DHF-TODIM: Dual Hesitant Fuzzy extension of TODIM

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

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

Edge cases

• •ties for max(Λ) take the lowest index.
• •empty-set convention used in Zhu 2012 (mean of ∅ = 0).
• •zero-pad shorter sets to the longer length using the standard hesitant pairing convention.

Common pitfalls

• •Hatalı: 'DHF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Dual Hesitant Fuzzy numbers/tuples
• •Hatalı: 'DHF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'DHF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: DHF-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: DHF-TODIM'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: Read DHFE matrix P=(ρ_ij)_{m×n} with ρ_ij=(h_ij,g_ij); validate DHFE constraints (E-2), weight simplex (E-3), and Φ>0 (E-4). Formül: P = (\rho_{ij})_{m\times n},\ \rho_{ij} = (h_{ij}, g_{ij}),\ \Lambda = (\Lambda_{1},\dots,\Lambda_{n}),\ \sum_{j}\Lambda_{j}=1,\ \Phi > 0 Anchor: Liu et al. 2023, §5 TODIM Steps 1-2 (Table 4 input → Table 5 CDHF matrix); validity from Zhu 2012 Def.1
2. 2.Adım 2 (F2): Step 2: Normalise the DHFE matrix by cost-complement (Liu 2023, Eq.5.1): for benefit criteria j∈B keep ρ_ij=(h_ij,g_ij); for cost criteria j∈C swap to ρ'_ij=(g_ij,h_ij). Formül: \rho'_{ij}=\begin{cases}(h_{ij},g_{ij}) & j\in B\ \text{(benefit)}\\(g_{ij},h_{ij}) & j\in C\ \text{(cost)}\end{cases} Anchor: Liu et al. 2023, §5 Step 2 / Eq.(5.1)
3. 3.Adım 3 (F3): Step 3: Reference criterion r=argmax_j Λ_j and relative weights w'_jr = Λ_j / Λ_r (Liu 2023, Eq.5.2). On ties for max(Λ) take the lowest index. Formül: r = \arg\max_{j}\Lambda_{j},\quad w'_{jr} = \dfrac{\Lambda_{j}}{\Lambda_{r}},\ j=1,\dots,n;\ 0\le w'_{jr}\le 1 Anchor: Liu et al. 2023, §5 Step 3 / Eq.(5.2)
4. 4.Adım 4 (F4): Step 4: DHFE score (DHFE special case of Liu 2023, Eq.3.2 with ω≡0): S(ρ_ij) = (1/|h_ij|) Σ_{γ∈h_ij} γ − (1/|g_ij|) Σ_{η∈g_ij} η, with the empty-set convention used in Zhu 2012 (mean of ∅ = 0). Formül: S(\rho_{ij}) = \dfrac{1}{|h_{ij}|}\sum_{\gamma\in h_{ij}}\gamma - \dfrac{1}{|g_{ij}|}\sum_{\eta\in g_{ij}}\eta\quad\text{(DHFE reduction of Liu 2023, Eq.3.2)} Anchor: Liu et al. 2023, Eq.(3.2) reduced to DHFE (ω≡0); equivalent to Zhu 2012 Def.2 score
5. 5.Adım 5 (F5): Step 5: DHFE divergence (DHFE special case of Liu 2023, Eq.3.1 with ω≡0). Order σ(j) inside each DHFE so that h, g elements are arranged in decreasing order before pairing; zero-pad shorter sets to the longer length using the standard hesitant pairing convention. Formül: d(\rho_{i1},\rho_{i2}) = \left[\dfrac{1}{4}\sum_{j=1}^{n}\Lambda_{j}\Big(|\mu_{i1\sigma(j)}-\mu_{i2\sigma(j)}|^{2}+|v_{i1\sigma(j)}-v_{i2\sigma(j)}|^{2}\Big)\right]^{1/2}\quad\text{(Liu 2023, Eq.3.1 reduced to DHFE)} Anchor: Liu et al. 2023, Eq.(3.1) reduced to DHFE (ω≡0)
6. 6.Adım 6 (F6): Step 6: Per-criterion dominance ℜ_j(a_i, a_p) using the divergence of F5 and the score of F4 (Liu 2023, Eq.5.3). Sign decided by ΔS_j = S(ρ_ij) − S(ρ_pj) on criterion j. Σ in denominator is Σ_{r=1}^{n} w'_jr. Formül: \Re_{j}(a_{i},a_{p}) = \begin{cases}\sqrt{\dfrac{w'_{jr}\,d(\rho_{ij},\rho_{pj})}{\sum_{r=1}^{n}w'_{jr}}}, & \Delta S_{j}>0\\[4pt] 0, & \Delta S_{j}=0\\[4pt] -\dfrac{1}{\Phi}\sqrt{\dfrac{(\sum_{r=1}^{n}w'_{jr})\,d(\rho_{ij},\rho_{pj})}{w'_{jr}}}, & \Delta S_{j}<0\end{cases} Anchor: Liu et al. 2023, Eq.(5.3); Tables 8-12 illustrate the resulting m×m dominance matrices per criterion
7. 7.Adım 7 (F7): Step 7: Total dominance Δ(a_i, a_p) = Σ_{j=1}^{n} ℜ_j(a_i, a_p) (Liu 2023, Eq.5.5-5.6). Produces an m×m matrix whose row i collects the comparison of alternative a_i against every partner a_p. Formül: \Delta(a_{i},a_{p}) = \sum_{j=1}^{n}\Re_{j}(a_{i},a_{p}),\quad [\Delta(a_{i},a_{p})]_{m\times m} Anchor: Liu et al. 2023, Eqs.(5.5)-(5.6); illustrated in Table 13
8. 8.Adım 8 (F8): Step 8: Positive-ideal value PI⁺_i = (Σ_p Δ(a_i, a_p) − min_{i'} Σ_p Δ(a_{i'}, a_p)) / (max_{i'} Σ_p Δ(a_{i'}, a_p) − min_{i'} Σ_p Δ(a_{i'}, a_p)) ∈ [0,1]; descending sort yields the ranking (Liu 2023, §5 Steps 7-8). Formül: PI^{+}_{i} = \dfrac{\sum_{p=1}^{m}\Delta(a_{i},a_{p}) - \min_{i'}\sum_{p=1}^{m}\Delta(a_{i'},a_{p})}{\max_{i'}\sum_{p=1}^{m}\Delta(a_{i'},a_{p}) - \min_{i'}\sum_{p=1}^{m}\Delta(a_{i'},a_{p})},\quad PI^{+}_{i}\in[0,1];\ \text{rank}=\arg\!\downarrow PI^{+} Anchor: Liu et al. 2023, §5 Steps 7-8 / Fig.3 / Tables 14-15

Commonly paired with

• •n_a + DHF-TODIM (common)

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