Numerical function optimization by conditionalized PSO algorithm

Abstract

A lot of research has been directed to the new optimizers that can find a suboptimal solution for any optimization problem named as heuristic black-box optimizers. They can find the suboptimal solutions of an optimization problem much faster than the mathematical programming methods (if they find them at all). Particle swarm optimization (PSO) is an example of this type. In this paper, a new modified PSO has been proposed. The proposed PSO incorporates conditional learning behavior among birds into the PSO algorithm. Indeed, the particles, little by little, learn how they should behave in some similar conditions. The proposed method is named Conditionalized Particle Swarm Optimization (CoPSO). The problem space is first divided into a set of subspaces in CoPSO. In CoPSO, any particle inside a subspace will be inclined towards its best experienced location if the particles in its subspace have low diversity; otherwise, it will be inclined towards the global best location. The particles also learn to speed-up in the non-valuable subspaces and to speed-down in the valuable subspaces. The performance of CoPSO has been compared with the state-of-the-art methods on a set of standard benchmark functions.

Keywords

Swarm intelligence black-box optimizer particle swarm optimization adaptive conditionalized particle swarm optimization

1 Introduction

Optimization plays an important role in engineering problems [1 –9]. In many applications, at least one part is involved in an optimization problem [10 –25]. Many problems in engineering fields are somehow involved in an optimization task [26, 27]. They are widely considered to be Black-Box Optimizers (BBO). An algorithm that is capable of optimizing any arbitrary fitness function is called an optimizer. Optimization problems can be divided into two subcategories (a) unconstraint and (b) constraint. If an optimization problem has some conditions to be satisfied during exploring the candidate solutions, it can be considered to be a constraint optimization problem; otherwise, it should be considered to be an unconstraint optimization problem. If an optimizer always finds the optimal solution (i.e. the global optimum), it is considered to be a global optimizer such as mathematical optimizers [26, 27]. Although mathematical optimizers (widely referred to as mathematical programming methods) are almost fast and optimal, some problems are too complex and hard to be modeled and then solved by them.

Therefore, it is sometimes desired to find a suboptimal solution (i.e. a local optimum) instead of the global optimum. Consequently, a lot of research has been directed to the new optimizers that can find suboptimal solutions for any optimization problem (they are named Heuristic Black-Box Optimizers (HBBO)). They have been appropriate for solving almost all optimization problems. The Population-Based Optimization (PBO) algorithms, such as Genetic Optimization (GO) algorithm, and Particle Swarm Optimization (PSO) algorithm, Harmony Search Optimization (HSO) algorithm are of this type. Several HBBOs have been proposed like GO algorithm, HSO algorithm, PSO algorithm, CuCkoo Search Optimization (CCSO) algorithm, Ant Colony Optimization (ACO) algorithm, Cat Swarm Optimization (CaSO) algorithm, and Artificial Bee Colony Optimization (ABCO) algorithm [28].

They contain two sub-fields: (a) Swarm Intelligence (SI) and (b) Evolutionary Algorithms (EA). They do not usually assume anything about the given problem and as a result they are considered to be black-box problem solvers. Therefore, they are very versatile in solving both industrial problems and also pioneering scientific problems [29].

They can find the suboptimal solutions of an optimization problem faster than the mathematical programming methods (if they find them at all). As mentioned, the PSO algorithm is an example of this type. The proposed PSO incorporates conditional learning behavior among birds into the PSO algorithm. Indeed, the particles, little by little, learn how they should behave in some similar conditions. The problem space is first divided into a set of subspaces. Any particle of a subspace will be inclined towards its best experience if its subspace has low diversity; otherwise, it will be inclined towards the global best. The particles also learn to speed-up in the non-valuable subspaces and to speed-down in the valuable subspaces.

Some problems have certain solutions; e.g. the best player in a sport, the longest day of the year and the solution of an ordinary differential equation of first degree are some examples of these simple problems. In contrast, some problems have multiple solutions known as optimal points or local optima; e.g. the best work of art, the most beautiful landscape and the most pleasant piece of music can be named as examples of such problems [30 –32].

In this paper, the objective function is of type cost function. It means its output is defined as a cost value. According to many articles related to the topic, any optimization problem can be considered to be minimization of a cost function. Indeed, to maximize cost function F(x), we can minimize -F(x). The new function can be viewed as a fitness function or objective function, and consequently through an optimizer we can find its optimum value. In the paper, a special case of the PSO algorithm has been proposed. While all of the state-of-the-art methods suffer from lack of adaptive learning of environment or locations, referred to as conditional learning here, the main idea of the proposed PSO algorithm is to integrate conditional learning behavior of birds into the original PSO algorithm. Conditional learning is a kind of association or consistent learning. The proposed method is named as Conditionalized Particle Swarm Optimization ( CoPSO ) algorithm.

In classic conditional learning, the animals learn to find an associative relationship connecting an environmental condition and a reward-punishment signal. This “conditional learning” expression was first introduced by a Russian physiologist, Ivan Pavlov. Assume a bird sees somebody many times hurting birds with a gun. It will flee if it sees the man again even without gun. This behavior is named conditional learning. This behavior has been employed in the proposed CoPSO algorithm. The particles (birds) in a swarm will try to reach the global optimum (the best solution irrespective to the swarm) if the diversity in their locality (in their swarm) at the search space is high; otherwise, they will try to reach their local optimum (the optimum of the whole swarm).

The particles (birds) in a swarm will try to reach the global optimum (the best solution irrespective to the swarm) if the diversity in their locality (i.e. in their swarm) at the search space is high; otherwise, they will try to reach their local optimum (i.e. the swarm optimum). Also if the diversity in their locality at the search space is high, then they will use larger speed parameters.

To make it more obvious, the paper contributions are as follows:

Conditional learning is introduced to be used into BBOAs.

A new PSO algorithm, which is able to converge to the optimal solution of any arbitrary objective function with a smaller number of fitness evaluations in comparison with state of the art methods, is introduced by applying conditional learning into a BBOA, i.e. PSO algorithm.

Section 2 presents related works. In Section 3, we introduce the proposed method, i.e. CoPSO algorithm. In Section 4, we show that this procedure is efficient for finding various types of optimal or nearly optimal solutions. Section 5 concludes the paper.

2 Related works

SI is a concept widely used in artificial intelligence established on the basis of collective behavior in decentralized and self-organized multi-agent systems. It is a kind of intelligence seen in the swarms of the dumb (i.e. simple) agents properly connected into a swarm. The behavior of these swarms yields smart results. It is a property of systems of non-intelligent agents showing cooperatively intelligent behavior. It is considered to be a decentralized, flexible, robust, and self-organized learning [33]. For example, the PSO algorithm and the ACO algorithm are among this type of intelligence.

EAs as a sub-field in Evolutionary Computation Algorithms (ECA) are among black-box PBO algorithms. Any EA algorithm is based on the evolution of an alive creature. EA algorithms should almost have the following steps (like GO algorithm): (a) re-producing step, (b) mutation step, (c) re-combination step, and finally (d) natural selection step. Any individual of the population stands for a candidate solution in these algorithms. The worth of an individual is computed based on the objective function. They evolve the population by their operators (each operator accomplishes one of the main steps). The most well-known algorithms of this type include GO algorithm [34, 35], Genetic Programming (GP) algorithm [36, 37], Evolutionary Strategy (ES) algorithm and Evolutionary Programming (EP) algorithm.

As mentioned, for general problem optimization, instead of traditional optimization approaches, ECAs can be considered to be an influential option because of their abilities that make them suitable for exploring search space. The PSO algorithm [29 –38], the GO algorithm [34, 35], the GP algorithm [36, 37], the ACO algorithm [39, 40], and the ABCO algorithm [41 –45] are extensively used in pattern recognition.

The ACO algorithm [46, 47] imitates the exploring behavior of ants [48]. Besides, there are also some other popular EC methods, such as the Learning Classifier Systems (LCS) [49, 50], the Differential Evolution Optimization (DEO) algorithm [51, 52], the Weighted Differential Evolution Optimization (WDEO) algorithm [53], the Artificial Immune System Optimization (AISO) algorithm [54, 55] and the Evolutionary Multi-Objective Optimization (EMOO) algorithms [56].

Kennedy and Eberhart introduced the PSO algorithm in mid-1990. The PSO algorithm considers each candidate solution to be a particle that is equal to exactly one bird in the whole flock. Initially, the particles are randomly created and positioned in the search space. After that, it updates each candidate solution in an iterative manner in accordance with his best experienced solution and the best solution experienced by the swarm [57]. It is an SI-based optimizer. These SI-based systems commonly consist of a population of simple agents that interact locally with each other and with their environment. Although there is not usually any centralized control on the behaviors of the agents, their local inter-actions result in the emergence of a complex collective behavior. It imitates the collective behavior of birds’ swarm (as an example see Fig. 1). It has been effectively employed to solve various problems emerged in different fields like power text partitioning, flow-analysis, bio-informatics, optimization of energy consumption in wireless sensor networks, feature selection [58 –63], data classification [64, 65], and data clustering [66]. The PSO algorithm has been an ordinary algorithm for solving complex optimization problems [67].

Fig. 1

Bird Swarm Movement.

Cellular-PSO (CPSO) algorithm partitions the exploration space through Cellular Automata (CA). Particles located inside a cell of the CA use some special rules of that CA. According to the position of each particle, its cell is determined. Particles of each cell are considered to be almost an isolated search space and in each of them a different exploration path is employed. Indeed, particles of each cell only see particles of the neighbor cells. Therefore, it guarantees the diversity and it will highly likely to find global optimum [68]. Kamosi et al. [69] have introduced a new PSO algorithm which is based on multi-swarm. Their multi-swarm PSO algorithm has guaranteed the diversity among population through employing a pair of populations [69]. In the CPSO, the number of cells becomes greater in an exponential manner by adding the problem size (or feature size) or by adding the number of partitions. This is the most important weakness of CPSO. Another important weakness in CPSO is that it cannot change number of the cells throughout run-time.

AISO algorithm is a heuristic optimization algorithm that is based on the real immune system (RIS) of the alive creatures. RIS is divided into two types: inborn and flexible. The AISO algorithm is organized according to the flexible RIS [70, 71]. Clonal Selection (ClS) algorithm is an innovative sub-category among the AISO algorithms. It has been introduced by DeCastro and VonZuben as a general optimizer [72].

There are different SI algorithms based on the behavior of bees in the nature. These algorithms are classified into two groups: those created based on foraging behavior and those created based on mating behavior. Examples of simulation algorithms derived from foraging behavior of bees include the ABCO algorithm. A Combinatorial ABCO (CABCO) algorithm was presented [73] with discrete coding to solve Travelling Salesman Problem (TSP). Knowing that original ABC is slow, a Dual-Population Crossover based ABCO (DPCABCO) [74] has been proposed to speed up ABC. This algorithm was proposed by Geem et al. [75]. The HSO algorithm is a meta-heuristic algorithm in the natural process of musical performance that searches a good condition during jazz improvisation. A Simplified Binary Harmony Search (SBHS) algorithm has been introduced to solve the binary knapsack problem [76]. Furthermore, an improved Adaptive Binary Harmony Search (ABHS) algorithm has been introduced to solve binary knapsack problem [77]. Ashrafi and Dariane [78] have proposed a new version of HSO algorithm named Melody Search Optimization (MSO) algorithm. Later, Shafique et al. [79] have extended it. The Bee Optimization (BO) algorithm was first introduced by [80] on a mathematical function. The BO algorithm was first used to solve adjoined optimization functions and after that to solve scheduling tasks [81] and then to solve binary data clustering [82].

The Imperialist Competitive Optimization (ICO) algorithm is an evolutionary optimization method inspired by imperialist competition imperial and is derived from imperialist behavior in an effort to overcome the colonies [83]. Other algorithms have been presented in recent years, most of which were in line with improving famous optimization algorithms, such as the PBI algorithm [84], the NPSO algorithm [85], the CCSO algorithm [86], the Differential Search Optimization (DSO) algorithm [87] and the Bird Mating Optimization (BMO) algorithm [88]. The sinusoidal Differential Evolution Optimization (sinDEO) algorithm [89], the Joint Operations Optimization (JOO) algorithm [90], and the Dynamic multi-swarm Particle Swarm Optimization with Cooperative learning strategy (DPSOC) algorithm were presented using a dynamic multi-population method to improve the particle swarm algorithm [91].

In another algorithm, the CSO algorithm has been enhanced with the use of chaos theory [77]. In 2015, the PSO algorithm has been enhanced using humanistic learning ideas to discover the solutions of different problems [92]. In 2016, the ACO algorithm has been combined with the GO algorithm, and a new optimization algorithm has been introduced [93]. In a study, authors have employed the GO algorithm in their efforts to find the closest general optimal solution to solve nonlinear multi-modal optimization problems [94].

3 CoPSO algorithm

All the notations of the proposed method, i.e. the CoPSO algorithm, are presented in Table 1. The CoPSO algorithm is a special type of the PSO algorithm. At first, it partitions the search space into a predefined number of regions. In each region, a specific set of parameters is used by the particles. It means that the particles of each region have an adaptive corresponding parameters’ set. By dividing each variable into PartNum parts we can have PartNum^ProbSize regions, where ProbSize stands as the number of variables. The particles of each region follow their corresponding parameters for the PSO algorithm, i.e. a special set of C₁, C₂ and C₃.

Table 1
The used notations for the parameters and their descriptions

Parameter Description Default Value

PartNum Number of fractions in each variable of the problem space 10

Pop Population –

PopSize Size of population 100

ProbSize Number of variables –

Pop _i The ith individual in the population –

${Pop}_{i}^{j}$ The jth variable in the ith individual of the population –

MaxValue ^j The upper limit of the jth variable –

MinValue ^j The lower limit of the jth variable –

l ^j The step level in the jth variable (feature) –

V _i The velocity of the ith individual in the population –

$V_{i}^{j}$ The jth variable of the velocity of the ith individual of the population –

b ^k The number of times that the best solution of the population is in the kth region 10

π ^k The relative number of times that the best solution of the population is in the kth region $\frac{1}{k}$

C ₁ First PSO parameter 2

C ₂ Second PSO parameter 2

C ₃ Third PSO parameter 2

Ψ ₁ Increment coefficient 1.1

Ψ ₂ Decrement coefficient $\frac{1}{Ψ_{1}}$

$C_{1}^{k}$ First PSO parameter for the kth region (part) –

$C_{2}^{k}$ Second PSO parameter for the kth region (part) –

$C_{3}^{k}$ Third PSO parameter for the kth region (part) –

G The global best of population –

G ^j The jth variable of the global best of population –

L _i The best place met by the ith individual in the population –

$L_{i}^{j}$ The jth variable in the best place met by the ith individual in the population –

R _k The best individual in the kth region –

$R_{k}^{j}$ The jth variable of the best individual in the kth region –

UpdateStep The number of fitness function evaluations per each updating of the PSO parameters –

Θ An inertia coefficient parameter in range [0, 1] 0.1

ϱ A random value extracted from uniform distribution in (0, 1) –

e A variable indicating current number of fitness function evaluations –

MaxItr The maximum iteration that algorithm should proceed –

θ _max 0.8 of the maximum variance seen so far in population –

θ _min 0.2 of the maximum variance seen so far in population –

σ _k the standard deviation of the particles belonging to the kth region –

$\bar{σ}$ the standard deviation of all particles –

σ ^Q20 The 20th quartile of the vector σ_k –

σ ^Q80 The 80th quartile of the vector σ_k –

Parameter	Description	Default Value
PartNum	Number of fractions in each variable of the problem space	10
Pop	Population	–
PopSize	Size of population	100
ProbSize	Number of variables	–
Pop _i	The ith individual in the population	–
${Pop}_{i}^{j}$	The jth variable in the ith individual of the population	–
MaxValue ^j	The upper limit of the jth variable	–
MinValue ^j	The lower limit of the jth variable	–
l ^j	The step level in the jth variable (feature)	–
V _i	The velocity of the ith individual in the population	–
$V_{i}^{j}$	The jth variable of the velocity of the ith individual of the population	–
b ^k	The number of times that the best solution of the population is in the kth region	10
π ^k	The relative number of times that the best solution of the population is in the kth region	$\frac{1}{k}$
C ₁	First PSO parameter	2
C ₂	Second PSO parameter	2
C ₃	Third PSO parameter	2
Ψ ₁	Increment coefficient	1.1
Ψ ₂	Decrement coefficient	$\frac{1}{Ψ_{1}}$
$C_{1}^{k}$	First PSO parameter for the kth region (part)	–
$C_{2}^{k}$	Second PSO parameter for the kth region (part)	–
$C_{3}^{k}$	Third PSO parameter for the kth region (part)	–
G	The global best of population	–
G ^j	The jth variable of the global best of population	–
L _i	The best place met by the ith individual in the population	–
$L_{i}^{j}$	The jth variable in the best place met by the ith individual in the population	–
R _k	The best individual in the kth region	–
$R_{k}^{j}$	The jth variable of the best individual in the kth region	–
UpdateStep	The number of fitness function evaluations per each updating of the PSO parameters	–
Θ	An inertia coefficient parameter in range [0, 1]	0.1
ϱ	A random value extracted from uniform distribution in (0, 1)	–
e	A variable indicating current number of fitness function evaluations	–
MaxItr	The maximum iteration that algorithm should proceed	–
θ _max	0.8 of the maximum variance seen so far in population	–
θ _min	0.2 of the maximum variance seen so far in population	–
σ _k	the standard deviation of the particles belonging to the kth region	–
$\bar{σ}$	the standard deviation of all particles	–
σ ^Q20	The 20th quartile of the vector σ_k	–
σ ^Q80	The 80th quartile of the vector σ_k	–

First let’s define the step level in each variable based on the Equation 1. $l^{j} = \frac{({MaxValue}^{j} - {MinValue}^{j})}{PartNum}$ (1) where l^j, MaxValue^j, MinValue^j and PartNum stand respectively for the step level in the jth variable, the maximum value of the jth variable, the minimum value of the jth variable, and the number of fractions in each variable of the problem space. Each region is defined based on Equation 2.

$\begin{matrix} S_{k} = {i | \forall j \in {1, 2, \dots, ProbSize} : \\ [(a_{j} - 1) \times l^{j} + 1 \leq {Pop}_{i}^{j} \leq a_{j} \times l^{j}]} \end{matrix}$ (2) where S_k is the set of all the population individuals that fall in kth region, and all values of a_j are defined based on Equation 3. $k = \sum_{r = 1}^{ProbSize} a_{r} \times {PartNum}^{r - 1} + 1$ (3) subject to a_r∈ { 0, 1, 2, …, ProbSize - 1 }

The updating equation of each particle is defined based on Equation 4. ${Pop}_{i}^{j} = {Pop}_{i}^{j} + V_{i}^{j}$ (4) where ${Pop}_{i}^{j}$ and $V_{i}^{j}$ stand respectively for the jth variable of the ith individual in the population and the jth variable of the velocity of the ith individual in the population. The $V_{i}^{j}$ is defined based on Equation 5.

$\begin{matrix} V_{i}^{j} = ϑ \times V_{i}^{j} + ϱ_{1} \times π^{k} \times C_{1}^{k} \times (L_{i}^{j} - {Pop}_{i}^{j}) \\ + ϱ_{2} \times π^{k} \times C_{2}^{k} \times Is (R_{k}) \times (R_{k}^{j} - {Pop}_{i}^{j}) \\ + ϱ_{3} \times π^{k} \times C_{3}^{k} \times (G^{j} - {Pop}_{i}^{j}) \end{matrix}$ (5) where k parameter is an integer indicating the index of region that ith individual is located, the ϑ parameter is an inertia coefficient value in range [0, 1], the parameters ϱ₁, ϱ₂, and ϱ₃ are three random values extracted from the uniform distribution U (0, 1), the G^j is the jth variable of the best individual in the population, the $R_{k}^{j}$ is the jth variable of the best individual in the kth region, Is (R_k) is Boolean function that returns one if R_k has value previously; otherwise, it returns zero, the $L_{i}^{j}$ is the jth variable of the best experience of the ith individual, the $C_{1}^{k}$ , $C_{2}^{k}$ , and $C_{3}^{k}$ parameters are the PSO parameters and are computed respectively based on Equations 6, 7 and 8, and finally the π^k parameter is the relative number of times (i.e. iterations) that the best solution of the population is in the kth region and is computed based Equation 15. $C_{1}^{k} = {\begin{matrix} Ψ_{1} \times C_{1}^{k} & (e = p \times UpdateStep) & (σ_{k} < σ_{k}^{Q 20}) \\ C_{1}^{k} & o . t . w . \\ Ψ_{2} \times C_{1}^{k} & (e = p \times UpdateStep) & (σ_{k} > σ_{k}^{Q 80}) \end{matrix}$ (6) $C_{2}^{k} = {\begin{matrix} Ψ_{1} \times C_{2}^{k} & (e = p \times UpdateStep) & (σ_{k} > σ_{k}^{Q 80}) \\ C_{2}^{k} & o . t . w . \\ Ψ_{2} \times C_{2}^{k} & (e = p \times UpdateStep) & (σ_{k} < σ_{k}^{Q 20}) \end{matrix}$ (7) $C_{3}^{k} = {\begin{matrix} Ψ_{1} \times C_{3}^{k} & (e = p \times UpdateStep) & (\bar{σ} > θ_{\max}) \\ C_{3}^{k} & (ite \neq p \times UpdateStep) \\ Ψ_{2} \times C_{3}^{k} & (e = p \times UpdateStep) & (\bar{σ} < θ_{\min}) \end{matrix}$ (8) where the p is a positive integer, the e is the number of fitness function evaluations, the UpdateStep is an algorithm parameter (and is better to be equal to PopSize), the θ_max (defined by Equation 13) is 0.8 of the maximum variance in population so far, the θ_min (defined by Equation 14) is 0.2 of the maximum variance in population so far, the σ_k (defined by Equation 9) is the standard deviation of the particles belonging to the kth region, the σ^Q20 and σ^Q80 are the 20th and 80th quartiles of the vector σ_k, $\bar{σ}$ (defined by Equation 11) is the standard deviation of all particles, and finally Ψ₁ and Ψ₂ are increment and decrement coefficients. The σ_k is computed based on Equation 9. $σ_{k} = {\begin{matrix} {(\frac{\sum_{i \in S_{k}} \sum_{j = 1}^{ProbSize} {({Pop}_{i}^{j} - μ_{k}^{j})}^{2}}{| S_{k} |})}^{0.5} & | S_{k} | \neq 0 \\ 0 & | S_{k} | = 0 \end{matrix}$ (9) where |S_k| is the number of the particles belonging to the kth region and μ_k and $μ_{k}^{j}$ are the average of the particles belonging to the kth region and the jth variable of μ_k. The $μ_{k}^{j}$ is computed based on Equation 10. $μ_{k}^{j} = \frac{\sum_{i \in S_{k}} {Pop}_{i}^{j}}{| S_{k} |}$ (10)

The $\bar{σ}$ is defined by Equation 11. $\bar{σ} = {(\frac{\sum_{i = 1}^{PopSize} \sum_{j = 1}^{ProbSize} {({Pop}_{i}^{j} - {\bar{μ}}^{j})}^{2}}{PopSize})}^{0.5}$ (11) where $\bar{μ}$ and ${\bar{μ}}^{j}$ are the average of the whole particles and the jth variable of $\bar{μ}$ . The ${\bar{μ}}^{j}$ is computed based on Equation 12. $bar μ^{j} = \frac{\sum_{i = 1}^{PopSize} {Pop}_{i}^{j}}{PopSize}$ (12) $θ_{\max} = 0.8 \times max (σ)$ (13) $θ_{\min} = 0.2 \times max (σ)$ (14) $π^{k} = 1 - \frac{b^{k}}{\sum_{r = 1}^{{PartNum}^{ProbSize}} b^{r}}$ (15)

According to flowchart of the proposed method, which is depicted by Fig. 2, the only difference with the original part is “Update Particles” block where it is accomplished “Adaptively”. For more details, the pseudo code of the proposed CoPSO algorithm is presented in Fig. 3. The MaxItr parameter is the maximum iteration that the algorithm is permitted to proceed. The sign “↦” is the assignment sign. In the first step, i.e. statement 01, initialization of parameters is done. If PopSize, i.e. population size has not taken a value by function argument, it is set to 100 by default. If Ψ₁ (or PartNum, i.e. number of fragments in each dimension or ϑ, i.e. the inertia parameter) has not taken a value by function argument, it is set by default to 1.1 (or correspondingly 10, or 0.1). Variable z′, an auxiliary temporary variable, and itr, i.e. iteration variable, are set to 0. Two empty sparse arrays, i.e. b^k and z_k, are created with $\frac{1}{{PartNum}^{ProbSize}}$ and 0 as default value respectively. A virtual function value is defined on the sparse variables. If the function value is called with a sparse variable and an index k, e.g. b^k, as its argument, it returns the corresponding value if it is available; otherwise, it returns the default value of the sparse variable. Then, location and velocity of each particle are randomly created. Each particle is also placed in its local optimum at the initialization step. For initialization step, a random population is also created and after that, a third of the worst population individuals are removed and replaced with new random individuals.

In the second step, i.e. statement 02, the local optimum of each particle is updated. A sparse list, i.e. S, is created. For each particle, its region is determined. Then, each particle, i.e. ith particle, that belongs to kth region is put into S_k by calling put (i, S_k). If key k is available in sparse vector S, then the function put (i, S_k) assigns S^k∪ { i } to S^k; otherwise: it first creates S^k and then assigns {i} to S^k.

Fig. 2

The flowchart of CoPSO.

Fig. 3

The pseudo code of the proposed CoPSO algorithm. ReLU stands for Rectified Linear Unit.

At the third step, i.e. statement 03, the best particle in each region is first detected and then if the number of particles in the region is more than 2, the best (Pop_j, value (R_k)) is assigned to R_k, i.e. a sparse array that maintains the best position in each so-far-explored region. The function value (R_k) returns R_k if R has a key k; otherwise, it returns null position. The function best (Pop_j, value (R_k)) returns the better (in terms of objective function) position between Pop_j and R_k (if available). Finally, π_k is computed for each region according to equation 15 (with the exception that we use value (b^k) instead of b^k as it is a sparse array with default value 0).

At the fourth step, i.e. statement 04, if the number of particles in any arbitrary region is more than 2, then we compute σ_k and store it in a sparse array (with a 0 default value). At the next step, i.e. statement 05, σ^Q20, σ^Q80, $\bar{σ}$ , θ_max and θ_min are computed. If $\bar{σ} > θ_{\max}$ , then we assigns z′ + 1 to z′. If $\bar{σ} < θ_{\min}$ , then we assigns z′ - 1 to z′. Note that the variable z′ shows number of the times the first condition is triggered minus number of the times the third condition is triggered in Equation 7.

At the sixth step, i.e. statement 06, if the number of particles in an arbitrary region is more than 2, if σ_k < σ^Q20, then value (z_k) + 1 is assigned to z_k (i.e. a sparse array (with a 0 default value) that maintains number of the times the first condition is triggered minus number of the times the third condition is triggered in Equation 6); otherwise, if σ_k > σ^Q80, then value (z_k) - 1 is assigned to z_k.

At the subsequent step, i.e. statement 07, the best particle and its region are found, its location is stored in G and then its b_k and the itr (iteration counter) are increased each by one. At the next step, velocity and location vectors of particles are updated. Finally, if itr is still less than MaxItr, S and σ are first removed (reinitialized with empty lists) and then the algorithm goes to statement 02.

In the proposed CoPSO algorithm, it is tried to maintain the maximum diversity. For example, if the diversity in a region is high, the exploitation should be followed; otherwise, the exploration is targeted. The proposed CoPSO algorithm has done this idea using conditioning learning behavior of birds. It is done through partitioning the environment into non-overlapping regions and defining different controlling parameters. Also, by emphasizing the exploration parameter (i.e. the C₁) and deemphasizing the exploitation parameters (i.e. the C₂ and C₃) in low-diversity regions the proposed CoPSO algorithm makes the population diverse. In contrary, by deemphasizing the exploration parameters (i.e. the C₂ and C₃) and emphasizing the exploitation parameter (i.e. the C₁) in high-diversity regions the proposed CoPSO algorithm puts pressure on the population to move towards the optimum.

Parameters of algorithm are as follows: PopSize, PartNum, ProbSize, Ψ₁, C₁, C₂, C₃, ϑ, MaxItr, and ϑ. Note that in parameters’ initialization, PopSize and MaxItr are not set by our side. They are used as they are defined in standard CEC problems and they are same for all methods. Also, C₁, C₂, C₃, and ϑ are not set by our side. They are used as recommended by the original versions of PSO. The parameter Ψ₁ is set to 1.1, because exploring its value among the set {1.01, 1.05, 1.1, 1.2, 1.3, 1.4, 1.5} on first objective function of CEC 2009 [95] shows its best value is 1.1. The PartNum is not analyzed, because its analyzing is not the target of this paper. This paper only targets to determine if it (conditional learning) has any positive effect on optimization. Besides, the conditional learning can be applied to other optimization algorithms and it is more than tuning this parameter.

Figure 4 shows how to value the problem space. The parts marked in bold have greater value. It means global optimum is probably located in these areas more frequently. In the proposed method, it is tried to move the particles in the high-valued spaces at a slower speed so that the particles could do more exploration those spaces.

Fig. 4

The values of the problem space.

4 Experimental study

The benchmarks used throughout all the paper, contain different cost functions. Therefore, we should multiply them by a coefficient (i.e. minus one) and optimize their multiplied versions, as our method is presented as a maximizer of objective functions.

4.1 Structure of experimentations

The implementations of the proposed method and the state-of-the-art methods have been accomplished using a MATLAB 2015a on a number of benchmarks. The experimental results have been classified into 3 parts. CEC 2009 is the first benchmark that has been used throughout all results reported in section 4.2. CEC 2005 is the second used benchmark (employed in section 4.3) and finally the CEC 2013 is used as the third benchmark (employed in Section 4.4). Section 4.2 compares the proposed CoPSO method with the state-of-the-art basic heuristic optimizers. Section 4.3 compares the proposed CoPSO method with the state-of-the-art more sophisticated heuristic optimizers. In the fourth subsection, the proposed CoPSO method is compared with the state-of-the-art hybrid optimizers.

4.2 CEC 2009

In the first section, CEC 2009 is used as the benchmark [95]. The details of this benchmark have been presented in Table 2.

Table 2
Introduction of CEC 2009 benchmark

Function Min Range D

F1(Beale) 0 [– 4.5,4.5] 2

F2(Easom) – 1 [– 100,100] 2

F3(Matyas) 0 [– 10,10] 2

F4(Colville) 0 [– 10,10] 4

F5(Zakharov) 0 [– 5,10] 10

F6(Schwefel 2.22) 0 [– 10,10] 30

F7(Schewefel 1.2) 0 [– 100,100] 30

F8(Dixon-price) 0 [– 10,10] 30

F9(Step) 0 [– 5.12,5.12] 30

F10(Sphere) 0 [– 100,100] 30

F11(SumSquares) 0 [– 10,10] 30

F12(Quartic) 0 [– 1.28,1.28] 30

F13(Schaffer) 0 [– 100,100] 2

F14(6H Camel) – 1.0316 [– 5,5] 2

F15(Bohachevsky2) 0 [– 100,100] 2

F16(Bohachevsky3) 0 [– 100,100] 2

F17(Shubert) – 186.73 [– 10,10] 2

F18(Rosenbrock) 0 [– 30,30] 30

F19(Griewank) 0 [– 600,600] 30

F20(Ackley) 0 [– 32,32] 30

F21(Bohachevsky1) 0 [– 100,100] 2

F22(Booth) 0 [– 10,10] 2

F23(Michalewicz2) – 1.8013 [0,π] 2

F24(Michalewicz5) – 4.6877 [0,π] 5

F25(Michalewicz10) – 9.6602 [0,π] 10

F26(Rastrigin) 0 [– 5.12,5.12] 30

Function	Min	Range	D
F1(Beale)	0	[– 4.5,4.5]	2
F2(Easom)	– 1	[– 100,100]	2
F3(Matyas)	0	[– 10,10]	2
F4(Colville)	0	[– 10,10]	4
F5(Zakharov)	0	[– 5,10]	10
F6(Schwefel 2.22)	0	[– 10,10]	30
F7(Schewefel 1.2)	0	[– 100,100]	30
F8(Dixon-price)	0	[– 10,10]	30
F9(Step)	0	[– 5.12,5.12]	30
F10(Sphere)	0	[– 100,100]	30
F11(SumSquares)	0	[– 10,10]	30
F12(Quartic)	0	[– 1.28,1.28]	30
F13(Schaffer)	0	[– 100,100]	2
F14(6H Camel)	– 1.0316	[– 5,5]	2
F15(Bohachevsky2)	0	[– 100,100]	2
F16(Bohachevsky3)	0	[– 100,100]	2
F17(Shubert)	– 186.73	[– 10,10]	2
F18(Rosenbrock)	0	[– 30,30]	30
F19(Griewank)	0	[– 600,600]	30
F20(Ackley)	0	[– 32,32]	30
F21(Bohachevsky1)	0	[– 100,100]	2
F22(Booth)	0	[– 10,10]	2
F23(Michalewicz2)	– 1.8013	[0,π]	2
F24(Michalewicz5)	– 4.6877	[0,π]	5
F25(Michalewicz10)	– 9.6602	[0,π]	10
F26(Rastrigin)	0	[– 5.12,5.12]	30

In this section, we compared the CoPSO algorithm with the state-of-the-art methods. These methods are the GO algorithm [96], the DEO algorithm [51], the PSO algorithm [97], the BO algorithm [80], the PBI algorithm [84], the NPSO algorithm [85], the MRPSO algorithm [98], the EPSDE algorithm [99], the CCABCO algorithm [100] and the FFO algorithm [101]. We implemented the results with equal parameters for all methods, including the population of 100, and 500000 fitness function evaluations. The algorithm-specific parameters have been as reported in Table 3.

Table 3

Parameter setting

Item No.	Value	Parameter
1	100	PopSize
2	Depends on problem	ProbSize
3	2	C₁, C₂, C₃
4	500000	Number of Fitness Evaluations
5	0.5	Percent of crossover
6	0.01	Percent of mutation
7	10	Harmony memory size
8	100	New harmony memory size
9	0.75	Harmony memory
10	0.05	Pitch adjustment rate
11	0.1	Fret width
12	10	Number of imperialists

The results of this section are in terms of the averaged objective function value and its standard deviation. After accomplishing a total execution of any algorithm, the fitness of the best optimal solution is recorded as the score of the algorithm. Indeed, each method is executed 30 times. After execution of 30 independent runs, 30 different scores for the algorithm are recorded. The average and standard deviation of them (denoted respectively by mean best cost (Mean) and the standard deviation (Std) in tables) are reported. The results of the algorithm on 26 functions of the CEC 2009 have been presented in Table 4. The results presented by Table 4 are a comprehensive experimental study.

Table 4

Comparison of the proposed method with the state-of-the-art methods on CEC 2009 benchmark

	Criteria	GA	DE	PSO	BA	PBA	FA	NPSO	MRPSO	CCABC	EPSDE	Proposed
F1	Mean	0§	0§	0§	1.88E-05‡	0§	0§	0§	0§	0§	1.48E-05‡	0
	Std	0	0	0	1.94E-05	0	0	0	0	0	1.64E-05	0
F2	Mean	–1§	–1§	– 1§	– 0.99994‡	– 1§	– 1§	– 1§	– 1§	– 1§	– 0.8965‡	– 1
	Std	0	0	0	4.50E-05	0	0	0	0	0	3.56E-05	0
F3	Mean	0§	0§	0§	0§	0§	0§	0§	0§	0§	0§	0
	Std	0	0	0	0	0	0	0	0	0	0	0
F4	Mean	0.01494‡	0.04091‡	0§	1.11760‡	0§	0§	0§	1.13560‡	1.31760‡	0§	0
	Std	0.00736	0.08198	0	0.46623	0	0	0	0.45823	0.76643	0	0
F5	Mean	0.01336‡	0§	0§	0§	0§	0§	0§	0.01125‡	0§	0§	0
	Std	0.00453	0	0	0	0	0	0	0.00365	0	0	0
F6	Mean	11.0214‡	0§	0§	0§	7.59E-10‡	2.73028E-10‡	0§	7.268E-1‡	0§	0§	0
	Std	1.38686	0	0	0	7.10E-10	1.1535E-11	0	7.298E-10	0	0	0
F7	Mean	7.40E+03‡	0§	0§	0§	0§	147.401395‡	8.51453‡	5.54478‡	6.56735‡	5.26235‡	0
	Std	1.14E+03	0	0	0	0	448.571186	8.768288	5.71233	6.1288	5.0354	0
F8	Mean	1.22E+03‡	0.66667‡	0.66667‡	0.66667‡	0.66667‡	0.66667‡	0.66667‡	0.4583‡	0.23467‡	0§	0
	Std	2.66E+02	0.98E-9	1.02E-8	1.16E-09	5.65E-10	0	0	0.99E-8	2.66E-10	0	0
F9	Mean	1.17E+03‡	0§	0§	5.12370‡	0§	0§	0§	3.54970‡	0§	0§	0
	Std	76.56145	0	0	0.39209	0	0	0	0.3120	0	0	0
F10	Mean	1.11E+03‡	0§	0§	0§	0§	0§	0§	1.12070‡	0§	0§	0
	Std	74.21447	0	0	0	0	0	0	0.44709	0	0	0
F11	Mean	1.48E+02‡	0§	0§	0§	0§	0§	0§	0§	0§	0§	0
	Std	12.40929	0	0	0	0	0	0	0	0	0	0
F12	Mean	0.18070‡	0.00136‡	0.00116‡	1.72E-06‡	0.00678‡	3.66E-03‡	9.70E-04‡	1.52E-05‡	0.00042‡	0.10233‡	1.011E-06‡
	Std	0.02712	0.00042	0.00028	1.85E-06	0.00133	0.001401347	0.00125	1.75E-05	0.00765	0.1555	1.36E-07
F13	Mean	0.00424‡	0§	0§	0§	0§	0§	0§	0§	0§	0§	0
	Std	0.00476	0	0	0	0	0	0	0	0	0	0
F14	Mean	– 1.03163§	– 1.03163§	– 1.03163§	– 1.0316§	– 1.0316§	– 1.03163§	– 1.0316§	– 1.0316§	– 1.0316§	– 1.0316§	– 1.03163
	Std	0	0	0	0	0	0	0	0	0	0	0
F15	Mean	0.06829‡	0§	0§	0§	0§	0§	0§	0§	0§	0§	0
	Std	0.07822	0	0	0	0	0	0	0	0	0	0
F16	Mean	0§	0§	0§	0§	0§	0§	0§	0§	0§	0§	0
	Std	0	0	0	0	0	0	0	0	0	0	0
F17	Mean	– 186.73§	– 186.73§	– 186.73§	– 186.73§	– 186.73§	– 186.73§	– 186.73§	– 186.73§	– 186.73§	– 186.73§	– 186.73
	Std	0	0	0	0	0	0	0	0	0	0	0
F18	Mean	1.96E+05‡	18.20394‡	15.088617‡	28.834‡	4.2831‡	2.02E+01§	1.04E-07†	16.3261‡	14.4386‡	16.66861‡	1.3187
	Std	3.85E+04	5.03619	24.170196	0.10597	5.7877	1.147947126	2.95E-07	22.4319	29.57019	21.87019	0.02677
F19	Mean	10.63346‡	0.00148‡	0.01739‡	0§	0.00468‡	0§	0§	0.45739‡	0.09739‡	0.08249‡	0
	Std	1.16146	0.00296	0.02081	0	0.00672	0	0	0.72081	0.09771	0.08691	0
F20	Mean	14.67178‡	0§	0.16462‡	0§	3.12E-08‡	6.56E-10‡	0§	0§	0.11432‡	0.21362‡	0
	Std	0.17814	0	0.49387	0	3.98E-08	1.24159E-09	0	0	0.32587	0.549387	0
F21	Mean	0§	0§	0§	0§	0§	0§	0§	0§	0§	0§	0
	Std	0	0	0	0	0	0	0	0	0	0	0
F22	Mean	0§	0§	0§	0.00053‡	0§	0§	0§	0§	0§	0.00132‡	0
	Std	0	0	0	0.00074	0	0	0	0	0	0.00149	0
F23	Mean	– 1.8013§	– 1.8013§	– 1.57287‡	– 1.8013§	– 1.8013§	– 1.8013§	– 1.8013§	– 1.8013§	– 1.8013§	– 1.8013§	– 1.8013
	Std	0	0	0.11986	0	0	0	0	0	0	0	0
F24	Mean	– 4.64483‡	– 4.68348‡	– 2.4908‡	– 4.6877§	– 4.6877§	– 4.6000‡	– 4.6877§	– 1.3577‡	– 3.64483‡	– 2.4468‡	– 4.6877
	Std	0.09785	0.01253	0.25695	0	0	0.09270	0	0.78615	0.00785	0.03597	0
F25	Mean	– 9.49683‡	– 9.59115‡	– 4.0071‡	– 9.6602§	– 9.6602§	– 9.2952172‡	– 9.65352‡	– 9.57525‡	– 9.11525‡	– 9.65352‡	– 9.6602
	Std	0.14112	0.06421	0.50263	0	0	0.2820193	0.014947	0.019947	0.028947	0.014947	0
F26	Mean	52.92259‡	11.71673‡	43.97714‡	0§	0§	47.884069‡	0§	33.5561‡	44.9664‡	38.7871‡	0
	Std	4.56486	2.53817	11.72868	0	0	16.132004	0	21.78868	11.8868	11.09868	0
The sign “‡” indicates the proposed algorithm outperforms the given algorithm, the sign “†” indicates the given algorithm outperforms
the proposed algorithm and finally “§” indicates the difference in performance of the given algorithm and that
of the proposed algorithm are meaningless.

According to the results presented in Table 4, the proposed method has been able to converge to the optimal point in almost all of 26 objective functions of the CEC 2009 benchmark. As the results are reported in the form of averages over 30 trials, it is fair to conclude that the results indicate the proposed CoPSO algorithm could fully converge to the optimum in all 26 functions. Also, the standard deviation of the proposed CoPSO algorithm is also very low, and it is another reason showing the algorithm is effective. In functions F11, F13 and F15 all methods except for genetic algorithm converged to the optimal point. In functions F3, F14, F17 and F21, all methods could converge to the optimal point. The proposed method alone could achieve the optimal solution in functions F8, F12 and F18. The Friedman test is also used for having more confidence in the results and the results have been confirmed by it. The diagrams of the quality of the solutions found by different optimization methods in terms of different dimension sizes (i.e. twenty, thirty, forty, fifty, eighty and hundred) on all 26 objective functions are shown in Fig. 5. The term “Dimension” is equal to “Variable” in Fig. 5 and throughout all this paper. The results shown in Fig. 5 indicate the proposed method is the most robust method against dimension size (or problem size).

Fig. 5

The quality of the solutions found by different optimization methods in terms of different dimension sizes on all 26 objective functions.

The objective function of the best solution among the population at the ith evaluation of the fitness function is depicted in different iterations of different optimization methods in Fig. 6 (the results are averaged on 30 trials). The i values spotted and reported in Fig. 6 include iterations 100, 200, 500, 1000, 100000 and 500000.

Fig. 6

Results of the compared methods in 100th, 200th, 500th, 1000th, 100000th and 500000th fitness function evaluations on 26 benchmark functions.

To measure the particles’ convergence in the proposed method, the first 9 objective functions have been selected and the standard deviation of the population particles is shown on those functions by Fig. 7. The particles’ convergence curves in the 9 functions indicate that the particles in the middle of the evaluation are very diverse and they converge to the optimal point at the end of the execution. These plots show the rationale of the proposed CoPSO in the problem space.

Fig. 7

The standard deviation of the proposed algorithm on 9 functions from benchmark functions.

4.3 CEC 2005

In this subsection, the proposed method was compared to the following methods: the Drop Rain (RD) algorithm [76], the CS algorithm [86], the TLBO algorithm [102], the DSO algorithm [87] and the BMO algorithm [88]. We tested these methods on the test series CEC 2005 [103]. This comparison has been conducted based on the Mean and the Std (averaged on 30 trials). The number of fitness function evaluations is set to 300000, and a population size is 40.

The results presented in Table 5 confirm the superiority of the proposed CoPSO to the state-of-the-art methods. In the functions F1 and F9, the proposed CoPSO method together with another algorithm could have the best performance. In the function F1, the CS algorithm and the proposed CoPSO method have the best performance due to reaching the optimal point. The same scenario is occurred for the function F9 where the proposed CoPSO method and the DSA method find the optimal solution of the problem. The proposed CoPSO method is often among the two methods with the least standard deviation values. The results presented by Tables 4 and 5 are validated by the statistical test available at their last rows. In Table 4, the Friedman test is presented to test the significance of the results for all methods. Considering each dataset in Table 4 as a sample and each column as a method, the Freidman test output is presented in the last row of Table 4. The p-value of Freidman test is 3.21E-8 which shows significant difference in methods’ performance. The same experiment is repeated for Table 5 and the Freidman test output is presented in the last row of Table 5. The p-value of Freidman test is 4.26E-18 which shows significant difference in methods’ performance.

Table 5
Results of the proposed CoPSO in comparison with some newer algorithms on 26 functions based on the Mean and Std

Criteria RD CS TLBO DSA BMO Proposed

F1 Mean 2.63E-05‡ 0§ 3.64E-27‡ 1.23E-28‡ 1.23E-28‡ 0

Std 5.15E-05 0 5.82E-27 0 0 0

F2 Mean 8.46E+02‡ 1.40E-03‡ 1.79E-09† 1.56E+03‡ 1.36E+00‡ 1.88E-09

Std 1.83E+02 2.49E-03 4.50E-09 5.64E+02 1.45E+00 4.69E-09

F3 Mean 1.90E+06‡ 2.06E+06‡ 1.05E+06‡ 1.19E+07‡ 5.48E+06‡ 1.01E+06

Std 6.20E+05 6.83E+05 6.64E+05 5.55E+06 2.42E+06 6.72E+05

F4 Mean 1.39E+04‡ 1.51E+03‡ 2.73E+02‡ 8.46E+03‡ 3.87E+03‡ 1.67E+02

Std 2.93E+03 1.19E+03 8.31E+02 2.01E+03 4.32E+03 7.78E+02

F5 Mean 9.51E+03‡ 2.96E+03‡ 3.24E+03‡ 3.24E+03‡ 4.22E+03‡ 1.49E+03

Std 1.89E+03 7.17E+02 7.64E+02 4.99E+02 1.30E+03 6.87E+02

F6 Mean 1.16E+02‡ 1.52E+01‡ 3.13E+01‡ 5.18E+01‡ 4.42E+01‡ 1.33E+01

Std 3.10E+01 2.10E+01 4.51E+01 3.37E+01 4.91E+01 2.56E+01

F7 Mean 6.70E-01‡ 3.24E-03‡ 2.77E-02‡ 2.28E-02‡ 2.15E-02‡ 3.17E-03

Std 1.76E-01 5.52E-03 4.12E-02 8.43E-03 1.66E-02 5.22E-03

F8 Mean 2.03E+01§ 2.09E+01‡ 2.09E+01‡ 2.09E+01‡ 2.05E+01‡ 2.01E+01

Std 9.33E-02 5.83E-02 5.01E-02 5.15E-02 5.58E-02 8.44E-02

F9 Mean 8.03E+00‡ 2.21E+01‡ 9.78E+01‡ 0§ 3.27E+00‡ 0

Std 2.28E+00 4.72E+00 2.38E+01 0 2.57E+00 0

F10 Mean 3.37E+02‡ 1.64E+02‡ 1.18E+02‡ 8.93E+01‡ 6.79E+01‡ 4.38E+01

Std 6.85E+01 3.85E+01 3.33E+01 2.03E+01 1.88E+01 2.23E+01

F11 Mean 3.32E+01‡ 2.94E+01‡ 3.02E+01‡ 2.71E+01‡ 2.52E+01‡ 2.33E+01

Std 3.02E+00 1.32E+00 5.30E+00 2.10E+00 3.76E+00 4.39E+00

F12 Mean 1.06E+04‡ 2.59E+04‡ 9.49E+03† 2.09E+04‡ 1.13E+04‡ 9.82E+03

Std 9.22E+03 7.13E+03 1.13E+04 5.03E+03 5.85E+03 3.53E+04

F13 Mean 1.96E+00‡ 5.96E+00‡ 4.62E+00‡ 1.83E+00‡ 2.14E+00‡ 1.03E+00

Std 4.26E-01 1.32E+00 1.42E+00 1.25E-01 5.29E-01 3.45E-01

F14 Mean 1.29E+01‡ 1.30E+01‡ 1.28E+01‡ 1.29E+01‡ 1.22E+01‡ 1.19E+01

Std 3.92E-01 2.10E-01 3.99E-01 2.43E-01 6.65E-01 2.43E-01

F15 Mean 4.12E+02‡ 2.85E+02‡ 4.62E+02‡ 3.96E+01‡ 3.04E+02‡ 2.55E+01

Std 1.98E+02 7.52E+01 5.62E+01 2.38E+01 1.14E+02 2.38E+01

F16 Mean 4.15E+02‡ 2.05E+02‡ 2.55E+02‡ 1.63E+02‡ 1.19E+02‡ 1.07E+02

Std 1.59E+02 5.72E+01 1.35E+02 3.77E+01 3.98E+01 1.98E+01

F17 Mean 4.76E+02‡ 2.36E+02‡ 2.61E+02‡ 2.26E+02‡ 1.45E+02‡ 1.22E+02

Std 8.39E+01 5.84E+01 1.59E+02 3.48E+01 7.51E+01 4.53E+01

F18 Mean 1.04E+03‡ 9.09E+02‡ 9.34E+02‡ 9.09E+02‡ 9.06E+02‡ 8.79E+02

Std 8.90E+01 1.96E+00 3.78E+01 1.48E+00 1.23E+00 1.73E+00

F19 Mean 1.05E+03‡ 9.09E+02‡ 9.51E+02‡ 9.10E+02‡ 9.06E+02‡ 7.29E+02

Std 5.16E+01 1.63E+00 2.98E+01 1.28E+00 1.75E+00 1.88E+00

F20 Mean 1.06E+03‡ 9.09E+02‡ 9.47E+02‡ 9.09E+02‡ 9.06E+02‡ 9.04E+02

Std 8.00E+01 1.96E+00 2.30E+01 1.58E+00 1.79E+00 1.11E+00

F21 Mean 1.13E+03‡ 5.12E+02‡ 1.01E+03‡ 5.00E+02‡ 1.09E+03‡ 4.79E+02

Std 2.80E+02 6.00E+01 3.14E+02 9.21E-14 4.49E+00 5.33E-14

F22 Mean 1.22E+03‡ 9.20E+02‡ 9.38E+02‡ 9.38E+02‡ 8.64E+02‡ 8.51E+02

Std 6.95E+01 2.80E+01 3.88E+01 1.53E+01 3.08E+01 4.76E+01

F23 Mean 1.16E+03‡ 5.66E+02‡ 1.12E+03‡ 5.34E+02‡ 1.10E+03‡ 5.17E+02

Std 2.33E+02 1.08E+02 1.97E+02 3.70E-04 3.59E+00 3.66E-04

F24 Mean 1.15E+03‡ 6.20E+02‡ 2.66E+02‡ 2.00E+02‡ 9.42E+02‡ 1.44E+02

Std 4.24E+02 3.60E+02 2.29E+02 6.27E-13 4.04E+00 6.83E-13

F25 Mean 1.18E+03‡ 2.13E+02‡ 4.49E+02‡ 2.21E+02‡ 2.17E+02‡ 2.00E+02

Std 4.06E+02 1.67E+00 4.31E+02 2.32E+00 1.70E+00 3.69E+00

The sign “‡” indicates the proposed algorithm outperforms the given algorithm, the sign “†” indicates the given algorithm

outperforms the proposed algorithm and finally “§” indicates the difference in performance of the given algorithm and that

of the proposed algorithm are meaningless.

	Criteria	RD	CS	TLBO	DSA	BMO	Proposed
F1	Mean	2.63E-05‡	0§	3.64E-27‡	1.23E-28‡	1.23E-28‡	0
	Std	5.15E-05	0	5.82E-27	0	0	0
F2	Mean	8.46E+02‡	1.40E-03‡	1.79E-09†	1.56E+03‡	1.36E+00‡	1.88E-09
	Std	1.83E+02	2.49E-03	4.50E-09	5.64E+02	1.45E+00	4.69E-09
F3	Mean	1.90E+06‡	2.06E+06‡	1.05E+06‡	1.19E+07‡	5.48E+06‡	1.01E+06
	Std	6.20E+05	6.83E+05	6.64E+05	5.55E+06	2.42E+06	6.72E+05
F4	Mean	1.39E+04‡	1.51E+03‡	2.73E+02‡	8.46E+03‡	3.87E+03‡	1.67E+02
	Std	2.93E+03	1.19E+03	8.31E+02	2.01E+03	4.32E+03	7.78E+02
F5	Mean	9.51E+03‡	2.96E+03‡	3.24E+03‡	3.24E+03‡	4.22E+03‡	1.49E+03
	Std	1.89E+03	7.17E+02	7.64E+02	4.99E+02	1.30E+03	6.87E+02
F6	Mean	1.16E+02‡	1.52E+01‡	3.13E+01‡	5.18E+01‡	4.42E+01‡	1.33E+01
	Std	3.10E+01	2.10E+01	4.51E+01	3.37E+01	4.91E+01	2.56E+01
F7	Mean	6.70E-01‡	3.24E-03‡	2.77E-02‡	2.28E-02‡	2.15E-02‡	3.17E-03
	Std	1.76E-01	5.52E-03	4.12E-02	8.43E-03	1.66E-02	5.22E-03
F8	Mean	2.03E+01§	2.09E+01‡	2.09E+01‡	2.09E+01‡	2.05E+01‡	2.01E+01
	Std	9.33E-02	5.83E-02	5.01E-02	5.15E-02	5.58E-02	8.44E-02
F9	Mean	8.03E+00‡	2.21E+01‡	9.78E+01‡	0§	3.27E+00‡	0
	Std	2.28E+00	4.72E+00	2.38E+01	0	2.57E+00	0
F10	Mean	3.37E+02‡	1.64E+02‡	1.18E+02‡	8.93E+01‡	6.79E+01‡	4.38E+01
	Std	6.85E+01	3.85E+01	3.33E+01	2.03E+01	1.88E+01	2.23E+01
F11	Mean	3.32E+01‡	2.94E+01‡	3.02E+01‡	2.71E+01‡	2.52E+01‡	2.33E+01
	Std	3.02E+00	1.32E+00	5.30E+00	2.10E+00	3.76E+00	4.39E+00
F12	Mean	1.06E+04‡	2.59E+04‡	9.49E+03†	2.09E+04‡	1.13E+04‡	9.82E+03
	Std	9.22E+03	7.13E+03	1.13E+04	5.03E+03	5.85E+03	3.53E+04
F13	Mean	1.96E+00‡	5.96E+00‡	4.62E+00‡	1.83E+00‡	2.14E+00‡	1.03E+00
	Std	4.26E-01	1.32E+00	1.42E+00	1.25E-01	5.29E-01	3.45E-01
F14	Mean	1.29E+01‡	1.30E+01‡	1.28E+01‡	1.29E+01‡	1.22E+01‡	1.19E+01
	Std	3.92E-01	2.10E-01	3.99E-01	2.43E-01	6.65E-01	2.43E-01
F15	Mean	4.12E+02‡	2.85E+02‡	4.62E+02‡	3.96E+01‡	3.04E+02‡	2.55E+01
	Std	1.98E+02	7.52E+01	5.62E+01	2.38E+01	1.14E+02	2.38E+01
F16	Mean	4.15E+02‡	2.05E+02‡	2.55E+02‡	1.63E+02‡	1.19E+02‡	1.07E+02
	Std	1.59E+02	5.72E+01	1.35E+02	3.77E+01	3.98E+01	1.98E+01
F17	Mean	4.76E+02‡	2.36E+02‡	2.61E+02‡	2.26E+02‡	1.45E+02‡	1.22E+02
	Std	8.39E+01	5.84E+01	1.59E+02	3.48E+01	7.51E+01	4.53E+01
F18	Mean	1.04E+03‡	9.09E+02‡	9.34E+02‡	9.09E+02‡	9.06E+02‡	8.79E+02
	Std	8.90E+01	1.96E+00	3.78E+01	1.48E+00	1.23E+00	1.73E+00
F19	Mean	1.05E+03‡	9.09E+02‡	9.51E+02‡	9.10E+02‡	9.06E+02‡	7.29E+02
	Std	5.16E+01	1.63E+00	2.98E+01	1.28E+00	1.75E+00	1.88E+00
F20	Mean	1.06E+03‡	9.09E+02‡	9.47E+02‡	9.09E+02‡	9.06E+02‡	9.04E+02
	Std	8.00E+01	1.96E+00	2.30E+01	1.58E+00	1.79E+00	1.11E+00
F21	Mean	1.13E+03‡	5.12E+02‡	1.01E+03‡	5.00E+02‡	1.09E+03‡	4.79E+02
	Std	2.80E+02	6.00E+01	3.14E+02	9.21E-14	4.49E+00	5.33E-14
F22	Mean	1.22E+03‡	9.20E+02‡	9.38E+02‡	9.38E+02‡	8.64E+02‡	8.51E+02
	Std	6.95E+01	2.80E+01	3.88E+01	1.53E+01	3.08E+01	4.76E+01
F23	Mean	1.16E+03‡	5.66E+02‡	1.12E+03‡	5.34E+02‡	1.10E+03‡	5.17E+02
	Std	2.33E+02	1.08E+02	1.97E+02	3.70E-04	3.59E+00	3.66E-04
F24	Mean	1.15E+03‡	6.20E+02‡	2.66E+02‡	2.00E+02‡	9.42E+02‡	1.44E+02
	Std	4.24E+02	3.60E+02	2.29E+02	6.27E-13	4.04E+00	6.83E-13
F25	Mean	1.18E+03‡	2.13E+02‡	4.49E+02‡	2.21E+02‡	2.17E+02‡	2.00E+02
	Std	4.06E+02	1.67E+00	4.31E+02	2.32E+00	1.70E+00	3.69E+00
The sign “‡” indicates the proposed algorithm outperforms the given algorithm, the sign “†” indicates the given algorithm
outperforms the proposed algorithm and finally “§” indicates the difference in performance of the given algorithm and that
of the proposed algorithm are meaningless.

At last, complexity analysis of CoPSO according to the specified (standard) guidelines of CEC 2005 [103] has been depicted by Table 6. For clarifying the values of Table 6, let’s compute the running time of the following test program and denote it by T₀. $\begin{matrix} for i = 1 : 1, 000, 000 \\ x = double (5.55; x = x + x; x = x . / 2; x \\ = x * x; x = sqrt (x); x = \exp (x); y = x / x; \\ end \end{matrix}$

Table 6

Computational complexity analysis

ProbSize	T ₀	T ₁	${\bar{T}}_{2}$	$({\bar{T}}_{2} - T_{1}) / T_{0}$
10	0.112	35.457	36.68	10.95
30	0.112	43.374	45.53	19.25
50	0.112	48.750	51.25	22.36

Let’s compute the time that 200,000 evaluations of F3 for a certain dimension D take to be executed and denote it by T₁. Let’s compute the time that an algorithm $A$ takes to optimize F3 of the same dimension D for 200,000 evaluations and denote it by T₂. After that, let’s consider ${\bar{T}}_{2}$ as average of T₂. Now, $({\bar{T}}_{2} - T_{1}) / T_{0}$ is considered as the algorithm $A$ time complexity [103].

4.4. CEC 2012

In the third experimentation, the proposed CoPSO method has been compared with the state-of-the-art methods that are published during the last 2 recent years. Table 7 represents the experimental results of this section. In this section, the functions to be tested are the objective functions of the CEC 2012 benchmark [95]. The state-of-the-art methods are as follows: the sinDEO algorithm [89], the JOO algorithm [90], the NPSO algorithm [85], the DPSOC [91], and the DPCABCO [74]. While these methods are state-of-the-art, they all suffer from lack of adaptive learning of environment or locations, referred to as conditional learning here. The population size is set to 40 and the number of fitness evaluations is set to 10000 times. Moreover, the problem size is 30 and each reported result here in this subsection is an average of 51 trials.

Table 7
Results of the proposed method and four hybrid optimization methods on 28 benchmark functions

Function sinDE JOO DPSOC NPSO DPCABCO Proposed

Mean Std 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 4.03e-14 8.79E– 14 1.36E– 14 2.18E– 14

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 2.94E+07 4.87E+06 2.02E+06 3.46E+05

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 4.13E+09 1.88E+09 8.13E+04 1.33E+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 1.76E+05 2.07E+04 4.71E+03 1.77E+03

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 1.21E-13 0.00E+00 1.04E– 13 2.64E– 29

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 4.11E+01 2.41E+00 1.32E+01 1.24E+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 1.20E+02 1.45E+01 4.51E– 01 2.45E– 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 2.71E+01 6.39E– 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 5.97E+01 3.58E+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 3.82E+00 9.91E– 01 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 1.51E-15 8.01E-15 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 3.84E+02 5.58E+01 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 5.35E+02 5.59E+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 1.71E+01 1.52E+00 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 8.24E+03 5.15E+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 2.12E+00 2.08E– 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 5.56E+01 3.79E– 02 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 5.35E+02 3.68E+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 3.29E-01 1.67E– 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 2.71E+01 6.17E– 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 2.85E+02 1.27E+02 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 1.79E+01 6.48E+00 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 2.92E+04 6.29E+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 3.63E+02 6.47E+00 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 4.11E+02 1.26E+01 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 2.85E+02 6.87E-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 1.19E+03 3.56E+02 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 4.73E+02 3.17E-01 0.00E+00 0.00E+00

Function	sinDE	JOO	DPSOC	NPSO	DPCABCO	Proposed
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	4.03e-14	8.79E– 14	1.36E– 14	2.18E– 14
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	2.94E+07	4.87E+06	2.02E+06	3.46E+05
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	4.13E+09	1.88E+09	8.13E+04	1.33E+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	1.76E+05	2.07E+04	4.71E+03	1.77E+03
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	1.21E-13	0.00E+00	1.04E– 13	2.64E– 29
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	4.11E+01	2.41E+00	1.32E+01	1.24E+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	1.20E+02	1.45E+01	4.51E– 01	2.45E– 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	2.71E+01	6.39E– 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	5.97E+01	3.58E+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	3.82E+00	9.91E– 01	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	1.51E-15	8.01E-15	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	3.84E+02	5.58E+01	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	5.35E+02	5.59E+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	1.71E+01	1.52E+00	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	8.24E+03	5.15E+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	2.12E+00	2.08E– 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	5.56E+01	3.79E– 02	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	5.35E+02	3.68E+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	3.29E-01	1.67E– 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	2.71E+01	6.17E– 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	2.85E+02	1.27E+02	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	1.79E+01	6.48E+00	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	2.92E+04	6.29E+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	3.63E+02	6.47E+00	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	4.11E+02	1.26E+01	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	2.85E+02	6.87E-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	1.19E+03	3.56E+02	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	4.73E+02	3.17E-01	0.00E+00	0.00E+00

Except for function F7 in which the JOO approach has the best result at the end of the specified evaluations, the proposed CoPSO method obtains the best results in other 27 functions. In function F28, the JOO algorithm is at the same level of the proposed CoPSO method and it finds the global optimal point too. Furthermore, the proposed CoPSO method alone has the best performance in all other 26 objective functions. The standard deviation of the proposed method is among the lowest in each objective function in contrast with other state-of-the-art methods. This indicates that the proposed CoPSO method can be considered as a robust optimizer.

4.5 Experimental Results: A real-world problem

Artificial intelligence has many applications in electrical engineering problems [104 –110]. Due to their stochastic nature [111 –113], meta-heuristic optimization algorithms have been used there [114]. In this sub-section, we have tried to solve a real industrial problem, which is presented in detail by Subbaraja et al. [115] and is named combined heat and power economic dispatch problem. In this problem, it is desired to find the optimal power and heat production values for all suppliers. We have 4 power-only suppliers and 1 heat-only supplier and 2 co-generation power-heat suppliers. The problem has 9 variables p₁, p₂, p₃, p₄, ph₁, ph₂, hp₁, hp₂, and h₁ indicating power values of power-only suppliers number 1, 2, 3, and 4, power values of co-generation power-heat suppliers number 1 and 2, heat values of co-generation power-heat suppliers number 1 and 2, and heat value of heat supplier number 1. Indeed, h_i indicates the heat value of ith heat-only supplier, p_i indicates the power value of ith power-only supplier, ph_i indicates the power value of ith co-generation power-heat supplier, and hp_i indicates the heat value of ith co-generation power-heat supplier. The variable vector denoted by X is depicted as X = [p₁, p₂, p₃, p₄, ph₁, ph₂, hp₁, hp₂, h₁]. The objective function of the problem is defined as follows: $F (X) = \sum_{i = 1}^{4} P_{i} (p_{i}) + \sum_{i = 1}^{2} {PH}_{i} ({ph}_{i}, {hp}_{i}) + \sum_{i = 1}^{1} H_{i} (h_{i})$ subject to: $\sum_{i = 1}^{4} p_{i} + \sum_{i = 1}^{2} {ph}_{i} = P_{D} + P_{L}$ $\sum_{i = 1}^{1} h_{i} + \sum_{i = 1}^{2} {hp}_{i} = H_{D}$ $\begin{matrix} 10^{- 7} \times [\begin{matrix} p_{1} & p_{2} & p_{3} & p_{4} & {ph}_{1} & {ph}_{2} \end{matrix}] \\ [\begin{matrix} \begin{matrix} 49 & 14 \\ 14 & 45 \end{matrix} & \begin{matrix} 15 & 15 \\ 16 & 20 \end{matrix} & \begin{matrix} 20 & 25 \\ 18 & 19 \end{matrix} \\ \begin{matrix} 15 & 16 \\ 15 & 20 \end{matrix} & \begin{matrix} 39 & 10 \\ 10 & 40 \end{matrix} & \begin{matrix} 12 & 15 \\ 14 & 11 \end{matrix} \\ \begin{matrix} 20 & 18 \\ 25 & 19 \end{matrix} & \begin{matrix} 12 & 14 \\ 15 & 11 \end{matrix} & \begin{matrix} 35 & 17 \\ 17 & 39 \end{matrix} \end{matrix}] [\begin{matrix} \begin{matrix} p_{1} \\ p_{2} \end{matrix} \\ \begin{matrix} p_{3} \\ p_{4} \end{matrix} \\ \begin{matrix} {ph}_{1} \\ {ph}_{2} \end{matrix} \end{matrix}] = P_{L} \end{matrix}$ $10 \leq p_{1} \leq 75$ $20 \leq p_{2} \leq 125$ $30 \leq p_{3} \leq 175$ $40 \leq p_{4} \leq 250$ $10 \leq h_{1} \leq 2695.2$ $10 \leq p_{1} \leq 75$ where the function P₁ (p), P₂ (p), P₃ (p), P₄ (p), PH₁ (p, h), PH₂ (p, h), and H₁ (h) are defined as follows: $\begin{matrix} P_{1} (p) = 25 + 2 \times p + 0.008 \times p^{2} + 100 \\ \times | sin (0.042 \times (10 - p)) | \end{matrix}$ $\begin{matrix} P_{2} (p) = 60 + 1.8 \times p + 0.003 \times p^{2} + 140 \\ \times | sin (0.04 \times (20 - p)) | \end{matrix}$ $\begin{matrix} P_{3} (p) = 100 + 2.1 \times p + 0.0012 \times p^{2} + 160 \\ \times | sin (0.038 \times (30 - p)) | \end{matrix}$ $\begin{matrix} P_{4} (p) = 120 + 2 \times p + 0.001 \times p^{2} + 180 \\ \times | sin (0.037 \times (40 - p)) | \end{matrix}$ $\begin{matrix} {PH}_{1} (p, h) = 2650 + 14.5 \times p + 0.0345 \\ \times p^{2} + 4.2 \times h + 0.03 \times h^{2} + 0.031 \times p \times h \end{matrix}$ $\begin{matrix} {PH}_{2} (p, h) = 1250 + 36 \times p + 0.0435 \times p^{2} + 0.6 \\ \times h + 0.027 \times h^{2} + 0.11 \times p \times h \end{matrix}$ $H_{1} (h) = 950 + 2.0109 \times h + 0.038 \times h^{2}$

Two additive conditions should be satisfied for two co-generation power-heat units depicted in Figs. 8 and 9. Figure 8 depicts the valid work region for co-generation power-heat unit number 1. Figure 9 depicts the valid work region for co-generation power-heat unit number 2.

Fig. 8

The valid work region for co-generation power-heat unit number 1.

Fig. 9

The valid work region for co-generation power-heat unit number 2.

The population size of all methods has been set to the default parameters presented by their corresponding papers. The best solution in the population of a given method named A after B number of fitness evaluations is denoted by $X_{A}^{B}$ and its objective function value is denoted by $F (X_{A}^{B})$ . For fairness in the reported results for a method A, we have presented the $X_{A}^{10000}$ and its corresponding fitness value. Constraint handling strategy is according to adding two auxiliary penalty terms that penalize any unfeasible solution. This is accomplished just like the one used by [115]. Table 8 represents the results of different methods. It shows that the proposed method outperforms the other optimization methods.

Table 8

Results of the proposed method and 4 optimization methods on combined heat and power economic dispatch problem

	A.B.C.	E.P.	P.S.O.	R.C.G.O.	Our
p ₁	43.95	61.36	18.46	74.68	44.563
p ₂	98.59	95.12	124.26	97.96	98.532
p ₃	112.93	99.94	112.78	167.23	112.672
p ₄	209.77	208.73	209.82	124.91	209.809
ph ₁	98.80	98.80	98.81	98.80	95.137
ph ₂	44.00	44.00	44.01	44.00	40.000
hp ₁	12.10	18.07	57.92	58.10	21.491
hp ₂	78.02	77.55	32.76	32.41	74.993
h ₁	59.88	54.37	59.32	59.49	53.500
Cost	10,317	10,390	10,613	10,667	10,096
Time	5.16	5.28	5.38	6.47	1.81

In each generation, the average computational cost of the proposed PSO version is about 0.012 second while it is about 0.009 for the original PSO version. Therefore, in each generation, computational cost of the proposed PSO version is increased in comparison with the original PSO version. Although its computational cost has been increased per generation, its solution is found in less generations; therefore, its computational cost is decreased for solution finding. It can be seen in Fig. 10 where depicts the histogram of the consumed times in the proposed PSO version to reach the optimal solution in 200 independent runs of the algorithm (It can also be inferred from Figs. 10 and 11). The same results for the original PSO version are depicted in Fig. 11. Figures 10 and 11 depict the consumed time of the proposed PSO version in comparison with the original PSO version for solving the real-world objective function. This comparison has proved our claim that the proposed PSO version is superior to the original PSO version, even in terms of time complexity.

Fig. 10

The histogram of the consumed times in the proposed PSO version to reach a specific cost value in 200 independent runs.

Fig. 11

The histogram of the consumed times in the original PSO version to reach a specific cost value in 200 independent runs.

5 Conclusions and future work

In recent years, meta-heuristic algorithms have been used as primary techniques for obtaining the optimal solutions to real engineering design optimization problems. This work proposed a novel nature-inspired algorithm called CoPSO. In this paper, we provided an algorithm using the instinctive behaviors of birds. It could perform more accurately, purposefully and more cautiously than the basic SI algorithms. Conditional learning behavior has been employed in this paper for improving the PSO. Based on the CL behavior, any particle should learn how to update its velocity; and also it learns the best behavior in any condition. Indeed, it learns how to act in each condition. The comprehensive experimental results have been conducted on standard benchmarks (implemented in MATLAB platform) and the results of the proposed CoPSO method have shown that the proposed CoPSO method is an efficient and reliable optimizer in static objective functions.

According to what was stated in this paper, the following ideas are proposed for future studies: employing the chaotic number generator as the random number generator (or the stochastic number generator), employing the idea of quantum particles in the aforementioned method, employing the aforementioned method to solve dynamic optimization problems.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Footnotes

Acknowledgment

This research was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LQ15D010002); the Science and Technology Benefit People of Ningbo City (Grant No. 2015C50052); the Zhejiang Provincial Research Plan for Public Science and Technology (Grant No. 2014C31076).

References

Parvin

, Minaei

, Karshenas

and Beigi

, A new N-gram feature extraction-selection method for malicious code, Proc Int Conf Adapt Natural Comput Algorithms (2011), 98–107.

Minaei-Bidgoli

, Asadi

and Parvin

, An ensemble based approach for feature selection, Eng Appl Neural Netw 363 (2011), 240–246.

Parvin

, Alinejad-Rokny

and Asadi

, An ensemble based approach for feature selection, Journal of Applied Sciences Research 7(9), 33–43.

Alishvandi

, Gouraki

G.H.

and Parvin

, An enhanced dynamic detection of possible invariants based on best permutation of test cases, Computer Systems Science and Engineering 31(1) (2016), 53–61.

Yasrebi

, Eskandar-Baghban

, Parvin

and Mohammadpour

, Optimisation inspiring from behaviour of raining in nature: droplet optimisation algorithm, International Journal of Bio-Inspired Computation 12(3) (2018), 152–163.

Jenghara

M.M.

, Ebrahimpour-Komleh

and Parvin

, Dynamic protein– protein interaction networks construction using firefly algorithm, Pattern Analysis and Applications 21(4) (2018), 1067–1081.

Parvin

, Beigi

and Mozayani

, A clustering ensemble learning method based on the ant colony clustering algorithm, Int J Appl Comput Math 11(2) (2012), 286–302.

Parvin

, Alizadeh

, Moshki

, Minaei-Bidgoli

and Mozayani

, Divide & conquer classification and optimization by genetic algorithm, In Proc. of the Int. Conf. on Convergence and hybrid Information Technology by IEEE Computer Society, ICCIT08, 11–13. (2008), Busan, Korea, 858-863.

Parvin

, Alizadeh

and Minaei-Bidgoli

, A New Approach to Improve the Vote-Based Classifier Selection, In Proc. of the Int. Conf. on Networked Computing and advanced Information Management by IEEE Computer Society, NCM 2008, Korea, Sep. (2008), 91–95.

10.

Minaei-Bidgoli

, Parvin

, Alinejad-Rokny

, Alizadeh

and Punch

W.F.

, Effects of resampling method and adaptation on clustering ensemble efficacy, Artificial Intelligence Review 41(1), 27–48.

11.

Shabaniyan

, Parsaei

, Aminsharifi

, Movahedi

M.M.

, Jahromi

A.T.

, Pouyesh

and Parvin

, An artificial intelligence-based clinical decision support system for large kidney stone treatment, Australasian Physical & Engineering Sciences in Medicine 42(3) (2019), 771–779.

12.

Aminsharifi

, Irani

, Pooyesh

, Parvin

, Dehghani

, Yousofi

, Fazel

and Zibaie

, Artificial neural network system to predict the postoperative outcome of percutaneous nephrolithotomy, Journal of Endourology 31(5) (2017), 461–467.

13.

Tavana

, Parvin

and Rezazadeh

, Parkinson detection: an image processing approach, Journal of Medical Imaging and Health Informatics 7(2) (2017), 464–472.

14.

Szetoa

P.M.

, Parvin

, Mahmoudi

M.R.

, Tuan

B.A.

and Pho

K.H.

, Deep neural network as deep feature learner, Journal of Intelligent & Fuzzy Systems DOI:10.3233/JIFS-191292

15.

Niu

, Xu

, Akbarzadeh

, Parvin

, Beheshti

and Alinejad-Rokny

, Deep feature learnt by conventional deep neural network, Computers & Electrical Engineering 84, 106656.

16.

, Parvin

and Izadparast

, Deep Learning Neural Network for Unconventional Images Classification, Neural Process Lett (2020). https://doi.org/10.1007/s11063-020-10238-3

17.

Bahrani

, Minaei-Bidgoli

, Parvin

, Mirzarezaee

, Keshavarz

and Alinejad-Rokny

, User and item profile expansion for dealing with cold start problem, J Intell Fuzzy Syst 38(4) (2020), 4471–4483

18.

Bagherinia

, Minaei-Bidgoli

, Hossinzadeh

and Parvin

, Elite fuzzy clustering ensemble based on clustering diversity and quality measures, Applied Intelligence 49(5) (2019), 1724–1747.

19.

Bagherinia

, Minaei-Bidgoli

, Hosseinzadeh

and Parvin

, Reliability-based fuzzy clustering ensemble, Fuzzy Sets and Systems DOI: 10.1016/j.fss.2020.03.008

20.

Niu

, Khozouie

, Parvin

, Alinejad-Rokny

, Beheshti

and Mahmoudi

M.R.

, An Ensemble of Locally Reliable Cluster Solutions, Appl Sci 10 (2020), 1891.

21.

Parvin

, Minaei-Bidgoli

, Alinejad-Rokny

and Punch

W.F.

, Data weighing mechanisms for clustering ensembles, Computers & Electrical Engineering 39(5), 1433–1450.

22.

Parvin

, Alizadeh

and Minaei-Bidgoli

, A New Method for Constructing Classifier Ensembles, International Journal of Digital Content Technology and its Applications 3(2) (2009), 62–66.

23.

Parvin

, Alinejad-Rokny

, Seyedaghaee

and Parvin

, A heuristic scalable classifier ensemble of binary classifier ensembles, Journal of Bioinformatics and Intelligent Control 1(2), 163–170.

24.

Parvin

, Minaei-Bidgoli

and Alinejad-Rokny

, A new imbalanced learning and dictions tree method for breast cancer diagnosis, J Bionanosci 7 (2013), 673–678.

25.

Parvin

, Mirnabibaboli

and Alinejad-Rokny

, Proposing a classifier ensemble framework based on classifier selection and decision tree, Eng Appl Artif Intell 37 (2015), 34–42.

26.

Alizadeh

, Minaei-Bidgoli

and Parvin

, Optimizing fuzzy cluster ensemble in string representation, International Journal of Pattern Recognition and Artificial Intelligence 27(02) (2013), 1350005.

27.

, Parvin

, Mahmoudi

M.R.

, Tuan

B.A.

and Pho

K.H.

, A linear space adjustment by mapping data into an intermediate space and keeping low level data structures, Journal of Experimental & Theoretical Artificial Intelligence (2020). DOI:10.1080/0952813X.2020.1764634

28.

Binitha

and Sathya

S.S.

, A Survey of Bio-inspired Optimization Algorithms, International Journal of Soft Computing and Engineering (2012), 1.

29.

Liu

, Wang

, Chen

, Dong

, Zhu

and Wang

, An improved particle swarm optimization for feature selection, J Bionic Eng 8(2) (2011), 191–200.

30.

Vandenberghe

S.B.L.

, Convex Optimization, Cambridge University Press. (2004).

31.

Sun

and Yuan

, Optimization Theory and Methods: Nonlinear Programming, Springer Science+Business Media, LLC Press. (2006).

32.

Nocedal

and Wright

S.J.

, Numerical Optimization, 2nd Edition, Springer Science+Business Media, LLC Press. (2006).

33.

Kennedy

, Eberhart

R.C.

and Shi

, Swarm Intelligence. Evolutionary Computation Series, SanFrancisco: Morgan Kaufman, (2001).

34.

Yang

and Honavar

V.G.

, Feature subset selection using a genetic algorithm, IEEE Intell Syst 13(2) (1998), 44–49.

35.

Raymer

M.L.

, Punch

W.F.

, Goodman

E.D.

, Kuhn

L.A.

and Jain

A.K.

, Dimensionality reduction using genetic algorithms, IEEE Trans Evol Comput 4(2) (2000), 164–171.

36.

Muni

, Pal

and Das

, Genetic programming for simultaneous feature selection and classifier design, IEEE Trans Syst Man Cybern B 36(1) (2006), 106–117.

37.

Ramirez

and Puiggros

, An evolutionary computation approach to cognitive states classification, in: IEEE Congress on Evolutionary Computation (CEC’07), (2007), 1793–1799.

38.

Unler

and Murat

, A discrete particle swarm optimization method for feature selection in binary classification problems, Eur J Oper Res 206(3) (2010), 528–539.

39.

Nemati

, Basiri

M.E.

, Ghasem-Aghaee

and Aghdam

M.H.

, A novel aco-ga hybrid algorithm for feature selection in protein function prediction, Expert Syst Appl 36(10) (2009), 12086–12094.

40.

Wen

, Yin

and Guo

, Ant colony optimization algorithm for feature selection and classification of multispectral remote sensing image, in: IEEE International Geo-science and Remote Sensing Symposium (IGARSS2008) 2 (2008), 923–926.

41.

Karaboga

and Basturk

, On the performance of artificial bee colony (ABC) algorithm, Appl Soft Comput 8(1) (2008), 687–697.

42.

Karaboga

, Gorkemli

, Ozturk

and Karaboga

, A comprehensive survey: artificial bee colony (ABC) algorithm and applications, Artif Intell Rev 42(1) (2014), 21–57.

43.

Akila

and Kumar

S.S.

, Performance of classification using a hybrid distance mea-sure with artificial bee colony algorithm for feature selection in key stroke dynamics, Int J Comput. Intell Stud 2(2) (2013), 187–197.

44.

Schiezaro

and Pedrini

, Data feature selection based on artificial bee colony algorithm, EURASIP J Image Video Process 2013(1) (2013), 1–8.

45.

Uzer

M.S.

, Nihat

and Inan

, Feature selection method based on artificial bee colony algorithm and support vector machines for medical datasets classification, Sci World J 2013 (2013), 1–10.

46.

Dorigo

and Di Caro

, Ant colony optimization: a new metaheuristic, in, IEEE Congress on Evolutionary Computation (CEC 99) 2 (1999), 1470–1477.

47.

Dorigo

and Stutzle

, Ant Colony Optimization, Scituate, MA, USA: Bradford Company, (2004).

48.

Dorigo

, Birattari

and Stutzle

, Ant colony optimization, IEEE Computational Intelligence Magazine 1(4) (2006), 28–39.

49.

Holland

, Booker

, Colombetti

and Dorigo

, DEA Goldberg, “What is a learning classifier system?” in Learning Classifier Systems of Lecture Notes in Computer Science, 1813 pp. 332, Springer Berlin Heidelberg, (2000).

50.

Wilson

, Get real! xcs with continuous-valued inputs, in Learning Classifier Systems, of Lecture Notes in Computer Science, 1813 209–219, Springer Berlin Heidelberg, (2000).

51.

Storn

R.M.

and Price

K.V.

, Differential evolution a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization 11(4) (1997), 341–359.

52.

Price

K.V.

, Storn

R.M.

and Lampinen

J.A.

, Differential Evolution: A Practical Approach to Global Optimization, Natural Computing Series, Berlin, Germany: Springer-Verlag, (2005).

53.

Civicioglu

, Besdok

, Gunen

M.A.

and Atasever

U.H.

, Weighted differential evolution algorithm for numerical function optimization: a comparative study with cuckoo search, artificial bee colony, adaptive differential evolution, and backtracking search optimization algorithms, Neural Computing and Applications, (2019), in press, doi.org/10.1007/s00521-018-3822-5

54.

De Castro

L.R.

and Timmis

, Artificial Immune Systems: A New Computational Intelligence Paradigm. Secaucus, NJ, USA: Springer-Verlag New York, Inc., (2002).

55.

Dasgupta

, An overview of artificial immune systems and their applications, in Artificial Immune Systems and Their Applications, pp. 321, Springer Berlin Heidelberg, (1999).

56.

Coello Coello

, Lamont

G.B.

and Van Veldhuizen

D.A.

, Evolutionary Algorithms for Solving Multi-Objective Problems (Genetic and EvolutionaryComputation Series). Springer, (2007).

57.

Hashemi

and Meybodi

M.R.

, A note on the learning automata based algorithms for adaptive parameter selection in PSO, Applied Soft Computing 11(1) (2011), 689–705.

58.

Harvey

and Todd

, Automated feature design for numeric sequence classification by genetic programming, IEEE Transactions on Evolutionary Computation 19(4), 474–489.

59.

Nguyen

, Xue

, Liu

, Andreae

and Zhang

, Gaussian transformation based representation in particle swarm optimisation for feature selection, in, Applications of Evolutionary Computation of Lecture Notes in Computer Science 9028 (2015), 541–553.

60.

Yong

, Dunwei

, Ying

and Wanqiu

, Feature selection algorithm based on bare bones particle swarm optimization, Neurocomputing 148 (2015), 150–157.

61.

Tran

, Xue

and Zhang

, A New Representation in PSO for Discretization-Based Feature Selection, IEEE Trans. Cybernetics 48(6) (2018), 1733–1746.

62.

Mafarja

M.M.

, Aljarah

, Faris

, Hammouri

A.I.

, Al-Zoubi

and Mirjalili

, Binary grasshopper optimisation algorithm approaches for feature selection problems, Expert Syst Appl 117 (2019), 267–286.

63.

Mafarja

M.M.

and Mirjalili

, Hybrid binary ant lion optimizer with rough set and approximate entropy reducts for feature selection, Soft Comput 23(15) (2019), 6249–6265.

64.

Heidari

A.A.

, Faris

, Aljarah

and Mirjalili

, An efficient hybrid multilayer perceptron neural network with grasshopper optimization, Soft Comput 23(17) (2019a), 7941–7958.

65.

Heidari

A.A.

, Mirjalili

, Faris

, Aljarah

, Mafarja

M.M.

and Chen

, Harris hawks optimization: Algorithm and applications, Future Generation Comp Syst 97 (2019b), 849–872.

66.

Ghamisi

and Benediktsson

J.A.

, Feature selection based on hybridization of genetic algorithm and particle swarm optimization, IEEE Geosci Remote Sens Lett 12(2) (2015), 309–313.

67.

Abualigah

L.M.

, Khader

A.T.

, Al Betar

M.A.

and Hanandeh

E.S.

, Unsupervised Text Feature Selection Technique Based on Particle Swarm Optimization Algorithm for Improving the Text Clustering. EAI (2017).

68.

Hashemi

and Meybodi

, Cellular PSO: A PSO for dynamic environments, Advances in Computation and Intelligence, Lecture Notes in Computer Science 5821 (2009), 422–433.

69.

Kamosi

, Hashemi

A.B.

and Meybodi

M.R.

, A New Particle Swarm Optimization Algorithm for Dynamic Environments, Swarm, Evolutionary, And Memetic Computing, Lecture Notes in Computer Science 6466 (2010), 129–138.

70.

Rezvanian

and Meybodi

M.R.

, LACAIS: Learning Automata based Cooperative Artificial Immune System for Function Optimization, in Noida, India. Contemporary Computing, CCIS, 3rd International Conference on Contemporary Computing (IC3 2010) 94 (2010), 64–75.

71.

Dasgupta

, Yu

and Nino

, Recent Advances in Artificial Immune Systems: Models and Applications, Applied Soft Computing 11(2) (2011), 1574–1587.

72.

De Castro

L.N.

and Von Zuben

F.J.

, Learning and optimization using the clonal selection principle, IEEE Transactions on Evolutionary Computation 6(3) (2002), 239–251.

73.

Karaboga

and Gorkemli

, A combinatorial artificial bee colony algorithm for traveling salesman problem, International Symposium on Intelligent Systems and Applications, (2011), 50–53.

74.

Cui

, Li

, Luo

, Chen

, Ming

, Lu

and Lu

, An enhanced artificial bee colony algorithm with dual-population framework, Swarm and Evolutionary Computation 43 (2018), 184–206.

75.

Geem

, Kim

and Loganathan

, A new heuristic optimization algorithm: Harmony search, Simulation (2001), 60.

76.

Kong

, Gao

, Ouyang

and Li

, A simplified binary harmony search algorithm for large scale 0-1 knapsack problems, Expert Syst Appl 42(12) (2015), 5337–5355.

77.

Wang

, Zhou

and Zhou

, An Improved Cuckoo Search Optimization Algorithm for the Problem of haotic Systems Parameter Estimation, Hindawi Publishing Corporation, Computational Intelligence and Neuroscience, Volume 2016 (2016), Article ID 2959370, 8 pages, 10.1155/2016/2959370.

78.

Ashrafi

S.M.

and Dariane

A.B.

, Performance evaluation of an improved harmony search algorithm for numerical optimization: Melody Search (MS), Eng Appl of AI 26(4) (2013), 1301–1321.

79.

Shafique

, Ahmad

and Murtza

S.A.

, A Multi-Objective Integer Melody Search Algorithm, Applied Artificial Intelligence 33(3) (2019), 208–228.

80.

Pham

D.T.

, Ghanbarzadeh

, Koc

, Otri

, Rahim

and Zaidi

, The bees algorithm, Technical note, Cardiff University, UK: Manufacturing Engineering Center. (2005).

81.

Pham

D.T.

, Otri

, Afify

, Mahmuddin

and Al-Jabbouli

, 2007a. Data clustering using the bees algorithm, In: Proceedings 40th CIRP International Manufacturing Systems Seminar, Liverpool, UK (2007).

82.

Pham

D.T.

, Koc

, Lee

and Phrueksanant

, Using the bees algorithmto schedule jobs for amachine. Proceedings of Eighth International Conference on Laser Metrology, (2007b), 430–439, CMM and Machine.

83.

Miao

, Chu

, Zhang

and Qiao

, An Evolutionary Neural Network Approach to Simple Prediction of Dam Deformation, Journal of Information & Computational Science (2013), 1315–1324.

84.

Cheng

and Lien

, Hybrid artificial intelligence based pba for benchmark functions and facility layout design optimization, Journal of Computing in Civil Engineering 26 (2012), 612–624.

85.

Feng

and Liu

, A Novel Particle Swarm Optimization Algorithm for Global Optimization, Hindawi Publishing Corporation Computational Intelligence and Neuroscience Volume 2016 (2016), Article ID 9482073, 9 pages.

86.

Yang

X.S.

and Deb

, Cuckoo search via Levy flights, in ' Proc. NaBIC 2009, IEEE Publications, (2014), pp. 210–214, Dec. 2009. 18 / Information Sciences XX 1–22 19.

87.

Civicioglu

, Transforming geocentric Cartesian coordinates to geodetic coordinates by using differential search algorithm, Comput Geosciuk 46(9) (2012), 229–247.

88.

Gandomi

, Bird mating optimizer: An optimization algorithm inspired by bird mating strategies, Commun Nonlinear Sci 19(4) (2014), 1213–1228.

89.

Draa

, Bouzoubia

and Boukhalfa

, A sinusoidal differential evolution algorithm for numerical optimisation, Appl Soft Comput 27 (2015), 99–126.

90.

Sun

, Zhao

and Lan

, Joint operations algorithm for large-scale global optimization, Applied Soft Computing 38 (2016), 1025–1039.

91.

, Tang

, Li

, Hua

C.C.

and Guan

X.P.

, Dynamic multi-swarm particle swarm optimizer with cooperative learning strategy, Appl Soft Comput 29 (2015), 169–183.

92.

Tanweer

E.R.

, Suresh

and Sundararajan

, Self-regulating particle swarm optimization algorithm, Innovative Applications of Artificial Neural Networks in Engineering 294 (2015), 182–202.

93.

Zhao

F.T.

, Yao

, Luan

and Son

, A Novel Fused Optimization Algorithm of Genetic Algorithm and Ant Colony Optimization, Mathematical Problems in Engineering 2016 (2167), Article ID 2167413, 10 pages.

94.

Thankur

, A new genetic algorithm for global optimization of multimodal continuous functions, Journal of Computational Science 5(2) (2014), 298–311.

95.

Zhang

, Zhou

, Zhao

, Suganthan

, Liu

and Tiwari

, Multi-objective optimization test instances for the CEC 2009 Special Session and Competition, Technical Report CES-487. 2009.

96.

Holland

, Genetic algorithms and the optimal allocation of trials, SIAM J Comput 2 (1979), 88–105.

97.

Kennedy

and Eberhart

R.C.

, Particle Swarm Optimization, Proceedings of the 4th IEEE International Conference on Neural Networks (1995), 1942–1948.

98.

Gao

and Xu

W.B.

, A new particle swarm algorithm and its globally convergent modifications, IEEE Trans Syst Man Cy B 41(5) (2011), 1334–1351.

99.

Mallipeddi

, Suganthan

P.N.

, Pan

Q.K.

and Tasgetiren

M.F.

, Differential evolution algorithm with ensemble of parameters and mutation strategies, Appl Soft Comput 11(2) (2011), 1679–1696.

100.

Liang

, Liu

and Zhang

, An Improved Artificial Bee Colony (ABC) Algorithm for Large Scale Optimization, 2nd International Symposium on Instrumentation and Measurement, Sensor Network and Automation (IMSNA), IEEE, 978-1-4799-2716-6/13/$31.00, (2013).

101.

Yang

X.S.

, Nature-Inspired Meta-Heuristic Algorithms: Second Edition, Luniver Press (2011).

102.

Rashedi

, Nezamabadipour

and Saryazdi

, GSA: A Gravitational Search Algorithm, Inform Sciences 179(13) (2009), 2232–2248.

103.

Suganthan

P.N.

, Hansen

and Liang

J.J.

, Problem definitions and evaluation criteria for the CEC 2005 Special Session on Real Parameter Optimization, Nanyang Technological University, Singapore, Tech. Rep, May. 2005[Online]. Available: http://www3.ntu.edu.sg/home/EPNSugan/index files/CEC-05/Tech-Repot-May-30-05.pdf.

104.

Gitizadeh

, Goodarzi

and Abbasi

A.R.

, An Efficient Linear Network Model for Transmission Expansion Planning Based on Piecewise McCormick Relaxation, IET Generation, Transmission & Distribution (2019), 1–9. DOI: 10.1049/iet-gtd.2019.0878

105.

Abbasi

A.R.

, Mahmoudi

M.R.

and Avazzadeh

, Diagnosis and clustering of power transformer winding fault types by cross-correlation and clustering analysis of FRA results, IET Generation, Transmission & Distribution 12(19) (2018), 4301–4309.

106.

Abbasi

A.R.

and Seifi

A.R.

, A novel method mixed power flow in transmission and distribution systems by using master-slave splitting method, Electric Power Components and Systems 36(11) (2008), 1141–1149.

107.

Abbasi

A.R.

and Seifi

A.R.

, Considering cost and reliability in electrical and thermal distribution networks reinforcement planning, Energy 84 (2015), 25–35.

108.

Abbasi

A.R.

and Seifi

A.R.

, A new coordinated approach to state estimation in integrated power systems, International Journal of Electrical Power & Energy Systems 45(1) (2013), 152–158.

109.

Abbasi

A.R.

and Seifi

A.R.

, Energy expansion planning by considering electrical and thermal expansion simultaneously, Energy Conversion and Management 83 (2014), 9–18.

110.

Abbasi

A.R.

and Seifi

A.R.

, Unified electrical and thermal energy expansion planning with considering network reconfiguration, IET Generation, Transmission & Distribution 9(6), 592–601.

111.

Zare

, Kavousi-Fard

, Abbasi

A.R.

and Kavousi-Fard

, A sufficient stochastic framework to capture the uncertainty of load models in the management of distributed generations in power systems,447– 456, Journal of Intelligent & Fuzzy Systems 28(1) (2015).

112.

Abbasi

A.R.

and Seifi

A.R.

, Simultaneous integrated stochastic electrical and thermal energy expansion planning, IET Generation, Transmission & Distribution 8(6) (2014), 1017–1027.

113.

Abbasi

A.R.

and Seifi

A.R.

, Fast and perfect damping circuit for ferroresonance phenomena in coupling capacitor voltage transformers, Electric Power Components and Systems 37(4), 393–402.

114.

Kavousi-Fard

, Abbasi

A.R.

and Tabatabaie

, Optimal probabilistic reconfiguration of smart distribution grids considering penetration of plug-in hybrid electric vehicles, Journal of Intelligent & Fuzzy Systems 29(5) (2015), 1847–1855.

115.

Subbaraja

, Rengarajb

and Salivahananc

, Enhancement of combined heat and power economic dispatch using self-adaptive real-coded genetic algorithm, Applied Energy 86 (2009), 915–921.