Fuzzy SPOTIS - Fuzzy extension of SPOTIS

Overview

fuzzy-spotis extends SPOTIS 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, lower 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 decision matrix x̃ij=(xijα,xijβ,xijγ). Define per-criterion stable preference bounds as TFNs: s̃_j^min=(s_j^{min,α},s_j^{min,β},s_j^{min,γ}) and s̃_j^max=(s_j^{max,α},s_j^{max,β},s_j^{max,γ}). Set fuzzy ideal Ĩ_j*=s̃_j^max for benefit criteria (j∈Ω+) and Ĩ_j*=s̃_j^min for cost criteria (j∈Ω−). Extract TFN weights w̃j=(wjα,wjβ,wjγ).

   $${\tilde{x}}_{ij}=(x_{ij}^{\alpha },x_{ij}^{\beta },x_{ij}^{\gamma });{\tilde{I}}_j^{\ast }=\{\begin{array}{ll}{\tilde{s}}_j^{\max }& j\in {\Omega }_{+}\\ {\tilde{s}}_j^{\min }& j\in {\Omega }_{-}\end{array}$$
2. 2

   Compute scalar distance from the fuzzy ideal using COA defuzzification on both the alternative TFN and the ideal TFN: d_{ij}=|COA(x̃_{ij})−COA(Ĩ_j*)|. Compute scalar span: span_j=COA(s̃_j^max)−COA(s̃_j^min). Normalize: p_{ij}=d_{ij}/span_j.

   $$d_{ij}=\left|\frac{x_{ij}^{\alpha }+x_{ij}^{\beta }+x_{ij}^{\gamma }}{3}-\frac{I_j^{\ast \alpha }+I_j^{\ast \beta }+I_j^{\ast \gamma }}{3}\right|;p_{ij}=\frac{d_{ij}}{\text{COA}({\tilde{s}}_j^{\max })-\text{COA}({\tilde{s}}_j^{\min })}$$
3. 3

   Compute weighted aggregated distance D_i=Σ_j COA(w̃_j)·p_{ij} where COA(w̃_j)=(w_j^α+w_j^β+w_j^γ)/3. Rank alternatives in ascending order of D_i (lower distance from ideal = better).

   $$D_i=\sum _{j=1}^n\frac{w_j^{\alpha }+w_j^{\beta }+w_j^{\gamma }}{3}\cdot p_{ij};\text{rank ascending by }{D}_i$$

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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