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

Read the result as a complete descending ranking of alternatives Y_i. The alternative with the highest m-PF score S(s_i) = (1/m) Σ_{r=1}^{m} p_r◦ζ_{i·} is the best choice. The aggregated m-PF preference value s_i (visible as a polygon in H3) shows the per-pole strength profile of each alternative - useful for interpreting WHY one alternative dominates. When two alternatives have identical scores, the accuracy H(s_i) (held internally) breaks ties: higher H wins. Switching operator_choice from mFDWA (arithmetic, default) to mFDWG (geometric) can reorder the ranking - Sec. 5/6 of the seminal paper shows a complete reversal on the agriculture case, so document the operator alongside the result.

Output

rank position, higher is better

Data

M-Polar Fuzzy, uncertainty tuples complete

Size

2+ alternatives, 3-10 criteria works best

Used for

M-Polar Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   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.

   $$S=(s_{ij}{)}_{t\times n},s_{ij}=(p_1\circ {\zeta }_{ij},\dots ,p_m\circ {\zeta }_{ij}),p_r\circ {\zeta }_{ij}\in [0,1];\Theta =({\Theta }_1,\dots ,{\Theta }_n),\sum _{j=1}^n{\Theta }_j=1,{\Theta }_j\in (0,1];k\ge 1.$$

   Sec. 4 (input preamble) / Algorithm 1 Step 1

2. 2

   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.

   $$\text{mFDWA: }{s}_i={\mathrm{mFDWA}}_{\Theta }(s_{i1},\dots ,s_{in})={\oplus }_{j=1}^n\left({\Theta }_j{s}_{ij}\right)=(1-\frac{1}{1+\{{\sum }_{j=1}^n{\Theta }_j(\frac{p_1\circ {\zeta }_{ij}}{1-p_1\circ {\zeta }_{ij}}{)}^k{\}}^{1/k}},\dots ,1-\frac{1}{1+\{{\sum }_{j=1}^n{\Theta }_j(\frac{p_m\circ {\zeta }_{ij}}{1-p_m\circ {\zeta }_{ij}}{)}^k{\}}^{1/k}}).\text{mFDWG: }{s}_i={\mathrm{mFDWG}}_{\Theta }(s_{i1},\dots ,s_{in})={\otimes }_{j=1}^n((s_{ij}{)}^{{\Theta }_j})=(\frac{1}{1+\{{\sum }_{j=1}^n{\Theta }_j(\frac{1-p_1\circ {\zeta }_{ij}}{p_1\circ {\zeta }_{ij}}{)}^k{\}}^{1/k}},\dots ,\frac{1}{1+\{{\sum }_{j=1}^n{\Theta }_j(\frac{1-p_m\circ {\zeta }_{ij}}{p_m\circ {\zeta }_{ij}}{)}^k{\}}^{1/k}}).$$

   Def. 5 / Eq. 6 (mFDWA); Def. 8 / Theorem 9 (mFDWG); Algorithm 1 Step 2

3. 3

   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.

   $$S(s_i)=\frac{1}{m}\sum _{r=1}^m{p}_r\circ {\zeta }_{i\cdot }\in [0,1];H(s_i)=\frac{1}{m}\sum _{r=1}^m(-1{)}^{r+1}(p_r\circ {\zeta }_{i\cdot }-1)\in [-1,1].$$

   Def. 2 / Eq. 1 (score) and Def. 3 / Eq. 2 (accuracy); Algorithm 1 Step 3

4. 4

   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).

   $$Y_{i_1}⪰Y_{i_2}⪰\dots ⪰Y_{i_t}⟺S(s_{i_1})\ge S(s_{i_2})\ge \dots \ge S(s_{i_t});\text{tie-break: if }S(s_i)=S(s_{i^{\prime }})\text{ then }{Y}_i\succ Y_{i^{\prime }}⟺H(s_i)>H(s_{i^{\prime }}).$$

   Def. 4 (ordering); Algorithm 1 Step 4

Fits when

Edge cases and pitfalls

• •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.

• Boundary values 0 and 1: the Dombi operations of Eq. 4 contain (p_r◦ζ)/(1−p_r◦ζ) (mFDWA) and (1−p_r◦ζ)/(p_r◦ζ) (mFDWG). When p_r◦ζ=1 (mFDWA) or p_r◦ζ=0 (mFDWG), the term diverges. The seminal paper restricts to the open interval (0,1) implicitly. Caller must either reject such matrices or apply a small clamp (e.g. ε=1e-6).

• Operator-choice sensitivity (Sec. 6 Tables 6-7): mFDWA and mFDWG produce different rankings on the same matrix (the agriculture case yields Y_2≻Y_5≻Y_4≻Y_1≻Y_3 under mFDWA but Y_3≻Y_1≻Y_4≻Y_2≻Y_5 under mFDWG). The two operators are not equivalent; arithmetic favours large pole memberships while geometric is more conservative. Document the choice explicitly.

• Dombi parameter k must be ≥ 1: although Dombi (1982) admits all k>0, the seminal paper restricts to k≥1 because for 0<k<1 the operator behaves erratically; the case studies use k=3. Sensitivity analysis on k is recommended.

• No criterion-direction handling: the seminal paper assumes all criteria are benefit-direction. Cost criteria (e.g. price) are encoded directly as m-PF numbers, not complemented. If the caller wants min-direction handling (e.g. price-as-cost), they must complement (1 − p_r◦ζ) upstream before passing to MPF-DOMBI-WA - this is OUT OF SCOPE for the method.

• Single-DM only: Algorithm 1 ingests one decision matrix from a single expert. Multi-DM group decision making (MCGDM) is NOT supported by the canonical paper. For MCGDM, an upstream m-PF averaging operator must reduce the per-DM matrices to a single matrix before MPF-DOMBI-WA is applied.

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