IV-TOPSIS - Interval-valued intuitionistic fuzzy extension of TOPSIS

Overview

IV-TOPSIS extends crisp TOPSIS to bounded uncertainty represented as Interval Numbers IN: [L, U]. The Jahanshahloo-Lotfi-Izadikhah (2006) algorithm keeps both bounds throughout normalisation and weighting; the positive- and negative-ideal solutions are then assembled by taking max/min over the appropriate bound depending on whether each criterion is a benefit or a cost. Distances use the opposite bound (lower for d⁺ on benefits, upper for d⁻ on benefits, swapped for costs) so that the worst-case separation is measured. The final relative closeness C* is a crisp scalar in [0, 1] - no defuzzification step is needed.

Output

utility, higher is better

Data

Interval Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Interval MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1
   $${\tilde{a}}_{ij}^c=([{\nu }_{ij}^L,{\nu }_{ij}^U],[{\mu }_{ij}^L,{\mu }_{ij}^U])\text{for cost criteria}$$

   IV-TOPSIS-base

2. 2
   $${\tilde{v}}_{ij}=w_j\otimes {\hat{a}}_{ij}=([1-(1-{\mu }_{ij}^L{)}^{w_j},1-(1-{\mu }_{ij}^U{)}^{w_j}],[{\nu }_{ij}^L{}^{w_j},{\nu }_{ij}^U{}^{w_j}])$$

   IV-TOPSIS-base

3. 3
   $$A_j^{+}=([\underset{i}{\max }{\mu }_{ij}^L,\underset{i}{\max }{\mu }_{ij}^U],[\underset{i}{\min }{\nu }_{ij}^L,\underset{i}{\min }{\nu }_{ij}^U]),A_j^{-}=([\underset{i}{\min }{\mu }_{ij}^L,\underset{i}{\min }{\mu }_{ij}^U],[\underset{i}{\max }{\nu }_{ij}^L,\underset{i}{\max }{\nu }_{ij}^U])$$

   IV-TOPSIS-base

4. 4
   $$d_i^{\pm }=\sqrt{\frac{1}{4}\sum _{j=1}^n[({\mu }_{ij}^L-{\mu }_j^{L\pm }{)}^2+({\mu }_{ij}^U-{\mu }_j^{U\pm }{)}^2+({\nu }_{ij}^L-{\nu }_j^{L\pm }{)}^2+({\nu }_{ij}^U-{\nu }_j^{U\pm }{)}^2]}$$

   IV-TOPSIS-base

5. 5
   $$C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},\text{rank descending}$$

   IV-TOPSIS-base

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Interval numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Edge cases and pitfalls

• Value-space violation: ensure all entries satisfy IN: L ≤ U before computation; a single inverted bound corrupts the column denominator.

• Cost/benefit direction confusion: the bound used for the ideal swaps with the criterion direction. Mis-assigning min/max collapses IV-TOPSIS to a crisp variant on the wrong bound.

• Treating [L, U] as a triangular fuzzy number: IV-TOPSIS uses interval (not TFN) arithmetic. There is no centroid step; if your data are TFNs use FUZZY-TOPSIS instead.

Works with

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