Outranking

Fuzzy ELECTRE III: Fuzzy extension of ELECTRE-III

Montazer, G. A., Qahri Saremi, H., Ramezani, M. · 2009

Method assistant

Grounded explanations: it explains the method, it does not compute.

Assumptions to verify

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

When not to use

•Small dataset (m<3): outranking machinery underutilised

Worked example

1.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.

Commonly paired with

•FUZZY-AHP + FUZZY-ELECTRE-III (common)

How to cite

Montazer, G. A.; Qahri Saremi, H.; Ramezani, M. (2009). Design a new mixed expert decision aiding system using fuzzy ELECTRE III method for vendor selection. Expert Systems with Applications. https://doi.org/10.1016/j.eswa.2009.01.019

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Worked example

1.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.Step 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.