PiF-COPRAS - Picture extension of COPRAS

Overview

pif-copras extends COPRAS to handle Picture uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Picture Fuzzy Set (PiFS: μ, η, ν; μ+η+ν ≤ 1) algebra. The final scores are defuzzified via score function S = μ − ν before ranking.

Output

utility, higher is better

Data

Picture Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Picture Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Aggregate per-DM matrices H^(k) (k=1,...,l) into group PF decision matrix H = [h_ij] via PFWA (Lu 2021 Eq.14): h_ij = (1 − ∏_k (1−μ^k)^ψ_k, ∏_k η^k^ψ_k, ∏_k (ν^k+η^k)^ψ_k − ∏_k η^k^ψ_k), where ψ_k is the expert weight (Σψ_k = 1). If only a single DM is provided, H = H^(1).

   $$h_{i}j=(1-\prod _{k=1}^l(1-{\mu }_{ij}^k{)}^{{\psi }_k},\prod _{k=1}^l({\eta }_{ij}^k{)}^{{\psi }_k},\prod _{k=1}^l({\nu }_{ij}^k+{\eta }_{ij}^k{)}^{{\psi }_k}-\prod _{k=1}^l({\eta }_{ij}^k{)}^{{\psi }_k})$$

   Lu 2021 Eqs.12-14; Wang 2017 PFWA

2. 2

   Determine objective criteria weights q_j via CRITIC (Lu 2021 Eqs.15-17) using score S(h_ij) = μ − ν. Compute Pearson correlation τ_jt between columns; standard deviation σ_j; then q_j = σ_j Σ_t (1−τ_jt) / Σ_j [σ_j Σ_t (1−τ_jt)].

   $${\tau }_{jt}={\Sigma }_i(S(h_{ij})-{\bar{S}}_j)(S(h_{it})-{\bar{S}}_t)/sqrt({\Sigma }_i(S(h_{ij})-{\bar{S}}_j{)}^2\cdot {\Sigma }_i(S(h_{it})-{\bar{S}}_t{)}^2);{\sigma }_j=sqrt((1/m){\Sigma }_i(S(h_{ij})-{\bar{S}}_j{)}^2);q_j={\sigma }_j{\Sigma }_t(1-{\tau }_{jt})/{\Sigma }_j[{\sigma }_j{\Sigma }_t(1-{\tau }_{jt})]$$

   Lu 2021 Eqs.15-17; Diakoulaki et al. 1995 (CRITIC origin)

3. 3

   Weighted normalized matrix D = [d_ij] via scalar multiplication (Lu 2021 Eq.18; Wang 2017 scalar-mult Eq.4): d_ij = q_j × h_ij = (1−(1−μ)^{q_j}, η^{q_j}, (ν+η)^{q_j} − η^{q_j}).

   $$d_{ij}=q_j\times h_{ij}=(1-(1-{\mu }_{ij}{)}^{q_j},{\eta }_{ij}^{q_j},({\nu }_{ij}+{\eta }_{ij}{)}^{q_j}-{\eta }_{ij}^{q_j})$$

   Lu 2021 Eq.18; Wang 2017 Eq.4 (scalar mult)

4. 4

   Sum weighted PFNs over benefit (max-direction) attributes → R_i^+, and over cost (min-direction) attributes → R_i^- via Wang 2017 ⊕ (Lu 2021 Eqs.19-20): R_i = (1−∏(1−μ_d), ∏η_d, ∏(ν_d+η_d)−∏η_d).

   $$R_i^{+}={\oplus }_{j:max}{d}_{ij}=(1-\prod _{j:max}(1-{\mu }_{d_{ij}}),\prod _{j:max}{\eta }_{d_{ij}},\prod _{j:max}({\nu }_{d_{ij}}+{\eta }_{d_{ij}})-\prod _{j:max}{\eta }_{d_{ij}});R_i^{-}analogousovercostattributes$$

   Lu 2021 Eqs.19-20; Wang 2017 Eq.2 (⊕)

5. 5

   Significance O_i (Lu 2021 Eq.21): O_i = S(R_i^+) + Σ_k S(R_k^-) / [S(R_i^-) · Σ_k 1/S(R_k^-)], where S(α)=μ−ν (Cuong 2014 Def.3, Eq.6).

   $$O_i=S(R_i^{+})+{\Sigma }_{k=1}^{m}S(R_k^{-})/[S(R_i^{-})\cdot {\Sigma }_{k=1}^m(1/S(R_k^{-}))]$$

   Lu 2021 Eq.21

6. 6

   Utility degree G_i = O_i / max_k O_k (Lu 2021 Eq.22; reported as percent). Higher G_i ⇒ better. Rank alternatives by descending G_i.

   $$G_i=O_i/ma{x}_{k=1,...,m}{O}_k(presentedas{G}_i\times 100$$

   Lu 2021 Eq.22

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Picture Fuzzy 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

• •If only a single DM is provided, H = H^(1).

• Value-space violation: ensure all entries satisfy PiFS: μ (membership), η (neutral), ν (non-membership); μ+η+ν ≤ 1 before computation.

• Defuzzification method affects ranking: score function S = μ − ν is the canonical choice but alternatives exist.

Works with

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