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Ransikarn Ngambusabongsopa 1 and Zhiyong Li 1 and Esraa Eldesouky 1, 2

Academic Editor:Julien Bruchon

1, College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China
2, Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt

Received 10 August 2015; Accepted 29 September 2015; 9 December 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Various optimization approaches have been used to solve existing global optimization problems in science, engineering, and related fields. Evolutionary algorithms (EAs), inspired by the biological evolution and social behavior of organisms, are the most powerful among optimization frameworks. Several EAs have been successfully used to solve global optimization problems. Genetic algorithms (GAs) [1, 2] were developed based on the natural process of evolution. Particle swarm optimization (PSO) [3] was inspired by the social behavior of bird flocking or fish schooling. Memetic algorithms (MAs) [4] are similar to GAs except that MAs initiate a local search before undergoing the evolutionary process [5]. Ant colony optimization [6] resembles PSO based on the capability of ants to find the shortest route between their nest and a food source. Artificial bee colony [7, 8] was inspired by the echolocation behavior of bees. Harmony search [9] was based on the natural musical performance of a musician searching for an optimal state of harmony.

A recent study has proposed a new chemical reaction-inspired EA named "chemical reaction optimization" [10, 11]. CRO is a low-latency algorithm that mimics molecular interactions in chemical reactions to reach a low-energy stable state. An effective version of CRO is the real-coded chemical reaction optimization (RCCRO) [12], which can process a large set of continuous problems [13-17]. HP-CRO, a hybrid algorithm based on PSO and CRO, produces more distinguished results compared with RCCRO [18]. However, orthogonal chemical reaction optimization (OCRO) algorithms for global numerical optimization problems [19] are the best when compared with RCCRO and HP-CRO. OCRO is efficient in solving high-dimension functions but not in solving low-dimensional functions.

The current study proposes a hybrid mutation chemical reaction optimization algorithm for global numerical optimization (MCRO). This strategy can be used to solve high-dimension and low-dimension functions better and faster. The proposed framework combines CRO with a turning operator and a mutation operator. Our algorithm was determined by choosing the best solution that estimates the performance of the combination with three types of mutation operators (uniform, nonuniform, and polynomial).

This paper is organized as follows. Section 2 provides a brief review of preliminary studies on CRO and mutation operator. Section 3 presents the proposed approach: MCRO. Section 4 describes the evaluation methods and simulation results and discusses the comparisons of algorithms. Section 5 concludes this paper and suggests proposals for future work.

2. Preliminary

2.1. Optimization

In science, mathematics, engineering, and other related fields, optimization is the selection of an optimal solution from a set of feasible alternatives. In general, an optimization problem includes minimizing or maximizing a function by systematically selecting input values from a given feasible set [20]. The problem function [figure omitted; refer to PDF] is scalar, where a variable [figure omitted; refer to PDF] demonstrates a particular solution and [figure omitted; refer to PDF] is usually a vector of [figure omitted; refer to PDF] components. The [figure omitted; refer to PDF] components specify the dimensions of [figure omitted; refer to PDF] . An optimization problem can be subjected to a vector of constraints [figure omitted; refer to PDF] that limits the feasible region, where [figure omitted; refer to PDF] corresponds to the total number of constraints [12]. The aim of optimization is to generate the optimal solution and manage the convergence speed of a problem function. Our research focuses on function minimization; the goal of minimum optimization is to find the minimum solution [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

2.2. CRO

In principle, chemical reactions are governed by two rules of thermodynamics. First, energy cannot be created or destroyed but can be transformed from one entity to another; second, the entropy of a system tends to increase, where entropy is the measure of the degree of disorder [21]. CRO was inspired by the nature of chemical reactions and mimics the interactions of molecules in a chemical reaction to reach a low-energy stable state. Potential energy is the energy stored in a molecule with respect to its molecular configuration; the system becomes disordered when potential energy is converted to other forms [11]. Molecules stored in a container are vital to the manipulation of agents. Each molecule contains a profile that includes several attributes, such as molecular structure ( [figure omitted; refer to PDF] ), current potential energy ( [figure omitted; refer to PDF] ), and current kinetic energy ( [figure omitted; refer to PDF] ). In CRO reaction, the initial reactants in high-energy states incur a sequence of collisions. Molecules collide either with other molecules or with the walls of the container, pass through energy barriers, and become the final products in low-energy stable states. Four types of elementary CRO reactions capture the transition of high-energy molecules to stable states [10].

On-wall ineffective collision is the reaction created when a molecule hits the wall of a container and then bounces back. This reaction only slightly changes the molecular structure [figure omitted; refer to PDF] when [figure omitted; refer to PDF] occurs. The modification of molecular structure [figure omitted; refer to PDF] is processed by neighborhood search operator as [figure omitted; refer to PDF] \in neighborhood ( [figure omitted; refer to PDF] ). The central energy buffer is updated by extracting and storing a certain portion of its KE. The profile of the molecule is updated as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a random number that [figure omitted; refer to PDF] . The new [figure omitted; refer to PDF] is calculated using the new [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .

Intermolecular ineffective collision refers to two or more molecules that collide with each other and then separate. The profiles of the molecules and the central energy buffer are updated when [figure omitted; refer to PDF] . The number of molecules is unaltered after the collision, but the molecular structures are produced from their own neighborhoods by neighborhood search operator. Similar to on-wall ineffective collision, this reaction also slightly changes the molecular structure.

Decomposition occurs when a molecule hits the wall of a container and then splits into two or more molecules. This elementary reaction is applied to finish local search and to explore other regions. The profiles of the molecules and the central energy buffer are updated when [figure omitted; refer to PDF] and when the energy buffer is sufficient. This reaction significantly alters the molecular structures of the resultant molecules.

Synthesis represents the situation when two or more molecules collide and mingle to a single molecule. The profiles of the molecules and the central energy buffer are updated when [figure omitted; refer to PDF] . This reaction strongly and significantly alters the resultant molecular structure.

The algorithm of CRO is shown as Algorithm 1 [12]. On-wall and intermolecular ineffective collisions are local searches, whereas decomposition and synthesis are global searches.

Algorithm 1: Chemical reaction optimization (CRO).

Input : Objective function [figure omitted; refer to PDF] , constraints, and the dimensions of the problem

(1) / [figure omitted; refer to PDF] initialization phase [figure omitted; refer to PDF] /

(2) Assign parameter value to PopSize , InitialKE , StepSize , buffer , KELossRate , On-wallColl ,

(3) DecThres, SynThres, The set of molecules in this Container are molecules [figure omitted; refer to PDF]

(4) for each of the molecule do

(5) Assign a random solution to the molecular structure [figure omitted; refer to PDF] . Calculate the PE by [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and evaluate.

(6) Assign the KE with InitialKE

(7) end for

(8) / [figure omitted; refer to PDF] Iterations phase [figure omitted; refer to PDF] /

(9) while (the stopping criteria not met) do

(10) Get [figure omitted; refer to PDF] randomly in interval [figure omitted; refer to PDF]

(11) if ( [figure omitted; refer to PDF] ) then

(12) select a molecule [figure omitted; refer to PDF] from Container Randomly

(13) if (Decomposition criterion met) then

(14) Decomposition

(15) if Success then Add a new molecule [figure omitted; refer to PDF] to Container

(16) else

(17) On-wallIneffectiveCollision

(18) end if

(19) else

(20) select two molecule [figure omitted; refer to PDF] and [figure omitted; refer to PDF] from Container Randomly

(21) if (Synthesis criterion met) then

(22) Synthesis

(23) if Success then Remove molecule [figure omitted; refer to PDF] from Container

(24) else

(25) Inter-molecularIneffectiveCollision

(26) end if

(27) end if

(28) evaluate and keep any new minimum solution

(29) end while

(30) / [figure omitted; refer to PDF] The Output phase [figure omitted; refer to PDF] /

(31) Output the best solution and its function value.

2.3. Mutation Operator

Mutation is a natural process that changes a DNA sequence to a new variation endowing individual survival advantages in biology. Mutation operator, which has been applied to GAs, simulates the natural biological evolution of species by exchanging information and producing offspring chromosomes [1, 2]. In GAs, mutation operator is designed to work on an individual chosen; this individual is randomly selected or chosen by defined condition. The new gene value is modified based on user-definable probability or other conditions such as a distribution. GAs containing mutation can widely maintain offspring chromosomes, instead of being limited to the genes available in the initial population. Therefor GAs with mutation can generate better solution and prevent premature convergence compared to GAs without mutation.

The concept of MCRO is initialed by supposing that advantage of mutation operator would empower global numerical optimization while combining with CRO. Then we reformed the mutation operator algorithm in accordance with CRO infrastructure, such as replacing an individual in GAs by a molecule ( [figure omitted; refer to PDF] ), assigning probability and distribution similar to user-definable probability or condition in GAs, and changing gene value of GAs with molecular structure ( [figure omitted; refer to PDF] ). In CRO, a molecular structure contains amount number of moles and a pair of moles is presented as a set of [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the number of moles. Mutation operator algorithm generates the new molecular structure ( [figure omitted; refer to PDF] ) based on mutation operator formula relating to mutation type. There are many types of mutation and these types depend on the representation itself. Several mutation types are proposed such as uniform, nonuniform [22], and polynomial mutations [23, 24]. The mutation operator algorithm is demonstrated as Algorithm 2.

Algorithm 2: Mutation operator algorithm.

Input: a molecular structure of solution [figure omitted; refer to PDF]

(1) Assign probability [figure omitted; refer to PDF] , distribution [figure omitted; refer to PDF] , [figure omitted; refer to PDF] = Number of moles

[figure omitted; refer to PDF] = function upper bound; [figure omitted; refer to PDF] = function lower Bound

(2) For [figure omitted; refer to PDF] to [figure omitted; refer to PDF] do

(3) Get [figure omitted; refer to PDF] randomly in interval [figure omitted; refer to PDF]

(4) If [figure omitted; refer to PDF] then

(5) [figure omitted; refer to PDF] = do mutation operator Formula

(6) If [figure omitted; refer to PDF] then [figure omitted; refer to PDF]

(7) If [figure omitted; refer to PDF] then [figure omitted; refer to PDF]

(8) End if

(9) End For

(10) Output a new molecular structure = [figure omitted; refer to PDF]

3. Proposed Approach: MCRO

Mutation chemical reaction optimization (MCRO), which is the improved hybrid of CRO and two adjustment operators, generates various molecular structures to produce high-quality results and accelerate convergence. The turning operator and the mutation operator will be explained in this section.

3.1. Turning Operator

Turning operator is a new operator that is merged into a subalgorithm named neighborhood search operator or [figure omitted; refer to PDF] . Neighborhood search operator is used in three types of elementary reactions (on-wall ineffective collision, intermolecular ineffective collision, and decomposition) to transform the molecular structure from the neighborhood of the operand [12]. The new molecular structures of the solution are calculated as Formula (1), and turning operator is generated once for an objective function ( [figure omitted; refer to PDF] ) by using Formula (2). This new operator can highly improve the optimal quality and reliability of the algorithm: [figure omitted; refer to PDF]

3.2. Mutation Operator

Mutation operator is migrated into MCRO to improve solution diversity and accelerate convergence by generating various molecular structures. These structures are generated by using the processing mutation formula. This action increases the probability of finding the optimal solution and avoids being trapped into a local optimal solution; therefore, it accelerates convergence. In our research, the best mutation operator that matches our algorithm was selected from three types of mutation operators (uniform, nonuniform, and polynomial).

Uniform and nonuniform mutations are presented in basic GAs and their extending development of algorithm [22, 25]. Polynomial mutation is a popular one that was first introduced in Nondominated Sorting Genetic Algorithm (NSGA) and NSGA-II [23, 24]. Uniform, nonuniform, and polynomial mutations are calculated as Formulas (3), (4), and (5), respectively. As mutation operator algorithm mentioned in Section 2.3, input data is a set of molecular structures presented as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the number of moles, [figure omitted; refer to PDF] is a member of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Likewise output data is a set of molecular structures presented as [figure omitted; refer to PDF] . According to opportuneness, the variable names such as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] maybe present differently when they appear in other algorithms such as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] : [figure omitted; refer to PDF] Generally, the optimization method consists of three phases: initialization, iteration, and output, which are all included in our proposed approach. We applied mutation operator to the initialization and iteration phases of the algorithm. A mutation operator was applied to each new molecule in the initialization phase. In the iteration phase, we processed mutation operator in all types of elementary reaction, namely, on-wall ineffective collision, intermolecular ineffective collision, decomposition, and synthesis. Mutation operator was incorporated to change the molecular structure. We compared the results before and after the performance of mutation operator and selected the better result. We believe mutation is one potential component to improve the performance of MCRO, since mutation operator can spread the search space by randomly sampling new points. Furthermore this mutation operator algorithm increases the chance of generating more powerful result not less than twice of original CRO for every elementary reaction. Following are the four elementary reactions' explanations.

In on-wall ineffective collision of MCRO, a molecule [figure omitted; refer to PDF] is chosen. The first new molecular structure ( [figure omitted; refer to PDF] ) of [figure omitted; refer to PDF] is generated by neighborhood search operator which includes turning operator as mentioned in Section 3.1 and then the first fitness or [figure omitted; refer to PDF] is calculated by [figure omitted; refer to PDF] . The new [figure omitted; refer to PDF] is generated and buffer is updated when the condition [figure omitted; refer to PDF] occurs. Next the second new molecular structure ( [figure omitted; refer to PDF] ) is produced by computing mutation operator algorithm and then the second fitness or temp PE is calculated by [figure omitted; refer to PDF] . The profile of molecule M is updated as the best solution by comparing the first fitness ( [figure omitted; refer to PDF] ) and the second fitness (temp PE). Therefore this action contains two probabilities and achieves the best solution for each time while there is only one choice for original CRO. The on-wall ineffective collision of MCRO is shown in Algorithm 3.

Algorithm 3: On-wall ineffective collision of MCRO.

Input : a molecule [figure omitted; refer to PDF] and buffer

(1) Obtain [figure omitted; refer to PDF] with turning operator

(2) Calculate [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

(3) if [figure omitted; refer to PDF] then

(4) Get [figure omitted; refer to PDF] randomly in interval [figure omitted; refer to PDF]

(5) [figure omitted; refer to PDF]

(6) Update [figure omitted; refer to PDF]

(7) do Mutation of [figure omitted; refer to PDF] to [figure omitted; refer to PDF]

(8) Calculate the temp PE by [figure omitted; refer to PDF]

(9) if temp PE better than [figure omitted; refer to PDF] then Replace [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] with temp PE

(10) Update the profile of [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

(11) end if

Output [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

Intermolecular ineffective collision of MCRO algorithm starts by selecting two molecules [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Subsequently, neighborhood search operator with turning operator transforms the molecular structures of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] from [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, and evaluates [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] by [figure omitted; refer to PDF] . The new [figure omitted; refer to PDF] are generated as [figure omitted; refer to PDF] when [figure omitted; refer to PDF] occurs. Furthermore, the algorithm executes mutation operator to build two new molecular structures, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and processes fitness of two molecules temp PE₁ by [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and temp PE₂ by [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ). Meanwhile, the best fitness between [figure omitted; refer to PDF] and temp PE₁ is selected to be the optimal solution of [figure omitted; refer to PDF] , similar to the optimal solution of [figure omitted; refer to PDF] is the best fitness between [figure omitted; refer to PDF] and temp PE₂ , and then the profiles of molecules [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are updated. Prove that there are two choices to estimate the most outstanding result for this reaction that acted to a molecule; therefor this elementary reaction of MCRO consists of four chances to meet the optimum solution for a time, while the same reaction of original CRO is just giving two choices (one choice for a molecule). Algorithm 4 illustrates intermolecular ineffective collision of MCRO algorithm.

Algorithm 4: Intermolecular ineffective collision of MCRO.

Input : molecules [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

(1) Obtain [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with turning operator

(2) Calculate [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

(3) [figure omitted; refer to PDF]

(4) if [figure omitted; refer to PDF] then

(5) Get [figure omitted; refer to PDF] randomly in interval [figure omitted; refer to PDF]

(6) [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

(7) end if

(8) do Mutation of [figure omitted; refer to PDF] to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to [figure omitted; refer to PDF]

(9) Calculate the [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

(10) If [figure omitted; refer to PDF] better than [figure omitted; refer to PDF] : Replace [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

(11) If [figure omitted; refer to PDF] better than [figure omitted; refer to PDF] : Replace [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

(12) Update the profile of [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

(13) Update the profile of [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

(14) Output [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

Decomposition of MCRO algorithm begins by selecting a molecules [figure omitted; refer to PDF] from container. Copy a molecular structure (ω ) of [figure omitted; refer to PDF] to two new variables [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Next the molecular structures [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are transformed to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, by computing neighborhood search operator (with turning operator), and furthermore evaluate the fitness [figure omitted; refer to PDF] as [figure omitted; refer to PDF] and the fitness [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . The profiles of molecules are updated when [figure omitted; refer to PDF] as follows: set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] when [figure omitted; refer to PDF] is randomly in interval [figure omitted; refer to PDF] and a new molecule [figure omitted; refer to PDF] is created. In case of not ( [figure omitted; refer to PDF] ) and the energy buffer is sufficient set [figure omitted; refer to PDF] = (temp 1 + buffer ) × [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are randomly in interval [figure omitted; refer to PDF] ; in addition the central energy buffer is updated and a new molecule [figure omitted; refer to PDF] is created. If there exists a new molecule [figure omitted; refer to PDF] (Success = TRUE), the two new molecular structures [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are established by operated mutation operator algorithm and then the fitness of temp PE₁ and temp PE₂ are evaluated by f (tempω ₁ ) and f (tempω ₂ ), respectively. The first minimum fitness chooses the best one from [figure omitted; refer to PDF] or temp PE₁ , and the second minimum fitness selects the best one between [figure omitted; refer to PDF] and temp PE₂ . Furthermore profiles of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are updated. This concludes that there are four opportunities to reach the best result in decomposition of MCRO similar to intermolecular ineffective collision of MCRO; in other words, they are twice the same reaction in original CRO. Decomposition of MCRO is represented as Algorithm 5.

Algorithm 5: Decomposition of MCRO.

Input: A molecule [figure omitted; refer to PDF] and buffer

(1) Obtain [figure omitted; refer to PDF] and [figure omitted; refer to PDF] from [figure omitted; refer to PDF]

(2) Obtain [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with turning operator

(3) Calculate [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

(4) [figure omitted; refer to PDF]

(5) [figure omitted; refer to PDF]

(6) if [figure omitted; refer to PDF] then

(7) Get [figure omitted; refer to PDF] randomly in interval [figure omitted; refer to PDF]

(8) [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

(9) Create new molecules [figure omitted; refer to PDF]

(10) else if [figure omitted; refer to PDF] then

(11) Get [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] randomly in interval [figure omitted; refer to PDF]

(12) [figure omitted; refer to PDF]

(13) [figure omitted; refer to PDF]

(14) Update [figure omitted; refer to PDF]

(15) Create new molecules [figure omitted; refer to PDF]

(16) end if

(17) else

(18) [figure omitted; refer to PDF]

(19) end if

(20) If Success = TRUE

(21) do Mutation of [figure omitted; refer to PDF] to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to [figure omitted; refer to PDF]

(22) Calculate the [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

(23) If [figure omitted; refer to PDF] better than [figure omitted; refer to PDF] then Replace [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

(24) If [figure omitted; refer to PDF] better than [figure omitted; refer to PDF] then Replace [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

(25) Assign [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] to the profile of [figure omitted; refer to PDF]

(26) Assign [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] to the profile of [figure omitted; refer to PDF]

(27) end if

(28) Output [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

In synthesis of MCRO algorithm, two molecules [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are chosen. Secondly, neighborhood search operator (with turning operator) generated the two molecular structures of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] from [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, and the first fitness or [figure omitted; refer to PDF] is calculated by [figure omitted; refer to PDF] . Subsequently, the new [figure omitted; refer to PDF] is computed when [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . Afterward mutation operator algorithm generates [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , and the fitness generated by mutation operator algorithm or temp PE is calculated by [figure omitted; refer to PDF] . Finally the profile of molecule [figure omitted; refer to PDF] is updated by choosing the best solution between the fitness ( [figure omitted; refer to PDF] ) and the fitness generated by mutation operator (temp PE). There are two probabilities included in this elementary reaction while synthesis of original CRO provides only one. Synthesis of MCRO algorithm is presented as Algorithm 6.

Algorithm 6: Synthesis of MCRO.

Input : molecules [figure omitted; refer to PDF]

(1) Obtain [figure omitted; refer to PDF] from [figure omitted; refer to PDF] and [figure omitted; refer to PDF] from [figure omitted; refer to PDF]

(2) Calculate [figure omitted; refer to PDF]

(3) if [figure omitted; refer to PDF] then

(4) [figure omitted; refer to PDF]

(5) [figure omitted; refer to PDF]

(6) do Mutation of [figure omitted; refer to PDF] to [figure omitted; refer to PDF]

(7) Calculate the temp PE by [figure omitted; refer to PDF]

(8) if temp PE better than [figure omitted; refer to PDF] then Replace [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with temp PE

(9) Assign [figure omitted; refer to PDF] [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to the profile of [figure omitted; refer to PDF]

(10) else

(11) [figure omitted; refer to PDF]

(12) end if

(13) Output [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

4. Simulation Results and Discussion

4.1. Benchmarks and Parameters

The proposed MCRO was simulated to solve the 23 objective functions based on RCCRO [12]. The benchmark objective functions are classified into three categories, as shown in Table 1.

Table 1: The benchmark objective functions.

Category I (High-Dimensional Unimodal Functions) . Category I is composed of high-dimensional functions [figure omitted; refer to PDF] and includes 30 molecules in each container. These functions are easy to solve because each objective function has only one global minimum.

Category II (High-Dimensional Multimodal Functions) . Category II provides high-dimensional functions [figure omitted; refer to PDF] and includes 30 molecules in each container. These functions consist of many local minimums and can solve the most difficult problem.

Category III (Low-Dimensional Multimodal Functions) . Category III includes low-dimensional functions [figure omitted; refer to PDF] . In each container of functions, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] contain 2 molecules; [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] contain 4 molecules; [figure omitted; refer to PDF] contains 3 molecules; and [figure omitted; refer to PDF] contains 6 molecules. These functions comprise several local minimums.

Several parameters contained in MCRO include the main algorithm and the mutation operator parameters. Table 2 illustrates the main parameters in our simulations: popsize InitialKE , buffer , KELossRate , On-wallColl , DecThres , SynThres , and StepSize .

Table 2: MCRO parameters.

Parameter	popsize	InitialKE	Buffer	KELossRate	On-wallColl	DecThres	SynThres
Category I ( [figure omitted; refer to PDF] )	10	1000	0	0.1	0.2	150000	10
Category II ( [figure omitted; refer to PDF] )	20	10000000	100000	0.1	0.2	150000	10
Category III ( [figure omitted; refer to PDF] )	100	1000	0	0.1	0.2	150000	10

The mutation operator parameters in our experiment are probability and DistributionIndex . We set probability = 0.6 and DistributionIndex = 0.9 for every objective function.

4.2. MCRO Solution

To obtain the most appropriate MCRO solution, we considered three types of mutation operators: polynomial, uniform, and nonuniform. Our approach estimates start with alloyed CRO with turning operator, consequently integrating mutation operator to the algorithm. The first, second, and third options are the algorithms with polynomial, uniform, and nonuniform mutations, respectively. We generated the results of these three solutions by running each objective function 25 times. The optimal quality of the solution of each function is the mean of the best global minimum at 25 repeats. If the means of the results are equal, the second key for comparison is the standard deviation. All simulations included in Section 4.3 are performed on the same personal computer and same computing environment with computer Intel Core i7, CPU 3.10 GHz, and Ram 4 GB.

Table 3 compares the results of Category I functions for the three solutions. All of the solutions yield good results for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The algorithm with polynomial mutation is the leader for [figure omitted; refer to PDF] , and the algorithm with uniform mutation generates the best result for [figure omitted; refer to PDF] . The average result of Category I functions is the same for the three solutions.

Table 3: Comparisons of the result of the three solutions for Category I ( [figure omitted; refer to PDF] ) (the best results are marked in bold).

Function	Result of	Algorithm with mutation
Polynomial	Uniform	Nonuniform
[figure omitted; refer to PDF]	Mean	0	0	0
StDev.	0	0	0
Rank	1	1	1

[figure omitted; refer to PDF]	Mean	0	0	0
StDev.	0	0	0
Rank	1	1	1

[figure omitted; refer to PDF]	Mean	0	0	0
StDev.	0	0	0
Rank	1	1	1

[figure omitted; refer to PDF]	Mean	0	0	0
StDev.	0	0	0
Rank	1	1	1

[figure omitted; refer to PDF]	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	1	3	2

[figure omitted; refer to PDF]	Mean	0	0	0
StDev.	0	0	0
Rank	1	1	1

[figure omitted; refer to PDF]	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	3	1	2

Average rank	( [figure omitted; refer to PDF] )	1.29	1.29	1.29

Rank	( [figure omitted; refer to PDF] )	1	1	1

Table 4 compares the results of Category II functions for the three solutions. The results of the three solutions are as good as those of the two other solutions in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The solution with polynomial mutation performs best for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The solution with nonuniform mutation generates the best result for [figure omitted; refer to PDF] . The average result of high-dimensional multimodal functions is led by the solution combined with polynomial mutation, nonuniform mutation, and then uniform mutation.

Table 4: Comparisons of the result of the three solutions for Category II ( [figure omitted; refer to PDF] ) (the best results are marked in bold).

Table 5 compares the results of Category III functions. The solution with polynomial mutation computes the best result for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Meanwhile, the solution with uniform mutation is the leader for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , and the solution with nonuniform mutation performs best for [figure omitted; refer to PDF] . The polynomial solution does not generate the best result for the remaining functions, but it produces close results to those of the leader for the first five functions. The average ranking of Category III functions for the three solutions in order of solution is polynomial mutation, uniform mutation, and nonuniform mutation.

Table 5: Comparisons of the result of the three solutions for Category III ( [figure omitted; refer to PDF] ) (the best results are marked in bold).

The overall ranking is shown in the bottom of Table 5. The solution with polynomial mutation derives the most excellent results, followed successively by uniform mutation and nonuniform mutation. Hence, we prefer polynomial mutation operator for MCRO.

4.3. Experimental Method and Results

The aim of the present experiment is to evaluate the performance of MCRO by comparing it with three previously published algorithms: RCCRO4 [12], the best version of RCCRO; HP-CRO2 [18], the best version of HP-CRO and a hybrid of CRO and PSO; and OCRO [19], which represents the best results among the three algorithms. We generated the result for MCRO, OCRO, HP-CRO2, and RCCRO4 by running each objective function 25 times.

We provided three experiment methods to evaluate the capability of comparison algorithms. These three methods were evaluated for optimal solution quality, convergence speed, and statistical hypothesis testing.

(1) Optimal Solution Quality Evaluation . For each objective function, the optimal solution or the best global minimum is the most important result in each runtime. In a computing time, each round of iteration contains the local minimum as a result. Consequently, if the current global minimum is worse than the local minimum, then the current global minimum is replaced by the local minimum. The best global minimum at each computing time is generated when the program meets the stop condition of MCRO and other optimization algorithms. The mean of the best global minimum at 25 computing times is the optimal solution of each function. If the optimal solutions of the competitors are equal, then the second key for comparison is the standard deviation. The following optimal solution quality evaluations are demonstrated.

Table 6 presents the optimal solution quality evaluation of MCRO, OCRO, HP-CRO2, and RCCRO4 for high-dimensional unimodal functions or Category I ( [figure omitted; refer to PDF] ). MCRO performs best for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] but not for [figure omitted; refer to PDF] . MCRO and OCRO are the best for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; HP-CRO2 and RCCRO4 derive the same results as MCRO and OCRO for [figure omitted; refer to PDF] . RCCRO4 generates the best result for [figure omitted; refer to PDF] . The ranking of optimal solution quality for Category I functions in descending order is as follows: MCRO, OCRO, HP-CRO2, and RCCRO4.

Table 6: Optimal solution quality for Category I ( [figure omitted; refer to PDF] ) (the best results are marked in bold).

Function	FEs	Result of	MCRO	OCRO	HP-CRO2	RCCRO4
[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	250000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Average	Rank	( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Rank		( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Table 7 compares the optimal solution quality for Category II functions. MCRO performs best for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; OCRO generates the same results as MCRO for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; HP-CRO2 is the best for [figure omitted; refer to PDF] ; RCCRO4 has the best result for [figure omitted; refer to PDF] . The ranking of optimal solution quality for Category II functions in descending order is the same as that for Category I.

Table 7: Optimal solution quality for Category II ( [figure omitted; refer to PDF] ) (the best results are marked in bold).

Function	FEs	Result of	MCRO	OCRO	HP-CRO2	RCCRO4
[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	250000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	150000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Average	Rank	( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Rank		( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Table 8 compares the results of Category III functions. MCRO is the best in this category because it generates the best results for 8/10 functions: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . MCRO is followed by OCRO, which is the best for [figure omitted; refer to PDF] . HP-CRO2, which generates the best result for [figure omitted; refer to PDF] , is in third rank, followed by RCCRO4 in fourth rank. The ranking of optimal solution quality for Category III functions in descending order is the same as that for Categories I and II.

Table 8: Optimal solution quality for Category II ( [figure omitted; refer to PDF] ) (the best results are marked in bold).

Function	FEs	Result of	MCRO	OCRO	HP-CRO2	RCCRO4
[figure omitted; refer to PDF]	7500	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	250000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	1250	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	5000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	10000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	4000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	7500	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	10000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	10000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

[figure omitted; refer to PDF]	10000	Mean	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
StDev.	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Rank	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Average	Rank	( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Rank		( [figure omitted; refer to PDF] )	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Average rank		( [figure omitted; refer to PDF] )	1.39	2.13	2.78	3.17

Overall Rank		( [figure omitted; refer to PDF] )	1	2	3	4

The overall ranking presented at the bottom of Table 8 shows that MCRO performs best in the optimal solution quality evaluation. MCRO is followed successively by OCRO, HP-CRO2, and RCCRO4.

(2) Convergence Speed Evaluation . Convergence speed is an essential issue that indicates the performance of algorithms in optimization. In our experiment, the convergence speed of the algorithm was calculated by counting the number of iterations (FEs) before the algorithm converges with the accepted result. The accepted result is the results given there are the average FEs needed to reach the threshold expressed as acceptable solutions for each objective function such as acceptable solution of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , the robust algorithm has to generate the result at lease not worse than accepted result. The algorithm with fewer FEs is more powerful than that with greater FEs. Table 9 compares the convergence speeds of the algorithms. MCRO demonstrates the fastest convergence speed for all functions, except for [figure omitted; refer to PDF] , where HP-CRO2 is the best, and for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where OCRO is the best. In addition, the distance between MCRO and the second best of each function is significant. For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , MCRO converges the accepted results with 691, 819, 338, 896, 1056, and 314 FEs, respectively, whereas OCRO, respectively, has 23253, 25053, 31245, 65200, 64947, and 2818 FEs for the same functions. For [figure omitted; refer to PDF] , MCRO and RCCRO4 undergo 28 and 4721 FEs, respectively. The convergence speed ranking of comparison algorithms for all objective functions from fastest to slowest is as follows: MCRO, OCRO, HP-CRO2, and RCCRO4.

Table 9: Convergence speed (the best results are marked in bold).

Function	Accept	MCRO	OCRO	HP-CRO2	RCCRO	Function	Accept	MCRO	OCRO	HP-CRO2	RCCRO
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	691	23253	138711	x	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	29767	95685	51597	81992
Rank	1	2	3	4	Rank	1	4	2	3

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	334	2620	105625	x	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	314	2818	2996	3354
Rank	1	2	3	4	Rank	1	2	3	4

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	819	25053	237038	x	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	2075	44535	56843	90405
Rank	1	2	3	4	Rank	1	2	3	4

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	338	31245	128103	x	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	166	651	752	937
Rank	1	2	3	4	Rank	1	2	3	4

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	1251	6732	23608	120316	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	1544	3849	3419	2935
Rank	1	2	3	4	Rank	1	4	3	2

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	704	4695	29235	97733	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	5239	8444	7555	x
Rank	1	2	3	4	Rank	1	3	2	4

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	349	9471	107379	48223	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	2324	2011	2331	2703
Rank	1	2	4	3	Rank	2	1	3	4

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	6741	53688	4306	11499	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	2469	6233	5643	3662
Rank	2	4	1	3	Rank	1	4	3	2

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	909	3928	103567	138790	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	8232	6143	7278	6220
Rank	1	2	3	4	Rank	4	1	3	2

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	895	65200	74078	x	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	28	5419	7604	4721
Rank	1	2	3	4	Rank	1	3	4	2

[figure omitted; refer to PDF]	1	1056	64947	x	x	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	3796	5974	7210	4745
Rank	1	2	3	3	Rank	1	3	4	2

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	15711	75733	28990	89666
Rank	1	3	2	4

Average	(Rank)	1.22	2.43	2.91	3.35

Rank				1	3	4	2

After evaluating the convergence speed on the basis of iteration number, we evaluated the same parameter in MCRO, OCRO, HP-CRO2, and RCCRO4 by drawing a convergence curve for specific functions ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ) in a particular run. MCRO remains the most advantageous among the four algorithms.

(3) Statistical Hypothesis Testing . Friedman test [26], a well-known nonparametric statistical test, was used to assess the statistical hypothesis. Friedman test evaluates the differences or equality of results among competitor algorithms. We calculated the Friedman statistic according to [figure omitted; refer to PDF] distribution with [figure omitted; refer to PDF] degrees of freedom. Then, we estimated the [figure omitted; refer to PDF] value according to normal approximations. The Friedman statistic [figure omitted; refer to PDF] is presented as Formula (6). Parameter [figure omitted; refer to PDF] is the number of objective functions, which is 23 in our test. [figure omitted; refer to PDF] represents the number of comparison algorithms. In this case, we compared our proposed algorithm MCRO with three previous algorithms (OCRO, HP-CRO2, and RCCRO4); hence [figure omitted; refer to PDF] . Friedman rank or [figure omitted; refer to PDF] is searching by transforming the quality results of MCRO, OCRO, HP-CRO2, and RCCRO4 to ranks for each objective function ( [figure omitted; refer to PDF] ). In cases of equal ranks, the average ranks are assigned: [figure omitted; refer to PDF]

Table 10 demonstrates the results of Friedman rank test in terms of the comparison of optimal solution quality and convergence speed. MCRO achieves the best rank in the statistic test comparing both the optimal solution quality and the convergence speed on the basis of the corresponding Friedman ranks 1.59 and 1.22, the statistic [figure omitted; refer to PDF] values 21.76398 and 35.96694, and the [figure omitted; refer to PDF] values [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In addition, the results explicitly display significant differences across the competing algorithms. In multiple comparisons, the post hoc test was utilized to compare a set of algorithms with a control algorithm (best algorithm). The post hoc test starts with finding the test statistic [figure omitted; refer to PDF] of the [figure omitted; refer to PDF] th and [figure omitted; refer to PDF] th algorithms, which is processed using Formula (7). Consider [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the Friedman ranks of the compared algorithms.

Table 10: Result of Friedman test.

Algorithm	Friedman ranks	Friedman ranks
Optimal result quality	Convergence
MCRO	1.59	1.22
OCRO	2.33	2.43
HP-CRO2	2.85	2.93
RCCRO4	3.24	3.41
Statistic [figure omitted; refer to PDF]	21.76398	35.96694
[figure omitted; refer to PDF] value	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

The unadjusted [figure omitted; refer to PDF] value was calculated using the [figure omitted; refer to PDF] value from the table of normal distribution [figure omitted; refer to PDF] . The [figure omitted; refer to PDF] value was then compared with an appropriate level of significance [figure omitted; refer to PDF] . In the present study, the following three computing procedures were applied to obtain the adjusted [figure omitted; refer to PDF] values (APVs): Bonferroni-Dunn procedure [27] presented as Formula (8), Holm procedure [28] assigned as Formula (9), and Hochberg procedure [29] expressed as Formula (10). Subscripts [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are indexes that each corresponds to a concrete hypothesis, and [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] -value obtained for the [figure omitted; refer to PDF] th hypothesis.

The Bonferroni-Dunn procedure adjusts the value [figure omitted; refer to PDF] through a single step: [figure omitted; refer to PDF]

The Holm procedure adjusts the value [figure omitted; refer to PDF] in a step-down manner: [figure omitted; refer to PDF]

The Hochberg procedure adjusts the value [figure omitted; refer to PDF] in a step-up direction: [figure omitted; refer to PDF]

The results of the APVs for the Friedman test are shown in Table 11, where the control algorithm is the proposed algorithm MCRO. These results comprise the [figure omitted; refer to PDF] value, unadjusted [figure omitted; refer to PDF] values, and APVs that were extracted using the post hoc procedures and computed using the three procedures. We compared the results for both optimal solution quality and convergence speed. MCRO is superior to RCCRO4, HP-CRO2, and OCRO in all post hoc procedures considered.

Table 11: Adjusted [figure omitted; refer to PDF] values for the Friedman test (control algorithm: MCRO).

Compare method	Algorithm	[figure omitted; refer to PDF]	Unadjusted [figure omitted; refer to PDF] value	Bonferroni-Dunn	Holm	Hochberg
Result quality (StDev.)	RCCRO4	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
HP-CRO2	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
OCRO	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Convergence	RCCRO4	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
HP-CRO2	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
OCRO	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

(4) Computation Time . Normally, the numbers of computation times are often proportional to capability of computer. Unlike the other results such as optimal solution quality and convergence speed, although the simulations perform on various personal computers, these results are very approximate. Hence our experiments disregard using the computation time as an evaluation method. However, the computation times appearing in Table 12 indicate that, in our simulated situation, the computer times for these four comparison algorithms are closed.

Table 12: Computation times.

Function	Time to reach the best global optimum (sec.)
MCRO	OCRO	HP-CRO2	RCCRO4
[figure omitted; refer to PDF]	0.0668	0.0050	0.0052	0.0063
[figure omitted; refer to PDF]	0.0069	0.0066	0.0056	0.0068
[figure omitted; refer to PDF]	0.0202	0.0073	0.0086	0.0095
[figure omitted; refer to PDF]	0.0070	0.0076	0.0065	0.0123
[figure omitted; refer to PDF]	0.0244	0.0102	0.0112	0.0165
[figure omitted; refer to PDF]	0.0105	0.0098	0.0149	0.0088
[figure omitted; refer to PDF]	0.0123	0.0274	0.0074	0.0553
[figure omitted; refer to PDF]	0.0133	0.0528	0.0621	0.0433
[figure omitted; refer to PDF]	0.0263	0.0324	0.0494	0.0335
[figure omitted; refer to PDF]	0.0162	0.0195	0.0295	0.0443
[figure omitted; refer to PDF]	0.0231	0.472	0.0547	0.0335
[figure omitted; refer to PDF]	0.0238	0.0275	0.0288	0.0553
[figure omitted; refer to PDF]	0.0255	0.0287	0.0301	0.0295
[figure omitted; refer to PDF]	0.0007	0.0009	0.0009	0.0308
[figure omitted; refer to PDF]	0.0165	0.0151	0.0156	0.0152
[figure omitted; refer to PDF]	<0.0001	0.0005	0.0005	0.0005
[figure omitted; refer to PDF]	<0.0001	0.0005	0.0009	0.0005
[figure omitted; refer to PDF]	0.0002	0.0007	0.0007	0.0007
[figure omitted; refer to PDF]	0.0001	0.0010	0.0010	0.0009
[figure omitted; refer to PDF]	0.0005	0.0012	0.0011	0.0013
[figure omitted; refer to PDF]	0.0005	0.0005	0.0005	0.0006
[figure omitted; refer to PDF]	0.0006	0.0008	0.0008	0.0006
[figure omitted; refer to PDF]	0.0008	0.0008	0.0009	0.0007

4.4. Discussion

Section 4.2 recommends that the best MCRO solution is the combination of CRO, turning operator, and polynomial mutation operator. The solution with polynomial mutation obtains the most appropriate results over uniform and nonuniform mutations.

The experimental results in Section 4.3 substantiate that MCRO is more powerful than OCRO, HP-CRO2, and RCCRO4. MCRO can generate the best result on most of the 23 objective functions (see Figure 1). We determined several issues, such as optimal solution quality, convergence speed, and statistical hypothesis testing, to evaluate algorithm performance. Table 6 shows that MCRO can generate the best optimal solution for all high-dimensional unimodal functions. We can obtain the optimal solution 0 (equal [figure omitted; refer to PDF] ) and StdDev 0 for five functions ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ). MCRO achieves the best average rank in solving high-dimensional multimodal functions (Table 7) and performs the optimal solution 0 (equal [figure omitted; refer to PDF] ) and StdDev 0 for two functions ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ). For low-dimensional functions (Table 8), the results of MCRO are close to the true optimal, indicating that this algorithm is outstanding and consistent. Moreover, some comparison algorithms can generate the same results as MCRO in solving several objective functions, but MCRO converges faster than the other algorithms (Table 10).

Figure 1: [figure omitted; refer to PDF]

5. Conclusions and Future Work

This paper presents a hybrid optimization algorithm MCRO that solves global numerical optimization problems by combining CRO with turning and mutation operators. The most excellent MCRO framework is the utilization of the solution with polynomial mutation. Therefore, we provide polynomial mutation operator into our proposed approach. Our algorithm is advantageous because the nature of CRO is suitable for global numerical optimization, and turning operator is a perfect method to improve optimal solution quality and algorithm reliability. Furthermore, polynomial mutation operator accelerates algorithm convergence. To analyze the evaluation results, we compared the results of MCRO and other previously published algorithms (OCRO, HP-CRO2, and RCCRO4) in terms of optimal solution quality, convergence speed, and statistical hypothesis testing. There results indicate that MCRO is superior to other algorithms in solving global numerical optimization problems. Several potential parameters such as Popsize , InitialKE , buffer , KELossRate , On-wallColl , DecThres , SynThre , StepSize , probability , and DistributionIndex were also discussed.

MCRO is useful in solving global numerical optimization problems with high-dimensional unimodal, high-dimensional multimodal, and low-dimensional functions. However, whether or not MCRO will also be excellent for solving other problem types is unclear. Hence, the applicability of MCRO to other problem types requires further research and consideration.

Our future work will focus on investigating the applicability of MCRO to other numerical optimization problems and practical engineering optimization problems. We also intend to develop a new hybrid metaheuristic approach to solve optimization problems.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 61173107 and 91320103), the National High Technology Research and Development Program of China (Grant no. 2012AA01A301-01), the Special Project on the Integration of Industry, Education and Research of Guangdong Province, China (Grant no. 2011A091000027), and the Project on the Integration of Industry, Education and Research of Guangdong Province, China.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2015 Ransikarn Ngambusabongsopa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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This paper proposes a hybrid metaheuristic approach that improves global numerical optimization by increasing optimal quality and accelerating convergence. This algorithm involves a recently developed process for chemical reaction optimization and two adjustment operators (turning and mutation operators). Three types of mutation operators (uniform, nonuniform, and polynomial) were combined with chemical reaction optimization and turning operator to find the most appropriate framework. The best solution among these three options was selected to be a hybrid mutation chemical reaction optimization algorithm for global numerical optimization. The optimal quality, convergence speed, and statistical hypothesis testing of our algorithm are superior to those previous high performance algorithms such as RCCRO, HP-CRO2, and OCRO.

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Title

A Hybrid Mutation Chemical Reaction Optimization Algorithm for Global Numerical Optimization

Author

Ngambusabongsopa, Ransikarn; Li, Zhiyong; Eldesouky, Esraa

Publication year

2015

Publication date

2015

Publisher

John Wiley & Sons, Inc.

ISSN

1024123X

e-ISSN

15635147

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1155/2015/375902

ProQuest document ID

1750365128

A Hybrid Mutation Chemical Reaction Optimization Algorithm for Global Numerical Optimization

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Objective function	Number of molecules	Lower-upper bound	[figure omitted; refer to PDF]	Name
Category I
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Sphere model
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Schwefel's problem 2.22
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Schwefel's problem 1.2
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Schwefel's problem 2.21
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Generalized Rosenbrock's function
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Step function quartic
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Function with noise

Category II
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Generalized Schwefel's problem 2.26
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Generalized Rastrigin's function
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Ackley's function
[figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Generalized Griewank function
[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0	Generalized penalized functions
[figure omitted; refer to PDF] [figure omitted; refer to PDF]	30	[figure omitted; refer to PDF]	0
Remark : in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Category III
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	1	Shekel's foxholes function
[figure omitted; refer to PDF]	4	[figure omitted; refer to PDF]	0.0003075	Kowalik's function
[figure omitted; refer to PDF]	2	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Six-hump camel-back function
[figure omitted; refer to PDF]	2	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Branin function
[figure omitted; refer to PDF] [figure omitted; refer to PDF]	2	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Goldstein-Price function
[figure omitted; refer to PDF]	3	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Hartman's family
[figure omitted; refer to PDF]	6	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	4	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Shekel's family
[figure omitted; refer to PDF]	4	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	4	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]