Extension card · Plithogenic
Plithogenic MAUT
This is the form of MAUT for situations where criterion values are given as a truth-indeterminacy-falsity triple and criteria carry a degree of contradiction relative to one another. Each criterion is converted to its own utility scale and summed with weights; the output remains a single aggregate utility score.
Base method
MAUT →
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?
Three things change; the logic of single-attribute utility followed by weighted summation does not.
Cells. In crisp MAUT every cell is a single number. Here every cell is three numbers: truth (T), indeterminacy (I), falsity (F). The sub-option idea from the plithogenic data-type card is carried, in this family, onto the criterion itself. One criterion is taken as dominant, with a contradiction degree of zero. How strongly the other criteria oppose this dominant criterion is given by a contradiction degree between 0 and 1. This degree belongs to the criterion, not the alternative: the same figure is used for every alternative under that criterion. Criterion weights are a separate input and come from outside.
Contradiction adjustment and score. Each cell is first adjusted by its own criterion's contradiction degree: truth is pulled upward towards the dominant criterion, while indeterminacy and falsity are shrunk by the same proportion. This adjusted triple is then reduced to a single score. One point deserves attention here: unlike P-VIKOR and P-TOPSIS, the cost criterion's triple is not complemented (F, I, T) at this step. Direction is not used at all here; it is left for the following step.
Single-attribute utility normalisation. In crisp MAUT, every cell is linearly normalised so that the criterion's worst value earns a utility of 0 and its best value earns 1. Here the same operation is carried out on the scores, but direction enters at this point. For a benefit criterion the score is scaled up relative to the column's smallest value; for a cost criterion the score is subtracted from the column's largest value and divided by the same column width. Direction is applied not by reversing the triple, but by changing this normalisation formula.
Summation. As in crisp MAUT, each criterion's utility norm is multiplied by its own weight and summed; the result is again a single aggregate utility score.
DecisionMind fixes, for this classical form, the contradiction adjustment, the application of direction through the normalisation formula rather than through a complement on the cost criterion, and the linear min-max utility norm. When independently recomputed in Python with three separate sets of contradiction degrees (zero, as given, and extreme), the contradiction degree was found never to change the aggregate utility score in this family. The reason is mathematical: because the same contradiction degree is applied to every cell in a given column, the adjustment shrinks the score by a constant proportion and adds a constant amount, and this cancels out completely under within-column min-max normalisation. This matches the finding already observed in P-VIKOR and P-COCOSO.
How to Read the Output
The aggregate utility score is read as in crisp MAUT: a measure of position between 0 (worst on every criterion) and 1 (best on every criterion), neither a percentage nor a probability. The difference is here: this utility scale is always built relative to the current set of alternatives, because min-max normalisation uses the smallest and largest score observed in the column. Where crisp MAUT can build a fixed utility curve independent of the decision-maker, no such fixed curve exists in this family; scores are recalculated whenever the alternative set changes and cannot be compared with an earlier analysis.
Thus instead of writing:
"Plithogenic MAUT showed that this alternative's true utility is 0.86"
the report should read:
"With these weights and these three alternatives, the highest aggregate utility belongs to A2 (0.86); the utility scale is built relative to these three alternatives and will be recalculated if a new alternative is added to the set"
When to Prefer This over the Base Method
Use this extension when your criterion scores are given as a truth-indeterminacy-falsity triple, and some criteria carry information that is more independent, or more in conflict, than others. If the T-I-F triple alone is sufficient and no such dominance-contradiction relationship can be established among the criteria, neutrosophic MAUT (n-maut, where available) is already sufficient. It has been shown above that the contradiction degree never changes the aggregate utility in this family, so choosing the plithogenic form merely "to take contradiction into account" produces no gain here.
Converting a measured criterion into a T-I-F triple manufactures indeterminacy rather than modelling it. The exit condition for crisp MAUT applies equally here: if the criteria are interdependent in the decision-maker's preference, additive summation is not valid.
Mistakes Specific to This Extension
Trying to take a complement on the cost criterion as in P-VIKOR. In this family, direction is applied through the formula of the utility normalisation, not through a complement on the triple. Taking the complement and also reversing the normalisation formula applies direction twice, which is as wrong as not applying it at all.
Raising the contradiction degree and assuming the result becomes more reliable. As shown above, the contradiction degree never changes the aggregate utility score in this family; raising or lowering it confers no additional reliability.
Comparing the utility score against a different set of alternatives. The utility norm is built relative to this set's own smallest and largest score; it cannot be placed side by side with a score from a different analysis.
Violating the value space. Every cell must be a valid T, I, F triple, and a contradiction degree must be declared for every criterion.
The governing principle is this:
In plithogenic MAUT the contradiction degree serves the fair representation of cells; it is not a dial that changes the aggregate utility score. What changes the score is still the construction of the utility function through min-max normalisation, together with the weights.
Cases
The first case is DecisionMind's validation example. Smarandache's founding 2018 source defines the plithogenic operations but includes no example decision table; DecisionMind has therefore built a small, hand-traceable table using the same formulas. The second case is an illustrative fiction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
Three alternatives are assessed on three criteria. The first two criteria are "more is better", the third is "less is better". C1 is taken as the dominant criterion, with a contradiction degree of zero; the contradiction degrees of C2 and C3 relative to C1 are 0.33 and 0.67 respectively.
| Alternative | C1 (more is better) | C2 (more is better) | C3 (less is better) |
|---|---|---|---|
| A1 | (0.70; 0.20; 0.10) | (0.50; 0.30; 0.20) | (0.60; 0.30; 0.20) |
| A2 | (0.80; 0.10; 0.10) | (0.60; 0.20; 0.20) | (0.40; 0.20; 0.30) |
| A3 | (0.60; 0.20; 0.20) | (0.70; 0.20; 0.10) | (0.50; 0.30; 0.20) |
| Contradiction degree | 0.00 | 0.33 | 0.67 |
| Weight | 0.40 | 0.35 | 0.25 |
The method adjusts every cell by its own criterion's contradiction degree, reduces it to a score, normalises within the column according to direction with min-max, then multiplies by the weights and sums.
| Alternative | Aggregate utility | Rank |
|---|---|---|
| A2 | 0.86 | 1 |
| A3 | 0.60 | 2 |
| A1 | 0.16 | 3 |
The result reads as follows. A2 holds the highest score on C1, the most heavily weighted criterion, and the lowest, that is best, score on the cost criterion C3; these two advantages more than compensate for its relative weakness on C2. A1 sits in exactly the opposite position and finishes last.
The board's hesitation concerns the contradiction degrees. Even if the contradiction degrees are pulled down to zero or raised to 0.9, the aggregate utility scores of A1, A2 and A3 come out exactly the same (0.16, 0.86, 0.60); the contradiction degree never changes the result in this family. The real hesitation concerns the weights: if C2's weight is raised from 0.35 to 0.61 while C1 and C3 are reduced in the same proportion (C1 = 0.24, C3 = 0.15), A3 moves ahead (0.760) and A2 drops to second (0.756). The difference is very small, and A2's first place is moderately robust to C2's weight.
In the report: "With the given weights, A2 has the highest aggregate utility (0.86); when C2's weight is raised to 0.61, A3 moves ahead (0.760 / 0.756). Changing the contradiction degrees does not affect the result."
Source: DecisionMind's P-MAUT manifest, validation example. The plithogenic operations rest on the formulas defined by Smarandache (2018); since the founding source gives no example decision table, the table has been constructed by DecisionMind faithfully to the formulas. The aggregate utility scores, contradiction-degree sensitivity and weight scenario were independently recomputed by this card's author using the same algorithm in Python.
2. Waste management: A municipality's choice of solid-waste treatment technology
A municipality is to choose among three technologies for solid-waste treatment: incineration, a recycling plant and landfill. Three criteria are used: an environmental-impact score, operating cost ("less is better") and processing capacity. The municipal council takes environmental impact as the dominant criterion; it judges cost to carry information that is somewhat independent of environmental impact, and capacity to be more independent still of both, and sets the contradiction degrees accordingly. Each technology is scored on each criterion with a T-I-F triple; the indeterminacy comes from field pilots that have not yet been completed.
The method computes the aggregate utility score of the three technologies. Suppose the technology with the best environmental-impact score is also the most expensive; it still comes first, because environmental impact carries a higher weight than cost.
The council's hesitation is this: even if one of the contradiction degrees has been entered incorrectly, the contradiction degree does not change the result in this family, as shown in the illustrative example above. What genuinely needs checking is the weights. If the weight on cost is raised substantially, whether the lowest-cost technology then moves ahead should be tested separately.
In the report: "With the high weight given to environmental impact, one technology reaches the highest aggregate utility; this result is sensitive to the weight of the cost criterion. The contradiction degrees do not change the result, because in this family the contradiction degree serves only the representation of cells."
3. What Not to Do
Had C3's direction been marked "more is better" in the illustrative example, the highest-cost alternative would have received the best utility norm and the ranking would have become meaningless. The second error is taking a complement (F, I, T) on the cost criterion as in P-VIKOR and then also reversing the normalisation formula; direction is then applied twice and the result is corrupted. The third error is raising the contradiction degrees and reporting that "the result is now more reliable"; the contradiction degree never changes the aggregate utility score in this family, so no such gain in reliability exists. The fourth error is comparing A2's score of 0.86 with a score computed from a different set of alternatives; the utility norm is built relative to that set's own alternatives every time.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/p-maut
Keeney, R. L., & Raiffa, H. (1976). Decisions with Multiple Objectives: Preferences and Value Trade-offs. Wiley. (no DOI)
Dyer, J. S. (2005). MAUT — Multiattribute Utility Theory. In J. Figueira, S. Greco, & M. Ehrgott (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys (pp. 265–292). Springer. DOI: 10.1007/0-387-23081-5_7
Smarandache, F. (2017). Plithogeny, Plithogenic Set, Logic, Probability, and Statistics. Pons Publishing House, Brussels. (no DOI)
Smarandache, F. (2018). Plithogenic Set, an Extension of Crisp, Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Sets – Revisited. Neutrosophic Sets and Systems, 21, 153–166. DOI: 10.5281/zenodo.1408740
Abdel-Basset, M., & Mohamed, R. (2020). A novel plithogenic TOPSIS-CRITIC model for sustainable supply chain risk management. Journal of Cleaner Production, 247, 119586. DOI: 10.1016/j.jclepro.2019.119586