DHF-TOPSIS: Dual Hesitant Fuzzy extension of TOPSIS

Overview

Dual Hesitant outranking/ranking: Dual Hesitant Fuzzy Element (DHFE: h(x) membership set, g(x) non-membership set). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Dual Hesitant outranking/ranking: Dual Hesitant Fuzzy Element (DHFE: h(x) membership set, g(x) non-membership set)
• •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 Dual 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 Dual 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

• •If weights are completely unknown, derive them via the maximising-deviation model (Wang et al. 2020, Eq.(27)) using the DHFE Hamming distance d_{h} (Eq.(17)).

Common pitfalls

• •Hatalı: 'DHF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Dual Hesitant Fuzzy numbers/tuples
• •Hatalı: 'DHF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'DHF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: DHF-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: DHF-TOPSIS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Generate the DHF decision matrix D=(a_{ij})_{m×n} whose entries are DHFEs a_{ij}=(h_{ij}, g_{ij}); no normalisation is applied (Wang et al. 2020, Eq.(28)). Formül: D = \big( a_{ij} \big)_{m \times n} = \big( \langle h_{ij}, g_{ij} \rangle \big)_{m \times n} Anchor: Wang et al. 2020, p.11 Eq.(28)
2. 2.Adım 2 (F2): Step 2: Retrieve attribute weights w_{j} (Σw_{j}=1, 0≤w_{j}≤1). If weights are completely unknown, derive them via the maximising-deviation model (Wang et al. 2020, Eq.(27)) using the DHFE Hamming distance d_{h} (Eq.(17)). Formül: w_{j} = \dfrac{\sum_{i=1}^{m}\sum_{k=1}^{m} d_{h}(a_{ij}, a_{kj})}{\sum_{j=1}^{n}\Big(\sum_{i=1}^{m}\sum_{k=1}^{m} d_{h}(a_{ij}, a_{kj})\Big)}, \quad j=1,\ldots,n Anchor: Wang et al. 2020, p.10 Eq.(27)
3. 3.Adım 3 (F3): Step 3: Acquire the positive-ideal A⁺ and negative-ideal A⁻ solutions by column-wise extrema on the DHFE components, direction-aware (benefit ⇒ max on h, min on g; cost ⇒ swap). Formül: A^{+} = \Big[\langle \max_{i}\{\gamma_{i j}\}, \min_{i}\{\eta_{i j}\} \rangle \Big]_{j=1}^{n}, \quad A^{-} = \Big[\langle \min_{i}\{\gamma_{i j}\}, \max_{i}\{\eta_{i j}\} \rangle \Big]_{j=1}^{n}, \quad \text{benefit; for cost criteria, swap A^{+} and A^{-} component-wise} Anchor: Wang et al. 2020, p.11 Eqs.(29)-(30)
4. 4.Adım 4 (F4): Step 4: Compute the weighted generalised DHF distance d_{wpg} (Wang et al. 2020, Eq.(21)) between each alternative A_{i} and the ideals A⁺, A⁻. The distance uses mean functions G_{m}, G_{n} and volatility functions V_{m}, V_{n} (Defs.8; Eqs.(9)-(12)) and does not require length-equalisation of DHFE sets. Formül: G_{m}(e) = \tfrac{1}{l(h)}\sum_{\gamma \in h}\gamma, \quad G_{n}(e) = \tfrac{1}{k(g)}\sum_{\eta \in g}\eta; \quad V_{m}(e) = \tfrac{1}{2}\Big[\tfrac{1}{l(h)}\sum_{\gamma \in h}(\gamma - G_{m})^{2} + \sqrt{1 - \tfrac{1}{1 + \ln l(h)}}\Big], \quad V_{n}(e) = \tfrac{1}{2}\Big[\tfrac{1}{k(g)}\sum_{\eta \in g}(\eta - G_{n})^{2} + \sqrt{1 - \tfrac{1}{1 + \ln k(g)}}\Big]; \quad d_{wpg}(A_{i}, A^{\pm}) = \Big(\tfrac{1}{2}\sum_{j=1}^{n} w_{j}\big[\alpha\,|G_{m}(a_{ij}) - G_{m}(a_{j}^{\pm})|^{\lambda} + \alpha\,|G_{n}(a_{ij}) - G_{n}(a_{j}^{\pm})|^{\lambda} + \beta\,|V_{m}(a_{ij}) - V_{m}(a_{j}^{\pm})|^{\lambda} + \beta\,|V_{n}(a_{ij}) - V_{n}(a_{j}^{\pm})|^{\lambda}\big]\Big)^{1/\lambda}, \quad \alpha + \beta = 1,\ \lambda \ge 1 Anchor: Wang et al. 2020, pp.6-7 Defs.8, Eqs.(9)-(12), (21)
5. 5.Adım 5 (F5): Step 5: Closeness coefficient CC_{i} per Eq.(31); rank alternatives in descending order of CC_{i}. Formül: CC_{i} = \dfrac{d_{wpg}(A_{i}, A^{-})}{d_{wpg}(A_{i}, A^{+}) + d_{wpg}(A_{i}, A^{-})}, \quad 0 \le CC_{i} \le 1, \quad i = 1, \ldots, m Anchor: Wang et al. 2020, p.11 Eq.(31)

Commonly paired with

• •n_a + DHF-TOPSIS (common)

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