Rough-MABAC - Rough extension of MABAC

Overview

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.

Output

utility, higher is better

Data

Intuitionistic Rough, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Rough MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Normalize IFRN group matrix: for benefit criteria keep [β_ij] = [(μ_ij,v_ij),(μ̄_ij,v̄_ij)]; for cost criteria swap μ↔v: [β_ij] = [(v_ij,μ_ij),(v̄_ij,μ̄_ij)].

   $$[{\beta }_{ij}]=\{\begin{array}{ll}[({\mu }_{ij},v_{ij}),({\bar{\mu }}_{ij},{\bar{v}}_{ij})]& c_j\in {\Omega }_{\max }\\ [(v_{ij},{\mu }_{ij}),({\bar{v}}_{ij},{\bar{\mu }}_{ij})]& c_j\in {\Omega }_{\min }\end{array}$$

   Jia-Liu-Wang 2019 (ESWA 127:241-255) Eq.(33)

2. 2

   Weighted IFRN matrix using IFWA power operation applied to lower and upper IFN components separately: [γ_ij] = w_j ⊗ [β_ij].

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

   Jia-Liu-Wang 2019 (ESWA 127:241-255) Eq.(34)

3. 3

   Intuitionistic Fuzzy Rough Border Approximation Area (IFRBAA) [g_j] per criterion: IFRG operator (geometric mean of lower/upper IFN components independently).

   $$[g_j]=\mathrm{IFRG}([{\gamma }_{1j}],\dots ,[{\gamma }_{mj}])=\left[\left(\prod _{i=1}^m({\mu }_{ij}^{\prime \prime }{)}^{1/m},1-\prod _{i=1}^m(1-v_{ij}^{\prime \prime }{)}^{1/m}\right),\left(\prod _{i=1}^m({\bar{\mu }}_{ij}^{\prime \prime }{)}^{1/m},1-\prod _{i=1}^m(1-{\bar{v}}_{ij}^{\prime \prime }{)}^{1/m}\right)\right]$$

   Jia-Liu-Wang 2019 (ESWA 127:241-255) Eq.(35)

4. 4

   Signed IFRN score distance: d_ij = S([γ_ij]) − S([g_j]), where S([α]) = ((μ−v)+(μ̄−v̄))/2 is the IFRN score function. Positive = above border, negative = below.

   $$d_{ij}=S([{\gamma }_{ij}])-S([g_j]);S([(\mu ,v),(\bar{\mu },\bar{v})])=\frac{(\mu -v)+(\bar{\mu }-\bar{v})}{2}$$

   Jia-Liu-Wang 2019 (ESWA 127:241-255) Section 4.4 Step 5; IFRN score analogous to Xu 2007

5. 5

   Appraisal score Q_i = Σ_j d_ij across all criteria; rank descending. Higher Q_i = better alternative.

   $$Q_i=\sum _{j=1}^n{d}_{ij};\mathrm{rank}\mathrm{by}{Q}_i\downarrow$$

   Jia-Liu-Wang 2019 (ESWA 127:241-255) Section 4.4 (Table 7 verification)

Fits when / Look elsewhere when

Assumptions to verify

• 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

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy Rough: L ≤ U; approximations defined by equivalence classes before computation.

• IFRN comparison error: when determining sign of d_ij, use Definition 5 Sum-based IFN comparison - NOT simple μ subtraction. Sum([α])=(μ,ν)⊕(μ̄,v̄) uses IFN addition before computing score.

Works with

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