NSS-CODAS - Neutrosophic Spherical extension of CODAS

Overview

NSS-CODAS fuses neutrosophic logic (T, I, F triples) with spherical-style squared-norm membership (0 ≤ T²+I²+F² ≤ 3) and applies CODAS-style closeness ratio ranking. Group judgements are pre-aggregated via SWAM or SWGM (Eqs.13-14). Criteria weights are NSS triples aggregated the same way. The revised closeness ratio ξ_i (Eq.28) is interpreted with lower-is-better - opposite to crisp CODAS - because the modification subtracts the NIS-normalised distance from the PIS-normalised distance.

Output

utility, lower is better

Data

Neutrosophic Spherical, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Neutrosophic Soft Set MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Decision-makers express judgements using NSS linguistic terms (Table 1 of seminal: PMS, ES, HP, RGS, EI, VMS, LP, ELS, DNI), each mapped to a (T, I, F) triple.

   $$K_j(S_i)=⟨T_{ij},I_{ij},F_{ij}⟩\in \text{NSS},T,I,F\in [0,1],0\le T^2+I^2+F^2\le \sqrt{3}$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 §4 Step 1; Table 1

2. 2

   Aggregate decision-maker assessments via SWAM (Eq.13) or SWGM (Eq.14) using DM weights z to produce a single NS decision matrix D = [K_j(S_i)]_{m×n}.

   $${\text{SWAM}}_z(A_1,\dots ,A_n)=⟨\sqrt{1-\prod _{j=1}^n(1-T_{A_j}^2{)}^{z_j}},\sqrt{1-\prod _{j=1}^n(1-I_{A_j}^2{)}^{z_j}},\sqrt{1-\prod _{j=1}^n(1-F_{A_j}^2{)}^{z_j}}⟩;{\text{SWGM}}_z=⟨\prod _j{T}_{A_j}^{z_j},\prod _j{I}_{A_j}^{z_j},\prod _j{F}_{A_j}^{z_j}⟩$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(13)-(14); §4 Step 2

3. 3

   Build the weighted aggregated NS decision matrix by NSS multiplication (Eq.4): K_j(S_iz) = ⟨T_ij·T_zj, I_ij·I_zj, F_ij·F_zj⟩ where z = (T_zj, I_zj, F_zj) is the aggregated criterion-weight triple.

   $$K_j(S_{iz})=⟨T_{ij}\cdot T_{zj},I_{ij}\cdot I_{zj},F_{ij}\cdot F_{zj}⟩$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 Eq.(4); §4 Step 3 Eq.(19)

4. 4

   De-neutrosophise the weighted NS matrix via the score function S(K_j(S_iz)) = (T_ijz − F_ijz)² − (I_ijz − F_ijz)² (Eq.15 / Eq.20).

   $$\text{Score}(K_j(S_{iz}))=(T_{ijz}-F_{ijz}{)}^2-(I_{ijz}-F_{ijz}{)}^2$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 Eq.(15); §4 Step 4 Eq.(20)

5. 5

   Identify NS-PIS S* (per-criterion arg-max of score) and NS-NIS S^- (per-criterion arg-min of score).

   $$S^{\ast }=\{K_j,\arg \underset{i}{\max }\text{Score}(K_j(S_{iz})){\}}_{j=1}^n;S^{-}=\{K_j,\arg \underset{i}{\min }\text{Score}(K_j(S_{iz})){\}}_{j=1}^n$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(21)-(22); §4 Step 5

6. 6

   Compute Euclidean-type distances from each alternative to NS-PIS and NS-NIS using the NSS spherical distance metric.

   $$D(S_i,S^{\ast })=\sqrt{\frac{1}{2}\sum _{j=1}^n[(T_{S_i}-T_{S^{\ast }}{)}^2+(I_{S_i}-I_{S^{\ast }}{)}^2+(F_{S_i}-F_{S^{\ast }}{)}^2]};D(S_i,S^{-})\text{ analogous}$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(23)-(24); §4 Step 6

7. 7

   Compute the per-alternative max distance to NS-NIS (Dmax) and min distance to NS-PIS (Dmin).

   $$D_{\max }(S_i,S^{-})=\underset{i\le m}{\max }D(S_i,S^{-});D_{\min }(S_i,S^{\ast })=\underset{i\le m}{\min }D(S_i,S^{\ast })$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(25)-(26); §4 Step 7

8. 8

   Compute the revised closeness ratio ξ_i (Eq.28). Lower ξ_i = better alternative.

   $$\xi (S_i)=\frac{D(S_i,S^{\ast })}{D_{\min }(S_i,S^{\ast })}-\frac{D(S_i,S^{-})}{D_{\max }(S_i,S^{-})}$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 Eq.(28); §4 Step 8

9. 9

   Rank alternatives in ascending order of ξ_i (lower closeness ratio = better alternative).

   $$\text{rank}={\text{argsort}}_{\uparrow }\xi (S_i)$$

   Bhuvaneshwari & Antony Crispin Sweety 2024 §4 Step 9

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Neutrosophic Soft Set 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

• NSS norm constraint is T²+I²+F² ≤ √3 (NOT ≤ 1 as in spherical fuzzy and NOT linear T+I+F ≤ 3 as in classical neutrosophic). Validate at input.

• Ranking direction is inverted vs. crisp CODAS: lower ξ_i = better. Failing to invert misranks all alternatives.

• Criterion weights here are NSS triples (T_z, I_z, F_z), not scalars; multiply via NSS multiplication Eq.(4), not scalar multiplication.

• Seminal Tables 17 (SWAM closeness values) and the in-text comparison chart show divergent magnitudes (12438 vs 4.146 for S2). The chart values match the SWGM table format; we adopt the chart values as the canonical numeric reference.

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