# Estimation of Distribution Algorithms with Fuzzy Sampling for Stochastic Programming Problems
A differential evolution variant is proposed in [12] equipped with three new algorithmic components, including a central tendency-based mutation, adopted blending crossover, and a new distance-based selection mechanism. To deal with the noise, their algorithm uses non-conventional mutation strategies. In [31], an extension of multi-objective optimization is proposed, based on an differential evolution algorithm to manage the effect of noise in objective functions. Their method applies an adaptive range of the sample size for estimating the fitness values. In [32], instead of using averages, the search policy considers the distribution of noisy samples during the fitness evaluation process. A number of different approaches to deal with noise are presented in [33]. Most sampling methods are based on the use of averages, and this motivates us to use different sampling techniques. One possible sampling alternative is the use of fuzzy logic, which is an important pillar of computational intelligence. The idea of the fuzzy set was first introduced in [34]; this enabled a member to belong to a set in a partitioned way, as opposed to in a definite way, as stated by classical set theory. In other words, membership can be assigned a value within the \([0,1]\) interval instead of the \(\{0,1\}\) set. Over the past four decades, the theory of fuzzy random variables [35] has been developed via a large number of studies in the area of fuzzy stochastic optimization [36; 37]. The noisy part of our problem can be considered to be randomness or fuzziness, and it can be understood as a fuzzy stochastic problem, which can be found in the literature [38; 39; 40; 41; 42]. In this paper, EDAs are used to solve nonlinear optimization problems that contain noise. The proposed EDA-based methods follow the class of EDAs proposed in [43]. The designed EDA-model is firstly combined with variable-sample methods (SPRS) [30]. Sampling techniques have a serious problem when the sample size is small, so estimating the objective function values with noise is accurate in these cases. Therefore, we propose a new sampling technique based on fuzzy systems to deal with small sample sizes. Another EDA-based method uses the proposed fuzzy sampling technique. Moreover, additive versions of the proposed methods are developed to optimize functions without noise in order to evaluate different efficiency levels of the proposed methods. In order to test the performance of the proposed methods, different numerical experiments were carried out using several benchmark test functions. Moreover, three real-world applications are considered to assess the performance of the proposed methods. The rest of the paper is structured as follows. In Section 2, we highlight the main structure and techniques for EDAs. The design elements and proposed methods are stated in Section 3. In Section 4, algorithmic implementations of the proposed methods and numerical experiments are discussed. The results for three stochastic programming applications are presented in Section 5. Finally, the paper is concluded in Section 6.