Extension card · Hesitant
Simplified Neutrosophic Hesitant Fuzzy TOPSIS (Akram, Naz and Smarandache, 2019)
This is the form of TOPSIS for situations where a criterion's truth, indeterminacy and falsity degrees are each hesitant in their own right, that is, each carries more than one plausible value. Weights are not taken from outside; the method itself derives them from the disagreement in the data.
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?
Five things change; the decision logic stays the same.
Cells. In crisp TOPSIS every cell is a single number. Here every cell consists of three components: truth, indeterminacy, falsity. What sets it apart from N-TOPSIS is this: here each component itself is not a single number but a hesitant set, that is, more than one plausible value is held together at once. For instance, a supplier's truth component on a criterion might be given as two values, {0.6; 0.7}. In the same cell, the truth, indeterminacy and falsity sets can each have a different length. Criterion weights are not taken from outside here; the method derives the weights from the data itself.
Length equalisation. All alternatives' truth sets on the same criterion are brought to the same length as one another, and likewise for their indeterminacy sets and their falsity sets. The missing slot is filled with a coefficient (λ): the largest and smallest value in the set are blended in the ratio λ. λ=0.5 is a neutral fill; λ=0 is cautious (close to the smaller value), λ=1 is assertive (close to the larger value).
Weighting. Crisp TOPSIS takes weights from outside. Here the method measures how large the T-I-F distance between alternatives is on each criterion; the criterion that most distinguishes the alternatives receives the highest weight. If the weights are already fully known, this calculation is not needed, and an extension that takes weights from outside, such as N-TOPSIS, should be used instead.
Ideal and anti-ideal. In crisp TOPSIS the ideal and anti-ideal are built from each column's own best and worst value. Here the ideal and anti-ideal are two fixed points, independent of the data: the ideal is the point where truth is complete (1) and indeterminacy and falsity are zero; the anti-ideal is its exact opposite. These two points do not change whatever data is entered.
Distance and result. Distance is calculated by combining the differences in the truth, indeterminacy and falsity sets through a power parameter (α); α=2 gives a Euclidean distance, α=1 gives something close to the average of absolute differences. The closeness coefficient is again a single number between 0 and 1.
DecisionMind fixes the length-equalisation rule, the fixed ideal points and the generalised distance formula for this extension. λ and α are 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, and it only ranks this particular set of alternatives. The difference is here: the score is the result of a calculation in which the weights, too, have been derived from the data. How the weights are distributed shows which criterion most distinguishes the alternatives, and this information should be shared separately in the report.
Thus instead of writing:
"SNHF-TOPSIS gives the most objective result because it derives both the uncertainty and the weights from the data"
the report should read:
"The weights have been derived from the size of the T-I-F difference between alternatives; this weight distribution should be shared in the report, together with the sensitivity of the result to the choice of the α parameter"
When to Prefer This over the Base Method
Use this extension where information is incomplete, inconsistent or conflicting, and where more than one defensible value exists among experts about that very incompleteness or conflict. If the weights are already fully known, this extension should not be used; instead N-TOPSIS, which takes weights from outside, should be used, because deriving weights from the data itself would mean disregarding a known weight. If every component (truth, indeterminacy, falsity) is a single number and there is no hesitancy, N-TOPSIS is again sufficient. If a criterion is measured, the base method should be kept; DecisionMind requires the table to use a single data type.
Mistakes Specific to This Extension
Calculating distance before length equalisation. If the truth, indeterminacy and falsity sets are compared before being brought to equal length, distances become undefined.
Trying to impose weights from outside. This extension derives the weights itself; if the weights are already known in crisp form, N-TOPSIS should be used.
Expecting an automatic direction reversal for a cost criterion. In this kernel the truth-indeterminacy-falsity degrees entered are used directly; no automatic complementing operation is applied for a cost criterion, unlike in crisp TOPSIS or N-TOPSIS. Even if the criterion direction is labelled, it does not affect the calculation. When entering the truth degree of a cost criterion, the user must already take this criterion's real direction into account.
Ignoring sensitivity to α. The closeness coefficient varies with the choice of α; which α was used must be stated in the report.
The governing principle is this:
Every value in every component must genuinely come from more than one defensible source; because the weights are derived from the data, they must be shared separately in the report, and it must not be forgotten that criterion direction is not automatically processed in this kernel.
Cases
The first case is Akram, Naz and Smarandache's (2019) supplier-selection example; the numbers are the paper's own and have been independently recomputed with DecisionMind's engine. The second case is an illustrative construction.
1. Supplier selection: Choosing among five CPU suppliers (Akram, Naz and Smarandache, 2019)
A manufacturer will choose among five CPU suppliers. There are four criteria: cost, technical competence, product quality and service quality. In every criterion the truth, indeterminacy and falsity degrees are given as sets, each carrying more than one plausible value. λ=0.5 (neutral fill), α=2 (Euclidean) are used. Because the weights are not fully known, they are derived by the method itself.
Every cell gives the truth (T), indeterminacy (I) and falsity (F) sets separately.
| Supplier | Cost | Technical competence | Product quality | Service quality |
|---|---|---|---|---|
| A1 | T{0.2} I{0.3;0.5} F{0.1;0.2;0.3} | T{0.6;0.7} I{0.1;0.3} F{0.2;0.4} | T{0.2;0.3} I{0.4} F{0.7;0.8} | T{0.4} I{0.1;0.3} F{0.5;0.7;0.9} |
| A2 | T{0.1} I{0.3} F{0.5;0.6} | T{0.4} I{0.3;0.5} F{0.5;0.6} | T{0.1;0.3} I{0.4} F{0.5;0.6;0.8} | T{0.6;0.8} I{0.2} F{0.3;0.5} |
| A3 | T{0.6;0.7} I{0.2;0.3} F{0.1;0.2} | T{0.1;0.2} I{0.3} F{0.6;0.7} | T{0.2;0.3} I{0.1;0.2} F{0.6;0.7} | T{0.2;0.3} I{0.4} F{0.2;0.5;0.6} |
| A4 | T{0.2;0.3} I{0.1;0.2} F{0.5;0.6} | T{0.3;0.4} I{0.2;0.3} F{0.5;0.6;0.7} | T{0.2;0.4} I{0.3} F{0.1;0.2} | T{0.6} I{0.2} F{0.3;0.5} |
| A5 | T{0.7} I{0.4;0.5} F{0.2;0.4;0.5} | T{0.6} I{0.1;0.7} F{0.3;0.5} | T{0.3} I{0.5} F{0.1;0.4} | T{0.5} I{0.1;0.2} F{0.3;0.4} |
The method brings the sets on each criterion to equal length, derives criterion weights from the difference between alternatives (cost 0.299, technical competence 0.237, product quality 0.252, service quality 0.212), calculates distance to the fixed ideal and anti-ideal points, and finds the closeness coefficient.
| Supplier | Closeness coefficient | Rank |
|---|---|---|
| A5 | 0.593 | 1 |
| A4 | 0.560 | 2 |
| A3 | 0.539 | 3 |
| A1 | 0.525 | 4 |
| A2 | 0.488 | 5 |
The result reads as follows. A5 has relatively narrow sets close to the ideal on all four criteria. A1 and A3 are close to one another; the gap between them is only 0.014. A2 has the sets furthest from the ideal on most criteria and finishes last.
The manufacturer's hesitation is this. When the distance power α is lowered from 2 to 1, the gap between A1 and A3 falls from 0.014 to 0.007; the ranking does not change, but the two suppliers become nearly tied. In other scenarios tried with λ (from 0.5 to 1) and α (from 2 to 6), the overall order (A5, A4, A3, A1, A2) is preserved; this order is relatively robust to the choice of parameters, but the A1–A3 gap is always narrow.
In the report: "With weights derived from the data, A5 has the highest closeness coefficient (0.593). The gap between A1 and A3 is small (0.014) and narrows further when the distance power α is changed; the choice between these two suppliers should be handled carefully."
Source: Akram, Naz and Smarandache (2019), §3, Tables 1 and 3, pp. 125–126. The closeness coefficients are taken from the paper's own table, verified by independently rerunning DecisionMind's SNHF-TOPSIS engine, and the engine's output matches the paper's published values exactly.
2. Care homes: Choosing among three outsourced care-service providers
A care-home chain will choose one of three outsourced care-service providers for a new branch. Criteria: service quality, staff experience and cost suitability. Because audit reports are incomplete or conflicting, the truth, indeterminacy and falsity degrees on every criterion have been gathered as sets holding together more than one value given by different auditors.
The method calculates each provider's distance to the fixed ideal and anti-ideal points, combines it with weights derived from the data, and finds the closeness coefficient. Suppose the provider with the highest truth set on service quality also has the widest indeterminacy set; it still comes out first, because service quality has received the highest weight.
The chain's hesitation is this: the width of this provider's indeterminacy set shows a serious disagreement among the auditors. The closeness coefficient dissolves this disagreement into a single number; the chain should request an additional independent audit before signing the contract.
In the report: "Because of the high weight of the service-quality criterion, this provider has the highest closeness coefficient. The indeterminacy set on the same criterion is wide, meaning the auditors disagree; an additional audit is recommended before contracting."
3. What Not to Do
In the first case, arbitrarily widening A1's truth set on the cost criterion from {0.2} to {0.2; 0.25}, as though it were a value given by the auditors, is wrong: the added value has no source at all. The second error is using this extension in a situation where the weights are already fully known from outside and taking the method's own weight estimate as authoritative; if the weight is known, N-TOPSIS should be used. The third error is entering data for the truth degree of a cost criterion as though it will be automatically reversed, as in crisp TOPSIS, without regard to its direction at all; no such reversal is applied in this kernel, and direction must already be accounted for when the data is entered.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/snhf-topsis
Akram, M., Naz, S., & Smarandache, F. (2019). Generalization of Maximizing Deviation and TOPSIS Method for MADM in Simplified Neutrosophic Hesitant Fuzzy Environment. Symmetry, 11(8), 1058. DOI: 10.3390/sym11081058
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
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Wang, H., Smarandache, F., Zhang, Y., & Sunderraman, R. (2010). Single valued neutrosophic sets. Multispace and Multistructure, 4, 410–413. (no DOI)
Ye, J. (2014). Multiple-attribute decision-making method under a single-valued neutrosophic hesitant fuzzy environment. Journal of Intelligent Systems, 24(1), 23–36. DOI: 10.1515/jisys-2014-0001