Extension card · Neutrosophic

Bipolar neutrosophic TOPSIS — the 2018 revised-ratio variant

A BN-TOPSIS profile that ranks six-component bipolar neutrosophic cells with the source's revised closeness and inferior ratio.

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?

Supply every decision cell as (T+,I+,F+,T−,I−,F−). The first three components lie in [0,1], and the last three in [−1,0]. All six must be finite and supplied independently. No missing pole is generated, signs are not discarded, and the components are not reduced to one score before calculation.

Criterion weights in this source are real nonnegative coefficients summing to one. They are not native six-component judgments. Supplied weights are used explicitly. If weights are omitted, the implemented maximizing-deviation profile computes each criterion's total pairwise componentwise absolute difference and normalizes these totals. This L1 interpretation reproduces the source's four printed weights when rounded to four decimals; the original notation does not expand the norm, so that interpretive boundary remains explicit. Euclidean distances are used later for separation from the ideals, not substituted into this weight calculation.

The two weight modes must be selected deliberately. An equal-weight default must not silently replace the omitted-weight mode. If every criterion has zero spread, data-derived weights are undefined. Supplying weights does not guarantee that the later revised ratios will be defined.

Calculation and direction.

The weighting operation keeps six native components. For a benefit criterion, the positive ideal uses (max T+, min I+, min F+, min T−, max I−, max F−) from the weighted matrix. The negative ideal reverses these choices; cost criteria reverse the two ideals. The negative pole's direction is part of the published formula and must not be copied from the positive pole.

The normalized Euclidean separation sums all six squared component differences over all criteria and divides by 6n inside the square root. The revised closeness is rho = d_minus/max(d_minus) − d_plus/min(d_plus). The final inferior ratio is IR = rho/min(rho).

Defined IR values lie in [0,1], and lower is better. The best value need not be zero. Exact equal values share a rank. A zero denominator is undefined and must be rejected; the ratio does not remove the zero-denominator problem. Recorded judgments must not be artificially perturbed just to make the calculation run.

For a zero criterion weight the implementation uses the constant tuple (0,1,1,−1,−1,0), making that criterion contribute zero separation. This is an explicit endpoint convention; the article does not prescribe 0^0. Shared ranks for exact ties are likewise a reporting convention. BN-specific rank reversal has not been independently established in this review; the source's general TOPSIS discussion does not prove immunity or a specific BN theorem.

How to Read the Output

The source e-commerce example supplies four alternatives and four criteria in Table 1. Its printed Step 4 ideals imply that T2 is a cost criterion, while T1, T3 and T4 are benefit criteria. This is an inference from the complete extrema vector, not an explicit direction label in Table 1.

Applying the published equations to those native cells, the explicitly stated directions, and the supplied rounded weights (0.2567,0.2776,0.2179,0.2478) gives the independent Decimal reconstruction below:

Alternative	Recomputed IR	Rank
S3	0.179200935589	1
S4	0.221290123137	2
S2	0.530184447589	3
S1	1.000000000000	4

These are equation results, not a claim of exact reproduction of the printed final values. For example, the source's T4 positive ideal prints F−=−0.062 although the applicable Table 2 extreme is −0.026. The printed distance/closeness/IR values also require separate consistency checks. No equation is changed to force those numbers.

The former implementation reversed the negative-pole extrema and treated all four example criteria as benefit. Its S4-first result and kernel-derived golden were therefore unsuitable evidence of source correctness. The original fixture remains archived; the current fixture is independently calculated, with unchanged absolute tolerance 1e−9. The automatic-weight profile is tested separately from this supplied rounded-weight example.

When to Prefer This over the Base Method

The kernel takes one native decision matrix. Expert aggregation is not part of this implemented profile; a tensor must not be silently reduced to the first expert. The actual weight mode, coefficients, directions and native components must survive the input, execution record and report. Group and post-hoc user flows are separately checked; the existence of an engine-level algebra fallback does not establish a native BN group operator.

Native component averages are not valid decision heatmaps. Original components remain available in tables; validated scalar IR and real criterion coefficients can be plotted without changing their meaning.

Mistakes Specific to This Extension

Applying the same extrema rule to both poles, treating a cost criterion as a benefit, or reducing the six components to one score before calculation produces a different method. Quietly substituting equal weights when weights are absent and hiding a zero denominator with epsilon also alter the published profile. The report must say that lower IR is better and identify the selected weight mode.

Cases

## 1. Reconstructing the e-commerce example from the source equations The four-alternative example above is recomputed from the published equations without imitating inconsistent printed intermediate values. It yields S3, S4, S2, S1; typographical and arithmetic discrepancies in the article remain a separate evidence note. The case demonstrates that source fidelity means applying the data and equations consistently, not forcing every printed number.

1. Reconstructing the e-commerce example from the source equations

The four-alternative example above is recomputed from the published equations without imitating inconsistent printed intermediate values. It yields S3, S4, S2, S1; typographical and arithmetic discrepancies in the article remain a separate evidence note. The case demonstrates that source fidelity means applying the data and equations consistently, not forcing every printed number.

2. A single-expert native assessment

The profile can be used directly when one expert supplies all six bipolar components and every criterion direction. Supplied weights are reported as supplied, while maximizing-deviation weights are reported as data-derived; the two routes are never presented as the same assumption.

3. What Not to Do

Do not erase negative signs, manufacture a missing pole from the positive pole, treat every criterion as a benefit, or make an undefined ratio run by adding epsilon. Reducing a multi-expert tensor to its first expert is not group decision-making and is rejected by this profile.

Formulae

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

1. Construct the native bipolar neutrosophic matrix
$K = [k_{ij}]_{m \times n}, k_{ij} = (T^{+}, I^{+}, F^{+}, T^{-}, I^{-}, F^{-}), (T^{+}, I^{+}, F^{+}) \in [0, 1]^{3}, (T^{-}, I^{-}, F^{-}) \in [- 1, 0]^{3}$
2. Use supplied weights or compute maximizing-deviation weights
$Δ_{j} = i = 1 \sum m l = 1 \sum m c = 1 \sum 6 ∣ k_{ij c} - k_{l j c} ∣, w_{j} = {w_{j}^{supplied}, Δ_{j} / \sum_{h = 1}^{n} Δ_{h}, if supplied, otherwise . w_{j} \geq 0, j \sum w_{j} = 1, j \sum Δ_{j} > 0 when derived .$
3. Apply the bipolar neutrosophic weight operation
$v_{ij} = (1 - (1 - T_{ij}^{+})^{w_{j}}, (I_{ij}^{+})^{w_{j}}, (F_{ij}^{+})^{w_{j}}, - (- T_{ij}^{-})^{w_{j}}, - (- I_{ij}^{-})^{w_{j}}, - [1 - (1 + F_{ij}^{-})^{w_{j}}]) (w_{j} > 0); v_{ij}_{w_{j} = 0} = (0, 1, 1, - 1, - 1, 0) (zero-weight convention) .$
4. Select direction-dependent native ideals
$A_{j}^{+} A_{j}^{-} = (i max v_{ij 1}, i min v_{ij 2}, i min v_{ij 3}, i min v_{ij 4}, i max v_{ij 5}, i max v_{ij 6}) = (i min v_{ij 1}, i max v_{ij 2}, i max v_{ij 3}, i max v_{ij 4}, i min v_{ij 5}, i min v_{ij 6}) A_{j}^{+} \leftrightarrow A_{j}^{-} (benefit), (benefit) . for cost criteria .$
5. Compute normalized six-component Euclidean distances
$d_{i}^{\pm} = \frac{1}{6 n} j = 1 \sum n c = 1 \sum 6 (v_{ij c} - A_{j c}^{\pm})^{2}$
6. Compute revised closeness
$ρ_{i} = \frac{d _{i}^{-}}{max _{h} d _{h}^{-}} - \frac{d _{i}^{+}}{min _{h} d _{h}^{+}}, h max d_{h}^{-} > 0, h min d_{h}^{+} > 0$
7. Compute inferior ratios and rank in ascending order
$I R_{i} = \frac{ρ _{i}}{min _{h} ρ _{h}}, h min ρ_{h} < 0, 0 \leq I R_{i} \leq 1, r_{i} = 1 + # {h : I R_{h} < I R_{i}}$

Sources

Akram, M., Shumaiza, and Smarandache, F. (2018). Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I. Axioms 7(2), 33. DOI: 10.3390/axioms7020033. Formula and example references use PDF pages 2–7 of the 17-page article.

What Changes from the Base Method?

Calculation and direction.

How to Read the Output

Alternative	Recomputed IR	Rank
S3	0.179200935589	1
S4	0.221290123137	2
S2	0.530184447589	3
S1	1.000000000000	4

When to Prefer This over the Base Method

Mistakes Specific to This Extension

Cases

1. Reconstructing the e-commerce example from the source equations

2. A single-expert native assessment

3. What Not to Do

Formulae

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

1. Construct the native bipolar neutrosophic matrix

K = [k_{ij}]_{m \times n}, k_{ij} = (T^{+}, I^{+}, F^{+}, T^{-}, I^{-}, F^{-}), (T^{+}, I^{+}, F^{+}) \in [0, 1]^{3}, (T^{-}, I^{-}, F^{-}) \in [- 1, 0]^{3}

2. Use supplied weights or compute maximizing-deviation weights

Δ_{j} = i = 1 \sum m l = 1 \sum m c = 1 \sum 6 ∣ k_{ij c} - k_{l j c} ∣, w_{j} = {w_{j}^{supplied}, Δ_{j} / \sum_{h = 1}^{n} Δ_{h}, if supplied, otherwise . w_{j} \geq 0, j \sum w_{j} = 1, j \sum Δ_{j} > 0 when derived .

3. Apply the bipolar neutrosophic weight operation

v_{ij} = (1 - (1 - T_{ij}^{+})^{w_{j}}, (I_{ij}^{+})^{w_{j}}, (F_{ij}^{+})^{w_{j}}, - (- T_{ij}^{-})^{w_{j}}, - (- I_{ij}^{-})^{w_{j}}, - [1 - (1 + F_{ij}^{-})^{w_{j}}]) (w_{j} > 0); v_{ij}_{w_{j} = 0} = (0, 1, 1, - 1, - 1, 0) (zero-weight convention) .

4. Select direction-dependent native ideals

A_{j}^{+} A_{j}^{-} = (i max v_{ij 1}, i min v_{ij 2}, i min v_{ij 3}, i min v_{ij 4}, i max v_{ij 5}, i max v_{ij 6}) = (i min v_{ij 1}, i max v_{ij 2}, i max v_{ij 3}, i max v_{ij 4}, i min v_{ij 5}, i min v_{ij 6}) A_{j}^{+} \leftrightarrow A_{j}^{-} (benefit), (benefit) . for cost criteria .

5. Compute normalized six-component Euclidean distances

d_{i}^{\pm} = \frac{1}{6 n} j = 1 \sum n c = 1 \sum 6 (v_{ij c} - A_{j c}^{\pm})^{2}

6. Compute revised closeness

ρ_{i} = \frac{d _{i}^{-}}{max _{h} d _{h}^{-}} - \frac{d _{i}^{+}}{min _{h} d _{h}^{+}}, h max d_{h}^{-} > 0, h min d_{h}^{+} > 0

7. Compute inferior ratios and rank in ascending order

I R_{i} = \frac{ρ _{i}}{min _{h} ρ _{h}}, h min ρ_{h} < 0, 0 \leq I R_{i} \leq 1, r_{i} = 1 + # {h : I R_{h} < I R_{i}}