Cubic-TOPSIS: Cubic extension of TOPSIS

Overview

Cubic Intuitionistic Fuzzy ranking: CIFN = (IVIFN, IFN): interval membership/non-membership ⟨[ζL,ζU],[ϑL,ϑU]⟩ (IVIFS part) + point membership/non-membership ⟨ζ,ϑ⟩ (IFS part); extended TOPSIS with weighted generalised distance. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Cubic Intuitionistic Fuzzy ranking: CIFN = (IVIFN, IFN): interval membership/non-membership ⟨[ζL,ζU],[ϑL,ϑU]⟩ (IVIFS part) + point membership/non-membership ⟨ζ,ϑ⟩ (IFS part); extended TOPSIS with weighted generalised distance
• •Preserves intuitionistic 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 Cubic 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 Cubic 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 TOPSIS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

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

Common pitfalls

• •Hatalı: 'CUBIC-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Cubic Set numbers/tuples
• •Hatalı: 'CUBIC-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'CUBIC-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: CUBIC-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: CUBIC-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): Step 1: Construct K decision matrices D^(k). Each entry α_ij^(k) = (A_ij, λ_ij) is a CIFN where A_ij = ⟨[ζ_ij^L, ζ_ij^U],[ϑ_ij^L, ϑ_ij^U]⟩ (IVIFN) and λ_ij = ⟨ζ_ij, ϑ_ij⟩ (IFN). Weight vector ω per criterion, expert priority λ = (λ_1,...,λ_K)^T. Formül: \alpha_{ij}^{(k)} = \bigl(\langle[\zeta_{ij}^{Lk},\zeta_{ij}^{Uk}],[\vartheta_{ij}^{Lk},\vartheta_{ij}^{Uk}]\rangle,\,\langle\zeta_{ij}^k,\vartheta_{ij}^k\rangle\bigr) \text{s.t. } 0\le\zeta^{Lk}\le\zeta^{Uk}\le 1,\; 0\le\vartheta^{Lk}\le\vartheta^{Uk}\le 1,\; \zeta^{Uk}+\vartheta^{Uk}\le 1 0\le\zeta^k,\vartheta^k\le 1,\; \zeta^k+\vartheta^k\le 1 Anchor: Garg & Kaur 2018, §4.1; Def. 2.5 (Kaur & Garg 2018 [41,42])
2. 2.Adım 2 (F2): Step 2: For each expert D^(k), compute CIF-PIA α^+ and CIF-NIA α^- component-wise over alternatives. For benefit criteria: PIA takes max ζ-intervals, min ϑ-intervals; NIA takes min ζ-intervals, max ϑ-intervals. IFN component: PIA uses min ζ_ij, max ϑ_ij; NIA uses max ζ_ij, min ϑ_ij. Formül: \alpha_j^+ = \Bigl(\langle[g_j^{L+},g_j^{U+}],[h_j^{L+},h_j^{U+}]\rangle,\langle r_j^+,s_j^+\rangle\Bigr) \quad\text{[Eq.(8)]} \alpha_j^- = \Bigl(\langle[g_j^{L-},g_j^{U-}],[h_j^{L-},h_j^{U-}]\rangle,\langle r_j^-,s_j^-\rangle\Bigr) \quad\text{[Eq.(9)]} g_j^{L+}=\max_i\{\zeta_{ij}^L\},\; g_j^{U+}=\max_i\{\zeta_{ij}^U\},\; h_j^{L+}=\min_i\{\vartheta_{ij}^L\},\; h_j^{U+}=\min_i\{\vartheta_{ij}^U\},\; r_j^+=\min_i\{\zeta_{ij}\},\; s_j^+=\max_i\{\vartheta_{ij}\} g_j^{L-}=\min_i\{\zeta_{ij}^L\},\; g_j^{U-}=\min_i\{\zeta_{ij}^U\},\; h_j^{L-}=\max_i\{\vartheta_{ij}^L\},\; h_j^{U-}=\max_i\{\vartheta_{ij}^U\},\; r_j^-=\max_i\{\zeta_{ij}\},\; s_j^-=\min_i\{\vartheta_{ij}\} Anchor: Garg & Kaur 2018, §4.2, Eqs.(8)-(9)
3. 3.Adım 3 (F3): Step 3: For each expert, compute weighted generalised distance (q=2 canonical) from A_i to CIF-PIA and CIF-NIA. Each CIFN contributes 6 component-wise |·|^q terms: two for ζ-interval, two for ϑ-interval, one for IFN membership, one for IFN non-membership. Formül: d_q(A_i,\alpha^+)=\left(\frac{1}{6}\sum_{j=1}^n\omega_j\Bigl\{|g_j^{L+}-\zeta_{ij}^L|^q+|g_j^{U+}-\zeta_{ij}^U|^q+|\vartheta_{ij}^L-h_j^{L+}|^q+|\vartheta_{ij}^U-h_j^{U+}|^q+|\zeta_{ij}-r_j^+|^q+|s_j^+-\vartheta_{ij}|^q\Bigr\}\right)^{1/q}\quad\text{[Eq.(10)]} d_q(A_i,\alpha^-)=\left(\frac{1}{6}\sum_{j=1}^n\omega_j\Bigl\{|\zeta_{ij}^L-g_j^{L-}|^q+|\zeta_{ij}^U-g_j^{U-}|^q+|h_j^{L-}-\vartheta_{ij}^L|^q+|h_j^{U-}-\vartheta_{ij}^U|^q+|r_j^--\zeta_{ij}|^q+|\vartheta_{ij}-s_j^-|^q\Bigr\}\right)^{1/q}\quad\text{[Eq.(11)]} Anchor: Garg & Kaur 2018, §4.3, Eqs.(7),(10)-(11)
4. 4.Adım 4 (F4): Step 4: Per-expert closeness coefficient C_i^(k) ∈ [0,1]. Used for per-expert ranking only; aggregation follows in F5. Formül: \mathfrak{C}_i^{(k)} = \frac{d_q\bigl((A_i)^{(k)},(\alpha^-)^{(k)}\bigr)}{d_q\bigl((A_i)^{(k)},(\alpha^+)^{(k)}\bigr)+d_q\bigl((A_i)^{(k)},(\alpha^-)^{(k)}\bigr)},\quad k=1,\ldots,K \quad\text{[Eq.(15)]} Anchor: Garg & Kaur 2018, §4.4 Step 4, Eq.(15)
5. 5.Adım 5 (F5): Step 5: Aggregate per-expert distances using expert priority weights λ = (λ_1,...,λ_K)^T with Σλ_k = 1. (For single-DM case K=1, λ_1=1, this step is trivial.) Formül: D_i^+ = \sum_{k=1}^K \lambda_k\, d_q\bigl((A_i)^{(k)},(\alpha^+)^{(k)}\bigr),\quad D_i^- = \sum_{k=1}^K \lambda_k\, d_q\bigl((A_i)^{(k)},(\alpha^-)^{(k)}\bigr) \quad\text{[Eq.(16)]} \lambda_k>0,\; \sum_{k=1}^K \lambda_k=1 Anchor: Garg & Kaur 2018, §4.4 Step 5, Eq.(16)
6. 6.Adım 6 (F6): Step 6: Overall closeness coefficient C_i using aggregated distances. Higher C_i → closer to ideal. Formül: \mathfrak{C}_i = \frac{D_i^-}{D_i^+ + D_i^-},\quad D_i^+ \neq 0,\quad 0\le\mathfrak{C}_i\le 1 \quad\text{[Eq.(17)]} Anchor: Garg & Kaur 2018, §4.4 Step 6, Eq.(17)
7. 7.Adım 7 (F7): Step 7: Rank alternatives in descending order of C_i. Highest C_i = best alternative. Formül: A_i \succ A_j \iff \mathfrak{C}_i > \mathfrak{C}_j Anchor: Garg & Kaur 2018, §4.4 Step 7

Commonly paired with

• •n_a + CUBIC-TOPSIS (common)

How to cite