Fuzzy CODAS - Fuzzy extension of CODAS

Overview

fuzzy-codas extends CODAS 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

   Extract TFN matrix x̃ij=(xijα,xijβ,xijγ), weights w̃j=(wjα,wjβ,wjγ).

   $$\tilde{X}=[{\tilde{x}}_{ij}];{\tilde{w}}_j=(w_j^{\alpha },w_j^{\beta },w_j^{\gamma })$$
2. 2

   Normalize: benefit r̃ij=(xijα/Mγj, xijβ/Mγj, xijγ/Mγj), Mγj=max_i(xijγ). Cost r̃ij=(mαj/xijγ, mαj/xijβ, mαj/xijα), mαj=min_i(xijα).

   $${\tilde{r}}_{ij}=\{\begin{array}{ll}(x_{ij}^{\alpha }/M_j^{\gamma },x_{ij}^{\beta }/M_j^{\gamma },x_{ij}^{\gamma }/M_j^{\gamma })& j\in {\Omega }_{+}\\ (m_j^{\alpha }/x_{ij}^{\gamma },m_j^{\alpha }/x_{ij}^{\beta },m_j^{\alpha }/x_{ij}^{\alpha })& j\in {\Omega }_{-}\end{array}$$
3. 3

   Weighted: ṽij=r̃ij⊗w̃j=(rα·wα, rβ·wβ, rγ·wγ).

   $${\tilde{v}}_{ij}=(r_{ij}^{\alpha }{w}_j^{\alpha },r_{ij}^{\beta }{w}_j^{\beta },r_{ij}^{\gamma }{w}_j^{\gamma })$$
4. 4

   NIS (negative ideal): n̊j=min_i COA(ṽij) per criterion. Euclidean Êi=√(Σj(COA(ṽij)-n̊j)²). Taxicab T̂i=Σj|COA(ṽij)-n̊j|.

   $${\hat{n}}_j=\underset{i}{\min }\frac{v_{ij}^{\alpha }+v_{ij}^{\beta }+v_{ij}^{\gamma }}{3};{\hat{E}}_i=\sqrt{\sum _j{\left(\text{COA}({\tilde{v}}_{ij})-{\hat{n}}_j\right)}^2};{\hat{T}}_i=\sum _j\left|\text{COA}({\tilde{v}}_{ij})-{\hat{n}}_j\right|$$
5. 5

   Pairwise H̃ik=(Êi-Êk)+ψ(|Êi-Êk|≥τ)·(T̂i-T̂k) where τ=0.02. Appraisal score hi=Σk H̃ik.

   $$H_{ik}=({\hat{E}}_i-{\hat{E}}_k)+\psi (|{\hat{E}}_i-{\hat{E}}_k|\ge \tau )\cdot ({\hat{T}}_i-{\hat{T}}_k);h_i=\sum _{k=1}^m{H}_{ik}$$
6. 6

   Rank alternatives descending by hi.

   $$\text{rank descending by }{h}_i$$

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

• 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.

Works with

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