Classical Traveling Salesman Problem Research Articles

Picker routing in scattered storage warehouses: an evaluation of solution methods based on TSP transformations

AbstractTo optimize their order fulfillment processes, many e-commerce warehouses employ a storage assignment strategy known as scattered or mixed-shelves storage. Under this approach, unit loads of homogeneous products are divided, and individual pieces are stored in various shelves throughout the warehouse. This arrangement ensures that products that appear together on unpredictable pick lists are stored in close proximity somewhere in the huge warehouses, reducing the travel distance for pickers. Despite these advancements, efficiently guiding pickers through the warehouse remains a significant planning challenge. Since the same products can be found in multiple storage positions, the traditional picker routing problem becomes more complex, as an additional selection task arises regarding which shelf to retrieve each requested product from. While previous research has developed several tailor-made solution algorithms, we demonstrate that known transformation schemes used for different variants of the well-known Traveling Salesman Problem (TSP) can be utilized to convert the single picker routing problem with scattered storage (SPRP-SS) into a classical TSP. This approach enables us to leverage the extensive array of state-of-the-art TSP solvers. The purpose of this paper is to explore the performance of these solvers when applied to solving the SPRP-SS. Through our computational study, we found that existing TSP solvers exhibit good performance, allowing near-optimal solutions to be obtained in less than a second for real-world scale SPRP-SS instances. Moreover, the efficiency of these TSP solvers remains unaffected by the number of cross aisles in the warehouse. Consequently, we exploit this flexibility to investigate the impact of cross aisles on picking performance in scattered storage warehouses.

Symmetric optical multipass matrix systems and the general rapid design methodology

We proposed an original type of multipass cell named symmetric optical multipass matrix system (SMMS), in which the same matrix patterns of various sizes can be formed on both sides. According to its special symmetric configurations, the SMMS design problem is modeled as a variant of the classical traveling salesman problem, which can be rapidly solved by evolutionary optimization algorithms. Two sets of 3-mirror SMMSs are designed, analyzed and built, which show superior characteristics of high stability, desirable beam quality and adjustable optical path lengths. Additionally, they can support simultaneous detection of multiple species with multi-laser channels. The proposed method is further extended to design a 4-mirror SMMS, which verifies the universality and robustness of the design methodology. The experimental observations are in consistent with the theoretical calculations. The newly proposed SMMSs have a broad application prospect in trace gas measurement.

On solving close enough orienteering problems with overlapped neighborhoods

The Close Enough Traveling Salesman Problem (CETSP) is a well-known variant of the classic Traveling Salesman Problem whereby the agent may complete its mission at any point within a target neighborhood. Heuristics based on overlapped neighborhoods, known as Steiner Zones (SZ), have gained attention in addressing CETSPs. While SZs offer effective approximations to the original graph, their inherent overlap imposes constraints on the search space, potentially conflicting with global optimization objectives. Here we show how such limitations can be converted into advantages in the Close Enough Orienteering Problem (CEOP) by aggregating prizes across overlapped neighborhoods. We further extend the classic CEOP with Non-uniform Neighborhoods (CEOP-N) by introducing non-uniform cost considerations for prize collection. To tackle CEOP (and CEOP-N), we develop a new approach featuring a Randomized Steiner Zone Discretization (RSZD) scheme coupled with a hybrid algorithm based on Particle Swarm Optimization (PSO) and Ant Colony System (ACS) — CRaSZe-AntS. The RSZD scheme identifies sub-regions for PSO exploration, and ACS determines the discrete visiting sequence. We evaluate the RSZD’s discretization performance on CEOP instances derived from established CETSP instances and compare CRaSZe-AntS against the most relevant state-of-the-art heuristic focused on single-neighborhood optimization for CEOP instances. We also compare the performance of the interior search within SZs and the boundary search on individual neighborhoods in the context of CEOP-N. Our experimental results show that CRaSZe-AntS can yield comparable solution quality with significantly reduced computation time compared to the single neighborhood strategy, where we observe an averaged 140.44% increase in prize collection and 55.18% reduction of algorithm execution time. CRaSZe-AntS is thus highly effective in solving emerging CEOP-N, examples of which include truck-and-drone delivery scenarios.

An efficient hybrid genetic algorithm for the traveling salesman problem with release dates

This paper deals with a generalization of the classic traveling salesman problem where each customer is associated with a release date, defined as the time at which the desired product becomes available at the depot. In this problem, a single uncapacitated vehicle is allowed to perform multiple trips in order to satisfy all demands. However, the vehicle cannot start a route unless all the products associated with the demands in the route are released. As a consequence, there might be a waiting time before starting the next route. The objective of the problem is to minimize the completion time of the service, defined as the time at which the vehicle returns to the depot after satisfying all demands. We propose a hybrid genetic algorithm that incorporates more advanced mechanisms to evaluate individuals and to ensure population diversity. We also introduce a novel dynamic programming splitting algorithm that converts a sequence of visits to customers, the so-called giant-tour, into the best set of routes that respects the sequence. Computational experiments performed on 522 benchmark instances show that our approach is able to find all 154 known optimal solutions from the literature. In addition, we were able to improve the best-known upper bounds for 364 instances in significantly shorter computational times when compared to the state-of-the-art.

Solving the single depot open close multiple travelling salesman problem through a multi-chromosome based genetic algorithm

The multiple travelling salesman problem (MTSP) extends the classical travelling salesman problem (TSP) by involving multiple salesman in the solution. MTSP has found widespread applications in various domains, such as transportation, robotics, and networking. Despite extensive research on MTSP and its variants, there has been limited attention given to the open close multiple travelling salesman problem (OCMTSP) and its variants in the literature. To the best of the author's knowledge, only one study has addressed OCMTSP, introducing an exact algorithm designed for optimal solutions. However, the efficiency of this existing algorithm diminishes for larger instances due to computational complexity. Therefore, there is a crucial need for a high-level metaheuristic to provide optimal/best solutions within a reasonable timeframe. Addressing this gap, this study proposes a first meta-heuristic called multi-chromosome-based Genetic Algorithm (GA) for solving OCMTSP. The effectiveness of the developed algorithm is demonstrated through a comparative study on distinct asymmetric benchmark instances sourced from the TSPLIB dataset. Additionally, results from comprehensive experiments conducted on 90 OCMTSP symmetric instances, generated from the renowned TSPLIB benchmark dataset, highlight the efficiency of the proposed GA in addressing the OCMTSP. Notably, the proposed multi-chromosome-based GA stands out as the top-performing approach in terms of overall performance. Further, solutions to symmetric TSPLIB benchmark instances are also reported, which will be used as a basis for future studies.

Machine Learning for Data-Driven Last-Mile Delivery Optimization

In the context of the Amazon Last-Mile Routing Research Challenge, this paper presents a machine-learning framework for optimizing last-mile delivery routes. Contrary to most routing problems where an objective function is clearly defined, in the real-world setting considered in the challenge, an objective is not explicitly specified and must be inferred from data. Leveraging techniques from machine learning and classical traveling salesman problem heuristics, we propose a “pool and select” algorithm to prescribe high-quality last-mile delivery sequences. In the pooling phase, we exploit structural knowledge acquired from data, such as common entry and exit regions observed in training routes. In the selection phase, we predict the scores of candidate sequences with a high-dimensional, pretrained, and regularized regression model. The score prediction model, which includes a large number of predictor variables such as sequence duration, compliance with time windows, earliness, lateness, and structural similarity to training data, displays good prediction accuracy and guides the selection of efficient delivery sequences. Overall, the framework is able to prescribe competitive delivery routes, as measured on out-of-sample routes across several data sets. Given that desired characteristics of high-quality sequences are learned and not assumed, the proposed framework is expected to generalize well to last-mile applications beyond those immediately foreseen in the challenge. Moreover, the method requires less than three seconds to prescribe a sequence given an instance and, thus, is suitable for very large-scale applications. History: This paper has been accepted for the Transportation Science Special Issue on Machine Learning Methods and Applications in Large-Scale Route Planning Problems. Funding: This research was funded by The Dutch Research Council (NWO) Data2Move project under [Grant 628.009.013] and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie [Grant 754462]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0029 .

Can language models be used for real-world urban-delivery route optimization?

Language models have contributed to breakthroughs in interdisciplinary research, such as protein design and molecular dynamics understanding. In this study, we reveal that beyond language, representations of other entities, such as human behaviors, that are mappable to learnable sequences can be learned by language models. One compelling example is the real-world delivery route optimization problem. We here propose a novel approach based on the language model to optimize delivery routes on the basis of drivers' historical experiences. Although a broad range of optimization-based approaches have been designed to optimize delivery routes, they do not capture the implicit knowledge of complex delivery operating environments. The model we propose integrates this knowledge in the route optimization process by learning from driving behaviors in experienced drivers. A real-world delivery route that preserves drivers' implicit behavioral patterns is first analogized to a sentence in natural language. Through unsupervised learning, we then learn the vector representations of words and infer the drivers' delivery chains on the basis of the tailored chain-reaction-based algorithm. We also provide insights into the fusion of language models and operations research methods. In our approach, language models are applied to learn drivers' delivery behaviors and infer new deliveries at the delivery zone level, while the classic traveling salesman problem (TSP) model is embedded into the hybrid framework for intra-zone optimization. Numerical experiments performed on real-world data from Amazon's delivery service demonstrate that the proposed approach outperforms pure optimization, supporting the effectiveness, efficiency, and extensibility of our model. As a versatile approach, the proposed framework can easily be extended to various disciplines in which the data follow certain grammar rules. We anticipate that our work will serve as a stepping stone toward the understanding and application of language models in tackling interdisciplinary research problems.

Formulations for the clustered traveling salesman problem with [formula omitted]-relaxed priority rule

In the classical Traveling Salesman Problem, the order in which the vertices are visited has no restrictions. The only condition imposed is that each vertex is visited only once. In some real situations this condition may not be sufficient to represent the problem, as there are cases in which the vertex visit order becomes extremely important. In other words, it is also necessary to consider a priority between the vertices. To deal with these situations, some formulations are proposed in the literature and are based on a rule called d-relaxed priority rule that captures the trade-off between total distance and vertex priorities. In this paper, the formulations from the literature are improved using valid inequalities and different formulations based on precedence variables are proposed. Computational results, based on data from the literature, are presented to demonstrate the competitiveness of the proposed approaches.

An Efficient Hybrid Genetic Algorithm for the Quadratic Traveling Salesman Problem

The traveling salesman problem (TSP) is the most well-known problem in combinatorial optimization which has been studied for many decades. This paper focuses on dealing with one of the most difficult TSP variants named the quadratic traveling salesman problem (QTSP) that has numerous planning applications in robotics and bioinformatics. The goal of QTSP is similar to TSP which finds a cycle visiting all nodes exactly once with minimum total costs. However, the costs in QTSP are associated with three vertices traversed in succession (instead of two like in TSP). This leads to a quadratic objective function that is much harder to solve. To efficiently solve the problem, we propose a hybrid genetic algorithm including a local search procedure for intensification and a new mutation operator for diversification. The local search is composed of a restricted double-bridge move (a variant of 4-Opt); and we show the neighborhood can be evaluated in O(n^2), the same complexity as for the classical TSP. The mutation phase is inspired by a ruin-and-recreate scheme. Experimental results conducted on benchmark instances show that our method significantly outperforms state-of-the-art algorithms in terms of solution quality. Out of 800 considered instances, it finds 437 new best-known solutions.

Solving the Min-Max Clustered Traveling Salesmen Problem Based on Genetic Algorithm.

The min-max clustered traveling salesmen problem (MMCTSP) is a generalized variant of the classical traveling salesman problem (TSP). In this problem, the vertices of the graph are partitioned into a given number of clusters and we are asked to find a collection of tours to visit all the vertices with the constraint that the vertices of each cluster are visited consecutively. The objective of the problem is to minimize the weight of the maximum weight tour. For this problem, a two-stage solution method based on a genetic algorithm is designed according to the problem characteristics. The first stage is to determine the visiting order of the vertices within each cluster, by abstracting a TSP from the corresponding cluster and applying a genetic algorithm to solve it. The second stage is to determine the assignment of clusters to salesmen and the visiting order of the assigned clusters. In this stage, by representing each cluster as a node and using the result of the first stage and the ideas of greed and random, we define the distances between each two nodes and construct a multiple traveling salesmen problem (MTSP), and then apply a grouping-based genetic algorithm to solve it. Computational experiments indicate that the proposed algorithm can obtain better solution results for various scale instances and shows good solution performance.

An Exact Approach for Selecting Pickup-Delivery Stations in Urban Areas to Reduce Distribution Emission Costs

This paper deals with a variant of the multifacility location-routing problem in urban areas. The distribution network is modelled by an undirected graph, in which the nodes are split into a set of pickup-delivery stations, a depot, and a set of customers. The arcs represent the minimum-cost connections between nodes. A customer is assigned to a pickup-delivery station if he or she can reach it at the lowest sustainable cost, i.e., on foot or by bicycle, without exceeding a predefined maximum distance. The goal is to minimise the goods’ total delivery cost, including pollutant emissions. In this perspective, both travel distance and means of transport play a key role. We present an exact novel approach based on partitioning the research space of the solutions of a Mixed Integer Linear Programming model. In the model, Boolean decisional variables, representing the selection of the locations for the pickup-delivery stations, are fixed simultaneously with the solution of the classical Travelling Salesman Problem. A branching constraint allows us to determine the route that serves the selected pickup-delivery stations and the route, if any, that serves customers who do not go to any pickup-delivery station. We conduct extensive experimentation to test the proposed approach’s computational efficiency and analyse the optimal solution’s robustness with respect to the maximum distance of customers from the stations, their activation cost and the pollutant emissions. The effectiveness of the proposed approach in terms of solution quality and computation time is certified by a set of computational tests based on randomly generated instances with up to 150 customers and 30 pickup-delivery stations. The application of the proposed exact method to a case study related to a district of the city of Genoa (Italy) confirms its validity also for sustainably addressing real-size urban delivery problems. An evaluation of incentives for customers using pickup-delivery stations, possibly by implementing discount policies on orders, is also proposed.

Inter-Object Navigation of Unmanned Aerial Vehicles to Increase the Efficiency and Accuracy of Inspection of Power Lines

The purpose of the work is to improve the accuracy and efficiency of a power line section inspection for a fault detection using unmanned aerial vehicles. The goal was achieved by using a unified computing and measurement platform on multicopter and aircraft drones and by simplifying the interaction between them and by using the inter-object navigation sensors. The most significant results were the development of a method of route planning by drones over different parts of the power grid and a method of inter-object navigation. The drone route planning problem was represented by a multiagent variation of the classical traveling salesman problem and was solved by the ant colony method. The method of inter-object navigation was distinguished by the representation of the power grid topology by high and low intensity graphs, involving a different number and types of drones in the inspection process. The application of the developed methods made it possible to increase the accuracy of power line inspections by 27-73%, and the efficiency by 2-8 times. Solving the problem of multicriteria optimization of the drone team flight route planning made it possible to reduce the cost of monitoring critical infrastructure facilities while improving its efficiency and accuracy. Thus, the conducted research has shown the effectiveness of the proposed approach for the monitoring of power facilities, route selection, number and composition of search teams. The direction for further research is to improve the ant algorithm.

A Novel Bias-TSP Algorithm for Maritime Patrol

This work aims to develop a search planning strategy to be used by a drone equipped with an inverse synthetic-aperture radar (ISAR) and an electro-optical sensor. After describing the specifics of our maritime scenario, we discuss four methodologies that can be used to find vessels involved in illegal fishing activities as quickly as possible. In addition to the clustering of the vessels, determined by the drone’s electro-optical sensor range, we introduce a novel technique to bias a traveling salesman problem (TSP) tour. This bias is based on deliberately increasing distances to vessels that are classified as probable fishing vessels. This increase in distance is meant to prioritize visits to probable fishing vessels. Vessels are classified based on length using the. The classification result and the vessel clustering are available before the actual planning of the tour. Simulations of scenarios in which we have a few vessels fishing illegally show that the novel technique, the bias-TSP, combined with a tour orientation based on operational considerations, outperforms the classic TSP: the mean distance traveled to find all the vessels involved in illegal fishing activities is reduced by at least 35–50%. We also show that different drone take-off locations significantly impact the results.

Solving a Multi-Conveyance Travelling Salesman Problem using an Ant Colony Optimization Method

Objectives: A well-known NP-complete problem is the travelling salesman problem (TSP). It has numerous engineering and scientific applications. In this article, we have proposed a multi-conveyance TSP where different conveyances are present to travel from one city to another city. This is an extension to classical TSP. In this TSP, the salesman visits all the cities only once during his/her tour, using different conveyances to travel from one city to another. The cost of travelling between cities using various modes of conveyance varies. The objective of this research is to find the minimum cost tour using an ant colony optimization (ACO) based approach by satisfying the constraints of the proposed multi-conveyance TSP. Method: The considered TSP has been solved using a novel ACO technique. The proposed ACO is adopted with the Roulette-wheel selection and “tuning solution” techniques. We have used a few benchmark datasets from TSPLIB to check the effectiveness of the proposed algorithm. The experimental findings for a few benchmarks TSP instances show that almost always, the proposed ACO is able to find a better result. Then, we used some redefined and randomly generated datasets for experiments. The experimental outcomes for various input datasets are also very encouraging. Findings: The goal of the proposed TSP is to find a complete tour with a minimum cost without exceeding the total travel cost and total travel time. Thus, there are two novel constraints in the classical TSP. Novelty: A unique aspect of the proposed research is the use of several conveyance facilities and a fixed total travel time and cost. This is new, as these three factors are integrated into a single TSP model. Keywords: Travelling Salesman Problem; Travel Cost; Travel Time; MultiConveyance TSP; Ant Colony Optimization

Approximations for many-visits multiple traveling salesman problems

A fundamental variant of the classical traveling salesman problem (TSP) is the so-called multiple TSP (mTSP), where a set of m salesmen jointly visit all cities from a set of n cities. The mTSP models many important real-life applications, in particular for vehicle routing problems. An extensive survey by Bektas (Omega34(3), 2006) lists a variety of heuristic and exact solution procedures for the mTSP that solve particular problem instances. In this work we consider a further generalization of mTSP, the many-visits mTSP, where each city v has a request r(v) of how many times it should be visited by the salesmen. This problem opens up new real-life applications such as aircraft sequencing, while at the same time it poses several computational challenges. We provide polynomial-time algorithms for several variants of the many-visits mTSP that compute constant-factor approximate solutions.

Совершенствование организации метрологического обеспечения на основе методов исследования операций

The article considers the problem of improving the methods of organizing the metrological support of enterprises based on the methods of operation research as well as the problem of optimizing the traveling routes of metrological groups mathematically reduced to the classical traveling salesman problem. A brief overview of the methods used in this task is presented, and the relationships between them are shown. Solving the problems based on contemporary geographic information systems and standard software modules from the Python application libraries are proposed. Much attention is paid to simple methods for checking the correctness of the results obtained by standard methods, used by metrological service employees not specialized in the field of programming and geographic information systems, which increases the likelihood of errors. To identify errors occurring when working with geoinformation systems, allowing the matrix of distances between points to be determined, a simple formula borrowed from topography is proposed for calculating the distances between points with known coordinates. With a developed road network, the distances calculated in this way are close to the results obtained in geoinformation systems taking into account the location of highways. The results of solving the problem are given on the example of the organization of verification and repair of measuring instruments at food industry enterprises in the city of Chimkent and he Turkestan region of the Republic of Kazakhstan

A Three-Layered Multifactorial Evolutionary Algorithm with Parallelization for Large-Scale Engraving Path Planning

Today, although laser engraving technology is widely used in 2D image engraving, when the image is larger and more complicated, most existing algorithms for engraving path planning have a huge computational burden and reduced engraving efficiency. Accordingly, this article addresses the trajectory optimization problem in large-scale image engraving. First, we formulate the problem as an improved model based on the large-scale traveling salesman problem (TSP). Then, we propose a three-layered algorithm called 3L-MFEA-MP, structured as follows: an upper layer, the genetic algorithm (GA); a middle layer, the GA; and a bottom layer, the parallel multifactorial evolutionary algorithm. Experiments on four classic large-scale TSP datasets show that our algorithm exhibits superior performance in terms of the path length and engraving time compared with other algorithms. In particular, compared with the single-thread algorithm, the proposed parallel algorithm reduced the engraving time by 80%. Moreover, the engraving machine experiment demonstrated that the engraving time of our algorithm on mona-lisa 100K, vangogh 120K, and venus 140K was approximately one tenth that of the traditional dot engraving method. The results indicate that the proposed algorithm can reduce the computational burden and improve engraving efficiency in engraving path planning.

AUV-Aided Hybrid Data Collection Scheme Based on Value of Information for Internet of Underwater Things

The current Internet of Underwater Things (IoUT) for marine observations and emergency responses suffers from two critical issues: 1) energy efficient and 2) timely data collection. Autonomous underwater vehicles (AUVs), serving as tools for collecting and forwarding distributed data, can deal with the unbalanced power consumption in a traditional multihop underwater communication network. However, the low speed of the AUV has not been able to guarantee the timeliness of delay-sensitive data. In this article, we introduce a hybrid data collection scheme (HDCS), taking both real-time data collection and energy efficiency (EE) issues into consideration. All sensor nodes (SNs) are first clustered based on their locations in the network. We develop an analytic expression to describe the attenuation of Value of Information (VoI), involving the relationship between the importance degree and timeliness; initial VoI could be measured by historical data. The emergency can be recognized by the presented criterion, and the transmission mode of cluster heads (CHs) in the same layer is judged by CHs themselves according to VoI. The selected CHs shall transmit the urgent data via multihop routing to avoid over attenuation of VoI. The normal data are collected by AUVs visiting all remaining CHs, and the shortest trajectory is achieved by addressing a variation of the classic traveling salesman problem (TSP). Our simulation experiments show that this mechanism can effectively increase long-term VoI while significantly improving EE.

A genetic algorithm for the close-enough traveling salesman problem with application to solar panels diagnostic reconnaissance

This paper addresses a variant of the classical Traveling Salesman Problem known as Close-Enough Traveling Salesman Problem . In this problem, there is a set of nodes (customers, targets), each of them associated with a region, denoted as neighborhood, that contains it. The goal is to determine the shortest tour that visits all the nodes, where a node is visited when the tour traverses or reaches the region associated with the node. We propose a genetic algorithm (GA), which uses several strategies to optimize the tour, such as 2opt, second-order cone programming, and a bisection algorithm. The proposed approach is tested on 62 benchmark instances. The results show that GA produces better or similar solutions compared to the ones produced by state-of-the-art algorithms in a reasonable computing time. Besides, GA found 32 new best solutions out of 62 instances. Furthermore, we propose different metrics to classify problem instances with the goal of detecting which problem characteristics have a larger impact on the difficulty of solving the problem. We also revised the already proposed metric, called Overlap Ratio, and correct its calculation done in previous contributions. Finally, we present a case study related to the diagnostic reconnaissance of solar panels. The case study is related to a research project developed at the University of Molise in collaboration with private IT companies. We show that the problem under study is properly modeled as a Close-Enough Traveling Salesman Problem and apply the GA to solve it, focusing on the benefits obtained by applying this solution approach.

A Double-Adaptive General Variable Neighborhood Search algorithm for the solution of the Traveling Salesman Problem

This work addresses a novel General Variable Neighborhood Search (GVNS) solution method, which integrates intelligent adaptive mechanisms to re-order the search operators during the intensification and diversification phases, in an effort to enhance its overall efficiency. To evaluate the performance of the new GVNS scheme, asymmetric and symmetric instances of the classic Traveling Salesman Problem (TSP) from the TSPLib were solved. The obtained results of the Double-Adaptive GVNS were compared with those achieved by two single-adaptive GVNS, which use an adaptive mechanism either for the intensification or the diversification phase and with a conventional GVNS. For a fair comparison, all GVNS schemes were structured using the same local search and shaking operators. Moreover, the novel GVNS algorithm was compared with some recent solution methods for the TSP, found in the open literature. The comparative studies revealed the high efficiency of the novel VNS scheme and underlined the significant impact of intelligent mechanisms on the performance of classic metaheuristic frameworks.