DHF-TODIM - Dual Hesitant Fuzzy extension of TODIM

Overview

DHF-TODIM implements the pure-DHFE projection of Liu et al. (2023) CDHF-TODIM. Inputs are Dual Hesitant Fuzzy Elements ρ=(h,g) with γ⁺+η⁺≤1. The 8-step pipeline (F1 input validation; F2 Eq.5.1 cost-complement normalisation; F3 Eq.5.2 relative weights against the heaviest criterion; F4 Eq.3.2 DHFE score; F5 Eq.3.1 DHFE divergence; F6 Eq.5.3 piecewise per-criterion dominance with loss attenuation 1/Φ; F7 Eq.5.5-5.6 cross-criterion aggregation; F8 PI⁺ min-max normalisation and descending sort) carries no probabilistic component and no complex argument. For non-zero complex memberships, use Liu's CDHF-TODIM directly - the manifest's DHFE-only ranking is not guaranteed to reproduce CDHF results.

Output

utility, higher is better

Data

Hesitant, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Dual Hesitant Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 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).

   $$P=({\rho }_{ij}{)}_{m\times n},{\rho }_{ij}=(h_{ij},g_{ij}),\Lambda =({\Lambda }_1,\dots ,{\Lambda }_n),\sum _j{\Lambda }_j=1,\Phi >0$$

   Liu et al. 2023, §5 TODIM Steps 1-2 (Table 4 input → Table 5 CDHF matrix); validity from Zhu 2012 Def.1

2. 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).

   $${\rho }_{ij}^{\prime }=\{\begin{array}{ll}(h_{ij},g_{ij})& j\in B\text{(benefit)}\\ (g_{ij},h_{ij})& j\in C\text{(cost)}\end{array}$$

   Liu et al. 2023, §5 Step 2 / Eq.(5.1)

3. 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.

   $$r=\arg \underset{j}{\max }{\Lambda }_j,w_{jr}^{\prime }=\frac{{\Lambda }_j}{{\Lambda }_r},j=1,\dots ,n;0\le w_{jr}^{\prime }\le 1$$

   Liu et al. 2023, §5 Step 3 / Eq.(5.2)

4. 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).

   $$S({\rho }_{ij})=\frac{1}{|h_{ij}|}\sum _{\gamma \in h_{ij}}\gamma -\frac{1}{|g_{ij}|}\sum _{\eta \in g_{ij}}\eta \text{(DHFE reduction of Liu 2023, Eq.3.2)}$$

   Liu et al. 2023, Eq.(3.2) reduced to DHFE (ω≡0); equivalent to Zhu 2012 Def.2 score

5. 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.

   $$d({\rho }_{i1},{\rho }_{i2})={\left[\frac{1}{4}\sum _{j=1}^n{\Lambda }_j(|{\mu }_{i1\sigma (j)}-{\mu }_{i2\sigma (j)}{|}^2+|v_{i1\sigma (j)}-v_{i2\sigma (j)}{|}^2)\right]}^{1/2}\text{(Liu 2023, Eq.3.1 reduced to DHFE)}$$

   Liu et al. 2023, Eq.(3.1) reduced to DHFE (ω≡0)

6. 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.

   $${\Re }_j(a_i,a_p)=\{\begin{array}{ll}\sqrt{\frac{w_{jr}^{\prime }d({\rho }_{ij},{\rho }_{pj})}{{\sum }_{r=1}^n{w}_{jr}^{\prime }}},& \Delta S_j>0\\ 0,& \Delta S_j=0\\ -\frac{1}{\Phi }\sqrt{\frac{({\sum }_{r=1}^n{w}_{jr}^{\prime })d({\rho }_{ij},{\rho }_{pj})}{w_{jr}^{\prime }}},& \Delta S_j<0\end{array}$$

   Liu et al. 2023, Eq.(5.3); Tables 8-12 illustrate the resulting m×m dominance matrices per criterion

7. 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.

   $$\Delta (a_i,a_p)=\sum _{j=1}^n{\Re }_j(a_i,a_p),[\Delta (a_i,a_p){]}_{m\times m}$$

   Liu et al. 2023, Eqs.(5.5)-(5.6); illustrated in Table 13

8. 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).

   $$P{I}_i^{+}=\frac{{\sum }_{p=1}^m\Delta (a_i,a_p)-{\min }_{i^{\prime }}{\sum }_{p=1}^m\Delta (a_{i^{\prime }},a_p)}{{\max }_{i^{\prime }}{\sum }_{p=1}^m\Delta (a_{i^{\prime }},a_p)-{\min }_{i^{\prime }}{\sum }_{p=1}^m\Delta (a_{i^{\prime }},a_p)},P{I}_i^{+}\in [0,1];\text{rank}=\arg \downarrow P{I}^{+}$$

   Liu et al. 2023, §5 Steps 7-8 / Fig.3 / Tables 14-15

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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.

• DHFE validity must be enforced before any DHFE arithmetic: γ⁺+η⁺ ≤ 1 with γ⁺=max(h), η⁺=max(g). Liu's Table 3 linguistic codings already satisfy this; user-supplied entries may not.

• Attention factor Φ controls loss aversion in Eq.5.3 (negative branch scales by 1/Φ). Liu §5 does not state a specific Φ for the Russia-Ukraine example; the manifest defaults to Φ=1.0. Higher Φ flattens loss impact and can compress PI⁺ separation.

• Reference criterion r=argmax_j Λ_j is ambiguous when multiple weights tie at the maximum (Liu §5 has Λ_1=Λ_2=0.25). Tie convention: pick the lowest index, so r=1 in the published example. Different tie conventions produce different w'_jr scales and may shift PI⁺ values.

• DHFE divergence (Eq.3.1 reduction) presumes h, g element-sets are paired in decreasing order σ(j); short sets are zero-padded. Different pairings yield different d values and downstream ℜ_j contributions.

• PI⁺ uses min-max normalisation across alternatives; when all row-sums are equal (max=min), denominator is zero and PI⁺ is undefined. Fallback: return PI⁺_i = 0 for all i and report a flat ranking.

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