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

Overview

FF-PROMETHEE (specifically 2TLFF-PROMETHEE per Akram-Bibi 2023) extends crisp PROMETHEE to MAGDM under Fermatean fuzzy uncertainty encoded as 2-tuple linguistic terms. Use when (i) multiple experts express criterion-by-criterion judgments in linguistic terms, (ii) hesitancy is high enough that μ²+ν² ≤ 1 (intuitionistic) or μ²+ν² > 1 but μ³+ν³ ≤ 1 (Fermatean) is needed, and (iii) outranking-style ranking is preferred over compensatory aggregation. The Gaussian preference function (σ ≈ 0.5) is the paper's default. Run Methodology 1 (Gaussian) and Methodology 2 (Usual) in parallel for robustness - paper Section 5 reports identical top alternative under both.

Output

preference flow, higher is better

Data

Fermatean Fuzzy, linguistic 2TLFFN tuples complete

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-10 criteria works best

Used for

Multi-attribute group decision-making under linguistic uncertainty, Personnel/manager selection (Akram-Bibi 2023 bank manager example), Supplier selection with expert linguistic judgments, Healthcare / engineering MAGDM with Fermatean-strength hesitancy

How it works

1. 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.

   $$c_{pq}^{(r)}=(({\beta }_l^{(r)},L_{pq}^{(r)}),({\beta }_m^{(r)},M_{pq}^{(r)})),p=\mathrm{1..}i,q=\mathrm{1..}j,r=\mathrm{1..}k$$

   Akram-Bibi 2023 §3 Steps 1-2; Table 2 linguistic scale

2. 2

   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).

   $$c_{pq}=2TLFFWA(c_{pq}^{(1)},\dots ,c_{pq}^{(k)})=\Delta (\tau \cdot ∛(1-\prod _{r=1}^k(1-({\Delta }^{-1}({\beta }_l^{(r)},L^{(r)})/\tau {)}^3{)}^{{\varpi }_r}),\tau \cdot \prod _{r=1}^k({\Delta }^{-1}({\beta }_m^{(r)},M^{(r)})/\tau {)}^{{\varpi }_r})-Akram-Bibi2023Eq.(3)$$

   Akram-Bibi 2023 §3 Step 3, Eq.(3); Table 8 example

3. 3

   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).

   $$x_q=2TLFFWA({\omega }_q^{(1)},\dots ,{\omega }_q^{(k)})perAkram-Bibi2023Eq.(4);defuzzifyviaS(x_q)beforeuseinpreferenceindexEq.(6)$$

   Akram-Bibi 2023 §3 Step 4, Eq.(4); Tables 9-10 example

4. 4

   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).

   $$S_{pq}=S(c_{pq})=\Delta ((\tau /2)\cdot (1+({\Delta }^{-1}({\beta }_l,L)/\tau {)}^3-({\Delta }^{-1}({\beta }_m,M)/\tau {)}^3))-Akram-Bibi2023Eq.(1)$$

   Akram-Bibi 2023 §2 Definition 5 Eq.(1); Table 11 example

5. 5

   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).

   $$D_q(T_g,T_l)={\Delta }^{-1}(S_q(T_g))-{\Delta }^{-1}(S_q(T_l))-Akram-Bibi2023Eq.(5)$$

   Akram-Bibi 2023 §3 Step 6, Eq.(5); Table 12 example

6. 6

   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.

   $$Gaussian(TypeV):P_q(T_g,T_l)=1-exp(-D_q^2/(2{\sigma }^2))if{D}_q>0else0.Usual(TypeI):P_q=1if{D}_q>0else0.$$

   Akram-Bibi 2023 §3 Step 7 + §2 Definitions 10-11; Brans-Vincke 1985 Table 1; Akram-Bibi Table 13 example

7. 7

   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.

   $$H(T_g,T_l)={\oplus }_{q=1}^j{x}_q\otimes P_q(T_g,T_l)=(({\beta }_{l,H},L_H),({\beta }_{m,H},M_H))-Akram-Bibi2023Eq.(6)$$

   Akram-Bibi 2023 §3 Step 8, Eq.(6); Tables 14, 18 examples

8. 8

   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).

   $${\Phi }^{+}(T_g)={\oplus }_{l\ne g}H(T_g,T_l);{\Phi }^{-}(T_g)={\oplus }_{l\ne g}H(T_l,T_g)-Akram-Bibi2023Eqs.(7-8)$$

   Akram-Bibi 2023 §3 Step 9(i), Eqs.(7-11); Table 15 example

9. 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.

   $$\Phi (T_g)={\Delta }^{-1}(S({\Phi }^{+}(T_g)))-{\Delta }^{-1}(S({\Phi }^{-}(T_g)))-Akram-Bibi2023Eq.(12);rankbydescending\Phi -Eq.(13)$$

   Akram-Bibi 2023 §3 Step 9(ii), Eqs.(12-13); Table 16 example

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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.

• Value-space violation: ensure all 2TLFFN entries satisfy μ³+ν³ ≤ 1 after aggregation; the 2TLFFWA operator (Eq.(3)) preserves the constraint but only when inputs do.

• Preference function choice changes intermediate Φ⁺/Φ⁻ values but not necessarily the top-1 alternative - paper §5 confirms Gaussian and Usual yield identical top alternative T5. For criteria with sharp indifference/preference thresholds, Type II (U-shape) or Type III (V-shape) may be more faithful than Gaussian.

• Cost-criteria sign convention: P_q(T_g, T_l) = F(−D_q(T_g, T_l)) for cost criteria (Akram-Bibi 2023 §3 Step 7); applying F to D_q directly without sign flip inverts cost-criterion preferences.

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