Hesitant fuzzy linguistic AHP (Yavuz, Öztayşi, Çevik Onar and Kahraman, 2015)

Five things change; the idea of weighting criteria through pairwise comparison is preserved.

Scope. Crisp AHP only weights criteria; it does not rank alternatives. This extension does both jobs at once. It first compares criteria two at a time, exactly as crisp AHP does, and extracts their weights. It then scores the alternatives against each criterion directly, on a seven-term scale rather than through pairwise comparison, and combines these scores with the weights to produce a ranking.

Cells. Every cell of the criterion comparison matrix is a term interval drawn from a seven-term linguistic scale (s0 to s6); an envelope such as "s4 to s5" represents a case where the decision-maker cannot settle on a single term. This differs from the numerical degree sets on the hesitant data-type card: here there is not more than one number, but an adjoining interval of terms on a fixed verbal scale. The alternatives' scores against each criterion are given on the same seven-term scale, in the same envelope form, but this is a direct score, not a comparison.

Scale conversion. Every term index (0 to 6) is divided by 6 to bring it between 0 and 1. This places both the criterion comparisons and the alternative scores on the same [0,1] footing.

Weight derivation (stays an interval). Crisp AHP solves an eigenvector, and fuzzy AHP descends to a single defuzzified weight. Here, the weight never descends to a single number. Every criterion's row sum is an interval (a lower and an upper end); this interval is turned into an INTERVAL weight by dividing it against the opposite end's total. The lower and upper ends are divided by the upper total and the lower total respectively, so the weight remains a two-ended band rather than a single number summing to 1.

Alternative scoring and ranking. The alternatives' scores are likewise converted to intervals and combined with the interval weights to give each alternative an interval score. The ranking is not produced by sorting these interval scores directly from largest to smallest, but by a preference-degree function that measures how far two intervals overlap; this function compares every pair of alternatives and gives a degree of superiority between 0 and 1, and the ranking follows from the sum of these degrees.

For this classical form, DecisionMind fixes the seven-term scale (g = 6), single-expert comparison, and the carrying of the interval weight through every stage without defuzzification. The method works only with benefit-direction criteria; if a cost criterion is supplied without first being converted, the engine stops.

The result is a ranking, but the score beneath it is not a crisp number, it is an interval. The preference degree shows how far one alternative outranks another, between 0 and 1: close to 1 means the superiority is decisive, close to 0.5 means the two alternatives' intervals overlap substantially. Criterion weights are also intervals and should not be presented as a single percentage; say "C1's weight lies between 0.54 and 0.73," not "C1's weight is 0.60."

Thus instead of writing:

"A1's preference score is higher than A2's, so A1 is definitively better"

the report should read:

"A1's interval score partly overlaps A2's; the preference degree (0.59) puts A1 ahead, but this superiority can reverse if a single term changes in either the alternative scores or the criterion weight"

This extension is appropriate when the decision needs to handle both the relative importance of the criteria and how the alternatives score against them, in the same exercise, with verbal judgement. Crisp AHP only generates weights; ranking the alternatives must then be handed to a separate method such as TOPSIS or VIKOR. This extension brings both under one roof, and lets the decision-maker stay within a term interval such as "s4 to s5."

Where a criterion is already measured (a past budget share, say), forcing it onto the seven-term scale manufactures uncertainty rather than modelling it. Crisp AHP's exit condition applies here too: the method works comfortably with two to twelve criteria. Where a criterion is cost-direction, this classical form does not accept it; the criterion must be converted to the benefit direction before it is supplied.

Presenting an interval weight as though it were a single crisp number. The weight is an interval; saying "C1's weight is 0.60" picks a single point from the interval and hides the rest.

Confusing the criterion comparison matrix with the alternative scoring matrix. The n×n matrix compares only the criteria; the alternatives' scores are a separate, directly supplied input. While criteria are compared two at a time, alternatives are not compared, they are scored.

Supplying a cost criterion without converting it. This classical form works only with benefit-direction criteria; the engine rejects a cost criterion rather than guessing and reversing its direction.

Presenting a single-expert result as though it were a group decision. This form has a single decision-maker; more than one expert's opinion must be handled separately, through a justified aggregation step.

The governing principle is this:

In hesitant fuzzy linguistic AHP, both the criterion weight and the alternative score remain intervals. The ranking follows from a preference degree that measures how far these intervals overlap; this degree is not a single, definitive claim of superiority.