COPRAS-IN: COPRAS with Interval Neutrosophic Numbers (INN)

Overview

Interval Neutrosophic ranking: INN: <[T_L,T_U],[I_L,I_U],[F_L,F_U]> with 0 ≤ T_U+I_U+F_U ≤ 3. Output typically ranking.

Strengths

• •Method-specific: Interval Neutrosophic ranking: INN: <[T_L,T_U],[I_L,I_U],[F_L,F_U]> with 0 ≤ T_U+I_U+F_U ≤ 3
• •Preserves interval_neutrosophic uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: inherited from crisp COPRAS base; cf. Belton-Gear 1983)

Edge cases

• •See F.steps and D.parameters for N-COPRAS-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Bkz. N-COPRAS F.steps citation_anchor'lar ve P.verification_status.

Worked example

1. 1.Adım 1 (F1): Step 1: Construct decision matrix X with INN entries x_ij = <[a_ij,b_ij],[c_ij,d_ij],[e_ij,f_ij]>. Formül: X = \left[\langle [a_{ij},b_{ij}],[c_{ij},d_{ij}],[e_{ij},f_{ij}]\rangle\right]_{m \times n} Anchor: Şahin 2019, Eq.(39)
2. 2.Adım 2 (F2): Step 2: Normalize decision matrix: benefit criteria keep INN as-is; cost criteria swap truth and falsity and complement indeterminacy. Formül: \bar{x}_{ij} = \begin{cases} \langle[a_{ij},b_{ij}],[c_{ij},d_{ij}],[e_{ij},f_{ij}]\rangle & \text{benefit} \\ \langle[e_{ij},f_{ij}],[1-d_{ij},1-c_{ij}],[a_{ij},b_{ij}]\rangle & \text{cost} \end{cases} Anchor: Şahin 2019, Eq.(40)
3. 3.Adım 3 (F3): Step 3: Determine combinative criterion weights wj_CWD by blending subjective weights (model M1 via minimum cross-entropy) and objective weights (model M2/M3 via Dice measure maximization). wj_CWD = (wj_sub^θ × wj_ob^(1-θ)) / Σ(wj_sub^θ × wj_ob^(1-θ)). Formül: w_{j,CWD} = \frac{(w_{j,sub})^{\theta} \times (w_{j,ob})^{1-\theta}}{\sum_{j=1}^{n}\left[(w_{j,sub})^{\theta} \times (w_{j,ob})^{1-\theta}\right]} Anchor: Şahin 2019, Eqs.(33)-(35),(38)
4. 4.Adım 4 (F4): Step 4: Compute weighted normalized matrix using INN scalar multiplication: w ⊗ x = <[1-(1-a)^w, 1-(1-b)^w], [c^w, d^w], [e^w, f^w]>. Formül: \tilde{x}_{ij} = w_j \otimes \bar{x}_{ij} = \langle[1-(1-a_{ij})^{w_j},\,1-(1-b_{ij})^{w_j}],\,[c_{ij}^{w_j},\,d_{ij}^{w_j}],\,[e_{ij}^{w_j},\,f_{ij}^{w_j}]\rangle Anchor: Şahin 2019, Def.3(3), Step 3
5. 5.Adım 5 (F5): Step 5: Aggregate cost-criterion values into P_i and benefit-criterion values into R_i per alternative using INWMSM^(k) operator (Interval Neutrosophic Weighted Maclaurin Symmetric Mean). Formül: P_i = \mathrm{INWMSM}^{(k)}_w(\tilde{x}_{i1},\ldots,\tilde{x}_{ik}),\quad R_i = \mathrm{INWMSM}^{(k)}_w(\tilde{x}_{i,k+1},\ldots,\tilde{x}_{in}) Anchor: Şahin 2019, Eqs.(41)-(42),(28)
6. 6.Adım 6 (F6): Step 6: Score all P_i and R_i using risk-indexed score function S_λ(x) = (1/6){λ(T_L+T_U) + (1-λ)(4-(I_L+I_U)-(F_L+F_U))}. Find Rmin = min_i S_λ(R_i). Formül: S_{\lambda}(x) = \frac{1}{6}\left[\lambda(T_L+T_U)+(1-\lambda)\left(4-(I_L+I_U)-(F_L+F_U)\right)\right] Anchor: Şahin 2019, Def.5, Eq.(16); Eq.(43)
7. 7.Adım 7 (F7): Step 7: Compute preference score Q_i for each alternative using COPRAS-IN formula. Formül: Q_i = S_{\lambda}(P_i) + \frac{\sum_{i=1}^{m}S_{\lambda}(R_i)}{S_{\lambda}(R_i)\cdot\sum_{i=1}^{m}\frac{1}{S_{\lambda}(R_i)}} Anchor: Şahin 2019, Eq.(45)
8. 8.Adım 8 (F8): Step 8: Rank alternatives descending by Q_i (higher Q_i = better). Compute utility degree N_i = (Q_i / max_i Q_i) × 100%. Formül: N_i = \frac{Q_i}{\max_i Q_i} \times 100\% Anchor: Şahin 2019, Eq.(46)

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