IF-ARAS - Intuitionistic Fuzzy ARAS

Overview

IF-ARAS extends crisp ARAS to settings where each rating is an Intuitionistic Fuzzy Number (μ, ν). The algorithm (i) IF-complements cost-criteria cells, (ii) constructs an optimal row R_0 in IFN space, (iii) defuzzifies each cell via Chen-Tan score s = μ − ν, shifts by +1, and column-sum-normalises across all m+1 rows, (iv) computes weighted sum S_i and the utility ratio K_i = S_i / S_0. Higher K_i = better. K_i ∈ [0, 1] when R_0 dominates every alternative cell-wise (the typical case). Algorithm is INVARIANT across pure IF-ARAS application papers (Mishra 2020 IT personnel selection, Mishra 2023 sustainable industrial buildings); criterion-weight source and score function are parametric choices (D.parameters).

Output

utility, higher is better

Data

Intuitionistic Fuzzy, ifn tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-15 criteria works best

Used for

IT personnel selection (Mishra 2020 original case), Sustainable industrial building selection (Mishra 2023), MAGDM under epistemic uncertainty, Expert-driven evaluation with linguistic IFN scales

How it works

1. 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}).

   $$R=(r_{ij}{)}_{m\times n},r_{ij}=({\mu }_{ij},{\nu }_{ij}),{\mu }_{ij}+{\nu }_{ij}\le 1;\text{IFWA aggregation if }K\ge 2$$

   Mishra 2020 Steps 1-3; Xu 2007 IFWA

2. 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.

   $${\hat{r}}_{ij}=\{\begin{array}{ll}({\mu }_{ij},{\nu }_{ij})& j\in J_b\\ ({\nu }_{ij},{\mu }_{ij})& j\in J_c\end{array}$$

   IF-ARAS.tex Step 1; Atanassov 1986 IFN complement

3. 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.

   $$r_{0j}=\left(\underset{i=1,\dots ,m}{\max }{\hat{\mu }}_{ij},\underset{i=1,\dots ,m}{\min }{\hat{\nu }}_{ij}\right)\forall j$$

   Mishra 2020 Step 5; Mishra 2023 Step 5; IF-ARAS.tex Step 2

4. 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.

   $$s(\alpha )={\mu }_{\alpha }-{\nu }_{\alpha };{\tilde{s}}_{ij}=s({\hat{r}}_{ij})+1;{\hat{s}}_{ij}=\frac{{\tilde{s}}_{ij}}{{\sum }_{i=0}^m{\tilde{s}}_{ij}}$$

   IF-ARAS.tex Step 3; Chen-Tan 1994 score; Mishra 2020 Steps 6-8

5. 5

   Weighted normalised matrix d_ij = w_j · ŝ_ij over all m+1 rows.

   $$d_{ij}=w_j\cdot {\hat{s}}_{ij}\forall i=0,1,\dots ,m,\forall j$$

   Zavadskas-Turskis 2010 Eq. (3) [crisp skeleton inherited]; Mishra 2020 Step 7

6. 6

   Optimality function S_i: sum of weighted normalised scores across all criteria, computed for every alternative including the optimal row R_0.

   $$S_i=\sum _{j=1}^n{d}_{ij}\forall i=0,1,\dots ,m$$

   Zavadskas-Turskis 2010 Eq. (4); Mishra 2020 Step 8; Mishra 2023 Step 8 (OPR M_i)

7. 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.

   $$K_i=\frac{S_i}{S_0},K_i\in [0,1]$$

   Zavadskas-Turskis 2010 Eq. (5); Mishra 2020 Step 9 (Q_i); Mishra 2023 Step 8 (UD); IF-ARAS.tex Step 4

8. 8

   Rank alternatives in descending order of K_i; the alternative with the largest K_i is the best.

   $$\text{rank}(A_i)={\text{argsort}}_{\downarrow }(K_i)$$

   Mishra 2020 Step 10; Mishra 2023 Step 9

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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.

• Confusing IFN (pair μ, ν) with TFN (triple a, b, c): IF-ARAS values MUST be IFN tuples satisfying μ+ν ≤ 1, NOT triangular fuzzy numbers.

• Applying crisp ARAS cost handling (1/x) instead of IFN complement (ν, μ): this breaks the IFN axioms and produces invalid (μ, ν) pairs.

• Omitting R_0 from the column-sum denominator in F4: must normalise over m+1 rows (alternatives + optimal row), otherwise K_i can exceed 1.

• Using the un-shifted Chen-Tan score s = μ − ν directly: scores can be negative, producing negative K_i. Always shift by +1 before column-sum normalisation.

Works with

How to cite