Salesman Problem Research Articles

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/ .

Application of an Improved Ant Colony Optimization Algorithm of Hybrid Strategies Using Scheduling for Patient Management in Hospitals

To balance the convergence speed and solution diversity and enhance optimization performance when addressing large-scale optimization problems, this research study presents an improved ant colony optimization (ICMPACO) technique. Its foundations include the co-evolution mechanism, the multi-population strategy, the pheromone diffusion mechanism, and the pheromone updating method. The suggested ICMPACO approach separates the ant population into elite and common categories and breaks the optimization problem into several sub-problems to boost the convergence rate and prevent slipping into the local optimum value. To increase optimization capacity, the pheromone update approach is applied. Ants emit pheromone at a certain spot, and that pheromone progressively spreads to a variety of nearby regions thanks to the pheromone diffusion process. Here, the real gate assignment issue and the travelling salesman problem (TSP) are chosen for the validation of the performance for the optimization of the ICMPACO algorithm. The experiment's findings demonstrate that the suggested ICMPACO method can successfully solve the gate assignment issue, find the optimal optimization value in resolving TSP, provide a better assignment outcome, and exhibit improved optimization ability and stability. The assigned efficiency is comparatively higher than earlier ones. With an assigned efficiency of 83.5%, it can swiftly arrive at the ideal gate assignment outcome by assigning 132 patients to 20 gates of hospital testing rooms. To minimize the patient's overall hospital processing time, this algorithm was specifically employed with a better level of efficiency to create appropriate scheduling in the hospital.

Hybrid ITÖ Algorithm for Large-Scale Colored Traveling Salesman Problem

Hybrid ITÖ Algorithm for Large-Scale Colored Traveling Salesman Problem

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.

A novel autonomous exploration algorithm via LiDAR/IMU SLAM and hierarchical subsystem for mobile robot in unknown indoor environments

Abstract Autonomous exploration in unknown environments is an essential capability for mobile robots. The complexity of autonomous exploration, however, means that existing algorithms struggle to balance efficiency and comprehensiveness, causing low mapping accuracy and redundant path planning. To perform accurate and efficient exploration tasks, we have proposed a novel autonomous exploration algorithm via LiDAR/IMU (Inertial Measurement Unit) Simultaneous Localization and Mapping (SLAM) and hierarchical subsystem for mobile robot in unknown environments. Firstly, to enhance mapping accuracy for mobile robot exploration, LiDAR/IMU SLAM is improved with the assistance of backward propagation and iterated Kalman filter, and Bidirectional Rapidly–Exploring Random Trees* (BI–RRT*) is applied for efficient frontier point detection. Secondly, we optimize local path planning by leveraging information theory through perceptual quality evaluation, which is then integrated with global path planning utilizing an enhanced Travelling Salesman Problem solver and a sparse grid map to amplify exploration efficiency. Thirdly, an enhanced hierarchical autonomous exploration method for mobile robots is proposed, which incorporates local path planning for seamless navigation around highly promising exploration spots, coupled with global path planning to effectively interconnect various sub–regions. Finally, simulations and field tests have demonstrated that the proposed method explores an unknown indoor environment with a 30.8% reduction in exploration time and a 29.9% reduction in exploration path in comparison with Dynamic Stage Viewpoint Planner. The map constructed in this paper has more accurate details and exploration paths have been shortened to ensure effective exploration.

Combining Genetic Algorithm with Local Search Method in Solving Optimization Problems

This research is focused on evolutionary algorithms, with genetic and memetic algorithms discussed in more detail. A graph theory problem related to finding a minimal Hamiltonian cycle in a complete undirected graph (Travelling Salesman Problem—TSP) is considered. The implementations of two approximate algorithms for solving this problem, genetic and memetic, are presented. The main objective of this study is to determine the influence of the local search method versus the influence of the genetic crossover operator on the quality of the solutions generated by the memetic algorithm for the same input data. The results show that when the number of possible Hamiltonian cycles in a graph is increased, the memetic algorithm finds better solutions. The execution time of both algorithms is comparable. Also, the number of solutions that mutated during the execution of the genetic algorithm exceeds 50% of the total number of all solutions generated by the crossover operator. In the memetic algorithm, the number of solutions that mutate does not exceed 10% of the total number of all solutions generated by the crossover operator, summed with those of the local search method.

Branch and Price for the Stochastic Traveling Salesman Problem with Generalized Latency

We consider an extension of the symmetric traveling salesman problem with generalized latency that explicitly models uncertainty. The stochastic traveling salesman problem with generalized latency (STSP-GL) aims to choose a subset of nodes of an undirected graph and determines a Hamiltonian tour among those nodes, minimizing an objective function that is a weighted combination of route design and passenger routing costs. These nodes are selected to ensure that a predefined percentage of uncertain passenger demand is served with a given probability. We formulate the STSP-GL as a stochastic program and propose a branch-and-price algorithm for solving its deterministic equivalent. We also develop a local search approach with which we improve the performance of the branch-and-price approach. We assess the efficiency of the proposed methods on a set of instances from the literature. We demonstrate that the proposed methods outperform a known benchmark, improving upper bounds by up to 85% and lower bounds by up to 55%. Finally, we show that solutions of the stochastic model are both more cost-effective and robust than those of the deterministic model. History: This paper has been accepted for the Transportation Science Special Issue on TSL Conference 2023. Funding: This work was supported by the Bundesministerium für Bildung und Forschung [Grant 03ZU1105FA]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0417 .

Determining Tourist Destination Priorities Using Website-Based Particle Swarm Optimization Methods (Case Study : North Padang Lawas Regency)

. Optimization of tourist routes is very important to minimize travel time and reduce travel costs. This study focuses on optimizing tourist routes in North Padang Lawas Regency using Multi Attribute Utility Theory (MAUT), and Particle Swarm Optimization (PSO) in the context of the Traveling Salesman Problem (TSP). This study discusses problems such as the lack of priority of tourist destinations and the need for shorter travel times. The research process includes problem identification, literature review, data collection through field observations and interviews, and data processing with MAUT to prioritize destinations. The identified priority destinations are Hotel Sapadia Gunung Tua, Barumun Nagari, Batik Sekar Najogi, Durian and Manggis Agrotourism, Rumah Makan Holat Alhamdulillah, Waterboom Gunung Tua, and Candi Bahal I, II, III. Furthermore, PSO is applied to determine the optimal travel route. PSO finds a route with a total travel time of 351 minutes, although the three-day travel time is extended to 375 minutes.

Reducing ACO Population Size to Increase Computational Speed

Ant Colony Optimization (ACO) is a powerful metaheuristic algorithm widely used to solve complex optimization problems in production and logistics. This paper presents a methodology for enhancing the ACO performance when applied to Traveling Salesman Problems (TSP). By reducing the number of ants in the colony, the algorithm's computational speed improves but solution quality is sacrificed. An optimal number of ants to produce the best results in the shortest time is specific to the problem at hand and can't be defined generally. This paper investigates the effect of ant population reduction relative to the problem size by measuring its impact on solution quality and execution time. Results show that for certain problem sizes ant population and execution time can be halved with practically no reduction in solution quality, or they can be reduced 5 times at the price of slightly worse solution quality. Reduction of ant population is much more impactful on reduction of execution time than it is on solution quality.

Using Online Algorithms to Solve NP-Hard Problems

This paper explores the application of online algorithms to tackle NP-hard problems, a class of computational challenges characterized by their intractability and wide-ranging real-world implications. Unlike traditional offline algorithms that have access to complete input data, online algorithms make decisions sequentially, often under constraints of incomplete information. We investigate various strategies, including greedy approaches, randomization, and competitive analysis, to assess their effectiveness in solving NP-hard problems such as the Traveling Salesman Problem, the Knapsack Problem, and the Set Cover Problem. Our analysis highlights the trade-offs between solution quality and computational efficiency, emphasizing the significance of the competitive ratio in evaluating algorithm performance. Additionally, we discuss the practical applications of online algorithms in dynamic environments, such as real-time systems and streaming data processing. Through a comprehensive review of existing literature and novel algorithmic designs, we aim to provide insights into the viability of online algorithms as a robust framework for addressing NP-hard problems in scenarios where immediate decision-making is crucial. The findings underscore the potential for future research to enhance these algorithms, making them increasingly applicable in complex, real-world contexts.