MPF-DOMBI-WA: m-Polar Fuzzy Dombi Weighted Averaging / Geometric MCDM (Akram, Yaqoob, Ali & Chammam 2020 / Akram & Adeel 2023 Ch. 8): single-DM m-PF MCDM ranking via Dombi t-conorm/t-norm-based aggregation operators (mFDWA primary, mFDWG companion) followed by m-PF score-based descending ordering

Overview

MPF-DOMBI-WA, Chen-Li-Ma (2014) tarafından önerilen m-Polar Fuzzy Set (mFS) kavramı üzerine, Dombi (1982) t-norm/t-konormunun parametre-kontrollü yumuşaklık esnekliği eklenmiş bir ağırlıklandırılmış aggregation operatörüdür. mFS'in özü şudur: tek bir alternatife AYNI ANDA m bağımsız kriterden gelen üyelik dereceleri (p₁∘ζ(u), ..., pₘ∘ζ(u)) ∈ [0,1]^m tanımlanır: multi-source / multi-agent değerlendirme tek bir yapısal nesnede tutulur (örn. m=4 → 4 farklı uzmanın 4 alternatif kriter üzerine üyelik dereceleri tek mFN olarak temsil edilir). Dombi t-normu D(a,b) = 1/(1+{(a/(1-a))^k+(b/(1-b))^k}^(1/k)) k≥1 parametresi ile AGGREGATION FİLTRESİNİN sertliğini kontrol eder: k→1 algebraic-toplam benzeri (yumuşak konsensüs), k→∞ minimum benzeri (en sert/pessimist). Bu manifestte yalnız mFDWA (Definition/Eq.6: weighted averaging) ele alınır: paper'da ayrıca tanımlı mFDOWA (ordered, Eq.16) ve mFDHA (hybrid, Eq.22) varianti F.steps kapsam dışıdır.

Strengths

• •Method-specific: Dombi-norm aggregation operator MCDM: Dombi sum/product on m-PF numbers (Eq. 4), weighted averaging (mFDWA, Eq. 6) and weighted geometric (mFDWG, Def. 9) operators, m-PF score function S(ζ)=(1/m)Σ p_i◦ζ (Eq. 1) ranking
• •Preserves m_polar uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: Akram-Yaqoob-Ali-Chammam-2020-Sec-6.1-Wang-Triantaphyllou-tests)

Edge cases

• •MPF-DOMBI-WA kritik edge case'leri: (1) p_r∘ζ_ij = 1 veya 0 → Dombi formülünde a/(1-a) bölünme: a=1 için ∞, a=0 için 0 → numerik stabilite için ε-clamp (typically ε=1e-9, [ε, 1-ε] aralığına bastır) önerilir (paper §2 Remark 1 işaret eder); engine bu default'u izler. (2) m=1 dejenerasyon → mFS klasik fuzzy set'e iner; mFDWA tek-kutuplu Dombi WA olur (geometric karşılığı mFDWG ile ayrı sonuç verir). (3) k→1 → algebraic toplam-üreten konsensüs (yumuşak); k→∞ → minimum-yakın aggregation (sert/pessimist): kullanıcı k duyarlılık testini Manifest verification step'inde yapmalı. (4) Tüm w_j eşit (w_j=1/n) → mFDWA simple averaging'e iner; ağırlık asimetrisi büyüdükçe Dombi non-linear etkisi belirgin olur.

Common pitfalls

• •Tipik öğrenci hataları: (a) Score Def.2'yi Atanassov IFS score (μ-ν) ile karıştırmak: mFS'te VETO değil ORTALAMA (1/m)Σ p_r∘ζ; (b) Accuracy Eq.2'deki (-1)^(r+1) işaret değişimini unutmak: bu Atanassov-style hesitation degree değil, multi-pole alternating sign: paper formülünü VERBATİM uygulayın; (c) k parametresini criterion weight gibi normalize etmeye çalışmak: k aggregation filtresi parametresidir, ağırlık değil; (d) mFDOWA ile mFDWA'yı karıştırmak: ordered varianti önce Score sıralaması yapar, weighted varianti sırasız w_j ağırlıklarını doğrudan kullanır; (e) m kutuplu yapıyı m kriterli skaler matris ile karıştırmak: burada her HÜCRE m-boyutlu vektördür, m-pole değil m-criteria değil.

Worked example

1. 1.Adım 1 (F1): Ingest the m-PF decision matrix S = (s_{ij})_{t×n} where s_{ij} = (p_1◦ζ_{ij},…,p_m◦ζ_{ij}) is an m-PF number, the criterion weight vector Θ = (Θ_1,…,Θ_n) with Σ_j Θ_j = 1 and Θ_j ∈ (0,1], and the Dombi parameter k ≥ 1 and operator choice ∈ {mFDWA, mFDWG}. Validate domain constraints E-1 through E-6. Formül: S = (s_{ij})_{t \times n}, \quad s_{ij} = (p_1 \circ \zeta_{ij}, \ldots, p_m \circ \zeta_{ij}), \quad p_r \circ \zeta_{ij} \in [0,1]; \quad \Theta = (\Theta_1,\ldots,\Theta_n), \quad \sum_{j=1}^n \Theta_j = 1, \quad \Theta_j \in (0,1]; \quad k \ge 1. Anchor: Sec. 4 (input preamble) / Algorithm 1 Step 1
2. 2.Adım 2 (F2): Apply the chosen Dombi aggregation operator row-wise to obtain the preference value s_i for each alternative Y_i (i=1,…,t). For mFDWA (Definition 5, Theorem 2, Eq. 6), pole-r coordinate of s_i is 1 − 1/(1 + {Σ_j Θ_j ((p_r◦ζ_{ij})/(1 − p_r◦ζ_{ij}))^k}^{1/k}). For mFDWG (Definition 8, Theorem 9), pole-r coordinate of s_i is 1/(1 + {Σ_j Θ_j ((1 − p_r◦ζ_{ij})/(p_r◦ζ_{ij}))^k}^{1/k}). Both produce m-PF numbers s_i = (p_1◦ζ_{i·}, …, p_m◦ζ_{i·}) ∈ [0,1]^m. Formül: \text{mFDWA: } s_i = \mathrm{mFDWA}_{\Theta}(s_{i1}, \ldots, s_{in}) = \oplus_{j=1}^{n}\bigl(\Theta_j\, s_{ij}\bigr) = \Bigl(1 - \tfrac{1}{1+\bigl\{\sum_{j=1}^{n}\Theta_j\,(\tfrac{p_1\circ\zeta_{ij}}{1-p_1\circ\zeta_{ij}})^k\bigr\}^{1/k}},\ \ldots,\ 1 - \tfrac{1}{1+\bigl\{\sum_{j=1}^{n}\Theta_j\,(\tfrac{p_m\circ\zeta_{ij}}{1-p_m\circ\zeta_{ij}})^k\bigr\}^{1/k}}\Bigr). \quad \text{mFDWG: } s_i = \mathrm{mFDWG}_{\Theta}(s_{i1}, \ldots, s_{in}) = \otimes_{j=1}^{n}\bigl((s_{ij})^{\Theta_j}\bigr) = \Bigl(\tfrac{1}{1+\bigl\{\sum_{j=1}^{n}\Theta_j\,(\tfrac{1-p_1\circ\zeta_{ij}}{p_1\circ\zeta_{ij}})^k\bigr\}^{1/k}},\ \ldots,\ \tfrac{1}{1+\bigl\{\sum_{j=1}^{n}\Theta_j\,(\tfrac{1-p_m\circ\zeta_{ij}}{p_m\circ\zeta_{ij}})^k\bigr\}^{1/k}}\Bigr). Anchor: Def. 5 / Eq. 6 (mFDWA); Def. 8 / Theorem 9 (mFDWG); Algorithm 1 Step 2
3. 3.Adım 3 (F3): Compute the m-PF score value S(s_i) ∈ [0,1] for each alternative Y_i using Definition 2 / Eq. 1: S(s_i) = (1/m) Σ_{r=1}^{m} p_r◦ζ_{i·}. The score is the unweighted arithmetic mean of pole memberships and serves as the ranking variable. When two alternatives have identical scores, the accuracy function H(s_i) = (1/m) Σ_{r=1}^{m} (−1)^{r+1} (p_r◦ζ_{i·} − 1) ∈ [−1, 1] (Definition 3 / Eq. 2) is used as a tie-break. Formül: S(s_i) = \frac{1}{m} \sum_{r=1}^{m} p_r \circ \zeta_{i\cdot} \in [0,1]; \quad H(s_i) = \frac{1}{m} \sum_{r=1}^{m} (-1)^{r+1} (p_r \circ \zeta_{i\cdot} - 1) \in [-1, 1]. Anchor: Def. 2 / Eq. 1 (score) and Def. 3 / Eq. 2 (accuracy); Algorithm 1 Step 3
4. 4.Adım 4 (F4): Rank the alternatives in descending order of S(s_i). Break ties (S(s_i)=S(s_{i'})) using descending H(s_i) per Definition 4 of the seminal paper. The best alternative is the one with the highest score (and, if tied, highest accuracy). Formül: Y_{i_1} \succeq Y_{i_2} \succeq \ldots \succeq Y_{i_t} \iff S(s_{i_1}) \ge S(s_{i_2}) \ge \ldots \ge S(s_{i_t}); \quad \text{tie-break: if } S(s_i) = S(s_{i'}) \text{ then } Y_i \succ Y_{i'} \iff H(s_i) > H(s_{i'}). Anchor: Def. 4 (ordering); Algorithm 1 Step 4

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