FF-PROMETHEE: 2-tuple Linguistic Fermatean Fuzzy PROMETHEE (Akram-Bibi 2023)

Overview

Fermatean fuzzy outranking: 2-tuple linguistic Fermatean fuzzy sets (2TLFFS), μ³+ν³ ≤ 1. Output typically preference_flow (higher value = preferred).

Strengths

• •Method-specific: Fermatean fuzzy outranking: 2-tuple linguistic Fermatean fuzzy sets (2TLFFS), μ³+ν³ ≤ 1
• •Preserves q_rung_orthopair 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 PROMETHEE; cf. Brans-Mareschal 2005 Springer Ch.5)
• •Assumes: All decision-maker linguistic assessments decode to 2TLFFN satisfying μ³+ν³ ≤ 1
• •Assumes: Decision-maker weight vector ϖ is on the simplex
• •Assumes: Choice of preference function (Gaussian/Usual/…) is justified for each criterion's nature

Assumptions to verify

• •All decision-maker linguistic assessments decode to 2TLFFN satisfying μ³+ν³ ≤ 1
• •Decision-maker weight vector ϖ is on the simplex
• •Choice of preference function (Gaussian/Usual/…) is justified for each criterion's nature

When not to use

• •Single decision-maker with crisp judgments: use crisp PROMETHEE instead
• •Hesitancy fits within intuitionistic constraint μ+ν ≤ 1: IF-PROMETHEE may be more parsimonious
• •Small dataset (m<3): outranking machinery underutilised

Edge cases

• •Default Type V Gaussian with σ=0.5 (§4 Methodology 1); Type I 'Usual' (§4 Methodology 2) and four other Brans-Vincke 1985 generalised forms are admissible.

Common pitfalls

• •Hatalı: 'FF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker linguistic assessments decode to 2TLFFN satisfying μ³+ν³ ≤ 1
• •Hatalı: 'FF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-maker weight vector ϖ is on the simplex
• •Hatalı: 'FF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Choice of preference function (Gaussian/Usual/…) is justified for each criterion's nature
• •Hatalı: FF-PROMETHEE'yi 'Single decision-maker with crisp judgments' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FF-PROMETHEE'yi 'Hesitancy fits within intuitionistic constraint μ+ν ≤ 1' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FF-PROMETHEE'yi 'Small dataset (m<3)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1-2 (Akram-Bibi 2023 §3): Experts express judgments via linguistic terms from Table 2 (9-term EP→EG scale); convert to 2TLFFNs c^{(r)}_{pq} = ((β_l, L), (β_m, M)). One D^(r) matrix per DM. Formül: c^{(r)}_{pq} = ((β_l^{(r)}, L_{pq}^{(r)}), (β_m^{(r)}, M_{pq}^{(r)})), p=1..i, q=1..j, r=1..k Anchor: Akram-Bibi 2023 §3 Steps 1-2; Table 2 linguistic scale
2. 2.Adım 2 (F2): Step 3 (Akram-Bibi 2023 §3): Aggregate the k per-DM matrices D^(r) into a single 2TLFFDM D = (c_{pq})_{ij} using the 2TLFFWA operator with DM weight vector ϖ = (ϖ_1, …, ϖ_k). Formül: c_{pq} = 2TLFFWA(c^{(1)}_{pq}, …, c^{(k)}_{pq}) = Δ( τ · ∛(1 − ∏_{r=1}^{k} (1 − (Δ^{-1}(β_l^{(r)},L^{(r)})/τ)^3)^{ϖ_r}) , τ · ∏_{r=1}^{k} (Δ^{-1}(β_m^{(r)},M^{(r)})/τ)^{ϖ_r} ): Akram-Bibi 2023 Eq.(3) Anchor: Akram-Bibi 2023 §3 Step 3, Eq.(3); Table 8 example
3. 3.Adım 3 (F3): Step 4 (Akram-Bibi 2023 §3): Per-DM criterion weight 2TLFFNs ω^{(r)}_q are aggregated via 2TLFFWA Eq.(4) to obtain the aggregated criterion weight vector x = (x_1, …, x_n) (entries still 2TLFFN; defuzzified before Step 8 weighting). Formül: x_q = 2TLFFWA(ω^{(1)}_q, …, ω^{(k)}_q) per Akram-Bibi 2023 Eq.(4); defuzzify via S(x_q) before use in preference index Eq.(6) Anchor: Akram-Bibi 2023 §3 Step 4, Eq.(4); Tables 9-10 example
4. 4.Adım 4 (F4): Step 5 (Akram-Bibi 2023 §3): Score function S(F) maps each aggregated 2TLFFN c_{pq} to a 2-tuple linguistic score using Eq.(1). Formül: S_{pq} = S(c_{pq}) = Δ( (τ/2) · (1 + (Δ^{-1}(β_l, L)/τ)^3 − (Δ^{-1}(β_m, M)/τ)^3 ) ): Akram-Bibi 2023 Eq.(1) Anchor: Akram-Bibi 2023 §2 Definition 5 Eq.(1); Table 11 example
5. 5.Adım 5 (F5): Step 6 (Akram-Bibi 2023 §3): Pairwise deviation D_q(T_g, T_l) = Δ^{-1}(S_q(T_g)) − Δ^{-1}(S_q(T_l)) for each criterion q and each ordered pair (g,l). Formül: D_q(T_g, T_l) = Δ^{-1}(S_q(T_g)) − Δ^{-1}(S_q(T_l)): Akram-Bibi 2023 Eq.(5) Anchor: Akram-Bibi 2023 §3 Step 6, Eq.(5); Table 12 example
6. 6.Adım 6 (F6): Step 7 (Akram-Bibi 2023 §3): Apply a generalised preference function P_q(T_g, T_l) = F(D_q(T_g, T_l)). For benefit criteria, P_q ≥ 0 when D_q > 0; for cost criteria, sign is reversed. Default Type V Gaussian with σ=0.5 (§4 Methodology 1); Type I 'Usual' (§4 Methodology 2) and four other Brans-Vincke 1985 generalised forms are admissible. Formül: Gaussian (Type V): P_q(T_g, T_l) = 1 − exp(−D_q² / (2σ²)) if D_q > 0 else 0. Usual (Type I): P_q = 1 if D_q > 0 else 0. Anchor: Akram-Bibi 2023 §3 Step 7 + §2 Definitions 10-11; Brans-Vincke 1985 Table 1; Akram-Bibi Table 13 example
7. 7.Adım 7 (F7): Step 8 (Akram-Bibi 2023 §3): Multi-criteria preference index H(T_g, T_l) = ⊕_{q=1}^{j} x_q ⊗ P_q(T_g, T_l), the weighted 2TLFFN aggregation of per-criterion preferences with criterion weight vector x. Formül: H(T_g, T_l) = ⊕_{q=1}^{j} x_q ⊗ P_q(T_g, T_l) = ((β_{l,H}, L_H), (β_{m,H}, M_H)): Akram-Bibi 2023 Eq.(6) Anchor: Akram-Bibi 2023 §3 Step 8, Eq.(6); Tables 14, 18 examples
8. 8.Adım 8 (F8): Step 9-i (Akram-Bibi 2023 §3): Leaving (positive) outranking flow Φ⁺(T_g) = ⊕_{l=1,l≠g}^{i} H(T_g, T_l) and entering (negative) outranking flow Φ⁻(T_g) = ⊕_{l=1,l≠g}^{i} H(T_l, T_g). PROMETHEE I partial ranking from intersection of P⁺ and P⁻ orders via Eq.(9-11). Formül: Φ⁺(T_g) = ⊕_{l≠g} H(T_g, T_l); Φ⁻(T_g) = ⊕_{l≠g} H(T_l, T_g): Akram-Bibi 2023 Eqs.(7-8) Anchor: Akram-Bibi 2023 §3 Step 9(i), Eqs.(7-11); Table 15 example
9. 9.Adım 9 (F9): Step 9-ii (Akram-Bibi 2023 §3): PROMETHEE II net outranking flow Φ(T_g) = Δ^{-1}(S(Φ⁺(T_g))) − Δ^{-1}(S(Φ⁻(T_g))); complete (linear) ranking in descending Φ. Formül: Φ(T_g) = Δ^{-1}(S(Φ⁺(T_g))) − Δ^{-1}(S(Φ⁻(T_g))): Akram-Bibi 2023 Eq.(12); rank by descending Φ: Eq.(13) Anchor: Akram-Bibi 2023 §3 Step 9(ii), Eqs.(12-13); Table 16 example

Commonly paired with

• •endogenous_2TLFFWA + FF-PROMETHEE (primary)

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