IF-TOPSIS: Intuitionistic Fuzzy TOPSIS

Overview

Distance-based ranking under Intuitionistic Fuzzy uncertainty. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Distance-based ranking under Intuitionistic Fuzzy uncertainty
• •Preserves intuitionistic 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 TOPSIS base (Belton-Gear 1983, Wang-Luo 2009))
• •Assumes: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •Assumes: All decision-makers use the same linguistic-to-IFN scale (e.g. Boran 2009 Table 1)
• •Assumes: Criterion directions (benefit/cost) are explicitly labelled

Assumptions to verify

• •Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •All decision-makers use the same linguistic-to-IFN scale (e.g. Boran 2009 Table 1)
• •Criterion directions (benefit/cost) are explicitly labelled

When not to use

• •Crisp data sufficient: use base TOPSIS directly (avoid unnecessary uncertainty layer)
• •Hesitation degree π must be modelled explicitly rather than implicitly: consider Pythagorean Fuzzy TOPSIS or q-ROF TOPSIS when μ+ν > 1 is observed in raw judgements

Edge cases

• •IF decision matrix R = (r_ij) where r_ij = (μ_ij, ν_ij) is the IFN rating of alternative A_i on criterion C_j. For group-decision settings (K≥2 DMs), collect K such matrices.
• •IF decision matrix R' = R ⊗ W using Atanassov's IFN multiplication operator, applied element-wise to each criterion column j with weight w_j = (μ_{w_j}, ν_{w_j}).
• •IF Positive-Ideal A* and Negative-Ideal A⁻ solutions per criterion direction. For benefit criteria (j ∈ J_b) maximise μ' and minimise ν'; for cost criteria (j ∈ J_c) the roles swap.
• •IF distance (3-component form with factor 1/(2n)).
• •IF ideal solution.

Common pitfalls

• •Hatalı: 'IF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •Hatalı: 'IF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-makers use the same linguistic-to-IFN scale (e.g. Boran 2009 Table 1)
• •Hatalı: 'IF-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion directions (benefit/cost) are explicitly labelled
• •Hatalı: IF-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IF-TOPSIS'yi 'Hesitation degree π must be modelled explicitly rather than implicitly' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Construct the IF decision matrix R = (r_ij) where r_ij = (μ_ij, ν_ij) is the IFN rating of alternative A_i on criterion C_j. For group-decision settings (K≥2 DMs), collect K such matrices. Formül: R = (r_{ij})_{m \times n},\ r_{ij} = (\mu_{ij}, \nu_{ij}),\ \mu_{ij}+\nu_{ij} \le 1 Anchor: Boran 2009 §3 input setup (matrix R definition; pre-Eq.5)
2. 2.Adım 2 (F2): Step 2: Aggregate decision-maker ratings into a single IFN per cell using the IFWA operator with DM importance weights λ_k. Formül: r_{ij} = \text{IFWA}_{\lambda}(r_{ij}^{(1)}, \ldots, r_{ij}^{(K)}) = \left(1 - \prod_{k=1}^{K}(1-\mu_{ij}^{(k)})^{\lambda_k},\ \prod_{k=1}^{K}(\nu_{ij}^{(k)})^{\lambda_k}\right) Anchor: Boran 2009 Eq. (6); Xu 2007 IFWA operator
3. 3.Adım 3 (F3): Step 3: Aggregate per-DM IFN criterion weights w_j^{(k)} into a single IFN weight vector W = (w_1,…,w_n) via IFWA over DMs. Formül: w_j = \text{IFWA}_{\lambda}(w_j^{(1)},\ldots,w_j^{(K)}) = \left(1 - \prod_{k=1}^{K}(1-\mu_{w_j}^{(k)})^{\lambda_k},\ \prod_{k=1}^{K}(\nu_{w_j}^{(k)})^{\lambda_k}\right) Anchor: Boran 2009 Eq. (7); Xu 2007 IFWA operator
4. 4.Adım 4 (F4): Step 4: Build the weighted IF decision matrix R' = R ⊗ W using Atanassov's IFN multiplication operator, applied element-wise to each criterion column j with weight w_j = (μ_{w_j}, ν_{w_j}). Formül: r'_{ij} = r_{ij} \otimes w_j = \left(\mu_{ij} \cdot \mu_{w_j},\ \nu_{ij} + \nu_{w_j} - \nu_{ij} \cdot \nu_{w_j}\right);\quad \pi'_{ij} = 1 - \mu'_{ij} - \nu'_{ij} Anchor: Boran 2009 Eqs. (8)-(9); Atanassov 1986 ⊗-operator (Eq. 4)
5. 5.Adım 5 (F5): Step 5: Determine the IF Positive-Ideal A* and Negative-Ideal A⁻ solutions per criterion direction. For benefit criteria (j ∈ J_b) maximise μ' and minimise ν'; for cost criteria (j ∈ J_c) the roles swap. Formül: A^{*} = \{(\mu^{*}_j, \nu^{*}_j)\}_{j=1}^{n},\ (\mu^{*}_j, \nu^{*}_j) = \begin{cases} (\max_i \mu'_{ij},\ \min_i \nu'_{ij}) & j \in J_b \\ (\min_i \mu'_{ij},\ \max_i \nu'_{ij}) & j \in J_c \end{cases};\ A^{-} \text{ swaps max} \leftrightarrow \min. Anchor: Boran 2009 Eqs. (10)-(14)
6. 6.Adım 6 (F6): Step 6: Compute separation measures S_i^{*} and S_i^{-} of each alternative from A* and A⁻ using the Szmidt-Kacprzyk normalised Euclidean IF distance (3-component form with factor 1/(2n)). Formül: S_i^{*} = \sqrt{\frac{1}{2n}\sum_{j=1}^{n}\left[(\mu'_{ij}-\mu^{*}_j)^2 + (\nu'_{ij}-\nu^{*}_j)^2 + (\pi'_{ij}-\pi^{*}_j)^2\right]};\quad S_i^{-} \text{ analogous with } A^{-}. Anchor: Boran 2009 Eqs. (15)-(16); Szmidt-Kacprzyk 2000 normalised Euclidean distance (specific eq not verified, anchor removed)
7. 7.Adım 7 (F7): Step 7: Compute the relative closeness coefficient C_i^{*} of each alternative to the IF ideal solution. Formül: C_i^{*} = \frac{S_i^{-}}{S_i^{*} + S_i^{-}},\quad 0 \le C_i^{*} \le 1 Anchor: Boran 2009 Eq. (17)
8. 8.Adım 8 (F8): Step 8: Rank alternatives in descending order of C_i^{*}; the alternative with the largest C* is the best. Formül: \text{rank}(A_i) = \text{argsort}_{\downarrow}(C_i^{*}) Anchor: Boran 2009 §3 Step 8 (descending sort by C*; no equation)

Commonly paired with

• •IF-AHP + IF-TOPSIS (common)
• •ENTROPY + IF-TOPSIS (common)

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