IVAIF-CODAS - Interval-Valued Atanassov Intuitionistic Fuzzy CODAS

Overview

IVAIF-CODAS extends CODAS to handle Interval-Valued Atanassov Intuitionistic Fuzzy (IVAIF) data. DM assessments are expressed as IVAIFNs ([μL,μU],[vL,vU]) using a linguistic scale, then aggregated via IIFWA. The method uses IVAIF Euclidean and Hamming distances from the negative ideal solution. Higher AS score = better alternative. Threshold θ (default 0.02) controls when the Hamming distance refinement is activated.

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

   Construct the IVAIF decision matrix X̃^l for each DM l. Each entry x̃_ijl = ([μL_ijl, μU_ijl], [vL_ijl, vU_ijl]) is assigned from a linguistic scale (Table 2). DM l also rates criterion importance using Table 3, giving weight vector W̃^l.

   $${\tilde{x}}_{ijl}=\left([{\mu }_{ijl}^L,{\mu }_{ijl}^U],[v_{ijl}^L,v_{ijl}^U]\right);0\le {\mu }_{ijl}^U+v_{ijl}^U\le 1$$

   Yeni & Özçelik 2018, p.437 Eq.(13)-(14)

2. 2

   Aggregate individual DM matrices into a single IVAIF decision matrix X̃ and aggregate criterion weights W̃ using IIFWA operator (Eq. 12). DM weights w_l are given (w1=0.40, w2=0.35, w3=0.25 in paper example).

   $${\text{IIFWA}}_w({\tilde{A}}_1,\dots ,{\tilde{A}}_k)=\left(\left[1-\prod _{l=1}^k(1-a_l{)}^{w_l},1-\prod _{l=1}^k(1-b_l{)}^{w_l}\right],\left[\prod _{l=1}^k{c}_l^{w_l},\prod _{l=1}^k{d}_l^{w_l}\right]\right)$$

   Yeni & Özçelik 2018, p.438 Eq.(12), Eq.(15)-(16)

3. 3

   Calculate the IVAIF weighted decision matrix R̃. Each weighted entry r̃_ij = w̃_j ⊗ x̃_ij uses the IVAIFN multiplication operation (Eq. 6).

   $${\tilde{r}}_{ij}={\tilde{w}}_j\otimes {\tilde{x}}_{ij}=\left([{\mu }_{wj}^L{\mu }_{ij}^L,{\mu }_{wj}^U{\mu }_{ij}^U],[v_{wj}^L+v_{ij}^L-v_{wj}^L{v}_{ij}^L,v_{wj}^U+v_{ij}^U-v_{wj}^U{v}_{ij}^U]\right)$$

   Yeni & Özçelik 2018, p.439 Eq.(17), based on Eq.(6)

4. 4

   Determine IVAIF negative ideal solution (NS̃). For each criterion j, the negative ideal is the IVAIFN with minimum membership and maximum non-membership across all alternatives.

   $${\tilde{n}}_j=\underset{i}{\min }{\tilde{r}}_{ij}=\left([\underset{i}{\min }{\mu }_{ij}^L,\underset{i}{\min }{\mu }_{ij}^U],[\underset{i}{\max }{v}_{ij}^L,\underset{i}{\max }{v}_{ij}^U]\right)$$

   Yeni & Özçelik 2018, p.439 Eq.(18)

5. 5

   Calculate normalized Euclidean distance (ED_i) and normalized Hamming distance (HD_i) of each alternative from NS̃, using IVAIF distance formulas (Park et al. 2008).

   $$E{D}_i=\sqrt{\frac{1}{4n}\sum _{j=1}^n\left[({\mu }_{ij}^L-{\mu }_{nj}^L{)}^2+({\mu }_{ij}^U-{\mu }_{nj}^U{)}^2+(v_{ij}^L-v_{nj}^L{)}^2+(v_{ij}^U-v_{nj}^U{)}^2\right]};H{D}_i=\frac{1}{4n}\sum _{j=1}^n\left[|{\mu }_{ij}^L-{\mu }_{nj}^L|+|{\mu }_{ij}^U-{\mu }_{nj}^U|+|v_{ij}^L-v_{nj}^L|+|v_{ij}^U-v_{nj}^U|\right]$$

   Yeni & Özçelik 2018, p.440 Eq.(19)-(20)

6. 6

   Construct the Relative Assessment matrix (RA). Entry p_ik = (ED_i − ED_k) + t(ED_i − ED_k) × (HD_i − HD_k), where t(x) = 1 if |x| ≥ θ, else 0. Threshold θ ∈ [0.01, 0.05]; default θ = 0.02.

   $$p_{ik}=(E{D}_i-E{D}_k)+t(E{D}_i-E{D}_k)\cdot (H{D}_i-H{D}_k);t(x)=\{\begin{array}{ll}1& |x|\ge \theta \\ 0& |x|<\theta \end{array}$$

   Yeni & Özçelik 2018, p.440 Eq.(21)-(22)

7. 7

   Calculate the Assessment Score (AS_i) for each alternative as the row sum of RA.

   $$A{S}_i=\sum _{k=1}^n{p}_{ik}$$

   Yeni & Özçelik 2018, p.440 Eq.(23)

8. 8

   Rank alternatives in decreasing order of AS_i. The alternative with the highest AS_i is the best.

   $$\text{rank}(A_i)=\text{rank by descending }A{S}_i$$

   Yeni & Özçelik 2018, p.440 Step 8

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

• •IF decision matrix X̃^l for each DM l. Each entry x̃_ijl = ([μL_ijl, μU_ijl], [vL_ijl, vU_ijl]) is assigned from a linguistic scale (Table 2). DM l also rates criterion importance using Table 3, givin
• •IF decision matrix X̃ and aggregate criterion weights W̃ using IIFWA operator (Eq. 12). DM weights w_l are given (w1=0.40, w2=0.35, w3=0.25 in paper example).
• •IF weighted decision matrix R̃. Each weighted entry r̃_ij = w̃_j ⊗ x̃_ij uses the IVAIFN multiplication operation (Eq. 6).
• •IF negative ideal solution (NS̃). For each criterion j, the negative ideal is the IVAIFN with minimum membership and maximum non-membership across all alternatives.
• •IF distance formulas (Park et al. 2008).

• IVAIFN validity: ensure μU + vU ≤ 1 for all entries; violating this makes the IVAIFN undefined.

• Threshold θ sensitivity: ranking can shift for different θ values (tested 0.01-0.05 in paper, stable in this example).

• IIFWA vs IIFWG: the paper uses IIFWA for aggregation; using geometric mean operator changes results.

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