Repairman Problem Research Articles

A GRASP with iterated local search for the traveling repairman problem with profits

The traveling repairman problem with profits is addressed.A new mixed-integer linear model is developed for the problem.A simple metaheuristic algorithm is developed for its solution.The algorithm integrates GRASP with ILS.46 best results are improved for the TRPP instances. Traveling Repairman Problem with profits (TRPP) is an extension of the Traveling Repairman Problem (TRP), where there is not an obligation to visit all vertices. A time-dependent profit is associated with each vertex and the objective is to maximize the total collected revenue. In this paper, we initially develop a new mixed-integer linear model capable of solving small size instances for the TRPP. To solve medium and large size instances, a simple and effective metaheuristic algorithm (GRASP-ILS) is introduced for the TRPP, which combines Greedy Randomized Adaptive Search Procedure (GRASP) for initial solution construction and Iterated Local Search (ILS) with an adaptive perturbation mechanism for solution improvement. The proposed GRASP-ILS is tested on the TRPP benchmark instances derived from the literature. The results indicate that the developed GRASP-ILS can produce efficient and effective solutions for the TRPP. Particularly, it has been observed that the developed GRASP-ILS has improved the best solutions for 46 instances and obtained all the best known results for the remaining instances in a reasonable computation time.

Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems

We consider the problem of constructing optimal decision trees: given a collection of tests that can disambiguate between a set of m possible diseases, each test having a cost, and the a priori likelihood of any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? This problem has been studied in several works, with O(log m)-approximations known in the special cases when either costs or probabilities are uniform. In this paper, we settle the approximability of the general problem by giving a tight O(log m)-approximation algorithm. We also consider a substantial generalization, the adaptive traveling salesman problem. Given an underlying metric space, a random subset S of vertices is drawn from a known distribution, but S is initially unknown—we get information about whether any vertex is in S only when it is visited. What is a good adaptive strategy to visit all vertices in the random subset S while minimizing the expected distance traveled? This problem has applications in routing message ferries in ad hoc networks and also models switching costs between tests in the optimal decision tree problem. We give a polylogarithmic approximation algorithm for adaptive TSP, which is nearly best possible due to a connection to the well-known group Steiner tree problem. Finally, we consider the related adaptive traveling repairman problem, where the goal is to compute an adaptive tour minimizing the expected sum of arrival times of vertices in the random subset S; we obtain a polylogarithmic approximation algorithm for this problem as well.

The A Priori Traveling Repairman Problem

The field of a priori optimization is an interesting subfield of stochastic combinatorial optimization that is well suited for routing problems. In this setting, there is a probability distribution over active sets, vertices that have to be visited. For a fixed tour, the solution on an active set is obtained by restricting the solution on the active set. In the well-studied a priori traveling salesman problem, the goal is to find a tour that minimizes the expected length. In the a priori traveling repairman problem (TRP), the goal is to find a tour that minimizes the expected sum of latencies. In this paper, we study the uniform model, where a vertex is in the active set with probability p independently of the other vertices, and give the first constant-factor approximation for a priori TRP.

Minimizing Latency of Capacitated k-Tours

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) (Chaudhuri et al. in 44th IEEE-FOCS, pp 36–45, 2003; Post and Swamy in 26th ACM-SIAM SODA, pp 512–531, 2015) to the capacitated vehicle setting, and while it has been previously studied (Lysgaard and Wholk in Eur J Oper Res 236(3):800–810, 2014), no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar (Approximation, randomization, and combinatorial optimization, algorithms and techniques, pp 72–83, 2004), and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it (Post and Swamy in 26th ACM-SIAM SODA, pp 512–531, 2015). With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem.

New integer programming formulation for multiple traveling repairmen problem

Abstract: The multiple traveling repairman problem (kTRP) is a generalization of the traveling repairman problem which is also known as the minimum latency problem and the deliveryman problem. In these problems, waiting time or latency of a customer is defined as the time passed from the beginning of the travel until this customer's service completed. The objective is to find a Hamiltonian Tour or a Hamiltonian Path that minimizes the total waiting time of customers so that each customer is visited by one of the repairmen. In this paper, we propose a new mixed integer linear programming formulation for the multiple traveling repairman problem where each repairman starts from the depot and finishes the journey at a given node. In order to see the performance of the proposed formulation against existing formulations, we conduct computational analysis by solving benchmark instances appeared in the literature. Computational results show that proposed model is extremely effective than the others in terms of CPU times.

Finding optimal tour schedules on transportation paths under extended time window constraints

This paper addresses time-critical routing on a given path under release dates and deadline restrictions, while the minimization of the total weighted completion time is pursued. Since there may be destinations with flexible picking area resources that enable a delivery of goods before the defined release date at no additional costs, the well-known Line-Traveling Repairman problem (Line-TRPTW) is extended into the Line-TRPeTW. The Line-TRPeTW has various practical applications, such as, for example, the tour planning of an inland container ship along a river or that of a vehicle along a coastline. Although the feasibility variant of the Line-TRPTW/Line-TRPeTW is known to be strongly $${\mathcal {NP}}$$NP-hard, a first practically applicable Branch&Bound procedure is generated. By making use of different dominance rules and lower bounds, a comprehensive computational study proves that instances comprising up to 200 requests and locations are solved to optimality in reasonable time. Moreover, the paper analyzes the complexity of the simpler variant with relaxed release dates at all customer locations. This relaxed variant provides tight lower bounds. Furthermore, the complexity analysis shows that the relaxed problem variant is binary $${\mathcal {NP}}$$NP-hard even if deadlines are ignored, but can be efficiently solved by a specific Dynamic Programming approach that runs in pseudo-polynomial time.

Online Algorithms for the Linear Tape Scheduling Problem

Even in today’s world of increasingly faster storage technologies, magnetic tapes continue to play an essential role in the market. Yet, they are often overlooked in the literature, despite the many changes made to the underlying tape architecture since they were conceived. In this article, we introduce the LINEAR TAPE SCHEDULING PROBLEM (LTSP), which aims to identify scheduling strategies for read and write operations in single-tracked magnetic tapes that minimize the overall response times for read requests. Structurally, LTSP has many similarities with versions of the Travelling Repairman Problem and of the Dial-a-Ride Problem restricted to the real line. We investigate several properties of LTSP and show how they can be explored in the design of algorithms for the online version of the problem. Computational experiments show that the resulting strategies deliver very satisfactory scheduling plans, which in most cases are clearly superior (potentially differing by one order of magnitude) to those produced by a strategy currently used in the industry.

The Repairman Problem Revisited

The Repairman Problem Revisited

3-approximation Algorithm for the Travelling Repairman Problem with Unit Time-windows

The travelling repairman problem (TRP) is a scheduling problem in which a repairman must visit locations to perform some task. Each location has a time window in which the repairman is allowed to arrive. The objective of this problem is to maximize the number of tasks performed. In this paper, we consider a special case in which all the locations are on a straight line, the tasks have no processing time, and all time-windows are of unit length. We present an improvement on the analysis of the 4-approximation algorithm presented by S. L. P erez P erez et al. in 2014.

Traveling Repairman Problem for Optical Network Recovery to Restore Virtual Networks After a Disaster [Invited

Virtual networks mapped over a physical network can suffer disconnection and/or outage due to disasters. After a disaster occurs, the network operator should determine a repair schedule and then send repairmen to repair failures following the schedule. The schedule can change the overall effect of a disaster by changing the restoration order of failed components. In this study, we introduce the traveling repairman problem to help the network operator make the schedule after a disaster. We measure the overall effect of a disaster from the damage it caused, and we define the damage as the numbers of disconnected virtual networks, failed virtual links, and failed physical links. Our objective is to find an optimal schedule for a repairman to restore the optical network with minimum damage. We first state the problem; then a mixed integer linear program (MILP) and three heuristic algorithms, namely dynamic programming (DP), the greedy algorithm (GR), and simulated annealing (SA), are proposed. Finally, simulation results show that the repair schedules using MILP and DP results get the least damage but the highest complexity; GR gets the highest damage with the lowest complexity, while SA has a good balance between damage and complexity.

Solving the traveling repairman problem on a line with general processing times and deadlines

Abstract This paper resolves the complexity status of the well-known Traveling Repairman Problem on a line (Line-TRP) with general processing times at the request locations and deadline restrictions. It has long remained an open research question whether an exact solution procedure with pseudo-polynomial running time can be developed for this version of the Traveling Repairman Problem that was known to be at least binary N P -hard. The presented proof of strong N P -completeness of the problem is provided by a reduction from 3-PARTITION. Since recent literature provides significant new results for further variants of the Line-TRP and the Line-TSP, a brief updated overview of the complexity status of the different variants is given. Another major contribution is that a practically applicable exact best-first search Branch&Bound approach that optimally solves instances of real-world size in reasonable time is proposed. By applying sophisticated dominance rules as well as lower bounds, the number of enumerated partial solutions is kept limited. The efficiency of the new approach and the applied instruments is validated by a computational study.

On combining machine learning with decision making

We present a new application and covering number bound for the framework of Learning with Operational Costs (MLOC), which is an exploratory form of decision theory. The MLOC framework incorporates knowledge about how a predictive model will be used for a subsequent task, thus combining machine learning with the decision that is made afterwards. In this work, we use the MLOC framework to study a problem that has implications for power grid reliability and maintenance, called the Machine Learning and Traveling Repairman Problem (ML&TRP). The goal of the ML&TRP is to determine a route for a crew, which repairs nodes on a graph. The repair crew aims to minimize the cost of failures at the nodes, but as in many real situations, the failure probabilities are not known and must be estimated. The MLOC framework allows us to understand how this uncertainty influences the repair route. We also present new covering number generalization bounds for the MLOC framework.

Productivity improvement by reduction of idle time through application of queuing theory

In Lean Six Sigma one of the primary objectives is to identify and eliminate the waiting or idle time for enhancing productivity and reducing production cost. However, in some situations it is better to replace ‘elimination’ by ‘optimization’ due to the presence of conflicting requirements. Optimization techniques in the form of queuing models are quite useful to resolve this conflict. In an Indian textile manufacturing plant, the waiting (idle) time in the winding process is identified as a reason for high production cost. In this situation, it is to be determined as to how many machines are to be assigned to an operator for reducing waiting or idle time of both the operators and the machines leading to reduced production cost. The situation can be compared and contrasted to a machine repairman problem aiming for optimizing the number of machines per operator. The various process parameters have been estimated and suitably modified or derived to serve as inputs to the Queuing Model. Sensitivity analysis has been carried out to identify and control the significant factors. The objective of this paper is to demonstrate through an application of queuing models the procedure of optimizing the waiting or idle time. Consequently, the focus is on the application of optimization techniques like Queuing Theory and Simulation while implementing lean six sigma.

Branch-and-price-and-cut for the multiple traveling repairman problem with distance constraints

In this paper, we extend the multiple traveling repairman problem by considering a limitation on the total distance that a vehicle can travel; the resulting problem is called the multiple traveling repairmen problem with distance constraints (MTRPD). In the MTRPD, a fleet of identical vehicles is dispatched to serve a set of customers. Each vehicle that starts from and ends at the depot is not allowed to travel a distance longer than a predetermined limit and each customer must be visited exactly once. The objective is to minimize the total waiting time of all customers after the vehicles leave the depot. To optimally solve the MTRPD, we propose a new exact branch-and-price-and-cut algorithm, where the column generation pricing subproblem is a resource-constrained elementary shortest-path problem with cumulative costs. An ad hoc label-setting algorithm armed with bidirectional search strategy is developed to solve the pricing subproblem. Computational results show the effectiveness of the proposed method. The optimal solutions to 179 out of 180 test instances are reported in this paper. Our computational results serve as benchmarks for future researchers on the problem.

Variable neighborhood search algorithm for heterogeneous traveling repairmen problem with time windows

The focus of this paper is generalized traveling repairman problem (TRP), a special case of the well known and well studied traveling salesman problem (TSP). Because of its specific objective function, that minimizes the sum of overall time all clients wait for until the end of a service, TRP has great applicability potential in client oriented practical problems. Therefore it has been known in literature as traveling deliveryman problem, minimum latency problem and cumulative capacitated vehicle routing problem. However, most studies that have treated TRP related problems have implied that only one repairman is present in the system and/or that all clients are available for service at the beginning of the planning horizon. In this paper we consider a TRP with a heterogeneous fleet of repairmen serving a set of clients whose arrival times are distributed over a planning horizon, i.e. heterogeneous TRPTW (hetTRPTW). For the hetTRPTW we present a mixed integer linear programming model, and a heuristic algorithm based on a variable neighborhood search (VNS) framework. Additionally, we propose a reduction strategy for neighborhoods in the VNS algorithm and test efficiency of implemented algorithms on four benchmark sets of problem instances. Results show that proposed algorithms could be used in real systems for solving small and moderate problem instances.

Heuristics for the traveling repairman problem with profits

In the traveling repairman problem with profits, a repairman (also known as the server) visits a subset of nodes in order to collect time-dependent profits. The objective consists of maximizing the total collected revenue. We restrict our study to the case of a single server with nodes located in the Euclidean plane. We investigate the properties of this problem, and we derive a mathematical model assuming that the number of nodes to be visited is known in advance. We describe a tabu search algorithm with multiple neighborhoods, we test its performance by running it on instances from the literature and compare the outcomes with an upper bound. We conclude that the tabu search algorithm finds good-quality solutions fast, even for large instances.

An Inter-Arrival Hyper-Exponential Machine Repairman problem with Two Heterogeneous Repairmen

An Inter-Arrival Hyper-Exponential Machine Repairman problem with Two Heterogeneous Repairmen

Scheduling Partially Ordered Jobs Faster than 2 n

In a scheduling problem, denoted by 1|prec|∑C i in the Graham notation, we are given a set of n jobs, together with their processing times and precedence constraints. The task is to order the jobs so that their total completion time is minimized. 1|prec|∑C i is a special case of the Traveling Repairman Problem with precedences. A natural dynamic programming algorithm solves both these problems in 2n n O(1) time, and whether there exists an algorithms solving 1|prec|∑C i in O(c n) time for some constant c<2 was an open problem posted in 2004 by Woeginger. In this paper we answer this question positively.

The flowtime network construction problem

Given a network whose edges need to be constructed, the problem is to find a construction schedule that minimizes the total recovery time of the vertices, where the recovery time of a vertex is the time when the vertex becomes connected to a special vertex (depot) that is the initial location of the construction crew. The construction speed is constant and is assumed to be incomparably slower than the travel speed of the construction crew in the already constructed part of the network. In this article, this new problem is introduced, its complexity is discussed, mixed-integer linear programming formulations are developed, fast and simple heuristics are proposed, and an exact branch-and-bound algorithm is presented which is based on specially designed lower bounds and dominance tests that exploit the problem’s combinatorial structure. The results of extensive computational experiments are also presented. Connections between the problem and the Traveling Repairman Problem, also known as the Delivery Man Problem, and applications in emergency restoration operations are discussed.

Two improved formulations for the minimum latency problem

In network problems, latency is associated with the metric used to evaluate the length of the path from a root vertex to each vertex in the network. In this work we are dealing with two applications or variations of the minimum latency problem known as the repairman problem and the deliveryman problem. We have developed two integer formulations for the minimum latency problem and compared them with other two formulations from the literature for the time-dependent traveling salesman problem. The present work highlights the similarities and differences between the different formulations. In addition, we discuss the convenience of including a set of constraints in order to reduce the computation time needed to reach the optimal solution. We have carried out extensive computational experimentation on asymmetrical instances, since they provide the characteristics of the deliveryman and repairman problems in a better way.