Appl Intell (2018) 48:2681–2697 https://doi.org/10.1007/s10489-017-1096-8
Improved cat swarm optimization algorithm for solving global optimization problems and its application
to clustering
Yugal Kumar1 · Pradeep Kumar Singh1
Published online: 19 December 2017
© Springer Science+Business Media, LLC, part of Springer Nature 2017
Abstract This paper presents a cat swarm optimization (CSO) algorithm for solving global optimization problems. In CSO algorithm, some modifications are incorporated to improve its performance and balance between global and local search. In tracing mode of the CSO algorithm, a new search equation is proposed to guide the search toward a global optimal solution. A local search method is incorpo- rated to improve the quality of solution and overcome the local optima problem. The proposed algorithm is named as Improved CSO (ICSO) and the performance of the ICSO algorithm is tested on twelve benchmark test functions. These test functions are widely used to evaluate the perfor- mance of new optimization algorithms. The experimental results confirm that the proposed algorithm gives better results than the other algorithms. In addition, the proposed ICSO algorithm is also applied for solving the clustering problems. The performance of the ICSO algorithm is evaluated on five datasets taken from the UCI repository. The simulation results show that ICSO-based clustering algorithm gives bet- ter performance than other existing clustering algorithms.
Keywords Cat swarm optimization · Clustering · Meta-heuristics · Numerical functions · Improved CSO
Yugal Kumar yugalkumar.14@gmail.com
Pradeep Kumar Singh pradeep 84cs@yahoo.com
1
1 Introduction
There are many optimization problems, including unimodal and multimodal, in the fields of engineering and science. These optimization problems can be further categorized into unimodal separable and inseparable problems, and multimodal separable and inseparable problems. In litera- ture, many methods have been reported for solving these problems; these methods either maximize or minimize the objective function. Moreover, these methods have also been applied for solving real-life problems such as clustering, classification, scheduling, path planning, resource alloca- tion, and many other problems. In recent years, large num- bers of optimization algorithms have been developed by researchers to find the solution for the above-mentioned optimization problems. These algorithms are divided into two categories, i.e., exact algorithms and approximation algorithms [1]. Exact algorithms find the optimal solution within the bounded time but have exponential compu- tational time, whereas approximation algorithms provide better results both in time and solution using heuristics. Meta-heuristic algorithms are the sub-branch of approxi- mation algorithms which have been applied for solving a wide range of optimization problems. Most of these algo- rithms are inspired by natural phenomena. Some examples of these algorithms are simulated annealing (SA) algorithm [2], genetic algorithm (GA) [3], particle swarm optimiza- tion (PSO) [4], ant colony optimization (ACO) [5], har- mony search (HS) [6], artificial bee colony (ABC) [7], firefly algorithm (FA) [8], league championship algorithm (LCA) [9], water cycle algorithm (WCA) [10], charge sys- tem search (CSS) algorithm [11, 12], magnetic charge system search (MCSS) [13–15], teaching learning based
Department of Computer Science and Engineering, Jaypee University of Information Technology, Waknaghat, Solan, Himachal Pradesh, India
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optimization (TLBO) [16, 17], and mine blast algorithm (MBA) [18].
Recently, the CSO algorithm has gained popularity among the scientific community due to its strong explo- ration ability and effective solving of complex optimiza- tion problems [19]. Due to its simple and straightforward implementation, this algorithm has been applied in diverse fields. Initially, Chu et al. presented the CSO algorithm for optimizing numerical test functions [19]. This algorithm is characterized by the behavior of cats, and each cat is described in terms of position and velocity. This algorithm measures the behavior of a cat in two modes – seeking mode and tracing mode. In the optimization process, the position of a cat is optimized by using its own experience (especially, in seeking mode) and velocity. However in trac- ing mode, the CSO algorithm can trap in local optima for some complex and intricate optimization problems, and also can have low convergence rate in last iterations. The CSO algorithm has weak diversity problem for a few benchmark functions.
Contribution of this work The main contribution of this study is that it handles the aforementioned shortcomings of the CSO algorithm. The aim of this research work is to pro- pose some modifications in the CSO algorithm in order to overcome the problems of weak diversity and convergence rate. To make the CSO algorithm more efficient, reliable and robust, these modifications are incorporated into indi- vidual cat memory and experience. The main contribution of this work is to deal with the following issues of CSO algorithm.
• Lack of balance between exploration (seeking mode) and exploitation (tracing mode) processes.
• Lack of diversity for fewer benchmark functions, espe- cially in tracing mode.
• Slow convergence problem in the last generations of the iterative process.
The following amendments are integrated in the CSO algorithm to deal with the above-mentioned issues. These amendments are as follows.
• A new enhanced and accelerated velocity equation is proposed for the CSO algorithm, especially for tracing mode.
• A new position update equation is proposed both for tracing and seeking modes.
• A local search method is incorporated to improve the quality of solution and handle local the optima problem.
The proposed algorithm is tested on several benchmark test functions for analyzing its performance and accuracy.
The experimental results showed that the performance of the proposed algorithm is improved in terms of global search and convergence rate.
The rest of the paper is organized as follows. Section 2 summarizes related works on CSO algorithm. The description of CSO algorithm is illustrated in Section 3. Section 4 presents the proposed ICSO algorithm and its steps for solving benchmark test functions. The experimen- tal results of the proposed algorithm and other state-of-the- art algorithms are given in Section 5. Section 6 explores the applicability of the proposed ICSO algorithm for solv- ing the real-world clustering problems. Section 7 reports the conclusion of the work done.
2 Related works
This section describes the related works in the direction of CSO algorithm for solving different optimization problems. Mohapatra et al. [20] developed a modified cat swarm opti- mization algorithm for improving the searching ability of cats. In this work, a mutation operator was applied to mutate the position of the best cat for achieving good results. The gene data analysis and classification tests showed that the modified CSO algorithm has better performance in compar- ison to other algorithms. To investigate the parallel structure of CSO algorithm, Tasi et al. [21] presented a parallel CSO algorithm and tested its performance on several benchmark functions. It was reported that the parallel CSO algorithm has better performance to PSO and CSO algorithms. In continuation of their work, Tasi et al. [22] developed an enhanced parallel cat swarm optimization (EPSO) algorithm to obtain higher accuracy and less computational time. In the EPSO algorithm, the Taguchi method was incorporated for achieving higher accuracy rate and optimal solutions. The performance of the EPSO algorithm was tested on five benchmark functions, and it was found that the EPSO algorithm has a higher accuracy rate. Orouskhani et al. intro- duced the concept of inertia weight in CSO algorithm for better convergence rate. The simulation results showed that the proposed algorithm achieves better convergence results in comparison to the original CSO algorithm [23]. For opti- mal design of the circular and concentric circular antenna arrays, Ram et al. [24] applied the CSO algorithm. Yang et al. [25] adopted the CSO algorithm for processing and analyzing the medical image. In their work, CSO algorithm was combined with limited memory Broyden–Fletcher– Goldfarb–Shanno with boundaries (L-BFGS-B) and called it HLCSO. The simulation results showed that the HLCSO method gives better accuracy and less computational times. To improve the performance of the CSO algorithm, Lin et al.
Improved cat swarm optimization algorithm for solving global optimization problems… 2683
[26] incorporated two improvements in CSO algorithm and called the resultant algorithm an improved CSO (ICSO). Results stated that the ICSO algorithm has higher accuracy rate in comparison to the traditional CSO algorithm. Guo et al. [27] applied the CSO algorithm to estimate the param- eters of solar cell model. Simulation results showed that the CSO is an effective tool for identification of parameters of solar cell model. To optimize the design of a channel cross-section, Liu et al. [28] applied the CSO algorithm for safety and stability of side walls in a hydraulic design. The CSO algorithm provides better results in comparison to GA and PSO algorithms. In continuation of their work, Ram et al. [29] also applied CSO algorithm for reducing the side lobe level of the concentric circular antenna and satisfactory results. To determine the location and control- ling parameters of SVC and TCSC, Nireekshana et al. [30] applied CSO algorithm and claimed that the CSO algo- rithm provides better results. Wang et al. [31] proposed a new algorithm based on CSO and called it NCSO. In their work, a new mechanism was introduced to make the MR parameter dynamic. Further, the Cauchy mutation operator was applied to enhance the global search of CSO algo- rithm. The performance of the NCSO algorithm was tested on twelve benchmark functions, and it was found that the NCSO is an effective and efficient algorithm. Kotekar and Kamath applied the CSO algorithm for clustering the web documents, and their results showed that the CSO algorithm enhances the web service process [32]. Yusiong applied the CSO algorithm to optimize the parameters of the artificial neural network (ANN). It was reported that the CSO is an effective tool for optimizing the parameters of ANN. Sharafi et al. [33] developed a binary version of CSO algorithm and tested it on several benchmark optimization functions. The results showed that the proposed algorithm obtains better quality results in comparison to the GA and Binary PSO algorithms. Pappula and Ghosh applied CSO algorithm for the synthesis of linear antenna arrays. The simulation results showed that the CSO algorithm gives better results than the PSO and ACO algorithms [34]. To improve the accu- racy rate, Orouskhani et al. [35] developed a new adaptive dynamic CSO algorithm, which was tested on six bench- mark test functions. It was revealed that it takes less time to converge. In literature, the CSO algorithm has also been applied for solving clustering problems [36–38].
3 Cat swarm optimization
Chu and Tasi developed CSO algorithm based on the two key characteristics of cats [19], i.e., hunting and resting skills. According to the hunting skill, a cat has strong
curiosity toward moving objects. In the resting skill, a cat spends most of its time in the resting position, even though it remains alert and slowly moves to different positions. But, if a target is identified, then cat quickly captures the target spending a lot of energy. So, on the behalf of these two key characteristics of cats, a mathematical model was formed to solve complex optimization problems and was named cat swarm optimization. In this model, two modes, i.e., seeking and tracing modes, are described to measure the behavior of cats. The working of these modes is explained in Sections 3.1 and 3.2.
3.1 Seeking mode
The seeking mode describes the resting skill of cats. In seek- ing mode, a cat moves to different positions in the search space, but remains alert. It can be interpreted as local search for the solutions. The following notations are used in this mode.
• Seeking Memory Pool (SMP): This parameter describes the number of copies of a cat to be replicated.
• Seeking Range of selected Dimension (SRD): It denotes the difference between new and old dimensions of cat selected for mutation.
• Counts of Dimension to Change (CDC): It represents the number of dimensions a cat position undergone for mutation.
The steps of seeking mode of CSO algorithm are given as follows.
1. Define the number of copies (T) of ith cat.
2. According to CDC parameter, do the following
i. Randomly add or subtract SRD values from current position of cats.
ii. Replace the old values for all copies.
3. Compute the fitness for all copies
4. Choose the best candidate solution and deploy at the
position of ith cat.
3.2 Tracing mode
This mode reflects the hunting skill of cats. When a cat hunts the prey, the position and velocity of cat are updated. So, a large difference occurs between new and old posi- tions of cats. The position (Xdj ) and velocity (Vdj ) of the
jth cat in the D-dimensional space can be defined as Xdj =
X1j ,X2j ,…….XDj ; Vdj = V1j ,V2j ,…….VDj ; where d = 1,2,…D.
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Y. Kumar, P. K. Singh
positions and velocities of cats. Sometime, the algorithm fails to explore promising solutions due to the lack of infor- mation regarding global best position of cat. Hence, to deal with these issues, the following amendments are proposed in the CSO algorithm.
=
The velocity and position of the jth cat are computed using (1) and (2).
dth dimension, w denotes a weight factor in the range of 0
and1,Vdj representstheoldvelocityofthejthcat,cisauser
defined constant, r denotes a random number in the range
of 0 and 1, Xd represents the best position achieved by jth jbest
cat in dth dimension, and Xdj denotes the current position of • the jth cat in dth dimension where d = 1,2,…D.
Xdjnew=Xdj+Vdj (2) where, Xdj new denotes the updated position of the jth cat in
12 D best Xbest,Xbest,…….Xbest .
Vdjnew =w∗Vdj +c∗r∗ Xdjbest −Xdj
(1) • where, Vdj new represents the updated velocity of jth cat in the
To explore more promising solution and enhance the convergence rate, the global best position of cat is used to guide the positions of cats in tracing mode. Hence, a new modified search equation is proposed for tracing mode of CSO algorithm which includes the global best position of cat.
Xd+1 =(1−β)∗Xd+β∗Pg+Vd (3) jnew j j
The CSO algorithm uses a velocity vector and previ- ous position of cat to update the position of a cat in tracing mode. The updated position of a cat is only influenced by velocity vector. Hence, to improve the diversity of CSO algorithm, especially in tracing mode, a new velocity updated equation is proposed, which is inspired from [39].
dth dimension, Xd denotes the current position of the jth cat j
in dth dimension and Vd represents the velocity of the jth cat. j
Mixture Ratio (MR) is used to combine the seeking and tracing modes of the CSO algorithm. MR is designed to determine the number of cats in seeking and tracing modes. The steps of the CSO algorithm are as follows.
1. Initialize the population of cats.
2. Define the user-defined parameters and numbers of cats
in seeking mode and tracing mode according to MR
parameter value.
3. Compute the fitness function for each cat and memorize
the best position.
4. According to flag:
• If cat is in seeking mode, apply the seeking mode process.
• Otherwise, apply tracing mode process.
5. Again set the number of cats in tracing and seeking modes according to the MR parameter.
6. Repeat steps 3–5 until the termination condition is satisfied.
4 Proposed ICSO algorithm
According to Orouskhani et al., the CSO algorithm suffers from premature convergence due to its weak diversity and introduced the concept of inertia weight to overcome this issue [23]. Moreover, it is also stated that in CSO algo- rithm, positions of cats are updated by using the current
•
•
Vd+1 =Vd +β Pg −Xd +α∗ε (4) jnewj j
where, ε is a random vector uniformly distributed in the range [0, 1]; α and β are acceleration parameters used to direct the position of a cat toward local and global best positions and Pg presents the global best position of a cat
To make the balance between the exploration and exploitation processes, both of acceleration parameters α and β act as control parameters. The α parameter acts as decreasing function, whereas β parameter serves as an increasing function. In this work, the values of both the parameters are adaptive and computed using the following equations.
α −α
α(t)=αmax− max min ∗t (5)
tmax
In (5), αmax and αmin present the upper and lower lim- its, tmax denotes the maximum number of iterations and t denotes the current iteration number. Hence, α (t) is a step function whose value ranges between upper and lower limits. The larger value of α supports exploration whereas small values support exploitation. The aim of α (t) parameter is to control the exploration process of cats in search space.
πt
β(t)=βmin +(βmax −βmin)sin t (6)
max
In (6), βmin and βmax denote the minimum and max- imum values of first and last iterations respectively
Improved cat swarm optimization algorithm for solving global optimization problems… 2685
tmax presents the maximum number of iterations and t denotes the current iteration number. The reason behind the incorporation of β (t) parameter is to influence the global exploration ability of the proposed algorithm. A large value of parameter strengthens the global best position of cat and also tends to the solution refinement.
4.1 Local search method
In this subsection, a local search method is summarized to improve the quality of solution. The proposed method is applied to the tracing phase of ICSO algorithm. The aim of this method is to guide the search direction and achieve the optimum solution. The need for local search method can be summarized as follows:
• To guide the search direction and obtain the optimum solution in search space
• To overcome the local optima problem through neigh- borhoods information
This search mechanism is applied to the current global best solution (Xgbest), and the neighborhood of best solution can be defined using (7).
Neighborof X =X −r,X +rn (7) gb gbest gbest
where, “r” describes the boundary of neighborhood, Xgb presents the current best solution and can be denoted as Xgb (0), and n presents the number of population. The con- cept of neighborhood can be explained with the help of Fig. 1 as depicted below.
and d = 1,2,……,N. In each iteration, the best agent during the search can be selected using (9).
Xgb (k) = MIN x1gb(k), . . . xLgb(k), . . . , xngb(k) (9)
The above steps are repeated until a better solution is obtained. The procedure of the local search method is illustrated in Algorithm 1.
Algorithm 1 Local search method For L 1 : n
Initialize the variables 0 1
End For
While(termination criteria not met), do the following ForL 1:n
Determine candidate solution using (8).
End For
Pick local best candidate solution using (9). End While
4.2 Steps of ICSO algorithm
In this subsection, the main steps of the proposed CSO algorithm are summarized.
Search Space
In ICSO, the candidate solution is defined as
X1j (k),X2j (k),…….XLj (k),………,XNj (k) , where
k = 1,2,……,P. Here, p denotes the number of
iterations in the local search method. The local search
method can visit N number of population in every itera-
tion such as XLj (k)(L = 1,2,……,N). It is assumed
thatXLgbest(k) = x1j(k),…xLj(k),…,xNj(k) andis determined using (8).
X Lg b ( k − 1 ) d ̸ = L
XLgb (k) = XLgb (k−1)+r∗cxd d = L (8)
where, XLgb (k) denotes the global best position of cat in kth iteration where k = 1,2,……,P and L = 1,2,…,N;cxd can be computed using 1−cxd (k−1)
Illustration of the local search method incorporated into ICSO algorithm
Fig. 1
Neighborhood search radius (r)
Candidate X (i)
Objective function values
Current global best X (0)
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Steps of ICSO algorithm
are considered to evaluate the efficiency of ICSO algorithm; first is standard benchmark test functions and another is clustering problems. In order to test the performance of ICSO algorithm, twelve bench mark test functions are con- sidered. These are the well-known test functions reported in the literature and widely used to assess the performance of algorithms. In this work, both the separable and inseparable unimodal and multimodal functions are taken to examine the performance of ICSO algorithm. The detailed descrip- tion of these test functions is given in Table 1. The proposed algorithm is implemented in Matlab 2010a environment using windows 7 operating system, Intel corei5 processor, 3.4 GHz and 8 GB RAM. The results are taken on an aver- age of 30 independent runs for each function. Further, the results of the proposed algorithm are compared with several other meta-heuristic algorithms.
5.1 Results and discussion
To start the experiment, it is necessary to set the different parameter values. The parameters of ICSO are as follows: SMP=10,MR=0.5,C=2,β ∈ 0.1 ∼ 0.7,α ∈ 0.1 ∼ 0.5, the size of population is 100 and ε in the range [01]. These parameters are kept constant throughout the exe- cution of the program. The results are taken on average of 30 independent runs. The average and standard devi- ation parameters are considered as performance measures to evaluate the performances of algorithms. The average parameter indicates the efficiency of algorithms, whereas robustness is computed using the standard deviation param- eter. The performance of ICSO is also compared with several popular meta-heuristic algorithms like PSO, GA, ABC, BBO, BAFA, FPA, HS, DE, SFLP, TLBO, DE, and CSO.
The performance of the proposed algorithm is also com- pared with several variants of GA, DE, HS, SFLP, and TLBO algorithms. Apart from these, the performance of the proposed ICSO algorithm is also compared with some CSO variants like CSO, BCSO, AICSO, and EPCSO. The parameter configurations of above-mentioned algorithms are taken as reported in the corresponding literature [40–47]. Table 2 demonstrates the results of CSO, BCSO, AICSO, EPCSO, and ICSO algorithms with twelve benchmark func- tions. The average and standard deviation parameters are taken to evaluate the performances of these algorithms. It is observed that ICSO algorithm provides better performance with all of benchmark functions in comparison to other CSO variants. For F1 and F6 functions, the proposed algorithm obtains optimum minimum value i.e. 0. For the rest of the functions, it is also seen that ICSO algorithm achieves a minimum value lower than that of the CSO variants.
Table 3 illustrates the results of the proposed ICSO and several popular meta-heuristic algorithms such as PSO,
Step 1:
Step 2: Step 3:
Step 4: Step 5:
Step 6: Step 7:
Initialize the different parameters of the pro- posed algorithm like number of cats (N), SMP, SRD, neighborhood structure, β,α and C and randomly placed N number of cats in random space search.
Initialize position and velocity of each cat into the D-dimensional search space.
Compute the fitness function of cats and store the best position of cat into memory.
While(i < maximumiteration )
According to the value of Flag, randomly dis- tribute cats into tracing and seeking modes
If (Flag==1); Cat in seeking mode
For each cat, apply seeking mode process
Step 7.1: Make j copies of each cat.
Step 7.2: Compute shifting bit value for each cat using SRD.
Step 7.3: Add or subtract each cat to shifting value.
Step 7.4: Compute the fitness function for each new position of cats.
Step 7.5: Compare the value of fitness function and keep the best position of cat into memory. Step 7.6: End for
Else, Cat in tracing mode
For each cat, apply tracing mode process
Step 9.1: Update the velocity of each cat using (3).
Step 9.2: Update the position of each cat using (4).
Step 9.3: Compute the fitness function for newly generated position of cat.
Step 9.4: Compare the fitness function value and keep the best position of cat in memory.
Step 9.5: End for
Update the position of cats and also determine the best position of cat.
IF (rand(0, 1) ≤ Fiti ) then
Apply Local Search Method (Algorithm 1). Update the position of cats and global best position.
End if
i=i++
End while
Obtain the final solution.
Step 8: Step 9:
Step 10:
Step 11: Step 12: Step 13:
Step 14: Step15: Step 16: Step 17:
5 Experimental settings
This section describes the experimental results of the pro- posed CSO algorithm. In this work, two types of problems
Improved cat swarm optimization algorithm for solving global optimization problems...
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Table 1 List of test functions used for experimentation
No. Function name
F1 Sphere
F2 Rosenbrock
F3 Rastrigin
F4 Griewank
F5 Ackley
F6 Step
F7 Powell
F8 Schwefel
F9 Schaffer
F10 Zakharov’s
F11 Michalewicz
F12 Quartic
Definition
Parameter [− 100, 100]
[− 30, 30]
[− 5.12, 5.12] [− 600, 600]
[− 32, 32] [− 100, 100] [−4,5]
[− 500, 500] [− 100, 100]
[− 5, 10] [0,π]
[− 1.28, 1.28]
ICSO 0.00E+00
D 2
xi
D22 2
F1 (x) =
F2(x)=
1 2 xi F4(x)=4000xi−cos√i+1
i=1i=1
i=0 i=1
100 xi+1−xi +(xi−1) i=1D D
F3(x)=D x2i −10cos(2πxi)+10
F5(x)=20+e−20exp −0.2 D xi −exp D i=1
cos(2πxi)
F6 (x) =
D 2 (xi + 0.5)
i=0 D/4
+(x4i−2 + 10x4i−1)4 + 10(x4i−3 + 10x4i)4
F10 (x) = (x4i−3 + 10x4i−2 )2 + 5(x4i−1 − 10x4i )2 i=1
1 D 2 1 D
i=1
F8 (x) = 418.9828D −
D √ xi sin |xi |
F9(x)=0.5+ 1 2 2 1+0.001 x21−x2
i=1
sin2 x2 −x2 −0.5
D D 2 D 4 F10(x)= (xi)2+ 1 ixi + 1 ixi
F11 (x) = F12(x)=
22 i=0 i=0
D 220
i=0
AICSO
7.01E-05 1.31E-04 2.54E+01 1.33E+01 6.63E-02 5.45E-02 6.91E-02 3.17E-02 1.98E-01 1.47E-02 4.91E-05 3.86E-05 2.15E+02 2.15E+01 9.63E-03 3.57E-03 1.84E+02 9.63E+01 5.36E-13 1.08E-13 4.67E+00 3.35E+00 1.43E-05 1.32E-05
Sinx1 sin ixi D 4
i=1 i=1
π
ixi +rand(0,1) BCSO
Comparison of results of CSO and proposed CSO
Table 2
algorithms for benchmark functions (using D=30)
2.00E-04 2.00E-04 2.96E+01 1.84E+01 1.83E-01 1.10E-01 1.19E-01 1.38E-02 2.46E-01 1.56E-02 4.00E-04 2.00E-04 2.29E+02 8.92E+01 7.96E-02 2.46E-02 2.09E+02 1.96E+02 9.22E-13 5.32E-14
− 5.76E+00 4.16E-01 3.46E-05 2.63E-05
1.05E-05 1.48E-04 2.68E+01 1.49E+01 1.73E-01 1.12E-01 9.77E-02 6.79E-02 2.38E-01 2.01E-01 5.98E-05 9.74E-06 2.20E+02 6.55E+01 6.13E-02 5.80E-02 1.95E+02 6.90E+01 3.37E-13 1.42E-14 4.56E+00 9.12E-01 3.18E-05 1.77E-06
Function Parameters
F1 Average SD
F2 Average SD
F3 Average SD
F4 Average SD
F5 Average SD
F6 Average SD
F7 Average SD
F8 Average SD
F9 Average SD
F10 Average SD
F11 Average SD
F12 Average SD
CSO
EPCSO
4.42E-06 3.32E-05 2.15E+01 2.01E+01 1.64E-02 1.19E-01 3.76E-02 2.47E-02 3.78E-03 8.72E-02 8.46E-06 7.24E-06 1.05E+02 1.68E+01 5.35E-03 2.78E-03 1.65E+02 5.60E+01 8.56E-14 5.21E-14 7.30E-01 4.33E-01 3.09E-06 2.99E-06
−
0.00E+00 1.65E+01 1.35E+01 5.36E-03 4.09E-02 3.93E-04 2.66E-05 6.48E-05 4.28E-04 0.00E+00 0.00E+00 7.84E+01 4.16E+00 5.49E-05 1.94E-06 7.30E+01 1.63E+01 2.68E-16 4.24E-17 6.33E-02 4.96E−3 7.84E-08 3.41E-07
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Table 3 Comparison of ICSO and other meta-heuristic algorithms for standard benchmark functions (using D=30)
Function No. F1
F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
Parameters
Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD
PSO
0.0003 0.0015 25.8537 16.2698 0.3582 0.6975 0.1045 0.0462 0.521 0.0406 0.0004 0.0026 385.901 161.14 0.5126 0.0034 9.82E+03 6. 412E+02 6.14E-10 1.35E-11
− 2. 96847 0.65249 0.004357 0.000571
ABC
5.92E-04 5.33E-05 2.35E+01 1.36E+01 3.29E-01 4.25E-02 1.51E-02 1.24E-02 9.63E-02 7.86E-02 1.46E-02 3.53E-03 1.38E+02 7.64E+01 7.44E-01 5.47E-01 2.43E+02 3.61E+01 8.54E-14 6.84E-15
− 1.44E+01 3.75E-01 5.34E-04 2.36E-04
BA
1
1
38.9874 15.6428 0.4266
1
0.8205 0.0814
1
1
1
1
113.897 86.0515 0.08245 0.0183 2.76E+02 1.12E+02 9.83E-14 4.86E-14 − 4.538246 0.71593 0.0041326 0.007865
FPA
0.4283 0.0826 36.9146 17.578 0.8916 0.7538
1
0.0209 0.3168 0.0736 0.2764 0.1967 86.1969 80.7233 0.07916 0.03816 3.53E+02 1.26E+03 5.22E-14 4.80E-14 − 3.861285 0.83529 0.0038632 0.001547
FA
3.79E-02 5.35E-02 1.72E+02 1.30E+02 2.29E+01 7.14E+00 1.05E-01 5.45E-02 2.05E+00 3.57E-01 3.11E-01 1.18E-01 2.49E+03 2.36E+02 6.33E-02 1.82E-02 2.86E+02 4.50E+01 5.34E-11 3.96E-12
− 9.63E+00 2.42E-02 8.55E-04 2.13E-04
ICSO
0
0
16.4721 13.4716 0.0536 0.0409 0.0393 0.0266 0.0648 0.0428
0
0
78.3548 41.562 0.005493 0.1938 1.43E+02 3.25E+01 2.68E-16 4.24E-17 − 1.8123 4.96E−2 7.84E-06 3.41E-07
ABC, BA, FPA, and FA. It is seen that the proposed algo- rithm provides better results for most of benchmark test functions using average and SD parameters. It is stated that the ICSO have good exploration capability to search the optimum solutions. Tables 4 and 5 depict the results of ICSO algorithm and some recent meta-heuristic algorithms like BBO, BBBO, HS, MHS, HSDE, SFLP, MSFLA, TLBO, ITLBO, DE, HSDE, GA, QGA-MPC, and GA-MPC. It is observed that the performance of ICSO algorithm is bet- ter than other algorithms being compared with most of benchmark functions. It is stated that the proposed algo- rithm obtains highly precise results due to enhanced solution search equations and also explore solution search space effectively due to local search method.
Figure 2a–f shows the convergence behavior of CSO and ICSO algorithms for six benchmark functions, i.e., F1 –F6 . It is seen that the proposed ICSO algorithm provides better results in terms of cost and number of iterations. Further, it is observed that the ICSO algorithm converges faster than the CSO algorithm. It is also seen that the local search method improves the efficiency of ICSO algorithm. The problem of trapping in local optima in last iterations is also solved.
5.2 Statistical test
In order to illustrate the efficacy of the proposed ICSO algorithm, some statistical tests are also applied to vali- date the performance of proposed algorithm. In this work, Friedman test is applied for statistical analysis. The Fried- man test is a nonparametric statistical test. In Friedman test, two hypotheses are created, i.e., null hypothesis (H0) and hypothesis (H1 ). The null hypothesis (H0 ) stands for no significant difference occurs between the performances of proposed algorithm and other algorithms. In contrast, hypothesis (H1) claimed that a significant difference occurs between the performances of algorithms [12]. Another important parameter of Friedman test is p-value, which is a probability value that shows whether a statistical test is significant or not; the smaller the p-value, the stronger is the evidence against H0. Table 6 presents the results of Friedman test and average ranking of algorithms. It is seen that the proposed algorithm obtains minimum rank, i.e., 1 among all other algorithms, whereas CSO algorithm has the worst rank, i.e., 5. The degree of freedom is 4. The p-value obtained from Friedman test is 1.354E-6, which strongly rejects the null hypothesis (H0). It can be seen that
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Table 4
Function F1
F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
Table 5
Function F1
F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
Comparison of Parameter
Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD
Comparison of Parameter
Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD Average SD
ICSO and
BBO
other meta-heuristic BBBO
for standard benchmark functions (using D=30)
2.40E-03 4.56E-04 5.79E+00 1.27E+00 7.08E-02 5.56E-02 1.89E-01 3.36E-02 1.01E-01 5.13E-02 1.71E+00 3.71E-01 1.38E+02 1.14E+02 7.50E-01 3.86E-02 2.46E+02 6.47E+01 1.30E-03 1.85E-03
− 3.92E+00 1.31E+00 2.00E-05 3.14E-06
1.97E-4 2.56E-5 1.79E+01 5.63E+00 3.07E-03 6.20E-04 1.08E-01 2.68E-02 4.04E-02 4.24E-03 3.67E-01 1.66E-01 8.71E+01 1.16E+01 6.61E-01 7.09E-02 1.57E+02 3.01E+01 2.34E-10 4.28E-11
− 2.16E+00 6.11E-01 4.94E-06 1.03E-06
algorithms HS
5.22E-4 3.2945E-5 1.90E+02 5.16E+01 1.69E+01 2.66E+00 1.60E-01 5.34E-02 1.09E+00 1.36E-01 4.37E+00 9.83E-01 4.59E+02 5.86E+01 1.53E-01 3.10E-02 2.83E+02 1.06E+02 1.03E-11 1.31E-12
− 4.32E+00 1.06E+00 8.44E-04 2.21E-05
MHS
1.63E-4 2.18E-5 2.73E+01 1.33E+01 8.01E-01 8.30E-01 1.43E-01 2.93E-02 7.71E-14 2.37E-14 0.00E+00 0.00E+00 1.17E+02 3.40E+01 3.63E-02 4.80E-03 1.84E+02 7.42E+01 4.93E-13 6.11E-14
− 1.72E+00 5.16E-01 2.18E-44 6.08E-44
HSDE
6.08E-5 5.33E-5 5.33E+02 3.90E+02 0.00E+00 0.00E+00 1.22E-02 8.30E-03 4.70E+01 1.16E+01 1.59E-01 7.97E-01 9.47E+01 3.82E+01 8.96E-02 2.14E-02 1.65E+02 6.53E+01 1.53E-15 1.06E-15
− 2.42E+00 1.23E+00 2.27E-03 1.06E-03
SFLP
1.09E-3 2.90E-21 3.69E+01 2.23E+01 2.44E+01 5.36E+00 3.52E-02 3.79E-03 9.76E-03 5.21E-02 2.36E-06 3.29E-06 1.65E+02 4.38E+01 9.76E-01 5.03E-01 2.62E+02 1.12E+02 3.29E-12 1.30E-12
− 2.95E+00 4.93E-01 3.79E-03 7.45E-04
MSFLA
5.79E-5 3.88E-6 2.73E-02 1.94E-02 2.55E+01 1.03E+01 1.64E-02 1.99E-02 6.67E+01 1.15E+01 5.70E-01 1.88E-01 8.95E+01 2.95E+01 3.67E-03 3.18E-03 1.77E+02 5.97E+01 2.28E-15 5.16E-15
− 2.43E+00 9.28E-01 2.84E-03 1.53E-04
GA-MPC 5.60E+06
5.64E+05 1.59E+09 3.12E+08 1.55E+04 1.55E+03 1.39E+03 1.08E+02 2.09E+01 3.32E-02 1.47E-02 1.58E-02 2.83E+03 5.12E+02 9.63E-01 5.33E-01 1.32E+00 5.37E+01 5.13E-14 2.44E-14
− 5.86E+00 1.05E+00 5.34E-05 3.87E-05
ICSO
0
0
16.4721 13.4716 0.0536 0.0409 3.93E-02 2.66E-03 6.48E-02 0.0428
0
0 7.84E+01 4.16E+00 5.49E-03 1.94E-03 1.43E+02 3.25E+01 2.68E-16 4.24E-17 − 1.8123 4.96E−2 7.84E-06 3.41E-07
ICSO
0
0
16.4721 13.4716 0.0536 0.0409 0.0393 0.0266 0.0648 0.0428
0
0
78.3548 41.562 0.005493 0.1938 1.43E+02 3.25E+01 2.68E-16 4.24E-17 − 1.8123 4.96E−2 7.84E-06 3.41E-07
ICSO and other meta-heuristic
for standard benchmark functions (using D=30)
TLBO
0
0
26.6567 19.456 0.2178 0.06325
0
0 3.55E−01 0.0836 2.74E−02 5.36E−03 124.1484 2.60E+02 0.0066 4.50E-03 2.17E+02 1.26E+02 4.64E-14 2.34E-14 − 4.352678 1.44E-02 3.25E-04 1.59E-04
ITLBO
0
0
22.7934 14.823 0.1632
0
0
0 1.42E−01 0.05618 1.17E−4 4.68E−5 83.7532 1.35E+01 0.0053 3.20E-03 1.59E+02 6. 832E+01 3.46E-16 1.89E-16
− 3.56815 1.53E−2 2.19E-05 1.22E-05
algorithms
DE
1.35E+02 7.03E+00 5.87E+04 8.69E+03 5.17E+02 1.32E+01 1.03E+00 8.17E-02 3.02E+02 5.20E+00 1.15E-01 1.66E-01 1.91E+04 3.69E+03 1.09E-01 9.85E-02 1.29E+03 4.27E+02 2.60E-10 1.43E-11
− 1.80E+01 4.20E-01 6.64E-03 4.72E-03
HSDE
6.08E–4 8.33E-5 5.33E+02 3.90E+02 0.00E+00 0.00E+00 1.22E-02 8.30E-03 4.70E+01 1.16E+01 1.59E-01 7.97E-01 9.47E+01 3.82E+01 8.96E-02 2.14E-02 1.65E+02 6.53E+01 1.53E-15 1.06E-15
− 2.42E+00 1.23E+00 2.27E-03 1.06E-03
GA
0.8378 0.5138 45.2843 21.6284
1
0.6921 0.8613 0.0648 0.7934 0.3216 0.7872 0.5643 830.075 393.68 0.5397 0.0073 4.62E+04 1.59E+03 5.50E-10 7.62E-10 − 5.86435 0.91578 0.235416 0.067134
QGA-MPC 1.43E+01
2.25E+00 1.54E+04 1.36E+04 5.19E+02 3.52E+01 9.56E-01 8.53E-02 2.12E+01 3.32E-02 9.08E-03 3.77E-02 1.65E+04 5.09E+02 7.83E-02 2.88E-02 1.74E+02 2.14E+01 3.27E-15 2.33E-16
− 3.60E+00 6.72E-01 4.56E-05 2.90E-05
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10
0 10
-2 10
-4 10
10-6
CSO
ICSO
10
10
-2
10
CSO ICSO
Ackley
22
Griewank
0
101
0 10
-1 10
-2 10
-3 10 0
5 10
100
-5 10
-10 10
CSO
ICSO
500
CSO ICSO
5
10
4
10
3
10
2
10
CSO
ICSO
400 500
CSO
ICSO
0 100 200 300 400 500
Iteration
(a)
Rastrigin
10-4
106
0 100 200 300 400 500 Iteration
(b)
Rosenbrock
100
200
300
Iteration
(c)
Sphere
400
1 10 0
100
200
300
2 10
100
-2
10
-4
10
Iteration
(d)
Step
Cost Cost Cost
Cost Cost Cost
0 100 200 300 400 500
Iteration
(e)
0 100 200 300 400 500
Iteration
(f)
Fig. 2 a–f shows the convergence of CSO and ICSO algorithms for functions F1 – F6
a significant difference occurs between the performances of algorithms. Multiple comparisons (N × N) of all algo- rithms are also carried out for benchmark test functions. Table 7 presents the results of (N × N) comparisons with adjusted p-values and test statistics. It is noticed that the proposed ICSO algorithm obtains minimum p-values than
other algorithms. Overall, it is concluded that the proposed algorithm is an effective and efficient algorithm for solv- ing the complex benchmark test functions than other CSO variants.
Further, Wilcoxon signed-ranks test is applied on the two best-performing algorithms. The two best-performing
Improved cat swarm optimization algorithm for solving global optimization problems... Table 6 Friedman test
2691
Mean of rank
5 3.75 3.25 2
1
statistics for all algorithm using benchmark test functions
Algorithm
CSO
BCSO
AICSO EPCSO Proposed ICSO
Functions
F1 F2 F3
555 344 433 222 1 1 1
Overall rank
60 45 39 24 12
Number of observation: 60 Sum of squares of ranks : 7462 Degree of freedom: 4
Number of algorithms: 5 Friedman test statistic: 32.73 Critical value: 9.4877
F4F5F6F7F8F9F10
5555555 4444443 3333334
F11 F12
5 5 3 4 4 3 2 2 1 1
222222 111111
Number of problems: 12 Correction factor: 540 p-value: 1.354E-6
2 1
Friedman Test Hypothesis: (i) H0: Algorithms are equally effective. (ii) H1: There is a significant difference in the performances of algorithms. (significance level = 0.05)
algorithms are EPCSO and proposed ICSO. Wilcoxon signed-ranks test is a non-parametric test applied in hypo- thetical testing involving two samples [13]. It is a pair- wise test that can be used to detect significant differences between the performance of EPCSO and ICSO algorithms. This test is closely related to t-test. The results of this test are mentioned in Tables 8-9. Table 8 provides the details of rank and signed rank using Wilcoxon test, whereas Table 9 shows the statistics of test. The p-value obtained for this test is 0.016113, and the median values for EPCSO and ICSO algorithms are 0.0045653and 5.985E-05, respectively. The statistics value (z) for Wilcoxon test is − 2.35. From the sta- tistical results, it is stated that the hypothesis strongly rejects the null hypothesis at the significance level of 0.05 and a significant difference occurs between the median values of EPCSO and ICSO algorithms.
6 Application of ICSO in data clustering
Data clustering is a powerful data analysis technique that can be used to group the data objects in the form of clusters using distance based similarity function [6]. The similarity function is used to compute the distance between data objects and cluster centers. The data clus- tering can be formulated as follows. Consider S = {s1,s2,s3 ......sn} is a set of n objects and Tn×m is a data matrix consisting of n rows and p columns, where n cor- responds to data objects and m represents the attributes or dimensions of a given dataset. In the given dataset, each ith data object is described by using a real valued m-dimensional vector xi {i = 1,2,3,......n} where each object xij denotes the jth feature (j = 1, 2, 3, . . . .m) of the
Table 7 Adjusted p-values and
Algorithm p value
CSO
BCSO 2.02E-02 AICSO 0.000143 EPCSO 4.81E-08 Proposed ICSO 3.29E-11
EPCSO
CSO 4.81E-08 BCSO 0.000143 AICSO 0.020159 Proposed ICSO 3.37E-02
ith object (i = 1, 2, 3, . . . .n). The goal test statistics for multiple comparisons (N × N comparison) among all algorithms
of the clustering
p value TS
0.000143 4.1642 8.65E-02 1.7533 0.020159 2.4109 3.54E-05 4.6025
TS Algorithm
BCSO
2.4109 CSO
4.1642 AICSO
6.5751 EPCSO
8.7667 Proposed ICSO
Proposed ICSO
6.58E+00 CSO 4.1642 BCSO 2.4109 AICSO 2.19E+00 EPCSO
p value
0.020159 0.086507 0.000143 1.01E-07
3.29E-11 1.01E-07 3.54E-05 0.033739
TS Algorithm
AICSO
2.4109 CSO
1.7533 BCSO
4.1642 EPCSO
6.3559 Proposed ICSO
8.7667 6.3559 4.6025 2.1917
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Table 8 Ranks of each function using Wilcoxon test for CSO and ICSO algorithms
Function CSO
ICSO D 0 −
Abs(D) Rank Signed-Rank 4.42E-06 3 −3
5.02 10 −10 0.01 7 −7 0.04 8 −8 3.72E-03 5 −5 8.46E-06 4 −4 26.32 11 −11 0.01 6 −6 92 12 −12 8.53E-14 1 −1 0.67 9 9 3.01E-06 2 −2
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11 − F12
4.42E-06 21.49 0.0164 0.0376 0.0037813 8.46E-06 104.72 0.0053493 165 8.56E-14 0.73 3.09E-06
4.42E-06 5.02 0.01 0.04 3.72E-03 8.46E-06 26.32 0.01
−
16.4721 −
0.00536 − 0.000393 − 6.48E-05 −
0 −
78.4 − 5.49E-05 −
73 −92 2.68E-16 − 8.53E-14 0.06328 0.67 7.84E-08 − 3.01E-06
problem is to compute a set of optimal partitions z = {z1,z2,.......zk} for the given data matrix Tn×m and can satisfy the following conditions.
i. ∀i̸=jzi∩zj=∅, ii. ki=1 zi = x,
iii. ∀izi=∅
It is observed that the Euclidean distance can be used as sim- ilarity function to compute the set of optimal partitions. It can be defined as the distance between data objects and the cluster centers. It is computed for every cluster center and assigned to all data objects. The data is clustered on the basis of minimum Euclidean distance. The objective function for clustering problems can be described as follows.
minimum distance. After partitioning the data into differ- ent clusters, a fitness function is applied to determine the goodness of the partitions. In this work, sum of square error (SSE) based fitness function is adopted to determine the fit- ness of each cluster. This fitness function is described in (11).
K S S E ( C )
F(Ck)= k (11)
Further, the proposed algorithm is also applied to solve the clustering problems. The main motive of the ICSO algorithm is to find optimal set of cluster for clustering problems. To evaluate the performance of the proposed algorithm, some benchmark datasets are taken from UCI repository. The results of the ICSO algorithm is also com- pared with well-known clustering algorithms reported in the literature.
6.1 Steps of ICSO algorithm for clustering
This subsection describes the algorithmic steps of the ICSO algorithm for solving clustering problems The main steps of ICSO algorithm are as follows.
K 2
min ∥Xi − Ck∥ (10)
minimize F(X, C) =
In (10), Xi denotes the ith data object, Ck represents the kth
k=1 x∈Di
and data objects are assigned to clusters according to the
Table 9 Results of Wilcoxon test (paired samples) on CSO and ICSO algorithms
K SSE(CK ) k∈1 k=1
Test Statistics
CSO
ICSO
Mean
Median
Significant value
p-value
Sum of signed Rank (W)
E (W)
SD (W)
Z
Hypothesis- H0: Median 1 is equal to Median 2; H1:
Median 1 is not equal to Median 2.
Hypothesis (H0) is rejected at the significance level 0.05
24.21 0.0045653 0.05 0.016113 60
13.98 5.985E-05
0.5 25.495098 0.019607
Dataset Cluster (K)
Iris 3 Wine 3 CMC 3 Cancer 2 Glass 6
Features Total Data Items
4 150 13 178 9 1473 9 683 9 214
Data in each cluster
(50, 50, 50)
(59, 71, 48) (629,334, 510)
(444, 239)
(70,17, 76, 13, 9, 29)
Table 10
Description of datasets used for experiment
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Table 11
Dataset
Iris
Cancer
CMC
Wine
Glass
Comparison of performance of the proposed ICSO and other
clustering algorithms
Parameters
Best Case
Avg. Case
Worst Case Standard Deviation F-Measure
Best Case
Avg. Case
Worst Case Standard Deviation F-Measure
Best Case
Avg. Case
Worst Case Standard Deviation F-Measure
Best Case
Avg. Case
Worst Case Standard Deviation F-Measure
Best Case
Avg. Case
Worst Case Standard Deviation F-Measure
Algorithms
K-means PSO
97.12 96.48 112.44 98.56 122.46 99.67 15.326 0.467 0.781 0.78 2989.46 2978.68 3248.25 3116.64 3566.94 3358.43 256.58 107.14 0.832 0.826 5828.25 5792.48 5903.82 5846.63 5974.46 5936.14 49.62 48.86 0.337 0.333 16768.18 16483.61 18061.24 16417.47 18764.49 16594.26 796.13 88.27 0.519 0.516 222.43 264.56 246.51 278.71 258.38 283.52 18.32 8.59 0.426 0.412
ACO CSO TLBO
96.89 96.94 96.56 98.28 97.86 96.84 99.34 98.58 98.08 0.426 0.392 0.546 0.778 0.781 0.782 2983.49 2985.16 2876.28 3178.09 3124.15 3084.74 3292.41 3443.56 3215.83 93.45 128.46 42.11 0.829 0.831 0.834 5756.42 5712.78 5778.61 5831.25 5804.52 5836.25 5929.36 5921.28 5921.32 44.34 43.29 38.96 0.332 0.334 0.331 16448.35 16431.76 16578.42 16530.53 16395.18 16360.04 16616.36 16589.54 16917.26 48.86 62.41 56.14 0.522 0.521 0.52 273.22 256.53 246.89 281.46 264.44 256.44 286.08 282.27 287.52 6.58 15.43 15.29 0.402 0.416 0.422
ICSO
95.78 97.05 97.63 0.213 0.784 2943.24 3036.41 3253.56 63.04 0.835 5654.11 5761.48 5896.25 45.16 0.339 16296.44 16342.21 16483.62 36.06 0.526 249.25 268.45 281.73 10.34 0.427
6.2 Performance evaluation
compared Figs. 3 and 4 show the clustering on iris data using ICSO algorithm.
To investigate the performance of the proposed algorithm, the results of ICSObased clustering algorithm are compared with well-known clustering algorithms such as K-Means, GA, PSO, TLBO, ACO and CSO algorithms [38, 48–51] Five real-life datasets are taken from the UCI database to evaluate the performance of all abovementioned algorithms. Table 10 presents the details of these datasets. The qual- ity of clusters is measured in terms of intra cluster distance (best, average and worst), standard deviation and f-measure parameters. Large values of these measures are required for better clustering. The parameters of the algorithms are the same as reported in the literature. Table 11 shows the com- parison between the proposed ICSO algorithm and the other techniques in terms of cluster-quality measures. The results are compared in terms of means and standard deviation over 1 independent runs in each case. For iris, cancer, CMC, wine and glass datasets, it is found that the ICSO algo- rithm provides better results than the other algorithms being
Iris Dataset
7 6 5 4 3 2 1
0 0.5 1 1.5 2 2.5 petal width
Clas Clas Clas
s Setosa
s Versicolour s Virginica
Fig. 3 Clustering of iris dataset using ICSO algorithm (2D View)
sepal width
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Iris Dataset
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8
6
4
2
0 8
Fig. 4
7 Conclusion
In this paper, an improved version of CSO algorithm is proposed using improved solution search equations for exploring optimal solution. In addition, two parameters are also introduced for better tradeoff between exploration and exploitation processes. Further, a local search method is also incorporated in the CSO algorithm for avoiding premature convergence problem and handling local optima problem. These improvements in CSO algorithm also improve the precision of solutions. The performance of the proposed algorithm is evaluated on twelve benchmark test functions and real life clustering problems. It is observed that the proposed algorithm gives better results in comparison to other existing algorithms. The simulation results demon- strate the ability and efficacy of the proposed algorithm for solving benchmark test functions. Moreover, the proposed algorithm is also implemented to solve some real life prob- lems. Clustering is one of the reallife problems in which determining the optimum cluster center is a tough task. In clustering problems, the proposed ICSO algorithm is used to determine optimum cluster centers. It is observed that ICSO algorithm provides stateof-the-art results for cluster- ing problems compared to other algorithms. The proposed ICSO algorithm performs better clustering in terms of qual- ity of clusters.
References
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Class Setosa Class Versicolour Class Virginica
73
6
sepal width
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Dr. Yugal Kumar is presently working as Assistant Profes- sor (Senior Grade) in Depart- ment of Computer Science & Engineering at Jaypee Uni- versity of Information Tech- nology (JUIT), Waknaghat, Himachal Pradesh, India. He has more than 10 years of teaching and research experi- ence at reputed colleges and universities of India. He has completed his Ph.D. in Com- puter Science & Engineering from Birla institute of Tech- nology, Mesra, Ranchi. His
primary area of research includes meta-heuristic algorithms, data clustering, swarm intelligence, pattern recognition, medical data inter- national journals and conferences of repute. He is serving as editorial review board member of various journals including Soft Computing, Neurocomputing, Computer Methods and Programs in Biomedicine, PLOSE ONE, Journal of Advanced Computational Intelligence and Intelligent Informatics and Journal of Information Processing System.
Dr. Pradeep Kumar Singh
is currently working as Assis- tant Professor (Senior Grade) in Department of Computer Science & Engineering at Jaypee University of Informa- tion Technology (JUIT), Wak- naghat, H.P. He is having 10 years of vast experience in academics at reputed colleges and universities of India. He has completed his Ph.D. in Computer Science & Engi- neering from Gautam Bud- dha University (State Gov- ernment University), Greater
Noida, UP, India. He is Senior Member of ACM Society. Dr. Singh is having life membership of Computer Society of India (CSI) as Senior Member Grade. He has worked as publicity chair of five IEEE Inter- national Conferences and associated as TPC member & reviewer of various Conferences & Journals too. He is Associate Editor of Inter- national Journal of Information Security and Cybercrime (IJISC) a scientific peer reviewed journal from Romania. He is currently guid- ing four Ph.D. scholars and having 176 Google scholar citations in his account. He has published 35 research papers in various International Journals and Conferences of repute. He is editorial review board mem- ber of various Scopus and SCI Journals including IEEE Transactions on Industrial Informatics. IJISP, IJSSE from IGI Global, USA. He has two sponsored research project from HPSCSTE & HIMCOSTE, HP Govt., INDIA.