PiF-EDAS - Picture extension of EDAS

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.

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

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).

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

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)).

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

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.

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

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

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

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

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

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

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