Outranking

Fuzzy PROMETHEE: Fuzzy extension of PROMETHEE

Goumas, M., Lygerou, V. · 2000

Overview

Fuzzy outranking/ranking: Triangular Fuzzy Number (TFN: l, m, u). Output typically preference_flow (higher value = preferred).

Strengths

•Method-specific: Fuzzy outranking/ranking: Triangular Fuzzy Number (TFN: l, m, u)
•Preserves fuzzy_TFN uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

•Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
•Assumes: Thresholds (q, p, v) can be expressed in Fuzzy (Triangular) scale
•Assumes: Veto + concordance semantics adapted to fuzzy arithmetic

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

Edge cases

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

Common pitfalls

•Hatalı: 'FUZZY-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Thresholds (q, p, v) can be expressed in Fuzzy (Triangular) scale
•Hatalı: 'FUZZY-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Veto + concordance semantics adapted to fuzzy arithmetic
•Hatalı: FUZZY-PROMETHEE'yi 'Small dataset (m<3)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1.Adım 1 (F1): Step 1: Choose preference function P_j and parameters (q,p) per criterion (d_j direction-aware). Formül: d_{j}(a,b) = \begin{cases} x_{aj}-x_{bj} & j\in J^{+} \\ x_{bj}-x_{aj} & j\in J^{-} \end{cases};\quad P_{j}(d): \mathbb{R}\to[0,1] Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (preference function definition)
2.Adım 2 (F2): Step 2: Pairwise preference π(a,b) = Σ w_j P_j(d_j(a,b)). Formül: \pi(a,b) = \sum_{j=1}^n w_j \cdot P_j(\tilde{x}_{aj} \ominus \tilde{x}_{bj}) \text{ (Fuzzy Strict Preference)} Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (pairwise preference index)
3.Adım 3 (F3): Step 3: Positive flow φ^+(a) = (1/(m-1)) Σ_b π(a,b). Formül: \phi^{+}(a) = \dfrac{1}{m-1} \sum_{b\neq a} \pi(a,b) Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (positive flow)
4.Adım 4 (F4): Step 4: Negative flow φ^−(a) = (1/(m-1)) Σ_b π(b,a). Formül: \phi^{-}(a) = \dfrac{1}{m-1} \sum_{b\neq a} \pi(b,a) Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (negative flow)
5.Adım 5 (F5): Step 5: Net flow φ(a) = φ^+(a) − φ^−(a); descending complete ranking (PROMETHEE II). Formül: \phi(a) = \phi^{+}(a) - \phi^{-}(a) Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (net flow & ranking)

Commonly paired with

•FUZZY-AHP + FUZZY-PROMETHEE (common)

How to cite

Goumas, M.; Lygerou, V. (2000). An extension of the PROMETHEE method for decision making in fuzzy environment: Ranking of alternative energy exploitation projects. European Journal of Operational Research. https://doi.org/10.1016/S0377-2217(99)00093-4

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Common pitfalls

•Hatalı: 'FUZZY-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Thresholds (q, p, v) can be expressed in Fuzzy (Triangular) scale

•Hatalı: 'FUZZY-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Veto + concordance semantics adapted to fuzzy arithmetic

•Hatalı: FUZZY-PROMETHEE'yi 'Small dataset (m<3)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1.Adım 1 (F1): Step 1: Choose preference function P_j and parameters (q,p) per criterion (d_j direction-aware). Formül: d_{j}(a,b) = \begin{cases} x_{aj}-x_{bj} & j\in J^{+} \\ x_{bj}-x_{aj} & j\in J^{-} \end{cases};\quad P_{j}(d): \mathbb{R}\to[0,1] Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (preference function definition)

2.Adım 2 (F2): Step 2: Pairwise preference π(a,b) = Σ w_j P_j(d_j(a,b)). Formül: \pi(a,b) = \sum_{j=1}^n w_j \cdot P_j(\tilde{x}_{aj} \ominus \tilde{x}_{bj}) \text{ (Fuzzy Strict Preference)} Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (pairwise preference index)

3.Adım 3 (F3): Step 3: Positive flow φ^+(a) = (1/(m-1)) Σ_b π(a,b). Formül: \phi^{+}(a) = \dfrac{1}{m-1} \sum_{b\neq a} \pi(a,b) Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (positive flow)

4.Adım 4 (F4): Step 4: Negative flow φ^−(a) = (1/(m-1)) Σ_b π(b,a). Formül: \phi^{-}(a) = \dfrac{1}{m-1} \sum_{b\neq a} \pi(b,a) Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (negative flow)

5.Adım 5 (F5): Step 5: Net flow φ(a) = φ^+(a) − φ^−(a); descending complete ranking (PROMETHEE II). Formül: \phi(a) = \phi^{+}(a) - \phi^{-}(a) Anchor: Goumas-Lygerou 2000, EJOR 123:606-613 (net flow & ranking)