Fuzzy ELECTRE I (Group, Trapezoidal)

Overview

Output is a partial order: each alternative is dominated by, dominates, or is incomparable/indifferent to every other. Read the Z matrix row-wise: z_gf = 1 means A_g outranks A_f. Equivalence classes appear as ties in the rank column; alternatives with z_gf = z_fg = 0 share no dominance relation and are flagged 'incomparable'. A choice problem typically selects from the top equivalence class (the kernel of the decision graph).

Output

boolean dominance, higher is better

Data

Fuzzy Tr FN, trapezoidal fuzzy ratings complete

Weights

Needs a weight source

Size

2+ alternatives, 3-8 criteria works best

Used for

Group choice problems under linguistic uncertainty, Supplier selection, Environmental impact assessment, Risk and project ranking with imprecise expert input

How it works

1. 1

   Form the group of K DMs and determine evaluation criteria (m alternatives, n criteria).

   $$E=\{e_1,e_2,\dots ,e_K\},A=\{A_1,\dots ,A_m\},C=\{C_1,\dots ,C_n\}$$

   Hatami-Marbini & Tavana 2011, p.375 §3 Step 1

2. 2

   Each DM k assigns linguistic performance ratings x̃_ijk to action A_i on criterion C_j.

   $${\tilde{x}}_{ijk}=(x_{ijk}^l,x_{ijk}^p,x_{ijk}^q,x_{ijk}^u),i=\mathrm{1..}m,j=\mathrm{1..}n,k=\mathrm{1..}K$$

   Hatami-Marbini & Tavana 2011, p.377 Step 2

3. 3

   Each DM k assigns linguistic weights w̃_jk to each criterion C_j.

   $${\tilde{w}}_{jk}=(w_{jk}^l,w_{jk}^p,w_{jk}^q,w_{jk}^u),j=\mathrm{1..}n,k=\mathrm{1..}K$$

   Hatami-Marbini & Tavana 2011, p.377 Step 3

4. 4

   Convert linguistic evaluations into trapezoidal fuzzy numbers via the predefined linguistic scale (Tables 1-2, Figs. 4-5).

   $$linguistic\_term\mapsto (l,p,q,u)$$

   Hatami-Marbini & Tavana 2011, p.379 Tables 1-2

5. 5

   Aggregate K DMs' ratings and weights using min-mean-mean-max scheme (Eqs.5-8).

   $${\tilde{x}}_{ij}=(x_{ij}^l,x_{ij}^p,x_{ij}^q,x_{ij}^u),x_{ij}^l=\underset{k}{\min }{x}_{ijk}^l,x_{ij}^p=\frac{1}{K}\sum _k{x}_{ijk}^p,x_{ij}^q=\frac{1}{K}\sum _k{x}_{ijk}^q,x_{ij}^u=\underset{k}{\max }{x}_{ijk}^u;{\tilde{W}}_j=(w_j^l,w_j^p,w_j^q,w_j^u)analogously$$

   Hatami-Marbini & Tavana 2011, p.375-376 Eqs.(5)-(8)

6. 6

   Construct the fuzzy decision matrix Ũ and the fuzzy weight vector W̃.

   $$\tilde{U}=[{\tilde{x}}_{ij}{]}_{m\times n},\tilde{W}=[{\tilde{W}}_1,{\tilde{W}}_2,\dots ,{\tilde{W}}_n]$$

   Hatami-Marbini & Tavana 2011, p.376 Eq.(9)

7. 7

   Linear-scale normalization to a comparable scale. Benefit criteria (Ω_B): r̃_ij = (l/d_j*, p/d_j*, q/d_j*, u/d_j*) with d_j* = max_i x_ij^u. Cost criteria (Ω_C): r̃_ij = (a_j^-/u, a_j^-/q, a_j^-/p, a_j^-/l) with a_j^- = min_i x_ij^l.

   $$j\in {\Omega }_B:{\tilde{r}}_{ij}=(\frac{x_{ij}^l}{d_j^{\ast }},\frac{x_{ij}^p}{d_j^{\ast }},\frac{x_{ij}^q}{d_j^{\ast }},\frac{x_{ij}^u}{d_j^{\ast }}),d_j^{\ast }=\underset{i}{\max }{x}_{ij}^u;j\in {\Omega }_C:{\tilde{r}}_{ij}=(\frac{a_j^{-}}{x_{ij}^u},\frac{a_j^{-}}{x_{ij}^q},\frac{a_j^{-}}{x_{ij}^p},\frac{a_j^{-}}{x_{ij}^l}),a_j^{-}=\underset{i}{\min }{x}_{ij}^l$$

   Hatami-Marbini & Tavana 2011, p.376 Eqs.(10)-(11)

8. 8

   Construct the weighted normalized fuzzy decision matrix Ṽ = [ṽ_ij] with ṽ_ij = W̃_j (·) r̃_ij (fuzzy product).

   $${\tilde{v}}_{ij}=(v_{ij}^l,v_{ij}^p,v_{ij}^q,v_{ij}^u)=(w_j^l{x}_{ij}^l,w_j^p{x}_{ij}^p,w_j^q{x}_{ij}^q,w_j^u{x}_{ij}^u)$$

   Hatami-Marbini & Tavana 2011, p.376 Eq.(12)

9. 9

   For each pair (g, f) and criterion j, compute the Hamming distances d(max(ṽ_gj, ṽ_fj), ṽ_gj) and d(max(ṽ_gj, ṽ_fj), ṽ_fj). Comparison rule: ṽ_gj ≥ ṽ_fj iff d(max, ṽ_fj) ≥ d(max, ṽ_gj).

   $$d(\tilde{A},\tilde{B})={\int }_R|{\mu }_{\tilde{A}}(x)-{\mu }_{\tilde{B}}(x)|dx$$

   Hatami-Marbini & Tavana 2011, p.375 Eq.(2) (Def.3), p.376 Step (Hamming comparison)

10. 10

    Construct the concordance matrix C̃. For each ordered pair (g, f), J_C = {j : ṽ_gj ≥ ṽ_fj} and c̃_gf = Σ_{j∈J_C} W̃_j (fuzzy sum of weights in the concordance set).

    $${\tilde{c}}_{gf}=(c_{gf}^l,c_{gf}^p,c_{gf}^q,c_{gf}^u)=(\sum _{j\in J_C}{w}_j^l,\sum _{j\in J_C}{w}_j^p,\sum _{j\in J_C}{w}_j^q,\sum _{j\in J_C}{w}_j^u)$$

    Hatami-Marbini & Tavana 2011, p.376 Eqs.(13)-(14)

11. 11

    Construct the discordance matrix D. d_gf is the ratio of the maximum Hamming distance over discordance criteria to the maximum Hamming distance over all criteria (yielding a crisp scalar in [0,1]).

    $$d_{gf}=\frac{{\max }_{j\in J_D}d(\max ({\tilde{v}}_{gj},{\tilde{v}}_{fj}),{\tilde{v}}_{fj})}{{\max }_{j}d(\max ({\tilde{v}}_{gj},{\tilde{v}}_{fj}),{\tilde{v}}_{fj})},J_D=\{j:{\tilde{v}}_{gj}<{\tilde{v}}_{fj}\}$$

    Hatami-Marbini & Tavana 2011, p.376 Eqs.(15)-(16)

12. 12

    Construct Boolean matrix B from the average concordance level C̄ = (c^l, c^p, c^q, c^u). b_gf = 1 iff c̃_gf ≥ C̄ (compared by Hamming distance, since both are TrFNs); else b_gf = 0.

    $$\bar{C}=(c^l,c^p,c^q,c^u),c^{\ast }=\frac{1}{m(m-1)}\sum _{f=1}^m\sum _{g=1}^m{c}_{gf}^{\ast }\text{for }\ast \in \{l,p,q,u\};b_{gf}=1[{\tilde{c}}_{gf}⪰\bar{C}]$$

    Hatami-Marbini & Tavana 2011, p.376-377 Eqs.(17)-(18)

13. 13

    Construct Boolean matrix H from the average discordance level D̄. h_gf = 1 iff d_gf < D̄; else 0.

    $$\bar{D}=\frac{1}{m(m-1)}\sum _{f=1}^m\sum _{g=1}^m{d}_{gf};h_{gf}=1[d_{gf}<\bar{D}]$$

    Hatami-Marbini & Tavana 2011, p.377 Eqs.(19)-(20)

14. 14

    Global matrix Z = B ⊗ H (Hadamard product, z_gf = b_gf · h_gf). Build the outranking decision graph G = (V, J): vertex per alternative; an arc A_g → A_f exists iff z_gf = 1. Pairs with z_gf = z_fg = 1 are indifferent; with z_gf = 0 = z_fg are incomparable.

    $$Z=B\otimes H,z_{gf}=b_{gf}\cdot h_{gf};A_{g}S{A}_f\iff z_{gf}=1$$

    Hatami-Marbini & Tavana 2011, p.377 Eq.(21) and Fig.1

Fits when / Look elsewhere when

Assumptions to verify

• Each linguistic term maps deterministically to a TrFN (Tables 1-2 of seminal)
• DMs are interchangeable (mean aggregation in Eqs.5-8 assumes equal authority)
• Hamming distance is the chosen TrFN dissimilarity (seminal §2 Def.3)

Edge cases and pitfalls

• •Ortalama concordance c̄ ve discordance d̄ eşiklerinin tek seçimi: paper §3.4 default 'ortalama' önerir; karar-verici daha katı outranking için percentile-based eşik (örn. 75th) kullanabilir, bu B matrisini seyrekleştirir.
• •ELECTRE-spesifik özellik: B matrisi simetrik OLABİLİR (b_gf=b_fg=1, A_g ile A_f birbirini outrank ediyor → 'indifferent'); ayrıca b_gf=b_fg=0 'incomparable'. Bu zenginlik TOPSIS/EDAS gibi total-order metoda dönüştürülürse kaybolur.
• •Trapezoidal fuzzy ≥ ilişkisi (J_C tanımı): Hamming distance en-küçük-üst-sınır (LUB) kıyası kullanılır (paper §3, Eq.14 öncesi). α-cut veya possibility-degree alternatif var ama paper Hamming seçer.
• •Grup uzman ağırlıkları eşit olmayabilir; paper §3 K-uzman için eşit ağırlık varsayar (Eq.8 mean), ağırlıklı genişleme (her uzmana λ_k) generik engine'de opsiyonel.
• •Çizge döngüsü (cycle): preferred-küme çıkarmak için strongly-connected component analizi gerekebilir; jenerik engine bunu raporlar (CC sayısı) ama final ranking için extra adım ister.

• Do not defuzzify the trapezoidal weights or matrix entries before Step 9. The Hamming-distance-based comparison in Eq.(2) requires the full membership function; centroid defuzzification destroys the asymmetry that drives the outranking decision.

• ELECTRE I is a choice method, not a complete ranking method. Incomparability is a feature, not a bug - it correctly signals that the imprecise data does not justify a strict ordering between certain pairs.

• Treating fuzzy weights as a probability simplex (Σ w_j = 1) is incorrect here. The concordance index sums TrFNs as fuzzy magnitudes per Eq.(14); the seminal weights in Table 5 do not sum to one.

• Aytaç et al. 2011 simplifies to triangular fuzzy numbers (3-tuple). This is a legitimate degenerate case (p = q) but loses one degree of freedom of the trapezoidal model. Stay with TrFN for the canonical algorithm.

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