Extension card · Plithogenic
Plithogenic ARAS
Plithogenic ARAS is the form of ARAS used when criterion assessment is given as degrees of truth, indeterminacy and falsity, and how contradictory this assessment is in itself is also known. It factors the contradiction share into the calculation beforehand, then computes the additive utility ratio.
Base method
ARAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Plithogenic →
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 does not.
Cells. In crisp ARAS every cell is a single number. Here every cell consists of three degrees: truth, indeterminacy and falsity. This triple looks the same as in Neutrosophic ARAS, but the plithogenic structure adds a contradiction degree to every criterion as well. The contradiction degree is a pre-set number showing how far that criterion stands from the "dominant feature" for this decision problem, and it varies from criterion to criterion, not from cell to cell.
Contradiction adjustment. This step is specific to the plithogenic structure and is absent from the other ARAS family members. Before entering the calculation, every cell is adjusted by its own criterion's contradiction degree: truth is pulled upward by the contradiction share, while indeterminacy and falsity are pulled downward by the same proportion. If the contradiction degree is zero (the criterion fully aligns with the dominant feature), the cell stays unchanged; the higher the contradiction degree, the further the cell's truth is strengthened. This step is absent from Neutrosophic ARAS and is the plithogenic structure's one new contribution.
Scale equalisation. In crisp ARAS, a cost criterion is inverted (1/x) and then divided by the column total. Here, inverting a cost criterion means taking its complement: truth and falsity swap places. After the contradiction adjustment, every cell is first reduced to a score (as in Neutrosophic ARAS: truth plus, twice indeterminacy and falsity minus), and these scores are then set as a proportion of the column total (including the optimal-alternative row).
Weighting and totalling. The method combines the contradiction-adjusted T-I-F cells with the criterion weight through the plithogenic summation rule (the same as the neutrosophic weighted sum), and sums every row (including the optimal alternative) across criteria.
Result and defuzzification. Every row's total is reduced to a single number with the same score function, truth plus, twice indeterminacy and falsity minus. The utility degree K carries the same meaning as in crisp ARAS: the ratio of a real alternative's score to the optimal alternative's score. DecisionMind applies the contradiction adjustment once, before normalisation; cells are never ranked directly at any stage, ranking is done only on K.
How to Read the Output
The utility degree K carries the same meaning here too. The best alternative is taken as 100, and the others receive a percentage relative to it; this percentage is valid only for this alternative set and these weights. The difference is here: beneath K lie both a three-degree judgement and the criterion's own contradiction share. Strong truth on a criterion with a high contradiction degree does not carry the same weight as the same truth on a criterion with a contradiction degree of zero; the contradiction adjustment amplifies this difference beforehand.
Thus instead of writing:
"According to Plithogenic ARAS, A2 is the best alternative"
the report should read:
"With the given weights (0.40; 0.35; 0.25), A2 has the highest utility degree (K=0.968); when the weight is concentrated on the first criterion (0.70; 0.15; 0.15), A1 and A3 swap places, but A2 keeps first place, so A2's lead is robust while the second and third ranks are sensitive to weight"
When to Prefer This over the Base Method
This method is suitable when how consistent or contradictory a criterion is with the decision problem is known in advance, and this information is meant to enter the assessment. Examples: multidimensional assessments where some criteria are complementary in nature, standing apart from the decision problem's "dominant feature"; expert panels where how far a criterion overlaps with the others is separately measured.
If no information about the contradiction degree exists, or every criterion is treated as equally "dominant," moving to the plithogenic structure adds nothing; Neutrosophic ARAS is sufficient in that case. Crisp ARAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Violating the value space. This is the manifest's own warning: every attribute value must be defined together with its own degree of membership and contradiction; the contradiction degree must not be added as an unjustified number outside the calculation.
Varying the contradiction degree from cell to cell. The contradiction degree is a property of the criterion itself; it does not change with the alternative. Applying a different contradiction share from alternative to alternative on the same criterion corrupts the comparison.
Applying the contradiction adjustment after normalisation. DecisionMind applies this adjustment first. Reversing the order makes the contradiction share enter the column total differently, and the results drift from the source formula.
Assigning a contradiction degree without justification, "to look more thorough." A contradiction degree other than zero must rest on a reason for why the criterion does not fully coincide with the decision problem; assigned without justification, it adds fabricated information.
The governing principle is this:
Plithogenic ARAS exists to factor in, beforehand, the information about how consistent a criterion is with the decision problem; any use that varies this information by alternative, or applies it without justification, corrupts the contradiction adjustment's one contribution.
Cases
The first case is DecisionMind's validation example. Smarandache's (2018) founding book defines the operations of plithogenic algebra (Equations 26, 44, 45) but gives no decision-table example; a small table, traceable by hand, was therefore produced under DecisionMind's 2026-05-25 protocol. The second case is an illustrative construction.
1. Illustrative example: Three candidates, three criteria (DecisionMind validation example)
An evaluation board is scoring three candidates on three criteria; the first and second criteria are "higher is better", the third is "lower is better". Every score is given as degrees of truth, indeterminacy and falsity. The board has assigned weights of 0.40, 0.35 and 0.25 to the three criteria respectively. The first criterion's contradiction degree is 0 (fully aligned), the second is 1/3, and the third is 2/3 (a feature more distant from the decision problem).
| Candidate | K1 (contradiction 0) | K2 (contradiction 1/3) | K3, lower is better (contradiction 2/3) |
|---|---|---|---|
| A1 | T=0.7 I=0.2 F=0.1 | T=0.5 I=0.3 F=0.2 | T=0.6 I=0.3 F=0.2 |
| A2 | T=0.8 I=0.1 F=0.1 | T=0.6 I=0.2 F=0.2 | T=0.4 I=0.2 F=0.3 |
| A3 | T=0.6 I=0.2 F=0.2 | T=0.7 I=0.2 F=0.1 | T=0.5 I=0.3 F=0.2 |
| Weight | 0.40 | 0.35 | 0.25 |
The method takes the complement on the third criterion, then adjusts every cell by its own criterion's contradiction degree, scores the contradiction-adjusted cells and sets them as a proportion of the column total (including the optimal alternative), then multiplies by the weights and sums.
| Candidate | Utility degree (K) | Rank |
|---|---|---|
| A2 | 0.968 | 1 |
| A3 | 0.864 | 2 |
| A1 | 0.820 | 3 |
The result reads as follows. A2 is best on K1, the heaviest criterion (contradiction degree zero, that is, the highest truth without any adjustment), and this lead is unaffected by any contradiction adjustment. A3 is best on K2, which has a contradiction degree of 1/3, but comes second because of its weakness on K1. A1 is best on no criterion and finishes last.
The board has one hesitation. When the weight is concentrated further on the first criterion (0.70; 0.15; 0.15) and recomputed, the utility degrees come out at 0.987 for A2, 0.816 for A1 and 0.771 for A3 (computed by running the same algorithm independently in Python). A2 keeps first place, but A1 and A3 swap places. This shows that A2's first place is robust, while the second and third ranks are sensitive to the weight distribution.
In the report: "With the given weights (0.40; 0.35; 0.25), A2 has the highest utility degree (K=0.968); when the weight is concentrated on the first criterion (0.70; 0.15; 0.15), A2 keeps first place, but A1 and A3 swap places. Not only first place, therefore, but also the second and third ranks should be reported together with the weight distribution."
Source: DecisionMind's validation example for the Plithogenic ARAS engine; the contradiction adjustment and scoring are faithful to Smarandache's (2018) definition of plithogenic algebra, but the decision matrix and results come not from the founding book but from a small table constructed by hand by this card's author and computed in Python. No named application paper specific to this method could be found in the manifest records; this point is noted separately in the verification note.
2. Transport: A municipality's choice of a new bus for its fleet
A municipal public-transport department will choose a new bus model to join its fleet from three candidates. Two criteria apply: passenger capacity and operating cost (lower is better). The technical commission has assessed every vehicle's "fitness for service" on these two criteria with degrees of truth, indeterminacy and falsity. The commission treats operating cost as the municipality's primary priority for this year, and has accordingly set this criterion's contradiction degree low (0). Because it treats passenger capacity as a secondary feature, it has given it a higher contradiction degree (1/3).
The method compares the three vehicles: it takes the complement on the cost criterion, adjusts every cell by its own criterion's contradiction degree, scores it and sets it as a proportion of the column total, then multiplies by the weights and sums. Suppose the vehicle with the lowest operating cost has middling passenger capacity. It still comes out first, because the cost criterion's low contradiction degree means the adjustment has not weakened its strength on this criterion.
The department has one hesitation. Passenger capacity's contradiction degree may be lowered if next year's budget priority changes; in that case, whether the vehicle strong on capacity but middling on cost moves ahead must be recomputed. The department should explain in the report the grounds on which these contradiction degrees were set.
In the report: "With the low contradiction degree and high weight given to operating cost, the cheapest vehicle comes out ahead; if the contradiction degrees are updated according to next year's budget priority, the ranking must be recomputed."
3. What Not to Do
The first error is treating all three criteria's contradiction degrees as zero in the illustrative example and running Neutrosophic ARAS instead; this erases the information that K2 and K3 overlap less with the decision problem, and although it may not visibly change A3's second place, it corrupts the reasoning behind it. The second error is applying the contradiction adjustment after normalisation; this makes the contradiction share enter the column total in the wrong proportion. The third error is reporting A2's utility degree of 0.968 as "a 97 per cent certainty of being the right choice"; K only shows these three candidates' proportional benefit relative to the optimal candidate, it is not a probability.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/p-aras
Smarandache, F. (2018). Plithogeny, Plithogenic Set, Logic, Probability, and Statistics. Pons Publishing, Brussels. (no DOI)
Smarandache, F. (1998). Neutrosophy: Neutrosophic Probability, Set, and Logic. American Research Press, Rehoboth. (no DOI)
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