Extension card · Fuzzy
Fuzzy MAUT
Fuzzy MAUT is the form of MAUT that works with triangular fuzzy numbers when criterion values, or the bounds of the utility function, are given approximately through expert judgement. It computes each criterion's utility in fuzzy terms and still ranks the result with a single aggregate utility score.
Base method
MAUT →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Fuzzy →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Three things change; the logic of "first single-attribute utility, then weighted sum" does not.
Cells. In crisp MAUT every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. Criterion weights take the same triangular form. The utility function here is not a curve negotiated with the decision-maker; in this classical form, DecisionMind uses a linear, range-normalised function (explained below); curved utility functions (saturating, or accelerating past a threshold) fall outside this extension's scope.
Single-attribute utility normalisation. In crisp MAUT every cell is linearly normalised so that the criterion's worst value receives a utility of 0 and its best value receives 1. Here the same normalisation is applied to the three components separately: for a benefit criterion, each component is normalised against its own (xjmin, xjmax) range; for a cost criterion, the components are normalised crosswise (the lowest component is computed from the highest end, the highest component from the lowest end). This crosswise application preserves the triangle's ordered structure (l≤m≤u) after the cost transformation.
Aggregation and defuzzification. In crisp MAUT the weighted sum directly gives a single utility score. Here the weighted sum runs on the three components separately (additive TFN summation); the result is a triangle, and only at the final step is it reduced to a single number by the centroid ((Vα+Vβ+Vγ)/3). DecisionMind holds this centroid defuzzification fixed for this classical Fuzzy MAUT.
How to Read the Output
Reading the output is the same as for crisp MAUT: it is a measure of position between 0 (worst on every criterion) and 1 (best on every criterion), not a percentage or a probability.
The difference is here. Behind the score lies a triangle width carried through the whole calculation. If the score gap between two alternatives is small, how sensitive that gap is to the verbal or approximate score on a given criterion must be tested separately, because the widening of a single criterion's triangle carries straight through to the final score via additive summation.
So instead of:
"Fuzzy MAUT showed that this alternative's true utility is 0.72"
the report should read:
"This utility of 0.72 has been reduced by the centroid from the triangular fuzzy input and linear range-normalisation of the criterion values; if the gap is small, which criterion the result is sensitive to in triangle width should be shown separately"
When to Prefer This over the Base Method
Use this extension when criterion values or expert scores are given verbally or approximately, and reducing them to a single number would create an artificial precision; crisp MAUT's own exit conditions (preference independence, the validity of additive summation) apply here in exactly the same way. Measured criteria should not be carried into this extension; if the matrix must be of a single type, a measured value is written as the same triangle across all three components. If the decision-maker's preference requires a non-linear utility curve (accelerating past a threshold, saturating), this classical Fuzzy MAUT form is not sufficient; a separate extension is needed where a non-linear utility curve and fuzzy input are both required together.
Mistakes Specific to This Extension
Defuzzifying first, then running crisp MAUT. Reducing the triangles to their centroid from the outset and running crisp MAUT is not Fuzzy MAUT; the width in the range-normalisation is skipped and score gaps are exaggerated. In the illustrative example below, this route gives A1 = 0.20, A2 = 0.825, A3 = 0.475; correct Fuzzy MAUT gives 0.30, 0.7167 and 0.4833. The ranking is the same (A2, A3, A1), but the A1-A2 gap inflates from 0.417 to 0.625.
Treating the raw triangle as "utility" without building a utility function. A criterion value's triangular fuzzy input is not the utility function itself; this classical form of DecisionMind uses range-normalisation in place of a utility function and holds this fixed explicitly. If a curved utility function is wanted, that is a separate assumption and must be stated in the report.
Forgetting crosswise normalisation on a cost criterion. If the benefit criterion's normalisation formula is applied to a cost criterion without crossing over (without swapping the end components), the triangle's ordered structure (l≤m≤u) breaks and the subsequent aggregation step becomes invalid.
Giving crisp weights but fuzzy scores. The method expects weights to be triangular too; if a crisp weight is to be given, its three components are written identically, and this is stated in the report.
The governing principle is this:
Fuzzy MAUT exists to carry the approximation in criterion judgement from single-attribute utility normalisation through to additive summation; any shortcut that defuzzifies the input from the outset, or skips crosswise normalisation on a cost criterion, distorts either the triangle's ordered structure or the size of the score gaps.
Cases
The first case is DecisionMind's validation example. It is a small three-alternative, three-criterion TFN table produced by the engine itself, faithful to the formula; it is not a page or table taken from the literature. The second case is an illustrative fiction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind's validation example)
This example is not a literature case; it is a formula-faithful validation fixture shared across the FUZZY-* family (the same matrix is also used on the FUZZY-CODAS card, because DecisionMind's fuzzy family is tested on a common synthetic fixture). Three alternatives are evaluated on three criteria with triangular fuzzy scores; the first two criteria are "more is better", the third is "less is better" (cost).
| Alternative | C1 | C2 | C3 (cost) |
|---|---|---|---|
| A1 | (0.65; 0.70; 0.75) | (0.45; 0.50; 0.55) | (0.55; 0.60; 0.65) |
| A2 | (0.75; 0.80; 0.85) | (0.55; 0.60; 0.65) | (0.35; 0.40; 0.45) |
| A3 | (0.55; 0.60; 0.65) | (0.65; 0.70; 0.75) | (0.45; 0.50; 0.55) |
| Direction | more is better | more is better | less is better |
| Weight | (0.35; 0.40; 0.45) | (0.30; 0.35; 0.40) | (0.20; 0.25; 0.30) |
The method normalises every triangle against the [worst, best] range, directly for the benefit criteria and crosswise for the cost criterion, multiplies and sums component by component with the weight triangles, and reduces the resulting aggregate-utility triangle to a single number by the centroid.
| Alternative | Total utility | Rank |
|---|---|---|
| A2 | 0.7167 | 1 |
| A3 | 0.4833 | 2 |
| A1 | 0.3000 | 3 |
The result reads as follows. A2 holds the highest score on C1, the most heavily weighted criterion, and the lowest (best) score on the cost criterion C3; these two advantages more than offset its relative weakness on C2. A1 carries the lowest total utility because it is weakest on C1 and most expensive on C3 (cost).
The board's hesitation is this. If C2's weight is raised from 0.35 to 0.62, with C1 and C3 reduced by the same proportion (C1≈0.20 and C3≈0.18), and the calculation redone, A3 moves ahead (0.64) and A2 drops to second (0.6267). A2's first place is a moderately-to-highly robust result, one that breaks down only if C2's weight is raised markedly (above roughly 0.60).
In the report: "With the given weights (C1=0.40, C2=0.35, C3=0.25), A2 has the highest total utility (0.7167); when C2's weight is raised to 0.62, A3 moves ahead (0.64 / 0.6267), so A2's first place is moderately-to-highly robust to C2's weight."
Source: Validation fixture for DecisionMind's Fuzzy MAUT engine; the steps are defined in the manifest, following an additive Fuzzy MAUT algorithm based on TFN range-normalisation. All numbers have been independently recomputed by this card's author with the same algorithm and verified against the engine output.
2. Logistics: A shipping company's choice of new warehouse automation system
A logistics company will choose among three warehouse automation systems (mixed conveyor-and-robotics solutions). Criteria: installation cost (less is better), daily throughput capacity, and maintenance/failure frequency (less is better). Since the company has not yet tested the systems on site, each criterion has been scored with triangular fuzzy numbers reflecting the range between the manufacturer's stated figures and an independent consultant's estimate; the consultant has also shared a "lowest, most likely, highest" estimate for installation cost.
The method normalises the three systems' triangles (crosswise for the cost and failure-frequency criteria), multiplies and sums them with the weights, and reduces the result by the centroid. Suppose the system with the highest throughput capacity also has the highest installation cost and the most frequent failures; it still comes first, because the capacity weight exceeds the sum of the other two.
The company's hesitation is this. Choosing the most expensive and most failure-prone system requires a trade-off between short-term budget and long-term operating risk; if the maintenance/failure-frequency criterion's weight is raised somewhat, taking share from the capacity weight, whether the second-ranked system moves ahead should be tested separately.
In the report: "With the high weight given to throughput capacity, one system reaches the highest total utility; this result is sensitive to the maintenance/failure-frequency criterion's weight, and this sensitivity must be shown to management separately before installation."
3. What Not to Do
The first error is reducing the triangles in the illustrative example to their centroid from the outset and running crisp MAUT. The ranking comes out the same (A2, A3, A1), but the A1-A2 gap inflates from 0.417 to 0.625; this is crisp MAUT, not Fuzzy MAUT, and it hides the contribution of the width in range-normalisation.
The second error, as in the logistics example, is mixing the manufacturer's statement and the consultant's estimate into a single triangle without justification, rather than treating the former as "most likely" and the latter as "lowest/highest". Each corner of the triangle must rest on its own source.
The third error is skipping crosswise normalisation on a criterion such as cost or failure frequency and applying the benefit criterion's formula directly. This breaks the triangle's ordered structure (l≤m≤u) and invalidates the subsequent aggregation step.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/fuzzy-maut
Keeney, R. L., & Raiffa, H. (1976). Decisions with Multiple Objectives: Preferences and Value Trade-offs. Wiley. (no DOI)
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X
Chen, C.-T. (2000). Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets and Systems, 114(1), 1–9. DOI: 10.1016/S0165-0114(97)00377-1 (The operational basis for this extension's triangular fuzzy number algebra; not an extension paper specific to MAUT, see the note above and the approval notes.)
Dyer, J. S. (2005). MAUT — Multiattribute Utility Theory. In J. Figueira, S. Greco, & M. Ehrgott (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys (pp. 265–292). Springer. DOI: 10.1007/0-387-23081-5_7