Extension card · Spherical
Spherical fuzzy TOPSIS — implemented scalar-weight profile
Spherical fuzzy TOPSIS is the form of TOPSIS used when criterion scores are given as three separate numbers stating a degree of support, rejection and hesitancy for a judgement. The calculation comes down to a single closure ratio; a small value in this ratio indicates preference under its two distance terms.
Base method
TOPSIS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Spherical →
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?
This card describes the implemented single-matrix profile. Its exact correspondence to the complete 2019 algorithm remains unverified because the original full text was unavailable in this review.
Cells. In crisp TOPSIS every cell is a single number. In the spherical fuzzy form every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π). The expert supplies all three directly. The only constraint is that the sum of the three degrees' squares cannot exceed 1. Hesitancy here is not computed as a residual share; this is what distinguishes it from intuitionistic or Pythagorean structures. Weights come from outside DecisionMind as crisp numbers. This kernel consumes one native matrix. The separate group API supports a native spherical weighted arithmetic aggregation, but the file-based (dataset) group route is unsupported. A long-format expert tensor must not be treated as a single expert's matrix.
Scale equalisation. Crisp TOPSIS divides every column by its own magnitude, because criteria are in different units: TL, days, points. In the spherical fuzzy form there is no separate scale-equalisation step. Every cell is already on the same 0–1 scale across three degrees, and the columns are comparable from the outset. DecisionMind instead moves directly to weighting. It scales every cell by the criterion's weight through spherical fuzzy multiplication. The scalar spherical operation updates the three components by different formulas; it is not componentwise multiplication or conversion to a crisp cell.
Distance. The ideal and anti-ideal points are built, as in crisp TOPSIS, from the best and worst observed value on each criterion. The difference is here: "best" is not a single number here. A score function evaluates and compares all three degrees together. For a benefit criterion, the positive ideal is the native triple with the highest score and the negative ideal has the lowest score; cost criteria reverse these choices. Distinct triples tied at an extreme are rejected until a verified tie-selection rule is available. Each criterion contributes an observed native triple. The assembled ideal can combine triples from different alternatives and need not be an observed alternative. Every alternative's distance to these two references is a Euclidean distance that jointly sums the squares of the differences of the three degrees.
Closure ratio. Crisp TOPSIS's closeness score is better the larger it is. SF-TOPSIS's closure ratio (ξ) runs the opposite way. Distance to the ideal is divided by one reference, distance to the anti-ideal is divided by another reference, and the difference between the two is taken. Smaller xi is better under this ratio. Defined xi is nonnegative and can exceed1. Zero xi means the same alternative attains the minimum positive-ideal distance and maximum negative-ideal distance; it does not mean zero distance to the ideal. DecisionMind holds the score function and the normalised Euclidean distance fixed in this family. Weights are crisp numbers. The single-matrix kernel does not aggregate experts.
How to Read the Output
The closure ratio only ranks alternatives within this particular set relative to one another. It cannot be compared with a different analysis and it changes when the alternative set changes; this is shared with crisp TOPSIS. What differs is the reading direction.
Thus instead of writing:
"In SF-TOPSIS too, the alternative with the highest score is first"
the report should read:
"SF-TOPSIS's output is the closure ratio ξ, and it runs the opposite way; the smallest ξ shows the one preferred by the combined revised ratio, and ranking runs from smallest to largest"
If ξ values are ordered from largest to smallest in the report and presented as "the highest score," the rank is effectively inverted. Readers, out of habit from crisp TOPSIS, look for the large number and take the wrong alternative for the winner.
When to Prefer This over the Base Method
Use this extension when experts give the degrees of support, rejection and hesitancy for a judgement separately. Choose spherical semantics at elicitation. Exceeding the Picture fuzzy sum constraint does not justify relabeling those judgments as spherical: the third component has a different meaning. A measured criterion (price, time) is not expanded into three degrees by force. The boundary that must not be confused with neighbouring types (Pythagorean, picture fuzzy) is set out on the data-type card.
The exit condition is the same as for crisp TOPSIS. The matrix must be of a single type; some criteria cannot be written as crisp and others as spherical fuzzy. If no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Confusing the value-space constraint. The constraint is the sum of the squares of the three degrees: μ²+ν²+π²≤1. It is not the sum of the three degrees themselves; that is the picture fuzzy structure's constraint. Nor is hesitancy a share derived as 1−μ−ν; that is the intuitionistic fuzzy structure's constraint. Confusing these three means either the data entered is invalid or the wrong family's calculation has been run.
Changing the score function. The ideal and anti-ideal points are built with the spherical fuzzy family's own score function. Carrying over the Pythagorean family's score function here leads to a different ideal point, and the rank changes.
Ranking the closure ratio from largest to smallest. This is the most commonly confused point in the method. Smaller ξ is better. Setting up the ranking in reverse out of crisp-TOPSIS habit shows the last-place alternative as first.
Computing hesitancy afterwards and writing it into the spherical structure. If the expert did not supply the third degree, the input is incomplete for this profile. Do not infer the missing component or assign another family from that omission.
The governing principle is this:
In SF-TOPSIS, uncertainty is carried through all three degrees to the very end of the calculation and gathered into a single closure ratio. A smaller value in this ratio is better, its constraint is a sum of squares, and its score function is specific to this family.
Cases
The first case is DecisionMind's validation example. It is a small table with three alternatives and three criteria; it is not a page taken from the literature. It was built to make the engine's steps traceable and has been verified by running the engine itself. The second case is an illustrative construction.
1. Illustrative example: Spherical fuzzy assessment of three alternatives on three criteria (DecisionMind validation example)
Three alternatives are assessed with spherical fuzzy triples on three criteria. All three are higher-is-better criteria. Weights are crisp numbers.
| Alternative | K1 | K2 | K3 |
|---|---|---|---|
| A1 | (0.70; 0.20; 0.50) | (0.80; 0.10; 0.40) | (0.60; 0.30; 0.50) |
| A2 | (0.90; 0.10; 0.30) | (0.60; 0.30; 0.60) | (0.80; 0.20; 0.40) |
| A3 | (0.50; 0.40; 0.60) | (0.70; 0.20; 0.50) | (0.70; 0.10; 0.50) |
| Direction | higher is better | higher is better | higher is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method scales every cell by the criterion's weight through spherical fuzzy multiplication. It then selects, for every criterion, the best and worst real triple according to the score function; these are the ideal and the anti-ideal. It measures every alternative's Euclidean distance to these two references and builds the closure ratio (ξ).
| Alternative | Closure ratio (ξ) | Rank |
|---|---|---|
| A2 | 0.0000 | 1 |
| A1 | 0.2958 | 2 |
| A3 | 1.1387 | 3 |
The result reads as follows. A2 has the highest support and the lowest rejection-hesitancy triple on K1 (0.40), the most heavily weighted criterion. Its ξ is0 because its two distance references coincide with the corresponding extrema; its distance to the positive ideal is0.13035018775052218, so it is not on the ideal. A1 is strong on the second criterion (0.80; 0.10; 0.40) but trails A2 on the most heavily weighted criterion; its ξ rises to 0.2958. A3 supplies the positive ideal on K3 but still ranks last by the combined revised ratio.
The decision's hesitation: if K2's weight is raised from 0.35 to 0.55 and K1's weight lowered from 0.40 to 0.20 (K3 held at 0.25), the ranking changes from top to bottom. The same calculation carries A1 to first with 0.0734, A2 to second with 0.2740, and A3 to third with 0.7994. A2 falls from first to second. This shows that the relative weight of K1 against K2 can, on its own, determine the ranking among these three alternatives. However the weights were set, the report must justify this.
In the report: "With the given weights (K1=0.40, K2=0.35, K3=0.25), A2 is the alternative preferred by the combined revised ratio (ξ=0.0000). If K2's weight is raised past K1's (K2=0.55, K1=0.20), A1 moves ahead (ξ=0.0734)."
Source of the numbers: DecisionMind's constructed native regression fixture. The current scalar-weight equations and the weight-change scenario are independently checked with70-digit decimal arithmetic. This is not an independently verified example from the 2019 paper; the complete source-to-implementation match remains open.
2. Aviation: A regional airline's choice of manufacturer offer for its narrow-body fleet
A regional airline will choose among three aircraft manufacturers' offers for a fleet-renewal decision. Three criteria have been set: suitability of the offer against total cost of ownership, suitability against fuel efficiency, and suitability against maintenance-network coverage. None of these is a directly measured quantity. Some of the underlying data (fuel-price projections, spare-parts lead time) is not yet settled at the offer stage. So the fleet-planning board has scored, for every offer, how much it supports, how much it rejects and how undecided it remains on the judgement "this offer suits the fleet," each separately.
The method scales every offer's spherical fuzzy triple on the three criteria by the weights. It builds the ideal and anti-ideal offer according to the score function and computes the closure ratio. Suppose the offer with the strongest support and the lowest hesitancy triple on fuel efficiency comes out preferred by the combined revised ratio, that is, its ξ is smallest. But the board's hesitancy on maintenance-network coverage has remained high.
The board's hesitation: the high hesitancy share (π) on the maintenance-network criterion has dissolved inside the offer's score. ξ alone does not show where this hesitancy comes from. Before choosing the offer with the smallest ξ, the board should separately report which criterion still carries a high hesitancy share. It should also consider adding an extra guarantee clause to the maintenance-network contract.
In the report: "With the given weights, this offer is preferred by the combined revised ratio, with the smallest ξ. The hesitancy share on the maintenance-network-coverage criterion has remained higher than on the other criteria and should be addressed separately at the contract stage."
3. What Not to Do
In the illustrative example, reading A2's result of ξ=0.0000 as "the lowest score, the worst alternative" is wrong. Looking for a large number out of crisp-TOPSIS habit would wrongly take A3 (ξ=1.1387) for first place. In SF-TOPSIS, however, a small ξ is good. The second error is reducing the triples on criterion K2 to a single number from the outset (for instance taking only μ) and running crisp TOPSIS; the hesitancy share is left out of the calculation and the ideal point comes out different. The third error is changing A1's triple on K2, (0.80; 0.10; 0.40), to a value such as (0.80; 0.10; 0.60) without checking the constraint. The sum of the squares becomes 0.64+0.01+0.36=1.01, exceeding the constraint.
Formulae
The computation step by step, with the formula behind each step.
1. Read the native spherical decision matrix and the criterion coefficients (real weights, or spherical fuzzy weights through extras.weights_sf).
2. Weight every spherical cell: w̃_j ⊗ α_ij for spherical fuzzy weights, the spherical scalar multiple w_j · α_ij for real weights.
3. Select the spherical positive and negative ideal cell of each criterion by the score; reverse the roles for cost criteria and reject distinct tied extreme cells.
4. Compute the normalized Euclidean spherical distances of every alternative to the ideal cells and the extreme reference distances.
5. Compute the closeness ratio ξ of Eq. (37) and rank increasing, assigning the same rank to equal ratios.
Sources
Kutlu Gündoğdu, F., & Kahraman, C. (2019). Spherical fuzzy sets and spherical fuzzy TOPSIS method. Journal of Intelligent & Fuzzy Systems, 36(1), 337–352. DOI: 10.3233/JIFS-181401
Hwang, C. L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications — A State-of-the-Art Survey. Lecture Notes in Economics and Mathematical Systems, Vol. 186. Springer-Verlag. DOI: 10.1007/978-3-642-48318-9
Ashraf, S., Abdullah, S., Mahmood, T., Ghani, F., & Mahmood, T. (2019). Spherical fuzzy sets and their applications in multi-attribute decision making problems. Journal of Intelligent & Fuzzy Systems, 36(3), 2829–2844. DOI: 10.3233/JIFS-172009