Computational Intelligence and Neuroscience

Volume 2016, Article ID 4525294, 18 pages

http://dx.doi.org/10.1155/2016/4525294

## Two-Swim Operators in the Modified Bacterial Foraging Algorithm for the Optimal Synthesis of Four-Bar Mechanisms

^{1}División Académica de Informática y Sistemas, Universidad Juárez Autónoma de Tabasco, 86690 Cunduacán, TAB, Mexico^{2}Centro de Investigación en Inteligencia Artificial, Universidad Veracruzana, Sebastián Camacho 5, Centro, 91000 Xalapa, VER, Mexico^{3}Instituto Politécnico Nacional (IPN-CIDETEC), U. Adolfo López Mateos, 07700 Ciudad de México, DF, Mexico

Received 9 November 2015; Revised 11 January 2016; Accepted 12 January 2016

Academic Editor: Saeid Sanei

Copyright © 2016 Betania Hernández-Ocaña et al. 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

This paper presents two-swim operators to be added to the chemotaxis process of the modified bacterial foraging optimization algorithm to solve three instances of the synthesis of four-bar planar mechanisms. One swim favors exploration while the second one promotes fine movements in the neighborhood of each bacterium. The combined effect of the new operators looks to increase the production of better solutions during the search. As a consequence, the ability of the algorithm to escape from local optimum solutions is enhanced. The algorithm is tested through four experiments and its results are compared against two BFOA-based algorithms and also against a differential evolution algorithm designed for mechanical design problems. The overall results indicate that the proposed algorithm outperforms other BFOA-based approaches and finds highly competitive mechanisms, with a single set of parameter values and with less evaluations in the first synthesis problem, with respect to those mechanisms obtained by the differential evolution algorithm, which needed a parameter fine-tuning process for each optimization problem.

#### 1. Introduction

Nature-inspired algorithms (NIAs) have been successfully used to solve Constrained Numerical Optimization Problems (CNOPs) using constraint-handling techniques [1] given that, originally, these algorithms were designed to deal solely with unconstrained search spaces. NIAs can be comprised of two groups: () Evolutionary Algorithms (EAs) [2] based on emulating the process of natural evolution and survival of the fittest and () Swarm Intelligence Algorithms (SIAs) [3] based on cooperative behaviors of simple organisms such as insects, birds, fish, or bacteria. EAs are one of the most used metaheuristics. However, SIAs have been gaining popularity among researchers and practitioners, mainly with the Particle Swarm Optimization (PSO) [4] and the Ant Colony Optimization (ACO) [5] algorithms.

Without loss of generality, a CNOP can be defined aswhere is the solution vector and each decision variable , is bounded by lower and upper limits , which define the search space ; is the number of inequality constraints and is the number of equality constraints (in both cases, the constraints can be linear or nonlinear). If denotes the feasible region, then it must be clear that . As it is commonly found in the specialized literature of nature-inspired algorithms to solve CNOPs [1, 6, 7] equality constraints are transformed into inequality constraints by using a small tolerance as follows: , .

In the context of mechanical engineering, synthesis is the design process of mechanical systems [8]. Four-bar mechanisms are widely used in machinery design, since they are the simplest articulated mechanisms for controlled movement with one degree of freedom. The synthesis of these mechanisms is a well-known CNOP, and originally two classical approaches were used for this synthesis: graphical and analytical methods. However, implementing such solutions is a complicated issue and their results are quite limited; for this reason, the design of these mechanisms is a case of hard numerical optimization. There are several types of syntheses; this work addresses the dimensional design of a mechanism, that is, to calculate the length of the necessary links for generating a specific movement [9].

The synthesis of four-bar mechanisms has been carried out with different nature-inspired metaheuristics. In [10] a modified Genetic Algorithm (GA) with a penalty function as constraint-handler was proposed. differential evolution (DE) and a variable control method for deviations were applied in [11]. In [12] the synthesis of a mechanism for tracking a trajectory of points, based on Simulated Annealing (SA), was developed. Regarding this same tracking problem, a performance comparison among three different metaheuristics, GA, DE, and PSO, was presented in [13], while the Artificial Bee Colony (ABC), PSO, a binary GA (BGA), and a hybrid GA-PSO approach were used in [14]. Finally, in [15] the synthesis of a planar four-bar mechanism for position control using the Harmony Search (HS) algorithm was carried out. From the abovementioned literature review, the diversity of nature-inspired algorithms to solve the synthesis of four-bar mechanisms is noticeable.

On the other hand, there are algorithms whose usage in this type of optimization problems has been less explored, as it is the case of the Bacterial Foraging Optimization Algorithm (BFOA), which is a SIA proposed by Passino in 2002 to solve unconstrained numerical optimization problems [16]. BFOA emulates the behavior of bacterium* E. coli* in the search of nutrients in its environment. The goal of each bacterium is to maximize the energy it obtains per each unit of time spent on the foraging process while avoiding noxious substances. BFOA is considered a SIA because bacteria can communicate among them. Such behavior can be summarized in four processes: () chemotaxis (swim and tumble movements are performed), () swarming (bacteria can communicate with each other to direct their search for nutrients), () reproduction (the best bacteria are duplicated and these replaced other bacteria), and () elimination-dispersal (the worst bacteria are eliminated and new bacteria are randomly dispersed).

Regarding unconstrained optimization, BFOA has been combined with other algorithms, particularly with EAs, to improve its performance, for example, with a GA in [17–19] and DE in [20]. Mutation operators have been added to BFOA in [21]. Moreover, hybrids with other SIAs [22], and particularly with PSO, are found in the specialized literature [23, 24]. Furthermore, BFOA was also combined with artificial immune system’s clonal selection and fuzzy logic within its chemotaxis process in [25].

When dealing with constrained search spaces there are different approaches [26]; for example, in [27], BFOA was adapted to solve CNOPs in a proposal called Modified-BFOA (MBFOA) which used a set of feasibility rules [28] as constraint-handler. Moreover, MBFOA was simplified and improved in its step size handling in the swim operator in [29]. Further modifications were proposed in [30, 31] to solve multiobjective CNOPs, where some four-bar mechanisms were tackled. More recently, MBFOA was further improved to solve CNOPs in the so-called Improved MBFOA (IMBFOA for short) [32] where the idea of two-swim movements within the chemotaxis process was initially explored. However, IMBFOA heavily depends on a local search based on sequential quadratic programming. IMBFOA is the starting point of this research, focused on the optimal synthesis of four-bar mechanisms. Finally, BFOA has been combined with two NIAs BFOA-DE-PSO in [33, 34], and the idea of using micropopulations was explored in [35].

From the above literature review, it was found that NIAs are a valid option to solve four-bar mechanisms. However, the usage of BFOA-based approaches is still scarce. Furthermore, the incorporation of additional variation operators in those BFOA-based approaches is more frequent in unconstrained optimization. The abovementioned is the main motivation of this work, where two improved swims (and different from those in IMBFOA) are proposed to enhance the capabilities of MBFOA to deal with four-bar mechanisms optimization. Furthermore, such two-swim mechanism performance allows eliminating the second-order local search operator. Therefore, the contribution of this work consists in getting knowledge about the type of operators which provide better results in those constrained search spaces defined by the four-bar mechanisms tackled in this research.

The proposed approach is compared against those BFOA-based approaches for constrained optimization (IMBFOA and MBFOA) and also against a DE-based approach designed to solve mechanical design problems. To the best of the authors’ knowledge, this is the first time that the variation operators of a BFOA-based approach are studied in such a way that they improve the capabilities of the algorithm to solve a set of four-bar synthesis problems.

The document is organized as follows: in Section 2 the general synthesis of a four-bar mechanism is explained, including the kinematics of the mechanism and its coupler, as well as the specifications of the three case studies to solve. In Section 3, brief descriptions of MBFOA and IMBFOA are presented and the new proposal “Two-Swim MBFOA” (TS-MBFOA) is introduced. Section 4 shows the results obtained by TS-MBFOA and their comparison against those obtained by other NIAs. Finally, Section 5 presents the conclusions and future work of this research.

#### 2. Synthesis of Four-Bar Mechanisms

Figure 1 shows a planar four-bar mechanism formed by a reference bar , an input bar (crank), a coupler , and an output bar (rocker). In order to analyze this mechanism two coordinate systems are established: a system that is fixed to the real world () and another for self-reference (). is the distance between the origin points of both systems, is the rotation angle of the reference system, and () corresponds to the angle for every bar in the mechanism; finally, the coordinate pair determines the position of the coupler.