Çift tereddütlü COPRAS (Rani, Mishra ve ark., 2020)

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Vakalar

Birinci vaka gerçek bir literatür vakasıdır. Rani, Mishra ve arkadaşlarının (2020) makalesindeki sürdürülebilir tedarikçi seçimi örneğidir; bu örnekte ret kümesi boştur (g=∅), yani düz tereddütlü verinin çift yapıdaki özel bir durumudur. İkinci vaka öğretici bir kurgudur.

Formüller

Hesabın adımları ve her adımın formülü. Adım açıklamaları İngilizcedir.

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).

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

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.

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

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).

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

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.

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

5. 5. Step V-prep (Rani & Mishra 2020 §3.2 identity tag) - In the seminal HF-SWARA-COPRAS recipe the cost direction is *not* normalised by an explicit complement; Eqs.12-13 aggregate the raw benefit and cost cells of P and the direction inversion is performed analytically by the (1−γ)·(Σ S(v_k))/(S(v_i)·Σ 1/S(v_k)) term in Eq.15 (Step VI). Therefore F5 is an identity pass-through P'≡P preserved as a structural anchor in the engine's F-sequence; downstream Step V (F6) uses P' verbatim and reproduces paper Table 8 values for S(σ_i) and S(v_i) within ≤0.005 (paper reports 3-decimal precision).

   $${\xi }_{ij}^{\prime }={\xi }_{ij}\forall (i,j)\in \{1,\dots ,m\}\times \{1,\dots ,n\}$$

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.

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

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).

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

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.

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

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%.

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

Kaynaklar