We presented a new hybrid method that combines cellular harmony search algorithms with the SmallestSmallWorld theory. A harmony search (HS) algorithm is based on musical performance processes that occur when a musician searches for a better state of harmony. Harmony search has successfully been applied to a wide variety of practical optimization problems. Most of the previous researches have sought to improve the performance of the HS algorithm by changing the pitch adjusting rate and harmony memory considering rate. However, there has been a lack of studies to improve the performance of the algorithm by the formation of population structures. Therefore, we proposed an improved HS algorithm that uses the cellular automata formation and the topological structure of SmallestSmallWorld network. The improved HS algorithm has a high clustering coefficient and a short characteristic path length, having good exploration and exploitation efficiencies. Nine benchmark functions were applied to evaluate the performance of the proposed algorithm. Unlike the existing improved HS algorithm, the proposed algorithm is expected to have improved algorithmic efficiency from the formation of the population structure.
Studies on network maps provide indepth understanding of the basic features and requirements of various systems. Many network connection topologies, assumed to be either completely regular or completely random, have been studied [
The harmony search (HS) algorithm [
So far, researches have been carried out to improve the performance of the HS algorithm by changing the parameters such as pitch adjusting rate (PAR) and harmony memory considering rate (HMCR). However, there has been a lack of studies on the performance improvement of the HS through the formation of population structures to transform it into a cGA. In this study, therefore, we proposed the improved HS algorithm, which is used the CA and has the topological structure of the SSWN. The improved HS algorithm has a high clustering coefficient and a short characteristic path length possessing good exploration and exploitation efficiencies.
A cellular automaton is a discrete model studied in computation theory, mathematics, physics, complexity science, theoretical biology, and microstructure modeling [
The cell’s dimension in the CA can be defined as one, two, and threedimensional cells. Typically, a 2D cell is the most widely used because it can deal with spatial phenomena. Talking about the shape of the cell, a square cell is usually used because of its highest computerprocessing efficiency, compared to the other types of cells. Neighbors in the CA can be defined depending on the form of the adjacent surrounding cells. In the form of a 2D square cell CA, it can be defined as a Von Neumann neighborhood with neighbors in four directions or as a Moore neighborhood having neighbors in eight directions as shown in Figure
Neighboring cells.
Neumann
Moore
The concept of the CA, as described in this subsection, has been applied to a variety of optimization techniques such as GAs [
The SmallWorld network models have received much attention since their introduction by Watts and Strogatz [
Classification of networks according to connectivity.
Nishikawa et al. [
Examples of shortcut configuration with a center node.
They defined the SSWN theory as follows.
A SSWN is composed of two parts: the underlying network (e.g., a regular lattice) and the subnetwork of shortcuts containing only the shortcuts and their nodes.
The nodes in the subnetwork of shortcuts must be uniformly distributed over the network.
Finally, among all possible configurations of connected subnetworks of shortcuts with uniformly distributed nodes, the ones with a single center involve the largest number of nodes.
These arguments indicate that, given a fixed number of shortcuts, the networks connected with a subnetwork of shortcuts having uniformly distributed nodes have smaller
The HS algorithm is based on musical performance processes that occur when a musician searches for a better state of harmony such as during jazz improvisation. Jazz improvisation seeks musically pleasing harmony (a perfect state) as determined by an aesthetic standard, just as an optimization process seeks a global solution (a perfect state) as determined by an objective function. The pitch of each musical instrument determines the aesthetic quality, just as a set of values assigned to each decision determines the value of the objective function. Figure
Relation between music improvisation and engineering optimization [
In musical 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 of making a good harmony is increased next the 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 the decision variables result in a good solution, that experience is stored in each variable’s memory, and the possibility of making a good solution is also increased at the next iteration. In brief, the steps of HS algorithm are given as follows [
The optimization problem is defined as minimize
The initial harmony memory (HM) is generated from a uniform distribution in the ranges
Generating a new harmony is called improvisation. The new harmony vector
elseif
If the fitness of the improvised harmony vector
If the stopping criterion (maximum number of iterations,
In the present paper, we proposed the improved HS algorithm that uses the CA concept and has the topological structure of the SSWN. A population is just a group of certain number of arbitrary objects in the same generation as for the generation in the original HS. Meanwhile, these objects are not related to each other. In the configuration of these populations as a kind of random network, the
Efficiency of average path length (
Property  Description 

Average path length ( 
Exploitation (local search) Low 
 
Clustering coefficient ( 
Exploration (Global Search) High 
The operation process of the cellular harmony search (CHS) is shown in Figure
Flowchart of the CHS.
Smallest Small World cellular harmony search (SWCHS) performs its operation between the center nodes by adding shortcuts in the CHS. The CHS operation between the center nodes is added in the calculation process as shown in Figure
Flowchart of the SSWCHS.
In this study, the proposed SSWCHS was applied for solving unconstrained benchmark functions widely examined in the literature. The SSWCHS, CHS, and original simple harmony search (SHS) problems were performed in this study to have comparisons among optimizers. The optimization task was carried out using 100 independent runs based on the results depending on the type of problem. Statistical values, including best, worst, and mean values, and mean iteration number were obtained to evaluate the performance of the reported algorithms. Benchmark functions were utilized to evaluate the performance of considered optimization techniques. Among benchmark functions, five benchmark functions have 2 design variables and the rest have 30 design variables [
Smallest Small World harmony search network.
In this study, the following problems were applied as shown in Table
Definitions and specifications of two design variables problems.
Functions  Definition 

Rosenbrock’s valley 

 
Branin’s function 

 
Easom’s function 

 
Goldstein price’s function 

 
Sixhump camel back function 

Table
Obtained statistical results of 2D benchmark problem.
SSWCHS  CHS  SHS  

1. Rosenbrock  
Mean 



Best 



Worst 



SD 



Feasible solution  100  100  89 
Mean iteration  698.68  1,102.25  20,142.62 
Iteration SD  574.89  1,030.11  1,091.92 
 
2. Branin's function  
Mean 



Best 



Worst 



SD 



Feasible solution  100  100  89 
Mean iteration  211.94  444.13  2,593.45 
Iteration SD  124.18  225.88  225.88 
 
3. Easom's function  
Mean 



Best 



Worst 



SD 



Feasible solution  67  67  49 
Mean iteration  1,501.27  2,461.70  10,817.10 
Iteration SD  564.37  969.64  1,133.84 
 
4. Goldstein price's function  
Mean 



Best 



Worst 



SD 



Feasible solution  100  100  40 
Mean iteration  257.89  423.43  1,009.08 
Iteration SD  150.64  177.71  280.99 
 
5. Sixhump camel back function  
Mean 



Best 



Worst 



SD 



Feasible solution  100  100  100 
Mean iteration  87.34  152.80  852.06 
Iteration SD  33.13  67.30  67.30 
“SD” stands for standard deviation.
In this study, the following benchmark problems were applied as shown in Table
Definitions and specification of 30D benchmark problems.
Functions  Definition 

Step function 

 
The Rastrigin function 

 
The Griewank function 

 
The Ackley function 

Table
Results of 30ND Problems.
SSWCHS  CHS  SHS  

1. Step function  
 
Mean 



Best 



Worst 



SD 



Feasible solution  100  36  1 
Mean iteration  30,673.72  148,717.39  91,353.00 
Iteration SD  41,902.01  49,240.37  — 
 
2. Rastrigin function  
Mean 



Best 



Worst 



SD 



Feasible solution  92  0  0 
Mean iteration  113,790.16  —  — 
Iteration SD  47,257.60  —  — 
 
3. Griewank function  
Mean 



Best 



Worst 



SD 



Feasible solution  10  0  3 
Mean iteration  73,642.00  —  208,726.00 
Iteration SD  43,912.52  —  26,918.84 
 
4. Ackley function  
Mean 



Best 



Worst 



SD 



Feasible solution  96  3  8 
Mean iteration  79,262.84  141,376.33  160,133.00 
Iteration SD  64,770.38  35,365.89  58,642.35 
In this study, an improved HS algorithm which combines the CA and the topological structure of SmallestSmallWorld network is proposed. Most of previous studies, there have been a lack of studies on the performance improvement of the harmony search algorithm by use of population or memory structures. A new hybrid harmony search algorithm having high clustering coefficient and short characteristic path length was required. The hybrid HS algorithm developed in this paper has good exploration and exploitation efficiencies. Nine benchmark functions were applied to assess the performance of the proposed algorithm. The applied benchmark functions consist of five 2D functions and four 30D functions. The evaluation indexes of the SSWCHS were better than those of CHS and SHS in terms of solution quality. The SSWCHS algorithm showed generally faster convergence and more stability than the CHS or SHS. It shows very competitive solutions with less number of iterations than other considered algorithms. It is recommended that the optimization techniques, as new algorithms became available, be used in a wide range of engineering optimization problems. However, the SSWCHS has so many of the HS structures that it can affect computation time. Therefore, it remains a complementary part. As a further research, parameter variations are expected to develop using the proposed SSWCHS.
This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean government (MSIP) (no. 2013R1A2A1A01013886).