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

Overview

iv-todim extends TODIM to handle Interval uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Interval Number (IN: [a, b]) algebra. The final scores are defuzzified via midpoint (a+b)/2 before ranking.

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-TODIM-base

2. 2
   $${\bar{w}}_{jr}=\frac{w_j}{w_r},w_r=\underset{j}{\max }{w}_j$$

   IV-TODIM-base

3. 3
   $${\varphi }_j(A_i,A_k)=\{\begin{array}{ll}{\bar{w}}_{jr}\sqrt{\frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\bar{w}}_{jr}}{{\bar{w}}_{jr}}}& \text{gain}\\ -\frac{1}{\theta }\sqrt{\frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\bar{w}}_{jr}}{{\bar{w}}_{jr}}}& \text{loss}\\ 0& \text{tie}\end{array}$$

   IV-TODIM-base

4. 4
   $$\delta (A_i,A_k)=\sum _{j=1}^n{\varphi }_j(A_i,A_k),d({\alpha }_1,{\alpha }_2)=\sqrt{\frac{1}{4}[({\mu }_1^L-{\mu }_2^L{)}^2+({\mu }_1^U-{\mu }_2^U{)}^2+({\nu }_1^L-{\nu }_2^L{)}^2+({\nu }_1^U-{\nu }_2^U{)}^2]}$$

   IV-TODIM-base

5. 5
   $${\xi }_i=\frac{{\sum }_k\delta (A_i,A_k)-{\min }_i{\sum }_k\delta (A_i,A_k)}{{\max }_i{\sum }_k\delta (A_i,A_k)-{\min }_i{\sum }_k\delta (A_i,A_k)},\text{rank descending}$$

   IV-TODIM-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: a ≤ b before computation.

• Defuzzification method affects ranking: midpoint (a+b)/2 is the canonical choice but alternatives exist.

Works with

How to cite