A new ABC-based multiobjective optimization algorithm with an improvement approach (IBMO: improved bee colony algorithm for multiobjective optimization)

Turkish Journal of Electrical Engineering & Computer Sciences http://journals.tubitak.gov.tr/elektrik/ Turk J Elec Eng & Comp Sci (2016) 24: 2349 – 2373 c TÜBİTAK ⃝ doi:10.3906/elk-1402-310 Research Article A new ABC-based multiobjective optimization algorithm with an improvement approach (IBMO: improved bee colony algorithm for multiobjective optimization) Tahir SAĞ1,∗, Mehmet ÇUNKAŞ2 Department of Computer Engineering, Faculty of Technology, Selçuk University, Konya, Turkey 2 Department of Electrical and Electronics Engineering, Faculty of Technology, Selçuk University, Konya, Turkey 1 Received: 27.02.2014 • Accepted/Published Online: 15.07.2014 • Final Version: 15.04.2016 Abstract: This paper presents a new metaheuristic algorithm based on the artificial bee colony (ABC) algorithm for multiobjective optimization problems. The proposed hybrid algorithm, an improved bee colony algorithm for multiobjective optimization called IBMO, combines the main ideas of the simple ABC with nondominated sorting strategy corresponding to the principal framework of multiobjective optimization such as Pareto-dominance and crowding distance. A fixed-sized external archive to store the nondominated solutions and an improvement procedure to promote the convergence to true Pareto front are used. The presented approach, IBMO, is compared with four representatives of the state-of-the-art algorithms: NSGA2, SPEA2, OMOPSO, and AbYSS. IBMO and the selected algorithms from specialized literature are applied to several multiobjective benchmark functions by considering the number of function evaluations. Then four quality indicators are employed for performance evaluations: general distance, spread, maximum spread, and hypervolume. The results show that the IBMO is superior to the other methods. Key words: Multiobjective optimization, artificial bee colony optimization, evolutionary algorithm, swarm intelligence 1. Introduction In the real world, many optimization problems contain more than one objectives that are generally conflicting with each other to be either minimized or maximized at the same time. For example, the good design for a bridge construction is represented by lower mass and higher strength. A plane design needs to optimize fuel efficiency, weight, and cargo simultaneously. It is not possible to find a single optimal solution that can ensure compliance with all objectives. The best solution means that it is not the worst one for each objective and it is better than the others for one objective at least. In other words, an optimal solution is not dominated by any other solutions in the search space. This is called a nondominated solution or Pareto optimum, while an image formed by Pareto-optimal solutions under the space of objective functions is known as a Pareto front [1]. Definition 1. Multiobjective optimization problem (MOP) is defined as: maximize/ minimize y = f (x) = {f1 (x), f2 (x), ...fM (x)} subject to g(x) = {g1 (x), g2 (x), ..., gJ (x)} ≤ 0 h(x) = {h1 (x), h2 (x), ..., hK (x)} = 0 (1) where x = {x1 , x2 , ..., xN } ∈ X y ∗ Correspondence: = {y1 , y2 , ..., yM } ∈ Y, 2349 SAĞ and ÇUNKAŞ/Turk J Elec Eng & Comp Sci where x is set of the decision vectors, X is the parameter space, y is the objective vector, and Y is the objective space. g(x)and h(x) are inequality and equality constraints, respectively. Definition 2. Pareto dominance: Let ⃗u = [u1 , u2 , . . . , un ] and ⃗v = [v1 , v2 , . . . , vn ] be decision vectors (solu∧ tions) and ⃗u dominates ⃗v is shown as ⃗u ≺ ⃗v , if and only if ∀i ∈ {1, 2, . . . , n} : ui ≤ vi ∃j ∈ {1, 2, . . . , n} : ui < vi and u ̸= v . The algorithms based on evolutionary computation (EC) are stochastic techniques that try to solve problems by using bio-inspired operators on a population of solutions. The population-based approaches can easily find multiple nondominated solutions and are able to approximate the whole Pareto front in a single run, which is an advantage for multiobjective optimization problems [2]. In the last few decades, EC has been studied for dealing with nondeterministic multiobjective optimization problems. The multiobjective evolutionary optimization algorithms (MOEAs) have become a very popular and fast-growing field [3–5]. Since Schaffer’s pioneering work in 1985 [6], various MOEAs have been proposed in this field to solve several domains of science and engineering problems. Some of them are as follows: scheduling heuristics, data mining, rule extraction, assignment and management, wireless sensor network, circuit design, bioinformatics, control systems and robotics, image processing, and neural network training [7]. There are several possible ways to classify MOEAs. Coello et al. in [8] categorize the algorithms in three groups: aggregating function approaches, population-based approaches, and Pareto-based approaches. Aggregating function combines all objectives into a single one using arithmetical operations and is the oldest mathematical approach for multiobjective optimization. However, it has a poor selection pressure to obtain the whole Pareto front. Population-based approaches try to diversify the search space by generating subpopulations for each objective but they are inefficient for the concept of Pareto dominance. Pareto-based approaches use the selection process based on the concept of Pareto optimality. The most advanced and commonly used algorithms take place in this group. Some of them are the niched Pareto genetic algorithm [9], the nondominated sorting genetic algorithm (NSGA) [10] and its improved version NSGA-II [11], the Pareto archive evolutionary strategy (PAES) [12], and the strength Pareto evolutionary algorithm (SPEA) [13] and its improved version SPEA2 [14]. In recent years, one of the other popular stochastic techniques is the algorithms of swarm intelligence, inspired by collective working behaviors of social insect colonies or animal swarms. These algorithms have become very popular owing to the fact that they are simple to implement and produce good results at low computational cost. The representative algorithms of swarm intelligence are particle swarm optimization (PSO), developed by Eberhart and Kennedy in 1995 and inspired by the social behavior of bird flocking or fish schooling [15]; ant colony optimization (ACO), developed by Dorigo in 1999 and mimicking the foraging behavior of social ants [16]; and artificial bee colony (ABC) optimization, developed recently by Karaboğa in 2005, which simulates the intelligence foraging behavior of honeybee swarms [17]. ACO is a preferred method for discrete optimization problems due to its nature. In consideration of multiobjective optimization, ACO is generally applied to traveling salesman problems [18], vehicle routing problems [19], flow-shop scheduling problems [20], and portfolio selection [21]. Research has especially focused on developing multiobjective particle swarm optimization (MOPSO) algorithms due to (...truncated)