NSS-CODAS: Neutrosophic Spherical extension of CODAS

Overview

Neutrosophic Spherical Set (NSS) outranking: T,I,F ∈ [0,1] independent with squared-norm bound 0 ≤ T²+I²+F² ≤ √3 (Bhuvaneshwari-Sweety 2024 Eq.(2)); combined with CODAS Euclidean-distance ranking. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Neutrosophic Spherical Set (NSS) outranking: T,I,F ∈ [0,1] independent with squared-norm bound 0 ≤ T²+I²+F² ≤ √3 (Bhuvaneshwari-Sweety 2024 Eq.(2)); combined with CODAS Euclidean-distance ranking
• •Preserves neutrosophic uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

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

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

When not to use

• •Crisp data sufficient: use base CODAS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for NSS-CODAS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'NSS-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Neutrosophic Soft Set numbers/tuples
• •Hatalı: 'NSS-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'NSS-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: NSS-CODAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: NSS-CODAS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: K_{j}(S_{i}) = \langle T_{ij}, I_{ij}, F_{ij}\rangle \in \text{NSS},\quad T,I,F \in [0,1],\ 0 \le T^{2}+I^{2}+F^{2} \le \sqrt{3} Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 §4 Step 1; Table 1
2. 2.Adım 2 (F2): Step 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}. Formül: \text{SWAM}_{z}(A_{1},\ldots,A_{n}) = \Bigl\langle \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}}}\Bigr\rangle;\quad \text{SWGM}_{z} = \Bigl\langle \prod_{j} T_{A_{j}}^{z_{j}},\ \prod_{j} I_{A_{j}}^{z_{j}},\ \prod_{j} F_{A_{j}}^{z_{j}}\Bigr\rangle Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(13)-(14); §4 Step 2
3. 3.Adım 3 (F3): Step 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. Formül: K_{j}(S_{iz}) = \langle T_{ij}\cdot T_{zj},\ I_{ij}\cdot I_{zj},\ F_{ij}\cdot F_{zj}\rangle Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 Eq.(4); §4 Step 3 Eq.(19)
4. 4.Adım 4 (F4): Step 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). Formül: \text{Score}\bigl(K_{j}(S_{iz})\bigr) = (T_{ijz} - F_{ijz})^{2} - (I_{ijz} - F_{ijz})^{2} Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 Eq.(15); §4 Step 4 Eq.(20)
5. 5.Adım 5 (F5): Step 5: Identify NS-PIS S* (per-criterion arg-max of score) and NS-NIS S^- (per-criterion arg-min of score). Formül: S^{*} = \{K_{j}, \arg\max_{i}\text{Score}(K_{j}(S_{iz}))\}_{j=1}^{n};\quad S^{-} = \{K_{j}, \arg\min_{i}\text{Score}(K_{j}(S_{iz}))\}_{j=1}^{n} Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(21)-(22); §4 Step 5
6. 6.Adım 6 (F6): Step 6: Compute Euclidean-type distances from each alternative to NS-PIS and NS-NIS using the NSS spherical distance metric. Formül: D(S_{i}, S^{*}) = \sqrt{\tfrac{1}{2}\sum_{j=1}^{n}\bigl[(T_{S_{i}}-T_{S^{*}})^{2}+(I_{S_{i}}-I_{S^{*}})^{2}+(F_{S_{i}}-F_{S^{*}})^{2}\bigr]};\quad D(S_{i}, S^{-})\text{ analogous} Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(23)-(24); §4 Step 6
7. 7.Adım 7 (F7): Step 7: Compute the per-alternative max distance to NS-NIS (Dmax) and min distance to NS-PIS (Dmin). Formül: D_{\max}(S_{i}, S^{-}) = \max_{i\le m} D(S_{i}, S^{-});\quad D_{\min}(S_{i}, S^{*}) = \min_{i\le m} D(S_{i}, S^{*}) Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 Eqs.(25)-(26); §4 Step 7
8. 8.Adım 8 (F8): Step 8: Compute the revised closeness ratio ξ_i (Eq.28). Lower ξ_i = better alternative. Formül: \xi(S_{i}) = \frac{D(S_{i}, S^{*})}{D_{\min}(S_{i}, S^{*})} - \frac{D(S_{i}, S^{-})}{D_{\max}(S_{i}, S^{-})} Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 Eq.(28); §4 Step 8
9. 9.Adım 9 (F9): Step 9: Rank alternatives in ascending order of ξ_i (lower closeness ratio = better alternative). Formül: \text{rank} = \text{argsort}_{\uparrow} \xi(S_{i}) Anchor: Bhuvaneshwari & Antony Crispin Sweety 2024 §4 Step 9

Commonly paired with

• •n_a + NSS-CODAS (common)

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