Z-number BWM (Aboutorab et al., 2018)

Three things change; the logic of comparing against the best and worst criterion does not.

Cells. In crisp BWM, every comparison is a single integer on a scale of 1 to 9. Here every comparison is given as two linguistic terms. One is the constraint term, stating how many times more important the criterion is ("equally important" through to "absolutely more important"); the other is the reliability term ("very low" through to "very high"). A table Aboutorab and colleagues prepared in advance converts each of these term pairs into a triangular fuzzy number. Comparing the best criterion with itself, and the worst criterion with itself, remains fixed as before.

Folding reliability into the comparison. This is the step that actually separates the extension from the base method. Every comparison's reliability term is first converted into a single number (α). The square root of this number scales the three components of the constraint triangle.

Where reliability is high, the constraint stays close to what it was. Where it is low, the constraint shrinks and that comparison drifts towards "equally important." This means a judgement the expert trusts less has less say in the weight calculation.

Solving for the weights and the consistency. In crisp BWM, the weights and the consistency indicator (ξ*) are solved from a single non-linear model. Here the same model is built on triangular fuzzy comparisons already scaled by reliability: it searches for triangular weights that make the ratios drawn from the best-to-others and others-to-worst comparisons deviate as little as possible. These triangular weights are then reduced to a single crisp number (a weighted average, l+4m+u divided by 6). The consistency indicator is likewise found within this same framework, by dividing by a reference value read off the linguistic class of the comparison between the best and the worst criterion.

DecisionMind holds the reliability scaling, the non-linear optimisation, and the weighted-average defuzzification fixed in this extension. The scale is taken from Aboutorab and colleagues' 25-combination table; no other conversion is used.

The weight shows a criterion's relative importance among the others within this set of comparisons, just as in crisp BWM; the weights always sum to 1. The consistency indicator (ξ*) measures how well the comparisons agree with one another, not how correctly the criteria's "true" importance has been captured.

The difference is here. In crisp BWM, the criterion chosen as best usually comes out with the highest weight, simply because of how the comparison is structured. Here reliability can upset this structure. If a criterion judged "slightly less important" than the best is compared with very high reliability, its weight can overtake that of the criterion chosen as best. This is an outcome crisp BWM never shows, and the report must explain it separately.

Thus instead of writing:

"The criterion chosen as best has also received the highest weight in Z-BWM, because it is already the most important criterion"

the report should read:

"The best criterion has received the highest weight at this reliability level; this ranking is sensitive to the reliability of the comparison linking the best to the other criterion, and it can change if that reliability falls"

Use this when the comparisons the expert gives against the best and worst criterion come from different sources, and those sources differ in reliability. If one comparison comes from a long-experienced expert and another from a participant giving an opinion for the first time, this difference ought to be reflected in the weight. If every comparison comes from a source of the same reliability, the reliability component adds no discriminating information, and Fuzzy BWM is sufficient.

The exit condition from crisp BWM applies here too. If the best and worst criterion cannot be chosen with confidence, or if the defuzzified consistency indicator lies above the widely accepted threshold of 0.30, the comparisons should be reviewed with the expert again before use.

Confusing the constraint scale with the reliability scale. In Aboutorab and colleagues' table, the constraint terms ("equally important" through to "absolutely more important") and the reliability terms ("very low" through to "very high") are two separate scales. The constraint is defined roughly between 1 and 4.5, reliability between 0 and 1. Writing two reliability terms, or two constraint terms, into a single cell makes that cell meaningless.

Feeding a Z-number straight into the optimisation. If the constraint and reliability terms are handed to the non-linear model before being converted into triangular fuzzy numbers, the model cannot be built; the conversion step must not be skipped.

Dividing the consistency indicator by the wrong reference value. The reference value is read off the linguistic class of the comparison between the best and worst criterion. Looking only at the constraint class and ignoring the reliability class gives a wrong consistency ratio.

Assuming the best criterion's weight will always come out highest. As shown above, low reliability can push the best criterion's weight below another's. This is a different outcome from crisp BWM, and the report must not conceal it.

The governing principle is this:

Z-BWM weights carry both the magnitude and the reliability of the comparisons given against the best and worst criterion. Low reliability can weaken a comparison, and can even remove the advantage of the criterion chosen as best.