Extension card · Spherical
Spherical fuzzy COPRAS
Spherical fuzzy COPRAS is the form of COPRAS used when criterion scores are given as three separate numbers: a judgement's degree of support, rejection and hesitancy. Every cell drops to a single score within its constraint in the first step; the benefit and cost totals are then built on these scores exactly as in crisp COPRAS.
Base method
COPRAS →
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?
Four things change. The benefit-cost ratio logic does not.
Cells. In crisp COPRAS every cell is a single number. In spherical fuzzy, every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π). The expert gives all three directly. The only constraint is that the sum of the three degrees' squares must not exceed 1. Hesitancy is not a leftover share; it is asked for separately. Weights come from outside in DecisionMind as crisp numbers. DecisionMind does not support group decisions in this family; a single, previously combined decision matrix is entered.
Early scoring. In SF-TOPSIS, SF-VIKOR and SF-EDAS the three degrees stay three-dimensional almost to the end of the calculation. In SF-COPRAS the situation is different. Every cell is reduced to a single number with a score function as early as the first step. The reason lies in the method's own structure: COPRAS's backbone is summing the values on the benefit and cost criteria separately, and summation works on single numbers. A spherical fuzzy way of summing three degrees does exist, but COPRAS's founding logic already rests on summing crisp numbers. SF-COPRAS does this summation early, through the score function; but this score is a rule that evaluates the three degrees together.
Scale equalisation. Crisp COPRAS converts every column into a share by dividing by its own total. Because the score function's result can come out negative (values in crisp COPRAS are always positive), SF-COPRAS first adds a fixed shift to every score and moves the result into the positive range. Only after this does division by the column total, as in crisp COPRAS, apply. This shifting step resolves crisp COPRAS's inability to work with negative values, for spherical fuzzy scores.
Benefit-cost total and benefit degree. From this step on, the method proceeds exactly as crisp COPRAS does. The scores, once converted to shares and weighted, are summed separately for the benefit criteria and separately for the cost criteria. The two totals are combined with the same term that rewards low cost, and the relative significance value is built. This value is divided by the highest one and converted to a percentage; the best alternative again scores 100. DecisionMind fixes the score function and the amount of the shift in this family.
How to Read the Output
The benefit degree carries the same meaning as in crisp COPRAS. The best alternative is taken as 100, the others receive a percentage relative to it, and this percentage is valid only for this alternative set and these weights. The difference is here: beneath this percentage now lies a three-degree assessment. The score function reduces these three degrees to a single number in the first step. Because of this, the gap between two alternatives' benefit degrees does not directly show how robust the balance of support, rejection and hesitancy in the input actually is.
Thus instead of writing:
"SF-COPRAS's benefit degree is read with the same certainty as crisp COPRAS's"
the report should read:
"The benefit degree carries the same percentage logic as crisp COPRAS's; but this percentage is built after every cell's three degrees have been reduced to a single score as early as the first step, and this score does not separately show on which criterion hesitancy was high"
When to Prefer This over the Base Method
This extension is used when experts give a judgement's degree of support, rejection and hesitancy separately. It is also suitable when crisp COPRAS's "percentage relative to the best" output is wanted under this uncertainty. The boundary that must not be confused with the spherical fuzzy identity and neighbouring types (Pythagorean, picture fuzzy) is on the data-type card.
The exit condition is the same as for crisp COPRAS. The matrix must be of a single type. If no compromise is acceptable on one criterion, this extension is also compensatory.
Mistakes Specific to This Extension
Confusing the value-space constraint. The constraint is the sum of the three degrees' squares. Deriving hesitancy as 1−μ−ν and writing it into the spherical structure is a different data type; this is the constraint of the intuitionistic fuzzy structure.
Confusing early scoring with the "defuzzify first, then run the crisp method" mistake. Applying the score function in the first step is SF-COPRAS's own design. The three degrees are evaluated together within the score function, not skipped over. What is wrong is ignoring the three degrees the expert gave and using only μ. Carrying another family's score function over here, for instance the Pythagorean one, is the same mistake.
Skipping the shifting step. If the score comes out negative and no shift is applied, the division-by-column-total step gives an undefined or misleading result. This is the spherical fuzzy counterpart of crisp COPRAS's inability to work with negative values.
Reading the benefit degree as an absolute quality percentage. The same mistake as in crisp COPRAS applies here too. The percentage is only a share relative to the best in this set; it is not an absolute measure of success.
The governing principle is this:
SF-COPRAS reduces the three degrees to a single number with the score function in the first step and applies crisp COPRAS's benefit-cost summation logic to these numbers exactly as it stands; early scoring is not a mistake but the method's design, though the benefit degree does not show which criterion's hesitancy it came from.
Cases
The first case is DecisionMind's validation example. It is a small, three-alternative, three-criterion table, not a page carried over from the literature. It was built to make the engine's steps traceable and was verified by running the engine itself. The second case is an illustrative construction.
1. Illustrative example: Three alternatives assessed on three criteria in spherical fuzzy form (DecisionMind validation example)
Three alternatives are assessed on three criteria with spherical fuzzy triples. All three are higher-is-better criteria.
| 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 reduces every cell to a single number with the score function, adds a fixed shift and divides by the column total. It multiplies these shares by the weights, combines them all into a single total since all are benefit criteria, and divides by the highest one to convert to a percentage.
| Alternative | Benefit degree | Rank |
|---|---|---|
| A2 | 100.00 | 1 |
| A1 | 89.29 | 2 |
| A3 | 83.98 | 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, and its score on this criterion is clearly higher than the other two. This advantage makes its total share the largest, and A2 reaches 100 per cent. A1 is second, A3 third; both trail A2 on K1.
The decision's hesitation: if K2's weight is raised from 0.35 to 0.80 and K1's weight is lowered from 0.40 to 0.10 (K3 falls from 0.25 to 0.10), the ranking changes. The same calculation carries A1 to first with 100.00 and A2 to second with 92.31. This shows that the relative weight of K1 and 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 weights given (K1=0.40, K2=0.35, K3=0.25), A2 has the highest benefit degree (100.00). If K2's weight is raised enough to overtake K1 (K2=0.80, K1=0.10), A1 moves ahead (100.00 against 92.31)."
Source: DecisionMind's SF-COPRAS validation example. The steps combine crisp COPRAS's (Zavadskas and Kaklauskas, 1996) benefit-cost summation logic with Kutlu Gündoğdu and Kahraman's (2019) score function. The benefit degrees and the weight-change scenario were obtained by running DecisionMind's SF-COPRAS engine directly.
2. Fire Service: A fire brigade's choice of tanker-truck supplier
A metropolitan fire brigade will choose among the tenders of three tanker-truck manufacturers for a fleet-renewal decision. Three criteria are set: the tender's suitability in terms of field durability, its suitability in terms of ease of staff training and maintenance, and its suitability in terms of delivery and commissioning time. None of these criteria can be given as a precise number at the contract stage. The technical board has separately scored a degree of support, rejection and hesitancy for the judgement "this vehicle can be used safely in the field" for every tender.
The method shifts every tender's score on the three criteria, converts it to a share, multiplies by the weights, and builds the benefit degree. Suppose the tender with the strongest support triple on the durability criterion reached 100 per cent, because that criterion carried the most weight. But on delivery time the board's hesitancy stayed high, because the manufacturer did not firmly commit to a delivery schedule.
The board's hesitation: the high hesitancy on delivery time carries the risk that the fleet renewal will miss its planned schedule. This risk may be masked within the benefit degree by the strength of the durability criterion. The board should consider setting a binding contractual ceiling on delivery time before selecting the tender with the highest degree.
In the report: "With the weights given, the tender strong on durability reaches the highest benefit degree. Because the hesitancy share on the delivery-time criterion has stayed high, a binding delivery-date clause in the contract is recommended."
3. What Not to Do
In the illustrative example, it is wrong to skip the score function and feed K1's triples directly into crisp COPRAS with their μ values alone (such as 0.70; 0.90; 0.50). The degrees of rejection and hesitancy are left out of the calculation, and A2's advantage appears larger than it actually is. The second mistake is skipping the shifting step and dividing negative scores directly by the column total; this corrupts the sign of the shares and renders the ranking meaningless. The third mistake is reporting A2's benefit degree of 100 per cent as "perfect" or "flawless"; this value only means it is the best among these three alternatives.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/sf-copras
Zavadskas, E. K., & Kaklauskas, A. (1996). Determination of an efficient contractor by using the new method of multicriteria assessment. In: International Symposium for the Organization and Management of Construction: Shaping Theory and Practice, Vol. 2: Managing the Construction Project and Managing Risk (CIB W65), 94–104. (no DOI)
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
Omerali, M., & Kaya, T. (2022). Augmented Reality Application Selection Framework Using Spherical Fuzzy COPRAS Multi Criteria Decision Making. Cogent Engineering, 9(1), 2020610. DOI: 10.1080/23311916.2021.2020610
Güleryüz, S. (2024). Sustainability Performance Evaluation in Faculties: A COPRAS-Based Assessment. In: Kahraman, C. et al. (Eds.), Intelligent and Fuzzy Systems — Intelligent Industrial Informatics and Efficient Networks: Proceedings of the INFUS 2024 Conference, Vol. 1 (Lecture Notes in Networks and Systems, Vol. 1088, pp. 137–146). Springer. DOI: 10.1007/978-3-031-70018-7_16