MHF-TOPSIS: m-Polar Hesitant Fuzzy extension of TOPSIS

Overview

MHF-TOPSIS, Akram, Adeel ve Alcantud (2019, Symmetry) tarafından önerilen m-POLAR HESITANT FUZZY TOPSIS yaklaşımıdır; m-polar fuzzy set (Chen et al. 2014) ile hesitant fuzzy set (Torra 2010) füzyonuyla doğan mHF-set kavramına dayanır. Bir mHF-element ℏ_m(z) = (P_1, P_2, ..., P_m) şeklinde m-tuple HFE'dir; her pole (boyut) için ayrı bir hesitant fuzzy değer kümesi tutulur. Bunun sezgisi şudur: tek bir kriter çoğu zaman TEK boyutlu (skalar) değil çok-kutuplu (örn. marka ismi: 'Articulate Core Identity' = Vision + Mission + Value + Direction, 4-pole 3HFE); her pole için DM hesitancy gösterir. Paper TWO fundamental novelty getirir: (i) Score Def.2: pole-bazlı aritmetik ortalama (m-tuple skor); Deviation Def.3: pole-bazlı standart sapma (tie-breaker); (ii) PIS/NIS pole-bazlı max/min ile tanımlanır: klasik TOPSIS'in 'closeness coefficient' fikri korunur, ama uzaklık Eq.3 m×r ÇİFTLİ TOPLAMA dönüşür (m pole × r hesitancy değeri). Brand-name uygulamasında DM 'optimistic spirit' seçer (eksik değerler MAX ile uzatılır); product-design'da 'pessimistic spirit' (MIN ile uzatılır).

Strengths

• •Method-specific: m-Polar Hesitant outranking/ranking: m-Polar Hesitant Fuzzy Element (mHFE: m-tuple of HFEs, ℏ_m: Z → P([0,1]^m))
• •Preserves hesitant 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: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •Assumes: Decision matrix entries are valid Multi-Hesitant Fuzzy numbers/tuples
• •Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• •Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

Assumptions to verify

• •Decision matrix entries are valid Multi-Hesitant Fuzzy numbers/tuples
• •Underlying crisp method's compensation assumption holds in uncertain space
• •All decision-maker(s) and experts use the same linguistic/uncertainty scale

When not to use

• •Crisp data sufficient: use base TOPSIS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •MHF-TOPSIS için kritik edge case'ler: (1) Pole uzunluğu eşitleme: DM farklı pole için farklı sayıda hesitancy değeri verirse, paper §3 'optimistic = add max' veya 'pessimistic = add min' politikası gerektirir; engine policy parametresi default = 'optimistic' (paper §3.1 brand-name kanonu). (2) m=1 özel durum: mHF-set indirgenir klasik HF-set; mHF-TOPSIS indirgenir HF-TOPSIS (Xu-Zhang 2013). (3) Pole sayısı m çok büyük (>5): Euclidean Eq.3'te m faktörü payda 'rm' içine girer, normalize edilir; ama hesaplama maliyeti q·m·r mertebesindedir. (4) w_k normalizasyonu Σ w_k = 1, eksik veri durumunda paper 'equal weights' önerir: engine bu default'u izler. (5) PIS/NIS pole-bazlı max: Eq.1 (ℏ_m^k')+ = (max_j sup{ζ_h ∈ p_1 ℏ_m^jk}, ..., max_j sup{ζ_h ∈ p_m}); her pole bağımsız extremum, tek-skalar değil VEKTÖREL.

Common pitfalls

• •Doktora öğrencisi tuzakları: (a) mHF-set ile interval-valued HF-set'i (IVHFS, Chen 2013) karıştırmak: IVHFS'te her hesitancy değeri bir [a,b] aralığı; mHF-set'te her kriter m boyutlu pole'a sahip, her pole HFE; (b) Optimistic/pessimistic fusion policy'i 'kullanıcı keyfine bağlı' sanmak: paper §3.1-3.2 SCENARIO-driven seçim yapar (brand kararı doğal optimistic; product reliability doğal pessimistic); (c) Pole-bazlı Score'u (s(ℏ_m) = m-tuple) skalar score ile karıştırmak: Remark 1'de skor-bazlı sıralama 'totally different' verdiğinde Deviation Δ tie-breaker olur (Remark 2); (d) Eq.3 payın çift toplamını (q kriter dış Σ + m pole iç Σ + r hesitancy iç-iç Σ) görmemek: toplam 3-katlı; (e) Weighted matrix Eq Step 3 ℏ_m^jk' = w_k · ℏ_m^jk: skalar w_k her hesitancy elemanına dağıtılır, m pole'a ayrı ayrı, ama w_k m'ye değil k'ye bağlıdır (kriter-bazlı tek ağırlık).

Worked example

1. 1.Adım 1 (F1): Step 1 (Akram et al. 2019, p.10): Assemble the mHF decision matrix H = (ℏ_m^{jk})_{p×q} whose entries are mHFEs ℏ_m^{jk}(z) = ({ζ_h | ζ_h ∈ p_1 ∘ ℏ_m^{jk}(z)}, …, {ζ_h | ζ_h ∈ p_m ∘ ℏ_m^{jk}(z)}). j indexes alternatives a_j ∈ A (j=1,…,p), k indexes criteria c_k ∈ C (k=1,…,q). Formül: H = \big( \hbar_{m}^{jk} \big)_{p \times q}, \quad \hbar_{m}^{jk}(z) = \Big( \{\zeta_{h} \mid \zeta_{h} \in p_{1} \circ \hbar_{m}^{jk}(z)\}, \ldots, \{\zeta_{h} \mid \zeta_{h} \in p_{m} \circ \hbar_{m}^{jk}(z)\} \Big) Anchor: Akram et al. 2019, p.10 Step 1 (Table 1)
2. 2.Adım 2 (F2): Step 1, continued (Akram et al. 2019, p.11 + §3.1 p.14 + §3.2 p.17): Prolong each pole-HFE until all HFEs within H share the same cardinality r. If prolongation_mode='optimistic', repeat the maximum value of each HFE; if 'pessimistic', repeat the minimum. r is set to the largest pole-HFE cardinality observed across H (so that the maximal information length is preserved). Formül: r = \max_{j,k,i} |p_{i} \circ \hbar_{m}^{jk}(z)|; \quad \text{optimistic: } p_{i} \circ \hbar_{m}^{jk}(z) \leftarrow p_{i} \circ \hbar_{m}^{jk}(z) \cup \{\max p_{i} \circ \hbar_{m}^{jk}(z)\}^{r - |p_{i} \circ \hbar_{m}^{jk}(z)|}; \quad \text{pessimistic: } \text{repeat } \min \text{ instead} Anchor: Akram et al. 2019, p.11 ¶3 + p.14 ¶1 (§3.1 optimistic) + p.17 ¶3 (§3.2 pessimistic)
3. 3.Adım 3 (F3): Step 3 (Akram et al. 2019, p.11 Table 2): Build the weighted mHF decision matrix H′ by scalar multiplication of each cell with its criterion weight w_k (no DHFE-style n·d power operator; literal element-wise scalar product of each ζ_h ∈ p_i ∘ ℏ_m^{jk}). Formül: \hbar_{m}^{jk'} = w_{k} \hbar_{m}^{jk} = \Big( w_{k}\{\zeta_{h} \mid \zeta_{h} \in p_{1} \circ \hbar_{m}^{jk}(z)\}, \ldots, w_{k}\{\zeta_{h} \mid \zeta_{h} \in p_{m} \circ \hbar_{m}^{jk}(z)\} \Big) = \Big( \{\zeta_{h}^{'} \mid \zeta_{h}^{'} \in p_{1} \circ \hbar_{m}^{jk}(z)\}, \ldots, \{\zeta_{h}^{'} \mid \zeta_{h}^{'} \in p_{m} \circ \hbar_{m}^{jk}(z)\} \Big) Anchor: Akram et al. 2019, p.11 Table 2 + p.11 unnumbered equation
4. 4.Adım 4 (F4): Steps 4-5 (Akram et al. 2019, pp.11-12 Eqs.(1)-(2)): Compute the mHF positive-ideal solution mHP_IS by taking the column-wise maximum on each pole component of H′, and the mHF negative-ideal solution mHN_IS by the column-wise minimum on each pole component. Formül: mHP_{IS} = \{(\hbar_{m}^{1'})^{+}, (\hbar_{m}^{2'})^{+}, \ldots, (\hbar_{m}^{q'})^{+}\}, \quad (\hbar_{m}^{k'})^{+} = \max_{j}(\hbar_{m}^{jk'}) = \Big( \{(\zeta_{h}^{'})^{+} \mid (\zeta_{h}^{'})^{+} \in p_{1} \circ \hbar_{m}^{jk}(z)\}, \ldots, \{(\zeta_{h}^{'})^{+} \mid (\zeta_{h}^{'})^{+} \in p_{m} \circ \hbar_{m}^{jk}(z)\} \Big); \quad mHN_{IS} = \{(\hbar_{m}^{1'})^{-}, \ldots, (\hbar_{m}^{q'})^{-}\}, \quad (\hbar_{m}^{k'})^{-} = \min_{j}(\hbar_{m}^{jk'}) Anchor: Akram et al. 2019, p.11 Eq.(1) + p.12 Eq.(2)
5. 5.Adım 5 (F5): Step 6 (Akram et al. 2019, p.12 Eqs.(3)-(4)): Compute the mHF Euclidean distance of each alternative a_j from mHP_IS and mHN_IS. The distance averages over all q criteria, all m poles, and all r prolonged HFE positions; the 1/(rm) normalisation factor is verbatim from the paper. Formül: D_{e}^{'}(a_{j}, mHP_{IS}) = \sqrt{ \tfrac{1}{rm} \sum_{k=1}^{q} \Big[ \sum_{i=1}^{m} \big\{ (\zeta_{h1}^{jk'} - (\zeta_{h1}^{k'})^{+})^{2} + (\zeta_{h2}^{jk'} - (\zeta_{h2}^{k'})^{+})^{2} + \cdots + (\zeta_{hr}^{jk'} - (\zeta_{hr}^{k'})^{+})^{2} \big\} \Big] }; \quad D_{e}^{'}(a_{j}, mHN_{IS}) = \sqrt{ \tfrac{1}{rm} \sum_{k=1}^{q} \Big[ \sum_{i=1}^{m} \big\{ (\zeta_{h1}^{jk'} - (\zeta_{h1}^{k'})^{-})^{2} + (\zeta_{h2}^{jk'} - (\zeta_{h2}^{k'})^{-})^{2} + \cdots + (\zeta_{hr}^{jk'} - (\zeta_{hr}^{k'})^{-})^{2} \big\} \Big] } Anchor: Akram et al. 2019, p.12 Eqs.(3)-(4)
6. 6.Adım 6 (F6): Steps 7-8 (Akram et al. 2019, p.12 Eq.(5)): Relative mHF closeness coefficient E_j′; rank alternatives in descending order of E_j′. The alternative with the highest E_j′ is the best. Formül: E_{j}^{'} = \dfrac{D_{e}^{'}(a_{j}, mHN_{IS})}{D_{e}^{'}(a_{j}, mHP_{IS}) + D_{e}^{'}(a_{j}, mHN_{IS})}, \quad j = 1, 2, \ldots, p; \quad 0 \le E_{j}^{'} \le 1 Anchor: Akram et al. 2019, p.12 Eq.(5) + p.13 Algorithm 1 Steps 7-8

Commonly paired with

• •n_a + MHF-TOPSIS (common)

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