IVAIF-CODAS: Interval-Valued Atanassov Intuitionistic Fuzzy CODAS

Overview

Interval-valued intuitionistic fuzzy outranking: IVAIFS. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Interval-valued intuitionistic fuzzy outranking: IVAIFS
• •Preserves intuitionistic uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

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

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

When not to use

• •Crisp data sufficient: use base CODAS directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •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).

Common pitfalls

• •Hatalı: 'IV-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Interval numbers/tuples
• •Hatalı: 'IV-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'IV-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: IV-CODAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IV-CODAS'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: \tilde{x}_{ijl} = \left([\mu^L_{ijl}, \mu^U_{ijl}], [v^L_{ijl}, v^U_{ijl}]\right); \quad 0 \le \mu^U_{ijl} + v^U_{ijl} \le 1 Anchor: Yeni & Özçelik 2018, p.437 Eq.(13)-(14)
2. 2.Adım 2 (F2): Step 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). Formül: \text{IIFWA}_w(\tilde{A}_1,\ldots,\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) Anchor: Yeni & Özçelik 2018, p.438 Eq.(12), Eq.(15)-(16)
3. 3.Adım 3 (F3): Step 3: Calculate the IVAIF weighted decision matrix R̃. Each weighted entry r̃_ij = w̃_j ⊗ x̃_ij uses the IVAIFN multiplication operation (Eq. 6). Formül: \tilde{r}_{ij} = \tilde{w}_j \otimes \tilde{x}_{ij} = \left([\mu^L_{wj}\mu^L_{ij},\; \mu^U_{wj}\mu^U_{ij}],\; [v^L_{wj}+v^L_{ij}-v^L_{wj}v^L_{ij},\; v^U_{wj}+v^U_{ij}-v^U_{wj}v^U_{ij}]\right) Anchor: Yeni & Özçelik 2018, p.439 Eq.(17), based on Eq.(6)
4. 4.Adım 4 (F4): Step 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. Formül: \tilde{n}_j = \min_i \tilde{r}_{ij} = \left([\min_i \mu^L_{ij},\; \min_i \mu^U_{ij}],\; [\max_i v^L_{ij},\; \max_i v^U_{ij}]\right) Anchor: Yeni & Özçelik 2018, p.439 Eq.(18)
5. 5.Adım 5 (F5): Step 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). Formül: ED_i = \sqrt{\frac{1}{4n}\sum_{j=1}^n \left[(\mu^L_{ij}-\mu^L_{nj})^2 + (\mu^U_{ij}-\mu^U_{nj})^2 + (v^L_{ij}-v^L_{nj})^2 + (v^U_{ij}-v^U_{nj})^2\right]}; \quad HD_i = \frac{1}{4n}\sum_{j=1}^n \left[|\mu^L_{ij}-\mu^L_{nj}| + |\mu^U_{ij}-\mu^U_{nj}| + |v^L_{ij}-v^L_{nj}| + |v^U_{ij}-v^U_{nj}|\right] Anchor: Yeni & Özçelik 2018, p.440 Eq.(19)-(20)
6. 6.Adım 6 (F6): Step 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. Formül: p_{ik} = (ED_i - ED_k) + t(ED_i - ED_k) \cdot (HD_i - HD_k); \quad t(x) = \begin{cases}1 & |x| \ge \theta \\ 0 & |x| < \theta \end{cases} Anchor: Yeni & Özçelik 2018, p.440 Eq.(21)-(22)
7. 7.Adım 7 (F7): Step 7: Calculate the Assessment Score (AS_i) for each alternative as the row sum of RA. Formül: AS_i = \sum_{k=1}^{n} p_{ik} Anchor: Yeni & Özçelik 2018, p.440 Eq.(23)
8. 8.Adım 8 (F8): Step 8: Rank alternatives in decreasing order of AS_i. The alternative with the highest AS_i is the best. Formül: \text{rank}(A_i) = \text{rank by descending } AS_i Anchor: Yeni & Özçelik 2018, p.440 Step 8

Commonly paired with

• •n_a + IV-CODAS (common)

How to cite