Extension card · Hesitant
Hesitant Fuzzy ELECTRE II (Chen and Xu, 2015)
This is the form of ELECTRE II for situations where performance scores are given as hesitant fuzzy sets. The fuzziness is carried through to the very last step and is never collapsed early into a single average; the output remains a complete ranking.
Base method
ELECTRE II →
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?
Four things change; the logic of the two-way distillation and incomparability does not.
Cells. In crisp ELECTRE II every cell is a single number. Here every cell becomes a set (a hesitant fuzzy element, HFE) in which several plausible membership degrees for the same judgement are held together. When a short set is compared with a longer one, the missing elements are filled in. This method uses the optimistic rule here: the missing elements are filled by repeating the set's own largest value. This is the exact opposite of the pessimistic rule (repeating the smallest value) used by the family's Hesitant ELECTRE I member.
Score and consistency instead of scale equalisation. Crisp ELECTRE II's vector normalisation is absent here. Instead, a score (the mean membership) and a deviation (the set's own internal consistency) are extracted from every HFE, and comparisons are made on these two numbers.
Concordance and discordance. Crisp ELECTRE II has a single concordance index and a single discordance index. Here three levels of concordance (strong, moderate, weak) and three levels of discordance (strong, moderate, weak) are built separately; the relative importance given to them is adjusted through seven user-set "attitude weights." One alternative being superior to another on both score and consistency counts as strong concordance, superiority on score alone counts as moderate concordance, and being equal on score but more consistent counts as weak concordance; a symmetric distinction is made for discordance. The discordance index is the ratio of the worst difference, computed with the hesitant distance measure, to the largest difference.
The form of the thresholds. In crisp ELECTRE II, two concordance thresholds (s1, s2) are chosen by the analyst. Here, too, thresholds are chosen by the user, but there are more of them: three concordance thresholds (c* > c' > c⁻) and two discordance thresholds (d' < d*). This is the point where it departs most from the family's Hesitant ELECTRE I member; there, the thresholds derive from the data's mean, whereas here they are set by the user, as in crisp ELECTRE II. Strong concordance requires the strictest of these thresholds, weak concordance the loosest. What DecisionMind fixes in this classical Hesitant ELECTRE II is that the strong and weak outranking graphs are built separately and then combined by averaging the forward (best-down) and backward (worst-up) distillations; this is exactly crisp ELECTRE II's own distillation logic.
How to Read the Output
What stays the same as in the base method is this: the output is a position, not a score; strict preference, equality and incomparability are three distinct relations.
The difference lies here: the concordance and discordance that determine this position come not from a single score comparison but from a three-level classification built jointly from score and consistency. Even when two alternatives share the same average score, the more consistent one can win weak concordance, and this can affect its position in the ranking.
Thus instead of writing:
"Hesitant ELECTRE II placed A1 first, with the rest strictly second and third"
the report should read:
"With the thresholds chosen, A1 comes out ahead in both the forward and the backward distillation; this result rests not only on the average score but also on which alternative was judged more consistently on which criterion"
When to Prefer This over the Base Method
Use this extension when a performance judgement is made up of several plausible values and a complete ranking is required, rather than a set of front-runners. Opening a measured performance out into a hesitant set manufactures uncertainty here too; a measured cell is written as the set's sole element.
Crisp ELECTRE II's exit condition (a complete ranking together with non-compensatory logic) applies here in exactly the same way. If reducing several plausible values to a core set, rather than a single ranking, is enough, the family's Hesitant ELECTRE I member, whose thresholds derive from the data and which asks for fewer parameters, may be more suitable.
Mistakes Specific to This Extension
Confusing the optimistic and pessimistic length-equalisation rules. This method extends a short HFE by repeating its own largest value (the optimistic rule). The family's Hesitant ELECTRE I member uses the rule that repeats the smallest value (the pessimistic rule). Confusing the two can make the alternative with the shorter set look better or worse than it is.
Acting as if the analyst does not choose the five thresholds (three concordance, two discordance). In this method, the thresholds are set by the user, as in crisp ELECTRE II; they do not derive from the data as they do in the family's Hesitant ELECTRE I member. Confusing the two produces the false justification that "the thresholds already came from the data."
Setting the seven attitude weights (strong/moderate/weak concordance and discordance) equal without thought. These weights determine how much distinguishing power strong and weak concordance carry in the ranking; setting them equal without consideration arbitrarily inflates or shrinks the effect of the consistency difference.
The governing principle is this:
In Hesitant ELECTRE II, fuzziness is carried through to the very last step, but the thresholds here are not the data's mean, they are chosen by the user, as in crisp ELECTRE II. The robustness of the ranking should be read against these five thresholds and seven attitude weights.
Cases
The first case is illustrative: because the PDF of Chen and Xu's (2015) paper is not accessible, the manifest uses a closed-form validation example based on the textbook chapter by Xu (2014) that describes the same method. The second case is fictional.
1. Health: Complete ranking of three medical-device suppliers (illustrative validation example)
A hospital wants to produce a complete ranking of three medical-device suppliers (A1, A2, A3). There are three criteria: device quality score, delivery reliability score and technical support score; all three are "higher is better." The evaluation team scored every supplier on every criterion with several plausible values. The criterion weights are 0.4, 0.3 and 0.3. The concordance thresholds are set at c* = 0.65, c' = 0.55, c⁻ = 0.45; the discordance thresholds at d* = 0.6, d' = 0.4.
| Supplier | Device quality | Delivery reliability | Technical support |
|---|---|---|---|
| A1 | {0.5; 0.7; 0.8} | {0.4; 0.6} | {0.6; 0.7} |
| A2 | {0.3; 0.4; 0.6} | {0.7; 0.8; 0.9} | {0.5; 0.6; 0.8} |
| A3 | {0.6; 0.8} | {0.5; 0.7} | {0.3; 0.4; 0.5} |
| Weight | 0.4 | 0.3 | 0.3 |
The method extracts a score and a deviation from every HFE, builds the strong/moderate/weak concordance and discordance sets, and weights them with the seven attitude weights. The concordance matrix C and the discordance matrix D are built from these weighted sets; A1's concordance value over A2 comes out at 0.58, and A3's concordance value over A1 at 0.61. These values are compared against the five thresholds to build the strong and weak outranking graphs; the ranking is derived from both the forward (best-down) and the backward (worst-up) distillation.
| Rank | Supplier |
|---|---|
| 1 | A1 |
| 2 | A2 |
| 3 | A3 |
The result reads as follows. A1 comes out ahead through a judgement on device quality that is both high and consistent (a narrow range between 0.5 and 0.8), and the forward and backward distillations agree on the same order; there is therefore no incomparable pair. A3 is the weakest supplier on technical support, and this is not enough to offset its advantage on device quality.
The hospital's hesitation is this: what would happen if A1's device-quality set became a much lower set, such as {0.2; 0.3}, instead of {0.5; 0.7; 0.8}? Running the same calculation independently again drops A1 to second place, and A2 rises to first; A3 stays third. A1's first place in the original result rests to a large extent on this strong and consistent judgement on this single criterion.
In the report: "With the thresholds set, the ranking is A1, A2, A3; A1's first place rests on a judgement on the device-quality criterion that is both high and consistent, and a marked drop on this criterion moves A1 down to second place."
Source: DecisionMind's own validation example, based on the formulas for Chen and Xu's (2015) method as given in Xu's (2014) textbook chapter (4.2.4(2)). Because the paper's PDF is not accessible, the numerical example was constructed independently from these formulas rather than from the paper itself. The concordance/discordance matrices and the sensitivity scenario were computed by this card's author by running the DecisionMind engine independently; the baseline result matches the manifest's own validation record exactly.
2. Logistics: An e-commerce company's complete ranking of courier choices
An e-commerce company will produce a complete ranking of three couriers. There are three criteria: delivery-speed score, damage-rate score (low damage is good, reversed so that "higher is better" applies) and customer-complaint resolution score. The operations team scored every courier on its past performance with several plausible values, and these values were gathered as hesitant sets.
The method extracts the scores and deviations, builds the concordance/discordance sets, and constructs the strong/weak outranking graphs with the five thresholds. Suppose the result shows that a courier scoring both high and consistent on delivery speed comes first in the forward distillation, but ties with a second courier in the backward distillation; the relationship between these two couriers is treated as incomparable.
The company's hesitation is this. The difference between the two couriers that come out incomparable stems from one of them having low consistency on the customer-complaint resolution criterion. If the evaluation period for this criterion were extended and more data collected, consistency could rise, and the incomparability could turn into a strict preference. This should be stated in the report.
In the report: "With these thresholds, the relationship between the courier that leads on delivery speed and the second courier comes out as incomparable; this stems from low consistency on the customer-complaint resolution criterion and may change once further data is collected."
3. What Not to Do
The first mistake is collapsing the hesitant sets in the medical-device table into single numbers by averaging from the outset and running crisp ELECTRE II on that; this erases the information about which supplier was judged more consistently on which criterion. The second mistake is confusing this method's optimistic length-equalisation rule (repeating the largest value) with the family's Hesitant ELECTRE I member's pessimistic rule. The third mistake is presenting the five thresholds (c*, c', c⁻, d*, d') as if they derived from the data; they are chosen by the user, as in crisp ELECTRE II, and the choice must be justified in the report.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/hf-electre-ii
Chen, N., & Xu, Z. S. (2015). Hesitant fuzzy ELECTRE II approach: A new way to handle multi-criteria decision making problems. Information Sciences, 292, 175–197. DOI: 10.1016/j.ins.2014.08.054
Liao, H., & Xu, Z. (2017). Hesitant Fuzzy Decision Making Methodologies and Applications. Uncertainty and Operations Research, Springer Singapore. DOI: 10.1007/978-981-10-3265-3
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Roy, B., & Bertier, P. (1973). La méthode ELECTRE II: une application au media-planning. In Operational Research '72: Proceedings of the Sixth IFORS International Conference on Operational Research (pp. 291–302). North-Holland. (no DOI)
Roy, B. (1991). The outranking approach and the foundations of ELECTRE methods. Theory and Decision, 31(1), 49–73. DOI: 10.1007/BF00134132