Plithogenic VIKOR

Four things change; the decision logic does not.

Cells. In crisp VIKOR every cell is a single number. Here every cell is three numbers: truth (T), indeterminacy (I) and falsity (F). The sub-alternative idea from the plithogenic data-type card is carried, in this family, to the criterion level: one criterion is taken as dominant, with a contradiction degree of zero. How far the other criteria oppose this dominant one is given as a contradiction degree between 0 and 1. This degree belongs to the criterion and does not vary from one alternative to another. Criterion weights and the compromise coefficient v are separate inputs and, as in crisp VIKOR, are supplied from outside.

Scale equalisation. Crisp VIKOR performs a linear normalisation against each criterion's best and worst value. In this family, for a cost criterion the triple is reversed first. Each cell is then adjusted by its own criterion's contradiction degree: truth is enlarged towards the dominant criterion, while indeterminacy and falsity shrink by the same proportion. This contradiction adjustment replaces crisp VIKOR's normalisation, which divides by the best/worst value.

Distance. For every criterion, the best (f*) and worst (f⁻) triples are built from the component-wise best and worst of the adjusted cells. Every alternative's ratio to these two points is measured by the vertex distance used for triangular numbers; this distance is computed as the square root of a third of the sum of the squared differences of the three components. The result is then divided by the distance between f* and f⁻ to bring it onto a 0-to-1 scale.

Group utility, individual regret and the compromise index. These three values (S, R, Q) are computed as in crisp VIKOR. S is the sum of the weighted distances, R is the largest weighted distance, Q combines the two with the coefficient v. Indeterminacy (I) and falsity (F) are not defuzzified in a separate step; they dissolve inside the f_ij ratio.

DecisionMind fixes, in this family, the cost complement, the contradiction adjustment, and the component-based construction of f* and f⁻. It should also be noted that the contradiction degree is applied identically to every alternative under a given criterion. Because of this, f* and f⁻ themselves are scaled by the same proportion, and this scale cancels out inside the f_ij ratio. Independent recomputation (tried with three different sets of contradiction degrees) has shown that the contradiction degree never changes the S, R or Q results in this family. This is a consequence of VIKOR's ratio-based structure and does not hold for the other P-family members (P-TOPSIS, P-EDAS, P-COPRAS, P-MARCOS).

The output is the same three columns as in crisp VIKOR, and the compromise decision drawn from them; it is read the same way: a small value is good, the Q ranking alone is not enough, and the two conditions, acceptable advantage and acceptable stability, are assessed together.

In the illustrative example below, A2 is first on both S and R, and the Q gap is above the threshold; the sole compromise solution is A2. When the weights are changed, A3 comes first on Q, but the gap now falls below the threshold, and the compromise becomes a set rather than a single alternative. The difference is here: in crisp VIKOR this sensitivity comes only from the weights; here there is also an assumption about the contradiction between criteria, but this assumption has no effect at all on S, R or Q; its effect concerns only how "fairly" the cells are represented in T-I-F form.

Thus instead of writing:

"According to Plithogenic VIKOR, the best alternative is A2"

the report should read:

"With these weights and v = 0.5, A2 is the sole compromise solution; it is first on both group utility and individual regret, and its gap with the second alternative is above the acceptance threshold"

If one of the conditions is not met, the compromise set is written out as it stands.

Use this extension when your criterion scores are given as a truth-indeterminacy-falsity triple and some criteria are a more independent or more contradictory source of information than others. If the T-I-F triple alone is sufficient and no such dominance-contradiction relationship can be established between the criteria, neutrosophic VIKOR (n-vikor) is already enough; the plithogenic form carries an extra assumption, but since this assumption has no effect on S, R or Q in this family, choosing the plithogenic form purely "to account for contradiction too" brings no gain in this method.

Turning a measured criterion into a T-I-F triple produces uncertainty rather than modelling it. If no compromise at all is acceptable on one criterion, this extension limits regret but does not eliminate it.

Assuming the contradiction degree changes the ranking. In P-TOPSIS, P-EDAS, P-COPRAS and P-MARCOS the contradiction degree does affect the ranking; in P-VIKOR it does not, because an adjustment applied in the same proportion to every alternative under a criterion cancels out inside the best-worst ratio on which S, R and Q rest. Not knowing this difference and thinking "I raised the contradiction degree and the result did not change, something must be wrong" is a mistake; it is the method's natural consequence.

Violating the value space. Every cell must be a valid T, I, F triple, and a contradiction degree must be declared for every criterion.

Skipping the two conditions and declaring the alternative with the smallest Q the winner. This is the same, most common mistake as in crisp VIKOR; it applies equally in the plithogenic form.

Running a cost criterion without taking its complement. Processing a cost criterion without taking the complement (F, I, T) of (T, I, F) is wrong. It pulls the most expensive or slowest alternative towards the ideal.

The governing principle is this:

In Plithogenic VIKOR the contradiction degree serves the fair representation of cells; it is not a dial that directly changes the result. What changes the result remains the weights, the coefficient v, and the two conditions.