BF-TOPSIS: Bipolar extension of TOPSIS

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

Edge cases

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

Common pitfalls

• •Hatalı: 'BF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Bipolar Fuzzy numbers/tuples
• •Hatalı: 'BF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'BF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: BF-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: BF-TOPSIS'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: Vector normalisation: scale each column to unit Euclidean norm. Formül: r_{ij} = \dfrac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^{2}}}, \quad i=1,\ldots,m;\ j=1,\ldots,n Anchor: Hwang & Yoon 1981, p.130 Eq.(4.1)
3. 3.Adım 3 (F3): Step 2: Weighted normalised matrix: multiply each column by its weight. Formül: v_{ij} = w_{j} \cdot r_{ij} Anchor: Hwang & Yoon 1981, p.130 Eq.(4.2)
4. 4.Adım 4 (F4): Step 3: Determine the positive-ideal A⁺ and negative-ideal A⁻ solutions per criterion direction. Formül: A^{+} = \{v_{1}^{+}, \ldots, v_{n}^{+}\},\ v_{j}^{+} = \begin{cases} \max_{i} v_{ij} & j \in J \\ \min_{i} v_{ij} & j \in J' \end{cases};\quad A^{-} = \{v_{1}^{-}, \ldots, v_{n}^{-}\},\ v_{j}^{-} = \begin{cases} \min_{i} v_{ij} & j \in J \\ \max_{i} v_{ij} & j \in J' \end{cases} Anchor: Hwang & Yoon 1981, p.130 Eqs.(4.3)-(4.4)
5. 5.Adım 5 (F5): Step 4: Euclidean separation of each alternative from A⁺ and A⁻. Formül: d_{i}^{+} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{+})^{2}},\quad d_{i}^{-} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{-})^{2}} Anchor: Hwang & Yoon 1981, p.130 Eqs.(4.5)-(4.6)
6. 6.Adım 6 (F6): Step 5: Relative closeness to the ideal solution; rank by C* descending. Formül: C_{i}^{*} = \dfrac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}},\quad 0 \le C_{i}^{*} \le 1 Anchor: Hwang & Yoon 1981, p.131 Eq.(4.7)

Commonly paired with

• •n_a + BF-TOPSIS (common)

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