DHF-TOPSIS - Dual Hesitant Fuzzy extension of TOPSIS

Overview

DHF-TOPSIS (Wang et al. 2020) extends classical TOPSIS to Dual Hesitant Fuzzy assessments. Operations are performed directly on DHFEs without length-equalisation: A⁺ and A⁻ are obtained from column-wise extrema of (h, g) components (direction-aware), the weighted generalised DHF distance d_wpg combines mean (G_m, G_n) and volatility (V_m, V_n) functions weighted by α and β (α+β=1), and alternatives are ranked in descending order of the closeness coefficient CC.

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

   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)).

   $$D=(a_{ij}{)}_{m\times n}=(⟨h_{ij},g_{ij}⟩{)}_{m\times n}$$

   Wang et al. 2020, p.11 Eq.(28)

2. 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)).

   $$w_j=\frac{{\sum }_{i=1}^m{\sum }_{k=1}^m{d}_h(a_{ij},a_{kj})}{{\sum }_{j=1}^n({\sum }_{i=1}^m{\sum }_{k=1}^m{d}_h(a_{ij},a_{kj}))},j=1,\dots ,n$$

   Wang et al. 2020, p.10 Eq.(27)

3. 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).

   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}

   Wang et al. 2020, p.11 Eqs.(29)-(30)

4. 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.

   $$G_m(e)=\frac{1}{l(h)}\sum _{\gamma \in h}\gamma ,G_n(e)=\frac{1}{k(g)}\sum _{\eta \in g}\eta ;V_m(e)=\frac{1}{2}[\frac{1}{l(h)}\sum _{\gamma \in h}(\gamma -G_m{)}^2+\sqrt{1-\frac{1}{1+\ln l(h)}}],V_n(e)=\frac{1}{2}[\frac{1}{k(g)}\sum _{\eta \in g}(\eta -G_n{)}^2+\sqrt{1-\frac{1}{1+\ln k(g)}}];d_{wpg}(A_i,A^{\pm })=\left(\frac{1}{2}\sum _{j=1}^n{w}_j\right[\alpha |G_m(a_{ij})-G_m(a_j^{\pm }){|}^{\lambda }+\alpha |G_n(a_{ij})-G_n(a_j^{\pm }){|}^{\lambda }+\beta |V_m(a_{ij})-V_m(a_j^{\pm }){|}^{\lambda }+\beta |V_n(a_{ij})-V_n(a_j^{\pm }){|}^{\lambda }]{)}^{1/\lambda },\alpha +\beta =1,\lambda \ge 1$$

   Wang et al. 2020, pp.6-7 Defs.8, Eqs.(9)-(12), (21)

5. 5

   Closeness coefficient CC_{i} per Eq.(31); rank alternatives in descending order of CC_{i}.

   $$C{C}_i=\frac{d_{wpg}(A_i,A^{-})}{d_{wpg}(A_i,A^{+})+d_{wpg}(A_i,A^{-})},0\le C{C}_i\le 1,i=1,\dots ,m$$

   Wang et al. 2020, p.11 Eq.(31)

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

• •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)).

• Value-space violation: ensure every DHFE entry satisfies γ⁺ + η⁺ ≤ 1 (γ⁺=max(h), η⁺=max(g), h,g ⊂ [0,1]) before computation.

• Parameter sensitivity: λ (distance order) and (α, β) (mean vs. volatility preference, α+β=1) materially affect CC magnitudes; Wang et al. (2020, §5.4) show ranking can change as λ varies - fix (α, β, λ) per the decision context.

• Direction-aware ideals: for cost criteria, A⁺ and A⁻ are swapped component-wise (Eqs.29-30 reverse roles), not derived from a normalisation step.

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