A Study on Fuzzy Systems

We use princiles of fuzzy logic to develop a general model representing several processes in a system’s operation characterized by a degree of vagueness and/or uncertainy. Further, we introduce three altenative measures of a fuzzy system’s effectiveness connected to the above model. An applcation is also developed for the Mathematical Modelling process illustrating our results.

💡 Research Summary

The paper proposes a fuzzy‑logic based framework for modeling and evaluating processes that exhibit vagueness and uncertainty. It begins by decomposing a system’s operation into three sequential stages (S₁, S₂, S₃) and introduces a linguistic scale of five labels—very low (a), low (b), intermediate (c), high (d), very high (e)—to describe the success level of each entity at each stage. For each stage a fuzzy subset Aᵢ of the label set U is constructed, with membership degrees defined piecewise according to the proportion of entities falling into each label. The membership function is discretized into five levels (1, 0.75, 0.5, 0.25, 0), which creates a simple but somewhat arbitrary quantization of performance.

A system profile s = (x, y, z) combines the three stage labels. The authors impose a “well‑ordered” constraint (x ≥ y ≥ z) to reflect the intuitive idea that performance should not improve after a decline. Under this constraint the profile’s overall membership m_R(s) is the product of the three stage memberships; otherwise it is set to zero. This product yields a fuzzy relation R on U³, from which two derived quantities are defined: a possibility value r_s (the ratio of m_R(s) to the maximal profile membership) and a pseudo‑frequency f(s) that averages memberships across k groups of entities. Probabilities p_s are then obtained by normalising these pseudo‑frequencies.

To assess system effectiveness three fuzzy‑based measures are introduced:

1. Total Possibilistic Uncertainty (T(r)) – drawn from possibility theory, it combines a “strife” component ST(r) (measuring conflict among ordered possibilities) and a “non‑specificity” component N(r) (measuring imprecision). Both are expressed as logarithmic sums over the ordered possibility distribution. Lower T(r) indicates a greater reduction of uncertainty and thus better performance.

2. Modified Shannon Entropy (H) – adapted to fuzzy environments via Dempster‑Shafer theory, it is computed as H = –∑ (m_s / n) ln(m_s / n), where m_s is the profile membership and n the total number of entities. The value is normalised by ln n to lie in