Traveling Salesman Problem Model Research Articles

Efficient Node Insertion Algorithm for Connectivity-Based Multipolling MAC Protocol in Wi-Fi Sensor Networks

Since low-power Wi-Fi sensors are connected to the Internet, effective radio spectrum use is crucial for developing an efficient Medium Access Control (MAC) protocol for Wi-Fi sensor networks. A connectivity-based multipolling mechanism was employed for Access Points to grant uplink transmission opportunities to Wi-Fi nodes with a reduced number of multipolling frame transmissions. The existing connectivity-based multipolling mechanism in IEEE 802.11 wireless LANs with many nodes may require excessive time to derive the optimal number of serially connected sequences due to the backtracking algorithm based on the Traveling Salesman Problem model. This limitation hinders the real-time implementation of the connectivity-based multipolling mechanism in Wi-Fi sensor networks. In this study, an efficient node insertion algorithm is proposed, by which the number of derived serially connected multipolling sequences that cover nodes in Wi-Fi sensor networks converges to only one as the number of Wi-Fi sensors increases in Wi-Fi sensor networks. As verified by simulation experiments for Wi-Fi sensor networks, the proposed node insertion algorithm produces a near-optimal number of multipolling sequences that cover the nodes in Wi-Fi sensor networks. This study proposes a node insertion algorithm for the real-time implementation of the connectivity-based multipolling mechanism in MAC protocol for Wi-Fi sensor networks.

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.

Multi-stage resilient operation strategy of urban electric–gas system against rainstorms

This paper proposes a multi-stage resilient operation strategy for an urban electric–gas system (UEGS) considering the impact of traffic under rainstorms. It guarantees that the defense budget in pre-disaster hardening can be assigned to cope with the worst scenario and that the load supply can be restored as soon as possible. For pre-disaster hardening, a more realistic multi-stage robust model (MSRM) of UEGS under rainstorms is proposed to accommodate the randomness and continuity of rainstorms. After transforming MSRM into the two-stage robust (TSR) form, the nested column-and-constraint generation (C&CG) method is applied for model solving, which is a mixed integer linear program with the min–max–min form. For the unacceptable computing time of the nested C&CG method in solving the equivalent TSR, a simplification is proposed using the sliding time window to provide defense priority for allocating the defense budget in pre-disaster hardening with a reduced calculation burden. In the recovery of UEGS, the traveling salesman problem model of repair crews routing with real-time traversal path information is proposed to meet the modeling demand of multi-period attacks and waterlogging on paths caused by rainstorms. Case studies on the two test systems validate the effectiveness of the proposed method and demonstrate the support of the gas distribution network to the power distribution network in rainstorm attacks.

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

An Algorithm for Painting Large Objects Based on a Nine-Axis UR5 Robotic Manipulator

An algorithm for automatically planning trajectories designed for painting large objects is proposed in this paper to eliminate the difficulty of painting large objects and ensure their surface quality. The algorithm was divided into three phases, comprising the target point acquisition phase, the trajectory planning phase, and the UR5 robot inverse solution acquisition phase. In the target point acquisition phase, the standard triangle language (STL) file, algorithm of principal component analyses (PCA), and k-dimensional tree (k-d tree) were employed to obtain the point cloud model of the car roof to be painted. Simultaneously, the point cloud data were compressed as per the requirements of the painting process. In the trajectory planning phase, combined with the maximum operating space of the UR5 robot, the painting trajectory of the target points was converted into multiple traveling salesman problem (TSP) models, and each TSP model was created with a genetic algorithm (GA). In the last phase, in conformity with the singularities of the UR5 robot’s motion space, the painting trajectory was divided into a recommended area trajectory and a non-recommended area trajectory and created by the analytical method and sequential quadratic programming (SQP). Finally, the proposed algorithm for painting large objects was deployed in a simulation experiment. Simulation results showed that the accuracy of the algorithm could meet the requirements of painting technology, and it has promising engineering practicability.

GPS: A New TSP Formulation for Its Generalizations Type QUBO

We propose a new Quadratic Unconstrained Binary Optimization (QUBO) formulation of the Travelling Salesman Problem (TSP), with which we overcame the best formulation of the Vehicle Routing Problem (VRP) in terms of the minimum number of necessary variables. After, we will present a detailed study of the constraints subject to the new TSP model and benchmark it with MTZ and native formulations. Finally, we will test whether the correctness of the formulation by entering it into a QUBO problem solver. The solver chosen is a D-Wave_2000Q6 quantum computer simulator due to the connection between Quantum Annealing and QUBO formulations.

Solving the TSP by the AALHNN algorithm.

It is prone to get stuck in a local minimum when solving the Traveling Salesman Problem (TSP) by the traditional Hopfield neural network (HNN) and hard to converge to an efficient solution, resulting from the defect of the penalty method used by the HNN. In order to mend this defect, an accelerated augmented Lagrangian Hopfield neural network (AALHNN) algorithm was proposed in this paper. This algorithm gets out of the dilemma of penalty method by Lagrangian multiplier method, ensuring that the solution to the TSP is undoubtedly efficient. The second order factor added in the algorithm stabilizes the neural network dynamic model of the problem, thus improving the efficiency of solution. In this paper, when solving the TSP by AALHNN, some changes were made to the TSP models of Hopfield and Tank. Say, constraints of TSP are multiplied by Lagrange multipliers and augmented Lagrange multipliers respectively, The augmented Lagrange function composed of path length function can ensure robust convergence and escape from the local minimum trap. The Lagrange multipliers are updated by using nesterov acceleration technique. In addition, it was theoretically proved that the extremum obtained by this improved algorithm is the optimal solution of the initial problem and the approximate optimal solution of the TSP was successfully obtained several times in the simulation experiment. Compared with the traditional HNN, this method can ensure that it is effective for TSP solution and the solution to the TSP obtained is better.

Pricing of parcel locker service in urban logistics by a TSP model of last-mile delivery

Parcel lockers offer a service for last-mile delivery in urban logistics. To price this service, we identified its values to consignees not at receiving sites and to couriers when delivering parcels to lockers instead of consignees. A travel salesman problem (TSP) model is built to analyze the cost change of last-mile delivery corresponding to the availability of consignees at designated receiving sites. Based on cost evaluation, fees that consignees and couriers should pay respectively are determined based on beneficiary-pay principle. The results of our empirical analyses show that the value of the locker service varies with the terms of express delivery and the availability uncertainty of whether consignees are present at the designated receiving sites, which provides a fundamental pricing basis for locker service providers to charge the couriers or the consignees.

Timeline Club: An optimization algorithm for solving multiple debris removal missions of the time-dependent traveling salesman problem model

Timeline Club: An optimization algorithm for solving multiple debris removal missions of the time-dependent traveling salesman problem model

New Method of Path Optimization for Medical Logistics Robots

The path planning problem of logistics robots is mainly subjected to the time cost of the operation of the mathematical model. To save the time of refilling process in the fast medicine dispensing system (FMDS), the optimization procedure is divided into two steps in this study. First, a new mathematical model called the multiple steps traveling salesman problem model (MTSPM) is proposed to optimize the replenishment quantity of each picking and establish picking sets. Second, an improved ant colony optimization (IACO) algorithm is employed, considering the effects of velocity, acceleration, and deceleration in the refilling route during the development of the new model. Simulation results and operational results demonstrated that MTSPM-IACO was better than both the order picking model (OPM) and MTSPM-ACO in terms of saving refilling time. Compared to the OPM, the optimization of the refilling time of MTSPM-IACO was more than 1.73% in simulation and 15.26% in operation. Compared to MTSPM-ACO, the optimization of the refilling time of MTSPM-IACO was more than 0.13% in simulation and 1.67% in operation.

Online time-optimal path and trajectory planning for robotic multipoint assembly

PurposeThe purpose of this paper is to propose an efficient path and trajectory planning method to solve online robotic multipoint assembly.Design/methodology/approachA path planning algorithm called policy memorized adaptive dynamic programming (PM-ADP) combines with a trajectory planning algorithm called adaptive elite genetic algorithm (AEGA) for online time-optimal path and trajectory planning.FindingsExperimental results and comparative study show that the PM-ADP is more efficient and accurate than traditional algorithms in a smaller assembly task. Under the shortest assembly path, AEGA is used to plan the time-optimal trajectories of the robot and be more efficient than GA.Practical implicationsThe proposed method builds a new online and efficient path planning arithmetic to cope with the uncertain and dynamic nature of the multipoint assembly path in the Cartesian space. Moreover, the optimized trajectories of the joints can make the movement of the robot continuously and efficiently.Originality/valueThe proposed method is a combination of time-optimal path planning with trajectory planning. The traveling salesman problem model of assembly path is established to transfer the assembly process into a Markov decision process (MDP). A new dynamic programming (DP) algorithm, termed PM-ADP, which combines the memorized policy and adaptivity, is developed to optimize the shortest assembly path. GA is improved, termed AEGA, which is used for online time-optimal trajectory planning in joints space.

A discrete chicken swarm optimization for traveling salesman problem

Traveling Salesman Problem (TSP) is a typical combinatorial optimization problem, and it is a NP-hard problem. The total number of routes increases exponentially with the number of cities, so it is great significance to design an effective algorithm to find the optimal solution accurately. Chicken Swarm Optimization (CSO) is a new intelligent optimization algorithm, which is mainly proposed for continuous problems. It has the advantages of fast convergence speed and high convergence accuracy. This paper proposed a Discrete Chicken Swarm Optimization (DCSO) for TSP. The CSO is discretized by introducing the methods of swap, order crossover and reverse order mutation, where the search space of the solution is enlarged, and the diversity of the solution is increased. The typical TSP models are simulated and compared with the Basic Ant Colony Optimization and Genetic Algorithm to verify the feasibility of the presented method.

Optimization for medical logistics robot based on model of traveling salesman problems and vehicle routing problems

Fast medicine dispensing system (FMDS) as a kind of medical logistic robot can dispense many drugs for one prescription at the same time. To guarantee the sustainability of drug dispensation, it is required that FMDS replenish drugs rapidly. The traditional order picking model (OPM) is difficult to meet the demand of prompt replenishment. To solve the problems of prolonged refilling route and inefficiency of drugs replenishment, a mixed refilling model based on multiple steps traveling salesman problem model (MTSPM) and vehicle routing problem model (VRPM) is proposed, and it is deployed in two circumstances of FMDS, including temporary replenishment mode (TRM) and concentrate replenishment mode (CRM). It not only meted the demand under different circumstances of drug replenishment but also shortened the refilling route significantly. First, the new pick sets were generated. Then, the orders of pick sets were optimized and the new paths were achieved. When the number of pickings is varied no more than 20, experiment results declared that the refilling route is the shortest by utilizing MTSPM when working under the TRM condition. Comparing MTSPM with OPM, the rate of refilling route length decreased up to 32.18%. Under the CRM condition, the refilling route is the shortest by utilizing VRPM. Comparing VRPM with OPM, the rate of refilling route length decreased up to 58.32%. Comparing VRPM with MTSPM, the rate of refilling route length has dropped more than 43.26%.

Research on Picking Route Optimization of Multi-zone Warehouse Based on Traveling Salesman Problem and Simulated Annealing Algorithm

The improvement of the national economy has led to the rapid development of e-commerce business, which has caused an explosive growth of warehouse orders. Picking is the core operation of the warehouse, which occupies a lot of time and capital costs. Therefore, optimizing the picking path has become an urgent problem for e-commerce companies to solve. This article calculates the distance matrix based on the characteristics of multi-zone warehouses, establishes a Traveling Salesman Problem (TSP) model to process 49 combined orders, and solves the model through the simulated annealing algorithm. In addition, add review stations at different locations to analyze the impact on delivery time. Finally, the optimal picking path, the optimal distribution method of orders, and the time it takes to deliver are obtained. The results indicate that the TSP model combined with the simulated annealing algorithm can optimize the picking path, adding a review station close to the center of the warehouse can reduce the time of delivery, and the model is universal applicability.

Two-layered ant colony system to improve engraving robot’s efficiency based on a large-scale TSP model

Laser engraving is an essential tool of automatic drawings and 3D printers. When the laser engraving tasks become large and complicated, engraving process will be time-consuming. To improve the time and energy efficiency, the trajectory optimization of laser engraving is studied. The trajectory of laser engraving robot is modelled as a large-scale traveling salesman problem (TSP), by converting grayscale images into halftone images. To solve the large-scale TSP, two-layered ant colony system (ACS) is newly proposed to combine k-means, top-layer ACS, and bottom-layer ACS. Finally, we use the presented algorithm to optimize the path of four engraving instances which include tens of thousands of discrete points. Experimental results show that this method can reduce laser engraving time by about 50% compared with traditional engraving methods.

Performance Comparison of Two-phase LP-based Heuristic Methods for Capacitated Vehicle Routing Problem with Three Objectives

This paper develops a two-phase LP-based heuristic for the Capacitated Vehicle Routing Problem (CVRP). It considers three objectives: (1) minimizing the total costs of fuel consumption and overtime, (2) maximizing the total personal relationships between customers and drivers, and (3) balancing the delivery weights of vehicles. The two-phase LP-based heuristic (cluster-first route-second) is proposed. First, in the clustering stage, three LP-based clustering models (denoted by C1, C2, and C3) are developed. Customers are grouped into clusters based on real distances between the customers for C1, personal relationships between the customers and drivers for C2, and the delivery weights of vehicles for C3. Second, in the routing stage, an LP-based traveling salesman problem model is used to form a route for each cluster, to minimize the total costs of fuel consumption and overtime labor. The experimental results from a case study of Thai SMEs show that when the C2 clustering model is applied, the performances are the best. Significant contributions of this paper include: (1) it is an original paper that proposes the C2 clustering model, and it has the best performances based on the experimental results, and (2) the proposed two-phase LP-based heuristic methods are suitable for practical use by SMEs since the required computational time is short, and it has multiple models with different objectives that can be selected to match a user's requirements.

Future directions in drone routing research

AbstractOptimization of routing problems using drones (unmanned aerial vehicles or UAVs) has become an important area of academic research. The purpose of this article is to look to the future and help stimulate drone routing research in directions we hope will prove interesting and fruitful. We discuss opportunities for better modeling of (1) drone capabilities for both existing drones and those likely to be used in the future, (2) constraints on drone performance and operations, (3) different objectives for various drone services, and (4) alternative delivery modes, as well as some areas for methodological advances and some possible new applications. While much of the research to date has leveraged existing TSP (traveling salesman problem), VRP (vehicle routing problem), and arc routing models, we look forward to new contributions from drone research that use better models of more realistic drone types and new drone applications.

Development of a dummy guided formulation and exact solution method for TSP

A traveling salesman problem (TSP) is a problem whereby the salesman starts from an origin node and returns to it in such a way that every node in the network of nodes is visited once and that the total distance travelled is minimized. An efficient algorithm for the TSP is believed not to exist. The TSP is classified as NP-hard and coming up with an efficient solution for it will imply NP = P . The paper presents a dummy guided formulation for the traveling salesman problem. To do this, all sub-tours in a traveling salesman problem (TSP) network are eliminated using the minimum number of constraints possible. Since a minimum of three nodes are required to form a sub-tour, the TSP network is partitioned by means of vertical and horizontal lines in such a way that there are no more than three nodes between either the vertical lines or horizontal lines. In this paper, a set of all nodes between any pair of vertical lines or horizontal lines is called a block. Dummy nodes are used to connect one block to the next one. The reconstructed TSP is then used to formulate the TSP as an integer linear programming problem (ILP). With branching related algorithms, there is no guarantee that numbers of sub-problems will not explode to unmanageable levels. Heuristics or approximating algorithms that are sometimes used to make quick decisions for practical TSP models have serious economic challenges. The difference between the exact solution and the approximated one in terms of money is very big for practical problems such as delivering household letters using a single vehicle in Beijing, Tokyo, Washington, etc. The TSP model has many industrial applications such as drilling of printed circuit boards (PCBs), overhauling of gas turbine engines, X-Ray crystallography, computer wiring, order-picking problem in warehouses, vehicle routing, mask plotting in PCB production, etc.

Case Article—DeLand Crayon Company: An Application of the Traveling Salesman Problem to Production Scheduling with Sequence-Dependent Setup Times

NP-complete problems such as the traveling salesman problem (TSP) play a prominent role in most advanced undergrad/graduate courses in discrete optimization modeling. Teaching such an important topic from a purely mathematical perspective without discussing specific applications may result in reduced student interest and motivation. The DeLand Crayon Company case introduces students to an application of the TSP to a practical production-planning scenario with realistic data. In crayon production, proper sequencing of colors in production can result in significant savings in total production time. The primary objective of the case is to provide students with hands-on experience in applying the TSP model. The case requires the students to consider variations of the basic model with the goal of encouraging them to think logically through the process of mathematical modeling and to consider the decision-making implications of their models. This case study can be used in undergraduate- and graduate-level courses in linear/integer programming or in production planning. It was implemented as part of a term project in a graduate operations research course with overwhelmingly positive feedback.

Combining Traveling Salesman and Traveling Repairman Problems: A multi-objective approach based on multiple scenarios

This paper analyzes a multi-objective variant of the well-known Traveling Salesman Problem (TSP) and the Traveling Repairman Problem (TRP) in order to address the classical conflict between cost minimization (represented by the TSP) and customer waiting time minimization (represented by the TRP). By simultaneously considering different scenarios with individual travel times, uncertainty in travel data is handled. We interpret each travel time scenario as an individual objective function and introduce deterministic multi-objective counterpart models denoted as the Multi-Objective TSP (MOTSP), Multi-Objective TRP (MOTRP), and the combined Multi-Objective TSP and TRP models (MOTSRP), respectively. Problems with and without additional deadline restrictions are considered, and the complexity status of computing the Pareto fronts of various problem variants for different underlying networks is resolved. As a particularly interesting case, we consider the MOTSRP with deadlines on a line and show that the problem is intractable even in this simple setting. Nevertheless, we propose a Dynamic Programming approach that solves random instances to optimality in reasonable time. Moreover, the computational study additionally evaluates the average complexity of the Line-MOTSRP with deadlines for different numbers of scenarios. The computational study also analyzes the Pareto fronts that are generated for specifically designed extremal instances.