An optimization algorithm based on behavior of see-see partridge chicks

Abstract

Nature is a huge source of inspiration for solving difficult and complex problems in science. Nature-based problem solvers always find an optimal solution (whether local or global) to any given problem; so they are sometimes considered as black-box problem solvers. Meta-heuristic algorithms that are inspired by nature (through imitating the nature) have opened a new approach to solving optimization problems. In the past decades, numerous research efforts have been focused in this specific area. In this paper, an optimization algorithm inspired by the nature has been introduced which is modeled from the behavior of the chicks of a type of bird called see-see partridge. We name the algorithm SSPCO. SSPCO optimization algorithm is an algorithm similar to the algorithm of particle swarm optimization (PSO) that the motion equation and the variables’ velocity are modeled from the behavior of this type of bird. Simulation of the algorithm was done with MATLAB software and the results of the 6 primary known algorithms and 8 advanced ones were compared with 14 unconstrained continuous functions. The results indicate that the proposed algorithm is an efficient algorithm comparing the others.

Keywords

Optimization see-see partridge chick SSPCO

1 Introduction

The optimization problem has been accompanied with the development of human existence; it is a very old problem, and the main idea is that from a practical point of view. Some specific problems from a set of candidate solutions to choose a most suitable are to achieve the optimal solution of the objective [1]. However, they can be transformed into each other. Optimization problem permeates all aspects of human life, and it widely exists in various fields [2]. There are some problems, namely, how to choose a reasonable plan to make the output maximization, the maximum output and profits. In resource allocation, distribution is to make the economic benefit obtained the best, which can meet different requirements of different aspects [3]. Almost all meta-heuristics share the following characteristics: they are nature-inspired (based on some principles from physics, biology or ethology); they make use of stochastic components (involving random variables); they do not use the gradient or Hessian matrix of the objective function; they have several parameters that need to be fitted to the problem at hand [41]. It literally means to find the best desirable possible solution. Optimization problems are in a wide range. Therefore, there should be solutions for these problems. For this reason, the subject of the research is active. It is an incentive to use random optimization algorithms inspired by nature as an alternative to deterministic computational methods [5]. Science has shown that the movement of living beings is dome based on a coherent social philosophy that accuracy in these behaviors can be useful to achieve a solution to many complex problems. The proposed algorithm in this paper is a type of meta-heuristic algorithm based on swarm intelligence inspired by nature. The algorithm is implemented in known continuous functions and its results were compared with 6 known optimization algorithms. This algorithm is a basic algorithm based on behavior of a bird, with a new and simple mechanism. SSPCO algorithm is similar to PSO algorithm, with this difference that in PSO algorithm [13], Particle motion is done based on two factors: local optimum, global optimum. But in SSPCO algorithm, Particle motion is done based on only a particle. This work will reduce the computing load and also will reduces particle memory load. This algorithm includes a new velocity equation, a parameter with priority name for the formation a queue for particles moving. The speed of this algorithm for to achieve optimal solution is very higher than similar algorithm, that is PSO algorithm. On this basis, in this paper, previous works have been introduced in Section 2 and the introduced algorithm was compared with all these works in the results section. In Section 3, the proposed algorithm has been introduced and described in detail and with all implementation details. In Section 4, the simulation results are presented that the simulation results are expressed in different phases and to analyze the performance of the algorithm and in Section 5, conclusions are provided.

2 Related works

In recent years, efforts were made to optimize swarm based [21]. The Genetic Algorithm (GA) is arguably the most well-known and mostly used evolutionary computation technique. It was originally developed in the early 1970 s at the University of Michigan by John Holland and his students, whose re-search interests were devoted to the study of adaptive systems [8]. A genetic algorithm is among one of the most successful evolutionary algorithms which were inspired by the evolution of nature. However, due to its outstanding performance in optimization, GA is considered as a function optimizer [5]. Similar to GA, Evolution Strategy (ES) imitates the principles of natural evolution as a method to solve optimization problems [22]. It was introduced in the 60ies by Rechenberg [36, 37]. Evolutionary Programming (EP) was first presented in the 1960s by L.J. Fogel as an evolutionary approach to artificial intelligence [38]. The Genetic Programming (GP) is an automated d method for creating a working computer program from a high-level problem statement of ‘what needs to be done’ [39]. Differential Evolution (DE) algorithm is one of the most popular algorithm for the continuo us global optimization problems. It was proposed by Storn and Price in the 90’s [40] in order to solve the Chebyshev polynomial fitting problem and has proven to be a very reliable optimization strategy for many different tasks. Particle Swarm Optimization (PSO) was initially introduced in 1995 by James Kennedy and Russell Eberhart as a global optimization technique [11]. In the particle swarm optimization, the particle is the population member which has lower mass and volume (small random mass) and it is a concept which refers to a state better than the behavior. Each particle in swarm represents a solution in a high aspect space with four vectors which finds the best position from its position and adjusts the best found position by its neighbors and its velocity and its position in the search space based on the best position itself (Pbest) and best position reached by its neighbors (Gbest) during the search process. To refer to some applications of the algorithm, [6 , 20] can be referred which have used the SF method to calculate the path in the binary search space to solve backpacks (KP) [13]. Meanwhile, to solve some of the problems [14] has used the combined PSO by with leading local search and some concepts from the genetic algorithm (GA). Artificial bee colony algorithm is the dominant simulation of the foraging behavior of the intelligent swarm of bees which is proposed by [9]. Derived from the main method ABC was proposed by [4] for JSSP by changing the behavior of supervisor bees and using SF to convert continuous values to binary values. A hybrid artificial bee algorithm was provided by discrete encryption for TSP by [10]. This algorithm was proposed by [7]. Harmony search algorithm is a meta-heuristic algorithm which searches in the normal process of a music function to find a good condition during jazz improvisation [16]. Bees Algorithm (BA) was first used for continuous optimization functions. It was used for scheduling work [18] and clustering binary data [17]. The imperialist competitive algorithm is an evolutionary optimization algorithm which is inspired by the imperialist competition [12, 15]. Evolution Strategies [42] was designed to solve technical optimization problem [12]. Students of University of Berlin, Rechenberg, Schwefel and Bienert did experiments to randomly change the criterion which are defining the shape of the bodies by taking the idea from mutation. In this whole process, evolutionary strategy was developed. Ant colony optimization algorithm takes its inspiration from the real world of ant colonies to solve optimization problems. It was first introduced by Marco Dorigo in 1992 [43], originally was applied to travelling salesman problem and then applied later to various hard optimization problems.

3 Proposed SSPCO optimization algorithm

The basic idea of this optimization algorithm is taken from the behavior of the chicks of a type of bird called See-see partridge. The chicks of this type of bird are located in a regular queue at the time of danger to reach a safe place and they start to move behind their mother to reach a safe point.

To simulate the behavior of the chicks of this bird in the form of an optimization algorithm, each chick is considered as a particle of the suboptimal problem. The state of each particle should be according to the behavior of this type of chicks in a regular queue that we know this queue takes us to the best optimal point and this does not mean that minimizing the search space, but also, it is converging particles after some searches in a regular queue to the best point answers (bird mother). According to Fig. 1, each chick in the search space seeks to find a chick with the priority of a unit higher than itself and it tries to adjust its motion equation based on this chick.

Fig.1

Chicks motion in proposed algorithm.

The value of priority variable is a number that causes the particles move in a regular convergence line to the global optimum after some moving in the search space.

According to Fig. 2, particle t for going to new position, X_t + 1, its velocity equation according to your previous velocity, V_t, and position of particle f its priority valuable is one until more than that of the particle. In each iteration of the algorithm, the particle that has a higher priority is located to be the base of other particles and particles adjust their movement based on these particles with higher priority and this automatically causes that the particle with a good optimum has a higher priority in each iteration and finally, the particle which is at the beginning of the line to the optimum solution will be the mother bird which has the best cost for the algorithm. In fact, the particle that has the best cost is the mother bird. We consider a variable for each particle entitled as priority variable as it has been presented in Fig. 3. For particle i, priority variable defined according to Equation 1: $i_{priority}$ (1)

Fig.2

Particles motion in proposed algorithm.

Fig.3

Sorted array of chicks priority.

In every assessment, when a particle was better than the best personal experience or local optimum; a unit is added to the priority variable of that particle:

$\begin{matrix} if i_{cost} > {pb}_{i}^{t} \to {pb}_{i}^{t} = p_{i}^{t} and i_{priority} \\ = i_{priority} + 1 \end{matrix}$ (2)

X_i.cost The cost of each particle in the benchmark, ${pb}_{i}^{t}$ is the best personal experience of each particle, and $p_{i}^{t}$ is the location of each particle. In every time of assessment, if the local optimum is better than the global optimum and vice versa, the particle’s priority variable goes higher and a unit is added to it:

$\begin{matrix} if {pb}_{i}^{t} > {gb}^{t} \to {gb}^{t} \\ = {pb}_{i}^{t} and i_{priority} = i_{priority} + 1 \end{matrix}$ (3)

gb^t is the global optimum. The motion equation of each particle is set almost similar to the particle swarm algorithm in the form of Equation 4: $p_{i}^{t + 1} = p_{i}^{t} + v_{i}^{t + 1}$ (4)

$v_{i}^{t + 1}$ is the velocity of each particle or chick. Then, Chickens sorted in array based on priority variable.

Now the particle velocity equation is calculated according to the Equation 5:

$\begin{matrix} v_{i}^{t + 1} & = & w * v_{i}^{t} + c * rand \\ * [p ({chick}_{(i + 1)_{priority}}) - p_{i}^{t}] \end{matrix}$ (5)

$v_{i}^{t}$ is the velocity of the particle, w is the coefficient impact of the previous velocity in the current velocity equation of particle, c is the coefficient impact of position of particle with upper priority in the current velocity equation of particle, rand () is a random number between 0 and one to create a random movement for particles, p (chick (_{i+1_priority}) is the location of the particle with one level higher priority than the current particle that the current particle tries to adjust its velocity according to the particle, $p_{i}^{t}$ is the current location of the particle. It can be seen that, according to Equation 5, each particle adjusts its movement based on a particle with one level higher priority. In this way, it does not matter the local and global optimums and at any point in time, it only moves to find a particle which is a unit ahead of that particle and for this reason, the calculation number, and time in this algorithm has a great benefit than the previous optimization algorithm. You can see the pseudo code of the algorithm in Fig. 4. Also the flowchart of the proposed algorithm has been presented in Fig. 5.

Fig.4

Pseudo code of SSPCO algorithm.

Fig.5

Flowchart of proposed algorithm.

4 Simulation results

Simulation of the algorithms was done in MATLAB software and in accordance with the parameters which were defined in Table 1. The standard form of algorithms is used for compare. Comparison of the algorithm was done in Table 2 based on the best cost with the introduced algorithm in the previous works. The comparison of the proposed algorithm was done with 6 optimization algorithm on 14 static benchmarks and the results have been obtained out of 30 runs. To have accurate results and the random results do not have a negative impact on output, we run the algorithm 30 times and then, we select the average of these 30 runs as the final result. Adjusting the parameters of the algorithm was in such a way that the results should be done in an equal condition and no algorithm has any advantage than others in term of the parameter. The results were based on three criteria; first, the best cost of each algorithm to reach the best answer of the benchmark has been shown. This is the best cost in the 100th iteration of each algorithm. In the next step, the standard deviation of the particles’ cost is shown in each iteration and finally, the standard deviation of 30 times running is shown that indicates the answers are much different in 30 runs of each algorithm. Variables and parameters are considered identical for algorithms. In the following, initially, the best cost diagram of each function is shown separately in 100 times running and then, the exact values have been shown in a table. Then, the standard deviation diagram of the cost of each algorithm is shown and finally, the analysis of the results is presented. The number of assessments for all algorithms is 100. The number of population is 100 and the number of variables is also 100.

Table 1
Parameter setting

Item No. Parameter Value

1 Population size 100

2 Variable 100

3 Xlow –10

4 Xhigh +10

5 C factor 2

6 Iteration 100

7 crossover 0.5

8 Mutation 0.01

10 New harmony memory size 100

11 Harmony memory 0.75

12 Pitch adjustment rate 0.05

13 Fret width 0.1

14 Number imperialist 10

Item No.	Parameter	Value
1	Population size	100
2	Variable	100
3	Xlow	–10
4	Xhigh	+10
5	C factor	2
6	Iteration	100
7	crossover	0.5
8	Mutation	0.01
10	New harmony memory size	100
11	Harmony memory	0.75
12	Pitch adjustment rate	0.05
13	Fret width	0.1
14	Number imperialist	10

Table 2

Experimental results of PSO, GA, ABC, BA, HS, ICA and SSPCO over 30 independent runs on 14 test functions of 100 variables with 100 FES

Function	Result	PSO	GA	ABC	BA	HS	ICA	SSPCO
F1	Best Cost	26.41^‡	92.65^‡	924.27^‡	191.29^‡	945.55^‡	315.16^‡	3.93e-05
	Mean	116.97	237.58	2658.62	3330.31	3333.38	1400.69	124.77
	STD	261.27	497.74	409.25	297.81	294.77	672.73	288.27
	Run Time	6.30	1.04	5.27	9.46	69.45	2.04	6.01
F2	Best Cost	205.85^‡	219.31^‡	4212.61^‡	2166.27^‡	5035.52^‡	2498.66^‡	1.19 e-05
	Mean	3610.68	5561.85	30033.16	158842.06	168897.98	3809.22	2627.30
	STD	16503.74	23646.75	16798.83	160117.37	178617.72	1303.33	5968.96
	Run Time	27.09	11.34	25.08	106.04	108.39	21.53	28.32
F3	Best Cost	229581^‡	2887569^‡	38653771^‡	848514^‡	7944091^‡	1180717^‡	1.20
	Mean	6254620	13233917	185444768	251509359	256545029	32391919	8834715
	STD	17179956	38497141	35902740	60397066	60758442	30608027	17044818
	Run Time	12.85	4.53	14.73	38.33	110.76	5.70	12.89
F4	Best Cost	473.13^‡	160.38^‡	8040.33^‡	3221.37^‡	4645.07^‡	3728.49^‡	5.33 e-05
	Mean	2967.37	6380.46	77228.14	216888.28	238289.25	6545.48	2703.78
	STD	10531.40	26471.95	40322.37	240474.74	279534.85	1016.66	6698.88
	Run Time	25.01	10.26	23.73	110.22	123.65	24.44	26.16
F5	Best Cost	1.89^‡	2.14^‡	9.17^‡	7.50^‡	7.69^‡	8.59^‡	0.00017
	Mean	2.60	2.92	9.89	9.89	9.90	9.07	1.39
	STD	1.22	1.82	0.10	0.11	0.08	0.20	1.95
	Run Time	7.40	1.71	6.53	17.83	82.57	3.26	6.74
	‡	5	5	5	5	5	5
	†	0	0	0	0	0	0
	§	0	0	0	0	0	0
F6	Best Cost	7312.05^‡	28652.70^‡	5780584.9^‡	136091.47^‡	2884928.2^‡	409932.53^‡	99.00
	Mean	173389.2	654985.6	14845029	20156460	20149319	6375948.3	221636.9
	STD	920134.0	2692011	3099406	2667490	2716132	3613297	980107.8
	Run Time	6.06	0.97	4.95	10.22	70.59	1.76	5.74
F7	Best Cost	0.41^‡	0.63^‡	1.30^‡	1.05^‡	1.24^‡	1.08^‡	6.68e-08
	Mean	0.66	0.90	1.66	1.83	1.83	1.35	0.43
	STD	0.27	0.17	0.09	0.07	0.07	0.16	0.48
	Run Time	7.43	1.20	5.68	12.63	76.04	2.46	6.52
F8	Best Cost	4.50^‡	4.94^‡	11.89^‡	6.60^‡	10.79^‡	8.99^‡	0.0017
	Mean	5.21	5.79	14.33	15.39	15.39	12.62	3.30
	STD	1.69	2.12	0.61	0.33	0.33	1.11	3.37
	Run Time	6.52	1.19	5.13	13.13	68.12	2.45	6.23
F9	Best Cost	535.85^‡	621.44^‡	2271.52^‡	736.17^‡	1820.48^‡	1263.01^‡	0.012
	Mean	1015.22	997.72	3503.70	4335.84	4329.21	2528.08	771.56
	STD	419.14	583.42	457.35	309.91	305.16	556.33	665.56
	Run Time	6.24	1.01	4.95	11.45	72.87	1.97	5.68
F10	Best Cost	477.59^‡	622.81^‡	2179.64^‡	796.56^‡	1886.28^‡	1423.63^‡	8.41e-05
	Mean	955.75	988.61	3496.23	4321.29	4324.2	2415.03	655.19
	STD	447.00	588.10	435.53	312.74	296.78	505.96	626.74
	Run Time	7.63	1.58	6.17	16.54	70.72	3.19	7.16
F11	Best Cost	0.48^‡	0.59^‡	1.34^‡	1.03^‡	1.20^‡	1.07^‡	6.44e-09
	Mean	0.70	0.85	1.63	1.83	1.83	1.30	0.45
	STD	0.24	0.19	0.10	0.07	0.07	0.17	0.49
	Run Time	9.04	2.58	8.98	29.38	78.26	6.19	10.36
F12	Best Cost	18.00^‡	34.53^‡	22.54^‡	9.19^‡	29.11^‡	14.85^‡	5.15e-05
	Mean	30.01	42.11	39.48	49.13	48.75	26.85	36.92
	STD	13.48	3.17	6.96	3.25	2.80	7.30	27.23
	Run Time	42.64	19.10	40.91	192.98	141.16	37.46	42.64
	‡	7	7	7	7	7	7
	†	0	0	0	0	0	0
	§	0	0	0	0	0	0
F13	Best Cost	855.62^‡	10482^‡	230902350^‡	1082214^‡	17143008^‡	22333320^‡	45.99
	Mean	10000343	64048235	1855030148	285121961	290002582	571645970	24690423
	STD	87680349	344360683	699959131	579378814	560805949	457465377	173990551
	Run Time	32.69	14.60	31.77	145.75	117.30	27.85	32.00
F14	Best Cost	0.0019^‡	0.0097^‡	0.0097^‡	0.0023^‡	0.0097^‡	0.0097^‡	8.37e-07
	Mean	0.36	0.04	0.45	0.48	0.48	0.01	0.40
	STD	0.33	0.096	0.29	0.31	0.31	0.0039	0.34
	Run Time	6.53	1.03	5.49	11.61	76.82	1.90	6.35
	‡	2	2	2	2	2	2
	†	0	0	0	0	0	0
	§	0	0	0	0	0	0
	‡	14	14	14	14	14	14
	†	0	0	0	0	0	0
	§	0	0	0	0	0	0

“‡”, “†”, and “§” denote that the performance of SSPCO is better than, worse than, and similar to that of the corresponding algorithm, respectively.

Proposed algorithm to conduct the mother particle to the best answer and this increases the diversity and the proposed algorithm has a higher standard deviation than the other algorithms and the answer has a high variation in these algorithms while it can achieve the best answer in less time than other algorithms.

For example, in The results in F1 function show the high performance of the algorithm in finding the best answer in shortest possible time and the algorithm in the 100th iteration has the best cost of 3.93e-05 while, the best next algorithm in the 100th iteration has the best cost of 26.41. The algorithm has a great distance with other algorithms in finding the best answer and it could reach a good answer after the 80th iteration. In the F2 function, the results of the algorithms in achieving the best answer are similar to each other but the particle swarm algorithm and the proposed algorithm are better than others especially, the proposed algorithm which has the best possible cost in the 100th iteration. In F5 function, the algorithms almost have similar results and the proposed algorithm has a much better cost in the 100th iteration than the other algorithms and in the 100th iteration, it has the cost of 0.00017 while other algorithms have the best cost in the range of 1.89 to 9.17. In the standard deviation, the particles’ cost of the proposed algorithm has a good convergence and has a standard deviation almost similar to the other algorithms. The results presented in Fig. 6 show that in the mentioned function, the proposed algorithm has the best cost than other algorithms in the 100th iteration and the algorithm has a high velocity to achieve the answer and almost, it has a very good cost than other algorithms from the 30th iteration onwards. In the section of the standard deviation, according to Fig. 7, the algorithm has a high standard deviation that reflects the mission of the algorithm which shows all particles are in the service of this issue. For the test of the Dependence of results to parameter, in Tables 3 and 4 measured the best cost in all functions with different population and dimension parameter values.

Fig.6

Evolution of the mean function best cost derived from PSO, GA, ABC, BA, HS, ICA and SSPCO versus the number of FES on 14 test problems. (a) F1. (b) F2. (c) F3. (d) F4. (e) F5. (f) F6. (g) F7. (h) F8. (i) F9. (j) F10. (k) F11. (l) F12. (m) F13. (n) F14.

Fig.7

Evolution of the mean function standard deviation derived from PSO, GA, ABC, BA, HS, ICA and SSPCO versus the number of FES on 14 test problems. (a) F1. (b) F2. (c) F3. (d) F4. (e) F5. (f) F6. (g) F7. (h) F8. (i) F9. (j) F10. (k) F11. (l) F12. (m) F13. (n) F14.

Table 3

Experimental results of SSPCO over 30 independent runs on F1–F14 test functions based on different dimension sizes with 100 FES

Function	Result	N = 100	N = 300	N = 500	N = 700	N = 800	N = 900	N = 1000
F1	Best Cost	3.93e-05	4.02e-05^‡	5.15e-05^‡	7.19e-05^‡	8.67e-05^‡	9.81e-05^‡	9.96e-05^‡
	Mean	124.77	399.23	854.96	917.13	802.72	1.32e+03	1.37e+03
	STD	288.27	887.38	2.07e+03	1.98e+03	1.95e+03	2.78e+03	2.65e+03
	Run Time	6.01	6.33	6.35	6.22	6.27	6.50	6.76
F2	Best Cost	1.19 e-05	0.012^‡	0.016^‡	0.017^‡	0.018^‡	0.019^‡	0.020^‡
	Mean	2627.30	7.15e+03	6.41e+04	1.94e+05	2.49e+05	1.30e+05	2.73e+05
	STD	5968.96	1.46e+04	2.34e+05	4.64e+05	4.88e+05	2.43e+05	5.19e+05
	Run Time	28.32	75.96	143.59	230.82	268.47	324.98	378.18
F3	Best Cost	1.20	1.00^†	1.33^‡	2.46^‡	5.04^‡	5.67^‡	8.79^‡
	Mean	8834715	3.45e+07	5.58e+07	5.91e+07	7.05e+07	1.00e+08	8.90e+07
	STD	17044818	6.28e+07	1.23e+08	1.46e+08	1.58e+08	2.31e+08	1.81e+08
	Run Time	12.89	12.71	17.73	20.46	23.81	26.03	27.84
F4	Best Cost	5.33e-05	1.09e-04^‡	2.66e-04^‡	1.81e-04^‡	0.011^‡	0.034^‡	0.035^‡
	Mean	2703.78	1.48e+04	6.61e+04	1.24e+05	1.40e+05	2.22e+05	3.04e+05
	STD	6698.88	4.21e+04	1.91e+05	2.29e+05	3.19e+05	4.66e+05	4.49+05
	Run Time	26.16	69.83	169.55	225.31	268.56	313.04	372.35
F5	Best Cost	0.00017	0.00035^‡	0.00040^‡	0.00086^‡	0.00088^‡	0.00089^‡	0.00095^‡
	Mean	1.39	2.92	2.89	9.89	9.90	9.07	1.39
	STD	1.95	1.82	2.10	1.11	2.08	1.20	1.95
	Run Time	6.74	6.71	6.53	5.83	9.57	6.26	6.74
F6	Best Cost	99.00	298.98^‡	498.95^‡	698.97^‡	798.99^‡	898.99^‡	989.99^‡
	Mean	221636.9	8.15e+05	8.86e+05	2.81e+06	1.51e+06	1.98e+06	2.29e+06
	STD	980107.8	3.97e+06	4.25e+06	1.13e+07	7.60e+06	9.45e+06	8.46e+06
	Run Time	5.74	9.29	9.23	10.57	10.49	9.89	10.34
F7	Best Cost	6.68e-08	6.09e-08^†	7.46e-08^‡	8.01e-08^‡	8.65e-08^‡	9.00e-08^‡	9.66e-08^‡
	Mean	0.43	0.52	0.54	0.63	0.61	0.73	0.74
	STD	0.48	0.64	0.69	0.83	0.96	1.04	1.14
	Run Time	6.52	12.66	17.07	20.22	23.99	24.30	26.17
F8	Best Cost	0.0017	0.0015^‡	0.0049^‡	0.0024^‡	0.0023^‡	0.019^‡	0.016^‡
	Mean	3.30	3.61	3.75	3.47	3.59	3.61	3.87
	STD	3.37	3.50	3.43	3.33	3.45	3.31	2.80
	Run Time	6.23	8.49	9.25	9.41	9.51	10.41	10.77
F9	Best Cost	0.012	0.0067^†	0.011^†	0.0094^†	0.0024^†	0.0064^†	0.26^‡
	Mean	771.56	2.17e+03	4.38e+03	5.66e+03	5.91e+03	5.86e+03	8.23e+03
	STD	665.56	1.92e+03	3.57e+03	4.70e+03	5.83e+03	5.96e+03	6.80e+03
	Run Time	5.48	8.49	8.96	8.94	9.08	10.03	10.04
F10	Best Cost	8.41e-05	0.009^‡	0.0010^‡	0.0027^‡	0.0093^‡	0.10^‡	0. 12^‡
	Mean	655.19	2.85e+03	3.97e+03	4.70e+03	6.78e+03	6.26e+03	7.24e+03
	STD	626.74	2.02e+03	2.94e+03	4.68e+03	6.00e+03	6.01e+03	6.90e+03
	Run Time	7.16	9.07	8.45	8.98	9.61	9.74	9.43
F11	Best Cost	6.44e-09	6.99e-09^‡	7.24e-09^‡	0.0018^‡	0.0025^‡	0.0068^‡	0.0099^‡
	Mean	0.45	0.55	0.36	0.40	0.65	0. 73	0.75
	STD	0.49	0.41	0.53	0.69	0.43	0.51	0.39
	Run Time	10.36	12.36	12.45	12.66	13.38	14.00	16.26
F12	Best Cost	5.15e-05	6.17e-05^‡	7.13e-05^‡	7.15e-06^†	0.0013^‡	0.0019^‡	0.0029^‡
	Mean	36.92	56.12	46.62	38.02	66.88	45.32	32.33
	STD	27.23	54.21	57.73	71.24	34.71	63.35	55.20
	Run Time	42.24	44.25	48.64	54.88	66.87	69.36	70.00
F13	Best Cost	45.99	137.97^‡	229.96^‡	321.49^‡	367.93^‡	413.98^‡	459.91^‡
	Mean	24690423	5.35e+07	2.36e+07	5.56e+07	1.17e+08	2.89e+08	2.46e+08
	STD	173990551	3.70e+08	1.58e+08	4.95e+08	1.11e+09	1.96e+09	1.83e+09
	Run Time	32.00	90.75	139.97	194.18	217.71	260.66	277.59
F14	Best Cost	8.37e-07	0.0018^‡	0.0029^‡	0.0023^‡	0.0097^‡	0.0099^‡	0.01^‡
	Mean	0.40	150.61	165.19	174.12	145.65	144.02	166.00
	STD	0.34	91.51	141.63	152.00	162.32	169.35	198.32
	Run Time	6.35	121.45	177.80	178.98	189.36	192.25	199.35
	‡	11	13	12	13	13	14
	†	3	1	2	1	1	0
	§	0	0	0	0	0	0

Table 4

Experimental results of SSPCO based on different populations over 30 independent runs on F1–F14 test functions of 100 variables with 100 FES

Function	Result	N = 100	N = 300	N = 500	N = 700	N = 800	N = 900	N = 1000
F1	Best Cost	3.93e-05	1.44e-05^†	9.42e-06^†	2.05e-05^†	6.14e-05^‡	8.74e-07^†	8.30e-06^†
	Mean	124.77	107.23	124.86	104.74	148.40	146.16	113.30
	STD	288.27	214.87	277.69	253.11	353.26	334.78	244.29
	Run Time	6.01	19.86	30.60	45.63	51.62	56.52	62.63
F2	Best Cost	1.19 e-05	3.66e-07^†	0.0018^‡	2.08e-05^‡	1.76e-06^†	1.22e-05^‡	7.82e-06^†
	Mean	2627.30	1.24e+03	1.82e+03	1.95e+03	1.95e+03	1.84e+03	2.35e+03
	STD	5968.96	2.48e+03	3.34e+03	3.78e+03	4.20e+03	4.46e+03	6.58e+03
	Run Time	28.32	83.82	129.21	170.05	198.26	213.58	231.94
F3	Best Cost	1.20	5.87^‡	3.75^‡	1.40^‡	0.49^†	0.35^†	0.27^‡
	Mean	8834715	9.15e+06	7.52e+06	7.70e+06	7.99e+06	1.09e+07	7.84e+06
	STD	17044818	1.95e+07	1.39e+07	1.38e+07	1.44e+07	1.84e+07	1.54e+07
	Run Time	12.89	26.36	44.62	60.76	72.41	78.03	88.17
F4	Best Cost	5.33e-05	0.0014^‡	3.39e-04^‡	1.12e-05^†	1.41e-05^†	2.06e-05^†	4.65e-05^†
	Mean	2703.78	2.31e+03	2.94e+03	2.82e+03	1.60e+03	1.55e+03	1.98e+03
	STD	6698.88	8.67e+03	6.00e+03	5.59e+03	4.12e+03	2.88e+03	4.22e+03
	Run Time	26.16	91.19	170.18	181.22	238.13	226.68	326.23
F5	Best Cost	0.00017	0.00015^†	0.00010^†	0.0008^†	0.0007^†	0.0003^†	0.0001^†
	Mean	1.39	2.12	2.80	9.89	8.00	8.67	1.39
	STD	1.95	2.82	1.10	2.19	2.08	2.60	2.65
	Run Time	6.74	7.71	8.13	11.53	15.37	22.06	26.14
F6	Best Cost	99.00	98.99^‡	98.96^‡	98.98^‡	98.98^‡	98.99^‡	98.99^‡
	Mean	221636.9	1.98e+05	1.03e+05	3.24e+05	1.98e+05	2.25e+05	1.65e+05
	STD	980107.8	1.12e+06	4.44e+05	1.80e+06	5.31e+05	1.07e+06	4.96e+05
	Run Time	5.74	20.95	35.15	67.65	51.58	65.26	73.33
F7	Best Cost	6.68e-08	2.26e-06^‡	5.87e-08^†	8.01e-08^‡	8.65e-08^‡	9.00e-08^‡	9.66e-08^‡
	Mean	0.43	0.45	0.45	0.63	0.61	0.73	0.74
	STD	0.48	0.49	0.50	0.83	0.96	1.04	1.14
	Run Time	6.52	28.37	25.07	20.22	23.99	24.30	26.17
F8	Best Cost	0.0017	8.59e-05^†	0.0025^‡	0.0021^‡	7.69e-07^†	5.62e-04^†	1.33e-04^†
	Mean	3.30	3.21	3.29	3.50	3.26	3.57	2.99
	STD	3.37	3.54	3.21	3.49	3.50	3.31	3.32
	Run Time	6.23	22.61	35.48	51.49	70.37	91.28	84.73
F9	Best Cost	0.012	0.0076^†	0.0019^†	1.98e-04^†	6.04e-04^†	1.94e-04^†	2.98e-04^†
	Mean	771.56	752.34	757.41	660.89	822.03	692.79	725.54
	STD	665.56	670.48	671.78	656.55	712.96	646.25	675.33
	Run Time	5.48	20.11	36.17	47.25	74.10	75.41	67.06
F10	Best Cost	8.41e-05	0.024^‡	0.0050^‡	1.22e-04^‡	4.91e-04^‡	0.0043^‡	0.0019^‡
	Mean	655.19	927.00	816.92	752.87	760.03	720.48	804.29
	STD	626.74	636.94	650.93	729.52	685.21	683.17	620.34
	Run Time	7.16	21.58	35.21	52.02	57.80	68.73	70.87
F11	Best Cost	6.44e-09	5.99e-09^†	4.24e-09^†	4.44e-09^§	6.24e-09^†	4.33e-09^†	3.24e-09^†
	Mean	0.45	0.55	0.66	0.60	0.46	0.73	0.75
	STD	0.49	0.41	0.43	0.59	0.53	0.69	0.39
	Run Time	10.36	12.36	15.39	19.65	25.03	39.00	42.08
F12	Best Cost	5.15e-05	4.17e-05^†	2.01e-05^†	3.19e-06^‡	4.21e-05^†	6.81e-05^‡	5.01e-05^†
	Mean	36.92	56.12	46.62	38.02	66.88	55.32	32.33
	STD	27.23	54.21	67.83	81.64	64.79	66.35	45.80
	Run Time	42.24	48.25	49.64	64.88	76.87	99.36	80.10
F13	Best Cost	45.99	45.99^§	45.97^†	45.97^†	45.97^†	45.99^§	45.98^†
	Mean	24690423	3.16e+06	1.06e+07	1.49e+07	2.09e+07	1.17e+07	5.02e+06
	STD	173990551	2.43e+07	7.70e+07	9.03e+07	1.45e+08	9.25e+07	2.99e+07
	Run Time	32.00	126.79	170.96	251.43	265.89	339.43	380.86
F14	Best Cost	8.37e-07	0.0038^‡	2.92e-04^‡	0.046^‡	0.0060^‡	1.12e-06^‡	0.017^‡
	Mean	0.40	38.86	33.61	43.12	44.70	27.50	42.01
	STD	0.34	27.13	25.66	20.40	29.76	27.23	21.14
	Run Time	6.35	147.18	227.73	317.77	339.18	370.03	406.90
	‡	6	7	8	5	6	5
	†	7	7	5	9	7	9
	§	1	0	1	0	1	0

In next section of simulation, SSPCO is compared with JADE [23], CMA-ES [24], CLPSO [25], EPSDE [26], GL-25 [27], jDE [28], MRPSO [29] and SaDE [30] on F₁ - F₁₄ in order to save runtime. In the experiments, the parameter settings of eight algorithms keep the same as in their original papers. The statistical results, in terms of F-mean and SD obtained in 25 independent runs under the same termination criterion by each algorithm, are reported in Table 5. The last three rows of Table 5 summarize the experimental results. From the last three rows of Table 5, we can see that SSPCO performs significantly better than other eight algorithms in 12 functions. While SSPCO algorithm significantly outperforms other eight algorithms, it has in two functions is drawbacks comparing with other algorithms. Its first drawback in function F6, and the second drawback in thefunction F13.

Table 5

Experimental results of JADE, CMA-ES, CLPSO, EPSDE, GL-25, jDE, MRPSO, SaDE, and SSPCO over 25 independent runs on F₁–F₁₄ test functions of 30 variables with 300,000 FES

Function	Result	JADE	CMA-ES	CLPSO	EPSDE	GL-25	jDE	MRPSO	SaDE	SSPCO
F1	Fmean	0.00E+00 ^§	1.43E-25 ^‡	0.00E+00 ^§	2.02E-30 ^‡	1.87E-27 ^‡	0.00E+00 ^§	2.31E-27 ^‡	0.00E+00 ^§	0.00E+00
	SD	0.00E+00	3.66E-26	0.00E+00	1.01E-29	4.16E-27	0.00E+00	3.41E-28	0.00E+00	0.00E+00
F2	Fmean	2.41E-28 ^‡	6.27E-25 ^‡	7.01E+02 ^‡	5.59E-26 ^‡	6.25E+01 ^‡	5.06E-08 ^‡	4.99E-10 ^‡	3.72E-05 ^‡	0.00E+00
	SD	1.61E-28	1.50E-25	1.18E+02	1.06E-25	1.10E+02	6.18E-08	7.72E-10	1.40E-04	0.00E+00
F3	Fmean	7.68E+03 ^‡	4.91E-21 ^‡	1.37E+07 ^‡	5.14E+06 ^‡	1.93E+06 ^‡	1.88E+05 ^‡	4.82E+05 ^‡	4.90+05 ^‡	1.08E-260
	SD	5.43E+03	1.06E-21	4.37E+06	2.07E+07	7.48E+05	1.54E+05	2.01E+05	1.83E+05	1.24E+06
F4	Fmean	1.62E-13 ^‡	7.07E+04 ^‡	6.01E+03 ^‡	2.48E+00 ^‡	8.59E+02 ^‡	2.67E-02 ^‡	1.25E+03 ^‡	1.09E+02 ^‡	5.54E-266
	SD	4.13E-13	2.01E+05	1.26E+03	7.60E+00	6.14E+02	4.14E-02	9.00E+02	1.08E+02	134.35
F5	Fmean	3.84E-07 ^‡	3.54E-10 ^‡	3.97E+03 ^‡	1.52E+03 ^‡	2.49E+03 ^‡	1.66E+02 ^‡	8.43E+03 ^‡	3.33E+03 ^‡	6.67E-19
	SD	8.28E-07	8.35E-11	4.88E+02	1.03E+03	1.86E+02	2.12E+02	1.98E+03	4.54E+02	9.68E-18
F6	Fmean	7.88E+00 ^†	6.38E-01 ^†	3.34E+00 ^†	9.57E-01 ^†	2.16E+01 ^†	1.62E+01 ^†	3.03E+01 ^†	5.38E+01 ^†	28.98
	SD	2.29E+01	1.49E+00	5.01E+00	1.74E+00	1.95E+00	2.29E+01	7.38E+01	3.22E+01	3.44E+04
F7	Fmean	5.12E-03 ^‡	6.80E-03 ^‡	4.40E-01 ^‡	1.68E-02 ^‡	1.48E-02 ^‡	1.10E-02 ^‡	2.31E-02 ^‡	1.58E-02 ^‡	0.00E+00
	SD	6.92E-03	7.29E-03	6.99E-02	1.40E-02	8.72E-03	8.81E-03	2.01E-02	1.08E-02	0.00E+00
F8	Fmean	2.07E+01 ^‡	2.07E+01 ^‡	2.09E+01 ^‡	2.09E+01 ^‡	2.10E+01 ^‡	2.09E+01 ^‡	2.09E+01 ^‡	2.10E+01 ^‡	0.00E+00
	SD	3.16E-01	4.77E-01	6.14E-02	4.54E-02	3.87E-02	5.69E-02	5.19E-02	4.81E-02	0.00E+00
F9	Fmean	0.00E+00 ^§	4.32E+02 ^‡	0.00E+00 ^§	0.00E+00 ^§	2.75E+01 ^‡	0.00E+00 ^§	1.17E+01 ^‡	1.19E-01 ^‡	0.00E+00
	SD	0.00E+00	1.13E+02	0.00E+00	0.00E+00	7.93E+00	0.00E+00	4.28E+00	3.30E-01	0.00E+00
F10	Fmean	3.73E+01 ^‡	4.78E+01 ^‡	9.40E+01 ^‡	4.65E+01 ^‡	1.46E+02 ^‡	8.01E+01 ^‡	1.13E+02 ^‡	4.76E+01 ^‡	0.00E+00
	SD	8.32E+00	1.54E+01	1.50E+01	9.44E+00	7.02E+01	1.03E+01	3.04E+01	1.34E+01	0.00E+00
F11	Fmean	2.37E+01 ^‡	6.24E+00 ^‡	2.55E+01 ^‡	3.42E+01 ^‡	3.27E+01 ^‡	3.12E+01 ^‡	1.64E+01 ^‡	1.68E+01 ^‡	0.00E+00
	SD	3.86E+00	2.47E+00	1.14E+00	3.48E+00	7.79E+00	1.26E+00	2.90E+00	2.08E+00	0.00E+00
F12	Fmean	1.20E+04 ^‡	7.90E+03 ^‡	1.42E+04 ^‡	3.53E+04 ^‡	6.53E+04 ^‡	2.00E+04 ^‡	5.79E+03 ^‡	2.87E+03 ^‡	0.00E+00
	SD	4.69E+03	8.13E+03	3.67E+03	6.95E+03	4.69E+04	1.23E+04	5.60E+03	3.26E+03	0.00E+00
F13	Fmean	1.87E+00 ^†	3.11E+00 ^†	1.97E+00 ^†	1.87E+00 ^†	6.97E+00 ^†	2.38E+00 ^†	2.32E+00 ^†	3.98E+00 ^†	7.63
	SD	2.60E-01	9.04E-01	2.01E-01	1.83E-01	5.62E+00	1.85E-01	7.61E-01	4.18E-01	1.25E-01
F14	Fmean	1.31E+01 ^‡	1.47E+01 ^‡	1.26E+01 ^‡	1.34E+01 ^‡	1.32E+01 ^‡	1.32E+01 ^‡	1.19E+01 ^‡	1.30E+01 ^‡	0.00E+00
	SD	2.43E-01	3.11E-01	2.04E-01	3.16E-01	2.02E-01	3.35E-01	4.82E-01	3.06E-01	0.00E+00
	‡	10	12	10	11	12	10	12	11
	†	2	2	2	2	2	2	2	2
	§	2	0	2	1	0	2	0	1

“‡”, “†”, and “§” denote that the performance of SRA is better than, worse than, and similar to that of the corresponding algorithm, respectively.

We compared proposed algorithm with sinDE [31], JOA [32], NPSO [33] and D-PSO-C [34]. This comparison is done with D = 30, pop size = 40, iteration = 10000*d in mean of 51 runs for 28 test benchmarks of CEC 2013 [35] in Table 6.

Table 6

Comparison among algorithms on CEC 2013 test functions

Function	SinDE		JOA		D-PSO-C		NPSO		SSPCO
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
1	2.23E–14	6.83E–14	1.78E–14	3.33E–14	2.44E–14	4.13E–11	2.52E–14	2.77E–12	0.00E+00	0.00E+00
2	2.16E+06	6.15E+05	2.09E+06	4.36E+05	2.76E+06	1.10E+05	2.22E+06	1.32E+05	0.00E+00	0.00E+00
3	8.49E+04	2.11E+05	8.19E+04	4.31E+05	9.11E+04	1.62E+05	8.57E+04	3.65E+05	8.03E+04	1.31E+05
4	6.38E+03	2.00E+03	5.82E+03	1.80E+03	6.50E+03	3.49E+03	6.47E+03	3.09E+03	0.00E+00	0.00E+00
5	1.14E–13	7.65E–29	1.11E–13	5.14E–29	2.74E–13	4.15E–29	1.22E–13	1.85E–29	0.00E+00	0.00E+00
6	1.46E+01	2.42E+00	1.52E+01	4.38E+00	3.54E+01	3.52E+00	1.76E+01	1.72E+00	1.32E+01	1.00E+00
7	1.21E–01	1.57E–01	1.09E–01	2.18E–01	1.81E–01	4.28E–01	1.34E–01	2.17E–01	1.11E–01	1.12E–01
8	2.09E+01	4.96E–02	1.78E+01	3.54E–02	2.67E+01	3.77E–02	2.13E+01	6.06E–02	1.47E+01	2.14E–02
9	1.52E+01	3.05E+00	1.48E+01	2.31E+00	1.78E+01	2.13E+00	1.50E+01	2.87E+00	1.33E+01	1.09E+00
10	2.04E–02	1.30E–02	1.66E–02	2.19E–02	2.65E–02	3.25E–02	2.19E–02	2.43E–02	1.52E–02	1.22E–02
11	1.95E–02	1.39E–01	1.98E–02	2.55E–01	2.34E–02	2.44E–01	2.00E–02	4.35E–01	0.00E+00	0.00E+00
12	3.02E+01	8.65E+00	2.87E+01	3.44E+00	3.66E+01	4.33E+00	3.18E+01	4.22E+00	2.54E+01	2.75E+00
13	7.33E+01	2.07E+01	7.14E+01	1.87E+01	7.76E+01	4.47E+01	7.40E+01	3.27E+01	7.02E+01	2.09E+01
14	5.04E+01	1.92E+01	5.00E+01	3.52E+01	5.78E+01	5.02E+01	5.16E+01	6.48E+01	4.78E+01	2.02E+01
15	2.95E+03	4.86E+02	2.82E+03	3.21E+02	3.13E+03	3.22E+02	2.97E+03	4.86E+02	2.48E+03	2.00E+02
16	1.74E+00	2.52E–01	1.59E+00	1.45E–01	1.87E+00	3.55E–01	1.75E+00	3.28E–01	1.46E+00	1.50E–01
17	3.37E+01	7.97E–01	3.22E+01	5.79E–01	4.09E+01	3.23E–01	3.59E+01	4.55E–01	3.08E+01	2.10E–01
18	7.86E+01	1.42E+01	7.48E+01	3.56E+01	7.89E+01	3.45E+01	7.55E+01	4.44E+01	7.16E+01	2.33E+01
19	2.24E+00	3.79E–01	2.64E+00	2.12E–01	3.75E+00	2.73E–01	2.33E+00	2.28E–01	2.06E+00	1.23E–01
20	9.99E+00	5.50E–01	9.75E+00	3.22E–01	9.83E+00	4.82E–01	9.81E+00	4.88E–01	9.34E+00	2.71E–01
21	2.87E+02	6.40E+01	2.81E+02	3.91E+01	2.99E+02	6.40E+01	2.91E+02	3.91E+01	2.77E+02	2.38E+01
22	1.49E+02	1.76E+01	1.32E+02	2.90E+01	1.55E+02	4.75E+01	1.57E+02	3.22E+01	1.20E+02	3.09E+01
23	3.14E+03	5.31E+02	3.28E+03	3.22E+02	3.83E+03	4.11E+02	3.20E+03	2.39E+02	3.12E+03	2.88E+02
24	2.00E+02	7.16E–03	2.11E+02	3.44E–03	2.67E+02	4.46E–03	2.52E+02	6.86E–03	1.89E+02	4.79E–03
25	2.49E+02	6.85E+00	2.37E+02	3.63E+00	2.74E+02	3.05E+00	2.51E+02	5.73E+00	2.08E+02	3.66E+00
26	2.02E+02	1.40E+01	1.88E+02	4.43E+01	2.37E+02	5.65E+01	2.12E+02	4.47E+01	1.79E+02	2.72E+01
27	3.01E+02	2.36E+00	3.09E+02	4.65E+00	3.45E+02	7.66E+00	3.20E+02	6.47E+00	2.81E+02	3.31E+00
28	3.00E+02	0.00E+00	0.00E+00	0.00E+00	3.23E+02	2.00E+00	3.54E+02	3.39E+00	0.00E+00	0.00E+00

In the end of comparison among algorithm, we compare SSPCO algorithm with three free search algorithms by Penev I [44], Penev II [45] and Penev III [46] with 100 dimensions on CEC 2013 test functions.

In the end of Section 4, we compare Time complexity of proposed algorithm and PSO algorithm. First, In Table 7, have come the best cost and run time for 14 tests with the time complexity. For proposed algorithm n based on populations and log n for iterations, that it can be said time complexity of proposed algorithm is n log n.

Table 7

Comparison among SSPCO algorithms and free search algorithms on CEC 2013 test functions

Function	Penev I		Penev II		Penev III		SSPCO
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
1	9.23E–14	13.83E–14	8.78E–14	11.33E–14	11.44E–14	11.13E–11	7.12E+04	4.77E+08
2	9.16E+06	13.15E+05	9.09E+06	12.36E+05	11.76E+06	8.10E+05	7.51E+04	5.44E+09
3	15.49E+04	8.11E+05	8.19E+04	13.31E+05	16.11E+04	9.62E+05	8.03E+04	8.31E+05
4	12.38E+03	5.16E+03	9.82E+03	6.82E+03	8.50E+03	8.49E+03	6.51E+03	8.84E+23
5	8.14E–13	7.45E–29	9.11E–13	12.14E–29	9.74E–13	4.23E–29	7.15E+00	0.00E+00
6	9.46E+01	9.42E+03	8.52E+01	11.38E+00	10.54E+01	3.34E+00	8.32E+01	8.41E+22
7	9.21E–01	8.57E–01	7.09E–01	2.18E–01	7.81E–01	4.55E–01	8.31E–01	9.12E–01
8	11.09E+01	6.16E–02	9.48E+01	6.32E–02	9.67E+01	6.47E–02	7.37E+01	6.74E–02
9	9.52E+01	8.25E+00	8.28E+01	6.81E+09	9.72E+01	6.12E+04	9.01E+01	4.79E+04
10	11.04E–02	4.35E–02	9.26E–02	9.19E–02	9.35E–02	4.65E–02	8.19E–02	6.62E–02
11	10.25E–02	5.69E–01	8.38E–02	6.54E–01	8.64E–02	6.44E–01	7.06E+06	5.10E+03
12	12.52E+01	9.05E+06	11.07E+01	6.48E+05	12.36E+01	6.36E+00	10.02E+01	7.07E+14
13	15.83E+01	9.02E+01	13.64E+01	6.97E+01	14.70E+01	6.48E+01	12.82E+01	6.09E+01
14	12.24E+01	6.72E+01	12.00E+01	9.02E+01	12.78E+01	6.02E+01	12.01E+01	6.82E+01
15	8.05E+03	9.36E+02	9.12E+03	6.91E+02	6.43E+03	9.12E+02	5.78E+03	2.66E+02
16	8.14E+00	9.50E–01	7.59E+00	6.44E–01	8.87E+00	4.56E–01	8.06E+00	8.50E–01
17	13.07E+01	8.97E–01	12.12E+01	6.79E–01	14.89E+01	8.23E–01	11.58E+01	5.14E–01
18	15.11E+01	6.32E+01	11.48E+01	6.36E+01	11.09E+01	6.45E+01	10.14E+01	6.83E+01
19	6.24E+00	4.19E–01	8.04E+00	6.52E–01	7.65E+00	6.23E–01	6.02E+00	5.13E–01
20	16.19E+00	8.44E–01	15.00E+00	8.20E–01	16.13E+00	9.02E–01	14.84E+00	6.91E–01
21	9.14E+02	8.60E+01	8.11E+02	6.81E+01	9.19E+02	4.99E+01	8.18E+02	9.35E+01
22	5.69E+02	7.71E+01	5.11E+02	6.453E+01	5.55E+02	6.85E+01	4.84E+02	6.19E+01
23	10.51E+03	8.37E+02	10.82E+03	6.22E+02	10.11E+03	6.11E+02	10.02E+03	7.80E+02
24	8.32E+02	4.76E–03	8.43E+02	8.06E–03	8.63E+02	7.46E–03	7.79E+02	6.70E–03
25	11.09E+02	8.85E+00	11.77E+02	4.63E+00	11.04E+02	7.09E+00	10.84E+02	9.66E+00
26	6.02E+02	5.40E+01	5.08E+02	6.13E+01	6.30E+02	4.65E+01	5.00E+02	1.70E+01
27	9.01E+02	5.36E+00	9.59E+02	7.45E+00	9.45E+02	7.16E+00	8.11E+02	4.81E+00
28	6.16E+02	5.08E+00	0.00E+00	0.00E+00	4.20E+02	6.40E+00	0.00E+00	0.00E+00

Table 8

Comparison of time complexity among PSO and SSPCO

Algorithm	Mean of Best Cost	Mean of Run Time	Time Complexity
PSO	17106.62728	14.53071429	O(N Log N)
SSPCO	1.46E+02	14.48571429	O(N Log N)

5 Conclusion

In this paper, an optimization algorithm was introduced which was inspired from the chicks of a type of a wild bird and the results of its implementation showed that it has a significant velocity and cost to achieve the best optimum. We considered a priority variable for the particles that this parameter led to the convergence of the particles around the mother particle and the chicks conduct their mother to the best answer. In fact, each particle just looks for a particle better than itself to move in the search space and it avoids dealing the local and global optimums and this feature greatly reduces the number of calculations and increases the velocity of the algorithm to achieve the optimum answer. Based on simulated results which was done with 6 known optimization algorithms on the 14 known functions, the proposed algorithm has a much better result than the previous algorithms in all functions so that in each 14 functions, the algorithms have a much best cost than the other algorithms and it could achieve the optimum answer in a much less time than the other algorithms and it was determined that the proposed algorithm is an efficient algorithm on the functions and static benchmarks. For future works, this algorithm can be used in dynamic functions and the performance of the algorithm can be investigated in this field.

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