# Self-Adaptive Differential Evolution with Gauss Distribution for Optimal Mechanism Design
## 1 Introduction  
A few recent decades witness a class of many powerful evolutionary algorithms and their applications. Like many other evolutionary algorithms, differential evolution is a population-based technique. This algorithm is developed by [PERSON] and [PERSON] [1]. It is inspired by the evolution phenomenon in nature and is a strong tool to solve engineering problems. There are three mechanisms (mutation, crossover, and selection) operating in DE in which three control parameters (population size, scaling factor, and crossover rate) have significant effect on exploration and exploitation of this algorithm.  
Parallel to the development of computer and industry 4.0, evolutionary algorithms gradually demonstrate its pros in comparison with a traditionally mathematical method, such as the gradient descent method [2] and the Newton-Raphson method [3]. It results that more and more researchers focus on improvement of evolutionary algorithms and differential evolution has not been laying out this trend.  
A literature review on the advance of the differential evolution algorithm, we cannot ignore some state of art developments, such as self-adaptive differential evolution algorithm (SaDE) [4], self-adaptive jDE algorithm (JDE) [5], and adaptive differential evolution with optional external archive (JADE) [6]. Recently, [PERSON] et al. incorporated the covariance matrix learning and by adjusting the bimodal distribution parameter to generate the control parameters of mutation and crossover operators, the effectiveness of DE is improved in solving problems that have a high correlation among variables [7]. [PERSON] and [PERSON] randomly selected two factors from the best and worst individuals in each current generation, and the third vector was chosen from the middle individuals. This mechanism effectively keeps the balance between the global exploration and local exploitation abilities for searching process [8]. [PERSON] et al. described an improved different evolution algorithm with opposition-based learning mechanism [9]. Eachof the above-mentioned algorithms may work best for one or some specific problems. Thus, study in this field is still on-going and this paper introduces an improved version of DE which solved all four groups of benchmark functions with outstanding convergence speed.  
Gaussian distribution, also called normal distribution, is a continuous function which is popularly used in optimization. It provides an excellent possibility to produce a new candidate for crossover operation based on a current position. Simultaneously, adaptive strategy of standard deviation parameter keeps good balance between exploitation and exploration process. This paper adopts a scaling factor generated by Gaussian distribution to improve mutation operation in a self-adaptive differential evolution algorithm (also named ISADE) proposed by [PERSON] et al. [10]. This factor will be applied to three types of mutation operation (DE/best/1, DE/best/2, and DE/rand to best/1) with equally random selection probability. In addition, standard deviation parameter of Gaussian distribution is updated by comparing the global best fitness value between the current and previous generations. The proposed algorithm is tested in 19 benchmark functions and 4 conventional engineering problems. Numerical simulation results confirm that the proposed algorithm express significant improvement in comparison to the original DE and some recent related works.  
The remain of this paper is composed of five sections. Section 2 describes the method of original DE. Section 3 reviews some recent related works on the applications of Gaussian distribution to DE. A modified Differential Evolution is introduced in Section 4. Section 5 describes comparisons of the proposed DE algorithm with the others and its application in real problem. Finally, Section 6 comprises some brief conclusions.