as with a scoring scheme that has no genuine zero), the result can be sensitive to that arbitrary choice"

WPM works with crisp, strictly positive (not zero or negative) numerical data: if a cell is zero or negative, the product becomes undefined or meaningless. You need alternatives in rows, criteria in columns, direction information ("higher is better" or "lower is better") for every criterion, no empty cells, and criterion weights that sum to 1. WPM does not produce weights, it asks for them; you can derive them from expert opinion (AHP, BWM, SWARA) or from the data itself (Entropy, CRITIC). Criteria need to be measured on a genuine ratio scale with a meaningful zero point (price, weight, duration, speed, and the like); a criterion measured only on an interval scale (with an arbitrary zero) can give a misleading result in WPM. DecisionMind holds nine WPM family members alongside the base method (across uncertainty types such as fuzzy, intuitionistic fuzzy, Fermatean fuzzy, Pythagorean fuzzy and q-rung orthopair); which fits your data situation depends on your data type. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably.

WPM is a sound choice if your criteria are measured with strictly positive numbers on a genuine zero point (price, weight, speed, capacity, and the like), your table has no gaps, and you accept that a weak criterion will pull the score down disproportionately (a harsher penalty than additive methods apply). It is also advantageous where you want to combine criteria in different units (kg, days, TL) directly, without setting an additional normalisation range, because of WPM's multiplicative, unit-cancelling structure.

WPM should not be used where your data has a zero or negative value (the product becomes undefined) or where a criterion is measured on a scale without a genuine zero point (for instance, a survey score whose zero represents an arbitrary starting point rather than "not satisfied at all"). If you want a weakness on one criterion to be fully offset by strength on others (that is, if WPM's harsh penalty is not wanted), an additive method such as SAW should be used instead.

All criteria positive, with a genuine zero point (ratio scale) → WPM

A criterion can be zero or negative → Not WPM; SAW or TOPSIS

A criterion has no genuine zero (only an interval scale, e.g. an arbitrarily anchored score) → Not WPM; SAW or TOPSIS

Full compensation wanted for a weak criterion, harsh penalty not wanted → SAW

A simple, multiplicative, unit-independent aggregation is wanted → WPM

WPM's greatest strength is its unit-independence: it can combine criteria in different units (kg, TL, days) directly, because its normalisation is itself a ratio and units cancel under multiplication. Its computation is very simple and needs few parameters. Because it does not forgive a weak criterion as easily as additive methods do, it makes it harder to hide alternatives that are "average at everything but very poor on one item" than additive methods do.

Its limitations stem from that same severity and from its old, simple structure. First, it is undefined for zero or negative values; this closes WPM off from some data situations. Second, it is unaffected when measurement changes PROPORTIONALLY (kilograms to tonnes), but if a criterion's measurement origin (its zero) is arbitrary, the result can be sensitive to that arbitrary choice; WPM assumes a genuine ratio scale. Third, applying additive (WSM/SAW) and multiplicative (WPM) methods to the same data can give different rankings; this has been termed, in the literature, the "decision-making paradox," and shows that WPM's ranking is sensitive to the choice of aggregation (Triantaphyllou and Mann, 1989). Fourth, being a fairly old and simple method, WPM specifically has less of the extensive rank-reversal and sensitivity literature that later methods such as TOPSIS or VIKOR have accumulated (Triantaphyllou, 2000).

The most common mistake is feeding WPM a table without noticing that a criterion holds a zero or negative value; this renders the product undefined or leads to a meaningless result. A second mistake is feeding WPM a criterion without a genuine zero point (for instance, a satisfaction score with an arbitrary starting point); ratios of such criteria are not meaningful, and the result becomes sensitive to the measurement's origin. A third mistake is comparing a WPM result directly with a SAW or TOPSIS result on the same data and asking "which is correct"; different aggregation logics can give different rankings, and this is not an error but the nature of the methods. A fourth mistake is failing to check that the criterion weights sum to 1; in the multiplicative formula the exponents must sum to 1, otherwise the scores become incomparable.

The governing principle is this:

A WPM result rests on the assumption that criteria are measured on a genuine ratio scale, with a meaningful zero point; where this assumption does not hold, the multiplicative result is not reliable either.