Cubic-TOPSIS - Cubic extension of TOPSIS

Overview

CUBIC-TOPSIS (Garg & Kaur 2018) extends TOPSIS to Cubic Intuitionistic Fuzzy Numbers (CIFNs). Each entry encodes both IVIFN ⟨[ζL,ζU],[ϑL,ϑU]⟩ and IFN ⟨ζ,ϑ⟩ simultaneously. The method computes weighted generalised distance (q=2) from alternatives to CIF-PIA and CIF-NIA - no defuzzification step. Supports multiple decision makers: per-expert closeness coefficients are aggregated via priority weights λ. Higher C_i = better alternative.

Output

utility, higher is better

Data

Cubic Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

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

How it works

1. 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. \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

   $${\alpha }_{ij}^{(k)}=(⟨[{\zeta }_{ij}^{Lk},{\zeta }_{ij}^{Uk}],[{\vartheta }_{ij}^{Lk},{\vartheta }_{ij}^{Uk}]⟩,⟨{\zeta }_{ij}^k,{\vartheta }_{ij}^k⟩)$$

   Garg & Kaur 2018, §4.1; Def. 2.5 (Kaur & Garg 2018 [41,42])

2. 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. \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}\}

   $${\alpha }_j^{+}=(⟨[g_j^{L+},g_j^{U+}],[h_j^{L+},h_j^{U+}]⟩,⟨r_j^{+},s_j^{+}⟩)\text{[Eq.(8)]}$$

   Garg & Kaur 2018, §4.2, Eqs.(8)-(9)

3. 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. 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)]}

   $$d_q(A_i,{\alpha }^{+})={\left(\frac{1}{6}\sum _{j=1}^n{\omega }_j\{|g_j^{L+}-{\zeta }_{ij}^L{|}^q+|g_j^{U+}-{\zeta }_{ij}^U{|}^q+|{\vartheta }_{ij}^L-h_j^{L+}{|}^q+|{\vartheta }_{ij}^U-h_j^{U+}{|}^q+|{\zeta }_{ij}-r_j^{+}{|}^q+|s_j^{+}-{\vartheta }_{ij}{|}^q\}\right)}^{1/q}\text{[Eq.(10)]}$$

   Garg & Kaur 2018, §4.3, Eqs.(7),(10)-(11)

4. 4

   Per-expert closeness coefficient C_i^(k) ∈ [0,1]. Used for per-expert ranking only; aggregation follows in F5.

   $${\mathfrak{C}}_i^{(k)}=\frac{d_q((A_i{)}^{(k)},({\alpha }^{-}{)}^{(k)})}{d_q((A_i{)}^{(k)},({\alpha }^{+}{)}^{(k)})+d_q((A_i{)}^{(k)},({\alpha }^{-}{)}^{(k)})},k=1,\dots ,K\text{[Eq.(15)]}$$

   Garg & Kaur 2018, §4.4 Step 4, Eq.(15)

5. 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.) \lambda_k>0,\; \sum_{k=1}^K \lambda_k=1

   $$D_i^{+}=\sum _{k=1}^K{\lambda }_k{d}_q((A_i{)}^{(k)},({\alpha }^{+}{)}^{(k)}),D_i^{-}=\sum _{k=1}^K{\lambda }_k{d}_q((A_i{)}^{(k)},({\alpha }^{-}{)}^{(k)})\text{[Eq.(16)]}$$

   Garg & Kaur 2018, §4.4 Step 5, Eq.(16)

6. 6

   Overall closeness coefficient C_i using aggregated distances. Higher C_i → closer to ideal.

   $${\mathfrak{C}}_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}},D_i^{+}\ne 0,0\le {\mathfrak{C}}_i\le 1\text{[Eq.(17)]}$$

   Garg & Kaur 2018, §4.4 Step 6, Eq.(17)

7. 7

   Rank alternatives in descending order of C_i. Highest C_i = best alternative.

   $$A_i\succ A_j⟺{\mathfrak{C}}_i>{\mathfrak{C}}_j$$

   Garg & Kaur 2018, §4.4 Step 7

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• CIFN validity: each entry must satisfy ζL ≤ ζU, ϑL ≤ ϑU, ζU+ϑU ≤ 1 (IVIFN part) and ζ+ϑ ≤ 1 (IFN part).

• Choice of q affects distances: q=2 (Euclidean) is canonical per Garg & Kaur 2018; q=1 gives Hamming-based distances.

• Expert weight vector λ must sum to 1. For single DM (K=1), set λ=(1.0) and the method reduces to standard CIFN-TOPSIS.

Works with

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