Rural Postman Problem Research Articles

Flexible Path Planning for Multi-Agent Field Observation

This paper proposes a flexible path-planning method for conducting agricultural observation tasks using multiple autonomous guided vehicles. The observation task was formulated as the rural postman problem by graphing the agricultural field to be observed and assigning costs to mandatory and optional paths. The observation area was re-divided according to the progress of each robot’s work, and the latest path was planned to minimize the difference between the working time of individual robots, thereby reducing the overall working time. Simulations were conducted to verify the influence of the parameters used in the proposed method. The effect of workload adjustment on progress delay was verified, and the increase in overall working time owing to delay was reduced by an average of 32.9%. A performance comparison was conducted using Google OR-Tools, software specialized for combinatorial optimization. Superior solutions with shorter computation times were obtained using the proposed method.

A branch-and-cut algorithm for the windy profitable location rural postman problem

A branch-and-cut algorithm for the windy profitable location rural postman problem

A heuristic algorithm for the drone rural postman problem

A heuristic algorithm for the drone rural postman problem

Hybrid Metaheuristic Approaches for the Multi-Depot Rural Postman Problem With Rechargeable and Reusable Vehicles

Hybrid Metaheuristic Approaches for the Multi-Depot Rural Postman Problem With Rechargeable and Reusable Vehicles

Theoretical and computational analysis of a new formulation for the Rural Postman Problem and the General Routing Problem

The Rural Postman Problem (RPP) is one of the most well-known problems in arc routing. Given an undirected graph, the RPP consists of finding a closed walk traversing and servicing a given subset of edges with minimum total cost. In the General Routing Problem (GRP), there is also a subset of vertices that must be visited. Both problems were introduced by Orloff and proved to be NP-hard. In this paper, we propose a new formulation for the RPP and the GRP using two sets of binary variables representing the first and second traversal, respectively, of each edge. We present several families of valid inequalities that induce facets of the polyhedron of solutions under mild conditions. Using this formulation and these families of inequalities, we propose a branch-and-cut algorithm, test it on a large set of benchmark instances, and compare its performance against the exact procedure that, as far as we know, produced the best results. The results obtained show that the proposed formulation is useful for solving undirected RPP and GRP instances of very large size.

Exact and Approximation Algorithms for the Multi-Depot Capacitated Arc Routing Problems

In this work, we investigate a generalization of the classical capacitated arc routing problem, called the Multi-depot Capacitated Arc Routing Problem (MCARP). We give exact and approximation algorithms for different variants of the MCARP. First, we obtain the first constant-ratio approximation algorithms for the MCARP and its nonfixed destination version. Second, for the multi-depot rural postman problem, i.e., a special case of the MCARP where the vehicles have infinite capacity, we develop a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(2-\frac{1}{2k+1})$</tex> -approximation algorithm ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> denotes the number of depots). Third, we show the polynomial solvability of the equal-demand MCARP on a line and devise a 2-approximation algorithm for the multi-depot capacitated vehicle routing problem on a line. Lastly, we conduct extensive numerical experiments on the algorithms for the multi-depot rural postman problem to show their effectiveness.

The single robot line coverage problem: Theory, algorithms, and experiments

AbstractLine coverage is the task of servicing a given set of one‐dimensional features in an environment. It is important for the inspection of linear infrastructure such as road networks, power lines, and oil and gas pipelines. This paper addresses the single robot line coverage problem for aerial and ground robots by modeling it as an optimization problem on a graph. The problem belongs to the broad class of arc routing problems and is closely related to the rural postman problem (RPP) on asymmetric graphs. The paper presents an integer linear programming formulation with proofs of correctness. Using the minimum cost flow problem, we develop approximation algorithms with guarantees on the solution quality. These guarantees also improve the existing results for the asymmetric RPP. The main algorithm partitions the problem into three cases based on the structure of the required graph, that is, the graph induced by the features that require servicing. We evaluate our algorithms on road networks from the 50 most populous cities in the world, consisting of up to 730 road segments. The algorithms, augmented with improvement heuristics, run within 3 s and generate solutions that are within 10% of the optimum. We experimentally demonstrate our algorithms with commercial UAVs on the UNC Charlotte campus road network.

A Divide and Conquer Strategy for Sweeping Coverage Path Planning

One of the main challenges faced by floor treatment service robots is to compute optimal paths that completely cover a set of target areas. Short paths are of noticeable importance because their length is directly proportional to the energy used by the robot. Such a problem is known to be NP-hard; therefore, efficient algorithms are needed. In particular, computation efficiency is important for mobile robots with limited onboard computation capability. The general problem is known as coverage path planning (CPP). The CPP has several variants for single regions and for disjoint regions. In this research, we are investigating the solutions for disjoint target regions (rooms) that have fixed connection points (doors). In particular, we have found effective simplifications for the cases of rooms with one and two doors, while the challenging case of an unbounded number of rooms can be solved by approximation. As a result, this work presents a divide and conquer strategy (DnCS) to address the problem of finding a path for a sweeping robot that needs to sweep a set of disjoint rooms that are connected by fixed doors and corridors. The strategy divides the problem into computing the sweeping paths for the target rooms and then merges those paths into one solution by optimising the room visiting order. In this strategy, a geometrical approach for room coverage and an undirected rural postman problem optimisation are strategically combined to solve the coverage of the entire area of interest. The strategy has been tested in several synthetic maps and a real scenario showing short computation times and complete coverage.

Local-Ideal-Points based Autonomous Space Decomposition framework for the Multi-objective Periodic Generalized Directed Rural Postman problem under Length Restrictions with Intermediate Facilities

This paper introduces a Multi-objective Periodic Generalized Directed Rural Postman Problem under Length Restrictions with Intermediate Facilities (MO-PGDRPPLRIF) applied in the metro track inspection routing problem, which is a variant of the Arc Routing Problem (ARP). In the MO-PGDRPPLRIF, arcs are partitioned into given clusters, and each cluster is associated with a service frequency. As the vehicle has limited working hours per day, several trips may be necessary to complete all tasks. The vehicle must stop at an intermediate facility after each trip such that the parking location where the vehicle ends one day is the same as the position where the vehicle starts the next day. The following objectives are considered in the problem: (i) minimizing the total distance; (ii) making time intervals between two adjacent services for clusters with multiple service frequencies as consistent as possible; and (iii) minimizing the number of trips. In this paper, a novel multi-objective evolutionary algorithm, namely Local-Ideal-Points based Autonomous Space Decomposition (LIP-ASD), is developed to address the proposed problem. LIP-ASD employs a space-decomposition scheme to narrow the search space and exclude inferior solutions. The Discrete Particle Swarm Optimization (DPSO) is embedded to find a satisfactory solution in each decomposed space. The proposed approach is tested through the modified Capacitated ARP (CARP) benchmark instances and a real-world application in Beijing Metro Corporation. Computational results confirm that our solution approach can solve the problem effectively and outperforms the other three state-of-the-art multi-objective evolutionary algorithms.

A two-phase hybrid algorithm for the periodic rural postman problem with irregular services on mixed graphs

In this article we address the periodic rural postman problem with irregular services (PRPP–IS), where some arcs and/or edges of a mixed graph must be traversed (to be serviced) a certain number of times in some subsets of days of a given time horizon. The goal is to define a set of minimum cost tours, one for each day or period of the time horizon, that satisfy the service requirements. For this problem we propose a two-phase algorithm that combines heuristics and mathematical programming. In the first phase, two different procedures are used to construct feasible solutions: a multi-start heuristic based on feasibility pump and a multi-start constructive heuristic. From these solutions, some fragments (parts of tours associated with the different days) are extracted. The second phase determines a solution for the PRPP–IS by combining the fragments by means of a mathematical model. We show the effectiveness of this solution approach through an extensive experimental phase on different sets of instances.

Improving a constructive heuristic for the general routing problem

AbstractThe general routing problem (GRP) is a fundamental ‐hard vehicle routing problem, first defined by Orloff in 1974. It contains as special cases the Chinese postman problem, the rural postman problem, the graphical TSP, and the Steiner TSP. We examine in detail a known constructive heuristic for the GRP, due to Christofides and others. We show how to speed it up, in both theory and practice, while obtaining solutions that are at least as good. Computational results show that, for large instances, our implementation is faster than the original by several orders of magnitude.

Polyhedral analysis and a new algorithm for the length constrained K–drones rural postman problem

Polyhedral analysis and a new algorithm for the length constrained K–drones rural postman problem

An Improved Branch and Cut Algorithm for the Windy Profitable Location Rural Postman Problem

An Improved Branch and Cut Algorithm for the Windy Profitable Location Rural Postman Problem

On path ranking in time-dependent graphs

In this paper we study a property of time-dependent graphs, dubbed “path ranking invariance”. Broadly speaking, a time-dependent graph is “path ranking invariant” if the ordering of its paths (w.r.t. travel time) is independent of the start time. In this paper we show that, if a graph is path ranking invariant, the solution of a large class of time-dependent vehicle routing problems can be obtained by solving suitably defined (and simpler) time-independent routing problems. We also show how this property can be checked by solving a linear program. If the check fails, the solution of the linear program can be used to determine a tight lower bound. In order to assess the value of these insights, the lower bounds have been embedded into an enumerative scheme. Computational results on the time-dependent versions of the Travelling Salesman Problem and the Rural Postman Problem show that the new findings enable to outperform state-of-the-art algorithms.

The target visitation arc routing problem

This paper studies the target visitation arc routing problem on an undirected graph. This problem combines the well-known undirected rural postman problem and the linear ordering problem. In this problem, there is a set of required edges partitioned into targets, which must be traversed and there are pairwise preferences for the order in which some targets are serviced, which generates a revenue if the preference is satisfied. The aim is to find a tour that traverses all required edges at least once, and offers a compromise between the revenue generated by the order in which targets are serviced, and the routing cost of the tour. A linear integer programming formulation including some families of valid inequalities is proposed. Despite the difficulty of the problem, the model can be used to solve to optimality around 62% of the test instances.

Time-dependent rural postman problem: time-space network formulation and genetic algorithm

In this paper, a new time-space network model is proposed for addressing the time-dependent rural postman problem (TDRPP) of a single vehicle. The proposed model follows the idea of arc-path alternation to form a feasible and complete route. Based on the proposed model, the time dependency of the TDRPP is better described to capture its dynamic process, compared to the existing methods using a piecewise constant function with limited intervals. Furthermore, the property of first-in-first-out (FIFO) can be satisfied with the time spent on each arc. We investigate the FIFO property for the considered time-dependent network and key optimality property for the TDRPP. Based on this property, a dedicated genetic algorithm (GA) is proposed to efficiently solve the considered TDRPP that suffers from computational intractability for large-scale cases. Comprehensive simulation experiments are conducted for various time-dependent networks to show the effectiveness of the proposed GA.

A partitioning column approach for solving LED sorter manipulator path planning problems

&lt;p style='text-indent:20px;'&gt;This study considers the path planning problem of picking light-emitting diodes on a silicon wafer. The objective is to find the shortest walk for the sorter manipulator covering all nodes in a fully connected graph. We propose a partitioning column approach to reduce the original graph's size, where adjacent nodes at the same column are seen as a required edge, and the connection of vertices at different required edges is viewed as a non-required edge. The path planning problem turns to find the shortest closed walk to traverse required edges and is modeled as a rural postman problem with a solvable problem size. We formulate a mixed-integer program to obtain the exact solution for the transformed graph. We compare the proposed method with a TSP solver, Concorde. The result shows that our approach significantly reduces the problem size and obtains a near-optimal solution. For large problem instances, the proposed method can obtain a feasible solution in time, but not for the benchmarking solver.&lt;/p&gt;

The Hierarchical Chinese Postman Problem: The slightest disorder makes it hard, yet disconnectedness is manageable

The Hierarchical Chinese Postman Problem is finding a shortest traversal of all edges of a graph respecting precedence constraints given by a partial order on classes of edges. We show that the special case with connected classes is NP-hard even on orders decomposable into a chain and an incomparable class. For the case with linearly ordered (possibly disconnected) classes, we get 5/3-approximations and fixed-parameter algorithms by transferring results from the Rural Postman Problem.

Modeling and Solving the Intersection Inspection Rural Postman Problem

Local governments inspect roads to decide which segments and intersections to repair. Videos are taken using a camera mounted on a vehicle. The vehicle taking the videos proceeds straight or takes a left turn to cover an intersection fully. We introduce the intersection inspection rural postman problem (IIRPP), which is a new variant of the rural postman problem (RPP) that involves turns. We develop integer programming formulations of the IIRPP based on two different graph transformations to generate least-cost vehicle routes. One formulation is based on a new idea of transforming a graph. A second formulation is based on a graph transformation idea from the literature. Computational experiments show that the formulation involving the new graph transformation idea performs much better than the other formulation. We also develop an RPP-based heuristic and a heuristic based on a modified RPP. Heuristic solutions are improved by solving integer programming formulations on an induced subgraph. Computational experiments show that the heuristics based on the modified RPP perform much better than the RPP-based heuristics. The best-performing heuristic generates very good quality IIRPP-feasible routes on large street networks quickly. Summary of Contribution. Our paper addresses a real-world problem faced by local governments during road inspections. The real-world problem that we solve and the methodologies that we use fall at the intersection of computing and operations research. We introduce the intersection inspection rural postman problem, which is a new variant of the rural postman problem that involves turns to capture this real-world scenario. The rural postman problem is an important problem in vehicle routing. Studying new variants of this problem is key to extending the practice and theory of vehicle routing. We develop an integer programming formulation based on a new idea of transforming a graph and also develop heuristics based on the rural postman problem.

Solving the length constrained [formula omitted]-drones rural postman problem

In this paper we address the Length Constrained K-Drones Rural Postman Problem (LC K-DRPP). This is a continuous optimization problem where a fleet of homogeneous drones have to jointly service (traverse) a set of (curved or straight) lines of a network. Unlike the vehicles in classical arc routing problems, a drone can enter a line through any of its points, service a portion of that line, exit through another of its points, then travel directly to any point on another line, and so on. Moreover, since the range of the drones is restricted, the length of each route is limited by a maximum distance. Some applications for drone arc routing problems include inspection of pipelines, railway or power transmission lines and traffic monitoring.To deal with this problem, LC K-DRPP instances are digitized by approximating each line by a polygonal chain with a finite number of points and allowing drones to enter and exit each line only at these points. In this way we obtain an instance of the Length Constrained K-vehicles Rural Postman Problem (LC K-RPP). If the number of points used to discretize the lines is large, the LC K-RPP instance can be extremely large and, hence, very difficult to solve optimally. Even heuristic algorithms can fail in providing feasible solutions in reasonable computing times. An alternative is to generate smaller LC K-RPP instances by approximating each line with few but “significant” segments.We present a formulation and some valid inequalities for the LC K-RPP. Based on this, we have designed and implemented a branch-and-cut algorithm for its solution. Moreover, in order to be capable of providing good solutions for large LC K-RPP instances, we propose a matheuristic algorithm that begins by finding good solutions for the LC K-RPP instance obtained by approximating each line by a single segment. Then, to find better solutions, some promising intermediate points are sequentially incorporated. Extensive computational experiments to assess the performance of both algorithms are performed on several sets of instances from the literature.