MPF-PROMETHEE: m-Polar Fuzzy extension of PROMETHEE I/II with AHP-derived crisp weights, six Brans-Vincke generalized criteria preference functions, and positive/negative/net outranking flows (Akram, Shumaiza & Alcantud 2020 / Akram & Adeel 2023 Ch. 7)

Overview

Outranking-flow PROMETHEE: six generalized criteria preference functions (I-VI), AHP-assisted scalar weights, positive/negative/net outranking flows, PROMETHEE I partial pre-order + PROMETHEE II complete order. Output typically net_outranking_flow_χ (higher value = preferred).

Strengths

• •Method-specific: Outranking-flow PROMETHEE: six generalized criteria preference functions (I-VI), AHP-assisted scalar weights, positive/negative/net outranking flows, PROMETHEE I partial pre-order + PROMETHEE II complete order
• •Preserves m_polar 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: PROMETHEE-De-Keyser-Peeters-1996)
• •Assumes: Every alternative-criterion-expert triple is represented by an m-polar fuzzy number ζ ∈ [0,1]^m (m independent positive features); no missing cells, no values outside [0,1]
• •Assumes: External AHP weights w(ϕ) satisfy w(ϕ) ≥ 0, Σ_ϕ w(ϕ) = 1, and the underlying AHP pairwise-comparison matrix has consistency ratio CR < 0.10 (Akram-Shumaiza-Alcantud 2020 Eqs. 11-12; case study CR = 0.087)
• •Assumes: Per-criterion preference function type (Brans-Vincke I-VI) and its parameters (k for Type II; q for III; l,m for IV; u,v for V; σ for VI) have been elicited from the decision makers and are consistent with the m-PF score scale [0,1]
• •Assumes: Criterion direction (max/min) is declared explicitly; for min-direction criteria the deviation argument is negated inside the preference function (Eq. 17): the implementation does NOT silently treat min as max

Assumptions to verify

• •Every alternative-criterion-expert triple is represented by an m-polar fuzzy number ζ ∈ [0,1]^m (m independent positive features); no missing cells, no values outside [0,1]
• •External AHP weights w(ϕ) satisfy w(ϕ) ≥ 0, Σ_ϕ w(ϕ) = 1, and the underlying AHP pairwise-comparison matrix has consistency ratio CR < 0.10 (Akram-Shumaiza-Alcantud 2020 Eqs. 11-12; case study CR = 0.087)
• •Per-criterion preference function type (Brans-Vincke I-VI) and its parameters (k for Type II; q for III; l,m for IV; u,v for V; σ for VI) have been elicited from the decision makers and are consistent with the m-PF score scale [0,1]
• •Criterion direction (max/min) is declared explicitly; for min-direction criteria the deviation argument is negated inside the preference function (Eq. 17): the implementation does NOT silently treat min as max
• •Expert weights α_ψ satisfy α_ψ ≥ 0 and Σ_ψ α_ψ = 1 (uniform α_ψ = 1/s in the canonical case study)
• •PROMETHEE rank-reversal sensitivity is acceptable for the use case: adding or removing alternatives may change the partial pre-order P̂/Î/R̂ and the complete order P̃/Ĩ (De Keyser & Peeters 1996 phenomenon, inherited verbatim by MPF-PROMETHEE)
• •Decision makers accept the meaning of the six Brans-Vincke generalized preference functions and the AHP-weighted multi-criteria preference index Π(R_φ, R_σ) as the outranking strength of R_φ over R_σ (Eq. 19)

When not to use

• •Only one decision maker is involved AND only crisp performance values are available: use crisp PROMETHEE I/II (Brans-Vincke 1985) directly without the m-PF wrapper
• •Performance ratings are bipolar fuzzy ⟨μ⁺ ∈ [0,1], μ⁻ ∈ [-1,0]⟩ (counter-properties of a single attribute) rather than m independent positive attributes: use BF-PROMETHEE (Akram-Shumaiza-Al-Kenani 2020, Symmetry 12(1):77) instead
• •AHP pairwise-comparison matrix has consistency ratio CR ≥ 0.10: the weights are unreliable and Π is biased; rebuild AHP comparisons or switch to an objective weighting (Shannon entropy) before running PROMETHEE
• •Ranking stability under alternative addition/removal is a hard requirement and cannot be relaxed: PROMETHEE inherits the classical rank-reversal phenomenon; consider methods with set-independent normalisation
• •The decision context demands explicit veto thresholds (a concordant outranking should still be vetoed by one criterion exceeding a strong-opposition gap): use MPF-ELECTRE-III/IV instead; PROMETHEE has no veto mechanism by construction

Edge cases

• •if χ^+(R_φ) > χ^+(R_σ) AND χ^-(R_φ) ≤ χ^-(R_σ) (with at least one strict); R_φ Î R_σ iff both equalities; otherwise R_φ R̂ R_σ (incomparable).

Common pitfalls

• •Hatalı: 'MPF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Every alternative-criterion-expert triple is represented by an m-polar fuzzy number ζ ∈ [0,1]^m (m independent positive features); no missing cells, no values outside [0,1]
• •Hatalı: 'MPF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: External AHP weights w(ϕ) satisfy w(ϕ) ≥ 0, Σ_ϕ w(ϕ) = 1, and the underlying AHP pairwise-comparison matrix has consistency ratio CR < 0.10 (Akram-Shumaiza-Alcantud 2020 Eqs. 11-12; case study CR = 0.087)
• •Hatalı: 'MPF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Per-criterion preference function type (Brans-Vincke I-VI) and its parameters (k for Type II; q for III; l,m for IV; u,v for V; σ for VI) have been elicited from the decision makers and are consistent with the m-PF score scale [0,1]
• •Hatalı: 'MPF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion direction (max/min) is declared explicitly; for min-direction criteria the deviation argument is negated inside the preference function (Eq. 17): the implementation does NOT silently treat min as max
• •Hatalı: 'MPF-PROMETHEE bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Expert weights α_ψ satisfy α_ψ ≥ 0 and Σ_ψ α_ψ = 1 (uniform α_ψ = 1/s in the canonical case study)
• •Hatalı: MPF-PROMETHEE'yi 'Only one decision maker is involved AND only crisp performance values are available' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-PROMETHEE'yi 'Performance ratings are bipolar fuzzy ⟨μ⁺ ∈ [0,1], μ⁻ ∈ [-1,0]⟩ (counter-properties of a single attribute) rather than m independent positive attributes' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-PROMETHEE'yi 'AHP pairwise-comparison matrix has consistency ratio CR ≥ 0.10' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1a: Construct s independent m-PF expert decision matrices T^{(ψ)} = [t_{φϕ}^{(ψ)}]_{l×n}, ψ=1,…,s, with each entry t_{φϕ}^{(ψ)} = (p_1◦ζ_{φϕ}^{(ψ)}, p_2◦ζ_{φϕ}^{(ψ)}, …, p_m◦ζ_{φϕ}^{(ψ)}) recording expert ψ's m-polar evaluation of alternative R_φ on criterion Q_ϕ. Formül: T^{(\psi)}=\bigl[t_{\varphi\phi}^{(\psi)}\bigr]_{l\times n},\;\; t_{\varphi\phi}^{(\psi)}=\bigl(p_{1}\!\circ\!\zeta_{\varphi\phi}^{(\psi)},\,\ldots,\,p_{m}\!\circ\!\zeta_{\varphi\phi}^{(\psi)}\bigr),\;\;\psi=1,\ldots,s Anchor: Akram-Shumaiza-Alcantud 2020 Eq. (13) preamble; Akram-Adeel 2023 §7.4 Step 1 (p.351)
2. 2.Adım 2 (F2): Step 1b: Aggregate the s expert matrices into a single m-PF decision matrix T = [t_{φϕ}]_{l×n} via the simple averaging operator (Eq. 7.13 / Eq. 13): for every pole i, p_i◦ζ_{φϕ} = (1/s) Σ_{ψ=1}^{s} p_i◦ζ_{φϕ}^{(ψ)}. The case study uses s=2 experts with uniform 1/s weights (i.e. the arithmetic mean), but the averaging is naturally extendable to non-uniform DM weights by replacing 1/s with α_ψ subject to Σ_ψ α_ψ=1. Formül: p_{i}\!\circ\!\zeta_{\varphi\phi}\;=\;\dfrac{1}{s}\sum_{\psi=1}^{s} p_{i}\!\circ\!\zeta_{\varphi\phi}^{(\psi)},\quad i=1,\ldots,m Anchor: Akram-Shumaiza-Alcantud 2020 Eq. (13); Akram-Adeel 2023 §7.4 Step 1 Eq. (7.13) (p.351)
3. 3.Adım 3 (F3): Step 2: Construct the crisp score matrix S = [ŝ_{φϕ}]_{l×n} by applying the m-PF score function to each cell of the aggregated T: ŝ_{φϕ} = (1/m) Σ_{i=1}^{m} p_i◦ζ_{φϕ} ∈ [0,1]. This compresses the m-polar vector to a single scalar surrogate suitable for the subsequent crisp PROMETHEE pipeline. Formül: \hat{s}_{\varphi\phi}\;=\;\dfrac{1}{m}\sum_{i=1}^{m} p_{i}\!\circ\!\zeta_{\varphi\phi} Anchor: Akram-Shumaiza-Alcantud 2020 Eq. (14); Akram-Adeel 2023 §7.4 Step 2 Eq. (7.14) (p.351)
4. 4.Adım 4 (F4): Step 3: For every ordered pair of alternatives (R_φ, R_σ), φ≠σ, and every criterion Q_ϕ, compute the pairwise deviation d_ϕ(R_φ, R_σ) = ŝ_ϕ(R_φ) − ŝ_ϕ(R_σ). The deviation is the elementary input of every generalized preference function in Step 4. Formül: d_{\phi}(R_{\varphi},R_{\sigma})\;=\;\hat{s}_{\phi}(R_{\varphi})-\hat{s}_{\phi}(R_{\sigma}),\quad \varphi,\sigma=1,\ldots,l,\;\varphi\neq\sigma Anchor: Akram-Shumaiza-Alcantud 2020 Eq. (15); Akram-Adeel 2023 §7.4 Step 3 Eq. (7.15) (p.351)
5. 5.Adım 5 (F5): Step 4: Evaluate the per-criterion preference function P_ϕ(R_φ, R_σ) = F_ϕ[d_ϕ(R_φ, R_σ)] for every pair, where F_ϕ is one of the six Brans-Vincke generalized criteria (Defs. 3-8 / Eqs. 2-7): Type I usual (0/1 step), Type II quasi (threshold k), Type III linear (q), Type IV level (l, m; 0/½/1 step), Type V linear-with-indifference (u, v), Type VI Gaussian (σ). For criteria to be minimised, the deviation is negated before passing to F_ϕ (Eq. 7.17). All preferences satisfy 0 ≤ P_ϕ(R_φ, R_σ) ≤ 1 and P_ϕ(R_φ, R_σ) > 0 ⇒ P_ϕ(R_σ, R_φ) = 0. Formül: P_{\phi}(R_{\varphi},R_{\sigma})\;=\;F_{\phi}\!\bigl[d_{\phi}(R_{\varphi},R_{\sigma})\bigr];\quad\text{Type I: }F(x)=\mathbf{1}[x>0];\;\;\text{II: }F(x)=\mathbf{1}[x>k];\;\;\text{III: }F(x)=\min(x/q,1)\,\mathbf{1}[x>0];\;\;\text{IV: }F(x)=\{0,\tfrac12,1\}\text{ for }x\le l,\,l<x\le l+m,\,x>l+m;\;\;\text{V: }F(x)=\{0,(x-u)/v,1\}\text{ for }x\le u,\,u<x\le u+v,\,x>u+v;\;\;\text{VI: }F(x)=\bigl(1-e^{-x^{2}/2\sigma^{2}}\bigr)\mathbf{1}[x>0];\;\;\text{minimised criteria: }P_{\phi}(R_{\varphi},R_{\sigma})=F_{\phi}\!\bigl[-d_{\phi}(R_{\varphi},R_{\sigma})\bigr] Anchor: Akram-Shumaiza-Alcantud 2020 Defs. 3-8 (Eqs. 2-7) and Eqs. (16)-(17); Akram-Adeel 2023 §7.2 Defs. 7.2-7.7 and §7.4 Step 4 Eqs. (7.16)-(7.17) (pp.346-352)
6. 6.Adım 6 (F6): Step 5: Compute the multi-criteria preference index (MCPI) Π(R_φ, R_σ) as the weighted average of per-criterion preferences using the AHP-derived (or otherwise externally supplied) normalised criterion weights w(ϕ): Π(R_φ, R_σ) = Σ_{ϕ=1}^{n} w(ϕ) P_ϕ(R_φ, R_σ) (since Σ w(ϕ)=1, the denominator in Eq. 18 vanishes). Π ∈ [0,1]: Π≈0 ⇒ weak global preference, Π≈1 ⇒ strong global preference. Π is the arc weight of the outranking graph whose vertices are alternatives. Formül: \Pi(R_{\varphi},R_{\sigma})\;=\;\sum_{\phi=1}^{n} w(\phi)\,P_{\phi}(R_{\varphi},R_{\sigma}),\quad \sum_{\phi=1}^{n} w(\phi)=1,\;\;\varphi\neq\sigma Anchor: Akram-Shumaiza-Alcantud 2020 Eqs. (18)-(19); Akram-Adeel 2023 §7.4 Step 5 Eqs. (7.18)-(7.19) (pp.352-353)
7. 7.Adım 7 (F7): Step 6a: PROMETHEE I partial pre-order. Compute the positive (outgoing/leaving) outranking flow χ^+(R_φ) = (1/(l−1)) Σ_{σ≠φ} Π(R_φ, R_σ) and the negative (incoming/entering) outranking flow χ^-(R_φ) = (1/(l−1)) Σ_{σ≠φ} Π(R_σ, R_φ). χ^+ measures how strongly R_φ dominates the rest; χ^- measures how strongly R_φ is dominated. Derive two pre-orders P^+/I^+ (χ^+: > / =) and P^-/I^- (χ^-: < / =). Their intersection yields the PROMETHEE I partial pre-order (P̂, Î, R̂) (Eq. 24): R_φ P̂ R_σ if χ^+(R_φ) > χ^+(R_σ) AND χ^-(R_φ) ≤ χ^-(R_σ) (with at least one strict); R_φ Î R_σ iff both equalities; otherwise R_φ R̂ R_σ (incomparable). Formül: \chi^{+}(R_{\varphi})\;=\;\dfrac{1}{l-1}\!\!\sum_{\sigma\neq\varphi}\!\!\Pi(R_{\varphi},R_{\sigma});\;\; \chi^{-}(R_{\varphi})\;=\;\dfrac{1}{l-1}\!\!\sum_{\sigma\neq\varphi}\!\!\Pi(R_{\sigma},R_{\varphi});\;\; R_{\varphi}\hat{P}R_{\sigma}\!:\,R_{\varphi}P^{+}R_{\sigma}\wedge R_{\varphi}(P^{-}\!\vee\!I^{-})R_{\sigma}\;\vee\; R_{\varphi}I^{+}R_{\sigma}\wedge R_{\varphi}P^{-}R_{\sigma};\;\; R_{\varphi}\hat{I}R_{\sigma}\!:\,R_{\varphi}I^{+}R_{\sigma}\wedge R_{\varphi}I^{-}R_{\sigma};\;\; R_{\varphi}\hat{R}R_{\sigma}\!:\text{otherwise} Anchor: Akram-Shumaiza-Alcantud 2020 Eqs. (20)-(24); Akram-Adeel 2023 §7.4 Step 6 (a) Eqs. (7.20)-(7.24) (pp.353-355)
8. 8.Adım 8 (F8): Step 6b: PROMETHEE II complete ranking. Compute the net outranking flow χ(R_φ) = χ^+(R_φ) − χ^-(R_φ). χ ∈ [−1, 1]: χ>0 ⇒ R_φ overall outranks; χ<0 ⇒ overall outranked. Sort alternatives by χ in descending order to obtain the complete pre-order (P̃, Ĩ): R_φ P̃ R_σ iff χ(R_φ) > χ(R_σ); R_φ Ĩ R_σ iff χ(R_φ) = χ(R_σ). The alternative with the largest χ is selected. Formül: \chi(R_{\varphi})\;=\;\chi^{+}(R_{\varphi})-\chi^{-}(R_{\varphi});\quad R_{\varphi}\tilde{P}R_{\sigma}\!\iff\!\chi(R_{\varphi})>\chi(R_{\sigma});\quad R_{\varphi}\tilde{I}R_{\sigma}\!\iff\!\chi(R_{\varphi})=\chi(R_{\sigma});\quad \text{rank}(R_{\varphi})=\text{rank of }\chi(R_{\varphi})\text{ in descending order} Anchor: Akram-Shumaiza-Alcantud 2020 Eqs. (25)-(26); Akram-Adeel 2023 §7.4 Step 6 (b) Eqs. (7.25)-(7.26) (p.355)

Commonly paired with

• •AHP + MPF-PROMETHEE (canonical (Akram-Shumaiza-Alcantud 2020 Sec. 3; Akram-Adeel 2023 §7.4.2: AHP is THE method's prescribed weighting partner; consistency ratio CR<0.10 mandatory))
• •Shannon entropy + MPF-PROMETHEE (alternative (when subjective AHP weights are unavailable or untrustworthy; replace Eq. 19 weights w(ϕ) with entropy-derived weights; algorithm itself unchanged))

How to cite