ELECTRE I: ELimination Et Choix Traduisant la REalité

Overview

Concordance-discordance (crisp outranking). Output typically preference_flow (higher value = preferred).

Strengths

• •Method-specific: Concordance-discordance (crisp outranking)

Limitations

• •Rank reversal known on alternative-set changes (ref: general MCDM literature)
• •Assumes: Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
• •Assumes: Non-compensatory preference structure

Assumptions to verify

• •Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
• •Non-compensatory preference structure

When not to use

• •Small dataset (m<3) → outranking benefit minimal
• •Compensatory preferences acceptable → simpler ranking method

Edge cases

• •if c_kl ≥ f̄; G_kl=1 if d_kl ≤ d̄ (A_k not strongly opposed on any criterion).
• •If the kernel is a singleton, that is the best alternative; otherwise a partial preorder is obtained.

Common pitfalls

• •Hatalı: 'ELECTRE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-maker can specify preference (p), indifference (q), and veto (v) thresholds
• •Hatalı: 'ELECTRE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Non-compensatory preference structure
• •Hatalı: ELECTRE'yi 'Small dataset (m<3) → outranking benefit minimal' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: ELECTRE'yi 'Compensatory preferences acceptable → simpler ranking method' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Euclidean normalisation: transform each criterion column to a dimensionless unit vector. Enables comparison across criteria with different units. Formül: x_{ij} = \frac{a_{ij}}{\sqrt{\sum_{k=1}^{m} a_{kj}^{2}}} Anchor: Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-6), p.14
2. 2.Adım 2 (F2): Step 2: Weight the normalised matrix: multiply each column of X by the corresponding criterion weight w_j to obtain the weighted normalised matrix Y. Formül: Y = X \cdot W,\quad y_{ij} = w_{j}\, x_{ij},\quad W = \operatorname{diag}(w_{1},\ldots,w_{n}),\quad \sum_{j}w_{j}=1 Anchor: Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.15
3. 3.Adım 3 (F3): Step 3: Determine concordance set C_kl and discordance set D_kl for each ordered pair (A_k, A_l): direction-aware via Δ_j(k,l); C_kl contains criteria where A_k is at least as good as A_l; D_kl is the complement. Formül: \Delta_{j}(k,l) = \begin{cases} y_{kj}-y_{lj} & j\in J^{+} \\ y_{lj}-y_{kj} & j\in J^{-} \end{cases};\quad C_{kl} = \{j : \Delta_{j}(k,l) \ge 0\},\quad D_{kl} = \{j : \Delta_{j}(k,l) < 0\},\quad C_{kl} \cup D_{kl} = \{1,\ldots,n\} Anchor: Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.15
4. 4.Adım 4 (F4): Step 4: Construct m×m concordance matrix C (index c_kl = sum of weights in C_kl) and discordance matrix D (index d_kl = max weighted-normalised gap where A_l outperforms A_k, normalised by the global max gap). Diagonal entries undefined. Formül: c_{kl} = \sum_{j \in C_{kl}} w_{j},\qquad d_{kl} = \frac{\max_{j \in D_{kl}} |y_{kj} - y_{lj}|}{\max_{j} |y_{kj} - y_{lj}|} Anchor: Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-7), p.16
5. 5.Adım 5 (F5): Step 5: Determine concordance dominance matrix F and discordance dominance matrix G using threshold values. Threshold f̄ = average c_kl; threshold d̄ = average d_kl (or user-supplied). F_kl=1 if c_kl ≥ f̄; G_kl=1 if d_kl ≤ d̄ (A_k not strongly opposed on any criterion). Formül: \bar{f} = \frac{1}{m(m-1)}\sum_{k \neq l} c_{kl},\quad F_{kl} = \mathbf{1}[c_{kl} \geq \bar{f}];\qquad \bar{d} = \frac{1}{m(m-1)}\sum_{k \neq l} d_{kl},\quad G_{kl} = \mathbf{1}[d_{kl} \leq \bar{d}] Anchor: Triantaphyllou 2000 Ch.2 §2.2.5 Eqs.(2-8)(2-9), p.17
6. 6.Adım 6 (F6): Step 6: Compute aggregate dominance matrix E = F ⊗ G (element-wise product). E_kl = 1 means A_k dominates A_l on both concordance and discordance grounds simultaneously. Formül: E_{kl} = F_{kl} \cdot G_{kl} \in \{0,1\} Anchor: Triantaphyllou 2000 Ch.2 §2.2.5 Eq.(2-10), p.17-18
7. 7.Adım 7 (F7): Step 7: Eliminate less favorable alternatives. Any alternative A_l whose column in E contains at least one 1 is dominated. The kernel (choice set) = alternatives with no 1 in their column. If the kernel is a singleton, that is the best alternative; otherwise a partial preorder is obtained. Formül: \text{Dominated} = \{A_l : \exists k,\, E_{kl}=1\},\quad \text{Kernel} = A \setminus \text{Dominated} Anchor: Benayoun et al. 1966; Triantaphyllou 2000 Ch.2 §2.2.5 p.18

Commonly paired with

• •AHP + ELECTRE (high)
• •BWM + ELECTRE (high)
• •ENTROPY + ELECTRE (high)
• •CRITIC + ELECTRE (high)
• •SWARA + ELECTRE (high)

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