Interactive fuzzy Bayesian search algorithm: A new reinforced swarm intelligence tested on engineering and mathematical optimization problems
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.
Section snippets
Interactive search algorithm (ISA)
The Interactive Search Algorithm (ISA) is a non-gradient population based search algorithm introduced by Mortazavi et al. (Mortazavi, Toğan, & Nuhoğlu, 2018). This algorithm applies two search patterns so-called tracking and interacting search patterns. In the tracking pattern, ISA conducts the agents using previous movement () and three main agents as: the weighted agent , the best agent and the prior best location of an agent . In the interactive pattern, ISA conducts the agents
Interactive fuzzy Bayesian search algorithm (IFBSA)
In the proposed Interactive Fuzzy Bayesian Search Algorithm (IFBSA) the performance of ISA method is enhanced via adding two auxiliary decision modules that use fuzzy and Bayesian mechanisms. The interacting search patterns in IFBSA is the same with ISA, but before explanation the methodology, it is required to assess the tracking search pattern in more detail. Such that, the first term of the tracking search pattern of ISA is named as the memory concept while its remaining part is called
Numerical tests
In the current section the search performance of the proposed Interactive Fuzzy Bayesian Search Algorithm (IFBSA) is assessed from different perspectives. To provide more insight into the framework or the proposed IFBSA, its performance is compared with some other well-stablished methods, selected methods and their properties are addressed in Table 2. These methods are selected, based on the author’s knowledge, to cover the last decade studies. It should be noted three first three methods have
Discussion on IFSBA’s behavior
In the proposed IFBSA the adjustment rate of the memory concept contribution is performed via the activation coefficient (k) of the Bayesian formulation and it plays an important role on the search behavior of the algorithm. So, in the current section the effect of this parameter on the optimization process is assessed. In this regard, F15 and F16 functions are selected as the benchmark problems. The lower value for k indicates that the memory concept contribution is adjusted more rapidly but
Conclusion
In the current study a new self-adaptive non-deterministic search technique named as Interactive Fuzzy Bayesian Search Algorithm (IFBSA) is introduced. Proposed IFBSA integrates the search patterns of the Interactive Search Algorithm (ISA) and employs a Fuzzy-Bayesian dual decision mechanism to adjust the trade-off between search behaviors (i.e. exploration and exploitation) of the algorithm. The developed fuzzy mechanism employs a nine-rule strategy to determine the agents’ interaction
Declaration of competing interest
Author of the manuscript entitled “Interactive Fuzzy Bayesian Search Algorithm: A New Reinforced Swarm Intelligence Tested on Engineering and Mathematical Optimization Problems” declares that he has no conflict of interests.
References (55)
- et al.
A generalized type-2 fuzzy logic approach for dynamic parameter adaptation in bee colony optimization applied to fuzzy controller design
Information Sciences
(2018) - et al.
A high-speed interval type 2 fuzzy system approach for dynamic parameter adaptation in metaheuristics
Engineering Applications of Artificial Intelligence
(2019) - et al.
Drosophila food-search optimization
Applied Mathematics and Computation
(2014) - et al.
Structural damage detection using artificial bee colony algorithm with hybrid search strategy
Swarm and Evolutionary Computation
(2016) - et al.
Monarch butterfly optimization: A comprehensive review
Expert Systems with Applications
(2021) - et al.
Harris hawks optimization: Algorithm and applications
Future Generation Computer Systems
(2019) - et al.
A novel hybrid method combining electromagnetism-like mechanism and firefly algorithms for constrained design optimization of discrete truss structures
Computers & Structures
(2019) - et al.
Slime mould algorithm: A new method for stochastic optimization
Future Generation Computer Systems
(2020) - et al.
An adaptive hybrid evolutionary firefly algorithm for shape and size optimization of truss structures with frequency constraints
Computers & Structures
(2018) - et al.
Two-layer particle swarm optimization with intelligent division of labor
Engineering Applications of Artificial Intelligence
(2013)
Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization
Applied Soft Computing
Thermal analysis of parabolic trough collectors via a swarm intelligence optimizer
Solar Energy
Large-scale structural optimization using a fuzzy reinforced swarm intelligence algorithm
Advances in Engineering Software
A new fuzzy strategy for size and topology optimization of truss structures
Applied Soft Computing
Enhanced butterfly optimization algorithm with a new fuzzy regulator strategy and virtual butterfly concept
Knowledge-Based Systems
Sizing and layout design of truss structures under dynamic and static constraints with an integrated particle swarm optimization algorithm
Applied Soft Computing
Solution of structural and mathematical optimization problems using a new hybrid swarm intelligence optimization algorithm
Advances in Engineering Software
Interactive search algorithm: A new hybrid metaheuristic optimization algorithm
Engineering Applications of Artificial Intelligence
Interval type-2 fuzzy logic for dynamic parameter adaptation in a modified gravitational search algorithm
Information Sciences
Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems
Computer-Aided Design
Oppositional teaching learning based optimization approach for combined heat and power dispatch
International Journal of Electrical Power & Energy Systems
Size, shape, and topology optimization of planar and space trusses using mutation-based improved metaheuristics
Journal of Computational Design and Engineering
Cost optimisation of glued laminated timber roof structures using genetic algorithms
Biosystems Engineering
Earthworm optimization algorithm: A bio-inspired metaheuristic algorithm for global optimization problems
International Journal of Bio-Inspired Computation
Self-adaptive learning based particle swarm optimization
Information Sciences
Solving structural optimization problems with discrete variables using interactive fuzzy search algorithm
Structural Engineering and Mechanics
A novel parameter estimation in dynamic model via fuzzy swarm intelligence and chaos theory for faults in wastewater treatment plant
Soft Computing
Cited by (14)
Optimization of Seismic Base Isolation System Using a Fuzzy Reinforced Swarm Intelligence
2022, Advances in Engineering SoftwareCitation Excerpt :No matter how effective these methods are, the mentioned specifications cause two main drawbacks for them. First, they are very sensitive to the initial condition of the starting process [11,12]. This means that if they start from an inappropriate place in the search domain, they can easily get stuck in local minima.
Data-driven allocation of smart grid-connected system based on ant colony optimization algorithm
2024, Journal of Intelligent and Fuzzy SystemsOptimizing base isolation system parameters using a fuzzy reinforced butterfly optimization: A case study of the 2023 Kahramanmaras earthquake sequence
2024, JVC/Journal of Vibration and ControlPERFORMANCE ASSESSMENT OF PARABOLIC TROUGH COLLECTORS UNDER CLIMATIC CONDITIONS OF IZMIR, TURKEY: A CASE STUDY
2024, Heat Transfer Research
- 1
ORCID iD: 0000-0002-6089-7046.