A Study of Global Numerical Maximization using Hybrid Chemical Reaction Algorithms

Several approaches are proposed to solve global numerical optimization problems. Most of researchers have experimented the robustness of their algorithms by generating the result based on minimization aspect. In this paper, we focus on maximization problems by using several hybrid chemical reaction optimization algorithms including orthogonal chemical reaction optimization (OCRO), hybrid algorithm based on particle swarm and chemical reaction optimization (HP-CRO), real-coded chemical reaction optimization (RCCRO) and hybrid mutation chemical reaction optimization algorithm (MCRO), which showed success in minimization. The aim of this paper is to demonstrate that the approaches inspired by chemical reaction optimization are not only limited to minimization, but also are suitable for maximization. Moreover, experiment comparison related to other maximization algorithms is presented and discussed. Research Article A Study of Global Numerical Maximization using Hybrid Chemical Reaction Algorithms Ransikarn Ngambusabongsopa1, Vincent Havyarimana2* and Zhiyong Li 1College of Computer Science and Electronic Engineering, Hunan University, 410082, Changsha, China 2Departemnt of Applied Sciences, Ecole Normale Supérieure, 6983, Bujumbura, Burundi Dates: Received: 28 February, 2017; Accepted: 06 March, 2017; Published: 07 March, 2017 *Corresponding author: Havyarimana Vincent, Departemnt of Applied Sciences, Ecole Normale Supérieure, 6983, Bujumbura, Burundi, Tel: (+257) 79830258; Fax: (+257) 22258948; E-mail:


Introduction
Optimization [1] is prevalent in various fi eld of science, engineering, economics and other related topics. Since the past few decades, plenty optimization frameworks were proposed to solve existing optimization problems. They depend on using a predefi ned constraint which is involved in the area of the search space to fi nd variables of the functions to be optimized.
In general, an optimization problem includes minimizing or maximizing a function systematically by selecting input values from a given feasible set [2]. The most famous framework is the evolution algorithm (EA), EA is heuristic algorithm which is inspired by the nature of the biological evolution and the social behavior of species. Several evolutionary algorithms have been addressed to optimization including Simulated Annealing (SA). It is inspired by annealing in metallurgy or physical process of increasing the crystal size of a material and reducing the defects through a controllable cooling procedure [3]. Furthermore, the genetic algorithm (GA) [4], is affected by Darwinian principle of the 'survival of the fi ttest' and the natural process of evolution through reproduction. Memetic algorithm (MA) [ 5], is inspired by Dawkins' notion of a meme. H o wever, particle swarm optimization (PSO) [6], is developed from social behavior of bird fl ocking or fi sh schooling by Eberhart and Kennedy. Ant colony optimization (ACO) [7], mimics ants which are able to discover the shortest route between their nest, and a source of food. Shuffl ed frog leaping algorithm (SFL) [8], is introduced to combine the benefi ts of the genetic based and the social behavior-based PSO. Bat algorithms (BAs) are inspired by the echolocation behavior of bats [9]. Harmony search (HS) [10], which is based on natural musical performance processes, occurs when a musician searches for an optimal state of harmony. Finally, chemical reaction optimization (CRO) is developed and proposed by Lam and Li [11,12], which simulate the effective drive to molecules in chemical reaction.
Although there are abundant approaches suggested to solve optimization problems, researchers still measure the capability of algorithms by comparing the optimal results generated with the previous methods in minimization aspect. They have corroborated that the published algorithms are suitable for solving minimization problems. Nevertheless, these are unwarranted to be appropriate for absolutely all optimization problems. As a result, the simulation based on maximization is considered in this paper. We also intend to solve maximization problems by experimenting several optimization algorithms based on CRO framework. These algorithms include an effective version of RCCRO (i.e., RCCRO4), HP-CRO2 a best version of hybrid algorithm based on PSO with CRO, OCRO which is a hybrid orthogonal crossover with CRO, and a recent established algorithm MCRO which is hybrid polynomial mutation operator with CRO.
The rest of this paper is organized as follows: a brief review of optimization algorithms based on CRO is introduced

Optimization algorithms based on CRO
CRO [11,12], is a framework that mimics molecular interactions in chemical reactions to reach a low-energy stable state. Potential energy is the energy stored in a molecule with respect to its molecular confi guration; 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 profi le that includes several attributes, such as molecular structure (), current potential energy (PE), and current kinetic energy (KE). 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 fi nal products in low-energy stable states.    Assign random solution to the molecular structure (particle position) w 6: Calculate the fi tness by f(w) 7: Set
Update buffer = buffer -KE ώ 7: do Mutation of ω to tempω 8: Calculate the tempPE by f(temp ω) 9: if tempPE better than PE ώ then Replace ώ with tempω , PE ώ with  The performance in terms of minimization among these four algorithms has been discussed in [16]. The ranking by the best powerful algorithm to the worst algorithm are MCRO, OCRO, HP-CRO2, and RCCRO4, respectively.

Problem functions and evaluation methods
Maximization Operation: As previously mentioned, discovering the most powerful optimal solution of any problem is the main purpose of optimization. In general, the function is called an objective function, cost function (i.e., minimization), or utility function (i.e., maximization). An optimal solution is considered to be the minimum / maximum of the objective function which is known as a global optimal solution. An problem, there may be more than one local minimal. A local minimum  is defi ned as a point for which the following expression holds [13,17].
The goal of minimization is to fi nd the minimum solution ś  S and (ś) ≤ f (s),  s S.
Mathematically, a minimization problem has the following form: Where R,E and I symbolize the real number set, the index set for equality constraints, and the index set for inequality constraints, respectively.
With the same concept, the aim of maximizing operation is to generate the maximum solution of f(x). ś  S and. (ś) ≥ f (s),  s S Mathematically, a maximization problem has the following form:

Benchmark functions and Parameters:
The benchmark functions in this paper are similar to the previous CRO publication [13][14][15][16], all experiments are simulated to solve the 23 objective problem functions. Such benchmark functions are classifi ed into three categories as shown in Table 1. Category I is the high-dimensional unimodal functions, category II is the High-Dimensional Multimodal Functions, and category III is the Low-Dimensional Multimodal Functions, More details are contrasted in [13][14][15][16].
This research obtains the main parameters provided in CRO framework [13], as shown in Table 2. Moreover, there are several individual parameters for each algorithm that are obtained as its original work, and more description are contained in [14][15][16].

Experimental results and discussion
The proposal for the conducted experiments is to evaluate the performance of competitor algorithm RCCRO4, HP-CRO2,   (1) The results of optimal solution quality evaluation are illustrated in Table 3-5. Table 3 represents the optimal solution quality of MCRO, OCRO, HP-CRO2, and RCCRO4 for category I which contains seven high-dimensional unimodal functions (f1-f7). MCRO conducts the best for f2, f3, f4, f5, f6 and f7, except the f1 that generates the best result by RCCRO4.The ranking of optimal solution quality for category I functions from best to worst as follows: MCRO, HP-CRO2, RCCRO4 and OCRO respectively. Table 4 compares the optimal solution quality for 6 functions in Category II which are high-dimensional multimodal functions. MCRO operates the best for f11, f12 and f13 while HP-CRO2 has the best results for f8, f9, and f10.
Therefore, the ranking of optimal solution quality of Category II functions is led by MCRO and HP-CRO2, followed by RCCRO4 and OCRO respectively. The overall ranking of optimal solution quality is represented in Table 6, showing that MCRO performs the best in the optimal solution quality evaluation. MCRO is followed successively by HP-CRO2, RCCRO4 and OCRO.
(2) The results of convergence speed evaluation for 4 comparison algorithms are presented in MCRO and HP-CRO2 report the best results for f16 .Moreover,   Besides, when evaluating the convergence speed based on the iteration number (FEs), we evaluate the similar results of competition algorithm: MCRO, OCRO, HP-CRO2, and RCCRO4 by drawing a convergence curve of specifi c functions which select 2 functions from each category: category I (f3, f7), category II (f11, f12) and category III (f16, f22) in a particular run. We note that appreciate curve for maximizing operation should be grown, as opposed to minimization. Figure 1-3 shows that MCRO remains the most outstanding among the four algorithms.
(3) As mentioned above, we provide Freidman test for statistical hypothesis testing .Friedman rank is processed by transforming the results of each function comparison algorithms to ranks. The average ranks are provided when ranks are equal. Table 8 presents the results of Freidman test for optimum solution quality and convergence speed. The results of Friedman rank test in terms of the comparison of optimal solution quality MCRO achieves the best rank, followed by HP-CRO2, RCCRO4 and OCRO respectively. In terms of the competition of convergence speed, MCRO also performs the outstanding and HP-CRO2 is the second best similar to optimal solution quality comparison, while OCRO is the third, then, RCCRO4 is the worst different from optimal solution quality comparison.    In the statistic test, comparing both the optimal solution quality and the convergence speed on the basis of the

Conclusion
This paper is concerned with solving optimization problem in maximization aspect. We provided several approaches   both optimal solution quality and convergence speed. OCRO, which is the second best on minimization, is the third in the part of convergence speed and the worst in optimal solution quality on maximization. Finally, RCCRO4, which the worst on minimization, is the third in the part of optimal solution quality and the worst in convergence speed of maximization.
The results forcefully verifi ed that MCRO is the best optimization approach to be considered as a promising algorithm for solving optimization (minimization or maximization) problems.