A Comprehensive Overview of Multi-Swarm Binary Arithmetic Optimization Algorithm and Its Application

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A Comprehensive Overview of Multi-Swarm Binary Arithmetic Optimization Algorithm and Its Application

Introduction

Meta-heuristic algorithms have emerged as a powerful tool for solving complex optimization problems across various domains, including signal processing, production scheduling, engineering optimization, and task allocation. Unlike traditional optimization methods such as Newton’s method or the simplex method, meta-heuristic algorithms demonstrate superior efficiency in handling high-complexity, highly constrained real-world engineering problems. However, conventional meta-heuristic algorithms are primarily designed for continuous optimization problems and cannot be directly applied to binary optimization problems, such as feature selection, workshop scheduling, and fault section localization in radial distribution networks.

To address this limitation, researchers have explored binary versions of meta-heuristic algorithms. Among these, the Arithmetic Optimization Algorithm (AOA) stands out due to its simplicity and high optimization accuracy. AOA mimics arithmetic operations (addition, subtraction, multiplication, and division) to efficiently explore and exploit the search space. Despite its effectiveness in continuous optimization, AOA lacks a binary version, limiting its applicability to discrete binary problems.

This paper introduces a novel binary variant of AOA, termed the Binary Arithmetic Optimization Algorithm (BAOA_S), which employs a modified sigmoid transfer function to convert continuous values into binary form. Additionally, to overcome the limitations of BAOA_S, such as slow convergence and susceptibility to local optima, a Multi-Swarm Binary Arithmetic Optimization Algorithm (MS-BAOA) is proposed. MS-BAOA incorporates a multi-swarm strategy and mutation mechanism to enhance population diversity and improve optimization performance.

The effectiveness of MS-BAOA is validated through extensive experiments on the CEC2013 benchmark functions, where it outperforms other binary meta-heuristic algorithms, including Binary Particle Swarm Optimization (BPSO), Binary Gray Wolf Optimization (BGWO), Binary Fish Migration Optimization (BFMO), and Binary Equilibrium Optimizer (BiEO). Furthermore, MS-BAOA is applied to the fault section localization problem in distribution networks, demonstrating its capability to accurately and efficiently identify single and multiple fault sections.

Background and Related Work

Meta-Heuristic Algorithms

Meta-heuristic algorithms, also known as intelligent optimization algorithms, are computational techniques inspired by natural phenomena, biological behaviors, and human intelligence. These algorithms include Particle Swarm Optimization (PSO), Genetic Algorithm (GA), and Differential Evolution (DE), which have been widely used in solving complex optimization problems.

Recent advancements have led to the development of new meta-heuristic algorithms, such as the Henry Gas Solubility Optimization (HGSO), Coronavirus Herd Immunity Optimizer (CHIO), Gannet Optimization Algorithm (GOA), and Arithmetic Optimization Algorithm (AOA). AOA, in particular, has gained attention due to its simplicity and efficiency in solving high-dimensional optimization problems.

Binary Meta-Heuristic Algorithms

To extend meta-heuristic algorithms to binary optimization problems, researchers have proposed binary versions using transfer functions. The Binary Particle Swarm Optimization (BPSO) was one of the earliest binary meta-heuristic algorithms, employing a sigmoid function to map continuous values to binary space. Subsequent studies introduced binary variants of other algorithms, such as the Binary Bat Algorithm (BBA), Binary Gray Wolf Optimization (BGWO), and Binary Butterfly Optimization Algorithm (BBOA).

Despite these advancements, existing binary meta-heuristic algorithms often suffer from slow convergence and premature convergence to local optima. To address these issues, this paper introduces a modified sigmoid transfer function and a multi-swarm strategy to enhance the performance of the binary AOA.

Binary Arithmetic Optimization Algorithm (BAOA_S)

Transfer Function Design

The core challenge in developing a binary version of AOA lies in converting continuous position updates into binary values. Traditional approaches use transfer functions to map continuous values to probabilities, which are then compared with random numbers to determine binary outcomes. Common S-shaped transfer functions include:

• S1: (1 / (1 + e^{-x}))

• S2: (1 / (1 + e^{-2x}))

• S3: (1 / (1 + e^{-x/3}))

• S4: (1 / (1 + e^{-x/4}))

However, these functions exhibit slow switching rates between 0 and 1, leading to suboptimal convergence. To overcome this, BAOA_S employs a modified sigmoid transfer function with a steeper slope, enabling faster transitions between binary states. The modified function is defined as:

[ text{trans_val} = frac{1}{1 + exp(-5 times (X_b – 0.5))} ]

This modification ensures that the probability of switching between 0 and 1 is higher, improving the algorithm’s convergence speed and ability to escape local optima.

Position Update Mechanism

In the original AOA, position updates are performed using arithmetic operations in continuous space. For binary optimization, BAOA_S modifies the exploration and exploitation phases to accommodate binary constraints:

  1. Exploration Phase:
    • If (r_2 < 0.5), the division search strategy is applied.

    • If (r_2 geq 0.5), the multiplication search strategy is applied.

  2. Exploitation Phase:
    • If (r_3 < 0.5), the subtraction strategy is applied.

    • If (r_3 geq 0.5), the addition strategy is applied.

The updated positions are then converted to binary values using the modified sigmoid transfer function.

Multi-Swarm Binary Arithmetic Optimization Algorithm (MS-BAOA)

Multi-Swarm Strategy

To enhance population diversity and prevent premature convergence, MS-BAOA divides the original population into two sub-swarms. Each sub-swarm independently explores the search space and periodically exchanges information through communication strategies:

  1. Communication Strategy 1:
    • Every (K) iterations, the best individuals from both sub-swarms are replaced by the global best individual.

  2. Communication Strategy 2:
    • Every (M) iterations, the best individuals from both sub-swarms are exchanged.

These strategies ensure that sub-swarms share valuable information, improving the overall search efficiency.

Mutation Strategy

Despite the multi-swarm approach, BAOA_S may still converge to local optima due to its reliance on the best individual for position updates. To address this, MS-BAOA introduces a mutation strategy:

• After each iteration, the best individual undergoes mutation by randomly altering half of its dimensions.

• The mutated individual is evaluated, and if it yields a better fitness value, it replaces the original individual.

This mutation mechanism further enhances population diversity and exploration capability.

Algorithm Workflow

The MS-BAOA workflow consists of the following steps:

  1. Initialization:
    • Initialize the population and parameters.

  2. Fitness Evaluation:
    • Evaluate the fitness of each individual.

  3. Sub-Swarm Division:
    • Split the population into two sub-swarms.

  4. Position Update:
    • Apply BAOA_S position updates within each sub-swarm.

  5. Communication and Mutation:
    • Execute communication strategies and mutation.

  6. Termination:
    • Repeat until the maximum iterations are reached.

Experimental Validation

Benchmark Function Testing

MS-BAOA is evaluated on 11 CEC2013 benchmark functions and compared with BAOA_S, BPSO, BGWO, BFMO, and BiEO. The results demonstrate that MS-BAOA achieves superior performance in terms of convergence speed and solution accuracy.

Fault Section Localization in Distribution Networks

Problem Formulation

Fault section localization in radial distribution networks is a binary optimization problem where the goal is to identify faulty sections based on switch status information. The key components include:

  1. Section State Encoding:
    • (s_i = 1) if the section is faulty; otherwise, (s_i = 0).

  2. Switch State Encoding:
    • (I_j = 1) if the switch detects a fault current; otherwise, (I_j = 0).

  3. Fitness Function:
    • Minimize the discrepancy between expected and actual switch states while penalizing excessive fault sections.

Experimental Results

MS-BAOA is tested on two radial distribution network models with single and multiple fault scenarios. The results show that MS-BAOA accurately identifies fault sections and outperforms other algorithms in terms of convergence speed and precision.

Conclusion

This paper presents a novel binary variant of the Arithmetic Optimization Algorithm (BAOA_S) and its enhanced version, the Multi-Swarm Binary Arithmetic Optimization Algorithm (MS-BAOA). The key contributions include:

  1. A modified sigmoid transfer function for faster binary switching.
  2. A multi-swarm strategy to improve population diversity.
  3. A mutation mechanism to escape local optima.

Extensive experiments on benchmark functions and real-world fault localization problems validate the superiority of MS-BAOA. Future work will explore MS-BAOA’s performance in large-scale binary optimization and its integration with surrogate models for expensive optimization problems.

doi.org/10.19734/j.issn.1001-3695.2024.04.0158

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