Extension card · Hesitant
Hesitant fuzzy VIKOR (Liao and Xu, 2013)
This is the form of VIKOR for situations where more than one plausible membership degree is held together for a single criterion. It computes group utility and individual regret using distances between sets, and ranks the result, once again, with a compromise index.
Base method
VIKOR →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Hesitant →
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 decision logic stays the same.
Cells. In crisp VIKOR every cell is a single number. Here every cell is a set that holds several plausible values for the same criterion together. Weights arrive fully from outside, as in base VIKOR; HF-VIKOR does not generate weights.
Scale equalisation. Sets can be different lengths within a criterion column. DecisionMind extends short sets with a fixed filler value. The default filler is 0.5, meaning a neutral degree, "neither good nor bad." This is a different design choice from HF-TOPSIS's approach, which generates its filler from the set's own extreme values: HF-VIKOR fills with a neutral constant, whereas HF-TOPSIS fills with a blend of the set's own minimum and maximum.
Distance, score and combination. For every criterion, the best and worst set are determined first by looking at the mean degree and, in case of a tie, by the set's variance. The lower-variance set is preferred, because it is less dispersed. This extends crisp VIKOR's single-number comparison to sets. The distance between two sets is found as follows: both are first extended to the same length and sorted from largest to smallest, then the absolute differences between values at matching positions are summed and averaged. This is called the hesitant Manhattan distance. Group utility S̃ and individual regret R̃ are built by scaling these set distances against the best-worst range and either summing them with the weights or taking the largest. The logic is the same as for S and R in crisp VIKOR; only now the distance being measured is between sets.
Result and defuzzification. The output is again S̃, R̃ and the Q̃ that combines them. The sets descend to a single number only once these three values have been calculated and the compromise conditions have been tested. Throughout the rest of the calculation, the hesitancy is preserved.
For classical HF-VIKOR, DecisionMind fixes the filler value at 0.5 and the compromise coefficient v at 0.5. Both can be changed by the user.
How to Read the Output
S̃, R̃ and Q̃ are read together, exactly as in crisp VIKOR, along with the two conditions. What differs is this: the length-harmonisation filler for short sets can change the ranking itself. The middle-ranked alternatives, in particular, are sensitive to this value. This is an additional source of sensitivity that crisp VIKOR does not have.
Thus instead of writing:
"According to HF-VIKOR, the best alternative is A2"
the report should read:
"With these weights, v = 0.5 and a filler value of 0.5, A2 is the single compromise solution; when the filler value is changed, the second- and third-ranked alternatives swap places"
When to Prefer This over the Base Method
This extension is appropriate when more than one plausible, defensible degree exists for the same criterion-alternative pair. The distinction on the data-type card applies here too: where a single measured value exists, a single number is written into the cell, and no set is built. DecisionMind requires the table to hold a single data type. Base VIKOR's exit condition applies exactly as before: if no compromise at all is acceptable on one criterion, this extension also limits regret but does not eliminate it.
Mistakes Specific to This Extension
Changing or assuming the length-harmonisation filler value without declaring it. As Case 1 shows, changing the filler between 0 and 1 alters the second, third and fourth ranks.
Comparing sets by mean alone and skipping the variance. When two sets share the same mean, DecisionMind prefers the one with lower variance. Skipping this step means the ideal and anti-ideal sets are chosen wrongly.
Choosing the compromise coefficient v as 0 or 1. The same mistake as in crisp VIKOR applies here: at these values the method stops being a compromise method.
The "more sophisticated" fallacy. Without a genuine source supporting more than one value, building a set adds no information.
The governing principle is this:
If the hesitant set's length-harmonisation filler and the between-set comparison rule are chosen without justification, S̃, R̃ and Q̃ come out differently from the same data. The filler value and the comparison rule must be stated explicitly in the report.
Cases
The first case is a genuine case from the literature. It is the Taiwanese domestic-airline example from Liao and Xu's (2013) paper, and the figures are taken from the paper. The second case is an illustrative construction.
1. Aviation: Comparing the service quality of four domestic airlines (Liao and Xu, 2013)
A civil aviation authority evaluates four domestic airlines on four service-quality criteria. All four criteria are of the higher-is-better kind. The scores from passenger surveys are not single numbers; they are sets made up of several plausible degrees observed across different survey periods. The weights have been set by the authority, and the filler value and v have been left at their defaults.
| Airline | ζ1 | ζ2 | ζ3 | ζ4 |
|---|---|---|---|---|
| UNI Air | {0.6; 0.7; 0.9} | {0.6; 0.8} | {0.3; 0.6; 0.9} | {0.4; 0.5; 0.9} |
| Transasia | {0.7; 0.8; 0.9} | {0.5; 0.8; 0.9} | {0.4; 0.8} | {0.5; 0.6; 0.7} |
| Mandarin | {0.5; 0.6; 0.8} | {0.6; 0.7; 0.9} | {0.3; 0.5; 0.7} | {0.5; 0.7} |
| Daily Air | {0.6; 0.9} | {0.7; 0.9} | {0.2; 0.4; 0.7} | {0.4; 0.5} |
| Weight | 0.10 | 0.20 | 0.40 | 0.30 |
The method determines the best and worst set for each criterion first by mean and, in case of a tie, by variance. It then measures each airline's distance to these two sets and converts this into weighted ratios. It calculates group utility S̃ and individual regret R̃, and combines them in Q̃.
| Airline | S̃ | R̃ | Q̃ | Rank |
|---|---|---|---|---|
| Transasia | 0.133 | 0.067 | 0.000 | 1 |
| Mandarin | 0.367 | 0.200 | 0.380 | 2 |
| UNI Air | 0.740 | 0.300 | 0.819 | 3 |
| Daily Air | 0.780 | 0.400 | 1.000 | 4 |
The result reads as follows. Transasia is best both overall and on its worst criterion. Both conditions are satisfied: under acceptable advantage, the gap is 0.380, above the threshold of 0.333; under acceptable stability, Transasia ranks first on both S̃ and R̃. Transasia is the single compromise solution.
The authority's hesitation is this. What happens if the filler value for short sets is taken as 0 or 1 instead of 0.5? Transasia remains first in every case, but the order of the remaining three airlines is sensitive to the filler. With a filler of 0, the order becomes Transasia, Daily Air, Mandarin, UNI Air. With a filler of 1, the order becomes Transasia, UNI Air, Mandarin, Daily Air. Only around a filler of 0.5 does Mandarin remain stable in second place.
In the report: "Transasia is the single compromise solution because it ranks first on both S̃ and R̃ (Q̃ = 0). The ranking from second place onward changes when the filler value is changed, so the filler value used must be stated in the report."
Source: Liao and Xu (2013), Tables 1-5, pp. 387-390. Example 5, Taiwanese domestic airlines, adapted from Liou et al.'s 2011 data. The S̃, R̃ and Q̃ values were obtained by independently rerunning DecisionMind's HF-VIKOR engine. The filler-sensitivity scenarios were calculated by this card's author using the same algorithm.
2. Librarianship: Choosing an electronic-database subscription
A university library, working with a limited budget, will subscribe to one of three electronic-database providers. There are four criteria: annual subscription fee, content-coverage score, interface-usability score and concurrent-access capacity. The fee is of the lower-is-better kind. The library committee has found that user feedback from different faculties gives different satisfaction degrees for every provider-criterion pair. It keeps these degrees together without collapsing them into an average.
The method calculates each provider's distance to the best and worst criterion set, and combines group utility and individual regret. Suppose the provider offering the most comprehensive content is also the most expensive. It nonetheless comes out as the single compromise solution, because the weight on content coverage is higher than the weight on fee.
The committee's hesitation is this. The engineering faculty's low score for interface usability has been kept in the same set as the other two faculties' high scores. The width of this set, for example {0.3; 0.8}, means that this provider's user experience varies considerably by faculty. The committee should investigate separately why one faculty's score differs, before reducing wide-set criteria to a single compromise score.
In the report: "With the high weight given to content coverage, this provider is the single compromise solution. The width of the interface-usability set reflects a difference in satisfaction across faculties and should be assessed separately."
3. What Not to Do
In the airline table, saying "the default was used" without stating the filler value is wrong. Because the filler gives a different result from second place onward as it varies between 0 and 1, which filler was used must be written in the report. The second error is marking two sets as "equal" when their means match, without comparing their variance. DecisionMind prefers the lower-variance set; skipping this step means the ideal set is chosen wrongly. The third error is reading Transasia's Q̃ = 0 as "a flawless airline." This only means it is in the best position among these four airlines.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/hf-vikor
Liao, H., & Xu, Z. (2013). A VIKOR-based method for hesitant fuzzy multi-criteria decision making. Fuzzy Optimization and Decision Making, 12(4), 373–392. DOI: 10.1007/s10700-013-9162-0
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
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Xu, Z., & Xia, M. (2011). Distance and similarity measures for hesitant fuzzy sets. Information Sciences, 181(11), 2128–2138. DOI: 10.1016/j.ins.2011.01.028