COPRAS-IN - Aralık Nötrosofik Sayılar ile COPRAS

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

Adım 1 — INN girdileri ile karar matrisini oluştur.

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

LaTeX
X = \left[\langle [a_{ij},b_{ij}],[c_{ij},d_{ij}],[e_{ij},f_{ij}]\rangle\right]_{m \times n}
Adım 2 — Yön hizalama: fayda ve maliyet hücreleri değişmeden girer (x̄_ij = x_ij). DM3 maliyet hücrelerini büyüklük olarak kodlar (yüksek doğruluk = yüksek maliyet) ve COPRAS bunları bölünen R_i teriminde (Adım 7) kullanır; bu zaten cezalandırır. Şahin 2019 Eq.(40) maliyet hücrelerini ⟨[e,f],[1−d,1−c],[a,b]⟩ ile tümler, çünkü makale Tablo 1'i memnuniyet olarak kodlar; makalenin Tablo 2'si DM3 kodlu matrise eşittir.

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

LaTeX
\bar{x}_{ij} = x_{ij} = \langle[a_{ij},b_{ij}],[c_{ij},d_{ij}],[e_{ij},f_{ij}]\rangle \quad (\text{DM3 magnitude coding; paper Eq.(40) complement not applied})
Adım 3 — Öznel (M1 modeli) ve nesnel (M2/M3 modeli) ağırlıkları birleştirerek CWD ağırlıklarını hesapla.

$w_{j, C W D} = \frac{(w_{j, s u b})^{θ} \times (w_{j, o b})^{1 - θ}}{\sum_{j = 1}^{n} [(w_{j, s u b})^{θ} \times (w_{j, o b})^{1 - θ}]}$

LaTeX
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]}
Adım 4 — INN skaler çarpımı ile ağırlıklı normalleştirilmiş matrisi hesapla.

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

LaTeX
\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
Adım 5 — Her alternatif için fayda kriteri değerlerini P_i'ye, maliyet kriteri değerlerini R_i'ye INWMSM^(k) operatörüyle (Aralık Nötrosofik Ağırlıklı Maclaurin Simetrik Ortalama) toplulaştır.

$P_{i} = {I N W M S M}_{w}^{(k)} ({\tilde{x}}_{i j} : j \in B), R_{i} = {I N W M S M}_{w}^{(k)} ({\tilde{x}}_{i j} : j \in C), B = benefit, C = cost$

LaTeX
P_i = \mathrm{INWMSM}^{(k)}_w\left(\tilde{x}_{ij} : j \in B\right),\quad R_i = \mathrm{INWMSM}^{(k)}_w\left(\tilde{x}_{ij} : j \in C\right),\quad B=\text{benefit},\ C=\text{cost}
Adım 6 — Risk indeksli skor fonksiyonu S_λ ile tüm P_i ve R_i'yi skorla. Rmin = min_i S_λ(R_i) bul.

$S_{λ} (x) = \frac{1}{6} [λ (T_{L} + T_{U}) + (1 - λ) (4 - (I_{L} + I_{U}) - (F_{L} + F_{U}))]$

LaTeX
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]
Adım 7 — Her alternatif için COPRAS-IN formülü ile Q_i tercih skorunu hesapla.

$Q_{i} = S_{λ} (P_{i}) + \frac{\sum_{i = 1}^{m} S_{λ} (R_{i})}{S_{λ} (R_{i}) \cdot \sum_{i = 1}^{m} \frac{1}{S_{λ} (R_{i})}}$

LaTeX
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)}}
Adım 8 — Q_i'ye göre azalan sırada alternatifler. N_i = (Q_i / max Q_i) × 100%.

$N_{i} = \frac{Q_{i}}{{max}_{i} Q_{i}} \times 100 %$

LaTeX
N_i = \frac{Q_i}{\max_i Q_i} \times 100\%

Sezgi

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.

Sonucu okuma: 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.

Sınırlılıklar

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

Sık yapılan hatalar

Risk indeksi duyarlılığı: farklı λ değerleri sıralamaları tersine çevirebilir. Kullanılan λ değerini her zaman raporla.
INN geçerliliği: hesaplamadan önce T_U+I_U+F_U ≤ 3 koşulunu kontrol et.
INWMSM k parametresi: k=1 ağırlıklı ortalamaya dejenere olur; k=2 literatürde en yaygın.

Hesap adımları ve dayanakları

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

Dayanak: Şahin 2019, Eq.(39)
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.

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

Dayanak: Şahin 2019, Eqs.(33)-(35),(38)
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]>.

Dayanak: Şahin 2019, Def.3(3), Step 3
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).

Dayanak: Şahin 2019, Eqs.(41)-(42),(28)
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).

Dayanak: Şahin 2019, Def.5, Eq.(16); Eq.(43)
Compute preference score Q_i for each alternative using COPRAS-IN formula.

Dayanak: Şahin 2019, Eq.(45)
Rank alternatives descending by Q_i (higher Q_i = better). Compute utility degree N_i = (Q_i / max_i Q_i) × 100%.

Dayanak: Şahin 2019, Eq.(46)