Computational Intelligence and Neuroscience

Volume 2018, Article ID 5865168, 26 pages

https://doi.org/10.1155/2018/5865168

## An Optimization Framework of Multiobjective Artificial Bee Colony Algorithm Based on the MOEA Framework

^{1}School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China^{2}Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou 730000, China^{3}College of Information Science and Technology, Gansu Agricultural University, Lanzhou 730070, China

Correspondence should be addressed to Jiuyuan Huo; moc.liamxof@yjouh

Received 11 June 2018; Revised 10 September 2018; Accepted 27 September 2018; Published 1 November 2018

Academic Editor: Daniele Bibbo

Copyright © 2018 Jiuyuan Huo and Liqun Liu. 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.

#### Abstract

The artificial bee colony (ABC) algorithm has become one of the popular optimization metaheuristics and has been proven to perform better than many state-of-the-art algorithms for dealing with complex multiobjective optimization problems. However, the multiobjective artificial bee colony (MOABC) algorithm has not been integrated into the common multiobjective optimization frameworks which provide the integrated environments for understanding, reusing, implementation, and comparison of multiobjective algorithms. Therefore, a unified, flexible, configurable, and user-friendly MOABC algorithm framework is presented which combines a multiobjective ABC algorithm named RMOABC and the multiobjective evolution algorithms (MOEA) framework in this paper. The multiobjective optimization framework aims at the development, experimentation, and study of metaheuristics for solving multiobjective optimization problems. The framework was tested on the Walking Fish Group test suite, and a many-objective water resource planning problem was utilized for verification and application. The experiment’s results showed the framework can deal with practical multiobjective optimization problems more effectively and flexibly, can provide comprehensive and reliable parameters sets, and can complete reference, comparison, and analysis tasks among multiple optimization algorithms.

#### 1. Introduction

The optimization problems in the real world are multiobjective in nature, which means that the optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. These problems are known as multiobjective optimization problems (MOPs) which can be found in many disciplines such as engineering, transportation, economics, medicine, and bioinformatics [1]. Most of the multiobjective techniques have been designed based on the theories of Pareto Sort [2] and nondominated solutions. Thus, the optimum solution for this kind of problem is not a single solution as in the mono-objective case, but rather a set of solutions known as the Pareto optimal set. This refers to when no element in the set is superior to the others for all the objectives.

By using the multiobjective optimization method, the conflicting objectives in these MOPs can acquire better trade-off, and satisfactory optimization results can be given. Therefore, with the complexity and nonlinearity of objectives and constraints, finding a set of good quality nondominated solutions becomes more challenging, and research of efficient and stable multiobjective optimization algorithms is also one of the key and major directions for scholars to study. Over the last few decades, the metaheuristics algorithms [3] have proven to be effective methods for solving MOPs. Among them, the evolutionary algorithms are very popular and widely used to effectively solve complex real-world MOPs [4]. Some of the most well-known algorithms belong to this class, such as the Nondominated Sorted Genetic Algorithm-II (NSGA-II) [5], Multiobjective *ε*-evolutionary Algorithm based on *ε* Dominance (*ε*-MOEA) [6], and Borg [7].

Nevertheless, the swarm intelligence algorithm [8] inspired by biological information is one important type of metaheuristic algorithms. With its unique advantages and mechanisms, it has become a popular and important field. The main algorithms include the particle swarm optimization (PSO) algorithm [9], ant colony optimization (ACO) algorithm [10], and shuffled frog leaping algorithm (SFLA) [11]. In 2005, Karaboga proposed an artificial bee colony (ABC) algorithm based on the foraging behavior of honeybees [12]. ABC has been demonstrated to have a strong ability to solve optimization problems, and its validity and practicality have been proven [13]. Because of achieving high convergence speed and strong robustness, it has been used in different areas of engineering and seems more suitable for multiobjective optimization. At present, the ABC algorithm and its application research mainly focuses on single-objective optimization. The study of multiobjective optimization has just begun.

However, because the multiobjective optimization needs to cope with real problems, there exists some inconvenience in practical applications. For instance, the multiobjective optimization algorithms are closely related to solving problems which are difficult to apply to other MOPs; a consistent model is needed to regulate and compare optimization strategies of different multiobjective optimization algorithms, and users have difficulty choosing the suitable optimization algorithm for their problems and also need to spend a lot of time learning the algorithms.

In this context, it is necessary to establish a unified, universal, and user-friendly multiobjective optimization framework which can be a valuable tool for understanding the behavior of existing techniques, for codes or modules that reuse existing algorithms, and for helping in the implementation and comparison of algorithms’ new ideas. Moreover, researchers have found that focusing on the study of one algorithm has a lot of limitations. If different heuristic algorithms can be effectively referred or integrated with each other, they can handle actual problems or large-scale problems more effectively and more flexibly [14].

Therefore, multiobjective optimization frameworks have been proposed to integrate optimization algorithms, optimization problems, evaluation functions, improvement strategies, adjustment methods, and output of results to provide an integrated environment for users to easily handle optimization problems, such as the jMetal [15], Paradiseo-MOEO [16], and PISA [17]. Among them, the MOEA framework [18] is a powerful and efficient platform which is a free and open source Java library for developing and experimenting with multiobjective evolutionary algorithms (MOEAs) and other general purpose multiobjective optimization algorithms.

However, in these integrated environments for MO algorithms, the multiobjective artificial bee colony (MOABC) algorithm has not been integrated yet, and the MOABC algorithm has been proven in our previous research to perform better than many state-of-the-art MO algorithms [19]. Therefore, a multiobjective ABC algorithm named RMOABC [19] was introduced to integrate with the MOEA framework for providing a flexible and configurable MOABC algorithm framework that is independent of specific problems in this paper.

The remainder of this paper is organized as follows. The related literatures are reviewed in Section 2. Section 3 provides the background concepts and related technologies of MO and introduces the RMOABC algorithm. Section 4 described the unified optimization framework for MOABC algorithm based on the MOEA framework. The case study is represented in Section 5. The experiment’s settings, results, and corresponding analyses are discussed in Section 6, and finally, the conclusions and future work are drawn in Section 7.

#### 2. Literature Review

In the past, methods based on metaheuristics developed by simulating various phenomena in the natural world have proven to be effective methods for solving MOPs [20]. Compared to traditional algorithms, modern heuristics are not tied to a specific problem domain and are not sensitive to the mathematical nature of the problem. They are more suitable for dealing with the practical MOPs. A subfamily of them in particular, the evolutionary algorithms, is now widely used to effectively handle MOPs in the real world [21]. In the mid-1980s, the genetic algorithm (GA) began to be applied to solve MOPs. In 1985, Schaffer [22] proposed a vector evaluation GA which realized the combination of the genetic algorithm and multiobjective optimization problems for the first time. In 1989, Goldberg proposed a new idea for solving MOPs by combining Pareto theory in economics with evolutionary algorithms, and it brought important guidance for the subsequent research on multiobjective optimization algorithms [23]. Subsequently, various multiobjective evolution algorithms (MOEAs) have been proposed, and some of them have been successfully applied in engineering [24]. For instance, Li et al. proposed a new multiobjective evolutionary method based on the differential evolution algorithm (MOEA/D-DE) to solve MOPs with complicated Pareto sets [25].

Since 2001, optimization algorithms based on swarm intelligence inspired by the cooperation mechanism of the biological populations have been developed [8]. Through the cooperation of intelligent individuals, the wisdom of the swarm can achieve breakthroughs beyond the optimal individual. Swarm intelligence algorithms have been successfully applied to handle the optimizing problems with more than one objective. For the multiobjective particle swarm optimization (MOPSO) [26] algorithms, a local search procedure and a flight mechanism that are both based on crowding distance are incorporated into the MOPSO algorithm [27]. Kamble et al. proposed a hybrid PSO-based method to handle the flexible job-shop scheduling problem [28]. Leong et al. integrated a dynamic population strategy within the multiple-swarm MOPSO [29].

Among the swarm intelligence algorithm, due to the high accuracy and satisfactory convergence speed, the ABC algorithm shows a greater advantage in problem representation, solving ability, and parameter adjustment [30]. Because research on the multiobjective ABC algorithm has just begun in recent years, there are relatively few studies on MOABC algorithms and its applications. For instance, Hedayatzadeh et al. designed a multiobjective artificial bee colony (MOABC) based on the Pareto theory and *ε*-domination notion [31]. The performance of Pareto-based MOABC algorithm has been investigated by Akbari et al., and the studies showed that the algorithm could provide competitive performance [32]. Zou et al. presented a multiobjective ABC that utilizes the Pareto-dominance concept and maintains the nondominated solutions in an external archive [33]. And Akbari designed a multiobjective bee swarm optimization algorithm (MOBSO) that can adaptively maintain an external archive of nondominated solutions [34]. Zhang et al. presented a hybrid multiobjective ABC (HMABC) for burdening optimization of copper strip production that solved a two-objective problem of minimizing the total cost of materials and maximizing the amount of waste material thrown into the melting furnace [35]. Luo et al. proposed a multiobjective artificial bee colony optimization method called *ε*-MOABC based on performance indicators to solve multiobjective and many-objective problems [36]. Kishor presented a nondominated sorting based multiobjective artificial bee colony algorithm (NSABC) to solve multiobjective optimization problems [37]. Nseef et al. put forward an adaptive multipopulation artificial bee colony (ABC) algorithm for dynamic optimization problems (DOPs) [38]. In our previous works, a multiobjective artificial bee colony algorithm with regulation operators (RMOABC) which utilizes the mechanisms of adaptive grid and regulation operator was proposed in [19]. The experimental results show that compared with the traditional multiobjective algorithms, these variants of multiobjective ABC can find solutions with competitive convergence and diversity within a shorter period of time.

To effectively integrate different heuristic algorithms to handle MOPs more effectively and flexibly, a number of optimization algorithm frameworks were presented and applied in industrial and other fields. For instance, Choobineh et al. proposed a methodology for management of an industrial plant considering the multiple objective functions of asset management, emission control, and utilization of alternative energy resources [39]. Khalili-Damghani et al. proposed an integrated multiobjective framework for solving multiperiod portfolio project selection problems in the investment managers to make portfolio decisions by maximizing profits and minimizing risks over a multiperiod planning horizon [40]. An evolutionary multiobjective framework for business process optimization was presented by Vergidis et al. to construct feasible business process designs with optimum attribute values such as duration and cost [41]. Charitopoulos and Dua presented a unified framework for model-based multiobjective linear process and energy optimization under uncertainty [42]. Tsai and Chen proposed a simulation-based solution framework for tackling the multiobjective inventory optimization problem to minimize three objective functions [43]. A multiobjective, simulation-based optimization framework was developed by Avci and Selim for supply chain inventory optimization to determine supplier flexibility and safety stock levels [44]. Golding et al. introduced a general framework based on ACO for the identification of optimal strategies for mitigating the impact of regional shocks to the global food production network [45]. And a multiobjective optimization framework for automatic calibration of cellular automata land-use models with multiple dynamic land-use classes was presented by Newland et al. [46].

A number of multiobjective optimization framework for more general purposes have also been developed. For example, jMetal is an object-oriented Java-based framework designed to multiobjective optimization using metaheuristics and is available to people interested in multiobjective optimization [47]. PISA is a C-based framework for multiobjective optimization which is based on separating the algorithm specific part of an optimizer from the application-specific part [17]. A framework for dynamic multiobjective big data optimization, jMetalSP combines the multiobjective optimization features of the jMetal framework with the streaming facilities of the Apache Spark cluster computing system that was presented to solve dynamic multiobjective big data optimization problems [48]. The MOEA framework [18] is a powerful and efficient platform that is a free and open source Java library for developing and experimenting with multiobjective evolutionary algorithms (MOEAs) and other general purpose multiobjective optimization algorithms.

In summary, the research of multiobjective ABC algorithms is still in the initial stage, and the MOABC algorithms are still not implemented in the common multiobjective optimization frameworks. Therefore, this paper focuses on introducing the RMOABC algorithm based on the Pareto dominance theory into the MOEA framework to establish a unified, universal, and user-friendly multiobjective optimization framework for the general optimization purpose.

#### 3. Background Concepts and Related Technologies

##### 3.1. Pareto Dominate Concepts

Multiobjective optimization often has to minimize/maximize two or more nonlinear objectives at the same time which are in conflict with each other. Thus, the trade-offs decisions should be taken between these objectives. Most of the multiobjective algorithms are proposed based on the Pareto Sort [2, 49] theory, so the optimization result is not usually a single solution but rather a set of solutions named as a Pareto nondominated set.

Generally, a multiobjective optimization problem is to optimize a set of objectives subjected to some equality/inequality constraints. The goal of multiobjective optimization is to guide the optimization process towards the true or approximate Pareto front and to generate a well-distributed Pareto optimal set. The basic concepts of the multiobjective method based on the Pareto theory can be found in [50].

##### 3.2. Artificial Bee Colony Algorithm

The artificial bee colony (ABC) algorithm is a meta-heuristic and swarm intelligence algorithm proposed by Karaboga [12]. It is inspired by the foraging behavior of honeybees. Each individual bee is taken as an agent, and the swarm intelligence can be guided by the cooperation among different individuals. For its excellent performance, the ABC algorithm has become an effective means for solving complex nonlinear optimization problems.

The three types of bees—employed bees, onlookers, and scouts—constitute the artificial bee colony in the ABC algorithm. The optimization process is changed to the searching process of the nectar foods. Each position of the nectar source represents a feasible solution for the problem, and the nectar amount from the nectar source corresponds to the quality or fitness of the feasible solution. The evolutionary iterations and global convergence are achieved by the cooperation of the three kinds of bees: (1) employed bees perform local random searches in the areas near their food sources; (2) onlookers make an optimum food source to further evolve in accordance with the specific mechanism; and (3) scouts update the stagnant food source according to the processing mechanism for stagnant solutions.

Overall, the employed bees and onlookers can work together to obtain better food sources through random and targeted searches. When the stagnant number of optimal food source reaches a certain value that is prone to fall into the local search, the scouts will start a new random exploration task for the global search. Thus, through the collaboration of the three kinds of bees, the ABC algorithm can quickly and effectively achieve global convergence.

##### 3.3. RMOABC Algorithm

A typical goal in a multiobjective optimization problem is to obtain a set of Pareto optimal solutions. As identified earlier, it is necessary to provide a wide variety among the set of solutions for the decision-maker to choose from. By utilizing the Pareto theory, the original ABC algorithm has been improved and extended to handle the MOPs, and the new algorithm is called the RMOABC algorithm [19]. The RMOABC algorithm adopted two mechanisms, regulation operators and adaptive grid, to improve the accuracy and keep the diversity, respectively. And an external archive is also integrated to maintain the historical values of nondominated solutions found in the evolution process.

In the evolution process of optimization algorithms, it is essential to properly control the exploration and exploitation capabilities of the bees to efficiently find the global optimum for the optimization problem. According to the main update equation (i.e., Equation (1)) of the original ABC algorithm, it can be found that more emphasis is taken on the exploration capability [12]:where (or ) denotes the *j*-th element of (or ); *j* is a random index; denotes another solution selected randomly from the population; and is a random number in [−1, 1]. It is well known that the exploration and exploitation capabilities of ABC heavily depend on the control parameters in the updated equation of the bees. Thus, to improve the exploitation capability of the ABC algorithm, Zhu et al. proposed a Gbest-guided artificial bee colony algorithm (GABC) in [51] to replace Equation (1) in the original ABC algorithm to obtainwhere is a random number in [0, 1.5] and is the optimal solution fitness value in the *j*-th dimensional space.

To balance the trade-offs between the exploration and exploitation capabilities of MOABC, we proposed a multiobjective artificial bee colony algorithm with regulation operators (RMOABC) in [19] to dynamically adjust the capabilities of exploration and exploitation in the algorithm’s evolution process. The local and global dynamic regulation operators were integrated with the GABC algorithm. The mechanisms are to improve the ability of exploitation and guide the search of candidate solutions based on the information of global optimal solutions. The updated Equation (2) in the GABC algorithm was changed into the following equation:where the local dynamic regulation operator *k* is set to , the global dynamic regulation operator *r* is set to , *i* is the current iteration number, and MFE is the maximum iteration number of algorithms. The details can be found in the literature [19].

In the design of multiobjective algorithms, the external archive is a typical method for maintaining the historical values of nondominated solutions found in the evolution process. The adaptive grid [52] mechanism proposed in the PAES (Pareto Archive Evolutionary Strategy) algorithm was utilized in the RMOABC to produce well-distributed nondominated Pareto solutions set in the external archive. Each nondominated solution can be mapped in a certain location in the grid according to the values of multiobjective functions. The grid can adaptively maintain the distribution of candidate solutions stored in the external archive in a uniform way in the evolution process. Thus, these two mechanisms adopted in the RMOABC algorithm can help the algorithm to quickly achieve global convergence. The details can be found in the literature [19].

##### 3.4. MOEA Framework

Researchers of optimization algorithms have consensus that there is NO optimum strategy or algorithm for all of the optimized problems, but there is an effective strategy or algorithm for particular optimization problems. Thus, how to efficiently choose the proper algorithm for the particular optimization problem is a challenge for users. As mentioned above, a unified multiobjective optimization framework is good solution that can help users understand the behavior of existing techniques. And it can reuse the codes or modules in existing algorithms and can facilitate the implementation and comparison of new algorithms.

In multiobjective optimization frameworks, the MOEA framework is an open-source evolutionary computation library for Java that specializes in multiobjective optimization [18]. It is also an extensible framework for rapidly designing, developing, executing, and statistically testing multiobjective evolutionary algorithms (MOEAs). The framework supports a variety of state-of-the-art multiobjective evolutionary algorithms (MOEAs) such as NSGA-II (Nondominated Sorting Genetic Algorithm II), NSGA-III (Nondominated Sorting Genetic Algorithm III), *ε*-MOEA (Multiobjective *ε*-evolutionary Algorithm Based on *ε* Dominance), GDE3 (The Third Evolution Step of Generalized Differential Evolution), MOEA/D (Multiobjective Evolutionary Algorithm Based on Decomposition), PISA (Platform and Programming Language Independent Interface for Search Algorithms), and Borg MOEA. It also includes dozens of analytical test problems such as Zitzler-Deb-Thiele (ZDT), Deb-Thiele-Laumanns-Zitzler (DTLZ), CEC2009 (unconstrained problems), and so on. Thus, it can support the multiobjective optimization algorithm to be tested against a suite of state-of-the-art algorithms across a large collection of test problems. The new problems can be conducted in numerous comparative studies to assess the efficiency, reliability, and controllability of state-of-theart MOEAs.

#### 4. The Unified Optimization Framework with MOABC Algorithm

The purpose of this paper is to present a unified optimization framework for the MOABC algorithm (UOF-MOABC), which combines the features of the MOEA framework [18] for multiobjective optimization metaheuristics with the RMOABC algorithm presented in [19]. Based on the advantages of a number of classic and modern state-of-the-art optimization, a wide set of benchmark problems and a set of well-known quality indicators assess the performance of the MO algorithms included in the MOEA framework. The UOF-MOABC can assist in multiobjective optimization research at the development, experimentation, comparison, and study of MOABC for solving multiobjective optimization problems.

##### 4.1. System Architecture

The architecture of optimization algorithms should be generic enough to allow much needed flexibility to implement most of the metaheuristic; thus, before establishing the optimization framework, the metaheuristic should be characterized by a common behavior that is shared by all its algorithms.

As shown in Algorithm 1, unity procedures of metaheuristics were concluded by Wu et al. in [53]. This algorithm template is similar to most of the optimization algorithms that are based on the metaheuristic search which can be used to implement popular multiobjective technique and foster code reusability.