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

Overview

MPF-ELECTRE-II is a group decision method: it expects v independent expert decision matrices Z^(1),…,Z^(v) plus expert weights α and per-expert m-PF criterion weights w_j^(u). Step 2 aggregates these via a probabilistic-sum operator BEFORE normalisation. The final result is a complete ranking via the forward+reverse+average iterative procedure on strong and weak outranking graphs. Five thresholds tune the strict/lenient mix: three concordance levels (c⁻, c°, c*) and two discordance levels (d°, d*). The relative weights γ^c, γ^i in Eq. 4.31 control how strongly the indifferent set contributes to concordance - the nuclear-plant case study uses γ^c=1, γ^i=0 (because every indifferent set is empty for distinct alternatives in that dataset).

Output

ranking position, higher is better

Data

M-Polar Fuzzy, m polar tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

MCGDM with multi-polar criteria (§4.5 nuclear power plant site selection over 8 multi-polar criteria, 3 experts), Outranking-based group decisions requiring full ranking via forward/reverse iterative procedures (not only best-alternative selection like MPF-ELECTRE-I), Strategic infrastructure siting under expert panels with weighted opinions, Settings where weak preferences (not meeting strict R^s) must still be captured via R^w

How it works

1. 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.

   $$Z^{(u)}=(z_{ij}^{(u)}),z_{ij}^{(u)}=(z_{ij}^{(u),1},\dots ,z_{ij}^{(u),m}),u=1,\dots ,v$$

   Akram-Adeel 2023, §4.4 Step 1 (p.260)

2. 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.

   $$z_{ij}^k=1-\prod _{u=1}^v(1-z_{ij}^{(u),k}{)}^{{\alpha }_u},k=1,\dots ,m$$

   Akram-Adeel 2023, §4.4 Step 2, Eq. (4.22) (p.260)

3. 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.

   $$w_j^k=1-\prod _{u=1}^v(1-w_j^{(u),k}{)}^{{\alpha }_u};s(w_j)=\frac{1}{m}\sum _{h=1}^m{w}_j^h;{\gamma }_j=\frac{s(w_j)}{{\sum }_{j=1}^{s}s(w_j)}$$

   Akram-Adeel 2023, §4.4 Step 3, Eqs. (4.23)-(4.25) (p.261)

4. 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.

   $$Y=(y_{ij}),y_{ij}=(z_{ij}^1{w}_j^1,z_{ij}^2{w}_j^2,\dots ,z_{ij}^m{w}_j^m)$$

   Akram-Adeel 2023, §4.4 Step 4, Eq. (4.26) (p.262)

5. 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})}.

   $$s(y_{ij})=\frac{1}{m}\sum _{h=1}^m{y}_{ij}^h;F_{pq}=\{j:s(y_{pj})>s(y_{qj})\};G_{pq}=\{j:s(y_{pj})<s(y_{qj})\};I_{pq}=\{j:s(y_{pj})=s(y_{qj})\}$$

   Akram-Adeel 2023, §4.4 Step 5, Eqs. (4.27)-(4.30) (pp.262-263)

6. 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.

   $$f_{pq}={\gamma }^c\sum _{j\in F_{pq}}{\gamma }_j+{\gamma }^i\sum _{j\in I_{pq}}{\gamma }_j$$

   Akram-Adeel 2023, §4.4 Step 6, Eq. (4.31) (p.263)

7. 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.

   $$g_{pq}=\frac{\underset{j\in G_{pq}}{\max }d(y_{pj},y_{qj})}{\underset{j}{\max }d(y_{pj},y_{qj})},d(y_{pj},y_{qj})=\sqrt{\frac{1}{m}\sum _{k=1}^m(y_{pj}^k-y_{qj}^k{)}^2}$$

   Akram-Adeel 2023, §4.4 Step 7, Eqs. (4.32)-(4.33) (p.264)

8. 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}].

   $$R^s:(f_{pq}\ge c^{\ast }\wedge g_{pq}\le d^{\ast }\wedge f_{pq}>f_{qp})\vee (f_{pq}\ge c^{\circ }\wedge g_{pq}\le d^{\circ }\wedge f_{pq}>f_{qp});R^w:f_{pq}\ge c^{-}\wedge g_{pq}\le d^{\circ }\wedge f_{pq}>f_{qp}$$

   Akram-Adeel 2023, §4.4 Step 8, Eqs. (4.34)-(4.35) (p.265)

9. 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.

   $${\beta }^R(x_p)=1+\underset{x_p\in A}{\max }{\beta }^{\prime }(x_p)-{\beta }^{\prime }(x_p);\beta (x_p)=\frac{{\beta }^F(x_p)+{\beta }^R(x_p)}{2}$$

   Akram-Adeel 2023, §4.4 Step 9, Eqs. (4.36)-(4.37) (pp.265-266); Figs. 4.10-4.11

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• •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^

• Confusing m-polar fuzzy with bipolar fuzzy. m-PF membership lives in [0,1]^m (independent features); bipolar fuzzy uses ⟨μ⁺∈[0,1], μ⁻∈[-1,0]⟩ (counter-properties of a single attribute). Different value spaces - BF-* manifests are not the m=2 special case of MPF-* manifests.

• Threshold ordering. The algorithm assumes 0 < c⁻ < c° < c* < 1 and 0 < d° < d* < 1 (Step 8, p.264). The Akram & Adeel (2023) case-study text on p.277 prints '(d*, d°) = (0.50, 0.60)', which is a notational typo: the values must be paired as d°=0.50 (the tighter veto) and d*=0.60 (the looser veto) to satisfy d° < d* and to reproduce the published outranking matrix (Table 4.26). Validate threshold ordering at input.

• Weak-relation threshold ambiguity. Eq. 4.35 as printed uses 'g_{pq} ≤ d°' for the weak outranking relation, but reproducing Table 4.26 (which lists x3 R^w x2 with g_{32}=0.5376 > d°=0.50) requires using d* instead. Classical Grolleau-Tergny ELECTRE-II uses the larger discordance threshold for the weak relation. Implementations targeting the published case study should treat Eq. 4.35 as 'g_{pq} ≤ d*'.

• Expert aggregation moment matters. Step 2's m-PF weighted averaging (Eq. 4.22) operates on RAW expert matrices before any criterion weighting. Reversing this order (weighting per-expert first, then aggregating) breaks the operator's symmetry and silently changes both the F and G matrices.

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