DHF-EDAS - Dual Hesitant Fuzzy extension of EDAS

Overview

DHF-EDAS (Ning et al. 2023) is the PDHF-MAGDM extension of EDAS: p decision-makers each supply a PDHF (Probabilistic Dual Hesitant Fuzzy) matrix, aggregated by PDHFWA (Eq.21); cost columns are complemented (Step 3); the score function s_{D_α}(Ʒ)=(1+s_ħ-s_λ)/2 (Eq.11) reduces each cell to a crisp value; the score matrix is min-max normalised per attribute (Eq.22); the combined attribute weight ϖ (Eq.20) fuses CRITIC objective weight ω, PDHF entropy weight η, and decision-maker subjective weight w via Lagrange relative-entropy minimisation; the PDHFEDAS pipeline (PDHFAV→PDA/NDA→SP/SN→NSP/NSN→PDHFAS) yields the final appraisal score, ranked in DESCENDING order. Pure DHFE inputs are the special case τ_i=1/#ħ, υ_j=1/#λ (uniform probabilities); the algorithm collapses to dual-hesitant EDAS without code change. The Atanassov hesitation split α defaults to 0.5; α=1 maximises membership contribution, α=0 maximises non-membership.

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

   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).

   $$\Im =({\Im }_{ij}^k{)}_{m\times n},k=1,\dots ,p;{\Im }_{ij}^k=⟨{\hslash }_{ij}^k|{\tau }_{ij}^k,{\lambda }_{ij}^k|{\upsilon }_{ij}^k⟩;\sum _{k=1}^p{\theta }_k=1;\sum _{j=1}^n{w}_j=1$$

   Ning et al. 2023 §4 Step 1

2. 2

   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.

   $$d_{ij}=\underset{k=1}{\overset{p}{⨁}}{\theta }_k{\Im }_{ij}^k=\bigcup _{{\gamma }_{ij}^k\in {\hslash }_{ij}^k,{\eta }_{ij}^k\in {\lambda }_{ij}^k}\left\{\left\{\left[1-\prod _{k=1}^p(1-{\gamma }_{ij}^k{)}^{{\theta }_k}\right]|\prod _{k=1}^p{p}_{{\gamma }_{ij}^k}\right\},\left\{\prod _{k=1}^p\left({\eta }_{ij}^k{)}^{{\theta }_k}\right|\prod _{k=1}^p{q}_{{\eta }_{ij}^k}\right\}\right\}$$

   Ning et al. 2023 §4 Step 2, Eq.(21)

3. 3

   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).

   $$n_{ij}=\{\begin{array}{ll}{d}_{ij}& j\in B(\text{benefit})\\ d_{ij}^C& j\in C(\text{cost})\end{array},d_{ij}^C=⟨{\lambda }_{ij}|{\upsilon }_{ij},{\hslash }_{ij}|{\tau }_{ij}⟩$$

   Ning et al. 2023 §4 Step 3

4. 4

   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).

   $$s_{D_{\alpha }}(\Im )=\frac{1+s_{\hslash }(\Im )-s_{\lambda }(\Im )}{2},s_{\hslash }(\Im )=\sum _i{\tau }_i{\gamma }_i,s_{\lambda }(\Im )=\sum _j{\upsilon }_j{\eta }_j$$

   Ning et al. 2023 §3 Eq.(11) (Def. 5), Huang & Li 2013 Eq.(8)

5. 5

   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})).

   $$s_{ij}=\{\begin{array}{ll}\frac{s_{D_{\alpha }}(n_{ij})-{\min }_i{s}_{D_{\alpha }}(n_{ij})}{{\max }_i{s}_{D_{\alpha }}(n_{ij})-{\min }_i{s}_{D_{\alpha }}(n_{ij})}& s_{D_{\alpha }}(n_{ij})\in B\\ \frac{{\max }_i{s}_{D_{\alpha }}(n_{ij})-s_{D_{\alpha }}(n_{ij})}{{\max }_i{s}_{D_{\alpha }}(n_{ij})-{\min }_i{s}_{D_{\alpha }}(n_{ij})}& s_{D_{\alpha }}(n_{ij})\in C\end{array}$$

   Ning et al. 2023 §4 Step 5, Eq.(22)

6. 6

   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.

   $${\varpi }_j=\frac{\sqrt{{\omega }_j{\eta }_j{w}_j}}{{\sum }_{j=1}^n\sqrt{{\omega }_j{\eta }_j{w}_j}},\text{s.t.}\sum _{j=1}^n{\varpi }_j=1,{\varpi }_j\ge 0$$

   Ning et al. 2023 §3.2 Eqs.(13)-(20)

7. 7

   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.

   $$PDHFAV=[PDHFA{V}_j{]}_{1\times n},PDHFA{V}_j=\frac{{\sum }_{i=1}^m{s}_{ij}}{m}$$

   Ning et al. 2023 §4 Step 7, Eq.(23)

8. 8

   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.

   $$\{\begin{array}{l}PDHFPD{A}_{ij}=\frac{\max (0,s_{ij}-PDHFA{V}_j)}{PDHFA{V}_j}\\ PDHFND{A}_{ij}=\frac{\max (0,PDHFA{V}_j-s_{ij})}{PDHFA{V}_j}\end{array}(j\in B);\{\begin{array}{l}PDHFPD{A}_{ij}=\frac{\max (0,PDHFA{V}_j-s_{ij})}{PDHFA{V}_j}\\ PDHFND{A}_{ij}=\frac{\max (0,s_{ij}-PDHFA{V}_j)}{PDHFA{V}_j}\end{array}(j\in C)$$

   Ning et al. 2023 §4 Step 8, Eqs.(24)-(26)

9. 9

   Weighted sums PDHFSP_i and PDHFSN_i across attributes using the combined weight ϖ from Step 6.

   $$PDHFS{P}_i=\sum _{j=1}^n{\varpi }_{j}PDHFPD{A}_{ij},PDHFS{N}_i=\sum _{j=1}^n{\varpi }_{j}PDHFND{A}_{ij}$$

   Ning et al. 2023 §4 Step 9, Eq.(27)

10. 10

    Normalise PDHFSP and PDHFSN by their maxima: PDHFNSP_i=PDHFSP_i/max_i PDHFSP_i; PDHFNSN_i=1-PDHFSN_i/max_i PDHFSN_i.

    $$PDHFNS{P}_i=\frac{PDHFS{P}_i}{{\max }_{i}PDHFS{P}_i},PDHFNS{N}_i=1-\frac{PDHFS{N}_i}{{\max }_{i}PDHFS{N}_i}$$

    Ning et al. 2023 §4 Step 10, Eq.(28)

11. 11

    Compute the final PDHF appraisal score PDHFAS_i=(PDHFNSP_i+PDHFNSN_i)/2, PDHFAS_i∈[0,1].

    $$PDHFA{S}_i=\frac{1}{2}(PDHFNS{P}_i+PDHFNS{N}_i),PDHFA{S}_i\in [0,1]$$

    Ning et al. 2023 §4 Step 11, Eq.(29)

12. 12

    Sort alternatives in descending order of PDHFAS_i; the alternative attaining the greatest PDHFAS is optimal.

    $$X^{\ast }=\arg \underset{i}{\max }PDHFA{S}_i;\text{rank descending in }PDHFA{S}_i$$

    Ning et al. 2023 §4 Step 12

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

• PDHFE vs DHFE confusion: the engine accepts both; pure DHFE inputs are auto-promoted to PDHFE with uniform probability tags (τ_i=1/#ħ, υ_j=1/#λ) before Eq.(21). Do not multiply membership probabilities by 1/#ħ a second time.

• Score function α: Eq.(11) s_{D_α}(Ʒ)=(1+s_ħ-s_λ)/2 carries an implicit Atanassov hesitation split α (Huang & Li 2013 Eq.8). Changing α away from 0.5 reweights how the residual probability mass π=1-γ^+-η^+ is divided between membership and non-membership; ranking can flip on near-ties. Document α used and run sensitivity for α∈{0.25, 0.5, 0.75}.

• Combined weight strategy: the default 'lagrange_combined' (Eq.20) fuses CRITIC, PDHF entropy, and subjective weights with equal Lagrange weight per source. If the user has high confidence in only one source (e.g., subjective expert opinion), switch weight_method to 'subjective_only' or 'objective_only' and document the reduction; do not silently zero out one component within Eq.(20) (numerically the Lagrange solution diverges on a zero argument).

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