Traveling Salesman Problem With Time Windows Research Articles

The Solution of Traveling Salesman Problem with Time Windows by MILP in Matlab Code

A delivery driver uses his truck to distribute ordered goods from the warehouse (depot) to n customers. Each customer determines the delivery point (by GPS coordinates) and time window for the distribution of goods. This problem can be called the traveling salesman problem with time windows (TSPTW). The objective of the solution is to select the sequence of delivery points so that the travel distance and the total travel time are minimal. The necessary condition is that the deliverer leaves the warehouse, visits all delivery points (where he arrives at the required time windows), and returns to the warehouse. In this article, one exact robust solution of the TSPTW is presented by using mixed integer linear programming implemented in Matlab code. The created algorithm can be used for any number n of customers.

Assessment of the activity scheduling optimization method using real travel data

AbstractNew mobility services are appearing with the support of technological developments. Part of them is related to activity scheduling of individuals and the optimization of their travel patterns. A novel method called Activity Chain Optimization (ACO) is an application of the Traveling Salesman Problem with Time Windows (TSP-TW) extended with additional assumptions about temporal and spatial flexibility of the activities, where the travelers can optimize the total travel time of their daily activity schedule. This paper aims to apply the ACO method and evaluate its performance using a real-world household survey dataset, where activity chains of up to 15 activities during a day are considered. The optimization is developed using the genetic algorithm (GA) metaheuristic with suitable parameters selected and the branch-and-bound exact algorithm. The findings demonstrate that the branch-and-bound solution exhibits superior performance for smaller activity chain sizes, while the GA outperforms computationally for activity chains with a size from nine. However, the GA found the solutions in only 2% of the time compared to the branch-and-bound method. By applying the ACO method, relevant time savings and emission reduction can be achieved for travelers, when realizing daily activities.

A general variable neighborhood search for the traveling salesman problem with time windows under various objectives

The traveling salesman problem with time windows (TSPTW) has wide practical applications in transportation and scheduling operations. We study the TSPTW in the deterministic case as well as under travel time uncertainty. We consider the three classical TSPTW variants with the objectives of minimizing cost, completion time, and tour duration. In addition, a new TSPTW variant that maximizes the minimum slack of a tour, i.e., the smallest time buffer between the arrival time and the end of the time window over all nodes visited on the tour, is introduced. We further address a variant of the TSPTW in which the arc travel times are uncertain. This problem is modeled by means of an uncertainty set, and the goal is to determine a tour that remains feasible in the worst case within a budget stipulating the sum of all travel time deviations from their nominal values. To solve all targeted problem variants, we develop a two-phase general variable neighborhood search (GVNS) that applies an efficient move evaluation approach within the local search. Extensive numerical experiments on benchmark instances from the literature show that our GVNS finds high-quality solutions that are competitive with those obtained by the state-of-the-art heuristics for all problem variants.

Improved Peel-and-Bound: Methods for Generating Dual Bounds with Multivalued Decision Diagrams

Decision diagrams are an increasingly important tool in cutting-edge solvers for discrete optimization. However, the field of decision diagrams is relatively new, and is still incorporating the library of techniques that conventional solvers have had decades to build. We drew inspiration from the warm-start technique used in conventional solvers to address one of the major challenges faced by decision diagram based methods. Decision diagrams become more useful the wider they are allowed to be, but also become more costly to generate, especially with large numbers of variables. In the original version of this paper, we presented a method of peeling off a sub-graph of previously constructed diagrams and using it as the initial diagram for subsequent iterations that we call peel-and-bound. We tested the method on the sequence ordering problem, and our results indicate that our peel-and-bound scheme generates stronger bounds than a branch-and-bound scheme using the same propagators, and at significantly less computational cost. In this extended version of the paper, we also propose new methods for using relaxed decision diagrams to improve the solutions found using restricted decision diagrams, discuss the heuristic decisions involved with the parallelization of peel-and-bound, and discuss how peel-and-bound can be hyper-optimized for sequencing problems. Furthermore, we test the new methods on the sequence ordering problem and the traveling salesman problem with time-windows (TSPTW), and include an updated and generalized implementation of the algorithm capable of handling any discrete optimization problem. The new results show that peel-and-bound outperforms ddo (a decision diagram based branch-and-bound solver) on the TSPTW. We also close 15 open benchmark instances of the TSPTW.

Reinforced Lin–Kernighan–Helsgaun algorithms for the traveling salesman problems

The Traveling Salesman Problem (TSP) is a classical NP-hard combinatorial optimization problem with many practical variants. The Lin–Kernighan–Helsgaun (LKH) algorithm is one of the state-of-the-art local search algorithms for the TSP. LKH-3 is a powerful extension of LKH that can solve many TSP variants. Both LKH and LKH-3 use the candidate sets to improve their efficiency, and have two different methods, α-measure and POPMUSIC, to decide the candidate sets. In this work, we first propose a Variable Strategy Reinforced LKH (VSR-LKH) algorithm, which incorporates three reinforcement learning methods (Q-learning, Sarsa, and Monte Carlo) with LKH, for the TSP. We further propose a new algorithm called VSR-LKH-3 that combines the variable strategy reinforcement learning method with LKH-3 for typical TSP variants, including the TSP with Time Windows (TSPTW) and the Colored TSP (CTSP). The proposed algorithms replace the inflexible traversal operations in LKH and LKH-3 and let the algorithms learn to make a choice at each search step by reinforcement learning. Both LKH and LKH-3, with either the α-measure or the POPMUSIC method, can be significantly improved by our methods. Extensive experiments on 236 widely-used TSP benchmarks with up to 85,900 cities demonstrate the excellent performance of VSR-LKH. VSR-LKH-3 also significantly outperforms the state-of-the-art heuristics for the TSPTW and CTSP.

Unconstrained binary models of the travelling salesman problem variants for quantum optimization

Quantum computing is offering a novel perspective for solving combinatorial optimization problems. To fully explore the possibilities offered by quantum computers, the problems need to be formulated as unconstrained binary models, taking into account limitation and advantages of quantum devices. In this work, we provide a detailed analysis of the travelling salesman problem with time windows (TSPTW) in the context of solving it on a quantum computer. We introduce quadratic unconstrained binary optimization and higher-order binary optimization formulations of this problem. We demonstrate the advantages of edge-based and node-based formulations of the TSPTW problem. Additionally, we investigate the experimental realization of the presented methods on a quantum annealing device. The provided results pave the path for utilizing quantum computer for a variety of real-world tasks which can be cast in the form of travelling salesman problem with time windows.

The Robust Traveling Salesman Problem with Time Windows Under Knapsack-Constrained Travel Time Uncertainty

We study the traveling salesman problem with time windows (TSPTW) under travel time uncertainty—modeled by means of an uncertainty set including all travel time vectors of interest. We consider a knapsack-constrained uncertainty set stipulating a nominal and a peak travel time for each arc and an upper bound [Formula: see text] on the sum of all deviations from the nominal times. Viewing the difference between the peak time and its nominal value as the maximum delay possibly incurred when traversing the corresponding arc, the problem we consider is thus to find a tour that remains feasible for up to [Formula: see text] units of delay. This differs from previous studies on robust routing under travel time uncertainty, which have relied on cardinality-constrained sets and only allow for an upper bound on the number of arcs with peak travel time. We propose an exact algorithm based on column generation and dynamic programming that involves effective dominance rules and an extension of the [Formula: see text]-tour relaxation proposed in the literature for the classical TSPTW. The algorithm is able to solve the robust TSPTW under both knapsack- and cardinality-constrained travel time uncertainty. Extensive computational experiments show that the algorithm is successful on instances with up to 80 customers. In addition, we study the impact of the two uncertainty sets on the trade-off between service quality and cost exhibited by the resulting solutions.

Evaluation of the size of time windows for the travelling salesman problem in delivery operations

A great challenge in operational research is to apply time-efficient algorithms to find the optimal solutions to the travelling salesman problem (TSP) and its many variations. The TSP with time windows (TSPTW) arises due to intense pressure for business to improve customer service. As online shopping becomes more popular, customer satisfaction increases if customers can decide when their orders are delivered to them. Customers may choose a time window, which is defined by an earliest delivery time and a latest delivery time, during which the package is delivered. Delivering packages to multiple customers is a typical TSPTW. One main challenge for a delivery business is to determine the size of the time window (i.e., the difference between the latest and earliest delivery times), which affects delivery cost and customer satisfaction. Although many previous studies investigated the TSPTW, none of those focused on the size of time windows. This study is the first that experiments with different time window sizes and determines their impact on tour duration, customer satisfaction, and solution time of the optimal delivery routes. The experiment results show that increasing the size of the time window decreases tour duration and customer satisfaction and increases solution time. Decreasing the size of the time window increases tour duration and customer satisfaction and decreases solution time. A small solution time is necessary for the scheduling of deliveries to many customers. A large solution time prevents a delivery business from delivering packages using optimal routes, which increases delivery cost and decreases customer satisfaction. The results of this study indicate that a general guideline for business is to allow customers to choose a time window size that is within the cost limit but is sufficiently small to maximize customer satisfaction and optimize delivery routes.

The traveling salesman problem with time windows in postal services

This article focuses on a variant of the travelling salesman problem with time windows (TSPTW) that arises in postal services and parcel deliveries. It differs from the classical TSPTW as follows. First, route duration is a significant concern, as human costs largely exceed vehicle costs, emphasising the importance of reducing waiting time. Second, customers are divided in two categories: Commercial ones with time windows and private customers without time window restrictions, making the NP-hard TSPTW even harder to solve. To support the first statement, we present a multi-objective approach based on a constraint programming formulation of the problem which allows to balance the optimisation of both human and material resources. To address the issue of large-scale sparsely time-constrained instances and reduce the size of the large real-world instances, we also introduce a cluster/solve/sequence approach that relies on a mathematical programming formulation to sequence the customer visits in the final step. This decomposition technique allows to produce high-quality solutions (with an average optimality gap of 1.31%) for industrial problems in less than one minute of computation time.

Unconventional GVNS for Solving the Garbage Collection Problem with Time Windows

GVNS, which stands for General Variable Neighborhood Search, is an established and commonly used metaheuristic for the expeditious solution of optimization problems that belong to the NP-hard class. This paper introduces an expansion of the standard GVNS that borrows principles from quantum computing during the shaking stage. The Traveling Salesman Problem with Time Windows (TSP-TW) is a characteristic NP-hard variation in the standard Traveling Salesman Problem. One can utilize TSP-TW as the basis of Global Positioning System (GPS) modeling and routing. The focus of this work is the study of the possible advantages that the proposed unconventional GVNS may offer to the case of garbage collector trucks GPS. We provide an in-depth presentation of our method accompanied with comprehensive experimental results. The experimental information gathered on a multitude of TSP-TW cases, which are contained in a series of tables, enable us to deduce that the novel GVNS approached introduced here can serve as an effective solution for this sort of geographical problems.

Hybrid simulated annealing and tabu search method for the electric travelling salesman problem with time windows and mixed charging rates

The electric travelling salesman problem with time windows (ETSPTW) is an extension of the well-known travelling salesman problem with time windows (TSPTW). The ETSPTW additionally considers recharging operations of the electric vehicle at identical charging stations. However, different charging technologies used at public or private stations result in different charging times of the electric vehicles. Therefore, this study extends the ETSPTW by additionally considering charging operations at customer locations with different charging rates, called hereafter the electric travelling salesman problem with time windows and mixed charging rates (ETSPTW-MCR). To the best of our knowledge, this is the first study that considers both private and public charging stations for the ETSPTW. In addition to the extended version of the ETSPTW, this paper introduces a new and effective hybrid Simulated Annealing/Tabu Search (SA/TS) algorithm to solve the ETSPTW-MCR problem efficiently. Distinct from the existing hybridization of SA and TS, the proposed hybrid SA/TS algorithm employs efficient search procedures based on the TSPTW restrictions, a modified solution acceptance criterion, and an advanced tabu list structure. Moreover, an improved dynamic programming procedure is integrated to optimally find the charging station visits in shorter computational times. The proposed hybrid SA/TS is tested on several TSPTW and ETSPTW benchmark problems and compared with well-known solution approaches. Results of these experiments show that the proposed algorithm outperforms the other considered competitor algorithms both with regard to solution quality and computational time. Furthermore, 26 new best results are obtained for the ETSPTW instances. In addition, the hybrid algorithm is applied to a new problem set generated for the ETSPTW-MCR by extending the ETSPTW problems found in the literature. Comparisons with the ETSPTW results show that significant distance savings are found for most of the instances by charging the electric vehicle at customer locations. As a result of the computational studies, it should be concluded that the proposed algorithm is capable of finding efficient and more realistic route plans for the electric vehicles.

Comments on: Perspectives on integer programming for time-dependent models

In the paper, “Perspectives on Integer Programming for Time-Dependent models”, Boland and Savelsbergh discuss integer programming (IP) models of optimization problems involving activities that have time restrictions, or resources that vary with time [both aspects are present in a recent paper by Hill et al. (2018)]. Such problems are common in many industrial applications and are often very hard to solve. The Traveling Salesman Problem with Time Windows (TSPTW) is an important “time-dependent model” that combines routing and scheduling issues, and Boland and Savelsbergh illustrate techniques to handle time-dependent models on the TSPTW.

Enhanced discrete bacterial memetic evolutionary algorithm - An efficacious metaheuristic for the traveling salesman optimization

In this paper we present a novel universal metaheuristic, Discrete Bacterial Memetic Evolutionary Algorithm (DBMEA), which is based on the combination of the Bacterial Evolutionary Algorithm and local search techniques, used for solving NP-hard optimization problems. The algorithm was tested on a series of symmetric Traveling Salesman Problems (TSP) and Traveling Salesman Problem with time windows (TSPTW) benchmarks.The size of the symmetric TSP benchmarks went up to 5 000 cities. In all cases the DBMEA algorithm produced optimal or near-optimal solutions and the difference from the known best values was within 0.16%. While for large size problems it was much faster than the Concorde solver, it was found to be slower compared to the Helsgaun-Lin-Kernighan heuristic, which is the most efficient TSP solver method.With some slight modifications the same algorithm was also tested on TSP with time windows (TSPTW) benchmark instances. In most cases the DBMEA procedure found the known best solutions, and it was again the second fastest method compared with the state-of-the-art techniques for the TSPTW.DBMEA is called efficacious because it is a universal method. It can be efficiently applied to various NP-hard optimization problems and, as in all cases, it results in the optimal or a very near-optimal solutions, while its runtime is very predictable in terms of the size of the problem, and the topology of the instance does not affect its runtime significantly. Even though heuristics developed for a particular type of problem might perform better for that restricted class, our novel method proposed here is universally applicable and may be deployed successfully for optimizing other discreet NP-hard graph search and optimization problems as well.

Formulations for Minimizing Tour Duration of the Traveling Salesman Problem with Time Windows

Abstract Traveling Salesman Problem with Time Windows (TSPTW) serves as one of the most important variants of the Traveling Salesman Problem (TSP). The main objective functions expressed in the literature of the TSPTW consist of the following: (1) to minimize total distance travelled (or to minimize total travel time spent on the arcs), (2) to minimize total cost of traveling on the arcs and cost of waiting before services, or (3) to minimize total time passed from depot to depot (tour duration). When the unit cost of traveling and unit cost of waiting appear equal, then one may solve the problem just minimizing tour duration. According to our best knowledge, only one formulation with nonlinear constraints considers the aim of minimizing tour duration for symmetric case. However, for just minimizing tour duration, we haven’t seen any linear formulation for symmetric and any general formulation for asymmetric TSPTW. In this paper, for minimizing tour duration, we propose polynomial size integer linear programming formulation for symmetric and asymmetric TSPTW separately. The performances of our proposed formulations are tested on well-known benchmark instances used to minimize total travel time spent on the arcs. For symmetric TSPTW, our proposed formulation sufficiently and capably finds optimal solutions for all instances (the biggest are with 400 customers) in an extremely short time with CPLEX 12.4. For asymmetric TSPTW, we used instances generated from a real life problem which were used for minimizing total arc cost. We used these instances for minimizing tour duration and observe that, optimal solutions of the most of the instances up to 232 nodes are obtained within seconds. In addition, the proposed formulation for asymmetric case obtained 7 new best known solutions.

Quantitative Analysis of an IPS2 Delivery Planning Approach

The delivery planning of Industrial Product-Service Systems (IPS2) is a complex task. A highly dynamic network of provider, customers and suppliers needs to be managed and coordinated. Due to this complexity, sophisticated IT-support for IPS2 resource planning is required. For this purpose, the adaptive IPS2 planning method (AIPM) for the scheduling of delivery processes has been conceptually developed in previous research. The method is based on a genetic algorithm, in which a population of planning solutions is evaluated with regard to a fitness value to identify the best plans. To be able to conduct quantitative evaluation studies on the effectiveness and efficiency of the planning algorithm, it has been implemented in a prototype of an IPS2-Execution System (IPS2-ES). It became apparent that some modifications of the algorithms were necessary in order to apply the approach to real planning cases. With these modifications, the algorithm was suitable to be used for further evaluation studies.In this paper, the results of these new evaluation studies with the modified algorithm are presented. A benchmarking problem set of the traveling salesman problem with time windows (TSPTW) and a real-life industrial planning setting were analyzed. The results show, that the revised version of the AIPM is capable of solving both problems satisfactorily. In particular, the planning outcomes in the industrial scenario indicate high potential for practical applications in service companies and IPS2 networks.

A variable iterated greedy algorithm for the traveling salesman problem with time windows

This paper presents a variable iterated greedy algorithm for solving the traveling salesman problem with time windows (TSPTW) to identify a tour minimizing the total travel cost or the makespan, separately. The TSPTW has several practical applications in both production scheduling and logistic operations. The proposed algorithm basically relies on a greedy algorithm generating an increasing number of neighboring solutions through the use of the idea of neighborhood change in variable neighborhood search (VNS) algorithms. In other words, neighboring solutions are generated by destructing a solution component and re-constructing the solution again with variable destruction sizes. In addition, the proposed algorithm is hybridized with a VNS algorithm employing backward and forward 1_Opt local searches to further enhance the solution quality. The performance of the proposed algorithm was tested on several benchmark suites from the literature. Experimental results confirm that the proposed algorithm is either competitive to or even better than the best performing algorithms from the literature. Ultimately, new best-known solutions are obtained for 38 out of 125 problem instances of the recently proposed benchmark suite, whereas 15 out of 31 problem instances are also further improved for the makespan criterion.

Efficient constraint handling in electromagnetism-like algorithm for traveling salesman problem with time windows.

The traveling salesman problem with time windows (TSPTW) is a variant of the traveling salesman problem in which each customer should be visited within a given time window. In this paper, we propose an electromagnetism-like algorithm (EMA) that uses a new constraint handling technique to minimize the travel cost in TSPTW problems. The EMA utilizes the attraction-repulsion mechanism between charged particles in a multidimensional space for global optimization. This paper investigates the problem-specific constraint handling capability of the EMA framework using a new variable bounding strategy, in which real-coded particle's boundary constraints associated with the corresponding time windows of customers, is introduced and combined with the penalty approach to eliminate infeasibilities regarding time window violations. The performance of the proposed algorithm and the effectiveness of the constraint handling technique have been studied extensively, comparing it to that of state-of-the-art metaheuristics using several sets of benchmark problems reported in the literature. The results of the numerical experiments show that the EMA generates feasible and near-optimal results within shorter computational times compared to the test algorithms.

New integer linear programming formulation for the traveling salesman problem with time windows: minimizing tour duration with waiting times

The travelling salesman problem, being one of the most attractive and well-studied combinatorial optimization problems, has many variants, one of which is called ‘travelling salesman problem with Time Windows (TSPTW)’. In this problem, each city (nodes, customers) must be visited within a time window defined by the earliest and the latest time. In TSPTW, the traveller has to wait at a city if he/she arrives early; thus waiting times directly affect the duration of a tour. It would be useful to develop a new model solvable by any optimizer directly. In this paper, we propose a new integer linear programming formulation having O(n2) binary variables and O(n2) constraints, where (n) equals the number of nodes of the underlying graph. The objective function is stated to minimize the total travel time plus the total waiting time. A computational comparison is made on a suite of test problems with 20 and 40 nodes. The performances of the proposed and existing formulations are analysed with respect to linear programming relaxations and the CPU times. The new formulation considerably outperforms the existing one with respect to both the performance criteria. Adaptation of our formulation to the multi-traveller case and some additional restrictions for special situations are illustrated.

New State-Space Relaxations for Solving the Traveling Salesman Problem with Time Windows

The traveling salesman problem with time windows (TSPTW) is the problem of finding in a weighted digraph a least-cost tour starting from a selected vertex, visiting each vertex of the graph exactly once according to a given time window, and returning to the starting vertex. This n𝒫-hard problem arises in routing and scheduling applications. This paper introduces a new tour relaxation, called ngL-tour, to compute a valid lower bound on the TSPTW obtained as the cost of a near-optimal dual solution of a problem that seeks a minimum-weight convex combination of nonnecessarily elementary tours. This problem is solved by column generation. The optimal integer TSPTW solution is computed with a dynamic programming algorithm that uses bounding functions based on different tour relaxations and the dual solution obtained. An extensive computational analysis on basically all TSPTW instances (involving up to 233 vertices) from the literature is reported. The results show that the proposed algorithm solves all but one instance and outperforms all exact methods published in the literature so far.

A Time Bucket Formulation for the Traveling Salesman Problem with Time Windows

The traveling salesman problem with time windows (TSPTW) is the problem of finding a minimum-cost path visiting a set of cities exactly once, where each city must be visited within a given time window. We present an extended formulation for the problem based on partitioning the time windows into subwindows that we call buckets. We present cutting planes for this formulation that are computationally more effective than the ones known in the literature because they exploit the division of the time windows into buckets. To obtain a good partition of the time windows, we propose an iterative linear programming (LP)-based procedure that may produce buckets of different sizes. The LP relaxation of this formulation yields strong lower bounds for the TSPTW and provides a good starting point for our branch-and-cut algorithm. We also present encouraging computational results on hard test problems from the literature, namely, asymmetric instances arising from a practical scheduling application, as well as randomly generated symmetric instances. In particular, we solve a number of previously unsolved benchmark instances.