DHF-COPRAS: Dual Hesitant Fuzzy extension of COPRAS

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

• •Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• •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 COPRAS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •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.

Common pitfalls

• •Hatalı: 'DHF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Dual Hesitant Fuzzy numbers/tuples
• •Hatalı: 'DHF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'DHF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: DHF-COPRAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: DHF-COPRAS'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 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). Formül: Z = (z_{ij}^{k})_{m\times n},\quad k=1,\dots,l;\ z_{ij}^{k}=\langle h_{ij}^{k},\ g_{ij}^{k}\rangle;\ \Upsilon_{1}\cup\Upsilon_{2}=\{1,\dots,n\} Anchor: Rani & Mishra 2020 §3.2 Step I
2. 2.Adım 2 (F2): 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. Formül: \lambda_{k} = \dfrac{1 - e(\hbar_{k})}{\sum_{k=1}^{l}\bigl(1 - e(\hbar_{k})\bigr)},\quad k=1,\dots,l;\quad \sum_{k=1}^{l}\lambda_{k}=1 Anchor: Rani & Mishra 2020 §3.2 Step II, Eq.(7); HFS entropy: Mishra et al. (2018), Def. 5 Eq.(6)
3. 3.Adım 3 (F3): 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). Formül: \xi_{ij} = \bigoplus_{k=1}^{l} \lambda_{k}\,z_{ij}^{k} = \Bigl\langle \bigcup_{\gamma_{ij}^{k}\in h_{ij}^{k}} \Bigl\{1 - \prod_{k=1}^{l}\bigl(1-\gamma_{ij}^{k}\bigr)^{\lambda_{k}}\Bigr\},\ \bigcup_{\eta_{ij}^{k}\in g_{ij}^{k}} \Bigl\{\prod_{k=1}^{l}\bigl(\eta_{ij}^{k}\bigr)^{\lambda_{k}}\Bigr\} \Bigr\rangle Anchor: Rani & Mishra 2020 §3.2 Step III, Eq.(8); DHFWA lift: Zhu et al. 2012 Eq.(4)
4. 4.Adım 4 (F4): 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. Formül: k_{j} = \begin{cases} 1 & j=1 \\ s_{j}+1 & j>1 \end{cases};\quad p_{j} = \begin{cases} 1 & j=1 \\ p_{j-1}/k_{j} & j>1 \end{cases};\quad w_{j} = \dfrac{p_{j}}{\sum_{j=1}^{n} p_{j}} Anchor: Rani & Mishra 2020 §3.2 Step IV-A..IV-F, Eqs.(9)-(11); DHFE score: Zhu et al. 2012 Def. 2
5. 5.Adım 5 (F5): 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. Formül: \xi'_{ij} = \begin{cases} \xi_{ij} & j\in\Upsilon_{1}\ (\text{benefit}) \\ \xi_{ij}^{c}=\langle g_{ij},\ h_{ij}\rangle & j\in\Upsilon_{2}\ (\text{cost}) \end{cases} Anchor: Zhu et al. 2012 Eq.(6) DHFE complement; Rani & Mishra 2020 §3.2 Step V (implicit cost handling)
6. 6.Adım 6 (F6): 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. Formül: \sigma_{i} = \bigoplus_{j\in\Upsilon_{1}} w_{j}\,\xi_{ij},\qquad v_{i} = \bigoplus_{j\in\Upsilon_{2}} w_{j}\,\xi_{ij},\quad i=1,2,\dots,m Anchor: Rani & Mishra 2020 §3.2 Step V, Eqs.(12)-(13); DHFE scalar-mult/addition: Zhu et al. 2012 Eqs.(5),(7)
7. 7.Adım 7 (F7): 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). Formül: S(\sigma_{i}) = \tfrac{1}{|h_{\sigma_{i}}|}\sum_{\gamma\in h_{\sigma_{i}}}\gamma - \tfrac{1}{|g_{\sigma_{i}}|}\sum_{\eta\in g_{\sigma_{i}}}\eta,\quad S(v_{i})\ \text{analogously};\ \text{mean}(\emptyset)\equiv 0 Anchor: Zhu et al. 2012 Def. 2 (score function); Rani & Mishra 2020 Eq.(2) (HFE special case)
8. 8.Adım 8 (F8): 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. Formül: \theta_{i} = \gamma\,S(\sigma_{i}) + (1-\gamma)\,\dfrac{\sum_{i=1}^{m} S(v_{i})}{S(v_{i})\,\sum_{i=1}^{m} \dfrac{1}{S(v_{i})}},\quad i=1,2,\dots,m Anchor: Rani & Mishra 2020 §3.2 Step VI, Eqs.(14)-(15); Zavadskas & Kaklauskas 1996 classical COPRAS Q-formula
9. 9.Adım 9 (F9): 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%. Formül: G^{*} = \arg\max_{i} \theta_{i};\quad \lambda_{i} = \dfrac{\theta_{i}}{\theta_{\max}}\times 100\%,\ i=1,\dots,m;\quad \text{rank descending in } \theta_{i} Anchor: Rani & Mishra 2020 §3.2 Steps VII-VIII, Eqs.(16)-(17)

Commonly paired with

• •n_a + DHF-COPRAS (common)

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