Extension card · Hesitant
Hesitant GRA (Li and Wei, 2014)
Hesitant GRA is the form of GRA that works for situations where several plausible values for one criterion — expert opinions, scenarios, repeated measurements — are preserved together rather than collapsed into a single number. It builds its reference from both the best and the worst end, and ranks the result with a closeness ratio.
Base method
GRA →
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?
Five things change; the formula for the grey relational coefficient does not.
Cells. In crisp GRA every cell is a single number. Here every cell is a set of variable length, that is, several plausible values: two values such as {0.6; 0.5}, or three such as {0.7; 0.6; 0.3}. Each value reflects its own source, an expert or a scenario. When there is more than one decision-maker, the sets are first merged with an aggregation operator (HFSA or HFWA); in a single-decision-maker application, this step passes through without changing the matrix itself.
Length equalisation. To compare two sets, both are first brought to the length of the longer one. DecisionMind uses Xia and Xu's (2011) pessimistic rule: the set's own smallest value is appended to the end of the shorter set; an optimistic completion (appending the largest value) is not performed. Each set is then sorted from largest to smallest. This sorting and equalisation is carried out before every distance calculation that follows.
The reference is built from two ends. In crisp GRA the reference sequence is a fixed single positive ideal, (1; 1; …; 1). Here there are two references: the positive ideal (PIS), the position-by-position largest of every column's length-equalised values; and the negative ideal (NIS), the position-by-position smallest in the same way. For a cost criterion these two references swap places. This differs from the family's other members (grey, intuitionistic, neutrosophic, Fermatean fuzzy GRA); all of these look at a single positive ideal, whereas here, as in TOPSIS, there are two end points.
Distance. The distance between two sets is found by taking the sum of the squared position-by-position differences of the length-equalised and sorted values, dividing by the set's length and taking the square root (the hesitant normalised Euclidean distance). This is the set-extended form of the single-number difference used in crisp GRA.
Grey relational coefficient and closeness ratio. In crisp GRA a single grey relational degree is computed and ranked directly. Here a separate grey relational degree is built against both the positive ideal (ξ⁺) and the negative ideal (ξ⁻); the discriminating coefficient is again ρ = 0.5. The result is a closeness ratio resembling TOPSIS's closeness coefficient: η = ξ⁺ / (ξ⁺ + ξ⁻). Alternatives are ranked from largest to smallest on this ratio.
DecisionMind fixes, for this extension, the pessimistic length equalisation, the two-ideal (PIS/NIS) structure and the η closeness ratio; weights are taken from outside as crisp numbers.
How to Read the Output
The closeness ratio η is read between 0 and 1, just like the grey relational degree in crisp GRA, and cannot be compared with a different analysis; it is derived from the positive and negative ideals specific to this analysis.
The difference lies here. η carries both closeness to the ideal and distance from the anti-ideal together; unlike crisp GRA's single-reference degree, an alternative can still get a low η even when close to the ideal, if it is also close to the anti-ideal. Also, the number of values (the length) in a set can differ between alternatives. Because a short set is completed with the pessimistic rule, the difference between a very short set (one value) and a very long one (three values) is affected not only by the average but also by this completion rule.
Thus instead of writing:
"According to Hesitant GRA, x3 came out as the most reliable candidate"
the report should read:
"x1's closeness ratio, at 0.6627, is the highest; this comes from x1 staying, across the sum of the five criteria, both close to the positive ideal and far from the negative ideal, but its gap to x3 (0.042) is small and sensitive to the weight of a single criterion"
When to Prefer This over the Base Method
Use this extension when several plausible values exist for one criterion and reducing them to an early average would lose information. Examples: separate scores several experts give the same criterion, separate outcomes produced by different scenarios, and decision-relevant differences from repeated measurements. Every value must have a defensible source; fabricating extra values around a crisp value does not produce hesitant data.
If criteria are measured with a single reliable value, stay with base GRA. Should the table be mixed, DecisionMind asks for a single data type; a crisp criterion is written as a set with one element. Base GRA's exit condition applies here too: if a criterion carries a threshold that can never be compromised, GRA's additive structure will not protect it.
Mistakes Specific to This Extension
Adding values to a set without justification. Adding artificial values around a crisp value by saying "a little lower, a little higher" does not produce hesitant data, it produces artificial uncertainty. Every value must show its own source.
Skipping length equalisation or applying it optimistically. Two sets of different length cannot be compared position by position directly; DecisionMind uses the pessimistic rule (repeating the short set's own smallest value). Repeating the largest value (an optimistic completion) produces a different, higher distance and a different ranking; whichever rule is used must be stated in the report.
Confusing it with the single-ideal family. Grey, intuitionistic, neutrosophic and Fermatean fuzzy GRA look only at the positive ideal and give a degree directly through a weighted sum. Hesitant GRA looks at both the positive and the negative ideal and gives a closeness ratio (η). Confusing the two calculations produces a different ranking.
Failing to swap the references on a cost criterion. If the positive and negative ideal are not swapped on a "lower is better" criterion, the reference is built from the wrong end.
Never questioning the discriminating coefficient. ρ = 0.5 is DecisionMind's fixed value; in tables where the distances are close to one another, this choice can affect the result.
The governing principle is this:
In Hesitant GRA, every value in every set must carry its own source, length equalisation must be done consistently with the pessimistic rule, and it must be remembered that the result is a two-ideal closeness ratio, not a single-ideal grey relational degree.
Cases
The first case comes from the literature: the recruitment candidate-selection example in Li and Wei's (2014) paper. This example carries an important caveat: the result in the paper's own table and the result produced by DecisionMind's engine differ in ranking, a difference arising from a different reading of the same formula chain; this difference is stated explicitly below. The second case is an illustrative fiction.
1. Recruitment: Three candidates evaluated on five criteria in hesitant form (Li and Wei, 2014)
An organisation is evaluating three candidates (x1, x2, x3) for a senior position on five criteria: emotional stability, verbal communication, personality fit, past experience and self-confidence (all five "higher is better"). Scores exist from four evaluating experts; DecisionMind's engine uses one representative of these four (e1) as input. The weights are W = (0.15; 0.20; 0.225; 0.20; 0.225).
| Candidate | Emotional stability | Verbal communication | Personality fit | Past experience | Self-confidence |
|---|---|---|---|---|---|
| x1 | {0.6; 0.5} | {0.7; 0.5; 0.3} | {0.7; 0.6; 0.3} | {0.7; 0.5} | {0.8; 0.5} |
| x2 | {0.7; 0.3} | {0.6; 0.3} | {0.7; 0.4; 0.3} | {0.8; 0.4; 0.2} | {0.8; 0.7; 0.3} |
| x3 | {0.7; 0.5; 0.2} | {0.8; 0.7; 0.2} | {0.8; 0.5; 0.3} | {0.7; 0.4} | {0.6; 0.5} |
The method equalises the lengths in every column with the pessimistic rule, builds the positive and negative ideal (PIS/NIS) from the position-by-position largest and smallest values, computes the hesitant normalised Euclidean distance, converts it into grey relational coefficients with ρ = 0.5, and finds the closeness ratio (η).
| Candidate | Closeness ratio (η) | Rank |
|---|---|---|
| x1 | 0.6627 | 1 |
| x3 | 0.6207 | 2 |
| x2 | 0.4829 | 3 |
The result reads as follows. x1 and x3 are close to one another (a gap of 0.042), while x2 trails the other two on most of the five criteria and finishes last by a clear margin.
The organisation has an important hesitation here. Li and Wei's (2014) own paper reports the order x3 ≻ x1 ≻ x2 on this data (η ≈ 0.608; 0.566; 0.423). DecisionMind's engine, applying the manifest's F1-F7 steps to the letter (a single representative expert's HFSA reading, PIS/NIS, hesitant Euclidean distance, ρ = 0.5 grey coefficient), instead gives x1 ≻ x3 ≻ x2; only x1 and x3 swap places, and x2 finishes last in both calculations. This difference appears to stem from the paper's own table possibly using a Choquet integral with the four experts' Shapley fuzzy measure; this extra layer is not present in the manifest's F steps. In line with the "no drift from the literature" principle, DecisionMind has stayed faithful to its own formula chain and adopted the engine-produced values as authoritative.
The organisation's second hesitation concerns the weights. Verified with an independent Python calculation, if the verbal-communication weight is raised from 0.20 to 0.55 and the other four criteria are lowered accordingly (0.10; 0.55; 0.15; 0.10; 0.10), x3 (0.678) overtakes x1 (0.639). So x1's lead is sensitive both to the aggregation method used and to the criterion weights.
In the report: "DecisionMind's engine, following the manifest's formula chain, places x1 first with the highest closeness ratio (0.6627); the source paper's own table gives x3 first instead, a difference that most likely stems from the paper using an additional Choquet-Shapley aggregation step. The gap between x1 and x3 is small (0.042), and the ranking shifts in x3's favour when the verbal-communication weight is raised; both findings should be read together and left to scientific review."
Source: Li, X., & Wei, G. (2014), Journal of Intelligent & Fuzzy Systems, 27(3), pp. 1095-1105, §5 (Table 1 input, the four experts' Shapley fuzzy measure, final η values). DecisionMind's engine was run independently on this input; x2 finishing last agrees between the two calculations, but the x1-x3 order does not. The closeness ratios and the weight sensitivity were computed independently by this card's author in Python; the results match the DecisionMind manifest's kernel-recomputation record exactly (see the sign-off notes).
2. Education: A foundation's evaluation of academic potential among scholarship candidates
An education foundation is evaluating three candidates for an overseas master's scholarship on four criteria: academic performance trajectory, research potential, language proficiency and strength of reference letters (all four "higher is better"). Three panel members evaluate each candidate's academic performance trajectory not with a single mark but with a set drawn from the student's grades over the last three terms; the other criteria carry a single consensus score.
The method equalises the lengths on the multi-valued academic-performance criterion with the pessimistic rule, builds the positive and negative ideal, computes the hesitant distance, and finds the closeness ratios. Suppose a candidate whose grades rise steadily comes out first, even though this candidate is not the strongest on research potential.
The foundation's hesitation lies here. If this candidate's academic-performance set spreads over a wide range such as {0.55; 0.70; 0.85}, the pessimistic equalisation rule pulls this breadth down somewhat when compared against the other candidates' single-valued criteria. The foundation must decide whether this candidate's most recent term or the three-term trajectory should form the basis of the decision.
In the report: "The candidate with a steadily rising academic-performance trajectory comes first on the closeness ratio, despite a relative disadvantage on research potential. This result stems from preserving all three terms together in the academic-performance set; had only the most recent term's score been taken, the ranking could have changed."
3. What Not to Do
In the recruitment example, the first mistake is reducing x1's personality-fit set {0.7; 0.6; 0.3} to a single number by taking only its average (0.533); this erases the three experts' disagreement and destroys the hesitant structure's contribution. The second mistake is presenting DecisionMind's engine-produced order (x1 ≻ x3 ≻ x2) as if it were the paper's own table, or the reverse; the two are the results of separate calculation chains, and which one is used must be stated in the report. The third mistake is repeating the largest value when length-equalising a short set (an optimistic completion); DecisionMind uses the pessimistic rule, and this produces a different distance and a different ranking.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/hf-gra
Li, X., & Wei, G. (2014). GRA method for multiple criteria group decision making with incomplete weight information under hesitant fuzzy setting. Journal of Intelligent & Fuzzy Systems, 27(3), 1095–1105. DOI: 10.3233/IFS-131073
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Xia, M., & Xu, Z. (2011). Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 52(3), 395–407. DOI: 10.1016/j.ijar.2010.09.002
Guan, X., Sun, G., Yi, X., & Zhao, J. (2018). Grey relational analysis for hesitant fuzzy sets and its applications to multiattribute decision-making. Mathematical Problems in Engineering, 2018, Article 7436054. DOI: 10.1155/2018/7436054
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)