# Self-Adaptive Differential Evolution with Gauss Distribution for Optimal Mechanism Design
## 4 Modified Differential Evolution  
Gaussian distribution is widely used in probability and statistics, and is often applied to create real-valued random variables. Gaussian distribution gives a good chance to exploit a new potential candidate in the optimization of many complex functions. In this paper, we the improved exploration ability of EOBL-DE by using Gauss distribution. Our introduced strategy will generate a new trial vectors in mutation operation by Gaussian perturbation mechanism which multiples Gaussian random variable to a vector difference \((X_{r1,j}^{iter}-X_{r2,j}^{iter})\) in Equation (15), \((X_{r1,j}^{iter}-X_{r2,j}^{iter})\) and \((X_{r3,j}^{iter}-X_{r4,j}^{iter})\) in Equation (16), and \((X_{r2,j}^{iter}-X_{r3,j}^{iter})\) in Equation (17).  
The equation for mutation operation schemes has been modified as the following Equations (15)-(17).  
\[DE/best/1:V_{i,j}^{iter}\ =\ X_{best,j}^{iter}+N(\mu,\sigma^{2})_{i}*F*(X_{r1,j}^{ iter}-X_{r2,j}^{iter}) \tag{15}\]  
\[DE/best/2:V_{i,j}^{iter}\ =\ X_{best,j}^{iter}+N(\mu,\sigma^{2})_{i}*[F*(X_{r1,j}^ {iter}-X_{r2,j}^{iter})+F*(X_{r3,j}^{iter}-X_{r4,j}^{iter})] \tag{16}\]  
\[DE/rand\ to\ best/1:V_{i,j}^{iter}\ =\ X_{r1,j}^{iter}+F*(X_{best,j}^{iter}-X_{r1,j}^{ iter})+N(\mu,\sigma^{2})_{i}*F*(X_{r2,j}^{iter}-X_{r3,j}^{iter}) \tag{17}\]  
where \(r_{1},r_{2}\), and \(r_{3}\) are randomized from \([1;NP]\); N(\(\mu\),\(\sigma^{2}\)) is a Gaussian random variable with mean \(\mu\) and standard deviation \(\sigma\). Recommended value for \(\mu\) is 1, whereas \(\sigma\) is defined in Equation (18). \(N(\mu,\sigma^{2})_{i}\) is calculated to be suitable for each \(i\)-th individual based on the fitness value.  
\[\sigma_{i}=\frac{1}{0.5*exp\bigg{(}\frac{fit_{i}^{iter}-fit_{i}^{iter}}{T} \bigg{)}} \tag{18}\]  
where T = \(T_{o}*iter\), recommended value for \(T_{o}\) is 1.5, \(iter\) is current number of iteration. The G-ISADE algorithm is described as in Algorithm 2.  
```
Initialization: Generate initial population of NP candidates and their corresponding opposition within boundary.  Generate initial Gauss variables. Evaluation and rank: Ranking all 2 NP population based on comparing fitness value. After that, NP particles will be selected for the next generation. While termination condition is not reached. Adaptive scaling factor: Calculating mutation adaptive scaling factor \(F\) by Equation (10). Mutation: Calculate mutation vector \(V_{i}\) by Equations (15)-(17). Crossover: Perform crossover to create \(U_{i}\) vector. Selection: Calculate oppositon of a certain top best number of particles. Select the NP candidates for the next generation based on fitness value. Update Gauss variables: Recalculate standard deviation of Gauss distribution and generate new Gauss variable. The details of this step can be seen in the procedure of G-ISADE in Figure 4. End. Return the best solution.
```  
**Algorithm 2**G-ISADE algorithm.  
Recently, [PERSON] et al. [18] presented an individual-dependent mechanism for DE, namely IDE. This approach deployed a similar scale factor (\(F^{new}\)) to mutation operation, this parameter is randomized from a normal distribution (\(F^{base}_{i},0.1\)), where \(F^{base}_{i}\) is original scale factor, associated with individual \(x_{i}\) and determined by Equation (19).  
\[F^{base}_{i}=\frac{b_{i}}{NP} \tag{19}\]  
where \(b_{i}\) is an index of the fitness value of individual \(x_{i}\) after raking; \(NP\) is number of population. Standard derivation is set to 0.1. This is different from G-ISADE, which used a strategy to calculate standard deviation at each iteration. The comparison of two these algorithms is implemented in Section 5.