Rough-MABAC: Rough extension of MABAC

Overview

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

Strengths

• •Method-specific: Rough outranking/ranking: Rough number (lower approximation L, upper approximation U)
• •Preserves rough uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •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

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

When not to use

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

Edge cases

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

Common pitfalls

• •Hatalı: 'ROUGH-MABAC bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Rough numbers/tuples
• •Hatalı: 'ROUGH-MABAC bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'ROUGH-MABAC bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: ROUGH-MABAC'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: ROUGH-MABAC'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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)]. Formül: [\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} Anchor: Jia-Liu-Wang 2019 (ESWA 127:241-255) Eq.(33)
2. 2.Adım 2 (F2): Step 2: Weighted IFRN matrix using IFWA power operation applied to lower and upper IFN components separately: [γ_ij] = w_j ⊗ [β_ij]. Formül: [\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] Anchor: Jia-Liu-Wang 2019 (ESWA 127:241-255) Eq.(34)
3. 3.Adım 3 (F3): Step 3: Intuitionistic Fuzzy Rough Border Approximation Area (IFRBAA) [g_j] per criterion: IFRG operator (geometric mean of lower/upper IFN components independently). Formül: [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] Anchor: Jia-Liu-Wang 2019 (ESWA 127:241-255) Eq.(35)
4. 4.Adım 4 (F4): Step 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. Formül: d_{ij} = S([\gamma_{ij}]) - S([g_j]);\quad S([(\mu,v),(\bar{\mu},\bar{v})]) = \frac{(\mu-v)+(\bar{\mu}-\bar{v})}{2} Anchor: Jia-Liu-Wang 2019 (ESWA 127:241-255) Section 4.4 Step 5; IFRN score analogous to Xu 2007
5. 5.Adım 5 (F5): Step 5: Appraisal score Q_i = Σ_j d_ij across all criteria; rank descending. Higher Q_i = better alternative. Formül: Q_i = \sum_{j=1}^{n} d_{ij};\quad \mathrm{rank\ by\ }Q_i \downarrow Anchor: Jia-Liu-Wang 2019 (ESWA 127:241-255) Section 4.4 (Table 7 verification)

Commonly paired with

• •n_a + ROUGH-MABAC (common)

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