Extension card · Hesitant
m-polar hesitant fuzzy TOPSIS (Akram, Adeel and Alcantud, 2019)
This is the form of TOPSIS for situations where a criterion is assessed from more than one independent viewpoint (pole), and each viewpoint itself is hesitant, that is, carries more than one plausible value.
Base method
TOPSIS →
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 decision logic stays the same.
Cells. In crisp TOPSIS every cell is a single number. Here every cell carries m independent poles, and each pole is itself a hesitant set, that is, it contains more than one plausible membership value. For example, in a product design's "appearance" criterion, three sub-features (shape, colour, texture) are treated as separate poles, and each sub-feature is written with its own hesitant set, holding several experts' values together. Different poles within the same cell can have hesitant sets of different lengths. DecisionMind completes these to a common length; the completion can be optimistic (repeating the largest value) or pessimistic (repeating the smallest value), and the user chooses which. Criterion weights are crisp numbers, taken from outside.
Scale equalisation. The values inside the hesitant sets are already degrees between 0 and 1, so crisp TOPSIS's division by column magnitude is absent here. In its place, the length-completion just described is applied; every pole's every hesitant set is brought to the same length.
Weighting. In crisp TOPSIS, weighting is applied to the whole column. Here every hesitant value in every pole is multiplied separately by its own criterion's weight; the pole and hesitant structure is preserved, only the values shrink.
Distance and result. The ideal and anti-ideal are built, for every criterion, from the best and worst value at every pole's every hesitancy level separately. The distance is an averaged Euclidean distance taken over all poles and all hesitancy levels together. The closeness coefficient is again a single number between 0 and 1.
DecisionMind fixes, for this extension, the number of poles, the rule for equalising hesitant length, and the averaged distance. The direction of completion is chosen by the user and must be stated in the report.
How to Read the Output
The closeness coefficient is read as in crisp TOPSIS, ranking only this set of alternatives. The difference is here: the direction of completion enlarges or shrinks the values of the chosen poles, and this choice can affect the result. The same raw data, completed optimistically versus pessimistically, can produce different closeness coefficients, and even a different ranking among alternatives that are close together.
Thus instead of writing:
"m-polar hesitant TOPSIS gives the most balanced result because it uses several viewpoints together"
the report should read:
"A direction of completion has been chosen, and this choice can affect the result; which direction was used, and how sensitive close alternatives are to this choice, must be stated in the report"
When to Prefer This over the Base Method
This extension is suitable when a criterion genuinely comes from more than one independent viewpoint, for example from several sub-features or stakeholder groups, and there is also hesitancy among experts within each viewpoint. If there is only a single viewpoint but hesitancy is present, HF-TOPSIS is sufficient; adding the pole layer would only add needless complexity. If a criterion is measured, the base method should be used; DecisionMind asks for a single data type in the table. The base TOPSIS's compensatory nature and its condition of accepting no trade-off on one criterion apply here exactly as before.
Mistakes Specific to This Extension
Changing the number of poles from cell to cell. Every cell must carry the same number of poles; three poles in one cell and four in another leaves the construction of the ideal and anti-ideal undefined.
Leaving the direction of completion unstated. Optimistic and pessimistic completion can give the same raw data different closeness coefficients; the order of close alternatives is sensitive to this choice. This is shown in Case 1.
Carrying out weighting with a different operation. In this kernel, weighting means multiplying every hesitant value directly by the criterion's weight; using a different aggregation rule will not reproduce the paper's figures.
Aggregating groups after weighting. If there is more than one decision-maker's matrix, these must be aggregated into a single m-polar hesitant matrix before weighting; the method does not define this aggregation itself.
The governing principle is this:
Every pole must carry its own independent viewpoint's data, the direction of completion must not be chosen without justification, and it must be stated in the report.
Cases
The first case is Akram, Adeel and Alcantud's (2019) product-design example; the figures are the paper's own and have been independently recomputed with DecisionMind's engine. The second case is an illustrative construction.
1. Product design: Choosing among four design proposals (Akram, Adeel and Alcantud, 2019)
A company will choose one of four product designs. There are four criteria, all "higher is better": appearance, material, dimensions and tolerance, and performance standard. Each criterion consists of three independent sub-features (poles), and each pole carries several experts' plausible values as a hesitant set. Weights: appearance 0.2012, material 0.2259, dimensions and tolerance 0.2631, performance standard 0.3098. Completion is pessimistic (the smallest value is repeated); every hesitant set is completed to length 4.
In every cell, the three poles are given one after another as three hesitant sets, separated by a slash.
| Design | Appearance | Material | Dimensions and tolerance | Performance standard |
|---|---|---|---|---|
| P_d1 | {0.25;0.45;0.47} / {0.30;0.31;0.36} / {0.20;0.25;0.26} | {0.45;0.49;0.51;0.59} / {0.67;0.68;0.71} / {0.50;0.56;0.63;0.64} | {0.85;0.86;0.87} / {0.53;0.59;0.66} / {0.72;0.75;0.76;0.78} | {0.55;0.65} / {0.40;0.48;0.60;0.61} / {0.80;0.85;0.86} |
| P_d2 | {0.46;0.48;0.49} / {0.47;0.49} / {0.55;0.60;0.61;0.63} | {0.49;0.50} / {0.71;0.74;0.79} / {0.35;0.59;0.61;0.65} | {0.66;0.68;0.69} / {0.47;0.50;0.51;0.64} / {0.65;0.66;0.81} | {0.54;0.58;0.59;0.61} / {0.77;0.79;0.84} / {0.55;0.60;0.68} |
| P_d3 | {0.51;0.53;0.57;0.60} / {0.46;0.52;0.70} / {0.29;0.30;0.51;0.52} | {0.71;0.73;0.77} / {0.46;0.52;0.70} / {0.29;0.30;0.51;0.52} | {0.51;0.55} / {0.66;0.68;0.75;0.76} / {0.39;0.40;0.58;0.62} | {0.81;0.83;0.87} / {0.56;0.62;0.70} / {0.69;0.70;0.76;0.82} |
| P_d4 | {0.39;0.41;0.43} / {0.60;0.68;0.71;0.73} / {0.50;0.67;0.69} | {0.53;0.54;0.56;0.58} / {0.60;0.63;0.73;0.79} / {0.40;0.47;0.49} | {0.59;0.61;0.73;0.74} / {0.26;0.38;0.41;0.43} / {0.51;0.77} | {0.37;0.48;0.49;0.59} / {0.26;0.38;0.41;0.43} / {0.60;0.67} |
| Direction | higher is better | higher is better | higher is better | higher is better |
The method completes every pole's every hesitant set to length 4 in the pessimistic manner, multiplies by the weights, builds the ideal and anti-ideal design, and computes each design's closeness coefficient.
| Design | Closeness coefficient | Rank |
|---|---|---|
| P_d2 | 0.567 | 1 |
| P_d3 | 0.551 | 2 |
| P_d1 | 0.495 | 3 |
| P_d4 | 0.372 | 4 |
The result reads as follows. P_d2 has strong, narrow sets on performance standard, the heaviest criterion. P_d3 follows immediately behind it, with a gap of only 0.017. P_d4 has lower values than the others across the whole of the fourth criterion and finishes last.
The company's hesitation is this. If the direction of completion is switched from pessimistic to optimistic (so that the largest value is repeated), the gap between P_d2 and P_d3 falls from 0.017 to 0.003; the ranking does not change, but the two designs become almost equal. This shows that the direction of completion can be decisive for close alternatives.
In the report: "With pessimistic completion, P_d2 has the highest closeness coefficient (0.567); the gap to P_d3 (0.551) is small, and this gap narrows to 0.003 once the direction of completion is switched to optimistic. The choice between the second- and third-ranked designs is sensitive to this decision."
Source: Akram, Adeel and Alcantud (2019), §3.2, Tables 6-8, pp. 17-20. The closeness coefficients are taken from the paper's own tables and have been verified by independently re-running DecisionMind's MHF-TOPSIS engine; the values the engine produces differ from the paper's published values by a few parts in a thousand, and the ranking is the same (detail in the sign-off note).
2. Public transport: Choosing among three vehicle models for a new bus fleet
A municipal public-transport operator will choose one of three vehicle models for a new bus fleet. There are three criteria, all "higher is better": passenger comfort, energy efficiency and ease of maintenance. Each criterion has been gathered as a pole from three independent stakeholder groups (passengers, drivers, maintenance staff), and within each group more than one survey or report forms a hesitant set.
The method computes each vehicle's distance to the ideal and anti-ideal vehicle, at the pole and hesitancy level, and converts this into a closeness coefficient. Say the vehicle with the lowest score on passenger comfort scored clearly highest on the maintenance-staff pole, and because this outweighed the rest, it came out first.
The operator's hesitation is this: the hesitant set for the maintenance-staff pole has been widened by completing it with values from different service periods. If the direction of completion is switched to optimistic, this vehicle's score falls, and the vehicle that scored better on passenger comfort could move ahead.
In the report: "Because of its lead in the maintenance-staff assessment, this vehicle is ranked first. The ranking has been found sensitive to a change in the direction of completion; this sensitivity should be separately tested before the selection."
3. What Not to Do
In the first case, arbitrarily narrowing the hesitant set of one of P_d1's performance-standard poles as if it were a single expert's indecision: this loses the real source of the set, that is, which expert gave which value. The second error is writing a report without stating the direction of completion; the 0.017 gap between P_d2 and P_d3 narrows to 0.003 once the direction of completion changes. The third error is mixing a criterion with four poles into the same table as one with three poles; the number of poles must stay fixed from cell to cell.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/mhf-topsis
Akram, M., Adeel, A., & Alcantud, J. C. R. (2019). Multi-Criteria Group Decision-Making Using an m-Polar Hesitant Fuzzy TOPSIS Approach. Symmetry, 11(6), 795. DOI: 10.3390/sym11060795
Chen, J., Li, S., Ma, S., & Wang, X. (2014). m-Polar fuzzy sets: An extension of bipolar fuzzy sets. The Scientific World Journal, 2014, 416530. DOI: 10.1155/2014/416530
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Hwang, C. L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications — A State-of-the-Art Survey. Springer-Verlag. DOI: 10.1007/978-3-642-48318-9