Version of control variables, such as for example scaling aspect (= where = 1,, NP. can self-adapt not merely the scaling aspect as well as the crossover price CR but also the populace size NP. This algorithm utilizes extra variables such as for example (0,1]) people. Furthermore, an exterior archive structure was proposed by storing the set of parameter vectors of recently discarded individuals. These parameter vectors provide the additional information about promising progress direction and increase the population diversity. The following equations represent the DE/current-to-is an individual randomly selected from the population or external archive. In terms of parameter adaptation, JADE adapts the crossover rate CRis the Gaussian distributed random number generator. After that, the crossover rate CRis truncated to [0,1]. Moreover, is modified as follows: is a constant value in [0,1], meanstands for the arithmetic mean, and is adapted as follows: is the Cauchy distributed random number generator. After that, the scaling factor is truncated to 1 1 if 1 or regenerated if 0. Also, that is a mean value to generate is modified as follows: is a constant value in [0,1], meanstands for the Lehmer mean, and contains the successfully evolved scaling factors of individuals after the selection operation. 2.5. MDE Ali and Pant  proposed a Modified Differential Evolution (MDE). This algorithm utilizes the Cauchy distribution as another mutation operation. In the selection operation, all individuals are monitored and 72559-06-9 the results of the selection operation are stored in the failure counter. If some individuals consequently fail to be selected as an individual for the next generation over MFC (Maximum Failure Counter), MDE assumes that these individuals were felled into some local minima. Therefore, the algorithm applies the Cauchy distributed mutation to these individuals instead of the mutation and the crossover operations to escape the local minima. After that, the failure counter is initialized by 0. MDE has shown the good performance for the higher dimensional problems, compared with DE/rand/1/bin. 3. Analysis of the Cauchy Distribution The Cauchy distribution is a continuous probability distribution and it has two parameters stands for the halfwidth at halfmaximum (HWHM) of the distribution. The value of determines the shape of the Cauchy distribution. If is assigned a lower value, the peak of the probability density function would be higher and its width would be narrower. On the other hand, if is assigned a higher value, the probability density function would have a lower peak and a wider width. The Cauchy distribution generates a large step from the peak with a higher probability. In general, many evolutionary algorithms have used this long tail property as an escaping local minima technique. The probability density function and the cumulative distribution function of the Cauchy distribution are defined by and denote the location (= 0 and = 1 generate the standard Cauchy distribution. Figure 1 The various probability density functions of the Cauchy distribution. 4. Adaptive Cauchy DE 4.1. When Parameter Adaptation Should Be Performed? Finding appropriate moments of adapting control parameters is important problem for improving the DE performance. In this section, we explain when parameter adaptation should be performed. Looking for previous studies, jDE utilizes self-adaptive method which allows 72559-06-9 each individual to maintain suitable control parameter values by itself. However, the parameter adaptation of jDE depends on the predefined probabilities (and CR, except for NP. The control parameter NP does not seriously affect the performance of DE more than the other two control parameters. Prior to explaining the adaptation procedures, the characteristics of these parameters are described. The control parameter is related to the convergence speed of DE. Therefore, a higher value of encourages the exploration power which is generally useful in the early stage of DE. On the other hand, a lower value of promotes the exploitation power 72559-06-9 that is usually desirable in the later stage of DE. Moreover, the value of control parameter CR is related RAD21 to the diversity of population. The parameter adaptation of proposed algorithm utilizes and CRwhere is the individual’s index. At the initialization stage, these parameters are initialized as 0.5.