Salesman Problem Research Articles

Logic-based benders decomposition algorithm for robust parallel drone scheduling problem considering uncertain travel times for drones

The integration of trucks and drones in last-mile delivery has introduced new capabilities to the transportation industry. These two vehicles simultaneously offer unique features, which have improved performance and efficiency in the process of delivering products. This paper investigates a robust parallel drone scheduling traveling salesman problem with supporting drone, where drone travel times are uncertain and products gradually arrive at a depot over time. In this problem, a truck, a supporting drone, and service drones are located in the depot to deliver products with the goal of minimizing the total completion time. A mathematical model is proposed which is improved using the earliest release dates rule, followed by the development of an exact logic-based benders decomposition algorithm to solve the problem. In this algorithm, customers are initially assigned to the service drones or the truck in a master problem, and subsequent auxiliary problems are addressed utilizing the earliest release dates rule and a dynamic programming algorithm. Finally, various cuts are enhanced through strengthening techniques and sequentially added into the master problem. Numerical experiments demonstrate the efficiency of the improved mathematical model and the proposed algorithm. Furthermore, sensitivity analysis has provided several managerial recommendations for enhancing the delivery system performance.

Application of Hybrid Algorithm Based on Ant Colony Optimization and Sparrow Search in UAV Path Planning

The Traveling Salesman Problem (TSP) is a classic problem in combinatorial optimization, aiming to find the shortest path that traverses all cities and eventually returns to the starting point. The ant colony optimization algorithm has achieved significant results, but when the number of cities increases, the ant colony algorithm is prone to fall into local optimal solutions, making it difficult to obtain the global optimal path. To overcome this limitation, this paper proposes an innovative hybrid ant colony algorithm. Our main motivation is to introduce other optimization strategies to improve the global search ability and convergence speed of the ant colony algorithm in solving TSP problems. We first incorporate the iterative solution of the sparrow search algorithm (SSA) into the ant colony algorithm to provide a better initial pheromone distribution. Second, we improve the pheromone update method to enhance the algorithm’s diversity during the search process and reduce the risk of falling into local optima. Finally, we define a dynamic pheromone evaporation factor to adjust the pheromone evaporation rate according to real-time changes in the search process. Through simulation tests on large-scale TSP problems and practical applications, we find that the hybrid ant colony algorithm outperforms the ant colony algorithm in both accuracy and running time. In Eg.2, the average accuracy of ISSA-ACO is improved by 12%, and the average running time is reduced by 45.6%. This study not only provides a new and effective method for solving large-scale TSP problems but also provides valuable references and insights for the application of ant colony algorithms in solving other complex optimization problems. At the same time, our research further verifies the effectiveness of improving heuristic algorithms by fusing different optimization strategies, providing new ideas and directions for future algorithm design and optimization.

Optimizing Genetic Algorithm by Implementation of An Enhanced Selection Operator

The Traveling Salesman Problem (TSP) represents an extensively researched challenge in combinatorial optimization. Genetic Algorithms (GAs), recognized for their nature-inspired approach, stand as potent heuristics for resolving combinatorial optimization problems. Nevertheless, GA exhibits inherent deficiencies, notably premature convergence, which diminishes population diversity and consequential inefficiencies in computational processes. Such drawbacks may result in protracted operations and potential misallocation of computational resources, particularly when confronting intricate NP-hard optimization problems. To address these challenges, the current study underscores the pivotal role of the selection operator in ameliorating GA efficiency. The proposed methodology introduces a novel parameter operator within the Stochastic Universal Selection (SUS) framework, aimed at constricting the search space and optimizing genetic operators for parent selection. This innovative approach concentrates on selecting individuals based on their fitness scores, thereby mitigating challenges associated with population sorting and individual ranking while concurrently alleviating computational complexity. Experimental results robustly validate the efficacy of the proposed approach in enhancing both solution quality and computational efficiency, thereby positioning it as a noteworthy contribution to the domain of combinatorial optimization.

Genetic Algorithm Incorporating Group Theory for Solving the General Travelling Salesman Problem

Genetic Algorithm Incorporating Group Theory for Solving the General Travelling Salesman Problem

Set Covering Routing Problems: A review and classification scheme

Cyclic routing problems are a well-researched category of combinatorial problems in Operations Research (OR). They involve finding an optimal cycle (route), starting from an initial point (e.g., depot), visiting some nodes (e.g., customer or demand points), and returning to the starting point while adhering to constraints and optimizing a specific objective. These problems are categorized into two main types: (i) Hamiltonian routing problems: All customer nodes must be visited (must be on a cycle), such as the Traveling Salesman Problem (TSP), seeking the shortest possible cycle visiting each node once. (ii) Non-Hamiltonian routing problems: At least one customer node is not on a route. This work mainly focuses on classifying one of the main sub-problems of non-Hamiltonian routing problems, known as Set Covering Routing Problems (SCRPs), in which customers may be directly visited or covered by another node on the cycle or route. The SCRPs also incorporate Steiner nodes (transition points) to cover the customers who are not directly visited, integrating covering and routing decisions. SCRPs have wide-ranging applications in logistics, transportation planning, healthcare, telecommunication network design, and disaster response, making them vital for theoretical exploration and practical solutions. This work reviews the studies on SCRPs, presenting a classification scheme and a generic mathematical model for these problems, enabling a unified analysis and comparison. Finally, the study gaps, future research directions, and the limitations and implications for practical applications are discussed.

Hybrid Heuristic for Solving the Euclidean Travelling Salesman Problem

Hybrid Heuristic for Solving the Euclidean Travelling Salesman Problem

Framework for Small Traveling Salesman Problems

We study small traveling salesman problems (TSPs) because current quantum computers can find optional solutions for TSPs with up to 14 cities. Also, we study small TSPs because TSPs have been recommended to be benchmarks to measure quantum optimization on all types of quantum hardware. This means comparisons of quantum data about small TSPs. We extent previous numerical results that were reported in “Small Traveling Salesman Problems” for 6, 8 and 10 cities. The new results in this paper are for 10 – 14 cities in symmetric TSPs. The data for this new range of cities is consistent with the previous data and can be the basis for estimates of results from quantum computers that are upgraded to handle more than 14 cities. The work and analysis suggest two conjectures that we discuss. The paper also contains an annotated survey of recent publications about TSPs.

Deep reinforcement learning combined with transformer to solve the traveling salesman problem

Deep reinforcement learning combined with transformer to solve the traveling salesman problem

Drone-Assisted Last-Mile Delivery Under Windy Conditions: Zero Pollution Solutions

As cities expand and the global push for zero pollution intensifies, sustainable last-mile delivery (LMD) systems are essential to minimizing environmental and health impacts. This study addresses the need for more sustainable LMD by examining the integration of wind conditions into drone-assisted deliveries, focusing on their effects on air and noise pollution in urban areas. We extend the flying sidekick traveling salesman problem (FSTSP) by incorporating meteorological factors, specifically wind, to assess drone delivery efficiency in varying conditions. Our results show that while drones significantly reduce greenhouse gas emissions compared to traditional delivery vehicles, their contribution to noise pollution remains a concern. This research highlights the environmental advantages of using drones, particularly in reducing CO2 emissions, while also emphasizing the need for further investigation into mitigating their noise impact. By evaluating the trade-offs between air and noise pollution, this study provides insights into developing more sustainable, health-conscious delivery models that contribute to smart city initiatives. The findings inform policy, urban planning, and logistics strategies aimed at achieving zero pollution goals and improving urban livability.

Nested benders decomposition for a deterministic biomass feedstock logistics problem

AbstractIn this paper, we address a biomass feedstock logistics problem to supply biomass from production fields to satellite storage locations (SSLs) and from there to bioenergy plants (BePs) and then to a biorefinery. It entails a new problem feature of routing load-out equipment sets among the SSLs to perform loading/unloading of biomass and/or its pre-processing operations. The ownership of the loading equipment is a very capital-intensive link of the ethanol production supply chain, which when loaded onto trucks and routed along the logistics chain significantly brings down the ethanol production costs. This will make ethanol a cost-competitive alternative to fossil fuels, lead to sustainable use of fossil fuels and add to the overall relevance of the bioenergy sector. In this regard, the objective of our problem is to minimize the total cost incurred due to the ownership of equipment sets, fixed setups, and land rental cost, as well as the cost of transporting biomass from the fields to the BePs and biocrude oil from the BePs to the refinery. A mixed-integer mathematical model of the problem is presented, and a nested Benders decomposition-based solution approach is developed which involves decomposing this large problem into three stages. Stage 1 deals with the selection of fields, BePs, and SSLs, and assignment of fields to the SSLs. The remaining model consists of multiple Capacitated Vehicle Routing Problems (CVRPs) that are separable over individual BePs. For each BeP, the CVRP is further decomposed into Stage 2 and Stage 3 sub-problems where the Stage 2 problem is an allocation problem that assigns SSLs to tours associated to each BeP, and the Stage 3 problem is a variant of Traveling Salesman Problem (TSP) that determines the sequence in which equipment is routed over the predesignated set of SSLs for each tour. These sub-problems are integer programs rather than linear programs. First novelty of our proposed approach is to effectively handle the integrality of variables arising due to the consideration of the routing of load-out equipment. Second is solution methodology and in the use of proposed multi-cut version of optimality cuts that capture the solution value at an integer solution for the sub-problems. These cuts aid in faster convergence and are shown to be stronger than those proposed in the literature. The applicability of the proposed methodology is demonstrated by applying it to a real-life problem that utilizes available GIS data for the catchment area of regions around Gretna and Bedford in Virginia. We then solved a set of varying problem size instances using the state-of-the-art CPLEX® Branch-and-Bound and Benders Strategy methods. The CPLEX® algorithms struggled to solve instances even 10 times smaller than the real-life problem size instances; with MIP optimality gaps ranging from 5.85% to 82.79% in the allowed time limit of 10,000 s. On the other hand, our proposed nested Benders decomposition algorithm was able to achieve faster convergence and provided optimal solutions for all the considered problem instances with an average CPU run-time of around 3,700 s. This validates the efficacy and superiority of our solution approach. Lastly, we summarize our work and point out some interesting potential future research opportunities.

Enhancing offshore parcel delivery efficiency through vessel-unmanned aerial vehicle collaborative routing

Offshore oil and gas production is vital for global energy supply, but it faces logistical challenges due to the high costs and inefficiencies of traditional supply methods. This paper introduces the vessel-unmanned aerial vehicle (UAV) routing problem with multiple visits in a single flight (VURP-M), a novel logistical model that addresses aimed at enhancing the replenishment of offshore platforms with small, essential items. The VURP-M allows the UAV to perform multiple visits during a single flight, optimising the delivery process. To tackle the VURP-M, we propose two mixed-integer second-order cone programs that capture the problem's complexities. Given its NP-hard nature, we employ an adaptive large neighbourhood search (ALNS) method, featuring a segmented initialisation process and problem-specific operators guided by a rule-based mechanism to improve solution efficiency. The ALNS formulates an initial solution by solving a travelling salesman problem to create a giant tour, which is then used to group sequential targets into multiple UAV flights. The subsequent optimal resolution of a SOCP determines the take-off and landing points for each flight. Subsequently, the ALNS refines the initial solution through destroy and repair operators, enhancing the search for superior sequences and allocation schemes. The effectiveness of our approach is demonstrated through a real-world case study and numerical experiments on random instances featuring up to 100 platforms. The results offer implications of the collaborative vessel-UAV model for the offshore logistics industry.

Hybrid genetic algorithm with Wiener process for multi-scale colored balanced traveling salesman problem

Colored traveling salesman problem (CTSP) can be applied to Multi-machine Engineering Systems (MES) in industry, colored balanced traveling salesman problem (CBTSP) is a variant of CTSP, which can be used to model the optimization problems with partially overlapped workspace such as the planning optimization (For example, process planning, assembly planning, productions scheduling). The traditional algorithms have been used to solve CBTSP, however, they are limited both in solution quality and solving speed, and the scale of CBTSP is also restricted. Moreover, the traditional algorithms still have the problems such as lacking theoretical support of mathematical physics. In order to improve these, this paper proposes a novel hybrid genetic algorithm (NHGA) based on Wiener process (ITÖ process) and generating neighborhood solution (GNS) to solve multi-scale CBTSP problem. NHGA firstly uses dual-chromosome coding to construct the solutions of CBTSP, then they are updated by the crossover operator, mutation operator and GNS. The crossover length of the crossover operator and the city number of the mutation operator are controlled by activity intensity based on ITÖ process, while the city keeping probability of GNS can be learned or obtained by Wiener process. The experiments show that NHGA can demonstrate an improvement over the state-of-art algorithms for multi-scale CBTSP in term of solution quality.

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems

Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the approximation set up to a multiplicative factor in each component, a convex approximation set only requires this multiplicative approximation to be achieved by some convex combination of finitely many images of solutions in the set. This makes convex approximation sets efficiently computable for a wide range of multiobjective problems: even for many problems for which (classic) approximations sets are hard to compute. In this article, we propose a polynomial-time algorithm to compute convex approximation sets that builds on an exact or approximate algorithm for the weighted sum scalarization and is therefore applicable to a large variety of multiobjective optimization problems. The provided convex approximation quality is arbitrarily close to the approximation quality of the underlying algorithm for the weighted sum scalarization. In essence, our algorithm can be interpreted as an approximate version of the dual variant of Benson’s outer approximation algorithm. Thus, in contrast to existing convex approximation algorithms from the literature, information on solutions obtained during the approximation process is utilized to significantly reduce both the practical running time and the cardinality of the returned solution sets while still guaranteeing the same worst-case approximation quality. We underpin these advantages by the first comparison of all existing convex approximation algorithms on several instances of the triobjective knapsack problem and the triobjective symmetric metric traveling salesman problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was supported by the German Research Foundation [Project 398572517]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0220 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0220 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Learning Feature Embedding Refiner for Solving Vehicle Routing Problems.

While the encoder-decoder structure is widely used in the recent neural construction methods for learning to solve vehicle routing problems (VRPs), they are less effective in searching solutions due to deterministic feature embeddings and deterministic probability distributions. In this article, we propose the feature embedding refiner (FER) with a novel and generic encoder-refiner-decoder structure to boost the existing encoder-decoder structured deep models. It is model-agnostic that the encoder and the decoder can be from any pretrained neural construction method. Regarding the introduced refiner network, we design its architecture by combining the standard gated recurrent units (GRU) cell with two new layers, i.e., an accumulated graph attention (AGA) layer and a gated nonlinear (GNL) layer. The former extracts dynamic graph topological information of historical solutions stored in a diversified solution pool to generate aggregated pool embeddings that are further improved by the GRU, and the latter adaptively refines the feature embeddings from the encoder with the guidance of the improved pool embeddings. To this end, our FER allows current neural construction methods to not only iteratively refine the feature embeddings for boarder search range but also dynamically update the probability distributions for more diverse search. We apply FER to two prevailing neural construction methods including attention model (AM) and policy optimization with multiple optima (POMO) to solve the traveling salesman problem (TSP) and the capacitated VRP (CVRP). Experimental results show that our method achieves lower gaps and better generalization than the original ones and also exhibits competitive performance to the state-of-the-art neural improvement methods.

Hybrid ITÖ Algorithm for Large-Scale Colored Traveling Salesman Problem

Hybrid ITÖ Algorithm for Large-Scale Colored Traveling Salesman Problem

A Scalable and Adaptable Supervised Learning Approach for Solving the Traveling Salesman Problems

A Scalable and Adaptable Supervised Learning Approach for Solving the Traveling Salesman Problems

Revolutionizing Optimization: The Game-Changing Power of Quantum Algorithms and Next-Generation Computational Technologies

Quantum algorithms offer promising advancements in solving complex optimization problems that are critical in various fields, including logistics, finance, and machine learning. Classical optimization techniques, while effective, often struggle with large-scale and high-dimensional data sets, leading to inefficiencies and long processing times.Quantum algorithms, leveraging the principles of superposition, entanglement, and quantum parallelism, provide a potential solution by exponentially speeding up certain computational tasks. This paper explores key quantum algorithms for optimization, including Grover's algorithm, the Quantum Approximate Optimization Algorithm (QAOA), and the Quantum Annealing method. We analyze their theoretical foundations, implementation challenges, and practical applications. Special attention is given to the performance comparisons between quantum and classical approaches in solving combinatorial optimization problems, such as the traveling salesman problem and portfolio optimization. We also discuss the limitations of current quantum hardware and the prospects for future improvements that could make quantum optimization algorithms more practical for real-world use.Our findings suggest that, while quantum algorithms are still in their early stages, they hold immense potential for transforming optimization in a variety of domains, offering new pathways toward more efficient problem-solving in both theoretical and applied contexts. we explore how hybrid quantum-classical approaches can enhance optimization results by combining the strengths of both paradigms. Future research directions are highlighted, focusing on improving quantum error correction, scaling quantum systems, and developing more robust quantum algorithms to tackle increasingly complex optimization problems.

Improved Floyd-Warshall Algorithm for Solving Travelling Salesman Problem

The shortest path problem is a fundamental challenge in graph theory, focused on identifying the most efficient routes between nodes in a network. It stands as one of the extensively researched combinatorial optimization problems. In this paper, we explained the fundamental concepts of shortest path algorithms, with a particular emphasis on Floyd Warshall algorithm. We apply the algorithm and showed how this algorithm can be effectively employed to determine the shortest path in various scenarios. Furthermore, this paper addresses real-life applications, particularly focusing on the traveling salesman problem. Individuals often have the option to travel along different routes to reach their destination. However, selecting suboptimal routes can result in time-consuming journeys. To tackle this issue, we apply the Floyd Warshall algorithm to solve the traveling salesman problem (TSP). Additionally, we demonstrate the enhancement of the Floyd Warshall algorithm's efficiency for TSP using the rectangular algorithm. This paper concludes with a comparative analysis between the original and improved Floyd Warshall algorithms, emphasizing the computational reduction achieved through the proposed enhancements. Jagannath University Journal of Science, Volume 11, Number 1, June 2024, pp. 45−52

Solving a real-world package delivery routing problem using quantum annealers.

Research focused on the conjunction between quantum computing and routing problems has been very prolific in recent years. Most of the works revolve around classical problems such as the Traveling Salesman Problem or the Vehicle Routing Problem. The real-world applicability of these problems is dependent on the objectives and constraints considered. Anyway, it is undeniable that it is often difficult to translate complex requirements into these classical formulations. The main objective of this research is to present a solving scheme for dealing with realistic instances while maintaining all the characteristics and restrictions of the original real-world problem. Thus, a quantum-classical strategy has been developed, coined Q4RPD, that considers a set of real constraints such as a heterogeneous fleet of vehicles, priority deliveries, and capacities characterized by two values: weight and dimensions of the packages. Q4RPD resorts to the Leap Constrained Quadratic Model Hybrid Solver of D-Wave. To demonstrate the application of Q4RPD, an experimentation composed of six different instances has been conducted, aiming to serve as illustrative examples.