BF-VIKOR: Bipolar extension of VIKOR

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 VIKOR directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}.

Common pitfalls

• •Hatalı: 'BF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Bipolar Fuzzy numbers/tuples
• •Hatalı: 'BF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'BF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: BF-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: BF-VIKOR'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: Determine the best f_j* and worst f_j^- value of every criterion across alternatives, respecting benefit/cost direction. Formül: f_{j}^{*} = \begin{cases}\max_{i} x_{ij} & j \in J \\ \min_{i} x_{ij} & j \in J'\end{cases},\quad f_{j}^{-} = \begin{cases}\min_{i} x_{ij} & j \in J \\ \max_{i} x_{ij} & j \in J'\end{cases} Anchor: Opricovic & Tzeng 2004, §2 Eq.(3)
3. 3.Adım 3 (F3): Step 2: Compute the group utility S_i and the individual regret R_i. S aggregates weighted normalised regret (L_1-metric); R is the maximum weighted regret (L_∞-metric). Formül: S_{i} = \sum_{j=1}^{n} w_{j}\,\dfrac{f_{j}^{*}-x_{ij}}{f_{j}^{*}-f_{j}^{-}},\quad R_{i} = \max_{j}\left[w_{j}\,\dfrac{f_{j}^{*}-x_{ij}}{f_{j}^{*}-f_{j}^{-}}\right] Anchor: Opricovic & Tzeng 2004, §2 Eq.(4)
4. 4.Adım 4 (F4): Step 3: Compute the VIKOR index Q_i as a convex combination of normalised S and R, weighted by the compromise coefficient v. Formül: Q_{i} = v\,\dfrac{S_{i}-S^{*}}{S^{-}-S^{*}} + (1-v)\,\dfrac{R_{i}-R^{*}}{R^{-}-R^{*}},\ S^{*}=\min_{i}S_{i},\ S^{-}=\max_{i}S_{i},\ R^{*}=\min_{i}R_{i},\ R^{-}=\max_{i}R_{i} Anchor: Opricovic & Tzeng 2004, §2 Eq.(5)
5. 5.Adım 5 (F5): Step 4: Propose A^(1) (the lowest-Q alternative) as compromise solution iff both C1 (acceptable advantage) and C2 (acceptable stability) hold. If C1 fails, return the maximum prefix A^(1)..A^(M) for which Q(A^(M)) − Q(A^(1)) < DQ. If only C2 fails, return {A^(1), A^(2)}. Formül: DQ = \dfrac{1}{m-1};\quad C1: Q(A^{(2)}) - Q(A^{(1)}) \ge DQ;\quad C2: A^{(1)} \text{ is best in } S \text{ or in } R Anchor: Opricovic & Tzeng 2004, §2 Eqs.(6)-(7)

Commonly paired with

• •n_a + BF-VIKOR (common)

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