N-GRA - Neutrosophic extension of GRA

Overview

n-gra extends GRA 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

   SVN matrix; cost complement for cost criteria.

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

   Biswas, Pramanik & Giri 2014, Sec.3 Step1

2. 2

   Neutrosophic ideal reference sequence.

   $${\alpha }_j^0=⟨\underset{i}{\max }{T}_{ij},\underset{i}{\min }{I}_{ij},\underset{i}{\min }{F}_{ij}⟩,j=1,\dots ,n$$

   Biswas, Pramanik & Giri 2014, Sec.3 Step2

3. 3

   Neutrosophic Euclidean distances to ideal; global extremes.

   $${\Delta }_{ij}=\sqrt{\frac{({\max }_k{T}_{kj}-T_{ij}{)}^2+({\min }_k{I}_{kj}-I_{ij}{)}^2+({\min }_k{F}_{kj}-F_{ij}{)}^2}{3}};{\Delta }_{\min }=\underset{i}{\min }\underset{j}{\min }{\Delta }_{ij};{\Delta }_{\max }=\underset{i}{\max }\underset{j}{\max }{\Delta }_{ij}$$

   Biswas, Pramanik & Giri 2014, Sec.3 Step3-4

4. 4

   Grey Relational Coefficient with ρ=0.5.

   $${\xi }_{ij}=\frac{{\Delta }_{\min }+\rho {\Delta }_{\max }}{{\Delta }_{ij}+\rho {\Delta }_{\max }},\rho =0.5$$

   Deng 1989, Eq.(3); Biswas, Pramanik & Giri 2014, Sec.3 Step5

5. 5

   Grey Relational Grade; rank descending.

   $${\Gamma }_i=\sum _{j=1}^n{w}_j{\xi }_{ij};\text{rank descending}$$

   Biswas, Pramanik & Giri 2014, Sec.3 Step6-7

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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