IF-TOPSIS - Intuitionistic Fuzzy TOPSIS

Overview

IF-TOPSIS extends crisp TOPSIS to settings where each rating is an Intuitionistic Fuzzy Number (μ, ν) representing membership/non-membership. The algorithm aggregates expert ratings and weights via IFWA (Steps 2-3), weights the matrix with Atanassov's ⊗ operator (Step 4), extracts IF positive/negative ideal solutions per criterion direction (Step 5), then ranks alternatives by their crisp closeness coefficient C* ∈ [0,1] (Steps 6-7). Higher C* = better. NO defuzzification step is involved; the ranking emerges directly from IF-distance separations.

Output

utility, higher is better

Data

Intuitionistic Fuzzy, ifn tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Supplier selection (Boran 2009 original case), MAGDM under epistemic uncertainty, Expert-driven evaluation with linguistic IFN scales

How it works

1. 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.

   $$R=(r_{ij}{)}_{m\times n},r_{ij}=({\mu }_{ij},{\nu }_{ij}),{\mu }_{ij}+{\nu }_{ij}\le 1$$

   Boran 2009 §3 input setup (matrix R definition; pre-Eq.5)

2. 2

   Aggregate decision-maker ratings into a single IFN per cell using the IFWA operator with DM importance weights λ_k.

   $$r_{ij}={\text{IFWA}}_{\lambda }(r_{ij}^{(1)},\dots ,r_{ij}^{(K)})=\left(1-\prod _{k=1}^K(1-{\mu }_{ij}^{(k)}{)}^{{\lambda }_k},\prod _{k=1}^K({\nu }_{ij}^{(k)}{)}^{{\lambda }_k}\right)$$

   Boran 2009 Eq. (6); Xu 2007 IFWA operator

3. 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.

   $$w_j={\text{IFWA}}_{\lambda }(w_j^{(1)},\dots ,w_j^{(K)})=\left(1-\prod _{k=1}^K(1-{\mu }_{w_j}^{(k)}{)}^{{\lambda }_k},\prod _{k=1}^K({\nu }_{w_j}^{(k)}{)}^{{\lambda }_k}\right)$$

   Boran 2009 Eq. (7); Xu 2007 IFWA operator

4. 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}).

   $$r_{ij}^{\prime }=r_{ij}\otimes w_j=\left({\mu }_{ij}\cdot {\mu }_{w_j},{\nu }_{ij}+{\nu }_{w_j}-{\nu }_{ij}\cdot {\nu }_{w_j}\right);{\pi }_{ij}^{\prime }=1-{\mu }_{ij}^{\prime }-{\nu }_{ij}^{\prime }$$

   Boran 2009 Eqs. (8)-(9); Atanassov 1986 ⊗-operator (Eq. 4)

5. 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.

   $$A^{\ast }=\{({\mu }_j^{\ast },{\nu }_j^{\ast }){\}}_{j=1}^n,({\mu }_j^{\ast },{\nu }_j^{\ast })=\{\begin{array}{ll}({\max }_i{\mu }_{ij}^{\prime },{\min }_i{\nu }_{ij}^{\prime })& j\in J_b\\ ({\min }_i{\mu }_{ij}^{\prime },{\max }_i{\nu }_{ij}^{\prime })& j\in J_c\end{array};A^{-}\text{ swaps max}\leftrightarrow \min .$$

   Boran 2009 Eqs. (10)-(14)

6. 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)).

   $$S_i^{\ast }=\sqrt{\frac{1}{2n}\sum _{j=1}^n\left[({\mu }_{ij}^{\prime }-{\mu }_j^{\ast }{)}^2+({\nu }_{ij}^{\prime }-{\nu }_j^{\ast }{)}^2+({\pi }_{ij}^{\prime }-{\pi }_j^{\ast }{)}^2\right]};S_i^{-}\text{ analogous with }{A}^{-}.$$

   Boran 2009 Eqs. (15)-(16); Szmidt-Kacprzyk 2000 normalised Euclidean distance (specific eq not verified, anchor removed)

7. 7

   Compute the relative closeness coefficient C_i^{*} of each alternative to the IF ideal solution.

   $$C_i^{\ast }=\frac{S_i^{-}}{S_i^{\ast }+S_i^{-}},0\le C_i^{\ast }\le 1$$

   Boran 2009 Eq. (17)

8. 8

   Rank alternatives in descending order of C_i^{*}; the alternative with the largest C* is the best.

   $$\text{rank}(A_i)={\text{argsort}}_{\downarrow }(C_i^{\ast })$$

   Boran 2009 §3 Step 8 (descending sort by C*; no equation)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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.

• Confusing IFN (pair μ, ν) with TFN (triple a, b, c): IF-TOPSIS values MUST be IFN tuples satisfying μ+ν ≤ 1, NOT triangular fuzzy numbers.

• Applying crisp multiplication v_ij = w_j · r_ij instead of Atanassov ⊗: this breaks IFN axioms and produces invalid (μ, ν) pairs.

• Forgetting the 1/(2n) factor in Szmidt-Kacprzyk normalised Euclidean distance - Boran 2009 uses the NORMALISED form (Eq. 14), not the un-normalised SK 2000 Eq. 58.

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