BWM (Best-Worst Method)

You have criteria and want to establish their relative importance through expert judgement, but you do not want to carry the full pairwise-comparison load AHP demands. BWM is a weighting method used in exactly this situation. Its output has two parts: a weight vector that sums to 1, and a consistency indicator (ξ*, "xi star") showing how well these weights agree with the comparisons the expert gave. BWM does not rank alternatives. Within DecisionMind it is a weight provider; the weights it produces feed into ranking methods such as TOPSIS and VIKOR. Rezaei proposed the method in 2015.

AHP asks for every pair of criteria to be compared. BWM takes a different route. It first asks the expert to select only the most important (best) and least important (worst) criterion. It then asks the expert to compare the best criterion against every other criterion, and every criterion against the worst criterion. For n criteria, AHP demands n(n-1)/2 comparisons, whereas BWM asks for only 2n-3. The assumption behind the idea is this: people can compare two extremes, the best and the worst, more clearly and more reliably than everything else. Comparisons in the middle ground are more uncertain. BWM never asks about them; instead it leaves the mathematical model to derive them.

This idea carries a philosophical consequence. BWM reduces the expert's fatigue by asking for fewer judgements. But in exchange, it does not directly test the relationships among criteria in the middle ground. These relationships are inferred mathematically, purely from the best and worst comparisons. BWM can be regarded as a middle path between AHP's intensive cross-checking and SWARA's sequential chain. It asks for more judgement than SWARA and less than AHP. It has its own consistency indicator (ξ*), but this does not offer as intensive a check as AHP's CR.

The method proceeds through three steps.

First, identify the best and worst criteria. The expert selects the most important (best) and least important (worst) criterion among the set.

Second, build the two comparison vectors. The expert compares the best criterion against every other criterion (the Best-to-Others vector) and every criterion against the worst criterion (the Others-to-Worst vector), on a scale of 1 to 9. Comparing the best criterion with itself and the worst criterion with itself always equals 1.

Third, solve for the weights and the consistency. The method searches for weights that let the ratios drawn from the two vectors deviate as little as possible. Every criterion's weight must fit both its comparison with the best criterion and its comparison with the worst criterion as closely as possible at the same time. This solution also yields a single number representing the worst mismatch (ξ*). ξ* = 0 means the comparisons are perfectly consistent with one another; a large ξ* shows that these comparisons contradict one another.

The DecisionMind method page gives the formulas behind each step, the intermediate tables and the citation formats; this card carries no formulas.

The weight shows a criterion's relative importance among the others within this set of comparisons; the weights always sum to 1. ξ* measures something separate: how consistent the expert's best and worst comparisons are with one another. The two must not be confused. A low ξ* does not show that the weights reflect the "true" importance correctly. It shows only that the expert's two comparison sets, best-to-others and others-to-worst, do not contradict each other.

Thus instead of writing:

"ξ came out close to zero, so these weights are certainly correct"*

the report should read:

"These comparisons are consistent with one another (ξ≈0); the accuracy of the weights still depends on whether the expert chose the best and worst criterion correctly"*

BWM works with crisp data: the choice of the best and worst criterion, and a single integer from a 1–9 scale in every cell of the two comparison vectors. If your data is contradictory across experts or comes in fuzzy terms, BWM has fuzzy, Bayesian-group and other extensions; DecisionMind holds five BWM family members alongside the base method.

You need: a list of criteria to compare, the choice of the most and least important criterion, and 1–9-scale comparisons between these two criteria and the rest. BWM produces weights, it does not require weights supplied from outside; it needs no alternative data. At least two criteria are required; three to twelve criteria work comfortably. BWM's advantage becomes clear precisely here: as the number of criteria grows, AHP's comparison load (n(n-1)/2) grows far faster than BWM's (2n-3).

BWM is a suitable choice if you need to establish criteria's relative importance through expert opinion, want a consistency check, but cannot bear AHP's full comparison load because the number of criteria is large. It works well in any situation where the expert can confidently pick the most and least important criterion.

There are three situations where it should not be used. Where there is no expert opinion at all, BWM should not be used; let the data itself determine the weight instead. Where the ξ* derived from the best/worst comparisons comes out markedly above the threshold (the common convention is 0.30) and cannot be corrected, BWM should again not be used. Where the expert cannot confidently pick the most or least important criterion, BWM should also not be used; in that case the method's underlying assumption is undermined.

Expert judgement + a light comparison load + a consistency check are wanted → BWM

Same need, but the number of criteria is small and a full cross-check is also wanted → AHP

Criteria are easily ordered, speed is the priority, an independent consistency check is not essential → SWARA

No expert available, weight should be derived from the data → Entropy, CRITIC (objective)

Not weights but a ranking is needed → TOPSIS, VIKOR, PROMETHEE, the ELECTRE family (which can take its weights from BWM)

BWM's most important advantage is its light comparison load: 2n-3 comparisons is far fewer than AHP's n(n-1)/2, especially as the number of criteria grows, and this makes it easier for the expert to stay consistent (Mi et al., 2019). Even so, unlike SWARA, it has its own consistency indicator (ξ*). This does not leave BWM without an internal check. The idea of focusing on the best and worst extremes rests on the observation that human judgement is clearer at the extremes than in the middle ground, and this in turn improves the reliability of the comparisons.

Its limitations follow from this same structure. First, in the original non-linear model, more than one optimal set of weights can emerge when the comparisons are not perfectly consistent. Rezaei proposed a linear model in 2016 to resolve this ambiguity. Second, how the consistency indicator should be measured and what its thresholds should be remains debated in the literature (Liang, Brunelli and Rezaei, 2020). Third, the expert answers directly only about the relationships involving the best and worst criterion. No one directly observes the relationship between the other pairs of criteria; the model derives this mathematically (Mi et al., 2019). Fourth, choosing the wrong best or worst criterion, or having two experts treat different criteria as best or worst, changes the entire weight distribution from the outset.

The most common mistake is confusing the best and worst criterion: if best and worst are labelled the wrong way round, the meaning of the two comparison vectors is reversed and the weights become meaningless.

A second mistake is entering a value other than 1 for the best criterion's comparison with itself or the worst criterion's comparison with itself; this breaks the method's basic rule (a_BB=1, a_WW=1). A third mistake is ignoring a ξ* that comes out above the threshold (the common convention is 0.30) and reporting the weights as a "definite result" regardless. A high ξ* means the comparisons need to be reviewed again with the expert. A fourth mistake is presenting BWM weights directly as a ranking of alternatives; BWM does not rank, it only produces criterion weights.

The governing principle is this:

BWM weights are an internally consistent summary of the comparisons the expert gave against the best and worst criterion. ξ checks this consistency, but it does not check whether the best and worst criterion were chosen correctly. That remains the decision-maker's responsibility.*