BF-MARCOS: Bipolar extension of MARCOS

Overview

Bipolar outranking/ranking: Bipolar Fuzzy Set (BFS: positive membership μ⁺ ∈ [0,1], negative μ⁻ ∈ [-1,0]). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Bipolar outranking/ranking: Bipolar Fuzzy Set (BFS: positive membership μ⁺ ∈ [0,1], negative μ⁻ ∈ [-1,0])
• •Preserves bipolar_fuzzy 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 Bipolar Fuzzy 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 Bipolar Fuzzy 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 MARCOS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

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

Common pitfalls

• •Hatalı: 'BF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Bipolar Fuzzy numbers/tuples
• •Hatalı: 'BF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'BF-MARCOS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: BF-MARCOS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: BF-MARCOS'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 0: Construct bipolar fuzzy decision matrix B = [(μ⁺_ij, ν⁻_ij)]. Normalize cost criteria via BFN complement: (1−μ⁺, |ν⁻|−1). Convert all BFNs to crisp scores using score function Δ(B)=(1+μ⁺+ν⁻)/2 ∈ [0,1]. All subsequent steps operate on the resulting crisp score matrix x_ij = Δ(B*_ij). Formül: \Delta(B) = \dfrac{1 + \mu^+ + \nu^-}{2} \in [0,1] \\[4pt] B^*_{ij} = \begin{cases}(\mu^+_{ij}, \nu^-_{ij}) & j \in J^+ \\ (1-\mu^+_{ij}, |\nu^-_{ij}|-1) & j \in J^- \end{cases} \\[4pt] x_{ij} = \Delta(B^*_{ij}) Anchor: Wei et al. 2018b: BFN score function Eq.(2); Jana & Pal 2021: cost normalization Eq.(7)
2. 2.Adım 2 (F2): Step 1: Form extended decision matrix with AAI (anti-ideal) and AI (ideal) rows. Formül: X^{*} = [X^{T}\,;\,AAI\,;\,AI]^{T},\quad AAI_{j}=\min/\max,\ AI_{j}=\max/\min Anchor: Stević 2020, p.6 Eq.(1)
3. 3.Adım 3 (F3): Step 2: Normalise vs ideal AI per criterion direction. Formül: n_{ij} = \begin{cases} x_{ij}/x_{AI,j} & j\in J^{+} \\ x_{AI,j}/x_{ij} & j\in J^{-} \end{cases} Anchor: Stević 2020, p.6 Eq.(2)
4. 4.Adım 4 (F4): Step 3: Weighted normalised matrix v_ij = w_j · n_ij. Formül: v_{ij} = w_{j}\,n_{ij} Anchor: Stević 2020, p.6 Eq.(3)
5. 5.Adım 5 (F5): Step 4: Sum each alternative's weighted matrix row: S_i = Σ v_ij. Formül: S_{i} = \sum_{j=1}^{n} v_{ij} Anchor: Stević 2020, p.7 Eq.(4)
6. 6.Adım 6 (F6): Step 5: Utility ratios vs ideal K+ and anti-ideal K− rows. Formül: K^{+}_{i}=\dfrac{S_{i}}{S_{AI}},\quad K^{-}_{i}=\dfrac{S_{i}}{S_{AAI}} Anchor: Stević 2020, p.7 Eqs.(5)-(6)
7. 7.Adım 7 (F7): Step 6: Utility functions f(K+_i) and f(K−_i) of the utility ratios. Formül: f(K^{+}_{i})=\dfrac{K^{-}_{i}}{K^{+}_{i}+K^{-}_{i}},\quad f(K^{-}_{i})=\dfrac{K^{+}_{i}}{K^{+}_{i}+K^{-}_{i}} Anchor: Stević 2020, p.7 Eqs.(7)-(8)
8. 8.Adım 8 (F8): Step 7: Final utility f(K_i) and descending ranking. Formül: f(K_{i})=\dfrac{K^{+}_{i}+K^{-}_{i}}{1+\dfrac{1-f(K^{+}_{i})}{f(K^{+}_{i})}+\dfrac{1-f(K^{-}_{i})}{f(K^{-}_{i})}} Anchor: Stević 2020, p.7 Eq.(9)

Commonly paired with

• •n_a + BF-MARCOS (common)

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