Cubic-EDAS - Cubic Pythagorean Fuzzy EDAS (CuP-EDAS)

Overview

CuP-EDAS extends EDAS into the Cubic Pythagorean Fuzzy environment. Each decision-maker assessment is a CuPyFN = (⟨[Y⁻,Y⁺],[F⁻,F⁺]⟩,⟨Y,F⟩) with Pythagorean constraint (Y⁺)²+(F⁺)² ≤ 1 (stronger than intuitionistic). Expert opinions are aggregated via CuPyFWG. Distance-based optimization determines criteria weights. PDA/NDA use score function comparison vs average solution (avoids ideal solution bias). Higher appraisal score AS = (NS⁺+NS⁻)/2 is better.

Output

utility, higher is better

Data

Cubic Pythagorean Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 5-15 criteria works best

Used for

Waste management technology selection, Environmental decision-making under uncertainty, MAGDM with incomplete weight information, Cubic Pythagorean Fuzzy MCDM

How it works

1. 1

   Aggregate K expert decision matrices A^(k) using CuPyFWG operator (Def.5.3) with expert weights ω_k (default 1/K): CuPyFWG(h̄_1,...,h̄_K) = (⟨[∏(Y⁻_k)^ωk, ∏(Y⁺_k)^ωk], [√(1−∏(1−(F⁻_k)²)^ωk), √(1−∏(1−(F⁺_k)²)^ωk)]⟩, ⟨∏Y_k^ωk, √(1−∏(1−F_k²)^ωk)⟩). Produces aggregated decision matrix A = (a_sj)_{m×n}.

   $$CuPyFWG({\bar{h}}_1,\dots ,{\bar{h}}_K)=(⟨[\prod _k(Y_k^{-}{)}^{{\omega }_k},\prod _k(Y_k^{+}{)}^{{\omega }_k}],[\sqrt{1-\prod _k(1-(F_k^{-}{)}^2{)}^{{\omega }_k}},\sqrt{1-\prod _k(1-(F_k^{+}{)}^2{)}^{{\omega }_k}}]⟩,⟨\prod _k{Y}_k^{{\omega }_k},\sqrt{1-\prod _k(1-F_k^2{)}^{{\omega }_k}}⟩)$$

   Paul-Jana-Pal 2023, Def.5.3 (CuPyFWG), Eq.(Step1)

2. 2

   Determine criteria weights ω_j via non-linear optimization. Objective: maximize total generalized distance F(ω*) = Σ_j ω*_j · Σ_{s<t} D^q_stj, where D^q_stj is the generalized distance between alternatives s and t on criterion j. Subject to partial weight constraint system W (expert-specified bounds). Convert ω*_j → ω_j = (ω*_j)² / Σ_j(ω*_j)² if needed, or directly ω_j = (ω*_j)² (when Σ(ω*_j)²=1 by constraint). Parameter β ∈ (0,1], default β=0.5; q ≥ 1, default q=3.

   $$\max F({\omega }^{\ast })=\sum _{j=1}^n{\omega }_j^{\ast \beta }\sum _{1\le s<t\le m}{D}^q(a_{sj},a_{tj}),\text{s.t. }{\omega }_j^{\ast }\in W,\sum _j({\omega }_j^{\ast }{)}^2=1$$

   Paul-Jana-Pal 2023, Eq.(1)-(2), Def.6.1, Box I (Eq.11-13)

3. 3

   Compute average solution AV_j for each criterion j using Pythagorean fuzzy arithmetic mean via ⊕ operator: AV_j = (⊕_{s=1}^{m} a_sj) / m. Score function S(h̄) = (1/2)[(1/2)((Y⁻)²+(Y⁺)²−(F⁻)²−(F⁺)²) + Y²−F²], S ∈ [−1,1].

   $$A{V}_j=\frac{1}{m}\underset{s=1}{\overset{m}{⨁}}{a}_{sj},S(\bar{h})=\frac{1}{2}\left[\frac{1}{2}((Y^{-}{)}^2+(Y^{+}{)}^2-(F^{-}{)}^2-(F^{+}{)}^2)+Y^2-F^2\right]$$

   Paul-Jana-Pal 2023, Eq.(3), Def.5.5

4. 4

   Compute PDA matrix Z = (ζ_sj) and NDA matrix E = (η_sj) using score function S(h̄): For benefit criterion j ∈ B: ζ_sj = max{0, S(a_sj)−S(AV_j)} / S(AV_j); η_sj = max{0, S(AV_j)−S(a_sj)} / S(AV_j). For cost criterion j ∈ L: ζ_sj = max{0, S(AV_j)−S(a_sj)} / S(AV_j); η_sj = max{0, S(a_sj)−S(AV_j)} / S(AV_j).

   $${\zeta }_{sj}=\{\begin{array}{ll}\frac{\max (0,S(a_{sj})-S(A{V}_j))}{S(A{V}_j)}& c_j\in B\\ \frac{\max (0,S(A{V}_j)-S(a_{sj}))}{S(A{V}_j)}& c_j\in L\end{array},{\eta }_{sj}=\{\begin{array}{ll}\frac{\max (0,S(A{V}_j)-S(a_{sj}))}{S(A{V}_j)}& c_j\in B\\ \frac{\max (0,S(a_{sj})-S(A{V}_j))}{S(A{V}_j)}& c_j\in L\end{array}$$

   Paul-Jana-Pal 2023, Eqs.(4)-(5)

5. 5

   Compute weighted sums of PDA and NDA across criteria: S⁺_s = Σ_j ω_j · ζ_sj; S⁻_s = Σ_j ω_j · η_sj.

   $$S_s^{+}=\sum _{j=1}^n{\omega }_j{\zeta }_{sj},S_s^{-}=\sum _{j=1}^n{\omega }_j{\eta }_{sj}$$

   Paul-Jana-Pal 2023, Eqs.(6)-(7)

6. 6

   Normalize S⁺ and S⁻ by their maxima: NS⁺_s = S⁺_s / max_s{S⁺_s}; NS⁻_s = 1 − S⁻_s / max_s{S⁻_s}.

   $$N{S}_s^{+}=\frac{S_s^{+}}{{\max }_s\{S_s^{+}\}},N{S}_s^{-}=1-\frac{S_s^{-}}{{\max }_s\{S_s^{-}\}}$$

   Paul-Jana-Pal 2023, Eqs.(8)-(9)

7. 7

   Compute appraisal score AS_s = (1/2)(NS⁺_s + NS⁻_s), rank alternatives in descending order. Best alternative has highest AS.

   $$A{S}_s=\frac{1}{2}(N{S}_s^{+}+N{S}_s^{-}),0\le A{S}_s\le 1;S_i\succ S_j⟺A{S}_i>A{S}_j$$

   Paul-Jana-Pal 2023, Eq.(10)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid CuPyFNs: (Y⁺)²+(F⁺)² ≤ 1 and Y²+F² ≤ 1
• Expert weight vector sums to 1
• S(AV_j) > 0 for all criteria (required for PDA/NDA denominator)

Edge cases and pitfalls

• •default 1/K): CuPyFWG(h̄_1,...,h̄_K) = (⟨[∏(Y⁻_k)^ωk, ∏(Y⁺_k)^ωk], [√(1−∏(1−(F⁻_k)²)^ωk), √(1−∏(1−(F⁺_k)²)^ωk)]⟩, ⟨∏Y_k^ωk, √(1−∏(1−F_k²)^ωk)⟩). Produces aggregated decision matrix A = (a_sj)_{m×n}.
• •default β=0.5; q ≥ 1, default q=3.

• Value-space violation: ensure all CuPyFN entries satisfy (Y⁺)²+(F⁺)² ≤ 1 (Pythagorean, not just intuitionistic Y⁺+F⁺ ≤ 1) and Y²+F² ≤ 1 for the scalar PyFN part.

• Weight optimization is non-linear: requires LINGO or equivalent solver for the paper's original formulation; alternatively supply weights directly as ωj = (ω*j)².

• S(AV_j) may be zero or negative: handle division by zero in PDA/NDA with a small epsilon guard.

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