IF-ARAS: Intuitionistic Fuzzy ARAS

Overview

Utility-ratio ranking under Intuitionistic Fuzzy uncertainty. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Utility-ratio ranking under Intuitionistic Fuzzy uncertainty
• •Preserves intuitionistic uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from crisp ARAS base (Zavadskas-Turskis 2010) due to optimal-row coupling: removing/adding alternatives can shift R_0 and therefore K_i.)
• •Assumes: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •Assumes: All decision-makers use the same linguistic-to-IFN scale
• •Assumes: Criterion directions (benefit/cost) are explicitly labelled
• •Assumes: Criterion weights form a simplex

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
• •Criterion directions (benefit/cost) are explicitly labelled
• •Criterion weights form a simplex

When not to use

• •Crisp data sufficient: use base ARAS directly (avoid unnecessary uncertainty layer)
• •Distance-based ranking is preferred (use IF-TOPSIS or IF-EDAS instead)
• •Hesitation degree π must be modelled explicitly with μ+ν > 1 raw judgements: consider Pythagorean Fuzzy ARAS or q-ROF ARAS

Edge cases

• •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 and aggregate cell-wise vi
• •default Chen-Tan 1994: s(α) = μ − ν ∈ [−1, 1]) and shift to non-negative range: s̃_ij = s(r̂_ij) + 1 ∈ [0, 2]. Then column-sum-normalise across all m+1 rows (alternatives + R_0): ŝ_ij = s̃_ij / Σ_i s̃
• •when S_0 = max S_i.

Common pitfalls

• •Hatalı: 'IF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid IFN tuples (μ, ν) with μ+ν ≤ 1
• •Hatalı: 'IF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-makers use the same linguistic-to-IFN scale
• •Hatalı: 'IF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion directions (benefit/cost) are explicitly labelled
• •Hatalı: 'IF-ARAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Criterion weights form a simplex
• •Hatalı: IF-ARAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IF-ARAS'yi 'Distance-based ranking is preferred (use IF-TOPSIS or IF-EDAS instead)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IF-ARAS'yi 'Hesitation degree π must be modelled explicitly with μ+ν > 1 raw judgements' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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 and aggregate cell-wise via IFWA with DM weights λ_k before entering F2: r_ij = (1 − Π(1−μ_ij^{(k)})^{λ_k}, Π (ν_ij^{(k)})^{λ_k}). Formül: R = (r_{ij})_{m \times n},\ r_{ij} = (\mu_{ij}, \nu_{ij}),\ \mu_{ij}+\nu_{ij} \le 1;\quad \text{IFWA aggregation if } K \ge 2 Anchor: Mishra 2020 Steps 1-3; Xu 2007 IFWA
2. 2.Adım 2 (F2): Step 2: Cost-criterion adjustment via IFN complement: for j ∈ J_c (cost criteria), replace r_ij = (μ_ij, ν_ij) with its complement (ν_ij, μ_ij). Benefit criteria pass through unchanged. Formül: \hat{r}_{ij} = \begin{cases} (\mu_{ij}, \nu_{ij}) & j \in J_b \\ (\nu_{ij}, \mu_{ij}) & j \in J_c \end{cases} Anchor: IF-ARAS.tex Step 1; Atanassov 1986 IFN complement
3. 3.Adım 3 (F3): Step 3: Construct the optimal alternative row R_0 in IFN space, taking max-μ and min-ν per criterion (over the complemented matrix). This row is appended to the matrix as a (m+1)-th alternative anchor. Formül: r_{0j} = \left(\max_{i=1,\ldots,m} \hat{\mu}_{ij},\ \min_{i=1,\ldots,m} \hat{\nu}_{ij}\right)\quad \forall j Anchor: Mishra 2020 Step 5; Mishra 2023 Step 5; IF-ARAS.tex Step 2
4. 4.Adım 4 (F4): Step 4: Defuzzify each cell via the IF score function (default Chen-Tan 1994: s(α) = μ − ν ∈ [−1, 1]) and shift to non-negative range: s̃_ij = s(r̂_ij) + 1 ∈ [0, 2]. Then column-sum-normalise across all m+1 rows (alternatives + R_0): ŝ_ij = s̃_ij / Σ_i s̃_ij. Formül: s(\alpha) = \mu_{\alpha} - \nu_{\alpha};\quad \tilde{s}_{ij} = s(\hat{r}_{ij}) + 1;\quad \hat{s}_{ij} = \frac{\tilde{s}_{ij}}{\sum_{i=0}^{m} \tilde{s}_{ij}} Anchor: IF-ARAS.tex Step 3; Chen-Tan 1994 score; Mishra 2020 Steps 6-8
5. 5.Adım 5 (F5): Step 5: Weighted normalised matrix d_ij = w_j · ŝ_ij over all m+1 rows. Formül: d_{ij} = w_j \cdot \hat{s}_{ij}\quad \forall i=0,1,\ldots,m,\ \forall j Anchor: Zavadskas-Turskis 2010 Eq. (3) [crisp skeleton inherited]; Mishra 2020 Step 7
6. 6.Adım 6 (F6): Step 6: Optimality function S_i: sum of weighted normalised scores across all criteria, computed for every alternative including the optimal row R_0. Formül: S_i = \sum_{j=1}^{n} d_{ij}\quad \forall i = 0, 1, \ldots, m Anchor: Zavadskas-Turskis 2010 Eq. (4); Mishra 2020 Step 8; Mishra 2023 Step 8 (OPR M_i)
7. 7.Adım 7 (F7): Step 7: Degree of utility K_i = S_i / S_0 for each alternative i = 1, …, m; K_i ∈ [0, 1] when S_0 = max S_i. Formül: K_i = \frac{S_i}{S_0},\quad K_i \in [0, 1] Anchor: Zavadskas-Turskis 2010 Eq. (5); Mishra 2020 Step 9 (Q_i); Mishra 2023 Step 8 (UD); IF-ARAS.tex Step 4
8. 8.Adım 8 (F8): Step 8: Rank alternatives in descending order of K_i; the alternative with the largest K_i is the best. Formül: \text{rank}(A_i) = \text{argsort}_{\downarrow}(K_i) Anchor: Mishra 2020 Step 10; Mishra 2023 Step 9

Commonly paired with

• •IF-AHP + IF-ARAS (common)
• •IF-SWARA + IF-ARAS (common)
• •IF-MEREC + IF-ARAS (emerging)

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