Fuzzy ARAS - Fuzzy extension of ARAS

Overview

fuzzy-aras extends ARAS to handle Fuzzy uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Triangular Fuzzy Number (TFN: l, m, u) algebra. The final scores are defuzzified via centroid (l+m+u)/3 before ranking.

Output

utility, higher is better

Data

Fuzzy (TFN), uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Augment matrix with optimal alternative A_0 (best per criterion). For benefit criteria take TFN-max over alternatives; for cost criteria take TFN-min.

   $${\tilde{x}}_{0j}=\{\begin{array}{ll}{\max }_{i=1}^m{\tilde{x}}_{ij}& j\in {\Omega }_{\max }\\ {\min }_{i=1}^m{\tilde{x}}_{ij}& j\in {\Omega }_{\min }\end{array},{\tilde{x}}_{ij}=(l_{ij},m_{ij},u_{ij})$$

   Turskis-Zavadskas 2010 (Transport 25:4, p.426) Eqs.(14)-(15)

2. 2

   Cost inversion then column-sum normalization. For cost criteria: invert each TFN as (1/u, 1/m, 1/l). Then for all criteria (including A_0 row): divide each TFN by the fuzzy column sum using reverse-pair division.

   $$\text{Cost inversion: }{\tilde{x}}_{ij}^{\prime }=\left(\frac{1}{u_{ij}},\frac{1}{m_{ij}},\frac{1}{l_{ij}}\right),j\in {\Omega }_{\min };{\bar{x}}_{ij}={\tilde{x}}_{ij}\oslash \sum _{i=0}^m{\tilde{x}}_{ij}=\left(\frac{l_{ij}}{{\sum }_i{u}_{ij}},\frac{m_{ij}}{{\sum }_i{m}_{ij}},\frac{u_{ij}}{{\sum }_i{l}_{ij}}\right)$$

   Turskis-Zavadskas 2010 (Transport 25:4, p.426-427) Eqs.(16)-(18)

3. 3

   Weighted normalised matrix: element-wise TFN multiplication of normalised value by (TFN) criterion weight.

   $${\hat{d}}_{ij}={\bar{x}}_{ij}\otimes {\tilde{w}}_j=\left(l_{ij}^{\bar{x}}\cdot l_j^w,m_{ij}^{\bar{x}}\cdot m_j^w,u_{ij}^{\bar{x}}\cdot u_j^w\right);\text{if crisp weight: }{\hat{d}}_{ij}=\left(w_j{l}_{ij}^{\bar{x}},w_j{m}_{ij}^{\bar{x}},w_j{u}_{ij}^{\bar{x}}\right)$$

   Turskis-Zavadskas 2010 (Transport 25:4, p.427) Eq.(21)

4. 4

   Optimality function S̃_i: fuzzy sum of weighted-normalised values across all criteria, for each row including A_0.

   $${\tilde{S}}_i=\underset{j=1}{\overset{n}{⨁}}{\hat{d}}_{ij}=\left(\sum _{j=1}^n{l}_{ij}^d,\sum _{j=1}^n{m}_{ij}^d,\sum _{j=1}^n{u}_{ij}^d\right),i=0,1,\dots ,m$$

   Turskis-Zavadskas 2010 (Transport 25:4, p.427) Eq.(22)

5. 5

   Defuzzify S̃_i by centroid (l+m+u)/3, compute utility K_i = S_i/S_0, rank descending.

   $$S_i=\frac{l_i^S+m_i^S+u_i^S}{3};K_i=\frac{S_i}{S_0},i=1,\dots ,m;\mathrm{rank}\mathrm{by}{K}_i\downarrow$$

   Turskis-Zavadskas 2010 (Transport 25:4, p.427) Eqs.(23)-(24)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid Fuzzy (Triangular) 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

• •FUZZY-ARAS edge case'leri: (1) Optimal alternatif A_0 hesabı için cost kriterde min işlemi TFN üzerinde α-bazlı veya centroid-bazlı yapılabilir - paper §4 α-bazlı min kullanır (her bileşen ayrı min) - engine bu kanonu izler; bu monotonik olmayan TFN için tutarsızlık yaratabilir (α > β anomaliside ek kontrol gerekir). (2) Cost normalizasyonun iki-aşaması Eq.18: x̃_ij^* = (1/γ_ij, 1/β_ij, 1/α_ij) INVERT işlemi pozitif TFN gerektirir; α_ij = 0 → bölünme hatası → ε-clamp gerekir. (3) Σ_i x̃_ij = 0 (tüm alternatifler 0) → normalization tanımsız → input validation gerekir. (4) Centroid defuzz simetrik TFN için aritmetik orta verir; asimetrik TFN'de bias yaratır - kullanıcı literatürde alternatif defuzz (signed distance, graded mean integration) test etmek isterse manifest dışında implement etmelidir.

• Value-space violation: ensure all entries satisfy TFN: l ≤ m ≤ u, all ≥ 0 before computation.

• Defuzzification method affects ranking: centroid (l+m+u)/3 is the canonical choice but alternatives exist.

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