MPF-ELECTRE-I - m-Polar Fuzzy extension of ELECTRE-I (Akram, Waseem & Liu 2019)

Overview

MPF-ELECTRE-I produces a directed outranking graph over alternatives, not a complete numerical ranking. Read the graph: a node with high out-degree and zero in-degree dominates the field. Cells of the input matrix are m-polar tuples - each pole represents one independent criterion-attribute (e.g. sub-criteria of Infrastructure: fuel/water/transport). The pole aggregation v_{ij}=Σ y_{ij}^k preserves the multi-polar information until the final concordance comparison.

Output

Outranking Graph, lower 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

M-Polar Fuzzy MCDM, site-selection / location problems with multi-attribute criteria, expert-driven evaluation with multi-polar attributes (e.g. sub-criteria of Infrastructure: fuel/water/transport)

How it works

1. 1

   Form the m-PF decision matrix Z=(z_{ij}), where z_{ij}=(z_{ij}^1, z_{ij}^2, …, z_{ij}^m).

   $$Z=(z_{ij}),z_{ij}=(z_{ij}^1,z_{ij}^2,\dots ,z_{ij}^m)$$

   Akram-Adeel 2023, §2.3 Step 1 (p.90)

2. 2

   Weighted m-PF decision matrix Y=(y_{ij}), y_{ij}=w_j·z_{ij}=(w_j z_{ij}^1, …, w_j z_{ij}^m).

   $$Y=(y_{ij})=(y_{ij}^1,y_{ij}^2,\dots ,y_{ij}^m),y_{ij}^k=w_j{z}_{ij}^k$$

   Akram-Adeel 2023, §2.3 Step 3 (p.90)

3. 3

   m-PF concordance set F_{pq}={j : v_{pj} ≥ v_{qj}}, where v_{ij}=Σ_{k=1}^m y_{ij}^k aggregates the m poles of the weighted cell.

   $$F_{pq}=\{1\le j\le n:v_{pj}\ge v_{qj},p\ne q\},v_{ij}=\sum _{k=1}^m{y}_{ij}^k$$

   Akram-Adeel 2023, §2.3 Step 4 (p.90)

4. 4

   m-PF discordance set G_{pq}={j : v_{pj} ≤ v_{qj}}.

   $$G_{pq}=\{1\le j\le n:v_{pj}\le v_{qj},p\ne q\}$$

   Akram-Adeel 2023, §2.3 Step 5 (p.90)

5. 5

   m-PF concordance indices f_{pq}=Σ_{j∈F_{pq}} w_j and concordance matrix F.

   $$f_{pq}=\sum _{j\in F_{pq}}{w}_j,F=[f_{pq}{]}_{r\times r}$$

   Akram-Adeel 2023, §2.3 Steps 6-7 (p.90)

6. 6

   m-PF discordance indices g_{pq} (normalised Euclidean-distance ratio over poles) and discordance matrix G.

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

   Akram-Adeel 2023, §2.3 Steps 8-9 (p.91)

7. 7

   Concordance level f̄ and discordance level ḡ (averages of off-diagonal F and G indices).

   $$\bar{f}=\frac{1}{r(r-1)}\sum _{p=1}^r\sum _{\begin{array}{c}q=1\\ q\ne p\end{array}}^r{f}_{pq},\bar{g}=\frac{1}{r(r-1)}\sum _{p=1}^r\sum _{\begin{array}{c}q=1\\ q\ne p\end{array}}^r{g}_{pq}$$

   Akram-Adeel 2023, §2.3 Step 10 (p.91)

8. 8

   Concordance dominance H and discordance dominance L (boolean threshold matrices).

   $$h_{pq}=1[f_{pq}\ge \bar{f}],l_{pq}=1[g_{pq}<\bar{g}]$$

   Akram-Adeel 2023, §2.3 Step 11 (p.91)

9. 9

   Aggregate dominance matrix M = H ⊙ L (peer-to-peer / Hadamard product).

   $$M=H\odot L,m_{pq}=h_{pq}\cdot l_{pq}$$

   Akram-Adeel 2023, §2.3 Step 12 (p.92)

10. 10

    Build outranking directed graph: edge p→q iff m_{pq}=1. Read off preferences (unique edge ⇒ x_p ≻ x_q; both edges ⇒ indifferent; no edge ⇒ incomparable).

    $$p\to q⟺m_{pq}=1$$

    Akram-Adeel 2023, §2.3 Step 13 (p.92)

Fits when / Look elsewhere when

Assumptions to verify

• Each decision cell is a true m-polar tuple in [0,1]^m (NOT a scalar fuzzy number).
• Number of poles m is uniform across all cells.
• Criterion weights w_j sum to 1 and are externally supplied (MPF-ELECTRE-I consumes weights, does not produce them).
• User accepts graph-valued output (incomparability and indifference are first-class outcomes; no forced linear ranking).

Edge cases and pitfalls

• Treating m-PF cells as scalars. Each cell is an m-tuple; the v_{ij} sum is an intermediate aggregation used only for the concordance/discordance set test, NOT the cell value itself.

• 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.

• Empty outranking graph (no edges anywhere): weights too uniform or pole values too clustered. Inspect Y matrix and v_{ij} aggregates before assuming a method failure.

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