N-TOPSIS: Neutrosophic extension of TOPSIS

Overview

Neutrosophic outranking/ranking: Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Neutrosophic outranking/ranking: Single-Valued Neutrosophic Set (SVNS: T, I, F; T,I,F ∈ [0,1], T+I+F ≤ 3)
• •Preserves single_valued_neutrosophic uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Single-Valued Neutrosophic 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 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

When not to use

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

Edge cases

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

Common pitfalls

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

Worked example

1. 1.Adım 1 (F1): Construct SVN decision matrix; apply cost complement. Formül: \mathbf{D}=(\langle T_{ij},I_{ij},F_{ij}\rangle)_{m\times n};\quad\text{cost: }\tilde{a}_{ij}^c=\langle F_{ij},1-I_{ij},T_{ij}\rangle Anchor: Biswas, Pramanik & Giri 2016, Sec.3 Step1
2. 2.Adım 2 (F2): Weighted SVNN via scalar multiplication operator. Formül: \tilde{v}_{ij}=w_j\cdot\tilde{a}_{ij}=\langle 1-(1-T_{ij})^{w_j},\;I_{ij}^{w_j},\;F_{ij}^{w_j}\rangle Anchor: Biswas, Pramanik & Giri 2016, Sec.3 Step2
3. 3.Adım 3 (F3): Neutrosophic PIS A⁺ and NIS A⁻ (component-wise extrema of weighted SVNNs). Formül: \tilde{v}_j^+=\langle\max_i T_{ij}^v,\min_i I_{ij}^v,\min_i F_{ij}^v\rangle;\quad\tilde{v}_j^-=\langle\min_i T_{ij}^v,\max_i I_{ij}^v,\max_i F_{ij}^v\rangle Anchor: Biswas, Pramanik & Giri 2016, Sec.3 Step3
4. 4.Adım 4 (F4): Euclidean separation from PIS/NIS using normalised neutrosophic distance. Formül: d(\alpha_1,\alpha_2)=\sqrt{\tfrac{(T_1-T_2)^2+(I_1-I_2)^2+(F_1-F_2)^2}{3}};\quad d_i^\pm=\sqrt{\sum_{j=1}^n[d(\tilde{v}_{ij},\tilde{v}_j^\pm)]^2} Anchor: Biswas, Pramanik & Giri 2016, Sec.3 Step4
5. 5.Adım 5 (F5): Closeness coefficient CC_i; rank descending. Formül: CC_i=\frac{d_i^-}{d_i^++d_i^-},\quad CC_i\in[0,1];\quad\text{rank descending} Anchor: Biswas, Pramanik & Giri 2016, Sec.3 Step5

Commonly paired with

• •n_a + N-TOPSIS (common)

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