Extension card · Neutrosophic

Neutrosophic TOPSIS — native criterion weights

A neutrosophic TOPSIS profile that processes native `(T,I,F)` decision cells and criterion weights with the source equations.

Base method

TOPSIS →

Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.

Data type (family)

Neutrosophic →

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?

Each decision cell and each criterion weight is a native(T,I,F) triple: truth, indeterminacy and falsity. Every component must be finite and lie in[0,1]. Their sum need not be1; no component is inferred from another. Criterion weights are also triples and do not form a scalar unit-sum vector.

Supply benefit/cost directions explicitly. Real measurements or scalar weights must not be expanded into synthetic triples. Scalar expert-importance coefficients used in group aggregation are a different input from the native criterion weights.

Native calculation.

The native weighted cell is (T*Tw, I+Iw-I*Iw, F+Fw-F*Fw). The kernel does not apply a pre-weight cost complement. On the weighted matrix, benefit ideals use maximum truth and minimum indeterminacy/falsity; cost ideals reverse all three choices. The negative ideal reverses the positive ideal.

Normalized Euclidean distance uses all three components and the factor 1/(3*n) inside the square root. The final coefficient is CC=d_minus/(d_plus+d_minus). Higher CC is better. Equal coefficients receive equal ranks; a zero total separation is undefined and is rejected. The method does not use (T-F+1)/2 to replace the input cells or the final coefficient.

How to Read the Output

The original Tables1–4 provide four experts' judgments for four alternatives and six criteria. Independently aggregating those inputs using equations16,17,20 and evaluating the native TOPSIS equations gives:

Alternative	Recomputed CC	Rank
A3	0.9436950843	1
A1	0.8849659206	2
A4	0.4436310323	3
A2	0.0418297121	4

These are independently reconstructed equation results, not the article's printed Table7. The published example contains discrepancies: all four A1/C4 judgments are G but the printed aggregate differs from G; a native product, one ideal component and one coefficient also disagree with the article's own inputs/equations. The implementation is tested against70-digit independent calculations without changing equations to fit those printed values.

The previous DecisionMind scalar-weight fixture described a different computation. It is preserved as historical evidence and no longer defines the canonical native method.

When to Prefer This over the Base Method

One complete native criterion weight is required for every criterion. The applied triples must remain visible in the execution record and report; a scalar default must not be substituted. The native group API can aggregate decision matrices with explicitly supplied expert coefficients and an already aggregated/common native criterion-weight vector. Per-expert criterion weights and file-based (dataset) group integration require their own supported input path; unsupported requests must fail explicitly.

Changing a missing third component into another fuzzy family, forcing T+I+F=1, or computing a crisp matrix from a score before running TOPSIS changes the problem and is not this implementation.

Mistakes Specific to This Extension

Replacing a native criterion weight with a scalar, deriving a missing component, collapsing each triple to one score, or applying an arbitrary cost complement before weighting breaks the source method. Adding epsilon to a zero total separation turns an undefined case into a seemingly valid result. Reports must preserve decision triples, weight triples and criterion directions as distinct inputs.

Cases

## 1. Recomputing the published four-alternative example The source example above aggregates the four experts through the stated equations and gives A3, A1, A4, A2. Inconsistent printed intermediate values are not fitted by changing the implementation; independently recomputed coefficients and the article's printed coefficients are never reported as though they were identical.

1. Recomputing the published four-alternative example

The source example above aggregates the four experts through the stated equations and gives A3, A1, A4, A2. Inconsistent printed intermediate values are not fitted by changing the implementation; independently recomputed coefficients and the article's printed coefficients are never reported as though they were identical.

2. A preaggregated native decision matrix

The kernel can evaluate a single (T,I,F) matrix with common native criterion weights when a committee has already combined its judgments through a documented external process. The report states that the matrix was preaggregated and that the kernel did not infer expert importance.

3. What Not to Do

Do not expand scalar measurements into synthetic triples, force T+I+F=1, pass expert-importance coefficients as criterion weights, or feed only the truth component into crisp TOPSIS. Missing native inputs must be rejected with an explanatory error rather than guessed.

Formulae

The computation step by step, with the formula behind each step.

1. Read the single-expert or preaggregated native matrix and native criterion weights.
$R = (r_{ij})_{m \times n}, r_{ij} = (T_{ij}, I_{ij}, F_{ij}) \in [0, 1]^{3}, W = (w_{j})_{j = 1}^{n}, w_{j} = (T_{j}^{w}, I_{j}^{w}, F_{j}^{w}) \in [0, 1]^{3} .$
2. Multiply each native rating by its native criterion weight.
$v_{ij} = r_{ij} \otimes w_{j} = (T_{ij} T_{j}^{w}, I_{ij} + I_{j}^{w} - I_{ij} I_{j}^{w}, F_{ij} + F_{j}^{w} - F_{ij} F_{j}^{w}) .$
3. Construct native positive and negative ideals according to criterion direction.
$v_{j}^{+} = {(max_{i} T_{ij}^{v}, min_{i} I_{ij}^{v}, min_{i} F_{ij}^{v}), (min_{i} T_{ij}^{v}, max_{i} I_{ij}^{v}, max_{i} F_{ij}^{v}), j \in J_{b} j \in J_{c} v_{j}^{-} = {(min_{i} T_{ij}^{v}, max_{i} I_{ij}^{v}, max_{i} F_{ij}^{v}), (max_{i} T_{ij}^{v}, min_{i} I_{ij}^{v}, min_{i} F_{ij}^{v}), j \in J_{b} j \in J_{c} J_{b} = {j : benefit}, J_{c} = {j : cost} .$
4. Compute normalized neutrosophic Euclidean separations.
$d_{i}^{\pm} = \frac{1}{3 n} j = 1 \sum n [(T_{ij}^{v} - T_{j}^{\pm})^{2} + (I_{ij}^{v} - I_{j}^{\pm})^{2} + (F_{ij}^{v} - F_{j}^{\pm})^{2}] .$
5. Calculate relative closeness and rank descending; equal coefficients receive equal ranks.
$C C_{i} = \frac{d _{i}^{-}}{d _{i}^{+} + d _{i}^{-}}, d_{i}^{+} + d_{i}^{-} > 0, 0 \leq C C_{i} \leq 1, r_{i} = 1 + k = 1 \sum m 1 [C C_{k} > C C_{i}] .$

Sources

Biswas, P., Pramanik, S., and Giri, B.C. (2016). TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Computing and Applications 27:727–737. DOI: 10.1007/s00521-015-1891-2.