Integrating the artificial bee colony and bees algorithm to face constrained optimization problems
Introduction
Most search and optimization problems used in science and engineering involve inequality and/or equality constraints that the obtained optimal solutions must satisfy. Because constrained optimization problems (COPs) are frequently encountered in real world applications, significant attention and efforts have been invested in their efficient and effective resolution. A COP is usually written as a nonlinear programming problem of the following type. For minimization problems:where f is a D-dimensional function; x represents function variables; g denotes less-than-equal-to inequality constraints; Ng is number of g constraints; h means equality constraints; and Nh is number of h constraints. All x variables have lower and upper bounds [xL, xU].
Evolutionary algorithms, especially genetic algorithms (GA), have been widely used in recent decades to solve COPs [24]. Powell and Skolnick mapped feasible and infeasible solutions into two different intervals within a GA environment [35]. Deb used GA to propose a constraint handling method that frees settings of penalty numbers [4]. Douglas et al. analyzed COPs with employed niched pareto GA and penalty methods [7]. Coello compiled a comprehensive survey of popular constraint handling techniques used with evolutionary algorithms [3]. Barkat Ullah et al. developed a hybrid evolutionary algorithm based on GA and devised a local search technique that emphasizes equality constraint problems [1].
Swarm intelligence (SI) has generated growing interest as an algorithm that replicates biological and other natural systems. Optimization algorithms have been proposed and employed in various applications, and applications related to SI have demonstrated great promise. SI is the property of a system, in which coherent functional global patterns emerge due to the collective behaviors of agents interacting locally with their environment. SI algorithms include ant colony optimization, particle swarm optimization (PSO), artificial fish swarm, artificial bee colony (ABC) and gravitational search algorithm, among others [6], [8], [9], [10], [11], [21], [22], [26], [27], [29], [36], [37], [40], [42], [43], [44], [45]. Sun et al. applied an improved vector PSO to solve COPs using a constrained-preserving method [39]. Kou et al. proposed a co-evolutionary PSO with infeasible degree to address COPs [19]. Masuda and Kurihara used a multi-objective PSO to minimize both objective functions and total amount of constraint violations [23]. ABC performance on COPs has also been investigated [17], [25]. However, as no single algorithm is able to achieve optimal results for all algorithm applications [46], researchers are currently exerting significant efforts to further improve existing SI algorithms and develop new algorithms inspired by natural phenomena.
Bee or bee-colony algorithms are fairly new members of SI. It is widely presumed that only some aspects of the nature/behavior of honeybees can be exploited and that new characteristics can be added to create new classes of algorithms [47]. In recent years, the unique nature of honeybee colonies has inspired numerous researchers to develop various new bee-inspired algorithms such as virtual bee, the bees, BeeAdHoc, the marriage in honeybees, the BeeHive, bee system, bee colony optimization, and artificial bee colony [18]. The bee colony optimization algorithm has since been further developed and applied to transportation problems [41]. Yang initially proposed the virtual bee algorithm and demonstrated how it could solve two-dimensional numerical problems [47]. The bees algorithm (BA) originally proposed by Pham et al. is used to solve unconstrained function optimization problems and train multi-layered perceptron networks to recognize different patterns in control charts [29], [30], [31], [32], [33], [34]. Basturk and Karaboga proposed the artificial bee colony (ABC) algorithm and used it to solve unconstrained and constrained function optimization problems [2], [12], [13], [14], [15], [16].
Among all bee-inspired algorithms, ABC is considered the most effective and currently accounts for over half of bee SI applications [18]. However, BA directly performs bee SI based on perceptions of the natural behavior of bees and its basic concepts are superior to those of other algorithms. Experience has demonstrated that neighborhood search (ngh) is a key BA parameter. This study used a stochastic self-adaptive ngh to free its setting and improve BA performance as well. While sharing common concepts, both ABC and BA provide unique advantages. This paper thus proposes to integrate ABC and BA (ABC–BA) and apply the resulting ABC–BA algorithm to COPs. Moreover, this paper uses an infeasible degree selection method similar to that of [19] to handle the inequality constraints. This paper tests and compares the efficacy of two methods that use, respectively, tolerance values and functional relationships to convert equality constraints into inequality constraints.
The remainder of this paper is organized as follows: Section 2 presents the ABC and BA algorithms; Section 3 introduces the designs of the ABC–BA algorithm; Section 4 conducts experiments on 6 well-known benchmark functions of COPs; and Section 5 provides a discussion and conclusions.
Section snippets
Artificial bee colony
Basturk and Karaboga [2] proposed the artificial bee colony (ABC) algorithm and used ABC in a wide range of applications [18]. The collective intelligence of bee swarms consists of three essential components: food sources, employed bees, and unemployed bees. Unemployed bees are further segregated into onlookers and scouts [12], [13], [14], [15], [16]. The three main phases of ABC are the employed phase, onlooker phase, and scout phase. Each employed bee searches for a new candidate location cxs
Integration of ABC and BA (ABC–BA)
Taking advantage of the overlap between the foundational concepts of ABC and BA, this study developed an ABC–BA algorithm that integrates ABC and BA. The integrated ABC–BA overcomes the limitations of ABC and BA in handing particular problems. The ABC–BA population (N) comprises selected bees (NS2 = 2N/3) and remaining bees (NR1 = N/3). The ABC–BA selected bees perform the functions of ABC employed bees and/or BA selected bees. The ABC–BA remaining bees are directly assigned as BA remaining bees,
Constraint handling method
For optimization algorithms, the fitness (fit) of a solution directly adopts the function value if the solution is feasible. Kou et al. used the amount of constraint violation and solutions at hand to propose the fit [19]:where fmax is the objective function value of the worst feasible solution in the population. When a solution is infeasible, fit is no longer set to an extreme value. This study suggests the fit:
Conclusion
The proposed ABC–BA algorithm is executed with both ABC and BA sub-swarms. Population sizes of sub-swarms vary dynamically according to the most recent ABCbestand BAbest. When one sub-swarm performs better than another, the former will gain most of the ABC–BA selected bees to improve itself in the next iteration and its formulation will be executed more frequently. The first two ranking ABC–BA selected bees perform as both ABC employed bees and BA selected bees to ensure sub-swarm survival and
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