Multiple Traveling Salesman Problem Research Articles

Sustainable and Adequate Home-Care Logistics Including Precedence Constraints

Abstract Development of community home-based care for servicing the growing number of older adults with declining functional capacities is the priority of EU Member states to keep sustainable social systems of long-term care. For optimal utilisation of human resources, it is necessary to balance the activities of caregivers and other professionals who perform services in a network and evaluate their services in combination with the strategic development of networks. The number of users is increasing. Since the optimal solution for servicing is changing, the system should dynamically adapt to the new requirements of building capacities, infrastructure and human resources. The purpose of this paper is to develop an innovative model to support these needs with a minimum number of caregivers who would balance their activities within the time window of their working day and combining investments and organisational decisions to achieve the above objectives at the minimal net present value of expenses. The number of caregivers and scheduling is decided by using a nonlinear assembly balancing algorithm in the manner of assembly-line balancing. This algorithm is combined with the multiple Travelling Salesman Problem and network investments planning, which provides a new tool written in the R programming language for decision making by evaluating the net present value of services and combining it with the investment costs to maintain a sustainable social system that provides adequate services. The case of planning in the East Slovenian region is presented.

An External Definition of the One-Hot Constraint and Fast QUBO Generation for High-Performance Combinatorial Clustering

Recently, a clustering method using a combinatorial optimization problem, called combinatorial clustering, has been drawing attention due to the rapid spreads of quantum annealing and simulated annealing. Combinatorial clustering is performed by minimizing an objective function under a condition to satisfy a one-hot constraint. The objective function and the constraint function are generally formulated to a unified objective function of a QUBO (Quadratic Unconstrained Binary Optimization) problem using the method of the Lagrange multiplier. The coefficients of the QUBO function can be represented by a square matrix, which is called the QUBO matrix. Although the Lagrange multiplier needs to be large enough to avoid violating the constraint, it is usually hard to be set appropriately due to the limitation of the bit precision. For example, the latest quantum annealer can handle values represented by only six or fewer bits. Even conventional computing systems cannot control the larger value of the Lagrange multiplier as the number of data points increases. Besides, the execution time for combinatorial clustering increases exponentially as the problem size increases. This is because the time for the QUBO matrix generation is long and a dominant factor of the total execution time when the problem size is large. To solve these problems, this paper proposes combinatorial clustering that overcomes the limitation of the method of the Lagrange multiplier. The proposed method uses a QUBO solver that can externally define the one-hot constraint independent from the objective function, and the externally-defined constraint is satisfied by the operations with multiple bit-flips. As the QUBO function contains only the objective function, the method of the Lagrange multiplier is not necessary. The proposed method can optimize the objective function sufficiently even when the problem size is large. Since the constraint function is not included in the QUBO function, the proposed method also reduces the time for the QUBO matrix generation. The experimental results obtained using the artificial and real data show that the proposed method can improve the quality of annealing-based clustering results in all data sets. The experimental results also clarify that the quality of the proposed method is almost equal to or better than that of quasi-optimal clustering methods such as K-means. The evaluation using the multiple traveling salesman problem shows that the proposed method can obtain shorter tour lengths than the conventional annealing-based clustering in all 13 cases and K-means++ in 6 out of 12 cases with a significant difference. Furthermore, the proposed method can accelerate the execution time for combinatorial clustering because there is no need to calculate the coefficients of the constraint function.

Multiuser Oriented Multi-UAV Mission Assignment With Cooperative Information Sharing

Due to the overlap of different users’ information requirements for reconnaissance targets, when unmanned aerial vehicles (UAVs), serving multiple users, perform field reconnaissance missions, it is likely to cause repetitive visit to the same targets. To deal with this problem, a reasonable cooperative mission assignment algorithm is needed. By introducing information sharing mechanism with air communication among UAVs, this letter models the mission assignment problem as a vehicle routing problem with air information sharing (VRPAIS), which can efficiently avoid the repetitive visit of targets. An improved piecewise genetic algorithm with integer splitting is proposed to solve VRPAIS in polynomial time. Simulation results show that in the case of high overlap (about 80%) of information requirements, the proposed method is much more energy efficient compared with the traditional Multiple Traveling Salesman Problem (MTSP) based method without information sharing, with an average energy savings rate of 34.2%.

The traveling salesman puts-on a hard hat – Tower crane scheduling in construction projects

Effective use of site equipment is an essential aspect of construction projects. In particular, tower cranes are critical in large scale projects. This research aims to optimize the cranes’ usage in construction projects by efficiently scheduling the sequence of activities that are processed by each crane on the site while addressing several constraints such as crane collisions and tasks precedence constraints. An integer linear program is developed that is based on the Multiple Travelling Salesmen Problem (MTSP). Two sets of valid inequalities have been added to improve the tractability of the model. This model is then tested on a real-world case study of a large construction site with two tower cranes. In addition, heuristic approaches are introduced to increase the tractability of the model and are compared to the integer program in terms of solution quality and computation time by generating a large data-set of problems with different sizes. The heuristics are based on the cluster-first, route-second idea, where an optimization-based clustering method is used to pre-assign activities to cranes.

Experimental Study of a Hybrid Genetic Algorithm for the Multiple Travelling Salesman Problem

The multiple travelling salesman problem (MTSP), an extension of the well-known travelling salesman problem (TSP), is studied here. In MTSP, starting from a depot, multiple salesmen require to visit all cities so that each city is required to be visited only once by one salesman only. It is NP-hard and is more complex than the usual TSP. So, exact optimal solutions can be obtained for smaller sized problem instances only. For large-sized problem instances, it is essential to apply heuristic algorithms, and amongst them, genetic algorithm is identified to be successfully deal with such complex optimization problems. So, we propose a hybrid genetic algorithm (HGA) that uses sequential constructive crossover, a local search approach along with an immigration technique to find high-quality solution to the MTSP. Then our proposed HGA is compared against some state-of-the-art algorithms by solving some TSPLIB symmetric instances of several sizes with various number of salesmen. Our experimental investigation demonstrates that the HGA is one of the best algorithms.

Makespan Minimizing on Multiple Travel Salesman Problem With a Learning Effect of Visiting Time

The multiple traveling salesman problem (MTSP) involves the assignment and sequencing procedure simultaneously. The assignment of a set of nodes to each visitors and determining the sequence of visiting of nodes for each visitor. Since specific range of process is needed to be carried out in nodes in commercial environment, several factors associated with routing problem are required to be taken into account. This research considers visitors’ skill and category of customers which can affect visiting time of visitors in nodes. With regard to learning-by-doing, visiting time in nodes can be reduced. And different class of customers which are determined based on their potential purchasing of power specifies that required time for nodes can be vary. So, a novel optimization model is presented to formulate MTSP, which attempts to ascertain the optimum routes for salesmen by minimizing the makespan to ensure the balance of workload of visitors. Since this problem is an NP-hard problem, for overcoming the restriction of exact methods for solving practical large-scale instances within acceptable computational times. So, Artificial Immune System (AIS) and the Firefly (FA) metaheuristic algorithm are implemented in this paper and algorithms parameters are calibrated by applying Taguchi technique. The solution methodology is assessed by an array of numerical examples and the overall performances of these metaheuristic methods are evaluated by analyzing their results with the optimum solutions to suggested problems. The results of statistical analysis by considering 95% confidence interval for calculating average relative percentage of deviation (ARPD) reveal that the solutions of proposed AIS algorithm has less variation and Its’ confidence interval of closer than to zero with no overlapping with that of FA. Although both proposed metaheuristics are effective and efficient in solving small-scale problems, in medium and large scales problems, AIS had a better performance in a shorter average time. Finally, the applicability of the suggested pattern is implemented in a case study in a specific company, namely Kalleh.

Using genetic algorithm to solve multiple traveling salesman problem and considering Carbon emissions

Objectives: The Multiple Travelling Salesman problem is a complex combinatorial optimization problem which is a variance of the Traveling Salesman Problem,where a lot of salesmen are utilized in the solution. In this work a cold chain logistics and route optimization model with minimum transport cost, carbon cost and Refrigeration cost are constructed. Methods: A genetic algorithm is then proposed to solve for the Multiple Travelling Salesman Problem with time windows while transport cost, carbon emission cost and refrigeration cost is minimized. Findings: It was observed that the algorithm evolved towards the direction of the optimal value of the fitness function. Novelty: There are a number of studies that considered tournament selection strategy but just a few have applied genetic algorithm considering insertion method to solve a Multiple Travelling salesman Problem. This study uses insertion method to obtain optimal solution. Also, the researcher considered time windows, transport cost, carbon emission cost and refrigeration cost. Keywords: Genetic algorithm method; cold-logistics; multiple travelling salesman problem

Load Balance and Trajectory Design in Multi-UAV Aided Large-Scale Wireless Rechargeable Networks

Due to the small size, low cost, and autonomous nature, unmanned aerial vehicles (UAVs) have attracted growing attention in improving traditional networks’ performance. This paper considers a multi-UAV enabled large-scale wireless recharge networks, where a group of UAVs is dispatched as mobile wireless power transfer, and information collection systems to serve a set of ground low-power sensor nodes. Specifically, the UAVs employ radio frequency (RF) wireless power transfer (WPT) to supply energy towards the sensor nodes, and the communication services are available via wireless information transfer (WIT) systems. To achieve balanced energy consumption among UAVs, we maximize the energy utilization efficiency of UAVs, and minimize the communication delay by optimizing the trajectory jointly with constraints of the energy capacity, and the area of the target region. The formulated problem is a mixed-integer programming problem that is a variation of multiple traveling salesman problem. Thus we present a heuristic algorithm that combines the evolutionary algorithm, and variable neighborhood search to achieve the optimal visited sequence of the sensor nodes. Finally, extensive numerical results are provided to evaluate the performance of the proposed algorithm. It draws new insights on the estimation of the feasibility of the given UAVs whose energy capacity is limited.

BESLEYİCİ OTOBÜS ROTALARININ ÖN PLANLAMASI İÇİN İKİ DÜZEYLİ BİR ALGORİTMA ÖNERİSİ

A sustainable urban transportation system uses different classes of transportation modes whose services should be well integrated. The Feeder Bus Route Network Problem (FBRNDP) is an important part of this integration. FBRNDP primarily deals with the provision of access to an existing mainline movement through feeder transit system usually to expand it’s the service coverage. The multiple traveling salesman problem (MTSP) has similar properties with FBRNDP, thus, making the formulation of MTSP to be adoptable for feeder bus routes. In this study, a bi-level heuristic algorithm is developed to solve this problem by clustering demand nodes around nearest destination and using genetic algorithm (GA) based on fixed start MTSP to optimize the shortest distance the salesmen will have to travel to cover the service area. The algorithm compares well to the results of a case study found in literature and shows a promising way of designing feeder bus routes strictly based on the shortest distance and variation of the number of routes required. The proposed method can be useful in the initial planning of an integrated transit system and it may serve as a seed solution in a multi-objective optimization.

Three-Step Metaheuristic for the Multiple Objective Multiple Traveling Salesmen Problem

In this article, the multiple objective multiple Traveling Salesman Problem is considered. m salesmen have to visit n cities to perform some tasks each taking a given processing time. Two objectives are considered: balance the working loads of different salesmen and minimize their total traveled distance. To solve this NP-hard problem, a 3-phase metaheuristic was developed. In the first 2 phases, the principle of center of mass and a neighborhood search technique are used to assign the n cities to the m salesmen. In the third phase, a TSP solver was used to generate an optimal tour to every salesman using its assigned cities in phase 2. The metaheuristic was tested using TSP benchmarks of different sizes. The obtained results showed almost optimal load balancing for all tested instances and optimal tours in term of total traveled distances. A conducted comparison study showed that the proposed metaheuristic outperforms a recently published clustering algorithm for the workload objective.

Task assignment strategy for multi-robot based on improved Grey Wolf Optimizer

Multi-robot task allocation (MRTA) is the basis of a multi-robot system to perform tasks automatically, which directly affects the execution efficiency of the whole system. A distributed cooperative task allocation strategy based on the algorithm of the improved Grey Wolf Optimizer (IGWO) was proposed to quickly and effectively plan the cooperative task path with a large number of working task points. The MRTA problem was transformed into multiple traveling salesman problems (MTSPs), and the task target points were clustered by the K-means clustering algorithm and divided into several traveling salesman problems (TSPs). The Grey Wolf Optimizer (GWO) was improved by the Kent chaotic algorithm to initialize the population and enhance the diversity of the population. Furthermore, an adaptive adjustment strategy of the control parameter $$\vec {a}$$ was proposed to balance exploration and exploitation. The individual speed and position updates in PSO were introduced to enable the gray wolf individual to preserve its optimal location information and accelerate the convergence speed. The IGWO was used to solve the optimal solutions to multiple TSP problems. Finally, the optimal solution space was integrated to get the optimal solution of MTSP, and 16 international classical test functions simulated the IGWO. The results showed that the IGWO algorithm has faster convergence speed and higher accuracy. The task allocation strategy is reasonable, with roughly equal path length, small planning cost, fast convergence speed, and excellent stability.

In-port multi-ship routing and scheduling problem with draft limits

ABSTRACT This paper addresses an in-port multi-ship routing and scheduling problem in maritime transportation. The key objective is to find an optimal routing and schedule for multiple ships that are planned to pick up or deliver cargos located in the various terminals of a port with different draft limits. Using the multiple traveling salesman problem (mTSP) as a base, we formulate a multi-ship routing and scheduling model with draft limits and time windows constraints. To solve the developed model, a two-stage solution method based on dynamic programming and branch-and-bound algorithms is developed. The solution approach is applied to 41 real-sized problem instances. Our experiments revealed that the proposed method outperforms CPLEX by accurately solving the problem instances in a reasonable time. The formulated problem and presented solution can be implemented in fleet operations of autonomous ships in smart maritime transportation systems of the future.

A reinforcement learning approach for optimizing multiple traveling salesman problems over graphs

This paper proposes a learning-based approach to optimize the multiple traveling salesman problem (MTSP), which is one classic representative of cooperative combinatorial optimization problems. The MTSP is interesting to study, because the problem arises from numerous practical applications and efficient approaches to optimize the MTSP can potentially be adapted for other cooperative optimization problems. However, the MTSP is rarely researched in the deep learning domain because of certain difficulties, including the huge search space, the lack of training data that is labeled with optimal solutions and the lack of architectures that extract interactive behaviors among agents. This paper constructs an architecture consisting of a shared graph neural network and distributed policy networks to learn a common policy representation to produce near-optimal solutions for the MTSP. We use a reinforcement learning approach to train the model, overcoming the requirement data labeled with ground truth. We use a two-stage approach, where reinforcement learning is used to learn an allocation of agents to vertices, and a regular optimization method is used to solve the single-agent traveling salesman problems associated with each agent. We introduce a S-samples batch training method to reduce the variance of the gradient, improving the performance significantly. Experiments demonstrate our approach successfully learns a strong policy representation that outperforms integer linear programming and heuristic algorithms, especially on large scale problems.

A Two Stage Method for the Multiple Traveling Salesman Problem

The variation of the traveling salesman problem (TSP) with multiple salesmen (m-TSP) has been studied for many years resulting in diverse solution methods, both exact and heuristic. However, the high difficulty level on finding optimal (or acceptable) solutions has opposed the many efforts of doing so. The proposed method regards a two stage procedure which implies a modified version of the p-Median Problem (PMP) alongside the TSP, making a partition of the nodes into subsets that will be assigned to each salesman, solving it with Branch & Cut (B&C), in the first stage. This is followed by the routing, applying an Ant Colony Optimization (ACO) metaheuristic algorithm to solve a TSP for each subset of nodes. A case study was reviewed, detailing the positioning of five vehicles in strategic places in the Mexican Republic.

Novel Graph Model for Solving Collision-Free Multiple-Vehicle Traveling Salesman Problem Using Ant Colony Optimization

In this paper, a novel graph model to figure Collision-Free Multiple Traveling Salesman Problem (CFMTSP) is proposed. In this problem, a group of vehicles start from different nodes in an undirected graph and must visit each node in the graph, following the well-known Traveling Salesman Problem (TSP) fashion without any collision. This paper’s main objective is to obtain free-collision routes for each vehicle while minimizing the traveling time of the slowest vehicle. This problem can be approached by applying speed to each vehicle, and a novel augmented graph model can perform it. This approach accommodates not only the position of nodes and inter-node distances, but also the speed of all the vehicles is proposed. The proposed augmented graph should be able to be used to perform optimal trajectories, i.e., routes and speeds, for all vehicles. An ant colony optimization (ACO) algorithm is used on the proposed augmented graph. Simulations show that the algorithm can satisfy the main objective. Considered factors, such as limitation of the mission successfulness, i.e., the inter-vehicle arrival time on a node, the number of vehicles, and the numbers of vehicles and edges of the graph are also discussed.

TWO PHASE HEURISTIC ALGORITHM (TPHA) PADA MULTIPLE TRAVELLING SALESMAN PROBLEM (MTSP) DAN IMPLEMENTASI PROGRAMNYA

Multiple Traveling Salesman Problem (MTSP) is one variant of Traveling Salesman Problem (TSP) which involves several salesmen in making a trip to visit several customers. In this article, the Two-Phase Heuristic Algorithm (TPHA) is used to solve MTSP problems. The algorithm classifies customers into several regions using the K-Means algorithm, which will then find a route solution for each region using a genetic algorithm. The MTSP problems that were resolved using TPHA were implemented into the Borland Delphi 7.0 programming language. Application testing was conducted using 21, 32, and 46 point cases.

Generalised formulations for minimum distance trajectory in patrolling problems

In this study, three general formulations are presented for trajectory optimisation in patrolling problems. In the traditional patrolling problem, some basic assumptions are made (often implicitly). For example, it is known how many robots and how many starting depots exist. Furthermore, the starting depots are assumed to be pre-specified. Each of the three formulations provided here relaxes some (or all) of these assumptions, hence generalising the patrolling problem. A group of robots are supposed to travel through a number of nodes (viewpoints) in such an order so that the total travel distance is minimised. This problem is, in fact, a variant of the Travelling Salesman Problem and is called Multidepot multiple Travelling Salesman Problem. The effectiveness of the approach is demonstrated by comparing the results with those in the literature.

Dynamic Multi-Objective Auction-Based (DYMO-Auction) Task Allocation

In this paper, we address the problem of online dynamic multi-robot task allocation (MRTA) problem. In the existing literature, several works investigated this problem as a multi-objective optimization (MOO) problem and proposed different approaches to solve it including heuristic methods. Existing works attempted to find Pareto-optimal solutions to the MOO problem. However, to the best of authors’ knowledge, none of the existing works used the task quality as an objective to optimize. In this paper, we address this gap, and we propose a new method, distributed multi-objective task allocation approach (DYMO-Auction), that considers tasks’ quality requirement, along with travel distance and load balancing. A robot is capable of performing the same task with different levels of perfection, and a task needs to be performed with a level of perfection. We call this level of perfection quality level. We designed a new utility function to consider four competing metrics, namely the cost, energy, distance, type of tasks. It assigns the tasks dynamically as they emerge without global information and selects the auctioneer randomly for each new task to avoid the single point of failure. Extensive simulation experiments using a 3D Webots simulator are conducted to evaluate the performance of the proposed DYMO-Auction. DYMO-Auction is compared with the sequential single-item approach (SSI), which requires global information and offline calculations, and with Fuzzy Logic Multiple Traveling Salesman Problem (FL-MTSP) approach. The results demonstrate a proper matching with SSI in terms of quality satisfaction and load balancing. However, DYMO-Auction demands 20% more travel distance. We experimented with DYMO-Auction using real Turtlebot2 robots. The results of simulation experiments and prototype experiments follow the same trend. This demonstrates the usefulness and practicality of the proposed method in real-world scenarios.

PENERAPAN ARTIFICIAL FISH SWARM ALGORITHM (AFSA) PADA MULTIPLE TRAVELLING SALESMAN PROBLEMS (m-TSP)

Selling or trading in the industrial 4.0 era as it can now be done by opening a shop online. Therefore, shopping at this time can also be done online also, so that the online shop owners require orders that do not allow for Cash On Delivery (COD) transactions using package delivery services. This research discusses about finding a solution for good shipping with a minimum total mileage of several couriers at PT. Titipan Kilat District Banyuwangi uses AFSA as a settlement algorithm. The experimental process is carried out by using several parameter values to determine the parameters that affect the final solution. Each parameter will be tested with a maximum of 1000 iterations, then the best results will be tested again with a maximum iteration of 2000, and 5000 and will be compared with the original distance traveled by the couriers. The final solution offered in the form of a delivery route by three couriers with the total distance (Z) of the third courier is 87.28 Km with the smallest iteration value reaching the local minimum in iteration 1169. Keywords: artificial fish swarm algorithm (afsa), multiple travelling salesman problem (m-tsp), route, total mileage.

Particle Swarm Optimization for Cooperative Multi-Robot Task Allocation: A Multi-Objective Approach

This letter presents a new Multi-Objective Particle Swarm Optimization (MOPSO) approach to a Cooperative Multi-Robot Task Allocation (CMRTA) problem, where the robots have to minimize the total team cost and, additionally, balance their workloads. We formulate the CMRTA problem as a more complex variant of multiple Travelling Salesman Problems (mTSP) and, in particular, address how to minimize the total travel distance of the entire robot team, as well as how to minimize the highest travel distance of an individual robot. The proposed approach extends the standard single-objective Particle Swarm Optimization (PSO) to cope with the multiple objectives, and its novel feature lies in a Pareto front refinement strategy and a probability-based leader selection strategy. To validate the proposed approach, we first use three benchmark functions to evaluate the performance of finding the true Pareto fronts in comparison with four existing well-known algorithms in continuous spaces. Afterwards, we use six datasets to investigate the task allocation mechanisms in dealing with the CMRTA problem in discrete spaces.