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)

Overview

Pseudo-criterion outranking: q/p/ν thresholds, Shannon-entropy objective weights, partial concordance/discordance and Li-Wang credibility-based net ranking. Output typically net_credibility_score (higher value = preferred).

Strengths

• •Method-specific: Pseudo-criterion outranking: q/p/ν thresholds, Shannon-entropy objective weights, partial concordance/discordance and Li-Wang credibility-based net 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 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)
• •Assumes: 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))
• •Assumes: 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
• •Assumes: 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])

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

When not to use

• •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

Edge cases

• •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

Common pitfalls

• •Hatalı: 'MPF-ELECTRE-III bu varsayımı kontrol etmeden uygulamak'. Doğrusu: 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)
• •Hatalı: 'MPF-ELECTRE-III bu varsayımı kontrol etmeden uygulamak'. Doğrusu: 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))
• •Hatalı: 'MPF-ELECTRE-III bu varsayımı kontrol etmeden uygulamak'. Doğrusu: 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
• •Hatalı: 'MPF-ELECTRE-III bu varsayımı kontrol etmeden uygulamak'. Doğrusu: 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])
• •Hatalı: 'MPF-ELECTRE-III bu varsayımı kontrol etmeden uygulamak'. Doğrusu: 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)
• •Hatalı: MPF-ELECTRE-III'yi 'Single-decision-maker problem with no need for pseudo-criterion semantics' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-III'yi 'Best-alternative selection only (Pα problematic)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-III'yi 'External criterion weights must be honoured (e.g. ANP/AHP/BWM elicitation already exists)' 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, §5.2 Step 1 (pp.286-287)
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 (mFWAO_α) with expert weights α_u. Each pole is aggregated independently as a probabilistic sum lifted with exponential expert weights. 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, §5.2 Step 2, Eqs. (5.2)-(5.3) (pp.286-287)
3. 3.Adım 3 (F3): Step 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). Formül: s_{ij}=\tfrac{1}{m}\sum_{h=1}^{m}z_{ij}^{h};\;\; k_{ij}=\dfrac{s_{ij}}{\sum_{i=1}^{r}s_{ij}};\;\; e_{j}=-\dfrac{1}{\log r}\sum_{i=1}^{r}k_{ij}\log(k_{ij});\;\; \partial_{j}=1-e_{j};\;\; w_{j}=\dfrac{\partial_{j}}{\sum_{j=1}^{s}\partial_{j}} Anchor: Akram-Adeel 2023, §5.2 Step 3, Eqs. (5.4)-(5.8) (pp.287-288)
4. 4.Adım 4 (F4): Step 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.) Formül: Y=(y_{ij}),\;\; y_{ij}=(z_{ij}^{1}\,w_{j},\,z_{ij}^{2}\,w_{j},\,\ldots,\,z_{ij}^{m}\,w_{j}) Anchor: Akram-Adeel 2023, §5.2 Step 4, Eq. (5.9) (p.288)
5. 5.Adım 5 (F5): Step 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). Formül: s(y_{ij})=\tfrac{1}{m}\sum_{h}y_{ij}^{h};\;\; x_{p}\mathbf{P}x_{q}\!:\,s(y_{pj})-s(y_{qj})>p_{j};\;\; x_{p}\mathbf{Q}x_{q}\!:\,q_{j}<s(y_{pj})-s(y_{qj})\le p_{j};\;\; x_{p}\mathbf{I}x_{q}\!:\,|s(y_{pj})-s(y_{qj})|\le q_{j} Anchor: Akram-Adeel 2023, §5.2 Step 5, Eqs. (5.10)-(5.13) (p.289)
6. 6.Adım 6 (F6): Step 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. Formül: C_{j}(x_{p},x_{q})=\begin{cases}1, & j\in\mathbb{R}^{\mathbf{P}}_{pq}\\ \dfrac{p_{j}-(s(y_{qj})-s(y_{pj}))}{p_{j}-q_{j}}, & j\in\mathbb{T}^{\mathbf{Q}}_{pq}\\ 0,&\text{otherwise}\end{cases};\quad C_{pq}=\sum_{j=1}^{s}w_{j}\,C_{j}(x_{p},x_{q}) Anchor: Akram-Adeel 2023, §5.2 Step 6, Eqs. (5.14)-(5.15) (pp.289-291)
7. 7.Adım 7 (F7): Step 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. Formül: D_{j}(x_{p},x_{q})=\begin{cases}1, & s(y_{qj})-s(y_{pj})\ge\nu_{j}\\ 0, & s(y_{qj})-s(y_{pj})\le p_{j}\\ \dfrac{(s(y_{qj})-s(y_{pj}))-p_{j}}{\nu_{j}-p_{j}}, & \text{otherwise}\end{cases} Anchor: Akram-Adeel 2023, §5.2 Step 7, Eq. (5.16) (p.291)
8. 8.Adım 8 (F8): Step 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}). Formül: \beta(x_{p},x_{q})=\begin{cases}C_{pq}, & \bar{\mathbb{J}}(x_{p},x_{q})=\varnothing\\ C_{pq}\displaystyle\prod_{j\in\bar{\mathbb{J}}(x_{p},x_{q})}\dfrac{1-D_{j}(x_{p},x_{q})}{1-C_{pq}}, & \text{otherwise}\end{cases};\;\; \bar{\mathbb{J}}(x_{p},x_{q})=\{j:D_{j}(x_{p},x_{q})>C_{pq}\} Anchor: Akram-Adeel 2023, §5.2 Step 8, Eq. (5.17) (p.291)
9. 9.Adım 9 (F9): Step 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. Formül: \gamma^{+}(x_{p})=\!\!\sum_{q\neq p}\!\beta(x_{p},x_{q});\;\; \gamma^{-}(x_{p})=\!\!\sum_{q\neq p}\!\beta(x_{q},x_{p});\;\; \gamma(x_{p})=\gamma^{+}(x_{p})-\gamma^{-}(x_{p});\;\; \text{rank}(x_{p})=\text{rank of }\gamma(x_{p})\text{ in descending order} Anchor: Akram-Adeel 2023, §5.2 Step 9, Eqs. (5.18)-(5.20) (pp.291-292); Fig. 5.1 flowchart (p.293)

Commonly paired with

• •n_a (internal Shannon entropy, Eqs. 5.4-5.8) + MPF-ELECTRE-III (canonical)

How to cite