PCA Weighting (Objective Weighting via Principal Component Analysis)

PCA Weighting is not a ranking method; it does not rank alternatives, it produces criterion weights. Given a numerical decision table of alternatives and criteria, it scans the table without consulting any expert and yields a weight vector summing to 1. These weights then become the input to a ranking method such as TOPSIS or VIKOR.

Its foundation is Principal Component Analysis (PCA), a statistical technique Pearson first put forward in 1901 and Hotelling gave its present computable form in 1933. PCA is not itself an MCDM method but a general statistical technique that reduces the shared pattern of variation in a multivariate dataset to a small number of "components." Using it to generate criterion weights in decision analysis is an application that became widespread long after PCA itself; there is no single founding MCDM paper, and many studies apply the same approach, component selection via the Kaiser rule, converting component contributions into weights, in broadly similar ways.

Most objective weighting methods, such as Entropy and CRITIC, look at criteria one at a time; PCA Weighting looks at criteria together. The idea is this: in a decision table, criteria are usually not entirely independent of one another; some rise and fall together. PCA captures this shared movement and reduces it to a small number of "principal axes" (components), each representing a common pattern among a subset of the criteria. A criterion earns more weight the more strongly it is tied to these axes that genuinely separate the decision (the higher its "loading").

This has a consequence: PCA Weighting takes CRITIC's "penalise repetition" logic one step further; where CRITIC looks at the relationship between two criteria at a time, PCA resolves the shared structure across all criteria at once. Where a strong internal structure exists among criteria (groups that move together), PCA makes it visible; where criteria are already independent of one another, the additional information PCA contributes shrinks, and the weights come to resemble each criterion's own spread more closely.

The method proceeds through three steps.

First, standardisation. Each column has its own mean subtracted and is divided by its own standard deviation, so that criteria measured in different units (currency, days, points) become comparable on the same scale.

Second, finding the principal axes. The correlation structure among criteria is resolved from the standardised table, and from this structure the principal axes (components) summarising the criteria's shared movement are extracted. How much information each axis carries is measured by an eigenvalue; DecisionMind follows the traditional Kaiser rule, discarding axes whose eigenvalue falls below 1 and retaining only axes that genuinely carry information.

Third, weighting. How strongly each criterion is tied to the retained axes (its loading) is computed; this attachment is weighted by the share of information the relevant axis carries and summed. The result is a weight vector summing to 1.

The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.

The PCA Weighting output measures not how important a criterion is in the decision-maker's eyes, but how central a position it occupies in the shared structure among criteria. A weight of 0.40 does not mean "this criterion is 40 per cent of the decision"; it means "this criterion is one of the strongest carriers of the principal axes that actually separate the alternatives." Where no strong shared structure exists among criteria (each criterion moves on its own), the number of principal axes approaches the number of criteria, and the weights can end up similar to one another and non-discriminating; this is not an error but a sign that the data itself does not offer PCA a rich shared structure.

Weights are tightly bound to the alternative set and the criterion list: when a criterion is added or removed, the entire correlation structure, and so the entire weight set, is recomputed. Where the number of alternatives is small, correlation becomes unreliable; in that case the principal axes may reflect not a genuine shared structure but a random pattern in a small sample.

Thus instead of writing:

"The PCA analysis proved income is the most important indicator"

the report should read:

"Income is the strongest carrier of the principal axes that actually separate the indicators in this alternative set; the weight of 0.40 reflects this central position, not the indicator's political or administrative priority"

Classical PCA Weighting works with crisp data: one number per cell. DecisionMind holds, alongside the base method, a fuzzy extension (two members in total); this extension is DecisionMind's own fuzzy adaptation, used where data is supplied as expert judgement in the form of triangular fuzzy numbers.

You need: alternatives in rows, criteria in columns, one number per cell, no empty cells. Criterion direction (more or less is better) does not affect the standardisation step but determines which direction the weights should be read in. Weights are not entered; the method produces them. For the correlation structure to be resolved meaningfully, the number of alternatives must clearly exceed the number of criteria; with few alternatives (close to or fewer than the number of criteria), the correlation matrix becomes unreliable and can even become impossible to compute.

Where the number of criteria is large and includes indicators that repeat one another or carry a shared pattern (financial ratios, sustainability indices, multidimensional performance indicators), PCA Weighting makes this shared structure visible. Where the number of alternatives clearly exceeds the number of criteria (dozens of alternatives against a few criteria, rather than the reverse), it works reliably.

The cases where it should not be used follow from its own philosophy. Where the number of alternatives is small or close to the number of criteria, the correlation structure becomes unreliable. Where criteria have already been chosen to be independent of one another, the additional information PCA contributes stays limited, and the result resembles one from Entropy. Where the decision-maker explicitly regards one criterion as a priority, PCA cannot see this. Since the result requires explaining to the decision-maker "what does this principal axis mean," the interpretive burden is heavier than for other objective methods when presenting to a non-technical board.

Many mutually patterned criteria, plenty of alternatives → PCA Weighting

Criteria independent, spread alone is sufficient → Entropy

Relationship between criteria matters but few in number → CRITIC

The decision-maker's priority should show in the result → AHP, BWM, SWARA (subjective)

Few alternatives, close to the number of criteria → a simpler objective method instead of PCA Weighting

PCA Weighting's greatest strength is resolving the shared structure among a large number of criteria in a single pass; where CRITIC looks at pairwise relationships, PCA evaluates all criteria together. It is objective: the same table gives everyone the same weight. Where the number of criteria is high (dozens of indicators), it reduces a complex indicator set to a small number of meaningful axes, and these axes offer reports a visible answer to the question "which indicators move together."

Its limitations stem from the same structure. First, interpreting the principal axes requires advanced technical knowledge; what an axis "means" is not always clear, and this makes the method harder to explain to non-technical decision boards (Naeimiasl, Vahidi and Soheili, 2025). Second, with few alternatives the correlation matrix becomes unreliable, or even singular (rank-deficient); the number of principal axes then shrinks artificially. Third, weights depend on the alternative set and the criterion list; adding or removing one criterion changes the entire structure. Fourth, where no strong shared structure exists among criteria, the additional information PCA contributes is limited and the result may not differ noticeably from other objective methods.

The most common mistake is skipping standardisation and running PCA on raw data; a criterion with large units (currency) dominates the correlation structure and the weights come out dependent on units.

A second mistake is running PCA with a small number of alternatives (close to or fewer than the number of criteria) and trusting the result; in such cases the correlation matrix is unreliable or singular. A third is reporting only the weights without explaining what the principal axes mean; the reader then cannot understand where the weight comes from. A fourth is using one study's PCA weights on a different alternative set. Fifth is choosing a different number of components than the Kaiser rule would give and failing to state this in the report; this choice changes the weights directly.

The governing principle is this:

PCA Weighting measures a criterion's central position on the principal axes that separate the decision; a high weight means the criterion is "a strong carrier of the shared structure among criteria," not "important," and this shared structure must be explained separately.