Extension card · Intuitionistic
Intuitionistic fuzzy SAW (Kaur and Kumar, 2013)
IF-SAW is the form of SAW that works with intuitionistic fuzzy numbers when criteria rest on a judgement (support/rejection). It runs the weighted aggregation on the support and rejection degrees separately, then reduces the result to a single score difference and ranks on it.
Base method
SAW →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Intuitionistic →
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 does not.
Cells. In crisp SAW every cell is a single number. Here every cell is a pair of numbers: a degree of support (μ) and a degree of rejection (ν), with μ+ν ≤ 1. Weights, however, remain crisp, ordinary numbers summing to 1; only the decision matrix is intuitionistic, not the weight vector.
Scale equalisation. Crisp SAW divides every column by its own best value. Here, instead, every cell in the "lower is better" columns is complemented: the support and rejection degrees swap places, (μ, ν) → (ν, μ). This is the intuitionistic fuzzy step corresponding to crisp SAW's normalisation. On a cost criterion the "rejection" direction is turned into a "support" direction, so that every column reads in the same direction.
Aggregation and defuzzification. Crisp SAW takes a weighted sum. Here, an intuitionistic fuzzy weighted average is taken of each alternative's values across the criteria: the support degrees are combined over the weighted product of their "not supported at all" probabilities, and the rejection degrees over a direct weighted product. This aggregation differs from crisp SAW's linear sum; the result is again a single support-rejection pair (μ_S, ν_S). For ranking, this pair is reduced to a single score by taking the difference (μ_S − ν_S); the hesitancy margin (π = 1 − μ_S − ν_S) does not enter the score, it stays behind it.
DecisionMind fixes this aggregation rule and the score difference (μ_S − ν_S) in classical IF-SAW. Weights are taken from outside, as crisp numbers; the method does not generate weights.
How to Read the Output
The score, like the total score in crisp SAW, only ranks this set of alternatives; it is not a percentage, and it is not compared with a different analysis.
The difference is this. The score is the difference between the same alternative's support and rejection degrees, and it hides the hesitancy margin. Two alternatives can reach the same score from very different hesitancy margins: one from a small gap between strong support and strong rejection, another from weak support and weak rejection. The report should therefore show not only the score, but the support and rejection degrees themselves.
Thus instead of writing:
"This alternative is definitely the best because its IF-SAW score is the highest"
the report should read:
"This score is the difference produced by the weighted aggregation of the support and rejection degrees; which alternative carries the larger hesitancy margin must be reported separately"
When to Prefer This over the Base Method
Use this extension when a criterion comes from a judgement and the opposing view of that judgement (the rejection degree) can also be measured from a separate source. Forcing a measured quantity into a judgement and inventing a support/rejection pair, for example writing ν as 1−μ, is not modelling uncertainty but manufacturing it. DecisionMind requires a single data type; if the table is mixed, a measured value is also written as a pair with zero hesitancy, that is, in the form (μ, 0), meaning full support.
SAW's exit condition applies unchanged: if no compromise is acceptable on one criterion, this extension also carries full-compensation logic and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Entering a pair with μ+ν > 1. If the intuitionistic fuzzy set's core constraint is violated (the sum of support and rejection exceeds 1), the calculation is invalid; a judgement whose sum exceeds 1 must either be reduced or moved to a broader structure such as Pythagorean fuzzy.
Leaving a cost criterion uncomplemented. If the support and rejection degrees in a "lower is better" column are not swapped, that criterion is treated as if it were "higher is better" too, and the ranking reverses.
Inventing ν as 1−μ. This zeroes the hesitancy margin and silently reduces the intuitionistic fuzzy structure to crisp data; the rejection degree must come from its own source.
Reading the score alone and ignoring the hesitancy margin. Two alternatives with the same score can carry very different reliability; where the hesitancy margin is large, the score should be read more cautiously.
The governing principle is this:
The IF-SAW score is a difference arising from the weighted combination of support and rejection degrees; this difference hides the hesitancy margin, and the report must show that margin separately.
Cases
The first case is DecisionMind's validation example: the manifest sets up this 3×3 table not from the literature but as a hand-traceable construction faithful to the formulas. The second case is an illustrative construction.
1. Illustrative example: Assessing three supplier candidates by a reliability judgement (DecisionMind validation example)
A procurement unit will choose one of three supplier candidates. There are three criteria: delivery reliability (C1, higher is better), quality compliance (C2, higher is better) and unit price (C3, lower is better). Each criterion has been turned into a judgement, "this supplier is adequate on this criterion", and the expert panel's votes have been recorded as degrees of support (μ) and rejection (ν).
| Candidate | C1 | C2 | C3 |
|---|---|---|---|
| A1 | (0.70; 0.20) | (0.60; 0.30) | (0.40; 0.50) |
| A2 | (0.80; 0.10) | (0.50; 0.40) | (0.30; 0.60) |
| A3 | (0.50; 0.40) | (0.70; 0.20) | (0.20; 0.70) |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method complements C3 (the cost criterion), combines the support and rejection degrees of the three criteria by weighting, and computes the score (μ_S − ν_S).
| Candidate | μ_S | ν_S | Score | Rank |
|---|---|---|---|---|
| A2 | 0.672 | 0.214 | 0.4584 | 1 |
| A3 | 0.632 | 0.264 | 0.3681 | 2 |
| A1 | 0.623 | 0.274 | 0.3489 | 3 |
The result reads as follows. A2 holds the highest support (0.80) and the lowest rejection (0.10) degree on delivery reliability, the heaviest criterion; this advantage offsets its middling performance on the other criteria. The gap between A3 and A1 (0.3681 against 0.3489) is small.
The unit's hesitation is this: what would happen if unit price (C3) were given a higher weight than its current 0.25, say 0.40, and delivery reliability (C1) a lower weight, say 0.25? In that case A3 moves ahead of A2 with a score of 0.4213 against 0.3842, because A3 offers the best price; this has been verified with an independent Python calculation. The report should therefore state that A2's lead is sensitive to the weight given to the price criterion.
In the report: "A2 ranks first with a score of 0.4584, thanks to its strong support and low rejection degree on delivery reliability; if the unit-price criterion is given a markedly higher weight than at present, A3 moves ahead, so the price-weight decision should be justified separately."
Source: DecisionMind IF-SAW manifest, validation example; the support/rejection degrees were constructed synthetically, and the calculation steps follow Atanassov's (1986) definition of the intuitionistic fuzzy set and Xu's (2007) weighted aggregation operator. No literature source is attached to this table in the manifest, so it is presented as an illustrative example.
2. Media: Choosing a new content-verification tool
A news organisation will choose one of three software tools for verifying user-generated content. The criteria are verification accuracy, ease of use, and integration cost (lower is better). Editors and the technical team have assessed each tool against a judgement, "this tool fits the daily workflow"; some editors supported it while others held reservations, and the support and rejection degrees have been derived from the proportions of participants.
The method complements the cost criterion and combines the team's support and rejection votes by weighting. Suppose the tool receiving the strongest support on verification accuracy, but also the highest integration cost, comes first, and the tool with slightly lower accuracy but a lower cost comes second, with a small gap between them.
The organisation's hesitation is this: the rejection degree on the first tool's integration-cost criterion, that is, the reservations of some technical team members, is high. This reservation has dissolved into the score difference and is no longer visible on its own. The reason for this reservation, the risk of incompatibility with the existing system, should be conveyed separately at the decision meeting.
In the report: "The ranking arises from the weighted combination of the team's support/rejection votes on verification accuracy, ease of use and integration cost; the first tool's high rejection degree on integration cost has dissolved into the score and should be assessed separately."
3. What Not to Do
If, in the illustrative example, A1's unit-price criterion had been left uncomplemented (that is, left as (0.40; 0.50), still pointing in the "support" direction while the rejection degree is high), the most expensive alternative would appear the best. The second error is writing the rejection degree as the complement of μ (1−μ) instead of collecting it separately from the expert panel; this zeroes the hesitancy margin and conceals the panel's genuine reservation. The third error is comparing A2's and A3's scores (0.4584 and 0.3681) on their own, without looking at their hesitancy margins (π), and declaring "A2 is definitely more reliable"; if the two alternatives' hesitancy margins differ, this difference must be written into the report.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/if-saw
Kaur, P., & Kumar, S. (2013). An intuitionistic fuzzy simple additive weighting (IFSAW) method for selection of vendor. IOSR Journal of Business and Management, 15(2), 78–81. DOI: 10.9790/487x-1527881
Fishburn, P. C. (1967). Additive utilities with incomplete product sets: Application to priorities and assignments. Operations Research, 15(3), 537–542. DOI: 10.1287/opre.15.3.537
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Xu, Z. (2007). Intuitionistic fuzzy aggregation operators. IEEE Transactions on Fuzzy Systems, 15(6), 1179–1187. DOI: 10.1109/tfuzz.2006.890678