Extension card · Hesitant
Hesitant Fuzzy COPRAS (Mishra, Rani and Pardasani, 2018)
The form of COPRAS for situations where more than one plausible degree of membership on a criterion must be held together. It builds the benefit and cost totals through set aggregation, then ranks the result, once again, with a relative-significance value.
Base method
COPRAS →
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 crisp COPRAS every cell is a single number. Here every cell is a set that holds more than one plausible value on the same criterion together. Weights are given fully from outside; HF-COPRAS, like the base method, does not generate weights.
Scale equalisation. Crisp COPRAS divides every column by its own total to convert it into a share. This step does not exist for hesitant sets, because the values already lie between 0 and 1 as degrees. Instead, DecisionMind aggregates the sets of the benefit criteria with one another using a special aggregation rule. This rule resembles the "either one or the other" logic of probability theory: when two sets are aggregated, the resulting set can be larger than the input sets. The same aggregation is done separately for the cost criteria. This gives each alternative one benefit set and one cost set.
Distance, score and aggregation. The benefit set and the cost set are each reduced to a number by averaging the values within the set. This is called the set score. The relative-significance value is built exactly as in crisp COPRAS: a correction term reflecting the balance of the cost scores against one another is added to the benefit score. DecisionMind computes this correction term with the same formula as crisp COPRAS, using the set scores as its input.
Result and defuzzification. As in crisp COPRAS, the output is a relative-significance value and a percentage derived from it; the best alternative receives one hundred per cent. Sets descend to a score, that is a single number, IMMEDIATELY AFTER the benefit and cost totals are built. This is a defuzzification that sits in the middle of the calculation; it is neither at the very start nor at the very end. The set-aggregation step can even enlarge the hesitancy, because the benefit set can come out more numerous than the input sets; the score then takes the average of this enlarged set.
DecisionMind holds the set-aggregation rule and the average-score function fixed in classical HF-COPRAS.
How to Read the Output
The relative-significance value and the percentage are read exactly as in crisp COPRAS. They show a share relative to the best alternative, not an absolute percentage of success. What differs is this: the set-aggregation step can multiply the number of input sets. So an alternative's benefit set can grow depending on how many expert opinions it draws from. The score is taken as the average of this enlarged set, and how many elements the set holds does not show up in the report.
Thus instead of writing:
"HF-COPRAS found this alternative 84 per cent successful"
the report should read:
"With these weights and this alternative set, this alternative has the best benefit-cost balance; the runner-up reaches 84 per cent of the benefit this alternative provides"
When to Prefer This over the Base Method
Use this extension when more than one plausible and defensible degree exists for the same criterion-alternative pair. The distinction on the data-type card applies here too: if a single measured value exists, a single number is written into the cell, and no set is built. DecisionMind requires the table to be of a single data type. The base COPRAS's requirement to avoid negative values applies here too; every value within the set must lie between 0 and 1.
Mistakes Specific to This Extension
Mistaking set aggregation for a simple average. The aggregation rule DecisionMind uses does not average the two sets directly. The resulting set can be larger than the inputs and follow a different distribution.
Marking the cost and benefit direction wrongly. If a cost criterion is placed on the benefit side, a high value on that criterion is rewarded.
Allowing a value in the set to fall outside the 0-to-1 range. The aggregation rule is not defined outside this range.
The "more advanced" fallacy. Where no genuine source supports more than one value, building a set adds no information; it only complicates the calculation.
The governing principle is this:
The set-aggregation step can enlarge hesitancy; because the score takes the average of this enlarged set, how many sources it came from must be shown in the report.
Cases
The first case is a genuine literature case. It is the insurance-company selection example from Mishra, Rani and Pardasani's (2018) paper, and the figures are taken from that paper. The second case is an illustrative fiction.
1. Insurance: Choosing motor-insurance service quality (Mishra, Rani and Pardasani, 2018)
A customer will choose among four insurance companies based on their motor-insurance service quality. There are four criteria: trust, responsiveness, reliability and tangibles. The first three are "more is better", the fourth is "less is better". Expert assessments give, for the same company-criterion pair, more than one plausible degree together rather than a single score. Weights were pre-computed using the Shapley value.
| Company | Trust | Responsiveness | Reliability | Tangibles |
|---|---|---|---|---|
| D1 | {0.1; 0.3; 0.6; 0.7; 0.9} | {0.2; 0.3; 0.6; 0.8} | {0.4; 0.5; 0.8} | {0.3; 0.5; 0.8} |
| D2 | {0.2; 0.5; 0.7; 0.8; 0.9} | {0.3; 0.4; 0.6; 0.7} | {0.2; 0.3; 0.6} | {0.5; 0.7; 0.8} |
| D3 | {0.3; 0.5; 0.7; 0.8; 0.9} | {0.3; 0.5; 0.6; 0.8} | {0.3; 0.6; 0.7} | {0.3; 0.5; 0.7} |
| D4 | {0.2; 0.4; 0.6; 0.7; 0.8} | {0.1; 0.2; 0.6; 0.8} | {0.3; 0.4; 0.6} | {0.4; 0.6; 0.8} |
| Direction | more is better | more is better | more is better | less is better |
| Weight | 0.4042 | 0.0461 | 0.0958 | 0.4539 |
The method first aggregates the sets of the first three criteria with the weights into a benefit set for each company, and builds a cost set from the fourth criterion's set. Both sets are averaged into scores. The scores are combined with crisp COPRAS's own formula to give the relative-significance value.
| Company | Relative significance | Rank |
|---|---|---|
| D3 | 0.840 | 1 |
| D1 | 0.736 | 2 |
| D2 | 0.688 | 3 |
| D4 | 0.665 | 4 |
The result reads as follows. D3 has the highest benefit score and the lowest cost score; these two advantages together put D3 first. D2 and D4 sit close to one another; the gap between them is only 0.023.
The customer's hesitation is this. What happens if the tangibles criterion's weight is raised from 0.4539 to 0.7581, and trust's weight is lowered from 0.4042 to 0.10? D3 still finishes first. But D2 and D4 swap places, with D4 moving ahead. This shows that the choice between the two lowest-ranked companies depends on the weight assumption.
In the report: "D3 has the highest relative significance (0.840), and this result is robust to changes in the weights. The order between D2 and D4, however, is sensitive to the weight of the tangibles criterion."
Source: Mishra, Rani and Pardasani (2018), Table 2 and Table 5, pp. 435-449. The relative-significance values and the weight-sensitivity scenario were obtained by independently re-running DecisionMind's HF-COPRAS engine. The ranking matches the paper's own Table 5 result (D3, D1, D2, D4) exactly.
2. Shipping: Choosing a port service provider
A ship operator will choose among three port operators for container loading and unloading services. There are four criteria: unit container handling fee, average vessel waiting time, port equipment capacity score and insurance claims-history score. Fee and waiting time are "less is better". The operator knows each port's performance from actual voyage records across different seasons (summer-winter). It therefore gives each port-criterion pair a set that varies by season rather than a single score. The weights come from the operator's own operations unit.
The method aggregates the benefit criteria's sets into a benefit set, and the cost criteria's sets into a cost set. It turns the scores into a relative-significance value. Suppose the lowest-fee port also has the longest waiting time. It still comes out first, because the weight on fee is higher than the weight on waiting time.
The operator's hesitation is this. The width of the waiting-time set, long in winter and short in summer, say, shows that this port's operation varies substantially by season. The relative-significance value dissolves this variability into a single number. If the operator has season-specific delivery commitments, it should look not only at the score but also at the set's seasonal distribution.
In the report: "Because of the high weight on fee, the cheapest port has the highest relative significance. This port's waiting-time set is wide and varies by season. This risk should be assessed separately."
3. What Not to Do
Had the tangibles criterion in the insurance table been marked "more is better", the company with the highest tangibles score would be rewarded, and the advantage D3 gains from its low cost would be reversed. The second error is skipping the aggregation step across the four companies' sets and instead taking each set's own average directly, running COPRAS with crisp numbers; this discards the information that set aggregation provides. The third error is reading D3's relative-significance value of 0.840 as "84 per cent quality service"; this value only compares these four companies against one another.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/hf-copras
Mishra, A. R., Rani, P., & Pardasani, K. R. (2018). Multiple-criteria decision-making for service quality selection based on Shapley COPRAS method under hesitant fuzzy sets. Granular Computing, 4(3), 435–449. DOI: 10.1007/s41066-018-0103-8
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. (no DOI)
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529–539. DOI: 10.1002/int.20418
Xia, M., & Xu, Z. (2011). Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 52(3), 395–407. DOI: 10.1016/j.ijar.2010.09.002