N-MULTIMOORA - Neutrosophic extension of MULTIMOORA

Overview

n-multimoora extends MULTIMOORA 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; score + vector normalisation.

   $${\hat{a}}_{ij}=⟨F_{ij},1-I_{ij},T_{ij}⟩(j\in {\Omega }_c);s_{ij}=\frac{1+{\hat{T}}_{ij}-2{\hat{I}}_{ij}-{\hat{F}}_{ij}}{2};{\bar{s}}_{ij}=s_{ij}/\sqrt{\sum _i{s}_{ij}^2}$$

   Stanujkic et al. 2017, Sec.3 Step1

2. 2

   Sub-method 1: Ratio System score.

   $$y_i^{\mathrm{RS}}=\sum _{j=1}^n{w}_j{\bar{s}}_{ij};\text{rank descending}$$

   Stanujkic et al. 2017, Sec.3 RS

3. 3

   Sub-method 2: Reference Point Chebyshev score.

   $$r_j^{\ast }=\underset{i}{\max }{\bar{s}}_{ij};y_i^{\mathrm{RP}}=\underset{j}{\max }{w}_j|r_j^{\ast }-{\bar{s}}_{ij}|;\text{rank ascending}$$

   Stanujkic et al. 2017, Sec.3 RP

4. 4

   Sub-method 3: Full Multiplicative Form via SVNWG operator.

   $${\tilde{U}}_i={\mathrm{SVNWG}}_w({\hat{a}}_{i1},\dots ,{\hat{a}}_{in})=⟨\prod _j{\hat{T}}_{ij}^{w_j},1-\prod _j(1-{\hat{I}}_{ij}{)}^{w_j},1-\prod _j(1-{\hat{F}}_{ij}{)}^{w_j}⟩;y_i^{\mathrm{FMF}}=s({\tilde{U}}_i);\text{rank descending}$$

   Stanujkic et al. 2017, Sec.3 FMF; Ye 2014 SVNWG

5. 5

   Theory of Dominance: final rank from pairwise dominance across RS, RP, FMF.

   $$A_i\succ A_k⟺A_i\text{ ranks better in }\ge 2\text{ of }\{RS,RP,FMF\};\text{count dominances for final order}$$

   Brauers & Zavadskas 2010; Stanujkic et al. 2017, Sec.3 Dominance

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

How to cite