DHF-COPRAS - Dual Hesitant Fuzzy extension of COPRAS

Overview

DHF-COPRAS (Rani & Mishra 2020 HF-SWARA-COPRAS lifted to DHFE) is the dual-hesitant extension of COPRAS with integrated SWARA criterion weighting and entropy-based DE weighting. l decision experts each supply a DHFE matrix; DHFWA (Eq.8 lift of Zhu 2012 Eq.4) aggregates them into the AHF-D matrix using crisp DE weights λ_k (Eq.7 HFS-entropy or direct). Each AHF-D cell is defuzzified via the DHFE score s(d)=mean(h)−mean(g) (Zhu 2012 Def. 2) for SWARA ordering; SWARA Eqs.(9)-(11) yield criterion weights w_j. Cost columns are complemented (Zhu 2012 Eq.6 swap h↔g) and benefit/cost aggregates σ_i, v_i are computed as DHFE-weighted sums (Eqs.12-13). Their crisp scores S(σ_i), S(v_i) feed the Zavadskas-Kaklauskas Q-formula θ_i=γ·S(σ_i)+(1−γ)·(Σ S(v_i))/(S(v_i)·Σ(1/S(v_i))) (Eq.15) with strategy parameter γ∈[0,1] (default 0.5; γ<0.5 pessimistic toward cost criteria, γ>0.5 optimistic toward benefit). Alternatives are ranked DESCENDING by θ_i; the degree of utility λ_i=(θ_i/θ_max)×100% (Eq.17) yields a percentage comparison against the best. Pure-HFE inputs (g=∅) collapse DHFWA→HFWA and reproduce the seminal Rani & Mishra ranking exactly.

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

   Step I (Rani & Mishra §3.2) - Originate the alternative set G=(G_1,…,G_m) and criteria set F=(F_1,…,F_n). Obtain l per-DE DHFE decision matrices Z=(z_{ij}^k)_{m×n}, k=1,…,l, with z_{ij}^k=⟨h_{ij}^k, g_{ij}^k⟩. The pure HFE inputs of the seminal HF-SWARA-COPRAS algorithm are the special case g_{ij}^k=∅ for all (i,j,k). Receive the criterion direction set Υ_1=benefit (max) / Υ_2=cost (min).

   $$Z=(z_{ij}^k{)}_{m\times n},k=1,\dots ,l;z_{ij}^k=⟨h_{ij}^k,g_{ij}^k⟩;{{\rm Y}}_1\cup {{\rm Y}}_2=\{1,\dots ,n\}$$

   Rani & Mishra 2020 §3.2 Step I

2. 2

   Step II (Rani & Mishra Eq.7) - Compute the crisp DE weight vector λ=(λ_1,…,λ_l). If weight_method='entropy_eq7', evaluate the HFS entropy e(ħ_k) (Eq.6 Mishra et al. 2018) on the k-th DE preference matrix and apply Eq.(7) λ_k=(1−e(ħ_k))/Σ_{k=1}^{l}(1−e(ħ_k)); the higher entropy ⇒ lower DE weight. If weight_method='direct', λ is supplied directly via expert_weight_vector additional_input.

   $${\lambda }_k=\frac{1-e({\hslash }_k)}{{\sum }_{k=1}^l(1-e({\hslash }_k))},k=1,\dots ,l;\sum _{k=1}^l{\lambda }_k=1$$

   Rani & Mishra 2020 §3.2 Step II, Eq.(7); HFS entropy: Mishra et al. (2018), Def. 5 Eq.(6)

3. 3

   Step III (Rani & Mishra Eq.8 → Zhu et al. 2012 DHFWA lift) - Aggregate the l per-DE DHFE matrices into the collective AHF-D matrix P=(ξ_{ij})_{m×n} via the Dual Hesitant Fuzzy Weighted Average (DHFWA) operator parameterised by λ_k. For pure HFE inputs (g=∅) this reduces verbatim to the HFWA Eq.(8) of Rani & Mishra; for non-empty g the non-membership component is aggregated geometrically by Zhu 2012 Eq.(4).

   $${\xi }_{ij}=\underset{k=1}{\overset{l}{⨁}}{\lambda }_k{z}_{ij}^k=⟨\bigcup _{{\gamma }_{ij}^k\in h_{ij}^k}\{1-\prod _{k=1}^l(1-{\gamma }_{ij}^k{)}^{{\lambda }_k}\},\bigcup _{{\eta }_{ij}^k\in g_{ij}^k}\{\prod _{k=1}^l\left({\eta }_{ij}^k{)}^{{\lambda }_k}\right\}⟩$$

   Rani & Mishra 2020 §3.2 Step III, Eq.(8); DHFWA lift: Zhu et al. 2012 Eq.(4)

4. 4

   Step IV (Rani & Mishra §3.2 Steps IV-A..IV-F, Eqs.9-11) - Compute SWARA criterion weights w=(w_1,…,w_n). (IV-A) Defuzzify each AHF-D cell via the DHFE score s(ξ_{ij})=mean(h)−mean(g) (Zhu 2012 Def. 2; reduces to Mishra Eq.(2) S(ħ)=mean(h) when g=∅). (IV-B) DE ranks criteria highest→lowest by significance. (IV-C) DE supplies comparative significance s_j for j>1. (IV-D) Comparative coefficient k_j: k_1=1, k_j=s_j+1 for j>1 (Eq.9). (IV-E) Recalculated weight p_j: p_1=1, p_j=p_{j−1}/k_j for j>1 (Eq.10). (IV-F) Normalised criterion weight w_j=p_j/Σ_{j=1}^{n} p_j (Eq.11), Σw_j=1.

   $$k_j=\{\begin{array}{ll}1& j=1\\ s_j+1& j>1\end{array};p_j=\{\begin{array}{ll}1& j=1\\ p_{j-1}/k_j& j>1\end{array};w_j=\frac{p_j}{{\sum }_{j=1}^n{p}_j}$$

   Rani & Mishra 2020 §3.2 Step IV-A..IV-F, Eqs.(9)-(11); DHFE score: Zhu et al. 2012 Def. 2

5. 5

   Step V-prep (Zhu et al. 2012 Eq.6 cost-criterion complement) - Convert the AHF-D matrix P to the direction-normalised matrix P'=(ξ'_{ij}) via the DHFE complement on cost columns: for j∈Υ_1 (benefit) ξ'_{ij}=ξ_{ij}; for j∈Υ_2 (cost) ξ'_{ij}=ξ_{ij}^{c}=⟨g_{ij}, h_{ij}⟩ (swap membership/non-membership sets). For pure HFE inputs (g=∅) the complement degenerates to ξ_{ij}^{c}=⟨∅, h_{ij}⟩; downstream Step V-VI then yields v_i with empty membership and full non-membership, and score s(v_i)=0−mean(h)≤0 - this preserves Rani & Mishra's pure-HFE cost handling because the seminal paper aggregates cost columns directly with negative sign in Eq.(15) Q-formula rather than complementing them.

   $${\xi }_{ij}^{\prime }=\{\begin{array}{ll}{\xi }_{ij}& j\in {{\rm Y}}_1(\text{benefit})\\ {\xi }_{ij}^c=⟨g_{ij},h_{ij}⟩& j\in {{\rm Y}}_2(\text{cost})\end{array}$$

   Zhu et al. 2012 Eq.(6) DHFE complement; Rani & Mishra 2020 §3.2 Step V (implicit cost handling)

6. 6

   Step V (Rani & Mishra Eqs.12-13) - Compute the per-alternative benefit-aggregate σ_i (Eq.12) and cost-aggregate v_i (Eq.13) as DHFE-valued weighted sums via Zhu 2012 Eqs.(5,7) (scalar multiplication λh and addition ⊕). σ_i aggregates over benefit columns Υ_1 of P; v_i aggregates over cost columns Υ_2 of P (using the direction-normalised P' from F5 when DHFE complement applies). The criterion weights w_j come from F4.

   $${\sigma }_i=\underset{j\in {{\rm Y}}_1}{⨁}{w}_j{\xi }_{ij},v_i=\underset{j\in {{\rm Y}}_2}{⨁}{w}_j{\xi }_{ij},i=1,2,\dots ,m$$

   Rani & Mishra 2020 §3.2 Step V, Eqs.(12)-(13); DHFE scalar-mult/addition: Zhu et al. 2012 Eqs.(5),(7)

7. 7

   Step VI-prep - Defuzzify σ_i and v_i into crisp scalars S(σ_i) and S(v_i) via the DHFE score function s(d)=mean(h)−mean(g) (Zhu 2012 Def. 2). For pure HFE inputs g=∅ both σ_i and v_i carry empty non-membership and S reduces to Mishra Eq.(2). For DHFE inputs the score may be negative (s(d)∈[−1,1]); document the convention and ensure subsequent Q-formula handles S(v_i)≤0 via the |·| convention or via adding a positive offset (see L.deviations_from_seminal).

   $$S({\sigma }_i)=\frac{1}{|h_{{\sigma }_i}|}\sum _{\gamma \in h_{{\sigma }_i}}\gamma -\frac{1}{|g_{{\sigma }_i}|}\sum _{\eta \in g_{{\sigma }_i}}\eta ,S(v_i)\text{analogously};\text{mean}(\varnothing )\equiv 0$$

   Zhu et al. 2012 Def. 2 (score function); Rani & Mishra 2020 Eq.(2) (HFE special case)

8. 8

   Step VI (Rani & Mishra Eqs.14-15) - Compute the relative weight θ_i of each alternative via the COPRAS compound ratio θ_i=γ·S(σ_i)+(1−γ)·(Σ_i S(v_i))/(S(v_i)·Σ_i (1/S(v_i))), γ∈[0,1]. Eq.(14) is the unweighted form (γ=1 limit), Eq.(15) is the strategy-parameterised form (γ=0.5 default). The compound ratio is the literature-standard Zavadskas-Kaklauskas 1996 COPRAS Q-formula applied on DHFE scores.

   $${\theta }_i=\gamma S({\sigma }_i)+(1-\gamma )\frac{{\sum }_{i=1}^{m}S(v_i)}{S(v_i){\sum }_{i=1}^m\frac{1}{S(v_i)}},i=1,2,\dots ,m$$

   Rani & Mishra 2020 §3.2 Step VI, Eqs.(14)-(15); Zavadskas & Kaklauskas 1996 classical COPRAS Q-formula

9. 9

   Steps VII-VIII (Rani & Mishra Eqs.16-17) - Determine the priority order and degree of utility. Step VII: the optimal alternative is G^*=arg max_i θ_i (Eq.16); alternatives are sorted descending by θ_i. Step VIII: degree of utility λ_i=(θ_i/θ_max)×100% (Eq.17), λ_i∈[0,100]%, λ_{i^*}=100%.

   $$G^{\ast }=\arg \underset{i}{\max }{\theta }_i;{\lambda }_i=\frac{{\theta }_i}{{\theta }_{\max }}\times 100\%,i=1,\dots ,m;\text{rank descending in }{\theta }_i$$

   Rani & Mishra 2020 §3.2 Steps VII-VIII, Eqs.(16)-(17)

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 weight_method='entropy_eq7', evaluate the HFS entropy e(ħ_k) (Eq.6 Mishra et al. 2018) on the k-th DE preference matrix and apply Eq.(7) λ_k=(1−e(ħ_k))/Σ_{k=1}^{l}(1−e(ħ_k)); the higher entropy ⇒ l
• •empty g the non-membership component is aggregated geometrically by Zhu 2012 Eq.(4).
• •when g=∅). (IV-B) DE ranks criteria highest→lowest by significance. (IV-C) DE supplies comparative significance s_j for j>1. (IV-D) Comparative coefficient k_j: k_1=1, k_j=s_j+1 for j>1 (Eq.9). (IV-E)
• •degenerates to ξ_{ij}^{c}=⟨∅, h_{ij}⟩; downstream Step V-VI then yields v_i with empty membership and full non-membership, and score s(v_i)=0−mean(h)≤0 - this preserves Rani & Mishra's pure-HFE cost h
• •when DHFE complement applies). The criterion weights w_j come from F4.

• Pure-HFE vs DHFE inputs: the engine accepts both; pure HFE inputs are auto-promoted to DHFE with g=∅ before Eq.(8) DHFWA. Do not add a synthetic g={1−max(h)} (Atanassov hesitation singleton) silently - that changes the score s(d)=mean(h)−mean(g) by −mean(g) and shifts the ranking; only do this if the user explicitly requests Atanassov-style intuitionistic embedding.

• Strategy parameter γ sensitivity: Eq.(15) ranking can flip at γ-thresholds reported in Rani & Mishra Table 9 (γ≈0.2-0.3 top swap, γ≈0.7-0.8 bottom swap in §4 SSS instance). Always document the γ used and run sensitivity for γ∈{0.0, 0.25, 0.5, 0.75, 1.0}; report the percentage of γ-grid points where the top alternative remains stable.

• S(v_i) sign and zero edge cases: when DHFE inputs are used (g≠∅), the cost score S(v_i)=mean(h)−mean(g) can be negative, making the Q-formula compound ratio Σ S(v_i)/(S(v_i)·Σ(1/S(v_i))) ill-defined (division by zero or by negative number changes sign). The engine handles this by (a) shifting all S(v_i) by adding a positive offset δ=max(0, −min_i S(v_i))+ε so that S'(v_i)=S(v_i)+δ>0, then applying Eq.(15) on S'(v_i); or (b) using |S(v_i)| in the ratio. Document the chosen convention. For pure HFE (g=∅) S(v_i)=mean(h)≥0 and the issue does not arise.

• Entropy-derived DE weight robustness: Eq.(7) λ_k=(1−e(ħ_k))/Σ(1−e(ħ_k)) becomes degenerate if every DE supplies maximally hesitant input (e(ħ_k)=1 ∀k makes the denominator 0). The engine falls back to uniform λ_k=1/l in that case and emits a warning; users with strong prior on DE expertise should switch weight_method='direct' to supply λ_k explicitly.

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