Fuzzy ELECTRE III - Fuzzy extension of ELECTRE-III

Overview

fuzzy-electre-iii extends ELECTRE-III to handle Fuzzy uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Triangular Fuzzy Number (TFN: l, m, u) algebra. The final scores are defuzzified via centroid (l+m+u)/3 before ranking.

Output

preference flow, higher is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Build/normalize decision matrix m×n (Montazer §6 Table 2 normalized 4 vendors × 6 criteria). Define per-criterion thresholds q_j ≤ p_j ≤ v_j (Montazer used identical 0.2 ≤ 0.5 ≤ 0.9 across all criteria) and weights w_j (Montazer used q_1=7,q_2=4,q_3=5,q_4=3,q_5=2,q_6=5 then normalize).

2. 2

   For each ordered pair (a,b), compute coalitions J^S = {j: g_j(b)-g_j(a) ≤ q_j} (strong support), J^Q = {j: q_j < g_j(b)-g_j(a) ≤ p_j} (weak support). Eqs.5-6.

3. 3

   Partial concordance per criterion c_j(a,b) - 1 if j∈J^S, linear ramp if j∈J^Q, 0 otherwise. Eq.7.

4. 4

   Aggregate to comprehensive concordance c(a,b) = Σ w_j · c_j(a,b). Eq.8. (Montazer §6 Table 3 shows c values mostly 1.0 with some 0.88-0.99.)

5. 5

   Per-criterion discordance d_j(a,b) - 0 if Δ ≤ p_j, linear ramp if p_j<Δ<v_j, 1 if Δ ≥ v_j. Eq.9. (Montazer §6 Table 4: ALL ZEROS for OIEC fixture - no veto triggered, comfortable thresholds.)

6. 6

   Compute V = {j: d_j > c}; if V empty, ρ = c (Eq.11). Else compute D_j = (1-d_j)/(1-c) for j∈V.

7. 7

   Build triangular fuzzy credibility ρ(a,b) = [C⊖D, C•D, C∩D] via three intersection methods: bounded subtraction (Eq.14, lower) gives a; algebraic product (Eq.13, middle) gives m; natural definition min (Eq.12, upper) gives b. Eq.16. Lemma 1: a ≤ m ≤ b. (Montazer §6 Table 5 has full ρ(a,b) for 4×4 = 16 pairs.)

8. 8

   Form outranking strength sets φ_1(a_i) = {ρ(a_i, a_j) : j≠i} (a_i outranking), φ_2(a_i) = {ρ(a_j, a_i) : j≠i} (a_i being-outranked). Eqs.17-18.

9. 9

   Triangular representatives Index1 = [Yager(min ρ over φ_1), Yager(avg ρ over φ_1), Yager(max ρ over φ_1)]; Index2 symmetric over φ_2. Eqs.19-20. (Montazer §6 Tables 6-7 give Min/Avg/Max with Yager defuzz per vendor.)

10. 10

    Net strength Q(a) = Index1 - Index2; Yager defuzzify Q(a) per Eq.21: Yager(triangle [a,m,b]) = ((3·a) - (m-a) + (b-m))/D where D is normalization factor (paper text). Rank by descending Yager(Q). (Montazer Table 8: Q(vendor 1)=0.008, Q(vendor 2)=0.009, Q(vendor 3)=0.026, Q(vendor 4)=-0.021 → ranking 3≻2≻1≻4.)

11. 11

    Optional final weighted score for multi-bid pipeline: A_v = Σq_i·A_i/Σq_i (Eq.23) where q_i = factor weights, A_i = match score per criterion. Used for system-level scoring in Montazer 2009 §6.1 fuzzy expert evaluation module.

Look elsewhere when

Assumptions to verify

• Thresholds (q, p, v) can be expressed in Fuzzy (Triangular) scale
• Veto + concordance semantics adapted to fuzzy arithmetic

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy TFN: l ≤ m ≤ u, all ≥ 0 before computation.

• Defuzzification method affects ranking: centroid (l+m+u)/3 is the canonical choice but alternatives exist.

Works with

How to cite