ELECTRE-TRI: Outranking-Based Sorting Method with Boundary Profiles (also ELECTRE TRI-B)

Overview

Outranking-based boundary-profile sorting. Output typically class_label (higher value = preferred).

Strengths

• •Method-specific: Outranking-based boundary-profile sorting

Limitations

• •Assumes: Pre-defined ordered classes/categories exist
• •Assumes: Reference profiles or examples are available

Assumptions to verify

• •Pre-defined ordered classes/categories exist
• •Reference profiles or examples are available

When not to use

• •No predefined classes → use a ranking method first

Edge cases

• •See F.steps and D.parameters for ELECTRE-TRI-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'ELECTRE-TRI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Pre-defined ordered classes/categories exist
• •Hatalı: 'ELECTRE-TRI bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Reference profiles or examples are available
• •Hatalı: ELECTRE-TRI'yi 'No predefined classes → use a ranking method first' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Define p boundary profiles {b_1,…,b_p} and the resulting p+1 ordered categories C_1 ≺ … ≺ C_{p+1}. Formül: C_{h+1} \succ C_h,\ h=1,\ldots,p Anchor: Mousseau-Slowinski-Zielniewicz 2000, Section 2.1
2. 2.Adım 2 (F2): Step 2: Compute partial concordance c_j(a, b_h) per criterion using indifference q_j and preference p_j thresholds (direction-aware via Δ_j). Formül: \Delta_{j}(a,b_h) = \begin{cases} g_j(b_h) - g_j(a) & j\in J^{+} \\ g_j(a) - g_j(b_h) & j\in J^{-} \end{cases};\quad c_j(a, b_h) = \begin{cases} 0, & \Delta_{j}(a,b_h) \ge p_j \\ 1, & \Delta_{j}(a,b_h) \le q_j \\ \dfrac{p_j - \Delta_{j}(a,b_h)}{p_j - q_j}, & \text{otherwise} \end{cases} Anchor: Mousseau et al. 2000, Eq.(2)
3. 3.Adım 3 (F3): Step 3: Compute the comprehensive concordance index c(a, b_h) = Σ w_j c_j(a, b_h) / Σ w_j. Formül: c(a, b_h) = \dfrac{\sum_{j} w_j\, c_j(a, b_h)}{\sum_{j} w_j} Anchor: Mousseau et al. 2000, Eq.(3)
4. 4.Adım 4 (F4): Step 4: Compute partial discordance d_j(a, b_h) per criterion using preference p_j and veto v_j thresholds (direction-aware via Δ_j). Formül: d_j(a, b_h) = \begin{cases} 1, & \Delta_{j}(a,b_h) \ge v_j \\ 0, & \Delta_{j}(a,b_h) \le p_j \\ \dfrac{\Delta_{j}(a,b_h) - p_j}{v_j - p_j}, & \text{otherwise} \end{cases},\quad \Delta_{j}(a,b_h) = \begin{cases} g_j(b_h) - g_j(a) & j\in J^{+} \\ g_j(a) - g_j(b_h) & j\in J^{-} \end{cases} Anchor: Mousseau et al. 2000, Eq.(4)
5. 5.Adım 5 (F5): Step 5: Compute the credibility (outranking) degree σ(a, b_h) combining concordance and discordance via the veto product. Formül: \sigma(a, b_h) = c(a, b_h) \cdot \prod_{j \in \overline{F}(a, b_h)} \dfrac{1 - d_j(a, b_h)}{1 - c(a, b_h)},\ \overline{F} = \{j : d_j(a, b_h) > c(a, b_h)\} Anchor: Mousseau et al. 2000, Eq.(5)
6. 6.Adım 6 (F6): Step 6: Pessimistic (conjunctive) assignment: scan h = p, p−1, …, 1; the first h where σ(a, b_h) ≥ λ assigns a to C_{h+1}; else assign to C_1. Formül: a \to C_{h+1}\ \text{where}\ h = \max\{h \,:\, \sigma(a, b_h) \ge \lambda\},\ \text{else}\ a \to C_1 Anchor: Mousseau et al. 2000, Section 2.2 (Pessimistic procedure)
7. 7.Adım 7 (F7): Step 7: Optimistic (disjunctive) assignment: scan h = 1, 2, …, p; the first h where b_h strictly outranks a (σ(b_h, a) ≥ λ AND NOT σ(a, b_h) ≥ λ) assigns a to C_h; else assign to C_{p+1}. Formül: a \to C_h\ \text{where}\ h = \min\{h \,:\, b_h \succ a\},\ \text{else}\ a \to C_{p+1} Anchor: Mousseau et al. 2000, Section 2.2 (Optimistic procedure)

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