Extension card · Intuitionistic
Intuitionistic fuzzy ARAS (Mishra, Sisodia, Pardasani and Sharma, 2020)
Intuitionistic fuzzy ARAS is the form of ARAS used when a criterion is assessed with a degree of support for, and a degree of rejection of, a judgement, and where more than one decision-maker's opinion can be combined. On cost criteria it reverses direction by swapping support and rejection, and calculates the ratio to the optimal alternative from these two degrees.
Base method
ARAS →
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?
Five things change; the decision logic stays the same.
Cells. In crisp ARAS every cell is a single, positive, non-zero number. Here every cell is a pair: a degree of supporting a judgement and a degree of rejecting it, two numbers whose sum does not exceed 1. Weights can still remain crisp. This extension's only new feature is that it can bring several decision-makers' pairs together. Every decision-maker's support-rejection pair is combined, along with the decision-makers' weights, into a single pair (weighted intuitionistic aggregation). Where a single decision-maker is used, this step is switched off and the calculation proceeds straight to the next step. Crisp ARAS has no such aggregation step.
Scale equalisation. In crisp ARAS, a cost criterion is reversed (1/x) and divided by the column total. Here, reversing a cost criterion means taking the complement of the intuitionistic fuzzy pair: the support and rejection degrees swap places. This must not be confused with crisp ARAS's 1/x operation; the intuitionistic fuzzy pair is reversed by a swap, not a division, because support and rejection are already defined on the [0,1] interval, and division would break that range.
Optimal alternative and defuzzification. In crisp ARAS, the optimal alternative is built from each criterion's best crisp value. Here the optimal-alternative row is likewise built from support-rejection pairs: the highest support and the lowest rejection on each criterion. The method then reduces every cell, real alternatives and the optimal alternative alike, to a single number through a score function: the score is the difference between support and rejection. Because this difference can come out negative, it is shifted first (1 is added). These shifted scores are then divided by the column total. Normalisation now runs on a crisp number, with the same logic as in crisp ARAS, with just one extra shifting step added.
Weighting and totalling. The method multiplies the shifted, normalised scores by the criterion weight and sums them across criteria for every row, the optimal alternative included; this is identical, step for step, to crisp ARAS's weighted-sum step, except the input is now a score derived from the support-rejection pair.
Utility degree. K carries the same meaning as in crisp ARAS: it is the ratio of the real alternative's total score to the optimal alternative's total score.
DecisionMind fixes the shifted score function (support minus rejection, plus one) and the construction of the optimal alternative in support-rejection space; the support-rejection pair is never ranked directly at any stage, and ranking is done only on K.
How to Read the Output
The utility degree K carries the same meaning here. The best alternative is treated as 100, and the others take a percentage relative to it; this percentage is valid only for this alternative set and these weights. The difference is this: beneath K, a two-degree judgement, support and rejection, first descends to a single score, then enters the weighted total of these scores. An alternative's K value can be more sensitive to how much weight is put on which criterion than in other extensions. The reason is that the shifting and normalisation step narrows the range of the scores. The report should therefore separately show how easily the weight distribution changes the K ranking.
Thus instead of writing:
"According to intuitionistic fuzzy ARAS, A1 is the best alternative"
the report should read:
"With the given weights (0.6; 0.4), A1 has the highest utility degree (K = 0.744); when the weights shift towards the cost criterion (0.4; 0.6), the order reverses completely and A1 falls to last place, so the weight distribution must be separately defended in the report"
When to Prefer This over the Base Method
This method is suitable where the criterion assessment comes from a judgement, and a separate source of information exists for the opposing view. Examples: cases where a board or expert panel records, separately, the proportion of votes or evidence both supporting and rejecting a proposition such as "this candidate is suitable"; board decisions where more than one decision-maker's opinion must be combined in a weighted way, without collapsing it into an average.
Where a criterion only answers the question "how suitable," and there is no separate evidence for rejection (rejection is written simply as 1 minus support), moving to the intuitionistic fuzzy structure adds nothing. In that case intuitionistic fuzzy data carries the same information as crisp or fuzzy data. Crisp ARAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory, and it does not eliminate anything below a threshold.
Mistakes Specific to This Extension
Confusing the intuitionistic fuzzy pair with a triangular fuzzy number. A cell consists of two numbers (support, rejection), not three (lowest, most likely, highest); the two are different data types, and confusing them makes both the swap step and the score step meaningless.
Applying crisp ARAS's 1/x operation instead of the complement on a cost criterion. This breaks the support-rejection pair's [0,1] bound and the condition that the sum must not exceed 1; it produces an invalid pair.
Failing to include the optimal alternative in the normalisation denominator. When the shifted scores are divided by the column total, the optimal-alternative row must be included in that total; if it is left out, utility degrees can exceed 1.
Using the unshifted score (support minus rejection) directly. This score can be negative; if the shifting step is skipped, the normalised values, and hence the utility degrees, can come out negative too.
The governing principle is this:
Intuitionistic fuzzy ARAS exists to carry evidence gathered separately for and against a judgement through to a ratio against the optimal alternative. Any implementation that confuses the complement with division, or leaves the optimal alternative out of the normalisation, corrupts the information the pair carries.
Cases
The first case is DecisionMind's validation example: a small table with a single decision-maker, three alternatives and two criteria, built by hand so that the algorithm can be traced step by step, not taken from a book or paper. The second case is an illustrative construction.
1. Illustrative example: Assessing three candidates on two criteria (DecisionMind validation example)
A board is assessing three candidates on two criteria: technical competence (higher is better) and expected cost (lower is better, given as a support-rejection pair from a single decision-maker's own assessment). The board has given technical competence a weight of 0.6 and cost a weight of 0.4.
| Candidate | Technical competence | Expected cost |
|---|---|---|
| A1 | support 0.8 / rejection 0.1 | support 0.7 / rejection 0.2 |
| A2 | support 0.6 / rejection 0.3 | support 0.5 / rejection 0.4 |
| A3 | support 0.4 / rejection 0.5 | support 0.3 / rejection 0.6 |
| Weight | 0.6 | 0.4 |
Because a single decision-maker is used, the aggregation step is skipped. The method swaps support and rejection on the cost criterion, builds the optimal-candidate row (the highest support and lowest rejection on each criterion), scales the shifted scores against the column total, multiplies by the weights, sums them, and divides by the optimal candidate's score.
| Candidate | Utility degree (K) | Rank |
|---|---|---|
| A1 | 0.7437 | 1 |
| A2 | 0.7346 | 2 |
| A3 | 0.7254 | 3 |
The result reads as follows. A1 has both the highest support and lowest rejection on technical competence, and the best position on the cost criterion too (after reversal); it comes first because it is strongest on both criteria. The gaps are small (only 0.009 between 0.7437 and 0.7346): the order among the three candidates is definite, but the margin is narrow.
The board has one hesitation. Had the weights been swapped, that is, 0.6 to cost and 0.4 to technical competence, the utility degrees would come out at 0.621 for A1, 0.720 for A2 and 0.819 for A3, and the order reverses COMPLETELY: A3 falls to first place, A1 to last (calculated by running the same algorithm independently in Python). Even with equal weights (0.5; 0.5), the order still reverses (A3 first at 0.773). This shows that the order of the three candidates depends entirely on which criterion is given priority; the 0.6 weight given to technical competence determines the decision itself.
In the report: "With a weight of 0.6 on technical competence, A1 has the highest utility degree (K = 0.744); when the weight shifts to cost (0.4; 0.6), the order reverses completely and A3 rises to first place. The weight distribution must therefore be separately approved by the board and justified in the report."
Source: DecisionMind's validation example for the intuitionistic fuzzy ARAS engine; a small, hand-built table with a single decision-maker, not the real IT-personnel-selection example with 5 candidates × 15 criteria × 3 decision-makers in Mishra et al.'s (2020) paper. The utility degrees and the weight-swap scenarios were independently recalculated by this card's author with the same algorithm.
2. Municipality: Choosing a recycling-facility operator
A district municipality will choose among three candidate firms to operate its solid-waste recycling facility. Two criteria apply: operating capacity (higher is better) and contract cost (lower is better). The municipal council's committee has separately stated, for every firm, how much it supports and how much reservation it holds about the proposition "this firm will carry out the contract without problems"; support comes from past references, and reservation comes from news reports of a delay the firm had with another municipality. The committee has given capacity a higher weight than cost.
The method compares the three firms: it swaps support and rejection on the cost criterion, builds the optimal-firm row, equalises the scores, and multiplies and sums them with the weights. Suppose the firm with the highest capacity support also carries a medium-level reservation from a past delay report. It nonetheless comes out first, because the weight on capacity exceeds that on cost. The lowest-cost firm, having weaker support on capacity, finishes second.
The council also has a hesitation. The reservation about the first firm rests on a single news report; signing the contract without verifying this report would mean ignoring the risk. The council should either confirm the reservation from a second, independent source, for example by speaking directly with the municipality concerned, or lower the weight on capacity and review the result again.
In the report: "With the high weight given to capacity, the first firm comes out ahead; because the reservation about this firm rests on a single news source, confirming this reservation from a second source before signing the contract is recommended."
3. What Not to Do
The first error, in the illustrative example, is applying crisp ARAS's 1/x operation when reversing the cost criterion, instead of swapping support and rejection. In that case, the support and rejection values spill outside the [0,1] interval, and an invalid pair is produced whose sum exceeds 1. The second error is normalising without including the optimal-candidate row in the column total; in that case, utility degrees can exceed 1, and the "what percentage relative to the optimal" interpretation collapses. The third error is reporting A1's utility degree of 0.744 as "a 74 per cent probability of being the right choice"; K shows only these three candidates' proportional benefit relative to the optimal candidate, not a probability.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/if-aras
Mishra, A. R., Sisodia, G., Pardasani, K. R., & Sharma, K. (2020). Multi-criteria IT personnel selection on intuitionistic fuzzy information measures and ARAS methodology. Iranian Journal of Fuzzy Systems, 17(4), 55–68. DOI: 10.22111/ijfs.2020.5406
Zavadskas, E. K., & Turskis, Z. (2010). A new additive ratio assessment (ARAS) method in multicriteria decision-making. Technological and Economic Development of Economy, 16(2), 159–172. DOI: 10.3846/tede.2010.10
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Chen, S. M., & Tan, J. M. (1994). Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 67(2), 163–172. DOI: 10.1016/0165-0114(94)90084-1