Interactive fuzzy Bayesian search algorithm: A new reinforced swarm intelligence tested on engineering and mathematical optimization problems

Highlights

• Developing a new Fuzzy decision-making mechanism.

• Developing a new Bayesian regulator strategy.

• Introducing a new compound Metaheuristic-Fuzzy-Bayesian search strategy.

Abstract

The current study deals with introducing a new probabilistic, self-adaptive, and gradient-free search algorithm. In the proposed method new Bayesian and Fuzzy auxiliary mechanisms are defined and simultaneously employed to extremely adjust the trade-off between exploration and exploitation search behaviors of the swarm-based technique so-called Interactive Search Algorithm (ISA). In this regard, a nine-rule fuzzy decision-making strategy and a hierarchical forecasting Bayesian formulation are developed. The integrated fuzzy and Bayesian mechanisms permanently monitor the search process, and try to dynamically tune the search behavior of each agent based on the governing conditions of the current problem and cause the proposed method to work as a self-adaptive search algorithm. This new search technique is named Interactive Fuzzy Bayesian Search Algorithm (IFBSA) and its performance is tested on a suit of unconstrained mathematical functions and constrained structural and mechanical optimization problems with different properties. Acquired outcomes demonstrate that the proposed IFBSA, thanks to its dual supplementary module, provides promising and superior results in the terms of accuracy, stability, and convergence rate.

Introduction

Optimization techniques are generally divided into gradient-based and non-gradient based methods. The gradient-based methods are rapid and accurate, but they have two main limitations. First, they require continuous and differentiable objective functions, while for many complex engineering problems, it is very difficult or even impossible to find such an objective function. Second, they are highly sensitive to the starting point of the optimization process, and determining the right starting point for many complex problems requires tedious and time-consuming trial and error procedures. Under these circumstances, non-gradient based methods provide an alternative approach to solve optimization problems. Metaheuristic algorithms are one of the most important non-gradient based optimization techniques. They are usually inspired by physical or natural phenomena or even social behaviors. These methods, contrary to the gradient-based approaches, do not require any continuous and differentiable objective functions and can be initiated from any arbitrary location in the search domain. One can list some this method as follows, Earthworm Optimization Algorithm (Wang, Deb, & Coelho, 2015), Harris Hawk Optimizer (Heidari, et al., 2019), Monarch Butterfly Optimization (Feng, Deb, Wang, & Alavi, 2021), Moth Search Algorithm (Wang, 2018) and Slime Mould Algorithm (Li, Chen, Wang, Heidari, & Mirjalili, 2020).

Accordingly, in the last two decades, these methods are broadly employed to get optimal (or near optimal) solutions for different mathematical, physical, mechanical, and structural optimization problems. For instance, Cheng and Jin developed a Reliability-based optimization algorithm for minimizing the weight of steel truss arch bridges subjected to probabilistic and deterministic constraints (J. Cheng & Jin, 2017). Mortazavi et al. utilized Interactive Search Algorithm (ISA) and iPSO techniques for solving engineering optimization problems with discrete variables (Ali Mortazavi, 2021a, Mortazavi, 2021b, Mortazavi et al., 2017). Cheng and Prayogo proposed a fuzzy adaptive Teaching And Learning Based Optimization (TLBO) algorithm to solve different type of optimization problems (M.-Y. Cheng & Prayogo, 2017). Tejani et al. investigated simultaneous size, shape, and topology optimization of planar and space truss systems using four mutation-based metaheuristic algorithms (Tejani, Savsani, Patel, & Savsani, 2018). Mortazavi et al. applied integrated particle swarm optimization (iPSO) for structural optimization problems of truss (Mortazavi, 2020a, Mortazavi et al., 2018, Mortazavi et al., 2019).

Ho– Ho-Huu et al. applied improved version of Differential Evolution (DE) constrained engineering problems with dynamic constraints (Ho-Huu, Nguyen-Thoi, Truong-Khac, Le-Anh, & Vo-Duy, 2018). Moloodpoor et al. (2019) utilized the iPSO algorithm for thermal optimizing of a solar collector system (Moloodpoor, Mortazavi, & Ozbalta, 2019). Mortazavi perfumed a comparative study to evaluate different performance of different metaheuristic techniques on solving structural optimization problems (Mortazavi, 2019). Sharma and Saha aaplied an improved Butterfly Optimization Algorithm (BOA) to solve optimization problems (Sharma & Saha, 2019). Yuan et al. proposed an adaptive Dragonfly Algorithm (DA) for the structural optimization of frame structures (Yuan, Lv, Wang, & Song, 2019). Sanchez et al. makes a comparative study for different fuzzy reinforced PSO method on human recognition process (Sánchez, Melin, & Castillo, 2020). Mortazavi applied a fuzzy strategy combined with a metaheuristic technique to solve structural size and topology optimization problems (Mortazavi, 2020b). Anter et al., combines Whale Optimization Algorithm (WOA) with chaos theory and fuzzy logic to generate a now model for optimizing different parameters of waste water treatment systems (Anter, Gupta, & Castillo, 2020). According to the given information, the metaheuristic techniques demonstrate a satisfactory search capability on some conventional optimization problems. However, they still show inadequate performance in the case of more complex problems (e.g. problems with complex domains and/or boundaries). Due to this fact, several research studies are performed to improve the search capability of metaheuristic methods. They offered different amendatory solutions like hybridizing affirmative features of two or more techniques or adding extra controller mechanism(s) to these algorithms. (Deep and Bansal, 2009, Ding et al., 2016, Le et al., 2019, Lieu et al., 2018, Liu et al., 2010, Miranda and Alves, 2013, Xin et al., 2010, Zhang et al., 2017). Peraza et al. employed fuzzy adapted Harmony Search (HS) algorithm for ball and beam controller (Peraza, Valdez, Castro, & Castillo, 2018).

To provide more insight about these attempts, a chronological list of the related works, based on author knowledge, Xin et al. Hybridized DE and PSO methods (Xin, et al., 2010); Wang et al. added simple probability formulation to improve PSO method (Y. Wang, et al., 2011); Deng et al. Combined PSO, Ant Colony Optimizer (ACO) and Genetic Algorithm (GA) (Deng, et al., 2012); Lim et al. Improved PSO using evolution swarm memory concept (Lim & Mat Isa, 2013); Roy et al. Combined TLBO method with opposed learning strategy (Roy, Paul, & Sultana, 2014); Olivas et al. Adjusted PSO with a fuzzy logic strategy (Olivas, Valdez, & Castillo, 2015); Mortazavi et al. introduced the improved fly-back approach to handle the constrained optimization problems (Mortazavi & Toğan, 2017); Mortazavi defined a fuzzy mechanism for adjusting the Butterfly optimization Algorithm (BOA) (Mortazavi & Moloodpoor, 2021); Castillo et al. employed the interval type-2 fuzzy mechanism to parameter adaptation of metaheuristic algorithms (Castillo, et al., 2019) Castillo and Angulo used the type-2 fuzzy approaches to parameter adaptation of the Bee Colony Optimization (BCO) (Castillo & Amador-Angulo, 2018) Olivas et al. applied the type-2 fuzzy approaches for parameter adaptation of Gravitational Search Algorithm (GSA) (Olivas, Valdez, Melin, Sombra, & Castillo, 2019).

The performance of a search algorithm is highly depended on its exploration and exploitation search behaviors and strategies applied to provide balance between them (Villar-García, Vidal-López, Rodríguez-Robles, & Guaita, 2019; Y. Zhang, Gong, & Cheng, 2017). Therefore, the mentioned improvements mostly try to somehow enhance the metaheuristic methods from these perspectives (i.e. enhancing exploration and/or exploitation and/or equilibrium between them). Although cited enhancements fulfill this aim to some extent, yet the affirmative attributes of the Artificial Intelligence (AI) leaves plenty of room for further improvements for metaheuristic algorithms. In this regard, Bayesian approach and fuzzy logic are two efficient methodologies that broadly are applied as the fundamental platforms for the AI-based systems. These methodologies can be employed to rising up the performance level of metaheuristic algorithms and convert them to more intelligent and robust search techniques.

On the one hand, fuzzy logic, which simulates the human way of thinking, allows metaheuristics to make logical decisions based on the governing conditions of the current problem. On the other hand, the Bayesian approach employing the probability principles equips the metaheuristic algorithms with a forecasting mechanism to predict and utilize more promising settings. So, the fuzzy logic and Bayesian approach can upgrade the conventional metaheuristic algorithms by dynamically adjusting their search behavior. Based on these facts, in the current study, the stated abilities of both fuzzy and Bayesian strategies are combined with the search capability of a swarm-based metaheuristic method so-called Interactive Search Algorithm (ISA) to gain an efficient, self-adaptive and decision-making search technique. The new method is named Interactive Fuzzy-Bayesian Search Algorithm (IFBSA). In the proposed method, the developed Fuzzy and Bayesian regulator mechanisms permanently monitor the optimization process and try to dynamically adjust the search behavior of each agent based on the governing conditions of the current problem. Subsequently, search performance of the IFBSA is tested on a suite of different unconstrained mathematical functions and constrained engineering problems, and acquired results are discussed and reported through the illustrative tables and diagrams.

The rest of this study is organized as follows. In Section 2, the standard Interactive Search Algorithm (ISA) is briefly explained. In Section 3 the proposed Interactive Fuzzy Bayesian Search Algorithm (IFBSA) and its fuzzy and Bayesian formulations are described. In Section 4, the proposed IFBSA is tested and assessed on a suite of optimization problems with different properties. In Section 5, two auxiliary modules of the given IFBSA method is discussed in more detail. Consequently, the last section is devoted to providing a brief conclusion about the present study.