Harmony search (HS) method is an emerging metaheuristic optimization algorithm. In this paper, an improved harmony search method based on differential mutation operator (IHSDE) is proposed to deal with the optimization problems. Since the population diversity plays an important role in the behavior of evolution algorithm, the aim of this paper is to calculate the expected population mean and variance of IHSDE from theoretical viewpoint. Numerical results, compared with the HSDE, NGHS, show that the IHSDE method has good convergence property over a testsuite of wellknown benchmark functions.
Most optimization algorithms are based on numerical linear and nonlinear programming methods that require substantial gradient information and usually seek to improve the solution in the neighborhood of an initial point. These algorithms, however, reveal a limited approach to complicated realworld optimization problems because gradient is often difficult to find out or does not exist. What is more, if there is more than one local optimum in the problem, the result may depend on the selection of the starting point, and the obtained optimal solution may not necessarily be the global optimum.
Recently, a new class of metaheuristics, named harmony search (HS), has been developed. The HS algorithm proposed in [
HS may be viewed as a simple realcoded genetic algorithm (GA), since it incorporates many important features of GA like mutation, recombination, and selection. HS has been successfully applied to a wide variety of practical optimization problems like designing controller [
Similar to the GA and particle swarm algorithms, the HS method is a random search technique. It does not require any prior domain knowledge, such as the gradient information of the objective functions. Unfortunately, empirical study has shown that the original HS method sometimes suffers from a slow search speed, and it is not suitable for handling the multimodal problems [
Recently, Omran and Mahdavi tried to improve the performance of HS by incorporating some techniques from swarm intelligence. The new variant called by them as global best harmony search (GHS) [
To overcome the shortcoming of premature convergence and stagnation, in this paper, we replace the pitch adjustment operation in classical HS (CHS) with a mutation strategy borrowed from the realm of the differential evolution (DE) algorithms, and we use
The new algorithm proposed in this paper, called IHSDE (improved harmony search methods based on differential mutation operator), has been extensively compared with the HSDE, and the classical HS. Mathematical analysis will show that the IHSDE, under certain conditions, possesses an increasing population variance (with generation) as compared to HSDE. The numerical experiments show that the proposed algorithm is effective in dealing with a test suite of wellknown benchmark functions.
The rest of the paper is organized in the following way. Section
Current metaheuristic algorithms imitate natural phenomena, and evolution in evolutionary algorithms. HS algorithm was conceptualized using the musical process of searching for a perfect state of harmony. In music improvisation, each player sounds any pitch within the possible range, together making one harmony vector. If all the pitches make a good harmony, that experience is stored in each player’s memory, and the possibility to make a good harmony is increased next time. Similarly, in engineering optimization, each decision variable initially chooses any value within the possible range, together making one solution vector. If all the values of decision variables make a good solution, that experience is stored in each variable’s memory, and the possibility to make a good solution is also increased next time. Figure
Analogy between music improvisation and engineering optimization.
The HS algorithm does not require initial values for the decision variables. Furthermore, instead of a gradient search, the HS algorithm uses a stochastic random search that is based on the harmony memory considering rate and the pitchadjusting rate so that derivative information is unnecessary. Compared to earlier metaheuristic optimization algorithms, the HS algorithm imposes fewer mathematical requirements and can be easily adopted for various types of engineering optimization problems.
The steps in the procedure of standard harmony search algorithm are as follows:
The harmony memory (HM) is a memory location where all the solution vectors (sets of decision variables) are stored. HMCR and PAR are parameters that are used to improve the solution vector.
Every component obtained by the memory consideration is examined to determine whether it should be pitch adjusted. This operation uses the PAR parameter, which is the rate of pitch adjustment as follows.
Pitch adjusting decision for
In Step 3, HM consideration, pitch adjustment, or random selection is applied to each variable of the new harmony vector in turn.
In the next section, we employ the differential mutation operator to improve the fitness of all the members in the HS memory so that the overall convergence speed of the original HS method can be accelerated.
Experiments with the CHS algorithm over the standard numerical benchmarks show that the algorithm suffers from the problem of premature and/or false convergence, slow convergence especially over multimodal fitness landscape.
To circumvent these problems of premature, Chakraborty et al. proposed harmony search algorithm with differential mutation operator (HSDE). They replaced the pitch adjustment operation (
In what follows, we reset
The pseudocode of IHSDE is described in Algorithm
Initiate_parameters
Initialize_HM
Theoretical analyses of the properties of HS algorithms are very important to understand their search behaviors and to develop more efficient algorithms [
The evolution of the expected population variance over generations provides a measure of the explorative power of the algorithm. In the following, we will estimate the expected mean and variance of the population obtained by applying mutation operator.
Our ideas are as follows, firstly we find an analytical expression for the population expected variance, and then we compare the expected population variance of IHSDE with HSDE to show that the IHSDE algorithm possesses greater explorative power.
In HS type algorithms, since each dimension is perturbed independently, without loss of generality, we can make our analysis for singledimensional population members.
Let us consider a population of scalars
If the elements of the population are perturbed with some random numbers or variables,
Let
Since
So,
According to [
Using (
Now,
Let
Thus,
According to [
Therefore,
Now,
Thus,
Let
Therefore, the variance of final population
The conclusion in [
The main analytical result is expressed in the form of the following theorem.
Let
In HSDE,
Since
Thus,
Improper parameters
The above mathematical analysis show that the IHSDE possesses an increasing population variance as compared to HSDE. This ensures that the explorative power of IHSDE is on average greater than that of HSDE, which in turn results into better accuracy of the IHSDE algorithm.
In the following section, we give some numerical experiments over standard test functions.
The effectiveness of the IHSDE algorithm has been evaluated on a test suite of wellknown benchmark functions (Table
Benchmark test functions.
Function name  Benchmark functions expression  Search range  Optimum value 

































































All the experiments were performed on Windows XP 64 System running on an Hp desktop with Intel(R) Xeon(R)
To judge the accuracy of different algorithms, 50 independent runs of each of the four algorithms were carried out and the best, the mean, the worst fitness values, and the standard deviation (Std) were recorded. Table
The statistical results for 50 runs tested on sixteen benchmark functions.
Function name  Algorithm  Best  Mean  Worst  Std 


CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 





 

CHS 




HSDE 





IHSDE 





NGHS 




Figure
The convergence and its boxplot.
In order to accurately give search process of each algorithm, we set the same parameters, and run CHS, HSDE, IHSDE method, respectively, for multimodal function F_{16}. Figure
Show search processes of CHS, HSDE, and IHSDE methods for
Contour line and Optimal Solution
Iterations of CHS Method
Iterations of HSDE Method
Iterations of IHSDE Method
We omitted plots for all the other functions (
Experimental results on benchmark functions show that the IHSDE method can outperform the other methods. From Figure
This paper has presented an improved harmony search algorithm by blending with it a different vectorbased mutation operator borrowed from the DE algorithms. The HM members are fine tuned by the DE’s mutation operator to improve their affinities so that enhanced optimization performances can be achieved. Mathematical analysis indicates that the IHSDE posses an increasing population variance as compared to HSDE. This ensures that the explorative power of IHSDE is on average greater than that of HSDE, which in turn results in better accuracy of the IHSDE algorithm. Several simulation examples of the unimodal and multimodal functions have been used to verify the effectiveness of the proposed methods. Compared with the HSDE and CHS, better optimization results are obtained using IHSDE approaches in most cases. Checking the effect of variation of the scale factor
This work is supported by National Natural Science Foundation of China under Grant no. 60974082, and Scientific Research Program Funded by Shaanxi Provincial Education Department under Grant no. 12JK0863.