Bayesian Fuzzy Optimization With Gaussian Fuzzy Process

Abstract

In many real-life problems, decision-making gets complicated due to dual sources of uncertainty, known as randomness and fuzziness or imprecision, which can be challenging for traditional optimization methods. Most existing fuzzy optimization techniques that optimize fuzzy-valued objective functions account for fuzziness but generally do not model randomness, whereas probabilistic optimization techniques account for randomness but generally assume crisp-valued objective functions and therefore do not capture fuzziness or imprecision. To handle this dual source of uncertainty, Kwakernaak introduced the concept of a fuzzy random variable as “random variables whose values are not real, but fuzzy numbers”. This work aims to derive a theoretical background for the Gaussian fuzzy process and fuzzy acquisition functions, which will be used to develop a novel Bayesian fuzzy optimization (BFO) technique that optimizes a fuzzy-valued objective function. The proposed Gaussian Fuzzy Process extends the classical Gaussian Process by modelling each function evaluation as a Gaussian fuzzy random variable, thereby simultaneously representing stochastic variability and imprecision within the surrogate model. Fuzzy acquisition functions are defined to act as a guide for the search process of BFO with the help of the posterior fuzzy mean and fuzzy variance. The proposed method demonstrated competitive performance in both fuzzy mean-variance portfolio allocation and Indian temperature data analysis. The proposed method can have broader applications in various fields like healthcare, material science, agriculture, etc.