تغییر الگوریتم مصنوعی کلونی زنبور عسل برای مسایل بهینه سازی مقید

Artificial Bee Colony (ABC) algorithm was firstly proposed for unconstrained optimization problems on where that ABC algorithm showed superior performance. This paper describes a modified ABC algorithm for constrained optimization problems and compares the performance of the modified ABC algorithm against those of state-of-the-art algorithms for a set of constrained test problems. For constraint handling, ABC algorithm uses Deb’s rules consisting of three simple heuristic rules and a probabilistic selection scheme for feasible solutions based on their fitness values and infeasible solutions based on their violation values. ABC algorithm is tested on thirteen well-known test problems and the results obtained are compared to those of the state-of-the-art algorithms and discussed. Moreover, a statistical parameter analysis of the modified ABC algorithm is conducted and appropriate values for each control parameter are obtained using analysis of the variance (ANOVA) and analysis of mean (ANOM) statistics.

Structural optimization, engineering design, VLSI design, economics, allocation and location problems are just a few examples of fields in which constrained optimization problems are encountered [1]. A constrained optimization problem (1) is defined as finding parameter vector View the MathML sourcex→ that minimizes an objective function View the MathML sourcef(x→) subject to inequality and/or equality constraints: equation(1) View the MathML sourceminimize   f(x→),x→=(x1,…,xn)∈Rnli≤xi≤ui,i=1,…,nsubject to:gj(x→)≤0,forj=1,…,qhj(x→)=0,forj=q+1,…,m Turn MathJax on The objective function f is defined on a search space, View the MathML sourceS, which is defined as a n-dimensional rectangle in View the MathML sourceRn (View the MathML sourceS⊆Rn). Domains of variables are defined by their lower and upper bounds. A feasible region View the MathML sourceF⊆S is defined by a set of m additional constraints (m ≥ 0) and View the MathML sourcex→ is defined on feasible space (View the MathML sourcex→∈F∈S). At any point View the MathML sourcex→∈F, constraints gj that satisfy View the MathML sourcegj(x→)=0 are called active constraints at View the MathML sourcex→. By extension, equality constraints hj are also called active at all points of View the MathML sourceS[2]. Constrained optimization problems are hard to optimization algorithms and also no single parameter (number of linear, nonlinear and active constraints, the ratio View the MathML sourceρ=F/S, type of the function, number of variables) is proved to be significant as a major measure of difficulty of the problem [3]. Since most of the optimization algorithms have been primarily designed to address unconstrained optimization problems, constraint handling techniques are usually incorporated in the algorithms in order to direct the search towards the feasible regions of the search space. Methods dealing with the constraints were grouped into four categories by Koziel and Michalewicz [4]: (i) methods based on preserving feasibility of solutions by transforming infeasible solutions to feasible ones with some operators [5] and [6]; (ii) methods based on penalty functions which introduce a penalty term into the original objective function to penalize constraint violations in order to solve a constrained problem as an unconstrained one. [7], [8], [9] and [10]; (iii) methods that make a clear distinction between feasible and infeasible solutions [11], [12], [13], [14] and [15]; (iv) other hybrid methods combining evolutionary computation techniques with deterministic procedures for numerical optimization [16], [17], [18] and [19]. Koziel and Michalewicz investigated a decoder based constraint handling approach which incorporates a homomorphous mapping, between n-dimensional cube and a feasible search space for solving constrained numerical optimization problems by evolutionary algorithms [4]. The method uses a decoder which transforms a constrained problem to an unconstrained problem via a homomorphous mapping (HM). Another study related with evolutionary algorithms is Adaptive Segregational Constraint Handling Evolutionary Algorithm (ASCHEA) proposed by Hamida and Schoenauer for constrained optimization problems based on a population level adaptive penalty function to handle constraints [20]. ASCHEA employs a constraint-driven mate selection for recombination and a segregational selection that favors a given number of feasible individuals and utilizes an equality constraint handling strategy which starts a large feasible domain and tightens it progressively [20]. Runarsson and Yao also investigated the performance of an evolution strategy using different constraint handling methods [21]. They described the over-penalty approach (OPA) which ranks feasible individuals according to their objective function value, followed by the infeasible solutions ranked according to penalty function value. Runarsson and Yao introduced an approach which balances the objective and penalty functions by stochastic ranking (SR) in Evolution Strategy (ES) [22]. The approach avoids setting a hard-to-set parameter penalty factor and treats constrained optimization as multi-objective optimization where constraints are regarded as an additional objective function. Moreover, they improved the performance of the evolution strategy (Improved Stochastic Ranking, ISR) by employing a search mechanism to overcome the problem of a search bias aligned with the coordinate axis [21]. Mezura-Montes and Coello Coello proposed an evolution strategy which is called Simple multi-membered Evolution Strategy (SMES) to solve global nonlinear optimization problems [23]. The approach uses a diversity mechanism based on allowing infeasible solutions to remain in the population instead of using a penalty function. A feasibility-based comparison mechanism is used to guide the process toward the feasible region of the search space. Also, the initial step size of the evolution strategy is reduced in order to perform a finer search and a combined panmictic recombination technique is applied to improve its exploitation capabilities [23]. In this work, ABC algorithm described by Karaboga [24] inspiring from the foraging behaviour of honey bees is modified to solve constrained optimization problems and its performance is investigated on constrained problems. ABC algorithm is extended by replacing the selection mechanism of the simple ABC algorithm with Deb’s selection mechanism [15] in order to cope with the constraints and introducing a probabilistic selection scheme that assigns probability values to feasible solutions based on their fitness values and to infeasible individuals based on their violations. Although it is a known fact that the handling technique employed may influence the performance of the algorithm, a handling technique using Deb’s rules, in the category of methods searching for feasible solutions on assumption that any feasible solution is better than any infeasible one [15] was adopted, since it has advantages in terms of simplicity, computational cost, fine tuning requirement etc. over other techniques. The performance of ABC algorithm has been tested on 13 well-known constrained optimization test problems described in [22] and compared with those of other previous studies [4], [20], [21], [22] and [23]. The remainder of the paper is organized as follows. In Section 2, the modified ABC algorithm adapted for solving constrained optimization problems is introduced and then in Section 3, the proposed algorithm is compared with other algorithms. Finally, in Section 4, conclusions and future works are provided.

A modified version of ABC algorithm for constrained optimization problems has been introduced and its performance has been compared with that of the state-of-the-art algorithms. A statistical analysis of the parameters of the modified ABC algorithm has been conducted and appropriate values have been recommended. It has been concluded that ABC algorithm can be efficiently used for solving constrained optimization problems. The performance of ABC algorithm can be also tested for real engineering problems existing in the literature and compared with that of other algorithms. Also, the effect of constraint handling methods on the performance of ABC algorithm can be investigated in future works.