PiF-EDAS - Picture extension of EDAS

Overview

pif-edas extends EDAS 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

   Form Picture Fuzzy decision matrix X̃ = [x̃_ij] with x̃_ij = (μ_ij, η_ij, ν_ij), μ_ij+η_ij+ν_ij ≤ 1.

   $$\tilde{X}=[{\tilde{x}}_{ij}{]}_{m\times n},{\tilde{x}}_{ij}=({\mu }_{ij},{\eta }_{ij},{\nu }_{ij}),{\mu }_{ij}+{\eta }_{ij}+{\nu }_{ij}\le 1$$

   Cuong 2013 Definition 1; Kamber 2026 Eq.(24)

2. 2

   Cost-complement normalize: for cost criteria, r_ij = (ν_ij, η_ij, μ_ij); for benefit criteria, r_ij = (μ_ij, η_ij, ν_ij) (unchanged).

   $$r_{ij}=({\nu }_{ij},{\eta }_{ij},{\mu }_{ij})\text{ if }j\in \text{Cost};r_{ij}=({\mu }_{ij},{\eta }_{ij},{\nu }_{ij})\text{ if }j\in \text{Benefit}$$

   Khan vd. 2019; Kamber 2026 Eq.(25)

3. 3

   Per-criterion average solution AV_j via PFWA with equal weights 1/m: AV_j = (1 − ∏(1−μ_ij)^(1/m), ∏ η_ij^(1/m), ∏ ν_ij^(1/m)).

   $$A{V}_j=\left(1-\prod _{i=1}^m(1-{\mu }_{ij}{)}^{1/m},\prod _{i=1}^m{\eta }_{ij}^{1/m},\prod _{i=1}^m{\nu }_{ij}^{1/m}\right)$$

   Wei 2017 Eq.(8) PFWA; Kamber 2026 Eq.(26)

4. 4

   Positive/Negative Distance from Average via Cuong score s(α) = μ − ν. PDA_ij = max(0, s(r_ij) − s(AV_j))/|s(AV_j)|; NDA_ij = max(0, s(AV_j) − s(r_ij))/|s(AV_j)|. NOTE: Kamber 2026 Eq.(27)/(28) print these with PDA and NDA labels swapped; canonical labeling is used here for compatibility with Step 6 normalization NSP=SP/maxSP, NSN=1−SN/maxSN.

   $$s(\alpha )=\mu -\nu ;PD{A}_{ij}=\frac{\max (0,s(r_{ij})-s(A{V}_j))}{|s(A{V}_j)|};ND{A}_{ij}=\frac{\max (0,s(A{V}_j)-s(r_{ij}))}{|s(A{V}_j)|}$$

   Cuong 2013 Eq.(5); Kamber 2026 Eqs.(27)-(28) (with corrected labels)

5. 5

   Weighted sums SP_i = Σ_j w_j · PDA_ij and SN_i = Σ_j w_j · NDA_ij.

   $$S{P}_i=\sum _{j=1}^n{w}_j\cdot PD{A}_{ij};S{N}_i=\sum _{j=1}^n{w}_j\cdot ND{A}_{ij}$$

   Kamber 2026 Eqs.(19)-(20); Keshavarz-Ghorabaee 2015 Eqs.(6)-(7)

6. 6

   Normalize: NSP_i = SP_i / max_i SP_i; NSN_i = 1 − SN_i / max_i SN_i.

   $$NS{P}_i=\frac{S{P}_i}{{\max }_{i}S{P}_i};NS{N}_i=1-\frac{S{N}_i}{{\max }_{i}S{N}_i};NS{P}_i:=0\text{ if }\underset{i}{\max }S{P}_i=0,NS{N}_i:=1\text{ if }\underset{i}{\max }S{N}_i=0$$

   Kamber 2026 Eqs.(21)-(22); Keshavarz-Ghorabaee 2015 Eqs.(8)-(9)

7. 7

   Appraisal Score AS_i = (NSP_i + NSN_i)/2; rank alternatives in descending order of AS_i (largest AS is best).

   $$A{S}_i=\frac{1}{2}(NS{P}_i+NS{N}_i);\text{rank desc by }A{S}_i$$

   Kamber 2026 Eq.(23); Keshavarz-Ghorabaee 2015 Eq.(10)

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

• 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.

• Bu manifestin B.extensions bloğunda kaynak gösterilen Wei (2017), 'Picture fuzzy aggregation operators...' (DOI 10.3233/JIFS-161798) GERİ ÇEKİLMİŞTİR (RETRACTED); bu kaynağı PFWA/agregasyon gerekçesi olarak KULLANMAYIN.

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