MPF-ELECTRE-IV: m-Polar Fuzzy extension of ELECTRE-IV with weight-free pseudo-criterion outranking, five dominance classes (quasi/canonical/pseudo/sub/veto), Vallée-Zielniewicz credibility levels and Belton-Stewart ascending+descending distillation (Akram & Adeel 2023)

Overview

Weight-free pseudo-criterion outranking: q/p/ν thresholds, dominance-class counts (NP, NQ, NI, NE), Vallée-Zielniewicz ζ ∈ {1, 0.8, 0.6, 0.4, 0.2, 0} credibility, two-way distillation pre-orders intersected to a final pre-order. Output typically preorder_rank (lower value = preferred).

Strengths

• •Method-specific: Weight-free pseudo-criterion outranking: q/p/ν thresholds, dominance-class counts (NP, NQ, NI, NE), Vallée-Zielniewicz ζ ∈ {1, 0.8, 0.6, 0.4, 0.2, 0} credibility, two-way distillation pre-orders intersected to a final pre-order
• •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 (§6.2 Step 1)
• •Assumes: Expert weights α_u ≥ 0 satisfy Σ_u α_u = 1 (Eq. 6.1 aggregation requires a stochastic α-vector; §6.3 uses α=(0.33, 0.33, 0.34))
• •Assumes: All criteria are benefit-oriented (or cost criteria have been transformed via componentwise complement 1 − z prior to Step 1); Eq. 6.1 probabilistic-sum aggregation assumes monotone-up semantics
• •Assumes: Per-criterion thresholds satisfy 0 ≤ q_j ≤ p_j ≤ ν_j (book §6.2 Step 3 prints non-strict ordering; q_j = p_j collapses weak preference Q to empty and is legal: DO NOT enforce strict q_j < p_j as in §5.2 MPF-ELECTRE-III)

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 (§6.2 Step 1)
• •Expert weights α_u ≥ 0 satisfy Σ_u α_u = 1 (Eq. 6.1 aggregation requires a stochastic α-vector; §6.3 uses α=(0.33, 0.33, 0.34))
• •All criteria are benefit-oriented (or cost criteria have been transformed via componentwise complement 1 − z prior to Step 1); Eq. 6.1 probabilistic-sum aggregation assumes monotone-up semantics
• •Per-criterion thresholds satisfy 0 ≤ q_j ≤ p_j ≤ ν_j (book §6.2 Step 3 prints non-strict ordering; q_j = p_j collapses weak preference Q to empty and is legal: DO NOT enforce strict q_j < p_j as in §5.2 MPF-ELECTRE-III)
• •Discrimination threshold is the ELECTRE-IV-specific constant 0.1 via α=0, β=0.1 (Akram & Adeel 2023 §6.2 last paragraph, p.322); deviating to α=-0.15, β=0.30 reverts to the ELECTRE-III ramp and is not the canonical ELECTRE-IV configuration
• •Group inputs are aggregated at the m-PF matrix level (Eq. 6.1) before any downstream step: per-expert ranking aggregation is not part of the §6.2 algorithm
• •User accepts a PRE-ORDER (not a strict linear order) as output: distillation can leave ties within a distillate, and ascending/descending disagreement yields legitimate incomparabilities

When not to use

• •External criterion weights are available, trusted and must be honoured: use MPF-ELECTRE-II (consumes external m-PF weights) or MPF-ELECTRE-III (internal Shannon entropy) instead; MPF-ELECTRE-IV will ignore weights by design
• •A strict linear ranking is required for downstream automation: MPF-ELECTRE-IV may return a partial pre-order with incomparable pairs (Akram & Adeel 2023 §6.4 Qazvin case happens to be a linear pre-order, but this is not guaranteed)
• •Best-alternative selection only (Pα problematic): use MPF-ELECTRE-I (kernel of non-dominated alternatives, no distillation overhead)
• •Single-decision-maker problem with m=1 (crisp): the method reduces algebraically to Roy & Hugonnard (1982) ELECTRE-IV; use the crisp pipeline directly

Edge cases

• •if s(x_p) > s(x_q) + p_j (Eq. 6.5); weak preference x_p Q x_q if s(x_q) + q_j < s(x_p) ≤ s(x_q) + p_j (Eq. 6.4); indifference x_p I x_q if s(x_q) < s(x_p) ≤ s(x_q) + q_j (Eq. 6.3). Equality x_p E x_q
• •if x_p Dq x_q; 0.8 if Dc; 0.6 if Dp; 0.4 if Ds; 0.2 if Dv (alternate condition only: when the primary Dv condition NP(q,p)=0 holds, the pair has already been classified under the stronger of Dq/Dc/Dp

Common pitfalls

• •Hatalı: 'MPF-ELECTRE-IV 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 (§6.2 Step 1)
• •Hatalı: 'MPF-ELECTRE-IV bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Expert weights α_u ≥ 0 satisfy Σ_u α_u = 1 (Eq. 6.1 aggregation requires a stochastic α-vector; §6.3 uses α=(0.33, 0.33, 0.34))
• •Hatalı: 'MPF-ELECTRE-IV bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All criteria are benefit-oriented (or cost criteria have been transformed via componentwise complement 1 − z prior to Step 1); Eq. 6.1 probabilistic-sum aggregation assumes monotone-up semantics
• •Hatalı: 'MPF-ELECTRE-IV bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Per-criterion thresholds satisfy 0 ≤ q_j ≤ p_j ≤ ν_j (book §6.2 Step 3 prints non-strict ordering; q_j = p_j collapses weak preference Q to empty and is legal: DO NOT enforce strict q_j < p_j as in §5.2 MPF-ELECTRE-III)
• •Hatalı: 'MPF-ELECTRE-IV bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Discrimination threshold is the ELECTRE-IV-specific constant 0.1 via α=0, β=0.1 (Akram & Adeel 2023 §6.2 last paragraph, p.322); deviating to α=-0.15, β=0.30 reverts to the ELECTRE-III ramp and is not the canonical ELECTRE-IV configuration
• •Hatalı: MPF-ELECTRE-IV'yi 'External criterion weights are available, trusted and must be honoured' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-IV'yi 'A strict linear ranking is required for downstream automation' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: MPF-ELECTRE-IV'yi 'Best-alternative selection only (Pα problematic)' 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, §6.2 Step 1 (pp.317-318)
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 (identical to the §5.2 Step 2 / Eq. 5.3 operator). 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, §6.2 Step 2, Eq. (6.1) (p.318)
3. 3.Adım 3 (F3): Step 3: Compute the score degrees s(z_{ij}) = (1/m)Σ_h z_{ij}^h directly on the aggregated matrix Z (no weighting). For every ordered pair (p,q), p≠q, and every criterion t_j, classify the pair-criterion into one of three pseudo-criterion preference relations against the thresholds (q_j, p_j): strict preference x_p P x_q if s(x_p) > s(x_q) + p_j (Eq. 6.5); weak preference x_p Q x_q if s(x_q) + q_j < s(x_p) ≤ s(x_q) + p_j (Eq. 6.4); indifference x_p I x_q if s(x_q) < s(x_p) ≤ s(x_q) + q_j (Eq. 6.3). Equality x_p E x_q is a fourth relation (s(x_p) = s(x_q)) tracked alongside P/Q/I. Note: MPF-ELECTRE-IV operates on the unweighted Z (not on the weighted Y of §5.2 Step 4): the thresholds in §6.3 (Table 6.7) are calibrated on the same scale as raw score degrees. Formül: s(z_{ij})=\tfrac{1}{m}\sum_{h=1}^{m}z_{ij}^{h};\quad x_{p}\mathbf{P}x_{q}\!:\,s(x_{p})>s(x_{q})+p_{j};\;\; x_{p}\mathbf{Q}x_{q}\!:\,s(x_{q})+q_{j}<s(x_{p})\le s(x_{q})+p_{j};\;\; x_{p}\mathbf{I}x_{q}\!:\,s(x_{q})<s(x_{p})\le s(x_{q})+q_{j} Anchor: Akram-Adeel 2023, §6.2 Step 3, Eqs. (6.2)-(6.5) (pp.318-319)
4. 4.Adım 4 (F4): Step 4: For every ordered pair (p,q), count the number of criteria in each preference mode: NP(x_p, x_q) (strong), NQ(x_p, x_q) (weak), NI(x_p, x_q) (indifferent), NE(x_p, x_q) (equal), with the partition constraint s = NP(p,q)+NQ(p,q)+NI(p,q)+NE(p,q)+NP(q,p)+NQ(q,p)+NI(q,p). Then assign each pair to at most one of five dominance classes by the following weight-free rules: (i) Quasi dominance Dq: NP(q,p)+NQ(q,p) = 0 AND NI(q,p) < NI(p,q)+NQ(p,q)+NP(p,q); (ii) Canonical dominance Dc: NP(q,p)=0 AND NQ(q,p) ≤ NP(p,q) AND NQ(q,p)+NI(q,p) < NP(p,q)+NQ(p,q)+NI(p,q); (iii) Pseudo dominance Dp: NP(q,p)=0 AND NQ(q,p) ≤ NP(p,q)+NQ(p,q); (iv) Sub-dominance Ds: NP(q,p)=0; (v) Veto dominance Dv: primary: NP(q,p)=0; OR alternate: NP(q,p)=1 AND there exists exactly one criterion j* with s(x_q) − s(x_p) ≤ ν_{j*} for that single strongly-preferred criterion AND NP(p,q) ≥ s/2. A pair may satisfy multiple classes; report the strongest (Dq > Dc > Dp > Ds > Dv) for credibility assignment in F5. Formül: s=N_{P}(p,q)+N_{Q}(p,q)+N_{I}(p,q)+N_{E}(p,q)+N_{P}(q,p)+N_{Q}(q,p)+N_{I}(q,p);\quad x_{p}D_{q}x_{q}\!\Leftrightarrow\! N_{P}(q,p)+N_{Q}(q,p)=0\,\wedge\,N_{I}(q,p)<N_{I}(p,q)+N_{Q}(p,q)+N_{P}(p,q);\quad x_{p}D_{c}x_{q}\!\Leftrightarrow\! N_{P}(q,p)=0\,\wedge\,N_{Q}(q,p)\le N_{P}(p,q)\,\wedge\,N_{Q}(q,p)+N_{I}(q,p)<N_{P}(p,q)+N_{Q}(p,q)+N_{I}(p,q);\quad x_{p}D_{p}x_{q}\!\Leftrightarrow\! N_{P}(q,p)=0\,\wedge\,N_{Q}(q,p)\le N_{P}(p,q)+N_{Q}(p,q);\quad x_{p}D_{s}x_{q}\!\Leftrightarrow\! N_{P}(q,p)=0;\quad x_{p}D_{v}x_{q}\!\Leftrightarrow\! N_{P}(q,p)=0\,\vee\,\bigl(N_{P}(q,p)=1\,\wedge\,s(x_{q})-s(x_{p})\le\nu_{j^{*}}\,\wedge\,N_{P}(p,q)\ge s/2\bigr) Anchor: Akram-Adeel 2023, §6.2 Step 4 (pp.319-321)
5. 5.Adım 5 (F5): Step 5: Assign Vallée-Zielniewicz credibility levels to each ordered pair according to its strongest dominance class: ζ(x_p, x_q) = 1 if x_p Dq x_q; 0.8 if Dc; 0.6 if Dp; 0.4 if Ds; 0.2 if Dv (alternate condition only: when the primary Dv condition NP(q,p)=0 holds, the pair has already been classified under the stronger of Dq/Dc/Dp/Ds and the corresponding ζ is used); 0 if no dominance relation holds. The ζ matrix (r×r, diagonal undefined) is the credibility-indexed outranking matrix for the distillation procedure in F6. Formül: \zeta(x_{p},x_{q})=\begin{cases}1,&x_{p}D_{q}x_{q}\\0.8,&x_{p}D_{c}x_{q}\\0.6,&x_{p}D_{p}x_{q}\\0.4,&x_{p}D_{s}x_{q}\\0.2,&x_{p}D_{v}x_{q}\text{ (alternate)}\\0,&\text{otherwise}\end{cases} Anchor: Akram-Adeel 2023, §6.2 Step 5, Vallée-Zielniewicz [57] (p.321)
6. 6.Adım 6 (F6): Step 6: Two-way Belton-Stewart distillation. Define the discrimination threshold s(ζ_λ) = α·ζ_λ + β (Vallée-Zielniewicz general α=-0.15, β=0.30; ELECTRE-IV-specific α=0, β=0.1: the discrimination threshold is the constant 0.1, the book's default), cut-off level ζ_{λ+1} = ζ_λ − s(ζ_λ), and the outranking-at-cut-off relation x_p D_A^{ζ_λ} x_q ⇔ ζ(x_p, x_q) > ζ_λ AND ζ(x_p, x_q) − ζ(x_q, x_p) > s(ζ(x_p, x_q)). For each alternative x_p compute strength S_A^{ζ_λ}(x_p) = Σ_{x_q ∈ M⁺} ζ(x_p, x_q) (Eq. 6.6) over the set M⁺={x_q : x_p D_A^{ζ_λ} x_q}, weakness S̄_A^{ζ_λ}(x_p) = Σ_{x_q ∈ M⁻} ζ(x_q, x_p) (Eq. 6.7) over M⁻={x_q : x_q D_A^{ζ_λ} x_p}, and qualification Q_A^{ζ_λ}(x_p) = S_A^{ζ_λ}(x_p) − S̄_A^{ζ_λ}(x_p) (Eq. 6.8). Ascending distillation: iteratively pick the set of alternatives with MINIMUM qualification at descending cut-off levels until A is exhausted; the order of removal (last-removed = best) yields the ascending pre-order. Descending distillation: same iteration but pick the set with MAXIMUM qualification; the order of removal (first-removed = best) yields the descending pre-order. The FINAL pre-order is the intersection of the two pre-orders: x_p ≻ x_q in the final pre-order iff x_p ≻ x_q in BOTH the ascending and descending pre-orders (Akram & Adeel 2023 §6.2 last sentence, p.324). Formül: s(\zeta_{\lambda})=\alpha\,\zeta_{\lambda}+\beta\;\;[\text{ELECTRE-IV: }\alpha=0,\beta=0.1];\;\; \zeta_{\lambda+1}=\zeta_{\lambda}-s(\zeta_{\lambda});\;\; x_{p}D_{A}^{\zeta_{\lambda}}x_{q}\!\Leftrightarrow\!\zeta(x_{p},x_{q})>\zeta_{\lambda}\,\wedge\,\zeta(x_{p},x_{q})-\zeta(x_{q},x_{p})>s\bigl(\zeta(x_{p},x_{q})\bigr);\;\; S_{A}^{\zeta_{\lambda}}(x_{p})=\!\!\sum_{x_{q}\in M^{+}}\!\!\zeta(x_{p},x_{q});\;\; \bar{S}_{A}^{\zeta_{\lambda}}(x_{p})=\!\!\sum_{x_{q}\in M^{-}}\!\!\zeta(x_{q},x_{p});\;\; Q_{A}^{\zeta_{\lambda}}(x_{p})=S_{A}^{\zeta_{\lambda}}(x_{p})-\bar{S}_{A}^{\zeta_{\lambda}}(x_{p});\;\;\text{final pre-order}=\text{ascending pre-order}\,\cap\,\text{descending pre-order} Anchor: Akram-Adeel 2023, §6.2 Step 6, Eqs. (6.6)-(6.8) and ascending/descending distillation algorithm (pp.322-324)

Commonly paired with

• •n_a (weight-free by design, §6.2) + MPF-ELECTRE-IV (canonical)

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