# Fuzzy Fireworks Algorithm Based on a Sparks Dispersion Measure
## 2 Literature Review
This is the so-called new or modern synthesis of evolutionary biology. The objective of the Genetic Algorithm is to maximize the payoff of candidate solutions in the population against a cost function from the problem domain. The strategy for the Genetic Algorithm is to repeatedly employ surrogates for the recombination and mutation genetic mechanisms on the population of candidate solutions, where the cost function (also known as objective or fitness function) applied to a decoded representation of a candidate governs the probabilistic contributions a given candidate solution can make to the subsequent generation of candidate solutions [5]. On the other hand, we should mention in a general way that fuzzy logic is a methodology that provides a simple way to obtain a conclusion from ambiguous, imprecise, or linguistic information inputs. Of course, it is well known that in 1965, [PERSON] proposed the formal definition of fuzzy sets [8]. A _fuzzy set_\(\mathbf{A}\) in \(\mathbf{X}\) is characterized by a _membership_ (_characteristic_) function\(\mathbf{f_{A}}(\mathbf{x})\), which associates with each point in \(\mathbf{X}\) a real number in the interval \([0,1]\), with the value of \(\mathbf{f_{A}}(\mathbf{x})\) at \(\mathbf{x}\) representing the \"degree of membership\" of \(\mathbf{x}\) in \(\mathbf{A}\). Thus, the closer the value of \(\mathbf{f_{A}}(\mathbf{x})\) to unity, the higher the degree of membership of \(x\) in \(\mathbf{A}\). When \(\mathbf{A}\) is a set in the ordinary sense of the term, its membership function can take on only two values, \(0\) or \(1\), with \(\mathbf{f_{A}}(\mathbf{x})=1\) or \(0\) according to whether \(\mathbf{x}\) does or does not belong to \(\mathbf{A}\). Thus, in this case, \(\mathbf{f_{A}}(\mathbf{x})\) reduces to the familiar characteristic function of \(\mathbf{a}\) in set \(\mathbf{A}\). When there is a need to differentiate between such sets and fuzzy sets, the sets with two-valued characteristic functions will be referred to as _ordinary sets_ or simply _sets_[9]. **3. Fireworks Algorithm (FWA)**  
The Fireworks Algorithm (FWA) is a swarm intelligence method, which was developed by [PERSON] and [PERSON] in 2009, based on the explosion behavior of fireworks [10]. For each firework, we begin a process of explosion, and a shower of sparks fill the local space around the firework. The recently generated sparks represent possible solutions in the search space [11]. To mimic the above mentioned process, the algorithm is represented mathematically as described below. **4. Number of Sparks**  
\[Minimizef(x_{i})\in R,\;\mathbf{x_{imin}}\leq x_{i}\leq\mathbf{x_{imax}} \tag{1}\]  
where \(x_{i},\;i=1,\;2,\ldots,d\) indicates a location in the search space, \(x_{imin}\leq x_{i}\leq x_{imax}\) represents the bounds of the same space, and \(f(x_{i})\) refers to the objective function [12]. After, using the following equation, the number of sparks for each firework is generated. \[S_{i}=m.\frac{y_{max}-f(x_{i})+\epsilon}{\sum_{i=1}^{n}\left(y_{max}-f(x_{i}) \right)+\epsilon} \tag{2}\]where \(m\) represents a constant parameter which controls the numbers of sparks of the \(n\) fireworks and \(y_{max}=max(f(x_{i}))(i=1,2,3,\ldots,n)\) is the maximum value, that is, the worst value of the objective function in the \(n\) fireworks and \(\epsilon\) indicates a constant representing the smallest number in the computer, and it is utilized with the goal that an error with a division by zero cannot occur [13]. With Equation (3), we can define the bounds for the number of sparks, with the goal of keeping a balance within a range of number of sparks. The bounds are defined for \(s_{i}\) as follows:  
\[\hat{S_{i}}=\left\{\begin{array