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

MPF-ELECTRE-IV is the weight-free sibling of MPF-ELECTRE-III. Three innovations distinguish it from MPF-ELECTRE-III: (1) it requires NO criterion weights at all - Shannon entropy and the weighted matrix Y are removed; criteria importance is encoded implicitly in the five dominance-class definitions (Dq stricter than Dc stricter than Dp stricter than Ds stricter than Dv); (2) it classifies each ordered pair into at most one of five dominance classes via counts NP/NQ/NI/NE of criteria in each preference mode, then assigns Vallée-Zielniewicz credibility levels ζ ∈ {1, 0.8, 0.6, 0.4, 0.2, 0} (Dq=1, Dc=0.8, Dp=0.6, Ds=0.4, Dv=0.2, none=0); (3) it ranks via two-way Belton-Stewart distillation (ascending = iteratively remove min-qualification alternatives; descending = iteratively remove max-qualification) and intersects the two pre-orders to obtain the final pre-order. The discrimination threshold is the constant 0.1 (α=0, β=0.1). Use this method when criterion weights are unavailable or unwanted, when the decision maker prefers an interpretable dominance-class hierarchy over a continuous credibility index, and when a pre-order output (with possible incomparabilities) is acceptable.

Output

preorder rank, higher is better

Data

M-Polar Fuzzy, uncertainty tuples complete

Size

2+ alternatives, 3-10 criteria works best

Used for

MAGDM where criterion weights are unavailable or politically contested - MPF-ELECTRE-IV's five dominance classes (Dq/Dc/Dp/Ds/Dv) encode criterion importance implicitly via NP/NQ/NI/NE counts, so no weighting workshop is required (Akram & Adeel 2023 §6.3 Islamic Azad University Qazvin contractor selection, 5 firms × 5 criteria × 3 experts × m=4), Outranking-based pre-order ranking when both an ascending (caution-pessimistic) and a descending (caution-optimistic) view should agree before declaring strict preference - final pre-order is the INTERSECTION (Eqs. 6.6-6.8 + §6.2 closing sentence p.324), Pseudo-criterion fuzzy ranking with veto information limited to the alternate Dv condition (NP(q,p)=1 ∧ s(x_q)-s(x_p)≤ν ∧ NP(p,q)≥s/2) - ν_j enters only there, not in concordance/discordance arithmetic (§6.2 Step 4 p.320), Settings where the decision maker prefers an interpretable discrete dominance-class hierarchy (Dq>Dc>Dp>Ds>Dv) over a continuous credibility index β ∈ [0,1] (the MPF-ELECTRE-III alternative, §5.2)

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, §6.2 Step 1 (pp.317-318)

2. 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).

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

   Akram-Adeel 2023, §6.2 Step 2, Eq. (6.1) (p.318)

3. 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.

   $$s(z_{ij})=\frac{1}{m}\sum _{h=1}^m{z}_{ij}^h;x_p\mathbf{P}{x}_q:s(x_p)>s(x_q)+p_j;x_p\mathbf{Q}{x}_q:s(x_q)+q_j<s(x_p)\le s(x_q)+p_j;x_p\mathbf{I}{x}_q:s(x_q)<s(x_p)\le s(x_q)+q_j$$

   Akram-Adeel 2023, §6.2 Step 3, Eqs. (6.2)-(6.5) (pp.318-319)

4. 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.

   $$s=N_P(p,q)+N_Q(p,q)+N_I(p,q)+N_E(p,q)+N_P(q,p)+N_Q(q,p)+N_I(q,p);x_p{D}_q{x}_q\iff N_P(q,p)+N_Q(q,p)=0\wedge N_I(q,p)<N_I(p,q)+N_Q(p,q)+N_P(p,q);x_p{D}_c{x}_q\iff N_P(q,p)=0\wedge N_Q(q,p)\le N_P(p,q)\wedge N_Q(q,p)+N_I(q,p)<N_P(p,q)+N_Q(p,q)+N_I(p,q);x_p{D}_p{x}_q\iff N_P(q,p)=0\wedge N_Q(q,p)\le N_P(p,q)+N_Q(p,q);x_p{D}_s{x}_q\iff N_P(q,p)=0;x_p{D}_v{x}_q\iff N_P(q,p)=0\vee (N_P(q,p)=1\wedge s(x_q)-s(x_p)\le {\nu }_{j^{\ast }}\wedge N_P(p,q)\ge s/2)$$

   Akram-Adeel 2023, §6.2 Step 4 (pp.319-321)

5. 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.

   $$\zeta (x_p,x_q)=\{\begin{array}{ll}1,& x_p{D}_q{x}_q\\ 0.8,& x_p{D}_c{x}_q\\ 0.6,& x_p{D}_p{x}_q\\ 0.4,& x_p{D}_s{x}_q\\ 0.2,& x_p{D}_v{x}_q\text{ (alternate)}\\ 0,& \text{otherwise}\end{array}$$

   Akram-Adeel 2023, §6.2 Step 5, Vallée-Zielniewicz [57] (p.321)

6. 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).

   $$s({\zeta }_{\lambda })=\alpha {\zeta }_{\lambda }+\beta [\text{ELECTRE-IV: }\alpha =0,\beta =0.1];{\zeta }_{\lambda +1}={\zeta }_{\lambda }-s({\zeta }_{\lambda });x_p{D}_A^{{\zeta }_{\lambda }}{x}_q\iff \zeta (x_p,x_q)>{\zeta }_{\lambda }\wedge \zeta (x_p,x_q)-\zeta (x_q,x_p)>s(\zeta (x_p,x_q));S_A^{{\zeta }_{\lambda }}(x_p)=\sum _{x_q\in M^{+}}\zeta (x_p,x_q);{\bar{S}}_A^{{\zeta }_{\lambda }}(x_p)=\sum _{x_q\in M^{-}}\zeta (x_q,x_p);Q_A^{{\zeta }_{\lambda }}(x_p)=S_A^{{\zeta }_{\lambda }}(x_p)-{\bar{S}}_A^{{\zeta }_{\lambda }}(x_p);\text{final pre-order}=\text{ascending pre-order}\cap \text{descending pre-order}$$

   Akram-Adeel 2023, §6.2 Step 6, Eqs. (6.6)-(6.8) and ascending/descending distillation algorithm (pp.322-324)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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

• 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 - non-strict in MPF-ELECTRE-IV. The book §6.2 Step 3 prints 'q ≤ p ≤ ν' (Chapter 6, p.318), explicitly weaker than §5.2's 'q < p ≤ ν' (Chapter 5, p.289). MPF-ELECTRE-IV permits q_j = p_j (the weak-preference interval collapses to empty and only strict P / indifference I survive). The Qazvin case keeps q_j < p_j ≤ ν_j throughout (Table 6.7), but implementations must NOT enforce strict q < p on MPF-ELECTRE-IV input - this would reject legal inputs that match the book's specification. E-3 uses ≤.

• ζ = 0 means 'no dominance relation', not an additional dominance class. The book lists six ζ values (1, 0.8, 0.6, 0.4, 0.2, 0) but only FIVE dominance classes (Dq, Dc, Dp, Ds, Dv). The sixth value 0 covers ordered pairs that fail every dominance condition - i.e., the credibility of the ordered pair is zero and the pair contributes nothing to the strength of x_p at any cut-off. Do NOT invent a 'D_0' dominance class.

• Veto dominance Dv has TWO alternative conditions. Primary: NP(q,p) = 0 (same as Ds - every alternative satisfying Ds also satisfies the primary Dv condition; the strongest-class rule selects Dq/Dc/Dp/Ds first). Alternate: NP(q,p) = 1 AND for that single strongly-preferred criterion j*, s(x_q) − s(x_p) ≤ ν_{j*} AND NP(p,q) ≥ s/2. Implementations that drop the alternate condition will miss the genuine Dv assignments - and assign ζ=0.2 (which is the LOWEST non-zero credibility level) to pairs that should have ζ=0. The alternate Dv condition is the only place where the veto threshold ν_j actually enters the algorithm.

• Discrimination threshold: ELECTRE-IV uses the constant 0.1, not the ELECTRE-III ramp. Akram & Adeel 2023 §6.2 (p.322): 'In ELECTRE IV procedure, the value of discrimination threshold is fixed as 0.1 by taking α = 0 and β = 0.1 to capture each dominance level.' Reusing the ELECTRE-III default (α=-0.15, β=0.30) silently increases the threshold spacing and may cause distillation to terminate earlier with coarser pre-orders. Always set discrimination_alpha=0 and discrimination_beta=0.1 unless you are deliberately replicating ELECTRE-III discrimination behaviour.

• Intersection of pre-orders, not their union or average. The final pre-order is the INTERSECTION of the ascending and descending distillation pre-orders. This means x_p ≻ x_q in the final pre-order requires strict agreement in BOTH distillations. Taking the union (≻ in either) or the average rank (mean of the two ranks) is a frequent implementation bug and produces a different - generally incorrect - output. The book §6.2 last sentence (p.324): 'The final ranking is derived by the intersection of both ranking lists.'

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