MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique)

MACBETH is a method used to determine criterion weights or alternative values, built entirely on verbal, non-numerical judgements. This card describes the method's use for criterion weighting: its output is a weight vector summing to one. AHP also makes pairwise comparisons, but asks the expert for a number such as "how many times more important"; MACBETH instead asks only "which category does the gap between these two reference profiles fall into," and generates the number itself through a linear-programming solution. Proposed by Bana e Costa and Vansnick in 1994, the method is used in fields such as health policy, environmental assessment and public investment, where decision-makers struggle to give numerical judgements but can make verbal comparisons.

MACBETH's underlying idea is that people can be inconsistent when giving ratio judgements such as "A is three times more important than B," but are more reliable in verbal comparisons such as "the gap between A and B is strong, the gap between B and C is weak." For criterion weighting, the method asks the question "how attractive is moving from the worst to the best on this criterion" through a set of reference profiles. This set consists of a neutral profile, at the lowest level on every criterion, together with profiles that each rise to the highest level on only one criterion. The expert assesses these gaps using verbal categories; MACBETH tests these categorical judgements through linear programming, searching for a numerical scale consistent with all of them.

This has one consequence: MACBETH tests consistency before generating a number. The expert's judgements may contradict one another; for instance, one judgement might state that the gap between A and B is stronger than that between C and D, while another implies the opposite. In that case the linear programme has no solution, and the method exposes this contradiction before producing any number. This makes MACBETH not merely a calculation tool but also a consistency checker.

The method proceeds through four steps.

First, building the verbal difference matrix. A neutral reference profile (at the lowest level on every criterion) is defined, along with one profile per criterion that rises to the highest level on only that criterion. The expert assesses every pairwise gap between these profiles using one of seven categories: no difference, very weak, weak, moderate, strong, very strong, extreme. A pair on which no judgement has been given is marked "no judgement."

Second, the consistency test. A linear programme tests whether at least one numerical scale exists that agrees with these verbal judgements. Pairs marked "no difference" must take equal values; pairs marked with a difference must carry a gap consistent with the category order. If no solution is found, the judgements are contradictory and must be revised.

Third, building the basic scale. Using consistent judgements, a numerical scale is chosen that maximises the interval between the most attractive reference profile (the high anchor) and the least attractive profile (the low anchor, neutral). This is MACBETH's "basic scale," usually reported after being rescaled to run between 0 and 100.

Fourth, converting to weights. For every criterion, the gap on this scale between the profile that rises on that criterion alone and the neutral profile (the swing) is computed. These swing values are divided by their own sum to convert them into criterion weights summing to one.

The formulas behind these steps are given on the DecisionMind MACBETH method page; this card carries no formulas.

The weight shows how large a swing moving from the worst to the best on a criterion creates on a numerical scale consistent with the expert's verbal judgements; it measures the attractiveness gap the expert attributes to that transition, not the criterion's "natural importance." If two criteria's weights are close, this means the expert attributed a similar magnitude of attractiveness to the worst-to-best gap on both. MACBETH's scale is not unique: more than one numerical scale can be consistent with the judgements, and the method selects only one of them (the basic scale); for this reason, the ratios and ordering between the numbers should be relied on, not the absolute figures.

For this reason, instead of writing:

"MACBETH calculated this criterion's weight exactly"

the report should read:

"On a scale consistent with the expert's verbal judgements, this criterion's swing came out largest; if the judgements were changed, or another consistent scale chosen, the ratios would stay similar but the absolute figures could change"

MACBETH works with crisp (categorical) data: every cell is either a category number from 0 to 6, or a mark showing that no judgement was given. DecisionMind carries no separately registered extension member within MACBETH's own family. You need at least three reference profiles (including the neutral profile), an above-diagonal verbal-judgement matrix between these profiles, information on which profile corresponds to which criterion, and which profiles are the neutral and the most attractive. The method produces weights, it does not take weights from outside. Three to twelve criteria is typical; as the number of judgements grows, so does the risk of the consistency test finding no solution.

MACBETH is a suitable choice if the expert struggles to give numerical ratios but can make verbal comparisons such as "this gap is small, that gap is large," and wants the consistency of these judgements tested from the outset. If the expert can already give numerical ratio judgements comfortably (as with AHP's nine-point scale), MACBETH's extra consistency test becomes an unnecessary step. If the number of criteria is very large, defining a separate reference profile for every criterion and collecting judgements between them becomes burdensome; methods requiring fewer judgements, such as LBWA or BWM, may be preferred in that case.

The expert can make verbal comparisons, consistency must be tested from the outset → MACBETH

The expert can give numerical ratios → AHP, BWM

Too many criteria, few judgements wanted → LBWA, BWM

Not weights but the data's own variability matters → Entropy, CRITIC

MACBETH's principal strength is that it can work with purely verbal categories, asking the expert for no number at all; this is a more natural input form for decision-makers who are hesitant to give numerical judgements. Before producing any number, the method tests whether the judgements are mutually consistent through linear programming, and clearly exposes contradictory judgements (cycles); this allows an error to be caught from the outset rather than noticed later (Bana e Costa, De Corte and Vansnick, 2012). Over the years the method has also been supported by software such as M-MACBETH, and has built up a broad body of applied literature in fields such as health, environment and public policy (Ferreira and Santos, 2018).

The method's greatest limitation is that the numerical counterpart of the seven verbal categories is chosen by the linear programme; because more than one scale can be consistent with the judgements, the selected "basic scale" is one possible solution, not the only correct one. Second, as the number of reference profiles grows, so does the number of pairwise judgements that must be collected, which can create a fatigue similar to AHP's pairwise-comparison burden. Third, when judgements turn out contradictory (when the linear programme has no solution), finding which judgement must be changed is left to the expert; the method exposes the contradiction but does not resolve it automatically. Fourth, because the numerical distance between categories (whether the gap between "moderate" and "strong" equals the gap between "strong" and "very strong") carries ordinal information rather than a cardinal distance, the result is sensitive to how the category boundaries are interpreted.

The most common mistake is mistakenly confusing one of the reference profiles with something other than the neutral (lowest) profile; the neutral profile must represent the lowest level on every criterion, otherwise the swing calculation loses its meaning. A second mistake is ignoring inconsistent judgements (the case where the linear programme has no solution) and declaring "I found a solution" after randomly changing one of the judgements; which judgement to revisit must be chosen carefully. A third mistake is presenting the basic scale MACBETH produces as an absolute reality; this scale is one of the possible scales consistent with the judgements. A fourth is confusing the criterion-weighting mode with the alternative-evaluation mode; a criterion's swing weight and an alternative's local value score are different concepts and cannot be substituted for one another.

The governing principle is this:

MACBETH's weight is the swing magnitude of a numerical scale found consistent with the expert's verbal judgements; if the judgements are consistent, the calculation is reliable, but the numbers produced are one of several possible scales.