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Outranking

MPF-ELECTRE-III - m-Polar Fuzzy extension of ELECTRE-III with pseudo-criterion thresholds, Shannon-entropy objective weights and Li-Wang net credibility ranking (Akram & Adeel 2023)

Pseudo-criterion outranking - q/p/ν thresholds, Shannon-entropy objective weights, partial concordance/discordance and Li-Wang credibility-based net ranking

Akram, M., Adeel, A.2023doi:10.1007/978-3-031-43636-9 ↗

Overview

MPF-ELECTRE-III is a group decision method with three innovations over MPF-ELECTRE-II: (1) it derives objective scalar criterion weights internally via Shannon's entropy formula (Eqs. 5.4-5.8), so no externally elicited weights are required; (2) it replaces binary concordance/discordance with pseudo-criterion partial indices governed by three per-criterion thresholds - indifference q_j, preference p_j and veto ν_j (Table 5.9 in the case study); (3) it ranks via the Li-Wang net credibility γ(x_p) = γ^+(x_p) − γ^-(x_p) in descending order - no Belton-Stewart distillation. The pseudo-criterion model is appropriate when small performance differences should not count as strict preference, and when an analyst wants explicit, criterion-specific veto power.

Output: net credibility score, higher is better
Data: M-Polar Fuzzy, uncertainty tuples complete
Weights: Derived internally, no weight source needed
Size: 2+ alternatives, 3-10 criteria works best
Used for: MAGDM with multi-polar expert evaluations under pseudo-criterion thresholds (Akram & Adeel 2023 §5.3 hazardous-waste-carrier firm selection case study, 5 alternatives × 5 criteria × 3 experts × m=3 poles), Outranking-based ranking when only weak/strict preference and veto information is reliable per criterion (q_j/p_j/ν_j thresholds, Eqs. 5.10-5.13), Group decision making where data alone (not expert elicitation) should determine criterion weights - Step 3 Shannon entropy (Eqs. 5.4-5.8) derives objective weights internally from the aggregated m-PF matrix, Full ranking γ-order extraction via Li & Wang (2007) net credibility (Eqs. 5.18-5.20) when Belton-Stewart ascending/descending distillation is undesired due to its non-uniqueness

How it works

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, §5.2 Step 1 (pp.286-287)
2
Aggregate the v expert matrices into a single m-PF decision matrix Z via the m-PF weighted averaging operator (mFWAO_α) with expert weights α_u. Each pole is aggregated independently as a probabilistic sum lifted with exponential expert weights.
$z_{ij}^{k} = 1 - u = 1 \prod v (1 - z_{ij}^{(u), k})^{α_{u}}, k = 1, \dots, m$
Akram-Adeel 2023, §5.2 Step 2, Eqs. (5.2)-(5.3) (pp.286-287)
3
Derive objective scalar criterion weights w_j by Shannon's entropy formula. (i) Score the aggregated cells s_{ij} = (1/m)Σ_h z_{ij}^h. (ii) Project column-wise k_{ij} = s_{ij} / Σ_i s_{ij}. (iii) Compute entropy e_j = −(1/log r) Σ_i k_{ij} log(k_{ij}). (iv) Divergence ∂_j = 1 − e_j. (v) Normalise w_j = ∂_j / Σ_j ∂_j. The weights are scalar (one number per criterion).
$s_{ij} = \frac{1}{m} h = 1 \sum m z_{ij}^{h}; k_{ij} = \frac{s _{ij}}{\sum _{i = 1}^{r} s _{ij}}; e_{j} = - \frac{1}{lo g r} i = 1 \sum r k_{ij} lo g (k_{ij}); \partial_{j} = 1 - e_{j}; w_{j} = \frac{\partial _{j}}{\sum _{j = 1}^{s} \partial _{j}}$
Akram-Adeel 2023, §5.2 Step 3, Eqs. (5.4)-(5.8) (pp.287-288)
4
Construct the aggregated weighted m-PF decision matrix Y by multiplying each pole of every aggregated cell by the scalar criterion weight w_j of its column: y_{ij}^k = z_{ij}^k · w_j. (Note: w_j is scalar - the same factor multiplies every pole - unlike MPF-ELECTRE-II where the m-PF weight is a vector that is multiplied pole-wise.)
$Y = (y_{ij}), y_{ij} = (z_{ij}^{1} w_{j}, z_{ij}^{2} w_{j}, \dots, z_{ij}^{m} w_{j})$
Akram-Adeel 2023, §5.2 Step 4, Eq. (5.9) (p.288)
5
For every ordered pair (p,q) p≠q and every criterion t_j, compare the score degrees s(y_{pj}) and s(y_{qj}) against the thresholds (q_j, p_j) and classify the pair-criterion into one of three pseudo-criterion preference modes: strict preference P (s(y_{pj}) − s(y_{qj}) > p_j), weak preference Q (q_j < s(y_{pj}) − s(y_{qj}) ≤ p_j), or indifference I (|s(y_{pj}) − s(y_{qj})| ≤ q_j).
$s (y_{ij}) = \frac{1}{m} h \sum y_{ij}^{h}; x_{p} P x_{q} : s (y_{p j}) - s (y_{q j}) > p_{j}; x_{p} Q x_{q} : q_{j} < s (y_{p j}) - s (y_{q j}) \leq p_{j}; x_{p} I x_{q} : ∣ s (y_{p j}) - s (y_{q j}) ∣ \leq q_{j}$
Akram-Adeel 2023, §5.2 Step 5, Eqs. (5.10)-(5.13) (p.289)
6
Compute the partial concordance index C_j(x_p,x_q) per criterion (Eq. 5.14): 1 if j∈ℝ^P_{pq} (the set of criteria on which x_p P x_q OR x_p I x_q OR x_p Q x_q AND the gap is at most q_j); (p_j − (s(y_{qj}) − s(y_{pj})))/(p_j − q_j) on the linear ramp j∈ℝ^Q_{pq} (the gap lies in (q_j, p_j]); 0 otherwise. The comprehensive concordance index C_{pq} = Σ_j w_j · C_j(x_p,x_q) (Eq. 5.15) aggregates partial indices with the Shannon-entropy weights w_j from Step 3.
$C_{j} (x_{p}, x_{q}) = ⎩ ⎨ ⎧ 1, \frac{p _{j} - ( s ( y _{q j} ) - s ( y _{p j} ))}{p _{j} - q _{j}}, 0, j \in R_{pq}^{P} j \in T_{pq}^{Q} otherwise; C_{pq} = j = 1 \sum s w_{j} C_{j} (x_{p}, x_{q})$
Akram-Adeel 2023, §5.2 Step 6, Eqs. (5.14)-(5.15) (pp.289-291)
7
Compute the partial discordance index D_j(x_p,x_q) per criterion (Eq. 5.16): 1 if the gap s(y_{qj}) − s(y_{pj}) ≥ ν_j (full veto); 0 if the gap ≤ p_j (no discordance); linearly interpolated (s(y_{qj}) − s(y_{pj}) − p_j)/(ν_j − p_j) when p_j < gap < ν_j. There is no comprehensive discordance matrix in MPF-ELECTRE-III: partial discordances D_j enter the credibility index directly in Step 8.
$D_{j} (x_{p}, x_{q}) = ⎩ ⎨ ⎧ 1, 0, \frac{( s ( y _{q j} ) - s ( y _{p j} )) - p _{j}}{ν _{j} - p _{j}}, s (y_{q j}) - s (y_{p j}) \geq ν_{j} s (y_{q j}) - s (y_{p j}) \leq p_{j} otherwise$
Akram-Adeel 2023, §5.2 Step 7, Eq. (5.16) (p.291)
8
Compute the credibility index β(x_p, x_q) (Eq. 5.17). Let 𝕁̄(x_p, x_q) = {j : D_j(x_p,x_q) > C_{pq}}. If 𝕁̄ is empty (no criterion's partial discordance exceeds the comprehensive concordance), β(x_p, x_q) = C_{pq}. Otherwise the comprehensive concordance is shrunk by each strongly discordant criterion: β(x_p, x_q) = C_{pq} · Π_{j∈𝕁̄}(1 − D_j(x_p,x_q))/(1 − C_{pq}).
$β (x_{p}, x_{q}) = ⎩ ⎨ ⎧ C_{pq}, C_{pq} j \in \overset{ˉ}{J} (x_{p}, x_{q}) \prod \frac{1 - D _{j} ( x _{p} , x _{q} )}{1 - C _{pq}}, \overset{ˉ}{J} (x_{p}, x_{q}) = \emptyset otherwise; \overset{ˉ}{J} (x_{p}, x_{q}) = {j : D_{j} (x_{p}, x_{q}) > C_{pq}}$
Akram-Adeel 2023, §5.2 Step 8, Eq. (5.17) (p.291)
9
Li & Wang net credibility ranking. Concordance credibility γ^+(x_p) = Σ_{q≠p} β(x_p, x_q) is the outranking strength of x_p over every other alternative. Discordance credibility γ^-(x_p) = Σ_{q≠p} β(x_q, x_p) is the strength with which other alternatives outrank x_p. Net credibility γ(x_p) = γ^+(x_p) − γ^-(x_p). Sort alternatives by γ in descending order: the highest γ is the optimal alternative.
$γ^{+} (x_{p}) = q \neq = p \sum β (x_{p}, x_{q}); γ^{-} (x_{p}) = q \neq = p \sum β (x_{q}, x_{p}); γ (x_{p}) = γ^{+} (x_{p}) - γ^{-} (x_{p}); rank (x_{p}) = rank of γ (x_{p}) in descending order$
Akram-Adeel 2023, §5.2 Step 9, Eqs. (5.18)-(5.20) (pp.291-292); Fig. 5.1 flowchart (p.293)

Fits when / Look elsewhere when

Fits when

•Preserves m_polar uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•Single-decision-maker problem with no need for pseudo-criterion semantics - use MPF-ELECTRE-I for choice or MPF-ELECTRE-II for ranking without q/p/ν machinery
•Best-alternative selection only (Pα problematic) - MPF-ELECTRE-I returns the kernel directly without the credibility-index overhead
•External criterion weights must be honoured (e.g. ANP/AHP/BWM elicitation already exists) - MPF-ELECTRE-III's Step 3 overrides external weights with internal Shannon entropy; pair MPF-ELECTRE-II with external weighting instead
•Crisp (m=1) data with no fuzziness - Akram & Adeel (2023) §5.2 reduces algebraically to Roy (1978) ELECTRE-III with Shannon-entropy weighting; use the crisp ELECTRE-III pipeline directly

Assumptions to verify

Each cell of every expert's decision matrix is a valid m-Polar Fuzzy tuple z_{ij}^{(u),k} ∈ [0,1]^m with a fixed pole count m across the whole problem (§5.2 Step 1)
Expert weights α_u ≥ 0 satisfy Σ_u α_u = 1 (Eq. 5.3 aggregation requires a stochastic α-vector; §5.3 uses α=(0.3654, 0.2885, 0.3462))
All criteria are benefit-oriented (or cost criteria have been transformed prior to Step 1 via componentwise complement 1 − z); Eq. 5.3 probabilistic-sum aggregation assumes monotone-up semantics
Pseudo-criterion thresholds satisfy 0 ≤ q_j < p_j ≤ ν_j and are commensurate with the score-degree scale s(y_{ij}) ∈ [0,1] (Eqs. 5.10-5.13, §5.3 Table 5.9 uses q ∈ [0.0005, 0.02], p ∈ [0.0015, 0.08], ν ∈ [0.003, 0.1])
Score function s(y_{ij}) = (Σ_h y_{ij}^h)/m and the column-projection k_{ij} = s_{ij}/Σ_i s_{ij} are non-degenerate (i.e. at least one criterion j has Σ_i s_{ij} > 0, otherwise Shannon entropy in Step 3 collapses)
Group inputs are aggregated at the m-PF matrix level (Eq. 5.3) before any downstream step - per-expert ranking aggregation is not part of the §5.2 algorithm and would invalidate Steps 3-9

Limitations

•Rank reversal known on alternative-set changes (ref: ELECTRE-family-Roy-1968)

Edge cases and pitfalls

•if j∈ℝ^P_{pq} (the set of criteria on which x_p P x_q OR x_p I x_q OR x_p Q x_q AND the gap is at most q_j); (p_j − (s(y_{qj}) − s(y_{pj})))/(p_j − q_j) on the linear ramp j∈ℝ^Q_{pq} (the gap lies in
•if the gap s(y_{qj}) − s(y_{pj}) ≥ ν_j (full veto); 0 if the gap ≤ p_j (no discordance); linearly interpolated (s(y_{qj}) − s(y_{pj}) − p_j)/(ν_j − p_j) when p_j < gap < ν_j. There is no comprehensive
•If 𝕁̄ is empty (no criterion's partial discordance exceeds the comprehensive concordance), β(x_p, x_q) = C_{pq}. Otherwise the comprehensive concordance is shrunk by each strongly discordant criterion

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 pseudo-criterion model requires 0 ≤ q_j < p_j ≤ ν_j for every criterion t_j. The book's Eq. 5.10 footnote prints 'p_j ≥ q_j', which is weakly correct but degenerate: when q_j = p_j the weak preference interval (q_j, p_j] is empty and the partial concordance C_j collapses to a 0/1 indicator (no linear ramp). Implementations should enforce strict q_j < p_j unless a deliberate hard-threshold model is intended.
Step 4 citation typo in the case study. §5.3 narrative point 4 (book p.298) says 'The aggregated weighted m-PF decision matrix, computed in the light of Eq. 4.7, is shown by Table 5.8.' The reference is a cross-chapter typo: the correct equation is Eq. 5.9 (y_{ij}^k = z_{ij}^k · w_j). Implementations should follow Eq. 5.9, not Eq. 4.7 (which refers to a different operator in Chapter 4).
Scalar vs vector weights between MPF-ELECTRE-II and MPF-ELECTRE-III. In MPF-ELECTRE-II the criterion weight w_j is itself an m-PF vector - the weighted matrix Y is built pole-wise: y_{ij}^k = z_{ij}^k · w_j^k. In MPF-ELECTRE-III the entropy weight w_j is scalar - the same factor multiplies every pole: y_{ij}^k = z_{ij}^k · w_j. Mixing the two conventions silently produces wrong Y values and downstream concordance/discordance indices.
Internal weights vs external weights. MPF-ELECTRE-III intentionally derives weights from the data (Shannon entropy on the score-degree projection) and does NOT consume externally elicited preferences. Feeding the method subjective weights bypasses Step 3 and removes its main advantage over MPF-ELECTRE-II (book §5.4.3 point 5). Conversely, very flat data (uniform score projection k_{ij}) drives entropy e_j toward 1 and divergence ∂_j toward 0, producing degenerate weights - always inspect the entropy and divergence rows of Table 5.6 before trusting the ranking.

Works with

Commonly takes its weights from

n_a (internal Shannon entropy, Eqs. 5.4-5.8)

How to cite

Akram, M.; Adeel, A. (2023). MCDM Methods with Multi-polar Fuzzy Information - Chapter 5, §5.2 An m-Polar Fuzzy ELECTRE III Method. Studies in Fuzziness and Soft Computing, vol. 430, Springer Nature. https://doi.org/10.1007/978-3-031-43636-9

BibTeX RIS APA (.txt)DOI ↗

System ID, as it appears in reports and the API

MPF-ELECTRE-III

Overview

Output

net credibility score, higher is better

Data

M-Polar Fuzzy, uncertainty tuples complete

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-10 criteria works best

Used for

MAGDM with multi-polar expert evaluations under pseudo-criterion thresholds (Akram & Adeel 2023 §5.3 hazardous-waste-carrier firm selection case study, 5 alternatives × 5 criteria × 3 experts × m=3 poles), Outranking-based ranking when only weak/strict preference and veto information is reliable per criterion (q_j/p_j/ν_j thresholds, Eqs. 5.10-5.13), Group decision making where data alone (not expert elicitation) should determine criterion weights - Step 3 Shannon entropy (Eqs. 5.4-5.8) derives objective weights internally from the aggregated m-PF matrix, Full ranking γ-order extraction via Li & Wang (2007) net credibility (Eqs. 5.18-5.20) when Belton-Stewart ascending/descending distillation is undesired due to its non-uniqueness

How it works

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, §5.2 Step 1 (pp.286-287)

Aggregate the v expert matrices into a single m-PF decision matrix Z via the m-PF weighted averaging operator (mFWAO_α) with expert weights α_u. Each pole is aggregated independently as a probabilistic sum lifted with exponential expert weights.

z_{ij}^{k} = 1 - u = 1 \prod v (1 - z_{ij}^{(u), k})^{α_{u}}, k = 1, \dots, m

Akram-Adeel 2023, §5.2 Step 2, Eqs. (5.2)-(5.3) (pp.286-287)

Derive objective scalar criterion weights w_j by Shannon's entropy formula. (i) Score the aggregated cells s_{ij} = (1/m)Σ_h z_{ij}^h. (ii) Project column-wise k_{ij} = s_{ij} / Σ_i s_{ij}. (iii) Compute entropy e_j = −(1/log r) Σ_i k_{ij} log(k_{ij}). (iv) Divergence ∂_j = 1 − e_j. (v) Normalise w_j = ∂_j / Σ_j ∂_j. The weights are scalar (one number per criterion).

s_{ij} = \frac{1}{m} h = 1 \sum m z_{ij}^{h}; k_{ij} = \frac{s _{ij}}{\sum _{i = 1}^{r} s _{ij}}; e_{j} = - \frac{1}{lo g r} i = 1 \sum r k_{ij} lo g (k_{ij}); \partial_{j} = 1 - e_{j}; w_{j} = \frac{\partial _{j}}{\sum _{j = 1}^{s} \partial _{j}}

Akram-Adeel 2023, §5.2 Step 3, Eqs. (5.4)-(5.8) (pp.287-288)

Construct the aggregated weighted m-PF decision matrix Y by multiplying each pole of every aggregated cell by the scalar criterion weight w_j of its column: y_{ij}^k = z_{ij}^k · w_j. (Note: w_j is scalar - the same factor multiplies every pole - unlike MPF-ELECTRE-II where the m-PF weight is a vector that is multiplied pole-wise.)

Y = (y_{ij}), y_{ij} = (z_{ij}^{1} w_{j}, z_{ij}^{2} w_{j}, \dots, z_{ij}^{m} w_{j})

Akram-Adeel 2023, §5.2 Step 4, Eq. (5.9) (p.288)

For every ordered pair (p,q) p≠q and every criterion t_j, compare the score degrees s(y_{pj}) and s(y_{qj}) against the thresholds (q_j, p_j) and classify the pair-criterion into one of three pseudo-criterion preference modes: strict preference P (s(y_{pj}) − s(y_{qj}) > p_j), weak preference Q (q_j < s(y_{pj}) − s(y_{qj}) ≤ p_j), or indifference I (|s(y_{pj}) − s(y_{qj})| ≤ q_j).

s (y_{ij}) = \frac{1}{m} h \sum y_{ij}^{h}; x_{p} P x_{q} : s (y_{p j}) - s (y_{q j}) > p_{j}; x_{p} Q x_{q} : q_{j} < s (y_{p j}) - s (y_{q j}) \leq p_{j}; x_{p} I x_{q} : ∣ s (y_{p j}) - s (y_{q j}) ∣ \leq q_{j}

Akram-Adeel 2023, §5.2 Step 5, Eqs. (5.10)-(5.13) (p.289)

Compute the partial concordance index C_j(x_p,x_q) per criterion (Eq. 5.14): 1 if j∈ℝ^P_{pq} (the set of criteria on which x_p P x_q OR x_p I x_q OR x_p Q x_q AND the gap is at most q_j); (p_j − (s(y_{qj}) − s(y_{pj})))/(p_j − q_j) on the linear ramp j∈ℝ^Q_{pq} (the gap lies in (q_j, p_j]); 0 otherwise. The comprehensive concordance index C_{pq} = Σ_j w_j · C_j(x_p,x_q) (Eq. 5.15) aggregates partial indices with the Shannon-entropy weights w_j from Step 3.

C_{j} (x_{p}, x_{q}) = ⎩ ⎨ ⎧ 1, \frac{p _{j} - ( s ( y _{q j} ) - s ( y _{p j} ))}{p _{j} - q _{j}}, 0, j \in R_{pq}^{P} j \in T_{pq}^{Q} otherwise; C_{pq} = j = 1 \sum s w_{j} C_{j} (x_{p}, x_{q})

Akram-Adeel 2023, §5.2 Step 6, Eqs. (5.14)-(5.15) (pp.289-291)

Compute the partial discordance index D_j(x_p,x_q) per criterion (Eq. 5.16): 1 if the gap s(y_{qj}) − s(y_{pj}) ≥ ν_j (full veto); 0 if the gap ≤ p_j (no discordance); linearly interpolated (s(y_{qj}) − s(y_{pj}) − p_j)/(ν_j − p_j) when p_j < gap < ν_j. There is no comprehensive discordance matrix in MPF-ELECTRE-III: partial discordances D_j enter the credibility index directly in Step 8.

D_{j} (x_{p}, x_{q}) = ⎩ ⎨ ⎧ 1, 0, \frac{( s ( y _{q j} ) - s ( y _{p j} )) - p _{j}}{ν _{j} - p _{j}}, s (y_{q j}) - s (y_{p j}) \geq ν_{j} s (y_{q j}) - s (y_{p j}) \leq p_{j} otherwise

Akram-Adeel 2023, §5.2 Step 7, Eq. (5.16) (p.291)

Compute the credibility index β(x_p, x_q) (Eq. 5.17). Let 𝕁̄(x_p, x_q) = {j : D_j(x_p,x_q) > C_{pq}}. If 𝕁̄ is empty (no criterion's partial discordance exceeds the comprehensive concordance), β(x_p, x_q) = C_{pq}. Otherwise the comprehensive concordance is shrunk by each strongly discordant criterion: β(x_p, x_q) = C_{pq} · Π_{j∈𝕁̄}(1 − D_j(x_p,x_q))/(1 − C_{pq}).

β (x_{p}, x_{q}) = ⎩ ⎨ ⎧ C_{pq}, C_{pq} j \in \overset{ˉ}{J} (x_{p}, x_{q}) \prod \frac{1 - D _{j} ( x _{p} , x _{q} )}{1 - C _{pq}}, \overset{ˉ}{J} (x_{p}, x_{q}) = \emptyset otherwise; \overset{ˉ}{J} (x_{p}, x_{q}) = {j : D_{j} (x_{p}, x_{q}) > C_{pq}}

Akram-Adeel 2023, §5.2 Step 8, Eq. (5.17) (p.291)

Li & Wang net credibility ranking. Concordance credibility γ^+(x_p) = Σ_{q≠p} β(x_p, x_q) is the outranking strength of x_p over every other alternative. Discordance credibility γ^-(x_p) = Σ_{q≠p} β(x_q, x_p) is the strength with which other alternatives outrank x_p. Net credibility γ(x_p) = γ^+(x_p) − γ^-(x_p). Sort alternatives by γ in descending order: the highest γ is the optimal alternative.

γ^{+} (x_{p}) = q \neq = p \sum β (x_{p}, x_{q}); γ^{-} (x_{p}) = q \neq = p \sum β (x_{q}, x_{p}); γ (x_{p}) = γ^{+} (x_{p}) - γ^{-} (x_{p}); rank (x_{p}) = rank of γ (x_{p}) in descending order

Akram-Adeel 2023, §5.2 Step 9, Eqs. (5.18)-(5.20) (pp.291-292); Fig. 5.1 flowchart (p.293)

Fits when / Look elsewhere when

Fits when

•Preserves m_polar uncertainty through the pipeline rather than premature crispification at elicitation
•Native group-decision support (multi-DM aggregation built into the pipeline)

Look elsewhere when

•Single-decision-maker problem with no need for pseudo-criterion semantics - use MPF-ELECTRE-I for choice or MPF-ELECTRE-II for ranking without q/p/ν machinery
•Best-alternative selection only (Pα problematic) - MPF-ELECTRE-I returns the kernel directly without the credibility-index overhead
•External criterion weights must be honoured (e.g. ANP/AHP/BWM elicitation already exists) - MPF-ELECTRE-III's Step 3 overrides external weights with internal Shannon entropy; pair MPF-ELECTRE-II with external weighting instead
•Crisp (m=1) data with no fuzziness - Akram & Adeel (2023) §5.2 reduces algebraically to Roy (1978) ELECTRE-III with Shannon-entropy weighting; use the crisp ELECTRE-III pipeline directly

Assumptions to verify

Each cell of every expert's decision matrix is a valid m-Polar Fuzzy tuple z_{ij}^{(u),k} ∈ [0,1]^m with a fixed pole count m across the whole problem (§5.2 Step 1)

Expert weights α_u ≥ 0 satisfy Σ_u α_u = 1 (Eq. 5.3 aggregation requires a stochastic α-vector; §5.3 uses α=(0.3654, 0.2885, 0.3462))

All criteria are benefit-oriented (or cost criteria have been transformed prior to Step 1 via componentwise complement 1 − z); Eq. 5.3 probabilistic-sum aggregation assumes monotone-up semantics

Pseudo-criterion thresholds satisfy 0 ≤ q_j < p_j ≤ ν_j and are commensurate with the score-degree scale s(y_{ij}) ∈ [0,1] (Eqs. 5.10-5.13, §5.3 Table 5.9 uses q ∈ [0.0005, 0.02], p ∈ [0.0015, 0.08], ν ∈ [0.003, 0.1])

Score function s(y_{ij}) = (Σ_h y_{ij}^h)/m and the column-projection k_{ij} = s_{ij}/Σ_i s_{ij} are non-degenerate (i.e. at least one criterion j has Σ_i s_{ij} > 0, otherwise Shannon entropy in Step 3 collapses)

Group inputs are aggregated at the m-PF matrix level (Eq. 5.3) before any downstream step - per-expert ranking aggregation is not part of the §5.2 algorithm and would invalidate Steps 3-9

Limitations

•Rank reversal known on alternative-set changes (ref: ELECTRE-family-Roy-1968)

Edge cases and pitfalls

•if j∈ℝ^P_{pq} (the set of criteria on which x_p P x_q OR x_p I x_q OR x_p Q x_q AND the gap is at most q_j); (p_j − (s(y_{qj}) − s(y_{pj})))/(p_j − q_j) on the linear ramp j∈ℝ^Q_{pq} (the gap lies in

•if the gap s(y_{qj}) − s(y_{pj}) ≥ ν_j (full veto); 0 if the gap ≤ p_j (no discordance); linearly interpolated (s(y_{qj}) − s(y_{pj}) − p_j)/(ν_j − p_j) when p_j < gap < ν_j. There is no comprehensive

•If 𝕁̄ is empty (no criterion's partial discordance exceeds the comprehensive concordance), β(x_p, x_q) = C_{pq}. Otherwise the comprehensive concordance is shrunk by each strongly discordant criterion

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 pseudo-criterion model requires 0 ≤ q_j < p_j ≤ ν_j for every criterion t_j. The book's Eq. 5.10 footnote prints 'p_j ≥ q_j', which is weakly correct but degenerate: when q_j = p_j the weak preference interval (q_j, p_j] is empty and the partial concordance C_j collapses to a 0/1 indicator (no linear ramp). Implementations should enforce strict q_j < p_j unless a deliberate hard-threshold model is intended.

Step 4 citation typo in the case study. §5.3 narrative point 4 (book p.298) says 'The aggregated weighted m-PF decision matrix, computed in the light of Eq. 4.7, is shown by Table 5.8.' The reference is a cross-chapter typo: the correct equation is Eq. 5.9 (y_{ij}^k = z_{ij}^k · w_j). Implementations should follow Eq. 5.9, not Eq. 4.7 (which refers to a different operator in Chapter 4).

Scalar vs vector weights between MPF-ELECTRE-II and MPF-ELECTRE-III. In MPF-ELECTRE-II the criterion weight w_j is itself an m-PF vector - the weighted matrix Y is built pole-wise: y_{ij}^k = z_{ij}^k · w_j^k. In MPF-ELECTRE-III the entropy weight w_j is scalar - the same factor multiplies every pole: y_{ij}^k = z_{ij}^k · w_j. Mixing the two conventions silently produces wrong Y values and downstream concordance/discordance indices.

Internal weights vs external weights. MPF-ELECTRE-III intentionally derives weights from the data (Shannon entropy on the score-degree projection) and does NOT consume externally elicited preferences. Feeding the method subjective weights bypasses Step 3 and removes its main advantage over MPF-ELECTRE-II (book §5.4.3 point 5). Conversely, very flat data (uniform score projection k_{ij}) drives entropy e_j toward 1 and divergence ∂_j toward 0, producing degenerate weights - always inspect the entropy and divergence rows of Table 5.6 before trusting the ranking.

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System ID, as it appears in reports and the API

MPF-ELECTRE-III