# A Shadowed Type-2 Fuzzy Approach for Crossover Parameter Adaptation in Differential Evolution
## 3 Differential Evolution  
The differential evolution (DE) algorithm is a population-based stochastic search strategy that was proposed in 1995 by [PERSON] and [PERSON]. Its main purpose is to solve optimization problems involving real parameters. In the literature, several works utilizing DE can be discussed as follows: the differential evolution (DE) algorithm is considered to be a practical and effective method for numerical optimization. It is known for its ease of understanding, simplicity of implementation, reliability, and efficiency. This article attempts to comprehensively describe DE, providing illustrations, valuable insights, and empirical advice. It acknowledges that globally optimal solutions are often challenging to obtain, with many problems being intractable [41]. When dealing with functions that have numerous interacting local optima, the application of fuzzy sets in the job-shop scheduling problem (JSP) with fuzzy processing time and completion time (FJSP) allows for a more comprehensive modeling of scheduling. Fuzzy relative entropy provides a method for evaluating the quality of a feasible solution  
Figure 3: Illustration of the shadowed set. by comparing the actual value with the ideal value, which is the due date. Consequently, the multi-objective FJSP can be transformed into a single-objective optimization problem that can be solved using a hybrid adaptive differential evolution (HADE) algorithm. This algorithm considers the maximum completion time, total delay time, and total energy consumption of jobs as optimization criteria. HADE employs a mutation strategy based on DE/current-to-best and makes its parameters (CR and F) adaptive and normally distributed. The selection of new individuals in HADE is based on the fitness value (FRE) obtained from a population consisting of N parents and N children [42]. Maximizing the traffic capacity of an intersection and reducing the delay rate of vehicles have always been challenges in traffic control research. The coordinated control of urban traffic signals is considered a multi-objective optimization problem. In this study, we examine a mathematical model for urban trunk traffic. We establish models for average delay, average queue length, total delay calculation at intersections, and vehicle exhaust emissions to formulate an optimization model for a new coordinated control system for traffic trunks. To address this, a study that combines fuzzy control theory with the adaptive sequencing mutation multi-objective differential evolution algorithm (FASM-MDEA) was offered in [43]. In our proposed model, we utilize two well-known statistical methods, the Pearson correlation coefficient and one-way analysis of variance, to select cost adjustment factors that have a high correlation with the actual effort. Subsequently, we employ the fuzzy C-means clustering algorithm with three cluster heads to group similar projects in the COCOMO-81 dataset. Furthermore, we utilize an improved self-adaptive differential evolution algorithm to optimize the parameters of the constructive cost model [44]. This paper focuses on the use of differential evolution to improve the approximation properties of function approximation models based on fuzzy partitions. Two cases are considered, fuzzy transform and fuzzy projection, and the design of hybrid evolutionary fuzzy systems is studied. Even though function approximation techniques based on fuzzy partitions have been well studied, few papers consider the problem of centroid selection of the basic functions. Thus, in most cases, uniform fuzzy partitions are considered. By using an evolutionary algorithm, a systematic approach on the selection of the partition is provided. The optimization problem involves the determination of the model parameters, which, in our case, are the fuzzy partition's membership functions' locations [45]. Type-3 fuzzy theory is a recent proposal in the literature.