MPF-ELECTRE-I: m-Polar Fuzzy extension of ELECTRE-I (Akram, Waseem & Liu 2019)

Overview

Outranking: m-PF concordance/discordance with aggregate dominance graph. Output typically outranking_graph (lower value = preferred).

Strengths

• •Method-specific: Outranking: m-PF concordance/discordance with aggregate dominance graph
• •Preserves m_polar uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: ELECTRE-family-Roy-1968)
• •Assumes: Each decision cell is a true m-polar tuple in [0,1]^m (NOT a scalar fuzzy number).
• •Assumes: Number of poles m is uniform across all cells.
• •Assumes: Criterion weights w_j sum to 1 and are externally supplied (MPF-ELECTRE-I consumes weights, does not produce them).
• •Assumes: User accepts graph-valued output (incomparability and indifference are first-class outcomes; no forced linear ranking).

Assumptions to verify

• •Each decision cell is a true m-polar tuple in [0,1]^m (NOT a scalar fuzzy number).
• •Number of poles m is uniform across all cells.
• •Criterion weights w_j sum to 1 and are externally supplied (MPF-ELECTRE-I consumes weights, does not produce them).
• •User accepts graph-valued output (incomparability and indifference are first-class outcomes; no forced linear ranking).

When not to use

• •User requires a complete linear ranking with scalar scores (use MPF-TOPSIS-LING or MPF-HF-TOPSIS instead).
• •Number of alternatives r < 3 (outranking graph degenerates).
• •Decision cells are scalar / interval / single-pole fuzzy (use crisp ELECTRE-I or a different fuzzy ELECTRE variant).

Edge cases

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

Common pitfalls

• •Hatalı: 'MPF-ELECTRE-I bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each decision cell is a true m-polar tuple in [0,1]^m (NOT a scalar fuzzy number).
• •Hatalı: 'MPF-ELECTRE-I bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Number of poles m is uniform across all cells.
• •Hatalı: 'MPF-ELECTRE-I bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights w_j sum to 1 and are externally supplied (MPF-ELECTRE-I consumes weights, does not produce them).
• •Hatalı: 'MPF-ELECTRE-I bu varsayımı kontrol etmeden uygulamak'. Doğrusu: User accepts graph-valued output (incomparability and indifference are first-class outcomes; no forced linear ranking).
• •Hatalı: MPF-ELECTRE-I'yi 'User requires a complete linear ranking with scalar scores (use MPF-TOPSIS-LING or MPF-HF-TOPSIS instead).' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-I'yi 'Number of alternatives r < 3 (outranking graph degenerates).' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-I'yi 'Decision cells are scalar / interval / single-pole fuzzy (use crisp ELECTRE-I or a different fuzzy ELECTRE variant).' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Form the m-PF decision matrix Z=(z_{ij}), where z_{ij}=(z_{ij}^1, z_{ij}^2, …, z_{ij}^m). Formül: Z=(z_{ij}),\;\; z_{ij}=(z_{ij}^{1}, z_{ij}^{2}, \ldots, z_{ij}^{m}) Anchor: Akram-Adeel 2023, §2.3 Step 1 (p.90)
2. 2.Adım 2 (F2): Step 2: Weighted m-PF decision matrix Y=(y_{ij}), y_{ij}=w_j·z_{ij}=(w_j z_{ij}^1, …, w_j z_{ij}^m). Formül: Y=(y_{ij})=(y_{ij}^{1}, y_{ij}^{2}, \ldots, y_{ij}^{m}),\;\; y_{ij}^{k}=w_{j}\,z_{ij}^{k} Anchor: Akram-Adeel 2023, §2.3 Step 3 (p.90)
3. 3.Adım 3 (F3): Step 3: m-PF concordance set F_{pq}={j : v_{pj} ≥ v_{qj}}, where v_{ij}=Σ_{k=1}^m y_{ij}^k aggregates the m poles of the weighted cell. Formül: F_{pq}=\{1\le j\le n : v_{pj}\ge v_{qj},\; p\ne q\},\;\; v_{ij}=\sum_{k=1}^{m} y_{ij}^{k} Anchor: Akram-Adeel 2023, §2.3 Step 4 (p.90)
4. 4.Adım 4 (F4): Step 4: m-PF discordance set G_{pq}={j : v_{pj} ≤ v_{qj}}. Formül: G_{pq}=\{1\le j\le n : v_{pj}\le v_{qj},\; p\ne q\} Anchor: Akram-Adeel 2023, §2.3 Step 5 (p.90)
5. 5.Adım 5 (F5): Step 5: m-PF concordance indices f_{pq}=Σ_{j∈F_{pq}} w_j and concordance matrix F. Formül: f_{pq}=\sum_{j\in F_{pq}} w_{j},\;\; F=[f_{pq}]_{r\times r} Anchor: Akram-Adeel 2023, §2.3 Steps 6-7 (p.90)
6. 6.Adım 6 (F6): Step 6: m-PF discordance indices g_{pq} (normalised Euclidean-distance ratio over poles) and discordance matrix G. Formül: g_{pq}=\dfrac{\displaystyle\max_{j\in G_{pq}}\sqrt{\tfrac{1}{m}\sum_{k=1}^{m}(y_{pj}^{k}-y_{qj}^{k})^{2}}}{\displaystyle\max_{j}\sqrt{\tfrac{1}{m}\sum_{k=1}^{m}(y_{pj}^{k}-y_{qj}^{k})^{2}}},\;\; G=[g_{pq}]_{r\times r} Anchor: Akram-Adeel 2023, §2.3 Steps 8-9 (p.91)
7. 7.Adım 7 (F7): Step 7: Concordance level f̄ and discordance level ḡ (averages of off-diagonal F and G indices). Formül: \bar f=\dfrac{1}{r(r-1)}\sum_{p=1}^{r}\sum_{\substack{q=1\\q\ne p}}^{r} f_{pq},\;\; \bar g=\dfrac{1}{r(r-1)}\sum_{p=1}^{r}\sum_{\substack{q=1\\q\ne p}}^{r} g_{pq} Anchor: Akram-Adeel 2023, §2.3 Step 10 (p.91)
8. 8.Adım 8 (F8): Step 8: Concordance dominance H and discordance dominance L (boolean threshold matrices). Formül: h_{pq}=\mathbb{1}[f_{pq}\ge \bar f],\;\; l_{pq}=\mathbb{1}[g_{pq}<\bar g] Anchor: Akram-Adeel 2023, §2.3 Step 11 (p.91)
9. 9.Adım 9 (F9): Step 9: Aggregate dominance matrix M = H ⊙ L (peer-to-peer / Hadamard product). Formül: M=H\odot L,\;\; m_{pq}=h_{pq}\cdot l_{pq} Anchor: Akram-Adeel 2023, §2.3 Step 12 (p.92)
10. 10.Adım 10 (F10): Step 10: Build outranking directed graph: edge p→q iff m_{pq}=1. Read off preferences (unique edge ⇒ x_p ≻ x_q; both edges ⇒ indifferent; no edge ⇒ incomparable). Formül: p\to q \iff m_{pq}=1 Anchor: Akram-Adeel 2023, §2.3 Step 13 (p.92)

Commonly paired with

• •AHP + MPF-ELECTRE-I (common)
• •ENTROPY + MPF-ELECTRE-I (occasional)

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