Rough-MABAC - MABAC yönteminin Rough uzantısı

Formül adımları

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

1. Adım 1 — IFRN grup matrisini normalleştir: fayda kriterleri aynı; maliyet kriterleri μ↔v değiştir.

   LaTeX

   [\beta_{ij}] = \begin{cases} [(\mu_{ij},v_{ij}),(\bar{\mu}_{ij},\bar{v}_{ij})] & c_j \in \Omega_{\max} \\ [(v_{ij},\mu_{ij}),(\bar{v}_{ij},\bar{\mu}_{ij})] & c_j \in \Omega_{\min} \end{cases}

2. Adım 2 — IFWA kuvvet operasyonu ile ağırlıklı IFRN matrisi: [γ_ij] = w_j ⊗ [β_ij].

   $$[{\gamma }_{ij}]=w_j\otimes [{\beta }_{ij}]=[(1-(1-{\mu }_{ij}^{\prime }{)}^{w_j},(v_{ij}^{\prime }{)}^{w_j}),(1-(1-{\overline{\mu }}_{ij}^{\prime }{)}^{w_j},({\overline{v}}_{ij}^{\prime }{)}^{w_j})]$$
   LaTeX

   [\gamma_{ij}] = w_j \otimes [\beta_{ij}] = \left[(1-(1-\mu'_{ij})^{w_j},\;(v'_{ij})^{w_j}),\;(1-(1-\bar{\mu}'_{ij})^{w_j},\;(\bar{v}'_{ij})^{w_j})\right]

3. Adım 3 — Her kriter için IFRG geometrik ortalaması ile IFRBAA [g_j].

   $$[g_j]=\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{G}([{\gamma }_{1j}],\dots ,[{\gamma }_{mj}])=[({\prod }_{i=1}^m({\mu }_{ij}^{″}{)}^{1/m},1-{\prod }_{i=1}^m(1-v_{ij}^{″}{)}^{1/m}),({\prod }_{i=1}^m({\overline{\mu }}_{ij}^{″}{)}^{1/m},1-{\prod }_{i=1}^m(1-{\overline{v}}_{ij}^{″}{)}^{1/m})]$$
   LaTeX

   [g_j] = \mathrm{IFRG}([\gamma_{1j}],\ldots,[\gamma_{mj}]) = \left[\left(\prod_{i=1}^m (\mu''_{ij})^{1/m},\;1-\prod_{i=1}^m (1-v''_{ij})^{1/m}\right),\;\left(\prod_{i=1}^m (\bar{\mu}''_{ij})^{1/m},\;1-\prod_{i=1}^m (1-\bar{v}''_{ij})^{1/m}\right)\right]

4. Adım 5 — IFRBAA'dan işaretli Öklid mesafesi. d_ij = +e([γ_ij]>[g_j]), 0, veya −e([γ_ij]<[g_j]). Karşılaştırma: Def.5 Sum fonksiyonu; mesafe: Denklem (18).

   $$\begin{gathered} d_{ij}=\{\begin{array}{cc}+e([{\gamma }_{ij}],[g_j])\hfill & \text{if }[{\gamma }_{ij}]>[g_j]\hfill \\ 0\hfill & \text{if }[{\gamma }_{ij}]=[g_j]\hfill \\ -e([{\gamma }_{ij}],[g_j])\hfill & \text{if }[{\gamma }_{ij}]<[g_j]\hfill \end{array} \\ [\mathrm{4pt}]e([{\alpha }_1],[{\alpha }_2])=\sqrt{\frac{1}{4}[({\mu }_1-{\mu }_2{)}^2+({\overline{\mu }}_1-{\overline{\mu }}_2{)}^2+(v_1-v_2{)}^2+({\overline{v}}_1-{\overline{v}}_2{)}^2+({\pi }_1-{\pi }_2{)}^2+({\overline{\pi }}_1-{\overline{\pi }}_2{)}^2]}\text{(Eq.18)} \\ [\mathrm{4pt}][{\alpha }_1]>[{\alpha }_2]⟺S(\mathrm{S}\mathrm{u}\mathrm{m}([{\alpha }_1]))>S(\mathrm{S}\mathrm{u}\mathrm{m}([{\alpha }_2]))\text{ where }\mathrm{S}\mathrm{u}\mathrm{m}([\alpha ])=(\mu ,v)\oplus (\overline{\mu },\overline{v})\text{ (Def.5)} \end{gathered}$$
   LaTeX

   d_{ij} = \begin{cases} +e([\gamma_{ij}],[g_j]) & \text{if } [\gamma_{ij}] > [g_j] \\ 0 & \text{if } [\gamma_{ij}] = [g_j] \\ -e([\gamma_{ij}],[g_j]) & \text{if } [\gamma_{ij}] < [g_j] \end{cases}\\[4pt]e([\alpha_1],[\alpha_2])=\sqrt{\tfrac{1}{4}\bigl[(\mu_1-\mu_2)^2+(\bar{\mu}_1-\bar{\mu}_2)^2+(v_1-v_2)^2+(\bar{v}_1-\bar{v}_2)^2+(\pi_1-\pi_2)^2+(\bar{\pi}_1-\bar{\pi}_2)^2\bigr]}\quad\text{(Eq.18)}\\[4pt][\alpha_1]>[\alpha_2]\iff S(\mathrm{Sum}([\alpha_1]))>S(\mathrm{Sum}([\alpha_2]))\text{ where }\mathrm{Sum}([\alpha])=(\mu,v)\oplus(\bar{\mu},\bar{v})\text{ (Def.5)}

5. Adım 5 — Q_i = Σ_j d_ij tüm kriterler; azalan sırala. Yüksek Q_i = daha iyi alternatif.

   $$Q_i=\sum _{j=1}^n{d}_{ij};\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{b}\mathrm{y}{Q}_i\downarrow$$
   LaTeX

   Q_i = \sum_{j=1}^{n} d_{ij};\quad \mathrm{rank\ by\ }Q_i \downarrow

Sezgi

Rough outranking/ranking - Rough number (lower approximation L, upper approximation U). Output typically utility (higher value = preferred).

Sonucu okuma: ROUGH-MABAC (Jia et al. 2019) uses Intuitionistic Fuzzy Rough Numbers (IFRN=[(μ_l,ν_l),(μ_u,ν_u)]), NOT plain rough intervals. Pipeline: (1) for cost criteria swap μ↔ν (normalization); (2) IFWA scalar power w_j⊗β_ij=(1-(1-μ)^{w_j}, ν^{w_j}) applied to each IFN component; (3) IFRG geometric mean gives IFRBAA [g_j] per criterion; (4) signed Euclidean distance d_ij from IFRBAA: positive if [γ_ij]>[g_j] (via Definition 5 Sum-based comparison), negative if below; (5) Q_i=Σ_j d_ij, rank descending. No midpoint defuzzification - ranking is purely score-based via Euclidean distances.

Varsayımlar

• Decision matrix entries are valid Rough numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Ne zaman kullanılmaz

• Classical data sufficient - use base MABAC directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

• 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 Rough 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

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin Rough: L ≤ U; approximations defined by equivalence classes koşulunu sağladığından emin olun.
• IFRN karşılaştırma hatası: d_ij işaretini belirlerken Tanım 5 Sum tabanlı IFN karşılaştırmasını kullanın - basit μ çıkarma değil.