IF-EDAS - Intuitionistic Fuzzy EDAS

Overview

IF-EDAS extends crisp EDAS to settings where each rating is an Intuitionistic Fuzzy Number (μ, ν). The algorithm (i) optionally aggregates K DMs via IFWA, (ii) defuzzifies cells via Chen-Tan score s = μ − ν, (iii) computes per-criterion Average solution AV and direction-aware PDA/NDA (cost criteria swap branches), (iv) weights and max-normalises into NSP, NSN, (v) averages to the Appraisal Score AS_i = (NSP_i + NSN_i)/2 ∈ [0, 1]. Higher AS_i = better. Algorithm is INVARIANT across pure IF-EDAS application papers (Yıldırım-Meydan 2021 BİST retail, Dumrul et al. 2024 renewable energy); score function is the only parametric choice (D.parameters).

Output

utility, higher is better

Data

Intuitionistic Fuzzy, ifn tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Financial performance evaluation (Yıldırım-Meydan 2021 BİST case), Sustainable renewable energy resource selection (Dumrul et al. 2024), MAGDM under epistemic uncertainty, Expert-driven evaluation with linguistic IFN scales

How it works

1. 1

   Construct the IF decision matrix R = (r_ij) where r_ij = (μ_ij, ν_ij) is the IFN rating of alternative A_i on criterion C_j. For group-decision settings (K≥2 DMs), collect K such matrices and aggregate cell-wise via IFWA with DM weights λ_k before entering F2: r_ij = (1 − Π(1−μ_ij^{(k)})^{λ_k}, Π (ν_ij^{(k)})^{λ_k}).

   $$R=(r_{ij}{)}_{m\times n},r_{ij}=({\mu }_{ij},{\nu }_{ij}),{\mu }_{ij}+{\nu }_{ij}\le 1;\text{IFWA aggregation if }K\ge 2$$

   Yıldırım-Meydan 2021 Steps 1-2; Dumrul 2024 Steps 1-2; Xu 2007 IFWA

2. 2

   Defuzzify each cell via the IF score function (default Chen-Tan 1994: s(α) = μ − ν ∈ [−1, 1]). This produces a crisp m×n score matrix that drives the PDA/NDA computation.

   $$s_{ij}={\mu }_{ij}-{\nu }_{ij},s_{ij}\in [-1,1]$$

   Yıldırım-Meydan 2021 Step 3; Dumrul 2024 Step 3; Chen-Tan 1994 score

3. 3

   Compute the Average Solution AV per criterion as the mean of crisp scores: AV_j = (1/m) Σ_i s_ij.

   $$A{V}_j=\frac{1}{m}\sum _{i=1}^m{s}_{ij}$$

   Yıldırım-Meydan 2021 Step 4; Dumrul 2024 Step 4; Keshavarz Ghorabaee 2015 Eq. (3) [crisp skeleton]

4. 4

   Direction-aware Positive and Negative Distance from Average (PDA, NDA). For benefit criteria j ∈ J_b: PDA_ij = max(0, s_ij − AV_j) / |AV_j|, NDA_ij = max(0, AV_j − s_ij) / |AV_j|. For cost criteria j ∈ J_c the branches swap: PDA_ij = max(0, AV_j − s_ij) / |AV_j|, NDA_ij = max(0, s_ij − AV_j) / |AV_j|.

   $$j\in J_b:PD{A}_{ij}=\frac{\max (0,s_{ij}-A{V}_j)}{|A{V}_j|},ND{A}_{ij}=\frac{\max (0,A{V}_j-s_{ij})}{|A{V}_j|};j\in J_c:\text{swap branches}$$

   Yıldırım-Meydan 2021 Step 5; Dumrul 2024 Step 4; Keshavarz Ghorabaee 2015 Eqs. (4)-(5)

5. 5

   Weighted Sums SP_i = Σ_j w_j · PDA_ij and SN_i = Σ_j w_j · NDA_ij across criteria.

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

   Yıldırım-Meydan 2021 Step 6; Dumrul 2024 Step 5; Keshavarz Ghorabaee 2015 Eqs. (6)-(7)

6. 6

   Normalise SP and SN by their respective maxima: NSP_i = SP_i / max_k SP_k and NSN_i = 1 − SN_i / max_k SN_k. NSP rewards positive deviation; NSN rewards low negative deviation. Both lie in [0, 1].

   $$NS{P}_i=\frac{S{P}_i}{{\max }_{k}S{P}_k};NS{N}_i=1-\frac{S{N}_i}{{\max }_{k}S{N}_k}$$

   Yıldırım-Meydan 2021 Step 7; Dumrul 2024 Step 6; Keshavarz Ghorabaee 2015 Eqs. (8)-(9)

7. 7

   Appraisal Score AS_i = (NSP_i + NSN_i) / 2 ∈ [0, 1].

   $$A{S}_i=\frac{1}{2}(NS{P}_i+NS{N}_i),A{S}_i\in [0,1]$$

   Yıldırım-Meydan 2021 Step 8; Dumrul 2024 Step 7; Keshavarz Ghorabaee 2015 Eq. (10)

8. 8

   Rank alternatives in descending order of AS_i; the alternative with the largest AS_i is the best.

   $$\text{rank}(A_i)={\text{argsort}}_{\downarrow }(A{S}_i)$$

   Yıldırım-Meydan 2021 Step 8; Dumrul 2024 Step 8

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• All decision-makers use the same linguistic-to-IFN scale
• Criterion directions (benefit/cost) are explicitly labelled
• Criterion weights form a simplex

Edge cases and pitfalls

• •IF decision matrix R = (r_ij) where r_ij = (μ_ij, ν_ij) is the IFN rating of alternative A_i on criterion C_j. For group-decision settings (K≥2 DMs), collect K such matrices and aggregate cell-wise vi
• •default Chen-Tan 1994: s(α) = μ − ν ∈ [−1, 1]). This produces a crisp m×n score matrix that drives the PDA/NDA computation.

• Confusing IFN (pair μ, ν) with TFN (triple a, b, c): IF-EDAS values MUST be IFN tuples satisfying μ+ν ≤ 1, NOT triangular fuzzy numbers.

• Failing to swap the PDA/NDA branches on cost criteria: for j ∈ J_c the positive branch tests AV_j − s_ij and the negative branch tests s_ij − AV_j. Reusing the benefit formula on cost columns inverts the ranking.

• Forgetting the 1 − ratio inversion in NSN: NSN_i = 1 − SN_i / max SN (so smaller SN gives larger NSN). Using NSN_i = SN_i / max SN directly produces the opposite ranking.

• Using |AV_j| = 0 in the F4 denominator. If every alternative scores identically on criterion j (AV_j = 0 in score domain), set PDA_ij = NDA_ij = 0 for that column instead of raising a division error.

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