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

Overview

COPRAS-IN extends the crisp COPRAS method to Interval Neutrosophic Numbers (INN). Each criterion value is expressed as <[T_L,T_U],[I_L,I_U],[F_L,F_U]>. Cost and benefit criteria are separated and aggregated via INWMSM operators: benefit cells form P_i (added), cost cells form R_i (divided). Enter cost-criterion INNs as the degree to which the cost or impact is high (higher truth = worse), the same convention as the other neutrosophic methods; do not pre-complement them. The risk index λ controls decision-maker attitude: λ=0 is pessimistic (focuses on indeterminacy/falsity), λ=1 is optimistic (focuses on truth-membership). The alternative with the highest Q_i score is best.

Data

Interval Neutrosophic

Weights

Needs a weight source

How it works

1. 1

   Construct decision matrix X with INN entries x_ij = <[a_ij,b_ij],[c_ij,d_ij],[e_ij,f_ij]>.

   $$X={\left[⟨[a_{ij},b_{ij}],[c_{ij},d_{ij}],[e_{ij},f_{ij}]⟩\right]}_{m\times n}$$

   Şahin 2019, Eq.(39)

2. 2

   Direction alignment: benefit and cost cells both enter unchanged (x̄_ij = x_ij). DM3 codes cost cells as magnitude (higher truth = higher cost) and COPRAS consumes them in the divided term R_i (Step 7), which already penalises them. Şahin 2019 Eq.(40) complements cost cells ⟨[e,f],[1−d,1−c],[a,b]⟩ because the paper codes its Table 1 as satisfaction; the paper's Table 2 equals the DM3-coded matrix.

   $${\bar{x}}_{ij}=x_{ij}=⟨[a_{ij},b_{ij}],[c_{ij},d_{ij}],[e_{ij},f_{ij}]⟩(\text{DM3 magnitude coding; paper Eq.(40) complement not applied})$$

   Şahin 2019, Eq.(40) - see implementation_review.literature_disambiguation

3. 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)).

   $$w_{j,CWD}=\frac{(w_{j,sub}{)}^{\theta }\times (w_{j,ob}{)}^{1-\theta }}{{\sum }_{j=1}^n\left[(w_{j,sub}{)}^{\theta }\times (w_{j,ob}{)}^{1-\theta }\right]}$$

   Şahin 2019, Eqs.(33)-(35),(38)

4. 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]>.

   $${\tilde{x}}_{ij}=w_j\otimes {\bar{x}}_{ij}=⟨[1-(1-a_{ij}{)}^{w_j},1-(1-b_{ij}{)}^{w_j}],[c_{ij}^{w_j},d_{ij}^{w_j}],[e_{ij}^{w_j},f_{ij}^{w_j}]⟩$$

   Şahin 2019, Def.3(3), Step 3

5. 5

   Aggregate benefit-criterion values into P_i and cost-criterion values into R_i per alternative using INWMSM^(k) operator (Interval Neutrosophic Weighted Maclaurin Symmetric Mean).

   $$P_i={\mathrm{INWMSM}}_w^{(k)}\left({\tilde{x}}_{ij}:j\in B\right),R_i={\mathrm{INWMSM}}_w^{(k)}\left({\tilde{x}}_{ij}:j\in C\right),B=\text{benefit},C=\text{cost}$$

   Şahin 2019, Eqs.(41)-(42),(28)

6. 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).

   $$S_{\lambda }(x)=\frac{1}{6}\left[\lambda (T_L+T_U)+(1-\lambda )\left(4-(I_L+I_U)-(F_L+F_U)\right)\right]$$

   Şahin 2019, Def.5, Eq.(16); Eq.(43)

7. 7

   Compute preference score Q_i for each alternative using COPRAS-IN formula.

   $$Q_i=S_{\lambda }(P_i)+\frac{{\sum }_{i=1}^m{S}_{\lambda }(R_i)}{S_{\lambda }(R_i)\cdot {\sum }_{i=1}^m\frac{1}{S_{\lambda }(R_i)}}$$

   Şahin 2019, Eq.(45)

8. 8

   Rank alternatives descending by Q_i (higher Q_i = better). Compute utility degree N_i = (Q_i / max_i Q_i) × 100%.

   $$N_i=\frac{Q_i}{{\max }_i{Q}_i}\times 100\%$$

   Şahin 2019, Eq.(46)

Fits when

Edge cases and pitfalls

• Risk index λ sensitivity: different λ values can reverse rankings (see Table 7 of Şahin 2019). Always report λ used.

• INN validity: ensure T_U+I_U+F_U ≤ 3 for all entries before computation.

• INWMSM k parameter: k=1 degenerates to weighted averaging; k=n uses geometric-type product. k=2 is most common in literature.

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