DHF-EDAS: Dual Hesitant Fuzzy extension of EDAS

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 EDAS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for DHF-EDAS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'DHF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Dual Hesitant Fuzzy numbers/tuples
• •Hatalı: 'DHF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'DHF-EDAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: DHF-EDAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: DHF-EDAS'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 (Ning §4): Obtain p per-expert decision matrices Ʒ=(Ʒ_{ij}^k)_{m×n} (k=1,…,p) of PDHFEs Ʒ_{ij}^k=⟨ħ_{ij}^k|τ_{ij}^k, λ_{ij}^k|υ_{ij}^k⟩. Receive the expert weight vector θ=(θ_1,…,θ_p) with Σθ_k=1, the subjective attribute weight vector w=(w_1,…,w_n) with Σw_j=1, and the attribute direction set (benefit B vs cost C). Pure DHFE inputs are the special case τ_i=1/#ħ, υ_j=1/#λ (uniform probability tags). Formül: \Im = (\Im_{ij}^{k})_{m\times n},\quad k=1,\dots,p;\ \Im_{ij}^{k}=\langle \hbar_{ij}^{k}|\tau_{ij}^{k},\ \lambda_{ij}^{k}|\upsilon_{ij}^{k}\rangle;\ \sum_{k=1}^{p}\theta_{k}=1;\ \sum_{j=1}^{n}w_{j}=1 Anchor: Ning et al. 2023 §4 Step 1
2. 2.Adım 2 (F2): Step 2 (Ning Eq.21): Aggregate the p per-expert PDHF matrices into the collective decision matrix D=(d_{ij})_{m×n} via the PDHFWA (Probabilistic Dual Hesitant Fuzzy Weighted Average) operator parameterised by expert weights θ_k. Formül: d_{ij} = \bigoplus_{k=1}^{p} \theta_{k}\,\Im_{ij}^{k} = \bigcup_{\gamma_{ij}^{k}\in \hbar_{ij}^{k},\ \eta_{ij}^{k}\in \lambda_{ij}^{k}} \left\{ \left\{\left[1-\prod_{k=1}^{p}\bigl(1-\gamma_{ij}^{k}\bigr)^{\theta_{k}}\right]\Big|\prod_{k=1}^{p} p_{\gamma_{ij}^{k}}\right\},\ \left\{\prod_{k=1}^{p}\bigl(\eta_{ij}^{k}\bigr)^{\theta_{k}}\Big|\prod_{k=1}^{p} q_{\eta_{ij}^{k}}\right\} \right\} Anchor: Ning et al. 2023 §4 Step 2, Eq.(21)
3. 3.Adım 3 (F3): Step 3 (Ning §4): Convert the collective PDHF matrix to the normalised matrix N=(n_{ij}). For benefit attributes (j∈B) n_{ij}=d_{ij}; for cost attributes (j∈C) n_{ij}=d_{ij}^{C} via the PDHFE complement (swap ħ and λ, preserve probability tags). Formül: n_{ij} = \begin{cases} d_{ij} & j\in B\ (\text{benefit}) \\ d_{ij}^{C} & j\in C\ (\text{cost}) \end{cases},\quad d_{ij}^{C} = \langle \lambda_{ij}|\upsilon_{ij},\ \hbar_{ij}|\tau_{ij}\rangle Anchor: Ning et al. 2023 §4 Step 3
4. 4.Adım 4 (F4): Step 4 (Ning Eq.11): Compute the score function matrix S=(s_{D_α}(n_{ij}))_{m×n} via s_{D_α}(Ʒ)=(1+s_ħ-s_λ)/2 where s_ħ=Σ_i τ_i γ_i and s_λ=Σ_j υ_j η_j; Atanassov hesitation split α∈[0,1] (Huang & Li 2013 Eq.8) governs how the residual probability mass is partitioned between membership and non-membership (α=0.5 default). Formül: s_{D_{\alpha}}(\Im) = \dfrac{1 + s_{\hbar}(\Im) - s_{\lambda}(\Im)}{2},\quad s_{\hbar}(\Im) = \sum_{i} \tau_{i}\gamma_{i},\ s_{\lambda}(\Im) = \sum_{j} \upsilon_{j}\eta_{j} Anchor: Ning et al. 2023 §3 Eq.(11) (Def. 5), Huang & Li 2013 Eq.(8)
5. 5.Adım 5 (F5): Step 5 (Ning Eq.22): Transform the score matrix S into the normalised score matrix Q=(s̃_{ij}) by min-max normalisation per attribute. For benefit attributes (j∈B): s̃_{ij}=(s_{D_α}(n_{ij})-min_i s_{D_α}(n_{ij}))/(max_i s_{D_α}(n_{ij})-min_i s_{D_α}(n_{ij})); for cost attributes (j∈C): s̃_{ij}=(max_i s_{D_α}(n_{ij})-s_{D_α}(n_{ij}))/(max_i s_{D_α}(n_{ij})-min_i s_{D_α}(n_{ij})). Formül: s_{ij} = \begin{cases} \dfrac{s_{D_{\alpha}}(n_{ij}) - \min_{i} s_{D_{\alpha}}(n_{ij})}{\max_{i} s_{D_{\alpha}}(n_{ij}) - \min_{i} s_{D_{\alpha}}(n_{ij})} & s_{D_{\alpha}}(n_{ij})\in B \\[6pt] \dfrac{\max_{i} s_{D_{\alpha}}(n_{ij}) - s_{D_{\alpha}}(n_{ij})}{\max_{i} s_{D_{\alpha}}(n_{ij}) - \min_{i} s_{D_{\alpha}}(n_{ij})} & s_{D_{\alpha}}(n_{ij})\in C \end{cases} Anchor: Ning et al. 2023 §4 Step 5, Eq.(22)
6. 6.Adım 6 (F6): Step 6 (Ning Eqs.13-20): Compute the combined attribute weight vector ϖ=(ϖ_1,…,ϖ_n) that minimises the Lagrange relative-entropy objective Eq.(19) min F=Σ ϖ_j ln(ϖ_j/ω_j)+Σ ϖ_j ln(ϖ_j/η_j)+Σ ϖ_j ln(ϖ_j/w_j) s.t. Σ ϖ_j=1, ϖ_j≥0, where ω is the CRITIC objective weight (Eqs.13-16), η is the PDHF entropy weight (Eqs.17-18), and w is the subjective weight; the closed-form Lagrange solution Eq.(20) reflects both subjective and objective information. Formül: \varpi_{j} = \dfrac{\sqrt{\omega_{j}\,\eta_{j}\,w_{j}}}{\sum_{j=1}^{n}\sqrt{\omega_{j}\,\eta_{j}\,w_{j}}},\quad \text{s.t.}\ \sum_{j=1}^{n}\varpi_{j}=1,\ \varpi_{j}\geq 0 Anchor: Ning et al. 2023 §3.2 Eqs.(13)-(20)
7. 7.Adım 7 (F7): Step 7 (Ning Eq.23): Compute the per-attribute average normalised score PDHFAV=[PDHFAV_j]_{1×n} via PDHFAV_j=(1/m) Σ_i s_{ij}; this is the column-wise mean of the normalised score matrix Q from Step 5. Formül: PDHFAV = [PDHFAV_{j}]_{1\times n},\quad PDHFAV_{j} = \dfrac{\sum_{i=1}^{m} s_{ij}}{m} Anchor: Ning et al. 2023 §4 Step 7, Eq.(23)
8. 8.Adım 8 (F8): Step 8 (Ning Eqs.24-26): Compute the Positive/Negative Distance from Average matrices PDHFPDA and PDHFNDA. For benefit attributes (Eq.25): PDHFPDA_{ij}=max(0, s_{ij}-PDHFAV_j)/PDHFAV_j, PDHFNDA_{ij}=max(0, PDHFAV_j-s_{ij})/PDHFAV_j. For cost attributes (Eq.26): PDHFPDA_{ij}=max(0, PDHFAV_j-s_{ij})/PDHFAV_j, PDHFNDA_{ij}=max(0, s_{ij}-PDHFAV_j)/PDHFAV_j. Formül: \begin{cases} PDHFPDA_{ij} = \dfrac{\max\bigl(0,\, s_{ij}-PDHFAV_{j}\bigr)}{PDHFAV_{j}} \\ PDHFNDA_{ij} = \dfrac{\max\bigl(0,\, PDHFAV_{j}-s_{ij}\bigr)}{PDHFAV_{j}} \end{cases}\ (j\in B);\quad \begin{cases} PDHFPDA_{ij} = \dfrac{\max\bigl(0,\, PDHFAV_{j}-s_{ij}\bigr)}{PDHFAV_{j}} \\ PDHFNDA_{ij} = \dfrac{\max\bigl(0,\, s_{ij}-PDHFAV_{j}\bigr)}{PDHFAV_{j}} \end{cases}\ (j\in C) Anchor: Ning et al. 2023 §4 Step 8, Eqs.(24)-(26)
9. 9.Adım 9 (F9): Step 9 (Ning Eq.27): Weighted sums PDHFSP_i and PDHFSN_i across attributes using the combined weight ϖ from Step 6. Formül: PDHFSP_{i} = \sum_{j=1}^{n} \varpi_{j}\, PDHFPDA_{ij},\quad PDHFSN_{i} = \sum_{j=1}^{n} \varpi_{j}\, PDHFNDA_{ij} Anchor: Ning et al. 2023 §4 Step 9, Eq.(27)
10. 10.Adım 10 (F10): Step 10 (Ning Eq.28): Normalise PDHFSP and PDHFSN by their maxima: PDHFNSP_i=PDHFSP_i/max_i PDHFSP_i; PDHFNSN_i=1-PDHFSN_i/max_i PDHFSN_i. Formül: PDHFNSP_{i} = \dfrac{PDHFSP_{i}}{\max_{i} PDHFSP_{i}},\quad PDHFNSN_{i} = 1 - \dfrac{PDHFSN_{i}}{\max_{i} PDHFSN_{i}} Anchor: Ning et al. 2023 §4 Step 10, Eq.(28)
11. 11.Adım 11 (F11): Step 11 (Ning Eq.29): Compute the final PDHF appraisal score PDHFAS_i=(PDHFNSP_i+PDHFNSN_i)/2, PDHFAS_i∈[0,1]. Formül: PDHFAS_{i} = \tfrac{1}{2}\bigl(PDHFNSP_{i} + PDHFNSN_{i}\bigr),\quad PDHFAS_{i}\in[0,1] Anchor: Ning et al. 2023 §4 Step 11, Eq.(29)
12. 12.Adım 12 (F12): Step 12 (Ning §4): Sort alternatives in descending order of PDHFAS_i; the alternative attaining the greatest PDHFAS is optimal. Formül: X^{*} = \arg\max_{i} PDHFAS_{i};\quad \text{rank descending in } PDHFAS_{i} Anchor: Ning et al. 2023 §4 Step 12

Commonly paired with

• •n_a + DHF-EDAS (common)

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