N-TOPSIS - Neutrosophic extension of TOPSIS

Overview

n-topsis extends TOPSIS to handle Neutrosophic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3) algebra. The final scores are defuzzified via score function S = (T − F + 1)/2 before ranking.

Output

utility, higher is better

Data

Single-Valued Neutrosophic, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Single-Valued Neutrosophic MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Construct SVN decision matrix; apply cost complement.

   $$\mathbf{D}=(⟨T_{ij},I_{ij},F_{ij}⟩{)}_{m\times n};\text{cost: }{\tilde{a}}_{ij}^c=⟨F_{ij},1-I_{ij},T_{ij}⟩$$

   Biswas, Pramanik & Giri 2016, Sec.3 Step1

2. 2

   Weighted SVNN via scalar multiplication operator.

   $${\tilde{v}}_{ij}=w_j\cdot {\tilde{a}}_{ij}=⟨1-(1-T_{ij}{)}^{w_j},I_{ij}^{w_j},F_{ij}^{w_j}⟩$$

   Biswas, Pramanik & Giri 2016, Sec.3 Step2

3. 3

   Neutrosophic PIS A⁺ and NIS A⁻ (component-wise extrema of weighted SVNNs).

   $${\tilde{v}}_j^{+}=⟨\underset{i}{\max }{T}_{ij}^v,\underset{i}{\min }{I}_{ij}^v,\underset{i}{\min }{F}_{ij}^v⟩;{\tilde{v}}_j^{-}=⟨\underset{i}{\min }{T}_{ij}^v,\underset{i}{\max }{I}_{ij}^v,\underset{i}{\max }{F}_{ij}^v⟩$$

   Biswas, Pramanik & Giri 2016, Sec.3 Step3

4. 4

   Euclidean separation from PIS/NIS using normalised neutrosophic distance.

   $$d({\alpha }_1,{\alpha }_2)=\sqrt{\frac{(T_1-T_2{)}^2+(I_1-I_2{)}^2+(F_1-F_2{)}^2}{3}};d_i^{\pm }=\sqrt{\sum _{j=1}^n[d({\tilde{v}}_{ij},{\tilde{v}}_j^{\pm }){]}^2}$$

   Biswas, Pramanik & Giri 2016, Sec.3 Step4

5. 5

   Closeness coefficient CC_i; rank descending.

   $$C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},C{C}_i\in [0,1];\text{rank descending}$$

   Biswas, Pramanik & Giri 2016, Sec.3 Step5

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Single-Valued Neutrosophic numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy SVNS: T,I,F ∈ [0,1]; 0 ≤ T+I+F ≤ 3 before computation.

• Defuzzification method affects ranking: score function S = (T − F + 1)/2 is the canonical choice but alternatives exist.

Works with

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