Article

Particle swarm optimization-based algorithms for TSP and generalized TSP

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

A novel particle swarm optimization (PSO)-based algorithm for the traveling salesman problem (TSP) is presented. An uncertain searching strategy and a crossover eliminated technique are used to accelerate the convergence speed. Compared with the existing algorithms for solving TSP using swarm intelligence, it has been shown that the size of the solved problems could be increased by using the proposed algorithm.Another PSO-based algorithm is proposed and applied to solve the generalized traveling salesman problem by employing the generalized chromosome. Two local search techniques are used to speed up the convergence. Numerical results show the effectiveness of the proposed algorithms.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... The commonly used path planning algorithms for vehicles could be divided into two types. One is traditional algorithms, such as artificial potential field method [1,2], rapidly exploring random tree (RRT) [3][4][5], and A* algorithm [6][7][8]; and the other is intelligent algorithms, such as genetic algorithm (GA) [9][10][11], ant colony optimization (ACO) [12,13], particle swarm optimization (PSO) [14][15][16], and neural network algorithm [17,18]. This study is about global path planning based on an environment model with known terrain and landcover. ...
... After a certain number of iterations (the number of iterations is determined by the number of target points), the optimal position is obtained [48,49]. Formula (14) is the speed update formula, and formula (15) is the position update formula [14]: ...
... After a certain number of iterations (the number of iterations is determined by the number of target points), the optimal position is obtained [48,49]. Formula (14) is the speed update formula, and formula (15) is the position update formula [14]: ...
Article
Full-text available
Path planning is widely used in many domains, and it is crucial for the advancement of map navigation, autonomous driving, and robot path planning. However, existing path planning methods have certain limitations for complex field scenes with undulating terrain and diverse landcover types. This paper presents an energy-efficient 3D path planning algorithm based on an improved A* algorithm and the particle swarm algorithm in complex field scenes. The evaluation function of the A* algorithm was improved to be suitable for complex field scenes. The slope parameter and friction coefficient were respectively used in the evaluation function to represent different terrain features and landcover types. The selection of expanding nodes in the algorithm depends not only on the minimum distance but also on the minimum consumption cost. Furthermore, the turning radius factor and slope threshold factor of vehicles were added to the definition of impassable points in the improved A* algorithm, so that the accessibility of path planning could be guaranteed by excluding some bends and steep slopes. To meet the requirements for multi-target path planning, the improved A* algorithm was used as the fitness function of the particle swarm algorithm to solve the traveling salesman problem. The experimental results showed that the proposed algorithm is capable of multi-target path planning in complex field scenes. Furthermore, the path planned by this algorithm is more passable and more energy efficient. In this experimental environment model, the average energy-saving efficiency of the path planned by the improved algorithm is 14.7% compared to the traditional A* algorithm. This would be beneficial to the development of ecotourism and geological exploration.
... The most common approaches developed in this category are based on imitating the natural phenomenon. Some of the most frequently utilized approaches include Genetic Algorithm [18], Ant Colony Optimization (ACO) [35,54,58,59], Particle Swarm Optimization (PSO) [49], Simulated Annealing (SA) [9], Artificial Neural Network (ANN) [4,20], Evolutionary Algorithms (EA) [27], Artificial Immune Algorithms (AIS) [41], self-organizing map [53], Cuckoo search [43], Discrete Symbiotic Organisms Search (DSOS) [21], Simulated Annealing based Symbiotic Organisms Search (SA-SOS) algorithm [22], a hybrid algorithm based on the Glowworm Swarm Optimization (GSO) [10], a parallel computation model for Self-Organizing Map (SOM) [61]. Collectively, these approaches are known as swarm intelligence optimization, which consists of a population that simulates animal behaviour in the real world. ...
... Bektas [6] employed an Ant Colony with a clonal selection principle to solve several standard benchmark problems within the NP-complete class. Shi et al. [49] combined GA with PSO to identify the global optimum of multimodal functions. Geng and Chen further explored hybrid methodologies by integrating a parallel GA with ACO, specifically for addressing the TSP [9,25]. ...
Article
Full-text available
This paper presents a novel Genetic Algorithm (GA) designed to tackle the Travelling Salesman Problem (TSP) with remarkable efficacy. It integrates group theory into population initialization, employs Partially Matched Crossover (PMX), and adopts a 2-optimal mutation strategy. The pioneering approach harnesses algebraic structures in constructing group tours, utilizing integer addition modulo n within Zn{\mathbb {Z}}_{n} to generate varied initial solutions. The initial population enhances diversity by ensuring that each individual/tour shares an identical node order but begins from a distinct starting node. This distinctiveness in starting nodes facilitates thorough exploration of the entire search space. The Partially mapped Crossover operator, grounded in order principles, is a crucial mechanism for transferring sequence and value characteristics from parental to offspring tour. This operation effectively guides a strong local search. Subsequently, applying a 2-opt optimal mutation seeks to refine the solution further, targeting a more globally optimal outcome. The effectiveness of this methodology is evaluated through experiments with TSP instances sourced from the widely recognized TSPLIB. Furthermore, the superiority of our proposed approach is demonstrated through comparisons with state-of-the-art methods developed within hybrid frameworks. Statistical analyses underscore the significance and effectiveness of the proposed methodology.
... The most common approaches developed in this category are based on imitating the natural phenomenon. Some of the most commonly used approaches are Genetic Algorithm (Dong & Cai, 2019), Ant Colony Optimization (ACO) (Jun-man & Yi, 2012;Tuba & Jovanovic, 2013;Ugur & Aydin, 2009;Stodola et al., 2020), Particle Swarm Optimization (PSO) (Shi et al., 2007), Simulated Annealing (SA) (Chen & Chien, 2011), Artificial Neural Network (ANN) (Durbin et al., 1989;Ali & Kamoun, 1993), Evolutionary Algorithms (EA) (Goldberg & Lingle Jr, 1985), Artificial Immune Algorithms (AIS) (Masutti & de Castro, 2009), self-organizing map (Somhom et al., 1997), Cuckoo search (Ouaarab et al., 2014), Discrete Symbiotic Organisms Search (DSOS) , Simulated Annealing based Symbiotic Organisms Search (SA-SOS) algorithm , a hybrid algorithm based on the Glowworm Swarm Optimization (GSO) (Chen et al., 2017), a parallel computation model for Self-Organizing Map (SOM) (Wang et al., 2017). Collectively, these approaches called swarm intelligence optimization, consist of a population that simulates animal behaviour in the real world. ...
... Bektas (Bektas, 2006) used an Ant colony with clonal selection principle for solving some standard benchmark problems belonging to a class of N P-complete problem. Shi et al. (Shi et al., 2007) combined GA with PSO to identify the global optimum of multimodal functions. Geng and Chen further explored hybrid methodologies by integrating a parallel GA with ACO, specifically for addressing the TSP (Geng et al., 2011;Chen & Chien, 2011). ...
Preprint
Full-text available
This paper presents a novel Genetic Algorithm (GA) designed to tackle the Travelling Salesman Problem (TSP) with remarkable efficacy. It integrates group theory into population initialization, employs Partially Matched Crossover (PMX), and adopts a 2-optimal mutation strategy. The pioneering approach harnesses algebraic structures in constructing group tours, utilizing integer addition modulo n within Zn to generate varied initial solutions. By ensuring that each individual/ tour shares an identical node order but begins from a distinct starting node, the initial population enhances diversity. This distinctiveness in starting nodes facilitates thorough exploration of the entire search space. The Partially Matched Crossover operator, grounded in principles of order, is a crucial mechanism for transferring sequence and value characteristics from parental to offspring tours. This operation effectively guides a strong local search. Subsequently, applying a 2-opt optimal mutation seeks to refine the solution further, targeting a more globally optimal outcome. The effectiveness of this methodology is evaluated through experiments with TSP instances sourced from the widely recognized TSPLIB. Furthermore, the superiority of our proposed approach is demonstrated through comparisons with state-of-the-art methods developed within hybrid frameworks. Statistical analyses underscore the significance and effectiveness of the proposed methodology.
... Researchers have developed various algorithms and heuristics to solve or approximate the TSP, such as branch and bound (Ali and Kennington 1986), dynamic programming (Xu et al. 2023), genetic algorithm (GA) (Goldberg and Richardson 1987), simulated annealing (SA) (Paydar et al. 2010;Johnson 1990), tabu search (Glover 1990(Glover , 1986, Ant Colony Optimization (ACO) (Dorigo 1997), Particle Swarm Optimization (PSO) (Shi et al. 2007), etc. Early investigations into TSP with subsequent research gradually expanding to incorporate new perspectives and approaches. ...
... Early investigations into TSP with subsequent research gradually expanding to incorporate new perspectives and approaches. Shi et al. (2007) discussed PSObased algorithms generalized TSP. Laporte (1992) described an overview of exact and approximate algorithms to solve TSP. ...
Article
Full-text available
The traveling salesman problem is a well-known combinatorial optimization problem. Solving the traveling salesman problem efficiently becomes more challenging when considering uncertainties in the problem parameters, which are prevalent in real-world scenarios. Pythagorean fuzzy uncertain variables combine the strengths of fuzzy logic with the principles of uncertainty theory, allowing for a more balanced and comprehensive representation of uncertainty. This paper defines the theoretical foundations of discrete, linear, and zigzag Pythagorean fuzzy uncertainty distributions, including their mathematical formulation and operational laws. Moreover, it proposes a novel approach to tackle the traveling salesman problem with Pythagorean fuzzy uncertainty distribution using the tabu search metaheuristic. By integrating this uncertainty representation into the tabu search metaheuristic, the proposed method effectively explores the solution space while considering the potential variations in the Pythagorean fuzzy distance matrix. The detailed steps of the algorithm are demonstrated via a numerical exemplification. Subsequently, a comprehensive case study is presented, wherein the objective is to determine the optimal touring sequence among the largest cities of China. This investigation is founded upon data source from Google Maps. We conduct a sensitivity analysis to assess the sensitivity of the model’s output by varying the input parameters. The proposed algorithm’s performance is compared with traditional tabu search and other existing methods. The results highlight the potential of incorporating Pythagorean fuzzy uncertainty distribution into metaheuristic algorithms for solving combinatorial optimization problems.
... It is one of the most commonly used and popular meta-heuristic methods. It can find effective solutions for a diverse range of complex optimization problems, such as the traveling salesperson [79], 0/1 Knapsack problem [80], finding max clique [81], and many others. In GA, a solution is perceived as a vector of bits. ...
Article
Full-text available
Interdependent systems confront rapidly growing cybersecurity threats. This paper delves into the realm of security decision-making within these complex interdependent systems. We design a resource allocation framework to improve the security of interdependent systems managed by a single defender. Our framework models these systems and their potential attack vulnerabilities using the notion of attack graphs. We propose four defense mechanisms, incorporating a popular network analysis algorithm called PageRank which is used to identify the importance of different critical assets in the system. These mechanisms stem from existing graph theories widely used in graphical models (including Adjacent Nodes, In-degree Nodes, Min-Cut Edges, and Markov Blanket). We adopt the PageRank algorithm to extract useful information about the attack graphs we use. Our approaches show low sensitivity to the number of concurrent attacks launched over interdependent systems. We evaluate our decision-making framework via ten attack graphs, which include multiple real-world interdependent systems. We quantify the level of security improvement under our defense methods compared to four well-known resource allocation algorithms and other proposed approaches. Our proposed framework leads to better resource allocations compared to these algorithms in most test cases. According to our results and statistical tests, our defense resource allocation framework enhances security decision-making under various circumstances. Moreover, We release the full implementation of our framework for the research community to leverage it and build on it with new methods and datasets.
... Over the years various techniques have been suggested to solve the TSP, such as Genetic Algorithm (GA) [7] [8], Hill Climbing [9], Nearest Neighbor and Minimum Spanning Tree algorithms [10], Simulated Annealing [11], Ant Colony [9], Tabu Search [12], Particle Swarm [13], Elastic Nets [14], Neural Networks [15], etc. Genetic algorithms are one of the algorithms that extensively applied to solve the TSP [16]. ...
Preprint
Full-text available
Genetic algorithm (GA) is an efficient tool for solving optimization problems by evolving solutions, as it mimics the Darwinian theory of natural evolution. The mutation operator is one of the key success factors in GA, as it is considered the exploration operator of GA. Various mutation operators exist to solve hard combinatorial problems such as the TSP. In this paper, we propose a hybrid mutation operator called "IRGIBNNM", this mutation is a combination of two existing mutations, a knowledge-based mutation, and a random-based mutation. We also improve the existing "select best mutation" strategy using the proposed mutation. We conducted several experiments on twelve benchmark Symmetric traveling salesman problem (STSP) instances. The results of our experiments show the efficiency of the proposed mutation, particularly when we use it with some other mutations. Keyword: Knowledge-based mutation, Inversion mutation, Slide mutation, RGIBNNM, SBM.
... According to their nature TSPs can be categorized into two types, namely, symmetric TSP (STSP) and asymmetric TSP (ATSP) (Dorigo et al. 1996;Majumdar and Bhunia 2011;Shi et al. 2007). For an STSP d i j = d ji , ∀ i, j ∈ V and for an ATSP d i j = d ji for at least one pair of vertices, i, j ∈ V . ...
Article
Full-text available
In this study, grey wolf optimizer (GWO), genetic algorithm (GA), and K-Opt operation are combined to develop a metaheuristic named GWO-GA for the traveling salesman problems (TSPs). The alteration of the positions of two nodes of a potential solution (sequence of nodes or route) of a TSP is defined as a swap operation and a sequence of such swap operations is defined as a swap sequence. Using swap operations and swap sequence, the perturbation rules of the basic GWO are modified for the perturbation of any potential route (solution) of a TSP. The proposed approach consists of two phases. In the first phase, the operations of GWO are applied to a randomly generated set of potential solutions of the target problem for a predefined number of iterations. At the end of each iteration, the 3-Opt operation is applied to the solutions for which better movement is not found using GWO operations. In the second phase of the algorithm, the GA is applied for another number of iterations on the output set of the GWO phase. In the GA phase, the Roulette wheel selection process and multi-point cyclic crossover operation are used in a different approach. K-Opt operation for K=3 is used in place of mutation operation. K-Opt operation is again applied to the best-found solution after the end of the second phase for a fixed number of iterations for a possible improvement. The efficiency of the algorithm is tested on a set of benchmark test instances of different sizes from the TSPLIB. The performance of the proposed GWO-GA is compared with a set of state-of-the-art algorithms on TSPs using statistical tests and its superiority is established using comparison tables of the results as well as statistical tests. Obtained optimal routes are also presented to support the concluding remarks. It is observed that the algorithm shows a 100% success rate for problem size up to 144 and optimum route up to problem size 264.
... Other combinatorial optimization algorithms have also been used, such as the combinatorial artificial bee colony algorithm [3], the intelligent optimization algorithm [13], the artificial bee colony algorithm [14], the particle swarm optimization algorithm [15], genetic algorithms [12,16], biogeography-based optimization [9,17], the memetic algorithm [18], discrete rat swarm optimization [19], and the farmland fertility algorithm [1]. However, many of these recent works exhibit some issues, such as low convergence speed, low result quality, and the need for tuning many parameters. ...
Article
Full-text available
The traveling salesman problem (TSP) is a classical optimization problem with practical applications in logistics, transportation, and network design. This research proposes an efficient mixed-integer linear programming (MILP) model based on the branch flow formulation which prevents the formation of sub-tours during the solution process and guarantees valid optimal routes. Implemented in Julia with the JuMP optimization package and the HiGHS solver, the model achieves high computational efficiency. Unlike classical models, the branch flow formulation ensures a quadratic constraint growth, rather than an exponential one, significantly enhancing scalability. Benchmark tests on various instances (Eil51, Eil76, KroA100) demonstrate results comparable to state-of-the-art combinatorial optimizers, and six new TSP instances, ranging from 50 to 300 cities, validate the model’s performance. This scalable and robust approach is well-suited for real-world applications in supply chain management, network optimization, and urban planning, and it shows potential for future extensions to dynamic or multi-objective TSP variants.
... These algorithms include construction algorithms, iterative improvement algorithms, branch-and-bound and branch-and-cut exact algorithms, and many meta heuristic algorithms such as simulated annealing (SA), tabu search (TS), ant colony (AC) [4] and genetic algorithm (GA). Some of the well-known tour construction procedures are the nearest neighbor procedure by Rosenkrantz et al. [20], the Clarke and Wright savings' algorithm, the insertion procedures, the partitioning approach by Karp [14] and the minimal spanning tree approach by Christofides etc. [5][6][7][8] [16][17][18][19][20][21], The branch exchange is perhaps the best-known iterative improvement algorithm for the TSP. The 2-opt and 3-optheuristics were described in Lin. ...
... It is one of the most commonly used and popular meta-heuristic methods. It can find effective solutions for a diverse range of complex optimization problems, such as the traveling salesperson [98], feature selection [73], cell formation [44], 0/1 Knapsack problem [86], finding max clique [14], and many others. In GA, a solution is perceived as a vector of bits. ...
Preprint
Full-text available
Interdependent systems confront rapidly growing cybersecurity threats. This paper delves into the realm of security decision-making within these complex interdependent systems. We design a resource allocation framework to improve the security of interdependent systems managed by a single rational (logical) defender. Our framework models these systems and their potential attack vulnerabilities using the notion of cyber attack graphs. We propose four defense mechanisms, incorporating a popular network analysis algorithm called PageRank which is used to identify the importance of different critical assets in the system. These mechanisms stem from existing graph theories widely used in graphical models (including Adjacent Nodes, In-degree Nodes, Min-Cut Edges, and Markov Blanket). We adopt two versions of the PageRank algorithm to extract useful information about the attack graphs we use. Our approaches show low sensitivity to the number of concurrent attacks launched over interdependent systems. We evaluate our decision-making framework via ten attack graphs, which include multiple real-world interdependent systems. We quantify the level of security improvement under our defense methods compared to four well-known resource allocation algorithms and other proposed approaches. Our proposed framework leads to better resource allocations compared to these algorithms in most test cases. According to our results and statistical tests, our defense resource allocation approach’s outcomes are superior. Our framework enhances security decision-making under various circumstances. Moreover, We release the full implementation of our framework for the research community to leverage it and build on it with new methods and datasets.
... o enhance system performance, the algorithm took into account load balancing as well as reaction time. In the suggested approach, nondependent containers were put on nearby hosts to minimize their connection [8], [36]. Compared to the existing and PSO algorithms, the results showed a 20-25 percent improvement. ...
Article
Full-text available
Big data analysis used by Internet of Things (IoT) objects is one of the most difficult issues to deal with today due to the data increase rate. Container technology is one of the many technologies available to address this problem. Because of its adaptability, portability, and scalability, it is particularly useful in IoT micro-services. The most promising lightweight virtualization method for providing cloud services has emerged owing to the variety of workloads and cloud resources. The scheduler component is critical in cloud container services for optimizing performance and lowering costs. Even though containers have gained enormous traction in cloud computing, very few thorough publications address container scheduling strategies. This work organizes its most innovative contribution around optimization scheduling techniques, which are based on three meta-heuristic algorithms. These algorithms include the particle swarm algorithm, the genetic algorithm, and the ant colony algorithm. We examine the main advantages, drawbacks, and significant difficulties of the existing approaches based on performance indicators. In addition, we made a fair comparison of the employed algorithms by evaluating their performance through Quality of Service (QoS) while each algorithm proposed a contribution. Finally, it reveals a plethora of potential future research areas for maximizing the use of emergent container technology.
... Compared with other algorithms, it has the characteristics of high accuracy, fast convergence, simple implementation and reliable solution to practical problems. It is widely used in the optimization of algorithms [25][26][27]. Therefore, we use PSO as a tool to find initial clustering centers in this work. ...
Article
Full-text available
As an extension of Fuzzy C-Means (FCM), Evidence C-Means (ECM) is proposed in the framework of Dempster–Shafer theory (DST) and has been applied to many fields. However, the objective function of ECM involves only the distortion between the object and the prototype, which relies heavily on the initial prototype. Therefore, ECM may encounter the problem of local optimization. To solve this problem, this paper introduces ECM with Particle Swarm Optimization (PSO) initialization to determine the initial clustering centroids, and proposes Particle Swarm Optimization-based Evidential C-Means (PSO-ECM), which reduces the influence of bad initial prototypes and improves the local optimality problem of ECM. PSO-ECM is compared with three other clustering algorithms in four experiments and with ECM on a noise-containing dataset. According to the experimental results, PSO-ECM performs well in terms of different clustering validity metrics compared with existing clustering algorithms, has high stability of clustering, and can effectively and stably cluster noise-containing datasets and accurately identify outlier points.
... In metaheuristics the new solution obtained by a modification of the current solution is usually named a neighbor solution. It is possible to see the application of many different metaheuristic algorithms to the TSP and its variants [1,16,18,21,25]. ...
Article
Full-text available
Turkish Cashier Problem (TCP) is a new application area of the traveling salesman problem that was introduced to the literature recently. In this problem, the cashier can use public transportation or take a taxi where the cashier must visit multiple customer locations while minimizing the total transportation cost. In this study, we introduce a more realistic version of this problem where time has been integrated. This aspect is achieved by imposing time intervals within which the cashier must visit the customers. We name this problem as the TCP with time windows (TCPwTW). We develop several matheuristic algorithms to solve the TCPwTW: a modified version of the Simplify and Conquer (SAC) algorithm that was suggested for the TCP, simulated annealing (SA), original and modified versions of the migrating birds optimization (MBO) algorithm coupled with mathematical programming. We also tried to find the exact optimum using a Solver where for complex problems, only lower bounds were found. Numerical experimentation reveals that while for problems with loose time intervals, an exact solver can be considered. Once the time intervals tighten up, the best solutions can be obtained using matheuristics involving SA and MBO.
... These algorithms include construction algorithms, iterative improvement algorithms, branch-and-bound and branch-and-cut exact algorithms, and many meta heuristic algorithms such as simulated annealing (SA), tabu search (TS), ant colony (AC) [4] and genetic algorithm (GA). Some of the well-known tour construction procedures are the nearest neighbor procedure by Rosenkrantz et al. [20], the Clarke and Wright savings' algorithm, the insertion procedures, the partitioning approach by Karp [14] and the minimal spanning tree approach by Christofides etc. [5][6][7][8] [16][17][18][19][20][21], The branch exchange is perhaps the best-known iterative improvement algorithm for the TSP. The 2-opt and 3-optheuristics were described in Lin. ...
Article
Full-text available
This paper presents Highest Suffix method for solving the classical symmetric traveling salesman problem. This concept is an alternative method for solving Traveling Salesman problem (TSP). It is possible to further improve a TSP tour that cannot be improved by other local search methods. To test the performance of the proposed method, two examples are solved here. This is a new approach to solve the classical symmetric travelling salesman problem by highest suffix method. So, this paper shows that the proposed algorithm is efficient for solving the Traveling Salesman problem (TSP).
... Many discrete optimization problems are used in realworld applications, e.g., the traveling salesman problem (TSP) [20,21], the graph coloring problem [22], the manufacturing cell formation problem [23], and the water pump switching problem [24]. Discrete problems of the TSP type are NP-hard complexity problems and, therefore, cannot be solved by any known method in polynomial time. ...
Article
Full-text available
This work addresses the problem of the development of a robotic system for the picking of parts cut by a CNC machine and the optimization of the sequencing of this picking process. An automated parts collection system is optimized to reduce the time required to perform the task of both picking and the subsequent classification by the type of part. The automated picking system, which is located at the end of a cutting machine, uses a robot equipped with an additional axis to expand its working space. Therefore, in this proposal, the industrial equipment necessary to automate this process is designed and the process to be optimized is computationally modeled. In particular, three discrete optimization algorithms are analyzed, with different evolution strategies and operators, but all of them are free of specific configuration parameters. The whole process is shown in this research, from the design of the procedure to the design of the tool, the algorithm selection, and elements validation. Finally, the first steps towards its industrial implementation are presented, and the hypothesis behind this project is validated.
... PIbased methods improve randomly generated initial solutions step by step such as k-opt, v-opt, and genetic algorithms [22,23]. While the ant system uses a PC-based strategy for finding the optimum tour of the traveling salesman problem, ABC [24] and PSO [25] try to improve solutions with a PI-based strategy. In recent years, some other swarm intelligence algorithms have also been proposed for solving traveling salesman problems. ...
Article
Full-text available
This paper presents a novel approach based on the ant system algorithm for solving discrete optimization problems. The proposed method is based on path construction, path improvement techniques, and the footprint mechanism. Some information about the optimization problem and collective intelligence is used in order to create solutions in the path construction phase. In the path improvement phase, neighborhood operations are applied to the solution, which is the best of the population and is obtained from the path construction phase. The collective intelligence in the path construction phase is based on a footprint mechanism, and more footprints on the arc improve the selection chance of this arc. A selection probability is also balanced by using information about the problem (e.g., the distance between nodes for a traveling salesman problem). The performance of the proposed method has been investigated on 25 traveling salesman problems and compared with state-of-the-art algorithms. The experimental comparisons show that the proposed method produced comparable results for the problems dealt with in this study.
... To represent the position and velocity of the particles and solve TSP, Pang et al. [26] presented a modified discrete PSO algorithm with integrated fuzzy matrices. A unique particle swarm optimization (PSO)-based approach for TSP was proposed by Shi et al. in a different paper [27]. They sped up convergence using a crossover elimination method and an ambiguous searching strategy. ...
Article
Full-text available
According to Moore's law, computer processing hardware technology performance is doubled every year. To make effective use of this technological development, the algorithmic solutions have to be developed at the same speed. Consequently, it is necessary to design parallel algorithms to be implemented on parallel machines. This helps to exploit the multi-core environment by executing multiple instructions simultaneously on multiple processors. Traveling Salesman (TSP) is a challenging non-deterministic-hard optimization problem that has exponential running time using brute-force methods. TSP is concerned with finding the shortest path starting with a point and returning to that point after visiting the list of points, provided that these points are visited only once. Meta-heuristic optimization algorithms have been used to tackle TSP and find near-optimal solutions in a reasonable time. This paper proposes a parallel River Formation Dynamics Optimization Algorithm (RFD) to solve the TSP problem. The parallelization technique depends on dividing the population into different processors using the Map-Reduce framework in Apache Spark. The experiments are accomplished in three phases. The first phase compares the speedup, running time, and efficiency of RFD on 1 (sequential RFD), 4, 8, and 16 cores. The second phase compares the proposed parallel RFD with three parallel water-based algorithms, namely the Water Flow algorithm, Intelligent Water Drops, and the Water Cycle Algorithm. To achieve fairness, all algorithms are implemented using the same system specifications and the same values for shared parameters. The third phase compares the proposed parallel RFD with the reported results of metaheuristic algorithms that were used to solve TSP in the literature. The results demonstrate that the RFD algorithm has the best performance for the majority of problem instances, achieving the lowest running times across different core counts. Our findings highlight the importance of selecting the most suitable algorithm and core count based on the problem characteristics to achieve optimal performance in parallel optimization.
... Algoritmos populacionais são frequentemente usados para solucionar o TSP e suas variações. Para o GTSP, autores também utilizaram o sistema de colônia de formigas [Pintea et al., 2017;Jun-man e Yi, 2012] e a otimização por enxame de partículas [Shi et al., 2007]. Ambos os algoritmos produzem tempos de execução relativamente altos e com menor número de soluçõeś otimas encontradas em comparação a algoritmos meméticos. ...
... Improved from [22] that solved a set of nodes divided into clusters for exploring the search space. ...
Article
Full-text available
span lang="EN-US">Nowadays, a significant number of researchers are focusing on utilizing soft computing approaches to address the issue of scheduling in applications concerned with hazardous waste management. In Malaysia, there is thoughtless awareness of the management of hazardous waste, even though the production of wastes in hazardous domains at the industrial and domestic levels has been rising lately. According to previous research findings, the location routing problem (LRP) can be designated as one of the models closer to the actual situation, evaluating the most suitable and optimal location for establishing facilities and utilizing transportation for pick-up and distribution. Recent studies have focused on enhancing the LRP model, and its methodologies approach to solve the waste management problem in hazardous domains. In this paper, a comprehensive review of the better promising and practicable mathematical model of LRP and its methodology approach is discussed, as well as an analysis of the publishing pattern and the trend of research over the preceding five years and more, as retrieved from the web of science (WoS) database. In conclusion, this research is significant in ensuring the effectiveness of reliable mathematical model development and suitable methodologies in the future for solving hazardous waste management problems.</span
... To solve the TSP problem, we employ the particle swarm algorithm [34]. Let P ¼ fp 0 ; p 1 ; . . ...
Article
Full-text available
Exploring underwater acoustic sensing networks to detect and monitor ocean characteristics provides a novel approach for marine environment monitoring, disaster warning and resource exploration. However, factors such as underwater acoustic communication, ocean currents and sensor power prevent the direct adoption of terrestrial wireless sensing networks. Moreover, most previous studies have simply increased the number of nodes while ignoring the interference caused by high node density. In this paper, data collection in an offshore tidal scenario is investigated by table forwarding based election (TFBE) scheme. To balance node energy and reduce energy consumption of autonomous underwater vehicles (AUVs), the improved algorithm (TFBE-A) is proposed. Meanwhile, in this scheme, we adopt AUVs to assist in collecting and put forward a 3D precision adjustable trajectory planning (3DPATP) algorithm. To better perform 3DPATP, the maximum minimum error estimation algorithm is presented to simplify the computer operations. Additionally, we study the probability distribution function of nodes in space, providing a foundation for arranging nodes and optimizing topology. Simulation results indicate that our algorithm outperforms compared strategies, particularly in latency and energy consumption.
... Relatively to the discrete domain, PSO has recently been proposed as an effective way to tackle NP-hard routing problems such as the TSP. Among the first contributions, we can cite the relevant works by [58,59]. However, the real burst of interest originated from [20,21], who showed the superiority of PSO in tackling these NP-hard problems. ...
Article
Full-text available
This paper proposes a new metaheuristic algorithm called Particle Swarm-based picking time minimization (Pkt_PSO), ideated for picking time minimization in manual warehouses. As the name suggests, Pkt_PSO is inspired by Particle Swarm Optimization (PSO), and it is specifically designed to minimize the picking time in order case picking contexts. To assess the quality and the robustness of Pkt_PSO, it is compared to five alternative algorithms used as benchmarks. The comparisons are made in nine different scenarios obtained by changing the layout of the warehouse and the length of the picking list. The results of the analysis show that Pkt_PSO has a slower convergence rate and suffers less of early stagnation in local minima; this ensures a more extensive and accurate exploration of the solution space. In fact, the solutions provided by Pkt_PSO are always better (or at least comparable) to the ones found by the benchmarks, both in terms of quality (closeness to the overall best) and reliability (frequency with which the best solution is found). Clearly, as more solutions are explored, the computational time of Pkt_PSO is longer, but it remains compatible with the operational needs of most practical applications.
... Although, the application domain is a bit different than ours (feature selection), in [7] it is possible to find a good review of a bunch of metaheuristics recently suggested in the literature. For the TSP and its variants there are many applications of metaheuristic algorithms some of which can be seen in [11][12][13][14][15][16]. ...
Article
Full-text available
In this study, we first introduce a new application of the asymmetric traveling salesman problem which is about a small restaurant with one cook and a single stove. Once a meal has started cooking on the stove, the cook prepares the next meal on the table where the preparation time is dependent on the previous meal prepared. For the solution of this problem, besides several simple construction algorithms and a new version of the simulated annealing (SA) algorithm, we focus on enhanced versions of the recently introduced migrating birds optimization (MBO) algorithm. The original MBO algorithm might suffer from early convergence. Here we introduce several different ways of handling this problem. The extensive numerical experimentation conducted shows the superiority of the enhanced MBO over the original MBO (about 2.62 per cent) and over the SA algorithm (about 1.05 per cent).
... These algorithms can find exact solutions, but the time and space complexity will increase sharply with the increase of the problem's scale, so they are only suitable for solving small-scale problems. Intelligent algorithms include the genetic algorithm [75], the ant colony algorithm [76], the particle swarm optimization algorithm [77], the hybrid algorithm [78,79], etc. Although these algorithms cannot guarantee exact solutions, they can quickly find approximate solutions within the allowable error range, achieving the best balance between solving accuracy and solving efficiency. ...
Article
Full-text available
Electronic equipment, including phased array radars, satellites, high-performance computers, etc., has been widely used in military and civilian fields. Its importance and significance are self-evident. Electronic equipment has many small components, various functions, and complex structures, making assembly an essential step in the manufacturing process of electronic equipment. In recent years, the traditional assembly methods have had difficulty meeting the increasingly complex assembly needs of military and civilian electronic equipment. With the rapid development of Industry 4.0, emerging intelligent assembly technology is replacing the original “semi-automatic” assembly technology. Aiming at the assembly requirements of small electronic equipment, we first evaluate the existing problems and technical difficulties. Then, we analyze the intelligent assembly technology of electronic equipment from three aspects: visual positioning, path and trajectory planning, and force–position coordination control technology. Further, we describe and summarize the research status and the application of the technology and discuss possible future research directions in the intelligent assembly technology of small electronic equipment.
... (1) Traveling Salesman Problem (TSP) Algorithm. TSP is a classical path planning algorithm, which can find the shortest path that traverses all nodes once [42]. ...
Article
Full-text available
Benefiting from the progress of microelectromechanical system (MEMS) technology, wireless sensor networks (WSNs) can run a large number of complex applications. One of the most critical challenges for complex WSN applications is the huge computing demands and limited battery energy without any replenishment. The recent development of UAV-assisted cooperative computing technology provides a promising solution to overcome these shortcomings. This paper addresses a three-tier WSN model for UAV-assisted cooperative computing, which includes several sensor nodes, a moving UAV equipped with computing resources, and a sink node (SN). Computation tasks arrive randomly at each sensor node, and the UAV moves around above the sensor nodes and provides computing services. The sensor nodes can process the computation tasks locally or cooperate with the UAV or SN for computing. In a life cycle of the UAV, we aim to maximize the energy efficiency of cooperative computing by optimizing the UAV path planning on the constraints of node energy consumption and task deadline. To adapt to the time-varying indeterminate environment, a deep Q network- (DQN-) based path planning algorithm is proposed. Simulation studies show that the performance of the proposed algorithm is better than the competitive algorithms, significantly improves the energy efficiency of cooperative computing, and achieves energy consumption balance.
... Developed by Eberhart and Kennedy in 1995, particle-swarm optimization (PSO) is a population-based stochastic optimization technique inspired by the swarming behavior characteristic of bird flocks or fish schools [29]. PSO has recently been applied to COPs, such as shop scheduling [30,31], the traveling-salesman problem [32], quality-of-service multicast routing [33], and vehicle routing [34,35]. ...
Article
Full-text available
The dynamic-scheduling problem of transmission tasks (DSTT) is an important problem in the daily work of radio and television transmission stations. The transmission effect obtained by the greedy algorithm for task allocation is poor. In the case of more tasks and equipment and smaller time division, the precise algorithm cannot complete the calculation within an effective timeframe. In order to solve this problem, this paper proposes a discrete particle swarm optimization algorithm (DPSO), builds a DSTT mathematical model suitable for the DPSO, solves the problem that particle swarm operations are not easy to describe in discrete problems, and redefines the particle motion strategy and adds random disturbance operation in its probabilistic selection model to ensure the effectiveness of the algorithm. In the comparison experiment, the DPSO achieved much higher success rates than the greedy algorithm (GR) and the improved genetic algorithm (IGA). Finally, in the simulation experiment, the result data show that the accuracy of the DPSO outperforms that of the GR and IGA by up to 3.012295% and 0.11115%, respectively, and the efficiency of the DPSO outperforms that of the IGA by up to 69.246%.
... In metaheuristics, the new solution obtained by a modification of the current solution is usually named as a neighbor solution. It is possible to see the application of many different metaheuristic algorithms to the TSP and its variants [7][8][9][10][11]. ...
Preprint
In this study, we first introduce a new application of the asymmetric traveling salesman problem (ATSP) which is about a small restaurant with one cook and a single stove. Once a meal has started cooking on the stove, the cook prepares the next meal on the table where the preparation time is dependent on the previous meal prepared. For the solution of this problem, besides several simple construction algorithms and a new version of the simulated annealing algorithm, we focus on enhanced versions of the recently introduced migrating birds optimization (MBO) algorithm. The original MBO algorithm might suffer from early convergence. Here we introduce several different ways of handling this problem. The extensive numerical experimentation conducted shows the superiority of the enhanced MBO over the original MBO (about 2.62 per cent) and over the simulated annealing algorithm (about 1.05 per cent).
... In 1991, Reinelt [15] introduced TSPLIB, which is still providing problem instances for TSP. In last few decades, many heuristic and meta-heuristic algorithms have been developed with some of the following notable algorithms: Ant colony optimization [16][17][18][19][20][21][22][23][24][25], Neural network [26][27][28][29], Self-organizing maps [30], Particle swarm optimization techniques [31,32], Simulated annealing [33], Weed optimization [34,35], Genetic algorithm [36][37][38][39]. ...
Chapter
Full-text available
In this chapter, we propose a novel algorithm that uses Genetic algorithm with group theory for initial population generation and also propose a novel crossover for solving Traveling Salesman Problem. In the group tour construction method, each individual/initial tour has distinct start city provided that population size is equal to total number of cities. In the initial population, each individual/tour has a distinct starting city. The distinct starting cites of each tour provide genetic material for exploration for the whole search space. Therefore, a heterogeneous starting city of a tour in initial population is generated to have rich diversity. Proposed crossover based on greedy method of sub-tour connection drives the efficient local search, followed by 2-opt mutation for improvement of tour for enhanced/optimal solution. The result of the proposed algorithm is compared with other standard algorithms followed by conclusion.
... This discrete CS has been tested on forty one (41) benchmark TSP instances from the TSPLIB library which is publicly available. The comparisons were carried out with basic discrete CS and with some other recent modifications (GSA-ACS-PSOT) [40] and discrete particle swarm optimization (DPSO) [41]. The results indicate that the proposed discrete CS outperforms the other two methods in solving TSP. ...
Article
Full-text available
Path finding is used to solve the problem of finding a traversable path through an environment with obstacles. This problem can be seen in many different fields of study and these areas rely on fast and efficient path finding algorithms. This paper aims to describe and review state of the art optimization techniques that are used on optimized path finding and compare their performances. Moreover, a special attention is paid on the proposed approaches to identify how they are tested on different test cases; whether the test cases are automatically generated or benchmark instances. The review opens avenues about the importance of automatic test case generation to test the different path finding algorithms.
Article
This study presents an integrated algorithm designed for real-time task assignment, collision-free path planning, and control. We introduce two algorithms: MPPI-GA for single-agent systems and generalized MPPI-GA for multi-agent systems. Our approach leverages model predictive path integral control (MPPI) and genetic algorithms (GA) as the foundational framework. Both MPPI and GA, being sampling-based algorithms, utilize a paired concept of samples and individuals, respectively. In contrast to the existing methods that adopt a hierarchical structure for task management, path planning, and control in a sequential manner, our proposed algorithm executes such operations concurrently, ensuring real-time performance. Moreover, the generalized MPPI-GA addresses the traveling salesperson problem with self-interest task selection and decentralized conflict resolution, even within multi-agent systems. Consequently, it can autonomously schedule tasks on each computational hardware unit without requiring computation of other agents’ behaviors. We validate the efficacy of MPPI-GA through numerical simulations implemented in Python. Furthermore, we conduct experiments employing two four-wheel-drive mobile robots equipped with the proposed algorithm, thus substantiating its real-time performance.
Article
Full-text available
In this paper, we present a new optimization algorithm based on the properties of quantum particles represented by their wavefunctions. This algorithm is called the “quantum wavefunction optimization algorithm (QWOA)”. We demonstrate the application of the QWOA to determine the optimal minimum distance for the traveling salesman problem (TSP). Specifically, we address the problem of traversing between cities in different countries using Google Maps, aiming to promote a real-time application of the proposed algorithm. To this end, we select cities from six different countries: Japan, India, Canada, China, Russia, and the United States of America. We use the QWOA to simulate and uncover the optimal shortest paths between these selected cities. The results of the QWOA are compared with those obtained using several well-known optimization algorithms, including the genetic algorithm (GA), simulated annealing (SA), particle swarm optimization (PSO), artificial bee colony (ABC), firefly algorithm (FA), and grey wolf optimizer (GWO). The experimental results, supported by statistical analysis, demonstrate the efficiency of the QWOA relative to these established optimization algorithms.
Article
In this study, the features of cyclic crossover process and K-opt are incorporated in the bat algorithm (BA) to solve the Travelling Salesman Problems (TSP) in different environments. Swap operation and swap sequence are applied for the modification of the different operations of the BA to solve the TSPs. The cyclic crossover operation is applied in a regular interval of iterations on the best found solution and each solution of the final population of BA for the enhancement of the exploration as well as exploitation of the search process. K-Opt operation is applied on the population in each iteration of the BA with some probability for the exploitation. The algorithm is tested with a set of benchmark test instances of the TSPLIB. The algorithm produces exact results for a set of significantly large size problems. For the TSPs in fuzzy environment, a fuzzy simulation approach is proposed to deal with the fuzzy data having linear as well as non-linear membership functions. Also, a rough simulation process is proposed to deal with the TSPs in the rough environment where rough estimation can be done following any type of rough measure. The performance of the algorithm is compared with the state-of-the-art algorithms for the TSPs with crisp cost matrices using different statistical tools.
Article
Planning an itinerary for travelling can be tedious, time-consuming, and challenging. This is especially true for tourists who have limited time budgets and are unfamiliar with a wide range of Points-of-Interest (POIs) in a city. To address this challenge, this paper proposes an Adaptive Genetic Algorithm (AGA) for personalized itinerary planning. This approach considers travelers' preferences, such as mandatory POIs, total number of POIs, POI popularity, POI cost, and POI rating. It views the itinerary planning problem as a multi-objective optimization problem and proposes an Adaptive Genetic Algorithm (AGA) to solve this problem. The results show that the AGAM algorithm is a promising approach for personalized itinerary planning. It is able to find itineraries that meet the traveler's preferences that are efficient in terms of time, cost, and overall rating.
Article
The traveling salesman problem is a classic combinatorial optimization challenge with profound implications for various industries. While significant progress has been made in solving traveling salesman problem instances, real-world applications often involve uncertainties that challenge the accuracy and robustness of traditional approaches. Pythagorean fuzzy uncertain variables combine the strengths of fuzzy logic with the principles of uncertainty theory, allowing for a more balanced and comprehensive representation of uncertainty. This paper defines the theoretical foundations of normal, lognormal, and empirical Pythagorean fuzzy uncertainty distributions, including their mathematical formulation and operational laws. Moreover, it presents a novel hybrid optimization approach that leverages the strengths of simulated annealing and genetic algorithms while incorporating Pythagorean fuzzy uncertain variables to address the traveling salesman problem under uncertain conditions. The synergy of these two techniques enables effective exploration and exploitation of solution candidates, leading to improved traveling salesman problem solutions. The detailed steps of the algorithm are demonstrated through a numerical example. A case study of a decision support system for optimizing a beverage logistics vehicle routing problem is discussed to find out the best possible route in the distribution zones. The incorporation of Pythagorean fuzzy uncertain variables enhances the algorithm’s robustness in uncertain environments, resulting in higher-quality solutions and improved adaptability to different levels of uncertainty
Chapter
The Traveling Salesman Problem (TSP) is one of the most studied problems within the transportation, manufacturing, and logistics industries. An optimal solution to the TSP consists of finding the sequence of paths of minimum cost, which starts at a specific location node, visits all other nodes only once, and finishes at the starting node. Finding the optimal or near-optimal solution to the TSP is a difficult task due to its NP-hard computational complexity. Although exact and approximate algorithms have been developed to solve the TSP to optimality, their performance is highly dependent on the dimensionality of the considered TSP instance. In this regard, Artificial Neural Networks (ANN) are proposed as a key component to reduce dimensionality and improve the performance of approximate algorithms for large TSP instances. The present work describes the development of a hybrid solving method that adapts ANN to reduce TSP dimensionality through clustering, uses the Clarke and Wright algorithm to solve the reduced TSP, expands the reduced TSP solution by considering all permutations of the ANN-clustered locations, and performs local search operators (flip/exchange) to refine the expanded TSP solution. The results with different TSP instances corroborate the finding that dimensionality reduction can improve the speed of the solving method and quality of the achieved solutions, particularly for large TSP instances (error gaps less than 10%).
Conference Paper
Given a set of cities and the distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. Note the difference between Hamiltonian Cycle and TSP. The Hamiltonian cycle problem is to find if there exists a tour that visits every city exactly once. Here we know that Hamiltonian Tour exists (because the graph is complete) and in fact, many such tours exist, the problem is to find a minimum weight Hamiltonian Cycle.
Chapter
With the rapid development of express delivery industry, more and more focus has been shifted to express delivery mechanism design. For door-to-door pickup scenes of couriers, we present the cooperative package pickup system to reduce the users’ express fee in form of sharing the express fee of users assigned the same express station. We formulate the Cooperative Package Pickup (CPP) problem to minimize all users’ comprehensive cost, which is the sum of express fee and pickup cost of courier. The Cooperative Package Pickup Mechanism (CPPM) is proposed to solve CPP problem. First, CPPM transforms CPP into Transformed Cooperative Package Pickup (TCPP) problem according to the properties of the triangle inequality. Then, an approximate algorithm, TCPPA, is proposed to solve TCPP problem based on submodular function minimization. Furthermore, according to the package assignments of TCPPA, we present a polynomial time algorithm, PPOA, to optimize pickup path of couriers. Additionally, in order to incentivize users to deliver their packages by this system, we propose a cooperative cost allocation scheme to incentive the user to deliver the package. Through extensive simulations, we demonstrate that CPPM reduces the comprehensive cost by 8.84% on average compared to the door-to-door behavior in the non-cooperative mode.KeywordsExpress deliverydoor-to-door pickupcooperative package pickupcooperation cost
Article
This study proposes a new loop-wise route representation in which a vehicle route is represented in terms of continuous loop variables and a base route. The proposed loop-wise representation can guarantee a local route connectivity at each node with no constraint function, thereby simplifying the optimization formulation. Based on the proposed loop-wise route representation, optimization formulation is expressed for the traveling salesman problems (TSPs) in the planar graph. The performance of the proposed method is compared with that of genetic algorithm (GA) methods in four different cases of the TSPs. Numerical results show that the proposed method can determine the optimal route for the TSPs with a much higher computational efficiency, compared with the conventional GA methods.
Article
Full-text available
In this paper, the Traveling Salesman Problem (TSP) is solved through the use of some approximation techniques where the results of the previous work showed some defects in solving the problem to obtain an optimal or close to optimal solution,so the use of hybrid algorithms to solve some results from the use of intuitive and exact algorithms. A hybrid algorithm has been proposed that combines the characteristics of the firefly algorithm (FA) and Particle Swarm Optimization (PSO) to obtain an algorithm that works effectively in overcoming some of the problems resulting from the use of each algorithm separately. Then using an improvement factor to improve each solution within the resulting community and to obtain solutions with a high diversity. The efficiency of the proposed method was measured by solving some standard problems TSP, and the results showed a high convergence of the algorithm towards the known optimal solution for each problem by solving 13 standard problems.
Chapter
Der Bereich des Transports gehört zu den wichtigsten Bereichen der Logistik. In diesem Kapitel werden einige Hauptvarianten von Transportproblemen näher betrachtet sowie Methoden, die in der Lage sind, sie zumindest mit ausreichender Qualität zu lösen. Wir diskutieren Zuordnungsprobleme, die eine grobe Transportplanung ermöglichen, Probleme der Ermittlung kürzester Wege, sowie die wichtigsten Arten von komplexeren Tourenplanungsproblemen: das Travelling-Salesman-Problem und das Vehicle-Routing-Problem. Insbesondere für das Vehicle-Routing-Problem werden verschiedene praktisch relevante Varianten betrachtet. Das Kapitel endet mit einer kurzen Diskussion von Netzwerkflussproblemen.
Chapter
Full-text available
In this chapter, we recollect some crucial and exciting fixed point theorems in the context of partial metric space. In addition, we underline the importance of the partial metric space in fixed point theory. Matthews [210] not only introduced the partial metric spaces but also obtained the first fixed point theorem in this new setting. More precisely, Matthews [210] showed that the famous Banach fixed point theory is valid in the framework of complete partial metric space. After this pioneering result of Matthews [210] in fixed point theory, a considerable number of researchers have investigated the partial metric spaces, and remarkable number of fixed point results in the context of partial metrics have appeared in the literature (see e.g. [1, 2, 7, 27–30, 34, 38–40, 42, 46, 49–53, 86, 92, 94, 102, 120, 125, 130, 136, 137, 146, 154, 155, 157, 164–166, 173, 175, 176, 178, 181, 188, 189, 202, 204, 210, 211, 222, 225, 230, 243, 245–250, 255, 258, 261, 262, 265, 266, 272–278] and the reference therein).
Article
Full-text available
The classical Particle Swarm Optimization is a powerful method to find the minimum of a numerical function, on a continuous definition domain. As some binary versions have already successfully been used, it seems quite natural to try to define a framework for a discrete PSO. In order to better understand both the power and the limits of this approach, we examine in detail how it can be used to solve the well known Traveling Salesman Problem, which is in principle very “bad” for this kind of optimization heuristic. Results show Discrete PSO is certainly not as powerful as some specific algorithms, but, on the other hand, it can easily be modified for any discrete/combinatorial problem for which we have no good specialized algorithm.
Article
Full-text available
In the Generalized Travelling Salesman Problem (GTSP), the aim is to determine a least cost Hamiltonian circuit or cycle through several clusters of vertices. It is shown that a wide variety of combinatorial optimization problems can be modelled as GTSPs. These problems include location-routeing problems, material flow system design, post-box collection, stochastic vehicle routeing and arc routeing.
Article
Full-text available
We consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster. This NP-hard problem is known in the literature as the symmetric generalized traveling salesman problem (GTSP), and finds practical applications in routing, scheduling and location-routing. In a companion paper [Networks 26, No. 2, 113-123 (1995; Zbl 0856.90116)], we modeled GTSP as an integer linear program, and studied the facial structure of two polytopes associated with the problem. Here we propose exact and heuristic separation procedures for some classes of facet-defining inequalities, which are used within a branch-and-cut algorithm for the exact solution of GTSP. Heuristic procedures are also described. Extensive computational results for instances taken from the literature and involving up to 442 nodes are reported.
Article
Full-text available
A parallel version of the particle swarm optimization (PPSO) algorithm is presented together with three communication strategies which can be used accord-ing to the independence of the data. The first strat-egy is designed for the parameters of solutions that are independent or are only loosely correlated such as the Rosenbrock and Rastrigrin functions. The second communication strategy can be applied to those pa-rameters that are more strongly correlated such as the Griewank function. In cases where the properties of the parameters are unknown, a third hybrid commu-nication strategy can be used. Experimental results demonstrate the usefulness of the proposed PPSO al-gorithm.
Article
Full-text available
Traveling salesman problems (TSP) and generalized traveling salesman problems (GTSP) are two kinds of well known and challenging combinatorial optimization problems with much diversified application fields. Between the two application problems the GTSP is more complex than TSP. Many researchers have studied TSP extensively, but relatively fewer studies pay attention to GTSP, and also its solution using genetic algorithm (GA). In this paper, the structure of conventional chromosome is generalized to be a chromosome termed as a generalized chromosome (GC). A genetic scheme named as generalized-chromosome-based genetic algorithm (GCGA) is also presented. The proposed GCGA enables GTSP and TSP to be solved under a uniform algorithm mode. Forty one benchmark test problems have been solved with the known optimal solutions using the proposed algorithm to verify its validity. The test results show that GCGA can directly solve GTSP without the need of intermediate transformation to TSP.
Article
Full-text available
The particle swarm is an algorithm for finding optimal regions of complex search spaces through the interaction of individuals in a population of particles. This paper analyzes a particle's trajectory as it moves in discrete time (the algebraic view), then progresses to the view of it in continuous time (the analytical view). A five-dimensional depiction is developed, which describes the system completely. These analyses lead to a generalized model of the algorithm, containing a set of coefficients to control the system's convergence tendencies. Some results of the particle swarm optimizer, implementing modifications derived from the analysis, suggest methods for altering the original algorithm in ways that eliminate problems and increase the ability of the particle swarm to find optima of some well-studied test functions
Article
The particle swarm optimization algorithm is analyzed using standard results from the dynamic system theory. Graphical parameter selection guidelines are derived. The exploration-exploitation tradeoff is discussed and illustrated. Examples of performance on benchmark functions superior to previously published results are given.
Article
Particle swarm optimization (PSO) is an alternative population-based evolutionary computation technique. It has been shown to be capable of optimizing hard mathematical problems in continuous or binary space. We present here a parallel version of the particle swarm optimization (PPSO) algorithm together with three communication strategies which can be used according to the independence of the data. The first strategy is designed for solution parameters that are independent or are only loosely correlated, such as the Rosenbrock and Rastrigrin functions. The second communication strategy can be applied to parameters that are more strongly correlated such as the Griewank function. In cases where the properties of the parameters are unknown, a third hybrid communication strategy can be used. Experimental results demonstrate the usefulness of the proposed PPSO algorithm.
Article
A hybrid approach based on ant colony algorithm for the traveling salesman problem is proposed, which is an improved algorithm characterized by adding a local search mechanism, a cross-removing strategy and candidate lists. Experimental results show that it is competitive in terms of solution quality and computation time.
Article
A survey and synthesis of research on the traveling salesman problem is given. The problem is defined and several theorems are presented. This is followed by a general classification of the solution techniques and a description of some of the proven methods. A summary of computational results is given.
Conference Paper
The time-dependent orienteering problem is dual to the time-dependent traveling salesman problem. It consists in visiting a maximum number of sites within a given deadline. The traveling time between two sites is in general dependent on the starting time. We provide a (2 + ε)-approximation algorithm for the time-dependent orienteering problem which runs in polynomial time if the ratio between the maximum and minimum traveling time between any two sites is constant. No prior upper approximation bounds were known for this time-dependent problem.
Article
The time-dependent orienteering problem is dual to the time-dependent traveling salesman problem. It consists of visiting a maximum number of sites within a given deadline. The traveling time between two sites is in general dependent on the starting time.For any ε>0, we provide a (2+ε)-approximation algorithm for the time-dependent orienteering problem which runs in polynomial time if the ratio between the maximum and minimum traveling time between any two sites is constant. No prior upper approximation bounds were known for this time-dependent problem.
Article
The particle swarm optimization algorithm is analyzed using standard results from the dynamic system theory. Graphical parameter selection guidelines are derived. The exploration–exploitation tradeoff is discussed and illustrated. Examples of performance on benchmark functions superior to previously published results are given.
Conference Paper
Particle Swarm Optimisation (PSO) is an optimisation algorithm that shows promise. However its performance on complex problems with multiple minima falls short of that of the Ant Colony Optimisation (ACO) algorithm when both algorithms are applied to travelling salesperson type problems (TSP). Unlike ACO, PSO can be easily applied to a wider range of problems than TSP. This paper shows that by adding a memory capacity to each particle in a PSO algorithm performance can be significantly improved to a competitive level to ACO on the smaller TSP problems.
Article
This paper contains the description of a traveling salesman problem library (TSPLIB) which is meant to provide researchers with a broad set of test problems from various sources and with various properties. For every problem a short description is given along with known lower and upper bounds. Several references to computational tests on some of the problems are given. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.
Article
In the generalized traveling-salesman problem (GTSP), we are given a set of cities that are grouped into possibly intersecting clusters. The objective is to find a closed path of minimum cost that visits at least one city in each cluster. Given an instance G of the GTSP, we first transform G into another instance G′ of the GTSP in which all the clusters are nonintersecting, and then transform G′ into an instance G″ of the standard traveling-salesman problem (TSP). We show that any feasible solution of the TSP instance G″ can be transformed into a feasible solution of the GTSP instance G of no greater cost, and that any optimal solution of the TSP instance G″ can be transformed into an optimal solution of the GTSP instance G.
Conference Paper
This paper investigates the philosophical and performance differences of particle swarm and evolutionary optimization. The method of processing employed in each technique are first reviewed followed by a summary of their philosophical differences. Comparison experiments involving four non-linear functions well studied in the evolutionary optimization literature are used to highlight some performance differences between the techniques.
Article
This paper defines and explores a somewhat different type of genetic algorithm (GA) - a messy genetic algorithm (mGA). Messy GAs process variable-length strings that may be either under- or over-specified with respect to the problem being solved. As nature has formed its genotypes by progressing from simple to more complex life forms, messy GAs solve problems by combining relatively short, well-tested building blocks to form longer, more complex strings that increasingly cover all features of a problem. This approach stands in contrast to the usual fixed-length, fixed-coding genetic algorithm, where the existence of the requisite tight linkage is taken for granted or ignored altogether. To compare the two approaches, a 30-bit, order-three-deceptive problem is searched using a simple GA and a messy GA. Using a random but fixed ordering of the bits, the simple GA makes errors at roughly three-quarters of its positions; under a worst-case ordering, the simple GA errs at all positions. In contrast to the simple GA results, the messy GA repeatedly solves the same problem to optimality. Prior to this time, no GA had ever solved a provably difficult problem to optimality without prior knowledge of good string arrangements. The mGA presented herein repeadedly achieves globally optimal results without such knowledge, and it does so at the very first generation in which strings are long enough to cover the problem. The solution of a difficult nonlinear problem to optimality suggests that messy GAs can solve more difficult problems than has been possible to date with other genetic algorithms. The ramifications of these techniques in search and machine learning are explored, including the possibility of messy floating-point codes, messy permutations, and messy classifiers.
Article
LNCS n°3106, http://dx.doi.org/10.1007/978-3-540-27798-9_32 The Quota Traveling Salesman Problem is a generalization of the well known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of the sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performance in the usual framework of the competitive analysis. In particular we present a 2-competitive algorithm that matches the lower bound for general metric spaces. In the case of the half-line metric space, we show that it is helpful not to move at full speed, and this approach is also used to derive the best on-line polynomial time algorithm known so far for the more general On-Line TSP problem (in the homing version). oui
Article
The optimal cooling schedule for simulated annealing is formulated to derive a differential equation for the time-dependent temperature T(t). Based on this equation, the long-term behavior of T(t), entropy production, and the Kullback-Leibler entropy are studied. For some simple examples, such as a many-level system and the small scale traveling salesman problem, the explicit time dependence of the temperature is obtained. Some comments are given on simulated annealing based on Tsallis statistics.
Conference Paper
This paper proposes a new application of particle swarm optimization for traveling salesman problem. We have developed some special methods for solving TSP using PSO. We have also proposed the concept of swap operator and swap sequence, and redefined some operators on the basis of them, in this way the paper has designed a special PSO. The experiments show that it can achieve good results.
Conference Paper
The particle swarm algorithm adjusts the trajectories of a population of “particles” through a problem space on the basis of information about each particle's previous best performance and the best previous performance of its neighbors. Previous versions of the particle swarm have operated in continuous space, where trajectories are defined as changes in position on some number of dimensions. The paper reports a reworking of the algorithm to operate on discrete binary variables. In the binary version, trajectories are changes in the probability that a coordinate will take on a zero or one value. Examples, applications, and issues are discussed
Conference Paper
eberhart @ engr.iupui.edu A concept for the optimization of nonlinear functions using particle swarm methodology is introduced. The evolution of several paradigms is outlined, and an implementation of one of the paradigms is discussed. Benchmark testing of the paradigm is described, and applications, including nonlinear function optimization and neural network training, are proposed. The relationships between particle swarm optimization and both artificial life and genetic algorithms are described, 1
Some applications ofthegeneralizedtravelingsalesmanproblem
  • G Laporte
  • A Asef-Vaziri
  • C Sriskandarajah
G. Laporte, A. Asef-vaziri, C. Sriskandarajah, Some applications ofthegeneralizedtravelingsalesmanproblem,JournaloftheOp-erational Research Society 47 (1996) 1461–1467.
Evolutionary optimization versus particle swarm optimization: philosophy and performance differences
  • Angeline
Some applications of the generalized traveling salesman problem
  • Laporte