We propose a method to improve the performance of evolutionary algorithms (EA). The proposed approach defines operators which can modify the performance of EA, including Levy distribution function as a strategy parameters adaptation, calculating mean point for finding proper region of breeding offspring, and shifting strategy parameters to change the sequence of these parameters. Thereafter, a set of benchmark cost functions is utilized to compare the results of the proposed method with some other wellknown algorithms. It is shown that the speed and accuracy of EA are increased accordingly. Finally, this method is exploited to optimize fuzzy control of truck backerupper system.
Evolutionary algorithms (EA) are usually exploited as the first candidates for hard optimization tasks. They can deal with many kinds of problems and cost functions such as multimodal, discrete, constraint on variables, high dimensionality, and stochastic cost functions; consequently, they are suitable for many applications. In the case of evolutionary computation, there are four historical paradigms that have served as the basis for much of the activity in this field, genetic algorithms (Holland, 1975) [
These methods have drawn much attention to the research community in conjunction with the parallel and/or distributed computations. EP, in particular, was studied initially as a method for generating artificial intelligence [
The classical evolutionary programming (CEP) can be presented as follows.
Generate the initial population of
Evaluate the fitness score for each individual
Each parent
Calculate the fitness of each offspring
Conduct pairwise comparison over the union of parents
Select the
Stop if the halting criterion is satisfied; otherwise,
Considering disadvantages arising in performance of the CEP results in searching for new methods. Although some methods have been introduced for dealing with these disadvantages, they have not attained considerable success yet. Moreover, EP family, as thought, has many advantages in dealing with multimodal cost functions, its applications in the real world are not up to mark in comparison with other evolutionary algorithms like genetic algorithm. Hence, and it is worthwhile to investigate new methods which are more successful in dealing with EP’s disadvantages. Many variants of EP have been developed to boost the performance of CEP by changing (
In EP, mutation is implemented by adding strategy parameters to variable vectors of parents in order to produce offspring. When one of the strategy parameters takes on a large value, adding it to the related variable causes abrupt change. Hence, the variable grows with large steps and deviates far from the optimum point whereas some of other variables do not sense considerable changes. If this event repeats for some iteration, the variable will go further, and consequently, it slows down EP in some iteration. To avoid such an occurrence, [
In this paper, speed and accuracy characterizations of EA are improved using cost and coordination information. The proposed approach defined operators which can modify performance of EA: Levy distribution function for strategy parameters adaptation, calculating mean point for finding proper region for breeding offspring, and shifting strategy parameters to change the sequence of these parameters. Thereafter, a set of benchmark cost functions are used to compare the results of the proposed method with some other known algorithms. Finally, this modified approach is exploited to optimize fuzzy control of truck backerupper system.
The organization of this paper is as follows: Section
The general form of CEP follows a twostep process of selection and variation in a population. Following initialization of a population, the fitness of each individual in the population is scored with respect to an arbitrary fitness function. In general, selection is applied as a tournament wherein the fitness of each individual in the population is compared against the fitness of a random set of other individuals in the same population. A “win” is recorded for an individual when individual’s fitness equals or exceeds that of another in the tournament set. Individuals are then ranked with respect to the number of wins, and those with the highest number of wins over some threshold are selected as parents for the next generation. Parents are randomly varied to generate offspring, and the fitness of each member in the population is reevaluated. This process is repeated for a userspecified number of generations [
Offspring (
In EP, mutation is performed by adding strategy parameters to the coordinate of parents. Classical evolutionary programming uses Gaussian distribution function for updating offspring and strategy parameters (mutation parameters). Since Gaussian distribution has some drawbacks, other distribution functions are alternatively used. In the following, some wellknown distribution functions will be introduced.
The strategy parameters play main role in deciding the place of offspring. Determining an optimal lower bound for the strategy parameter is essential for the EP algorithm in most applications. The optimal setting of the lower bound depends on the problem and cannot be the same throughout the evolution process. In [
It is obvious that the basic advantages of any algorithm is in deciding the most suitable place for breeding offspring and finding the route toward the global minimum. This goal is implemented by mean point operator in MCEP [
Therefore, three methods are utilized simultaneously in order to enhance the capabilities of EP family.
Yao et al. [
LEP is a variant of EP which is similar to CEP and FEP. The difference comes from defining mutation function. LEP uses a Levy distribution function in place of Gaussian for mutation function. However, unlike FEP, in LEP only distribution function in (
All three distribution functions, Gaussian, Cauchy, and Levy, are special cases of the stable distributions. These distribution functions can be produced by the following equation:
In evolutionary computational techniques, solution of problem and the values of the individuals finally converge to a unique point. This convergence is slowly seen over the generations, and the algorithm gradually approaches the optimum point as the number of generation increases. This procedure gives us the idea of adding the average of the individuals in each generation to the algorithm to enhance the convergence speed of the EP. This technique is known as inertia weight method [
Location of offspring which is produced by CEP and WSLEP. Offspring is somewhere inside of the dashed circles.
In the above equations,
In addition, Levy distribution function is used in mutation operator presented in (
In EP, mutation is implemented by adding strategy parameters to variable vectors of parents in order to produce offspring. When one of the strategy parameters takes on a big value, adding it to the related variable causes abrupt changes in the variable. So, the variable grows with big steps and deviates far from the optimum point, whereas some of other variables do not sense considerable changes. If this event continues for some iteration, the variable will go further. This event slows down EP in some iteration. To avoid such an occurrence, [
There is a question that if the three explained approaches can enhance EP separately, whether the compound method of them (when they are used simultaneously in EP) can be more helpful or not?
Here, the LEP with weighted mean point and optimized shifting strategy parameters approach is used. The pseudo code for used method is given as follows:
Choose the initial population of individuals
Produce strategy parameters by Levy mutation function
Evaluate the cost of each individual in that population.
Repeat this generation until termination: (time limit, sufficient cost achieved, etc.)
Update the weighted mean point by (
Shift strategy parameters
Breed individuals through mutation to give birth to the offspring
Evaluate the cost of the offspring
Raise tournament to decide the next generation members
End.
The main results of this paper are presented in two parts. In the first part, the seven algorithms of CEP, FEP, LEP, EEP, MCEP, SCEP, and WSLEP are compared. In this part, it will be shown that the speed and accuracy of the proposed EP improved in terms of reaching the global minimum can improve via the proposed approach. In second part, the proposed algorithm is compared with 4 algorithms: jumping gene (JG) [
The 12 benchmark functions used in our experimental study; the second column introduces name of functions.

Sphere model 

 

Schwefel's problems 2.22 

 

Schwefel's problems 1.2 

 

Schwefel's problems 2.21 

 

Generalized Schwefel's problem 2.26 

 

Generalized Rastrigin's function 

 

Ackley's function 

 

Generalized Griewank function 

 

Shekel's Foxholes function 

 

Sixhump camelback function 

 

Hartman's family 2 

 

Shekel's family 3 

These cost functions can be categorized in 3 subgroups: unimodal high dimensional, multimodal high dimensional and low dimensional.
Functions f1–f4 are high dimensionalunimodal problems, which have only one global minimum that is also their only local minimum as well. Functions f5–f8 are high dimensionalmultimodal functions where the number of local minima increases exponentially with the problem dimension. They are regarded as the most difficult class of problems for many optimization algorithms of which CEP has slow convergence on these functions [
The ranges of the variables and dimensions of the cost functions are chosen according to [
Parameters of the algorithms.
Tournament size 
10 
Population size  100 
Range bound of variables  Mentioned in last column of Table 
Number of repetition  20 
Number of generation  Mentioned in second column of Table 
The best algorithm among WSLEP, CEP, FEP, EEP, MCEP, SCEP, and LEP is going to be selected on 12 test functions. To avoid any concurrence in the results, all of the algorithms have been run 20 times, and the averages of the obtained results are presented in Table
Comparison algorithms on
Number of generation  CEP  FEP  EEP  LEP  SCEP  MCEP  WSLEP 


Range bound  


1500  155.67  25.194  27.34  26.685 



30  0 


2000  0.418  0.596  0.746  0.707 



30  0 


5000  1161  4438  2314  650 



30  0 


5000  0.607  34.38  2.14  1.07  0.0012  0.0002 

30  0 


3000  −8373 

−12154  −11167  −9500  −8230  −12162  30  −12569.5 


5000  44.275  1.864  0.124  13.435  30.34 


30  0 


1500  8.43  3.773  0.21  5.36  0.006  0.026 

30  0 


2000  2.373  0.971  1.03  0.694  0.009 


30  0 


100  3.432 

1.32  1.458  1.889  2.23  2.884  2  1 


100 







2  −1.031 


200 

−3.201  −3.21  −3.164  −3.30 


6  −3.32 


100  −8.578  −8.311  −8.823  −9.145  −10.17  −9.86 

4  −10.5 

All of the algorithms are run until a prespecified generation is reached. The number of generations is shown in the second column of Table
Multimodal functions make it difficult for EP families to find global minimum as they have many local and several global minimums. WSLEP has the best answer in this group too. Good results in this group are very important as there are few methods that have acceptable performance.
Essential parameters of the algorithms.
General  Population size  100 
Number of repetition  50  
 
WSLEP  Tournament size 
10 
Initial standard deviation  1  
 
JG  Number of transposon  1 
Length of transposon  2  
Crossover  Uniform  
Mutation rate  0.1  
 
IWPSO  Acceleration coefficients  2 
Linearly increasing inertia weight  From 0.5 to 1.5  
Maximum velocity 

Similar to the previous part, the algorithms have been tested on 12 cost functions. The test has been repeated twenty times, and average results have been considered. Table
Comparison algorithms on
Number of generation  BEA  JG  IWPSO  WSLEP  


1500  5 




2000  0.002 




5000  0.68 


0.021 

5000  0.02  0.57  0.0074 


3000  −2572  −7040  −8917  −12020 

5000  8.06 




1500 



0.0001 

2000  0.5 

0.103  0.0049 

100  2.03  1.74 

2.23 

100  −0.57 




200 

−2.86 



100  −9.63  −9.95  −10.32 

In the current experimental analysis, the set of estimators of medians is directly calculated from the average error results. Table
Comparison of estimation method results for different algorithms.
CEP  FEP  LEP  EEP  SCEP  MCEP  WSLEP  

CEP  0  −2.23  −1.03  −1  0.57  0.58  0.59 
FEP  2.23  0  1.19  1.23  2.81  2.8  2.82 
LEP  1.03  −1.19  0  0.033  1.61  1.63  1.63 
EEP  1  −1.23  −0.03  0  1.58  1.58  1.59 
SCEP  −0.57  −2.81  −1.61  −1.58  0  0.1  0.01 
MCEP  −0.58  −2.8  −1.63  −1.58  −0.1  0 

WSLEP  −0.59  −2.82  −1.63  −1.59  −0.01 

0 
In this section, the proposed EP method (WSLEP) is exploited to optimize fuzzy control of truck backerupper system. Truck backerupper problem is an excellent test bed for fuzzy control systems. Fuzzy controller, formulated on the basis of human understanding of the process or identified from measured control actions, can be regarded as an emulator of human operator. Controller design, however, may become difficult, especially if the numbers of state variables are large [
Simulated truck backerupper benchmark system.
The truck position is determined by three state variables,
For simulating purposes, we need a mathematical model of the truck. We use the following approximate model [
In step
Membership functions.
Now, rules for fuzzy system must be designed. The cost function is
The error is summed for all five initial points. The five algorithms CEP, FEP, EEP, MCEP, and SCEP along with the proposed algorithm (WSLEP) are used for minimizing the defined cost function by finding proper rules. Figure
Cost via iteration result for optimized system using WSLEP.
The minimum founded cost is
Final fuzzy rule base for the truck backerupper control problem.
S3  S1  S2  CE  B1  S2  
S2  CE  S3  S2  S2  S2  
S1  S2  B2  S3  S2  S2  
Φ  CE  B2  B1  CE  S2  S3 
B1  S1  B1  B3  B1  CE  
B2  S1  CE  B1  CE  S1  
B3  B1  CE  B2  CE  CE  
S2  S1  CE  B1  B2  
X 
Some initial points are chosen to test the designed controller. Figure
The truck trajectory using the designed fuzzy system as the controller for different initial conditions.
In this paper, we proposed a modified approach to increase speed and accuracy of evolutionary algorithms by designing controller using cost and coordination information. The proposed method defined operators which can improve performance of evolutionary algorithms: Levy distribution function for strategy parameters adaptation, calculating mean point for finding proper region for breeding offspring, and shifting strategy parameters to change the sequence of these parameters. Thereafter, a set of benchmark cost functions were used to compare the results of the proposed method with some other known algorithms. It was intuitively obvious that the proposed algorithm was more accurate and fast in finding the value and location of the global minimum in all three groups of the cost functions. Finally, this method was exploited to optimize fuzzy control of truck backerupper system.