Recently developed chaotic charged system search was combined to feasiblebased method to solve constraint engineering optimization problems. Using chaotic maps into the CSS increases the global search mobility for a better global optimization. In the present method, an improved feasiblebased method is utilized to handle the constraints. Some constraint design examples are tested using the new chaoticbased methods, and the results are compared to recognize the most efficient and powerful algorithm.
The charged system search (CSS) is known as one of the efficient optimization algorithms among metaheuristic algorithms in recent years. The large number of researches on applying and enhancing this method proves this [
In physics, the space surrounding an electric charge has a property known as the electric field. This field exerts a force on other electrically charged objects. The electric field surrounding a point charge is specified by Coulomb’s law. Coulomb confirmed that the electric force between any two small charged spheres is inversely proportional to the square of the separation distance between the particles directed along the line joining them and proportional to the product of the charges of the two particles. Also, the magnitude of the electric field at a point inside a charged sphere can be obtained using Gauss’s law that is proportional to the separation distance between the particles. In the CSS, charged particles (CPs) or solution candidates are treated as a charged sphere that can exert electrical forces on each other according to the Coulomb and Gauss laws of electrostatics. The resultant force acts on each CP creating an acceleration according to Newton's second law by which, in combination with Newtonian mechanics, the position of each CP can be determined [
Recently, Talatahari et al. [
The charged system search contains a number of charged particle (CPs) where each one is treated as a charged sphere and can insert an electric force to the others. The pseudocode for the CSS algorithm is summarized as follows [
The magnitude of charge for each CP is defined as
A number of the best CPs and the values of their corresponding fitness functions are saved in the charged memory (CM).
The probability of moving each CP toward the others is determined using the following function:
The resultant force vector for each CP is calculated as
Each CP moves to the new position as
If each CP swerves off the predefined bounds, its position is corrected using the harmony searchbased handling approach.
The better new vectors are included in the CM, and the worst ones are excluded from the CM.
Steps
To the best of our knowledge, random initialization of the CSS and the adjusted limit parameters may affect the performance of the algorithm and reduce or increase its convergence speed. In other words, the parameters of the algorithm such as
The chaotic CSS algorithms, denoted by CCSS, can be obtained by using the values generated by a chaotic map instead of one or more random parameters needed in the CSS algorithm. Therefore, considering which parameter is defined chaotically, we can specify different algorithms. Table
Different chaoticbased CSS algorithms.
Algorithm  Equation  Condition 

CCSS1 

The initial positions of CPs are determined chaotically. 
CCSS2 

The kind of the force is determined chaotically. 
CCSS3 

The probability of moving each CP toward the others is determined chaotically. 
CCSS4  CCSS2 + CCSS3  The 
CCSS5 

The coefficient of the force is determined chaotically. 
CCSS6 

The coefficient of the velocity is determined chaotically. 
CCSS7 

The coefficients of the force and velocity are determined chaotically. 
CCSS8  CCSS4 + CCSS7  The initial positions of CPs are determined randomly, and the rest random generators are placed to chaotic maps. 
CCSS9  CCSS1 + CCSS8  All random generators are placed to chaotic maps. 
Different utilized chaotic maps.
Name  Equation  Condition 

Logistic map [ 


Tent map [ 


Sinusoidal map [ 


Gauss map [ 


Circle map [ 


Sinus map 


Hénon map 


Ikeda map [ 


Liebovtech map [ 


Zaslavski map [ 


In [
Best maps for each CCSS.
Best maps  CCSS  

1  2  3  4  5  6  7  8  9  
1  Tent  Tent  Sinusoidal  Tent  Liebovtech  Liebovtech  Liebovtech  Sinus  Sinus 
2  Circle  Gauss  Ikeda  Ikeda  Sinus  Sinusoidal  Zaslavski  Liebovtech  Tent 
3  Sinus  Ikeda  Sinus/Gauss  Circle  Tent  Zaslavski  Sinus  Tent  Gauss 
On the other hand, it is necessary to handle the constraints of the problem by using a suitable method [
Any feasible solution is preferred to any infeasible solution.
Infeasible solutions with slight violations of the constraints are treated as feasible ones.
Between two feasible solutions, the one with better objective function value is preferred.
Between two infeasible solutions, the one having smaller sum of constraint violations is preferred.
By using the first and fourth rules, the search tends to the feasible region rather than infeasible region, and by employing the third rule, the search tends to the feasible region with good solutions [
If the location of CPs become out of the variable boundaries, the solutions cannot be used. In this paper, using the harmony searchbased handling approach, this problem is dealt with. According to this mechanism, any component of the CP’s vector violating the variable boundaries can be generated randomly from CM as
The population size
Three engineering design problems which have been previously solved using a variety of other techniques are considered to perform investigation on efficiency of the proposed algorithms. The description of these examples is as the following.
This problem consists of minimizing the weight of a tension/compression spring subject to constraints on shear stress, surge frequency, and minimum deflection as shown in Figure
Schematic of tension/compression string.
The welded beam structure, shown in Figure
Schematic of welded beam structure.
A cylindrical vessel is capped at both ends by hemispherical heads as shown in Figure
Schematic of pressure vessel.
The variables contain
For this example, performances are assessed on the basis of the best fitness values and the statistics results of the new approaches as reported in Figure
The results obtained by the CCSS methods for the string problem: (a) best results (b) mean of results, (c) worst result, and (d) Standard deviation.
Figure
The results obtained by the CCSS methods for the welded beam problem: (a) best results (b) mean of results, (c) worst result, (d) Standard deviation.
The obtained results using 9 variants of the presented algorithms are shown in Figure
The results obtained by the CCSS methods for the pressure vessel problem: (a) best results (b) mean of results, (c) worst result, and (d) Standard deviation.
As suggested in [
Performances are assessed on the basis of the best fitness values and the statistics results of the new approaches from 50 runs with different seeds. Simulation results show that for all examples, the proposed methods perform satisfactorily. Almost all of the proposed methods improve the reliability of the algorithm by reducing the standard deviation values. From numerical results, it is clear that CCSS6, in which the coefficient of the velocity is determined chaotically, is the most reliable algorithm having the smallest standard deviation values, while, the algorithm with chaotic coefficients for the force and velocity (CCSS7) is the worst one. Meanwhile, the CSS8 and CCSS6 methods have better performance in relation to the best, mean, and worst values. In CCSS8, all parameters of the algorithm are determined chaotically, but the initial positions of agents are defined randomly. To sum up, the coefficient of the velocity plays a key role in reliability of the algorithm and the results show that a chaotic velocity coefficient can improve the performance of the algorithm; in addition, using all chaotic parameters of the algorithm can improve the performance, as well; however, chaotic initialization has very small or even no influence on the final results. This chaotic charged model, which is a coupled system, for constraint optimization problems in future could also benefit from other coupled systems such as the kinetic models of competition and the corresponding hybrid competition models [