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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (31 March 2021) | Viewed by 32545

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Prof. Dr. Erik Valdemar Cuevas

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Department of Electronics, Universidad de Guadalajara, Avenue Revolucion 1500, Guadalajara, Mexico
Interests: computer vision; evolutionary computation; artificial intelligence; bio-inspired computation
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Prof. Dr. Francisco G. Montoya

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Department of Engineering, Electrical Engineering Section, University of Almeria, 04120 Almeria, Spain
Interests: geometric algebra; power quality; power theory; power engineering; optimization techniques
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Prof. Dr. Alfredo Alcayde

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Department of Engineering, Universidad de Almería, La Cañada de San Urbano s/n, 04120 Almería, Spain
Interests: power engineering; optimization techniques; ICT
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Special Issue Information

Dear Colleagues,

Metaheuristic computation is one of the most important emerging technologies of recent times. Over the last few years, there has been an exponential growth of research activity in this field. Despite the fact that the concept itself has not been precisely defined, metaheuristic methods have become the standard term that encompasses several stochastic, population-based, and system-inspired approaches.

Metaheuristic schemes use as inspiration our scientific understanding of biological, natural, or social systems, which at some level of abstraction can be represented as optimization processes. They are intended to serve as general-purpose easy-to-use optimization techniques capable of reaching globally optimal or at least nearly optimal solutions. Some common features clearly appear in most of the metaheuristic approaches, such as the use of diversification to force the exploration of regions of the search space, rarely visited until now, and the use of intensification or exploitation, to investigate thoroughly some promising regions. Another common feature is the use of memory to archive the best solutions encountered. Due to their robustness, metaheuristic techniques are well-suited options for industrial and real-world tasks. They do not need gradient information and they can operate on each kind of parameter space (continuous, discrete, combinatorial, or even mixed variants). Essentially, the credibility of metaheuristic algorithms relies on their ability to solve difficult, real-world problems reaching a better performance in terms of accuracy and robustness.

This Special Issue aims to provide a collection of high-quality research articles that address broad challenges in both theoretical and application aspects of metaheuristic algorithms. We invite colleagues to contribute original research articles as well as review articles that will stimulate the continuing effort on metaheuristic approaches to solving problems in different domains. In the Special Issue, the contributions are mainly divided into two groups: (A) foundations, improvements, or hybrid approaches and (B) applications. Potential topics for this Special Issue include, but are not limited to:

(A) Foundations, improvements or hybrid approaches:

- Analysis or comparison of metaheuristic methods (single or multi-objective)

- New stochastic search strategies

- Enhanced versions of existent metaheuristic schemes (single or multi-objective)

- New metaheuristic techniques generated through the combination of different paradigms

(B) Applications:

- In communications

- In control processes

- In decision making

- In signal and image processing

- In power systems

Dr. Erik Cuevas
Prof. Dr. Francisco G. Montoya
Dr. Alfredo Alcayde
Guest Editors

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Published Papers (12 papers)

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20 pages, 458 KiB

Open AccessArticle

A Shuffle-Based Artificial Bee Colony Algorithm for Solving Integer Programming and Minimax Problems

by Ivona Brajević

Mathematics 2021, 9(11), 1211; https://doi.org/10.3390/math9111211 - 27 May 2021

Cited by 15 | Viewed by 2390

Abstract

The artificial bee colony (ABC) algorithm is a prominent swarm intelligence technique due to its simple structure and effective performance. However, the ABC algorithm has a slow convergence rate when it is used to solve complex optimization problems since its solution search equation is more of an exploration than exploitation operator. This paper presents an improved ABC algorithm for solving integer programming and minimax problems. The proposed approach employs a modified ABC search operator, which exploits the useful information of the current best solution in the onlooker phase with the intention of improving its exploitation tendency. Furthermore, the shuffle mutation operator is applied to the created solutions in both bee phases to help the search achieve a better balance between the global exploration and local exploitation abilities and to provide a valuable convergence speed. The experimental results, obtained by testing on seven integer programming problems and ten minimax problems, show that the overall performance of the proposed approach is superior to the ABC. Additionally, it obtains competitive results compared with other state-of-the-art algorithms. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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22 pages, 5031 KiB

Open AccessArticle

Advanced Metaheuristic Method for Decision-Making in a Dynamic Job Shop Scheduling Environment

by Hankun Zhang, Borut Buchmeister, Xueyan Li and Robert Ojstersek

Mathematics 2021, 9(8), 909; https://doi.org/10.3390/math9080909 - 19 Apr 2021

Cited by 14 | Viewed by 2610

Abstract

As a well-known NP-hard problem, the dynamic job shop scheduling problem has significant practical value, so this paper proposes an Improved Heuristic Kalman Algorithm to solve this problem. In Improved Heuristic Kalman Algorithm, the cellular neighbor network is introduced, together with the boundary handling function, and the best position of each individual is recorded for constructing the cellular neighbor network. The encoding method is introduced based on the relative position index so that the Improved Heuristic Kalman Algorithm can be applied to solve the dynamic job shop scheduling problem. Solving the benchmark example of dynamic job shop scheduling problem and comparing it with the original Heuristic Kalman Algorithm and Genetic Algorithm-Mixed, the results show that Improved Heuristic Kalman Algorithm is effective for solving the dynamic job shop scheduling problem. The convergence rate of the Improved Heuristic Kalman Algorithm is reduced significantly, which is beneficial to avoid the algorithm from falling into the local optimum. For all 15 benchmark instances, Improved Heuristic Kalman Algorithm and Heuristic Kalman Algorithm have obtained the best solution obtained by Genetic Algorithm-Mixed. Moreover, for 9 out of 15 benchmark instances, they achieved significantly better solutions than Genetic Algorithm-Mixed. They have better robustness and reasonable running time (less than 30 s even for large size problems), which means that they are very suitable for solving the dynamic job shop scheduling problem. According to the dynamic job shop scheduling problem applicability, the integration-communication protocol was presented, which enables the transfer and use of the Improved Heuristic Kalman Algorithm optimization results in the conventional Simio simulation environment. The results of the integration-communication protocol proved the numerical and graphical matching of the optimization results and, thus, the correctness of the data transfer, ensuring high-level usability of the decision-making method in a real-world environment. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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30 pages, 505 KiB

Open AccessArticle

An Approach Integrating Simulated Annealing and Variable Neighborhood Search for the Bidirectional Loop Layout Problem

by Gintaras Palubeckis

Mathematics 2021, 9(1), 5; https://doi.org/10.3390/math9010005 - 22 Dec 2020

Cited by 6 | Viewed by 2312

Abstract

In the bidirectional loop layout problem (BLLP), we are given a set of machines, a set of locations arranged in a loop configuration, and a flow cost matrix. The problem asks to assign machines to locations so as to minimize the sum of the products of the flow costs and distances between machines. The distance between two locations is calculated either in the clockwise or in the counterclockwise direction, whichever path is shorter. We propose a hybrid approach for the BLLP which combines the simulated annealing (SA) technique with the variable neighborhood search (VNS) method. The VNS algorithm uses an innovative local search technique which is based on a fast insertion neighborhood exploration procedure. The computational complexity of this procedure is commensurate with the size of the neighborhood, that is, it performs

O (1)

operations per move. Computational results are reported for BLLP instances with up to 300 machines. They show that the SA and VNS hybrid algorithm is superior to both SA and VNS used stand-alone. Additionally, we tested our algorithm on two sets of benchmark tool indexing problem instances. The results demonstrate that our hybrid technique outperforms the harmony search (HS) heuristic which is the state-of-the-art algorithm for this problem. In particular, for the 4 Anjos instances and 4 sko instances, new best solutions were found. The proposed algorithm provided better average solutions than HS for all 24 Anjos and sko instances. It has shown robust performance on these benchmarks. For 20 instances, the best known solution was obtained in more than

50 %

of the runs. The average time per run was below 10 s. The source code implementing our algorithm is made publicly available as a benchmark for future comparisons. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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18 pages, 10133 KiB

Open AccessArticle

Multi-Objective Evolutionary Algorithms to Find Community Structures in Large Networks

by Manuel Guerrero, Consolación Gil, Francisco G. Montoya, Alfredo Alcayde and Raúl Baños

Mathematics 2020, 8(11), 2048; https://doi.org/10.3390/math8112048 - 17 Nov 2020

Cited by 7 | Viewed by 2409

Abstract

Real-world complex systems are often modeled by networks such that the elements are represented by vertices and their interactions are represented by edges. An important characteristic of these networks is that they contain clusters of vertices densely linked amongst themselves and more sparsely connected to nodes outside the cluster. Community detection in networks has become an emerging area of investigation in recent years, but most papers aim to solve single-objective formulations, often focused on optimizing structural metrics, including the modularity measure. However, several studies have highlighted that considering modularityas a unique objective often involves resolution limit and imbalance inconveniences. This paper opens a new avenue of research in the study of multi-objective variants of the classical community detection problem by applying multi-objective evolutionary algorithms that simultaneously optimize different objectives. In particular, they analyzed two multi-objective variants involving not only modularity but also the conductance metric and the imbalance in the number of nodes of the communities. With this aim, a new Pareto-based multi-objective evolutionary algorithm is presented that includes advanced initialization strategies and search operators. The results obtained when solving large-scale networks representing real-life power systems show the good performance of these methods and demonstrate that it is possible to obtain a balanced number of nodes in the clusters formed while also having high modularity and conductance values. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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36 pages, 1803 KiB

Open AccessArticle

A New Hybrid BA_ABC Algorithm for Global Optimization Problems

by Gülnur Yildizdan and Ömer Kaan Baykan

Mathematics 2020, 8(10), 1749; https://doi.org/10.3390/math8101749 - 12 Oct 2020

Cited by 8 | Viewed by 2268

Abstract

Bat Algorithm (BA) and Artificial Bee Colony Algorithm (ABC) are frequently used in solving global optimization problems. Many new algorithms in the literature are obtained by modifying these algorithms for both constrained and unconstrained optimization problems or using them in a hybrid manner with different algorithms. Although successful algorithms have been proposed, BA’s performance declines in complex and large-scale problems are still an ongoing problem. The inadequate global search capability of the BA resulting from its algorithm structure is the major cause of this problem. In this study, firstly, inertia weight was added to the speed formula to improve the search capability of the BA. Then, a new algorithm that operates in a hybrid manner with the ABC algorithm, whose diversity and global search capability is stronger than the BA, was proposed. The performance of the proposed algorithm (BA_ABC) was examined in four different test groups, including classic benchmark functions, CEC2005 small-scale test functions, CEC2010 large-scale test functions, and classical engineering design problems. The BA_ABC results were compared with different algorithms in the literature and current versions of the BA for each test group. The results were interpreted with the help of statistical tests. Furthermore, the contribution of BA and ABC algorithms, which constitute the hybrid algorithm, to the solutions is examined. The proposed algorithm has been found to produce successful and acceptable results. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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17 pages, 1515 KiB

Open AccessArticle

Energy-Efficient Clustering Routing Protocol for Wireless Sensor Networks Based on Yellow Saddle Goatfish Algorithm

by Alma Rodríguez, Carolina Del-Valle-Soto and Ramiro Velázquez

Mathematics 2020, 8(9), 1515; https://doi.org/10.3390/math8091515 - 4 Sep 2020

Cited by 58 | Viewed by 3347

Abstract

The usage of wireless sensor devices in many applications, such as in the Internet of Things and monitoring in dangerous geographical spaces, has increased in recent years. However, sensor nodes have limited power, and battery replacement is not viable in most cases. Thus, energy savings in Wireless Sensor Networks (WSNs) is the primary concern in the design of efficient communication protocols. Therefore, a novel energy-efficient clustering routing protocol for WSNs based on Yellow Saddle Goatfish Algorithm (YSGA) is proposed. The protocol is intended to intensify the network lifetime by reducing energy consumption. The network considers a base station and a set of cluster heads in its cluster structure. The number of cluster heads and the selection of optimal cluster heads is determined by the YSGA algorithm, while sensor nodes are assigned to its nearest cluster head. The cluster structure of the network is reconfigured by YSGA to ensure an optimal distribution of cluster heads and reduce the transmission distance. Experiments show competitive results and demonstrate that the proposed routing protocol minimizes the energy consumption, improves the lifetime, and prolongs the stability period of the network in comparison with the stated of the art clustering routing protocols. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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11 pages, 344 KiB

Open AccessArticle

Insights into Exploration and Exploitation Power of Optimization Algorithm Using DSCTool

by Peter Korošec and Tome Eftimov

Mathematics 2020, 8(9), 1474; https://doi.org/10.3390/math8091474 - 1 Sep 2020

Cited by 9 | Viewed by 3107

Abstract

When making statistical analysis of single-objective optimization algorithms’ performance, researchers usually estimate it according to the obtained optimization results in the form of minimal/maximal values. Though this is a good indicator about the performance of the algorithm, it does not provide any information about the reasons why it happens. One possibility to get additional information about the performance of the algorithms is to study their exploration and exploitation abilities. In this paper, we present an easy-to-use step by step pipeline that can be used for performing exploration and exploitation analysis of single-objective optimization algorithms. The pipeline is based on a web-service-based e-Learning tool called DSCTool, which can be used for making statistical analysis not only with regard to the obtained solution values but also with regard to the distribution of the solutions in the search space. Its usage does not require any special statistic knowledge from the user. The gained knowledge from such analysis can be used to better understand algorithm’s performance when compared to other algorithms or while performing hyperparameter tuning. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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24 pages, 1147 KiB

Open AccessArticle

An Improved Grey Wolf Optimizer for a Supplier Selection and Order Quantity Allocation Problem

by Avelina Alejo-Reyes, Erik Cuevas, Alma Rodríguez, Abraham Mendoza and Elias Olivares-Benitez

Mathematics 2020, 8(9), 1457; https://doi.org/10.3390/math8091457 - 31 Aug 2020

Cited by 9 | Viewed by 2849

Abstract

Supplier selection and order quantity allocation have a strong influence on a company’s profitability and the total cost of finished products. From an optimization perspective, the processes of selecting the right suppliers and allocating orders are modeled through a cost function that considers different elements, such as the price of raw materials, ordering costs, and holding costs. Obtaining the optimal solution for these models represents a complex problem due to their discontinuity, non-linearity, and high multi-modality. Under such conditions, it is not possible to use classical optimization methods. On the other hand, metaheuristic schemes have been extensively employed as alternative optimization techniques to solve difficult problems. Among the metaheuristic computation algorithms, the Grey Wolf Optimization (GWO) algorithm corresponds to a relatively new technique based on the hunting behavior of wolves. Even though GWO allows obtaining satisfying results, its limited exploration reduces its performance significantly when it faces high multi-modal and discontinuous cost functions. In this paper, a modified version of the GWO scheme is introduced to solve the complex optimization problems of supplier selection and order quantity allocation. The improved GWO method called iGWO includes weighted factors and a displacement vector to promote the exploration of the search strategy, avoiding the use of unfeasible solutions. In order to evaluate its performance, the proposed algorithm has been tested on a number of instances of a difficult problem found in the literature. The results show that the proposed algorithm not only obtains the optimal cost solutions, but also maintains a better search strategy, finding feasible solutions in all instances. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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20 pages, 631 KiB

Open AccessArticle

A Partial Allocation Local Search Matheuristic for Solving the School Bus Routing Problem with Bus Stop Selection

by Herminia I. Calvete, Carmen Galé, José A. Iranzo and Paolo Toth

Mathematics 2020, 8(8), 1214; https://doi.org/10.3390/math8081214 - 23 Jul 2020

Cited by 5 | Viewed by 2986

Abstract

This paper addresses the school bus routing problem with bus stop selection, which jointly handles the problems of determining the set of bus stops to visit, allocating each student to one of these bus stops and computing the routes that visit the selected bus stops, so that the total routing cost is minimized and the walking distance of the students is limited by a given value. A fast and efficient matheuristic is developed based on an innovative approach that first partially allocates the students to a set of active stops that they can reach, and computes a set of routes that minimizes the routing cost. Then, a refining process is performed to complete the allocation and to adapt the routes until a feasible solution is obtained. The algorithm is tested on a set of benchmark instances. The computational results show the efficiency of the algorithm in terms of the quality of the solutions yielded and the computing time. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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25 pages, 1018 KiB

Open AccessArticle

Settings-Free Hybrid Metaheuristic General Optimization Methods

by Héctor Migallón, Akram Belazi, José-Luis Sánchez-Romero, Héctor Rico and Antonio Jimeno-Morenilla

Mathematics 2020, 8(7), 1092; https://doi.org/10.3390/math8071092 - 3 Jul 2020

Cited by 7 | Viewed by 2144

Abstract

Several population-based metaheuristic optimization algorithms have been proposed in the last decades, none of which are able either to outperform all existing algorithms or to solve all optimization problems according to the No Free Lunch (NFL) theorem. Many of these algorithms behave effectively, under a correct setting of the control parameter(s), when solving different engineering problems. The optimization behavior of these algorithms is boosted by applying various strategies, which include the hybridization technique and the use of chaotic maps instead of the pseudo-random number generators (PRNGs). The hybrid algorithms are suitable for a large number of engineering applications in which they behave more effectively than the thoroughbred optimization algorithms. However, they increase the difficulty of correctly setting control parameters, and sometimes they are designed to solve particular problems. This paper presents three hybridizations dubbed HYBPOP, HYBSUBPOP, and HYBIND of up to seven algorithms free of control parameters. Each hybrid proposal uses a different strategy to switch the algorithm charged with generating each new individual. These algorithms are Jaya, sine cosine algorithm (SCA), Rao’s algorithms, teaching-learning-based optimization (TLBO), and chaotic Jaya. The experimental results show that the proposed algorithms perform better than the original algorithms, which implies the optimal use of these algorithms according to the problem to be solved. One more advantage of the hybrid algorithms is that no prior process of control parameter tuning is needed. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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29 pages, 15240 KiB

Open AccessArticle

A Competitive Memory Paradigm for Multimodal Optimization Driven by Clustering and Chaos

by Jorge Gálvez, Erik Cuevas and Krishna Gopal Dhal

Mathematics 2020, 8(6), 934; https://doi.org/10.3390/math8060934 - 8 Jun 2020

Cited by 1 | Viewed by 1737

Abstract

Evolutionary Computation Methods (ECMs) are proposed as stochastic search methods to solve complex optimization problems where classical optimization methods are not suitable. Most of the proposed ECMs aim to find the global optimum for a given function. However, from a practical point of view, in engineering, finding the global optimum may not always be useful, since it may represent solutions that are not physically, mechanically or even structurally realizable. Commonly, the evolutionary operators of ECMs are not designed to efficiently register multiple optima by executing them a single run. Under such circumstances, there is a need to incorporate certain mechanisms to allow ECMs to maintain and register multiple optima at each generation executed in a single run. On the other hand, the concept of dominance found in animal behavior indicates the level of social interaction among two animals in terms of aggressiveness. Such aggressiveness keeps two or more individuals as distant as possible from one another, where the most dominant individual prevails as the other withdraws. In this paper, the concept of dominance is computationally abstracted in terms of a data structure called “competitive memory” to incorporate multimodal capabilities into the evolutionary operators of the recently proposed Cluster-Chaotic-Optimization (CCO). Under CCO, the competitive memory is implemented as a memory mechanism to efficiently register and maintain all possible optimal values within a single execution of the algorithm. The performance of the proposed method is numerically compared against several multimodal schemes over a set of benchmark functions. The experimental study suggests that the proposed approach outperforms its competitors in terms of robustness, quality, and precision. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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14 pages, 227 KiB

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Multi-Objective Optimization Benchmarking Using DSCTool

by Peter Korošec and Tome Eftimov

Mathematics 2020, 8(5), 839; https://doi.org/10.3390/math8050839 - 22 May 2020

Cited by 8 | Viewed by 2706

Abstract

By performing data analysis, statistical approaches are highly welcome to explore the data. Nowadays with the increases in computational power and the availability of big data in different domains, it is not enough to perform exploratory data analysis (descriptive statistics) to obtain some prior insights from the data, but it is a requirement to apply higher-level statistics that also require much greater knowledge from the user to properly apply them. One research area where proper usage of statistics is important is multi-objective optimization, where the performance of a newly developed algorithm should be compared with the performances of state-of-the-art algorithms. In multi-objective optimization, we are dealing with two or more usually conflicting objectives, which result in high dimensional data that needs to be analyzed. In this paper, we present a web-service-based e-Learning tool called DSCTool that can be used for performing a proper statistical analysis for multi-objective optimization. The tool does not require any special statistics knowledge from the user. Its usage and the influence of a proper statistical analysis is shown using data taken from a benchmarking study performed at the 2018 IEEE CEC (The IEEE Congress on Evolutionary Computation) is appropriate. Competition on Evolutionary Many-Objective Optimization. Full article

(This article belongs to the Special Issue Advances of Metaheuristic Computation)

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