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

MPF-PROMETHEE is an AHP-assisted group decision method that operates on m-polar fuzzy performance ratings. Its three defining advantages are: (1) the six Brans-Vincke generalized preference functions (Types I-VI) let the analyst express, per criterion, exactly how a performance gap should translate into a preference intensity - from binary (Type I) through threshold-step (Types II, IV) to linear-ramp (Types III, V) and Gaussian (Type VI); (2) the multi-criteria preference index Π(R_φ,R_σ) (Eq. 19) is the AHP-weighted average of these preferences, naturally interpretable as the outranking strength of one alternative over another; (3) PROMETHEE I keeps incomparabilities explicit (small differences in χ^+ and χ^- that point in opposite directions yield R̂), while PROMETHEE II compresses χ into a single net flow χ = χ^+ − χ^- for a forced complete ranking. Use PROMETHEE I when transparency about incomparability matters; use PROMETHEE II when a single best-to-worst order is required.

Output

net outranking flow χ, higher is better

Data

M-Polar Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Hydroelectric power station site selection under multi-expert m-polar performance evaluation (canonical case study: Akram-Shumaiza-Alcantud 2020 Sec. 4; Akram-Adeel 2023 §7.4.3, 7 sites × 6 criteria × 3 poles × 2 experts), Multi-criteria group decision making (MAGDM) where each expert evaluates each alternative on m independent positive attributes (m-polar features) of every criterion, Renewable-energy project ranking and other infrastructure-siting problems where criteria mix linear preferences (Type V), thresholds (Types II, IV), and Gaussian smooth ramps (Type VI) - domains where the analyst wants per-criterion control of the deviation-to-preference mapping, AHP-assisted decision frameworks where weights are obtained from pairwise comparison (consistency ratio CR<0.10) and the analyst needs both partial pre-order (PROMETHEE I, P̂/Î/R̂) and complete ranking (PROMETHEE II, P̃/Ĩ)

How it works

1. 1

   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.

   $$T^{(\psi )}=[t_{\phi \varphi }^{(\psi )}{]}_{l\times n},t_{\phi \varphi }^{(\psi )}=(p_1\circ {\zeta }_{\phi \varphi }^{(\psi )},\dots ,p_m\circ {\zeta }_{\phi \varphi }^{(\psi )}),\psi =1,\dots ,s$$

   Akram-Shumaiza-Alcantud 2020 Eq. (13) preamble; Akram-Adeel 2023 §7.4 Step 1 (p.351)

2. 2

   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.

   $$p_i\circ {\zeta }_{\phi \varphi }=\frac{1}{s}\sum _{\psi =1}^s{p}_i\circ {\zeta }_{\phi \varphi }^{(\psi )},i=1,\dots ,m$$

   Akram-Shumaiza-Alcantud 2020 Eq. (13); Akram-Adeel 2023 §7.4 Step 1 Eq. (7.13) (p.351)

3. 3

   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.

   $${\hat{s}}_{\phi \varphi }=\frac{1}{m}\sum _{i=1}^m{p}_i\circ {\zeta }_{\phi \varphi }$$

   Akram-Shumaiza-Alcantud 2020 Eq. (14); Akram-Adeel 2023 §7.4 Step 2 Eq. (7.14) (p.351)

4. 4

   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.

   $$d_{\varphi }(R_{\phi },R_{\sigma })={\hat{s}}_{\varphi }(R_{\phi })-{\hat{s}}_{\varphi }(R_{\sigma }),\phi ,\sigma =1,\dots ,l,\phi \ne \sigma$$

   Akram-Shumaiza-Alcantud 2020 Eq. (15); Akram-Adeel 2023 §7.4 Step 3 Eq. (7.15) (p.351)

5. 5

   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.

   $$P_{\varphi }(R_{\phi },R_{\sigma })=F_{\varphi }[d_{\varphi }(R_{\phi },R_{\sigma })];\text{Type I: }F(x)=1[x>0];\text{II: }F(x)=1[x>k];\text{III: }F(x)=\min (x/q,1)1[x>0];\text{IV: }F(x)=\{0,\frac{1}{2},1\}\text{ for }x\le l,l<x\le l+m,x>l+m;\text{V: }F(x)=\{0,(x-u)/v,1\}\text{ for }x\le u,u<x\le u+v,x>u+v;\text{VI: }F(x)=(1-e^{-x^2/2{\sigma }^2})1[x>0];\text{minimised criteria: }{P}_{\varphi }(R_{\phi },R_{\sigma })=F_{\varphi }[-d_{\varphi }(R_{\phi },R_{\sigma })]$$

   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

   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.

   $$\Pi (R_{\phi },R_{\sigma })=\sum _{\varphi =1}^{n}w(\varphi )P_{\varphi }(R_{\phi },R_{\sigma }),\sum _{\varphi =1}^{n}w(\varphi )=1,\phi \ne \sigma$$

   Akram-Shumaiza-Alcantud 2020 Eqs. (18)-(19); Akram-Adeel 2023 §7.4 Step 5 Eqs. (7.18)-(7.19) (pp.352-353)

7. 7

   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).

   $${\chi }^{+}(R_{\phi })=\frac{1}{l-1}\sum _{\sigma \ne \phi }\Pi (R_{\phi },R_{\sigma });{\chi }^{-}(R_{\phi })=\frac{1}{l-1}\sum _{\sigma \ne \phi }\Pi (R_{\sigma },R_{\phi });R_{\phi }\hat{P}{R}_{\sigma }:R_{\phi }{P}^{+}{R}_{\sigma }\wedge R_{\phi }(P^{-}\vee I^{-})R_{\sigma }\vee R_{\phi }{I}^{+}{R}_{\sigma }\wedge R_{\phi }{P}^{-}{R}_{\sigma };R_{\phi }\hat{I}{R}_{\sigma }:R_{\phi }{I}^{+}{R}_{\sigma }\wedge R_{\phi }{I}^{-}{R}_{\sigma };R_{\phi }\hat{R}{R}_{\sigma }:\text{otherwise}$$

   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

   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.

   $$\chi (R_{\phi })={\chi }^{+}(R_{\phi })-{\chi }^{-}(R_{\phi });R_{\phi }\tilde{P}{R}_{\sigma }⟺\chi (R_{\phi })>\chi (R_{\sigma });R_{\phi }\tilde{I}{R}_{\sigma }⟺\chi (R_{\phi })=\chi (R_{\sigma });\text{rank}(R_{\phi })=\text{rank of }\chi (R_{\phi })\text{ in descending order}$$

   Akram-Shumaiza-Alcantud 2020 Eqs. (25)-(26); Akram-Adeel 2023 §7.4 Step 6 (b) Eqs. (7.25)-(7.26) (p.355)

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• •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).

• Confusing m-polar fuzzy with bipolar fuzzy. m-PF membership lives in [0,1]^m (m independent positive features per cell); bipolar fuzzy uses ⟨μ⁺∈[0,1], μ⁻∈[-1,0]⟩ (counter-properties of a single attribute). The Akram-Adeel 2023 book Chapter 7 in-text reference [9] for the m-PF PROMETHEE introduction is the BF-PROMETHEE paper of Akram-Shumaiza-Al-Kenani (Symmetry 12(1), 77, 2020), not the m-PF paper. Always cite the Math. Comput. Appl. 25:26 (2020) paper as the seminal source of MPF-PROMETHEE.

• Direction of minimised criteria. The preference function P_ϕ is defined on positive deviations; for criteria to be minimised the deviation is negated (Eq. 17): P_ϕ(R_φ,R_σ) = F_ϕ[−d_ϕ(R_φ,R_σ)]. Implementations that forget the sign flip for min-direction criteria silently invert the contribution of cost-like criteria (Q_3 Government incentives, Q_5 Climate changes, Q_6 Cost in the case study) - Π becomes wrong without any error signal.

• Per-criterion preference-function types are independent. A single PROMETHEE run may (and the case study does) mix all six types across criteria - Q_1 V, Q_2 III, Q_3 VI, Q_4 II, Q_5 IV, Q_6 I. Forcing a single type for all criteria (the usual-criterion analysis of §7.5.1 / Sec. 5.1) is a comparative-analysis variant, NOT the canonical method. The book's §7.5.1 explicitly contrasts the two configurations and shows that the all-Type-I ranking diverges from the six-types ranking at the third position.

• Rank reversal under alternative add/remove. The (l−1)^{-1} normalisation in Eqs. 20-21 implies that adding or removing an alternative changes the magnitude of every χ^+ and χ^-, hence potentially the partial pre-order P̂/Î/R̂ and the complete order P̃/Ĩ. This is the classical PROMETHEE rank-reversal phenomenon documented by De Keyser & Peeters 1996 and is inherited verbatim by MPF-PROMETHEE. For decisions that need ranking stability under set changes, accompany PROMETHEE with a sensitivity test that re-runs the method on each (l−1)-subset.

• AHP weights are external; method does not produce them. MPF-PROMETHEE consumes a normalised weight vector w(ϕ) (sum=1) and does NOT derive it internally. The seminal paper computes w by AHP (Sec. 3) and warns to check the consistency ratio CR<0.10 before trusting them (Akram-Shumaiza-Alcantud 2020 Eqs. 11-12; case study CR=0.087). If a different weight-derivation method is used (Shannon entropy, BWM, direct elicitation), the algorithm remains correct but the citation to AHP should be removed; conversely, blindly accepting inconsistent AHP weights (CR>0.10) propagates bias into Π without warning.

Works with

How to cite