MPF-ELECTRE-II: m-Polar Fuzzy extension of ELECTRE-II for multi-criteria group decision making (Akram & Adeel 2023)

Overview

Outranking: m-PF strong/weak relations with five concordance & discordance thresholds and forward/reverse/average iterative ranking. Output typically ranking_position (lower value = preferred).

Strengths

• •Method-specific: Outranking: m-PF strong/weak relations with five concordance & discordance thresholds and forward/reverse/average iterative ranking
• •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: ELECTRE-family-Roy-1968)
• •Assumes: Each criterion is uniformly m-polar across all alternatives and experts (constant pole count m)
• •Assumes: Expert weights α=(α_1,…,α_v) satisfy Σα_u=1 (probabilistic-sum aggregation Eq.4.22)
• •Assumes: All m-PF tuples are benefit-oriented (algorithm assumes higher pole value = better; cost criteria must be pre-transformed)
• •Assumes: Five threshold values satisfy 0 < c⁻ < c° < c* < 1 and 0 < d° < d* < 1 (paper p.264, Step 8)

Assumptions to verify

• •Each criterion is uniformly m-polar across all alternatives and experts (constant pole count m)
• •Expert weights α=(α_1,…,α_v) satisfy Σα_u=1 (probabilistic-sum aggregation Eq.4.22)
• •All m-PF tuples are benefit-oriented (algorithm assumes higher pole value = better; cost criteria must be pre-transformed)
• •Five threshold values satisfy 0 < c⁻ < c° < c* < 1 and 0 < d° < d* < 1 (paper p.264, Step 8)
• •Concordance/indifference set weights γ^c, γ^i are non-negative (Eq.4.31)
• •Group decisions are aggregated at the matrix-level moment (per-expert Z^(u) merged via Eq.4.22 BEFORE outranking is computed)

When not to use

• •Only a single decision-maker is available: MPF-ELECTRE-I or MPF-TOPSIS is simpler when v=1
• •Best-alternative selection is sufficient and a full ranking is not required: MPF-ELECTRE-I avoids the iterative graph-ranking overhead
• •Pseudo-criteria with explicit indifference/preference thresholds (q, p) are needed: use MPF-ELECTRE-III instead
• •All criteria are crisp (m=1): fallback to classical Grolleau-Tergny 1971 ELECTRE-II to avoid spurious pole averaging

Edge cases

• •If G_{pq}=∅, set g_{pq}=0.
• •if x_p∈S(k); delete S(k) and repeat until A(k+1)=∅. Reverse ranking applies the same procedure to the graph with all arc directions reversed, then β^R(x_p)=1+max_p β'(x_p)−β'(x_p). Average: β(x_p)=(β^

Common pitfalls

• •Hatalı: 'MPF-ELECTRE-II bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each criterion is uniformly m-polar across all alternatives and experts (constant pole count m)
• •Hatalı: 'MPF-ELECTRE-II bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Expert weights α=(α_1,…,α_v) satisfy Σα_u=1 (probabilistic-sum aggregation Eq.4.22)
• •Hatalı: 'MPF-ELECTRE-II bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All m-PF tuples are benefit-oriented (algorithm assumes higher pole value = better; cost criteria must be pre-transformed)
• •Hatalı: 'MPF-ELECTRE-II bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Five threshold values satisfy 0 < c⁻ < c° < c* < 1 and 0 < d° < d* < 1 (paper p.264, Step 8)
• •Hatalı: 'MPF-ELECTRE-II bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Concordance/indifference set weights γ^c, γ^i are non-negative (Eq.4.31)
• •Hatalı: MPF-ELECTRE-II'yi 'Only a single decision-maker is available' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-II'yi 'Best-alternative selection is sufficient and a full ranking is not required' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-II'yi 'Pseudo-criteria with explicit indifference/preference thresholds (q, p) are needed' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Construct independent m-PF decision matrices Z^{(u)} for each expert e_u, u=1,…,v. Entry z_{ij}^{(u)} = (z_{ij}^{(u),1},…,z_{ij}^{(u),m}) records expert u's m-polar evaluation of alternative x_i on criterion t_j. Formül: Z^{(u)}=(z_{ij}^{(u)}),\;\; z_{ij}^{(u)}=(z_{ij}^{(u),1},\ldots,z_{ij}^{(u),m}),\;\; u=1,\ldots,v Anchor: Akram-Adeel 2023, §4.4 Step 1 (p.260)
2. 2.Adım 2 (F2): Step 2: Aggregate the v expert matrices into a single m-PF decision matrix Z via the m-PF weighted averaging operator with expert weights α_u. Each pole is aggregated independently with a probabilistic-sum form. Formül: z_{ij}^{k}\;=\;1-\prod_{u=1}^{v}\bigl(1-z_{ij}^{(u),k}\bigr)^{\alpha_{u}},\quad k=1,\ldots,m Anchor: Akram-Adeel 2023, §4.4 Step 2, Eq. (4.22) (p.260)
3. 3.Adım 3 (F3): Step 3: Aggregate per-expert m-PF criterion weights w_j^{(u)} into a single m-PF criterion weight w_j via Eq. 4.23 (same operator as Step 2). Compute scalar normalised weights γ_j = s(w_j) / Σ s(w_j) using the score function s(w_j) = (Σ_h w_j^h)/m. Formül: w_{j}^{k}=1-\prod_{u=1}^{v}\bigl(1-w_{j}^{(u),k}\bigr)^{\alpha_{u}};\;\; s(w_{j})=\tfrac{1}{m}\sum_{h=1}^{m}w_{j}^{h};\;\; \gamma_{j}=\dfrac{s(w_{j})}{\sum_{j=1}^{s}s(w_{j})} Anchor: Akram-Adeel 2023, §4.4 Step 3, Eqs. (4.23)-(4.25) (p.261)
4. 4.Adım 4 (F4): Step 4: Construct the aggregated weighted m-PF decision matrix Y by pole-wise multiplication of the aggregated Z with the m-PF criterion weights w_j: y_{ij}^k = z_{ij}^k · w_j^k. Formül: Y=(y_{ij}),\;\; y_{ij}=(z_{ij}^{1}w_{j}^{1},\,z_{ij}^{2}w_{j}^{2},\ldots,z_{ij}^{m}w_{j}^{m}) Anchor: Akram-Adeel 2023, §4.4 Step 4, Eq. (4.26) (p.262)
5. 5.Adım 5 (F5): Step 5: Partition criteria indices for each ordered pair (p,q), p≠q, into three non-intersecting sets using score s(y_{ij})=(Σ_h y_{ij}^h)/m: concordance F_{pq}={j: s(y_{pj})>s(y_{qj})}, discordance G_{pq}={j: s(y_{pj})<s(y_{qj})}, indifferent I_{pq}={j: s(y_{pj})=s(y_{qj})}. Formül: s(y_{ij})=\tfrac{1}{m}\sum_{h=1}^{m}y_{ij}^{h};\;\; F_{pq}=\{j:s(y_{pj})>s(y_{qj})\};\;\; G_{pq}=\{j:s(y_{pj})<s(y_{qj})\};\;\; I_{pq}=\{j:s(y_{pj})=s(y_{qj})\} Anchor: Akram-Adeel 2023, §4.4 Step 5, Eqs. (4.27)-(4.30) (pp.262-263)
6. 6.Adım 6 (F6): Step 6: Build the m-PF concordance matrix F=[f_{pq}], where f_{pq} = γ^c · Σ_{j∈F_{pq}} γ_j + γ^i · Σ_{j∈I_{pq}} γ_j. γ^c and γ^i are user-supplied relative weights of the concordance and indifferent sets. Formül: f_{pq}=\gamma^{c}\!\sum_{j\in F_{pq}}\gamma_{j}+\gamma^{i}\!\sum_{j\in I_{pq}}\gamma_{j} Anchor: Akram-Adeel 2023, §4.4 Step 6, Eq. (4.31) (p.263)
7. 7.Adım 7 (F7): Step 7: Build the m-PF discordance matrix G=[g_{pq}]. g_{pq} is the per-pole Euclidean distance restricted to G_{pq} normalised by the global maximum distance: g_{pq} = max_{j∈G_{pq}} d(y_{pj},y_{qj}) / max_{j} d(y_{pj},y_{qj}); d(y_{pj},y_{qj}) = √((1/m)Σ_k(y_{pj}^k−y_{qj}^k)²). If G_{pq}=∅, set g_{pq}=0. Formül: g_{pq}=\dfrac{\displaystyle\max_{j\in G_{pq}}d(y_{pj},y_{qj})}{\displaystyle\max_{j}d(y_{pj},y_{qj})},\;\; d(y_{pj},y_{qj})=\sqrt{\tfrac{1}{m}\sum_{k=1}^{m}(y_{pj}^{k}-y_{qj}^{k})^{2}} Anchor: Akram-Adeel 2023, §4.4 Step 7, Eqs. (4.32)-(4.33) (p.264)
8. 8.Adım 8 (F8): Step 8: Establish strong (R^s) and weak (R^w) outranking relations with five thresholds (c⁻, c°, c*) and (d°, d*). Strong: x_p R^s x_q iff [f_{pq}≥c* ∧ g_{pq}≤d* ∧ f_{pq}>f_{qp}] OR [f_{pq}≥c° ∧ g_{pq}≤d° ∧ f_{pq}>f_{qp}]. Weak: x_p R^w x_q iff [f_{pq}≥c⁻ ∧ g_{pq}≤d° ∧ f_{pq}>f_{qp}]. Formül: R^{s}:\;\bigl(f_{pq}\ge c^{*}\wedge g_{pq}\le d^{*}\wedge f_{pq}>f_{qp}\bigr)\vee\bigl(f_{pq}\ge c^{\circ}\wedge g_{pq}\le d^{\circ}\wedge f_{pq}>f_{qp}\bigr);\quad R^{w}:\;f_{pq}\ge c^{-}\wedge g_{pq}\le d^{\circ}\wedge f_{pq}>f_{qp} Anchor: Akram-Adeel 2023, §4.4 Step 8, Eqs. (4.34)-(4.35) (p.265)
9. 9.Adım 9 (F9): Step 9: Compute the final ranking β as the average of forward β^F and reverse β^R rankings. Forward ranking iterates: V(k)={vertices with no precedent arc in G^s}; Ẽ(k)={arcs of G^w with both endpoints in V(k)}; G̃(k)=(V(k), Ẽ(k)); S(k)={vertices with no precedent arc in G̃(k)}; rank x_p at position k if x_p∈S(k); delete S(k) and repeat until A(k+1)=∅. Reverse ranking applies the same procedure to the graph with all arc directions reversed, then β^R(x_p)=1+max_p β'(x_p)−β'(x_p). Average: β(x_p)=(β^F(x_p)+β^R(x_p))/2. Formül: \beta^{R}(x_{p})=1+\max_{x_{p}\in A}\beta'(x_{p})-\beta'(x_{p});\;\; \beta(x_{p})=\dfrac{\beta^{F}(x_{p})+\beta^{R}(x_{p})}{2} Anchor: Akram-Adeel 2023, §4.4 Step 9, Eqs. (4.36)-(4.37) (pp.265-266); Figs. 4.10-4.11

Commonly paired with

• •AHP + MPF-ELECTRE-II (common)
• •ENTROPY + MPF-ELECTRE-II (occasional)
• •MPF-DOMBI-WA + MPF-ELECTRE-II (occasional)

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