Extension card · Fuzzy
Fermatean Fuzzy VIKOR (Gül, 2021)
This is the form of VIKOR for situations where the cubes of the support and rejection degrees given to a judgement sum to no more than 1. This constraint lets the two degrees be high together over a region wider than Pythagorean fuzzy allows; the method reduces every cell to a distance straight away, and runs the rest exactly as crisp VIKOR does.
Base method
VIKOR →
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.
Cells. In crisp VIKOR every cell is a single number. Here every cell is a support (μ) and rejection (ν) degree pair given to a judgement. The constraint is μ³+ν³≤1. The sum of cubes grows more slowly than the sum of squares, so this constraint permits some pairs (both at 0.85, say) that Pythagorean fuzzy does not allow. Criterion weights are taken from outside as crisp numbers.
Scale equalisation (direction reversal). On a cost criterion, crisp VIKOR swaps the best and worst value. Here, on a cost criterion, the support and rejection degrees swap places: μ and ν exchange, and because the constraint (μ³+ν³≤1) is symmetric, this complementing remains valid.
Best/worst value, distance and combination. The method builds the best pair (highest support, lowest rejection) and the worst pair (lowest support, highest rejection) on every criterion by gathering them together. It computes each alternative's Euclidean distance to these two reference points; the ratio of this distance to the distance between the best and worst pair corresponds to crisp VIKOR's normalised difference. The method weights and sums this ratio (S) and separates out its largest value (R); it then computes v and Q. From this point on, the calculation proceeds entirely with crisp numbers.
DecisionMind fixes v at its default of 0.5 in this extension; as with the base method, it does not test crisp VIKOR's two compromise conditions here (acceptable advantage, acceptable stability), and produces only a ranking by Q. The user must check this for themselves by looking at the S and R columns. Weights come from outside; the method does not generate weights.
How to Read the Output
A lower Q is better. However, the two conditions described on the base VIKOR card, acceptable advantage and acceptable stability, are not tested by DecisionMind in this extension; the user must check this for themselves by looking at the S and R columns. Two alternatives' Q values can come out close or equal; this is a normal and honest VIKOR result, not a computational error. The cube constraint only permits a wider (support, rejection) region; the result is not automatically more accurate or more reliable.
Thus instead of writing:
"According to Fermatean fuzzy VIKOR, the best alternative is A2"
the report should read:
"A2 is the alternative with the smallest Q under these weights and at v = 0.5; because DecisionMind does not test the compromise conditions in this extension, whether this is a single compromise solution or a compromise set must be assessed separately by looking at the S and R columns"
When to Prefer This over the Base Method
Consider this extension when experts' support-rejection pairs exceed even the Pythagorean fuzzy structure's squared bound (for example μ=0.9, ν=0.7: the sum of squares, 1.30, exceeds the Pythagorean bound, and if the sum of cubes, 0.729+0.343=1.07, still exceeds the bound too, q-Rung orthopair is needed; where it does not, Fermatean is sufficient); the detail is on the Pythagorean fuzzy data-type card. Where the two degrees already fit the Pythagorean bound, moving to Fermatean is not necessary, and only discrimination is lost. Converting a measured value directly into a support-rejection pair is not modelling uncertainty but manufacturing it. Crisp VIKOR's own exit condition applies here too: if no compromise is acceptable on one criterion, this extension also limits regret but does not eliminate it.
Mistakes Specific to This Extension
Domain violation. Every cell must satisfy μ³+ν³ ≤ 1; this is checked from the sum of cubes, not squares.
Writing ν as 1 − μ. In that case the sum is always exactly 1, and the extra room the cube constraint provides is never used.
Assuming the two conditions are tested. DecisionMind does not compute the acceptable-advantage and acceptable-stability conditions in this extension; it produces only a Q ranking. S and R must be read separately before using the phrase "single compromise solution."
Choosing Fermatean for pairs that already fit the intuitionistic or Pythagorean bound. If the pairs already fit the narrower bound, the wider region adds no further information, only reduces discrimination.
The governing principle is this:
Fermatean fuzzy VIKOR's cube constraint adds something only when the experts' pairs genuinely exceed the Pythagorean bound; in this extension, crisp VIKOR's two compromise conditions are not tested by the engine and must be assessed separately in the report by looking at the S and R columns.
Cases
The first case is DecisionMind's validation example; the synthetic 3x3 table in the manifest is built faithfully to the formulas and carries no literature source. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Choosing a corporate ERP cloud provider
A company compares three cloud providers on three criteria for migrating its enterprise resource planning (ERP) system: service uptime rate (more is better), integration quality (more is better), and support-ticket rejection rate (less is better). Each provider's support and rejection on every criterion is given as a Fermatean pair (μ,ν).
| Provider | Uptime rate | Integration quality | Support-ticket rejection rate (less is better) |
|---|---|---|---|
| A1 | (0.70, 0.40) | (0.50, 0.50) | (0.60, 0.50) |
| A2 | (0.80, 0.30) | (0.60, 0.40) | (0.40, 0.60) |
| A3 | (0.60, 0.50) | (0.70, 0.40) | (0.50, 0.50) |
| Weight | 0.40 | 0.35 | 0.25 |
The method swaps support and rejection on the cost criterion (support-ticket rejection rate), determines the best and worst Fermatean pair on every criterion, and measures and normalises each provider's distance to these two references. It weights and sums the distances (S) and separates out the largest (R). It computes Q at v = 0.5.
| Provider | S | R | Q |
|---|---|---|---|
| A2 | 0.228 | 0.228 | 0.000 |
| A3 | 0.552 | 0.400 | 0.766 |
| A1 | 0.836 | 0.350 | 0.854 |
The result reads as follows. A2 is both the provider closest to the ideal overall, first on S, and the one with the smallest distance on its worst criterion, first on R. Because DecisionMind does not test the conditions in this extension, S and R must be looked at separately before saying "single compromise solution": A2's clear lead on both measures suggests the result is not fragile, but this reading is not a condition test the engine itself produces.
The company's hesitation is sensitive to the weighting. If the weight on integration quality is raised from 0.35 to 0.60 and the other two criteria are each brought down to 0.20, the ranking reverses completely: A3 comes first (Q=0), A2 second (Q=0.298), A1 last (Q=1.0). This shows that a single weight trade-off in a three-criterion table can change the ranking from end to end.
In the report: "With the given weights (uptime 0.40, support-ticket rejection rate 0.25), A2 has by far the smallest Q (0.000); once the weight on integration quality is raised to 0.60, the ranking reverses completely in A3's favour. Because DecisionMind does not test the compromise conditions in this extension, which ranking is robust must be assessed separately by looking at the S and R columns."
Source: DecisionMind's Fermatean fuzzy VIKOR validation example. The distance and VIKOR skeleton rest on Senapati and Yager's (2020) Fermatean fuzzy set definition and on Gül's (2021) paper adapting VIKOR to the Fermatean fuzzy set. The evaluation values and the weight-trade-off scenario were independently computed by this card's author by running the kernel directly; the manifest's own decision table does not carry Gül's paper's numerical example but a synthetic table built faithfully to the formulas (detail in the approval note).
2. Logistics: A company's choice of last-mile delivery operator
An e-commerce company will hand its last-mile delivery business to one of three operators. There are three criteria: on-time delivery rate, service-area coverage, and customer complaint rate (less is better). The operations team records, for each operator, both strong evidence in its favour and serious reservations at the same time; for some operators the two values are high together, because the team has observed both strong delivery performance and recurring complaints in certain areas simultaneously.
The method computes each operator's Fermatean distance, normalises it against the best and worst value on each criterion, and separates out the weighted sum (S) and the distance on the worst criterion (R). Suppose the operator strongest on on-time delivery also came out above average, unfavourably, on customer complaint rate; it still emerged with the smallest Q, because the weight on on-time delivery had been kept high.
The company's hesitation is this: for an operator whose support and reservation are both high together, this two-sided information disappears once it collapses into a single Q figure. The company should carry out a separate field audit for operators whose Q comes out low but whose reservation degree is also high; because DecisionMind does not test the compromise conditions in this extension, it must assess for itself, by looking at the S and R columns, how robust the ranking is.
In the report: "One operator has the smallest Q because the weight on on-time delivery was kept high; a reservation about this operator's customer complaint rate has also been noted, and a separate field audit is recommended."
3. What Not to Do
Had support-ticket rejection rate been marked "more is better" in the illustrative example, the provider with the most rejections would have been rewarded, and the ranking would become meaningless. A second error is declaring A2's Q=0 value a "single compromise solution" and basing this on a condition test; DecisionMind performs no such test in this extension. A third error is entering a pair such as μ=0.9 and ν=0.9 without checking the μ³+ν³ ≤ 1 constraint (the sum of cubes, 1.458, exceeds the constraint); this invalidates the calculation.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ff-vikor
Gül, S. (2021). Fermatean fuzzy set extensions of SAW, ARAS, and VIKOR with applications in COVID-19 testing laboratory selection problem. Expert Systems, 38(8), e12769. DOI: 10.1111/exsy.12769
Senapati, T., & Yager, R. R. (2020). Fermatean fuzzy sets. Journal of Ambient Intelligence and Humanized Computing, 11, 663–674. DOI: 10.1007/s12652-019-01377-0
Yager, R. R. (2014). Pythagorean membership grades in multicriteria decision making. IEEE Transactions on Fuzzy Systems, 22(4), 958–965. DOI: 10.1109/TFUZZ.2013.2278989
Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems (Višekriterijumska optimizacija sistema u građevinarstvu). Doctoral dissertation, University of Belgrade, Faculty of Civil Engineering. (no DOI)
Opricovic, S., & Tzeng, G.-H. (2004). Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2), 445–455. DOI: 10.1016/S0377-2217(03)00020-1