Extension card · Hesitant
Probabilistic hesitant TOPSIS
This is the form of TOPSIS for situations where more than one plausible value for a criterion is given together with its own probability of occurrence. Every cell is first reduced to its expected value, after which the remaining steps of classical TOPSIS are applied.
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?
Three things change; the decision logic stays the same.
Cells. In classical TOPSIS every cell is a single number. Here every cell is a set in which more than one plausible value for the same criterion is given together with its own probability of occurrence. A value might, for instance, be written as "0.7 with 60 per cent probability, 0.5 with 40 per cent probability." The probabilities within a cell do not sum to more than 1. Criterion weights are crisp numbers, taken from outside.
Scale equalisation. The method first reduces every cell to its expected value: every possible value is multiplied by its own probability and summed within the criterion. This step summarises the distribution into a single number and precedes classical TOPSIS's vector normalisation. Once the expected values have been obtained, DecisionMind applies classical TOPSIS's own vector normalisation, dividing every column by the square root of the sum of the squares of its own values.
Result and defuzzification. The closeness coefficient is again a single number between 0 and 1. The difference lies here: reducing to the expected value discards the information of which values were how probable, leaving only the distribution's mean. Two different distributions can produce the same expected value. One might say "0.6 with 100 per cent probability" while another says "0.4 with 50 per cent probability, 0.8 with 50 per cent probability," and both descend to the same number.
DecisionMind fixes this reduction to the expected value, and the classical TOPSIS vector normalisation that follows it, in this extension.
How to Read the Output
The closeness score is read as in classical TOPSIS: it is not a percentage, and it is not compared against a different analysis. The difference is this: beneath the score lies a probability distribution, and this distribution becomes invisible once it is reduced to the expected value. The score gap between two alternatives depends not only on which value was observed, but also on the expert's probability distribution across the values.
Thus instead of writing:
"Probabilistic hesitant TOPSIS uses probability information, so its result is the most accurate"
the report should read:
"The closeness score, calculated over expected values, has produced this ranking; whether the probability distribution beneath the score is concentrated or spread out should be checked separately"
When to Prefer This over the Base Method
This extension is suitable when an expert or data source distributes a criterion across more than one plausible value, each with its own probability of occurrence, rather than a single value. If there is no probability information, and only more than one plausible value exists, HF-TOPSIS is sufficient; adding a probability would present information that does not exist as though it did. If the criterion is measured, the base method should be kept; DecisionMind requires the table to hold a single data type throughout. Classical TOPSIS's compensatory nature and its condition of allowing no compromise on one criterion apply here in exactly the same way.
Mistakes Specific to This Extension
Violating the value-domain constraint. Every possible value must lie between 0 and 1, and the probabilities within a cell must not sum to more than 1; violating this makes the expected value meaningless.
Changing the defuzzification method without stating it. The expected value is the canonical choice, but other defuzzification rules have also been defined. If the report does not state which one was used, a comparison against a different calculation becomes misleading.
Inventing probabilities without justification. Splitting a crisp value into two possible values as "50-50," with no basis in a genuine source, adds no information and only manufactures artificial uncertainty.
The "more sophisticated" fallacy. Without a genuine source supporting more than one plausible value, arbitrarily opening a crisp number into several nearby values and probabilities does not change the result; it only complicates the calculation.
The governing principle is this:
Every possible value and its probability must rest on a genuine source; the defuzzification method and the chosen weights must be stated in the report.
Cases
PHF-TOPSIS is an internal DecisionMind extension that combines the probabilistic hesitant expected-value calculation with the remaining steps of classical TOPSIS; its founding paper cannot be tied to a single author-year citation (details in the verification note). The first case is therefore DecisionMind's validation example. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Three alternatives assessed on three criteria
Three alternatives are assessed on three criteria; C1 and C2 are "higher is better," C3 is "lower is better." Every cell carries more than one possible value together with its own probability of occurrence. Weights: C1 0.40, C2 0.35, C3 0.25.
| Alternative | C1 | C2 | C3 (cost) |
|---|---|---|---|
| A1 | {0.7 (0.6); 0.5 (0.4)} | {0.9 (1.0)} | {0.4 (0.5); 0.6 (0.5)} |
| A2 | {0.5 (0.7); 0.3 (0.3)} | {0.6 (0.4); 0.8 (0.6)} | {0.7 (1.0)} |
| A3 | {0.8 (1.0)} | {0.5 (0.5); 0.4 (0.5)} | {0.3 (0.6); 0.5 (0.4)} |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method reduces every cell to its expected value (A1's expected value on C1 is, for instance, 0.7×0.6+0.5×0.4=0.62), applies classical TOPSIS's vector normalisation, builds the ideal and anti-ideal alternatives, and computes the closeness coefficient.
| Alternative | Closeness coefficient | Rank |
|---|---|---|
| A1 | 0.678 | 1 |
| A3 | 0.550 | 2 |
| A2 | 0.318 | 3 |
The result reads as follows. A1 holds a single, high expected value (0.9) on C2, the heaviest criterion. A3 holds the highest expected value on C1 and the lowest, that is the most advantageous, cost on C3, but falls lower on C2. A2 holds middling or weak expected values on all three criteria and finishes last.
The decision's hesitation is this. If C1's weight is raised from 0.40 to 0.60 and C2's and C3's weights each lowered to 0.20, A3 moves ahead, leaving A1 behind at 0.561 against 0.740. A3, which holds the highest expected value on C1, offsets its weakness on C2 once this criterion's weight is raised enough.
In the report: "With the given weights, A1 holds the highest closeness coefficient (0.678). Once C1's weight is raised to 0.60, A3 moves ahead. The preference between A1 and A3 depends on the relative weight of criteria C1 and C2."
Source: DecisionMind's PHF-TOPSIS validation example. The figures are taken from the manifest's synthetic fixture and have been recomputed independently with the expected-value and TOPSIS formulas.
2. Cybersecurity: Choosing among three security software packages
An organisation will choose one of three software providers for network security. Criteria: threat-detection rate, false-alarm rate and installation cost; the latter two are "lower is better," and threat-detection rate is "higher is better." Each provider's threat-detection rate is given as a probability distribution over more than one result from past penetration tests.
The method builds the ideal and anti-ideal provider by taking into account every provider's expected detection rate, expected false-alarm rate and cost, and arrives at the closeness coefficient. Suppose the provider with the highest expected detection rate also has the highest expected false-alarm rate, and still comes out first, because the detection-rate weight has been kept higher.
The organisation's hesitation is this: this provider's detection-rate distribution is wide, meaning the penetration-test results differ substantially from one another. The expected value conceals this variability; the organisation should assess the width of the distribution as a separate risk indicator.
In the report: "This provider holds the highest closeness coefficient because of the high weight on threat-detection rate. Its detection-rate distribution is wide; this variability should be monitored as a separate risk indicator."
3. What Not to Do
In the illustrative example, inventing unjustified probabilities (say, 0.9 and 0.1) instead of the given ones for A1's two possible values on C1 (0.7 and 0.5) distorts the expected value unrealistically. The second error is disregarding that C3 is a cost criterion and assessing its expected value as a benefit; this mistakenly rewards the most expensive alternative. The third error is writing the ranking between A1 and A3 as a definite advantage independent of the weight assumption; once C1's weight is raised enough, this ranking reverses.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/phf-topsis
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
Zhang, S., Xu, Z., & He, Y. (2017). Operations and integrations of probabilistic hesitant fuzzy information in decision making. Information Fusion, 38, 1–11. DOI: 10.1016/j.inffus.2017.02.001
Xu, Z., & Zhou, W. (2017). Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optimization and Decision Making, 16(4), 481–503. DOI: 10.1007/s10700-016-9257-5