Postman Problem Research Articles

Optimization of Chinese Postman Problem Using Fuzzy-Based Priority Weighted Graph

: In the early 1960s, Chinese mathematician Mei-Ko Kwan (M. Guan) introduced a method aimed at minimizing mail carriers’ route lengths. This method led to the exploration of various formulations of the Chinese Postman Problem (CPP), resulting in at least eight distinct formulations. In response to a practical concern, a new problem formulation called the Priority Constrained Chinese Postman Problem (PCCPP) emerged. In PCCPP, a linear order is provided for a set of significant nodes, and the objective is to traverse all edges at least once while prioritizing the prompt visitation of higher-priority nodes. This paper presents a modified approach to Chinese Postman Problems (CPP) using priorities in a fuzzy environment ranking function applied to priority nodes in CPP to get the optimal result. Finally, this paper also illustrates and justifies the implementation of the modified approach with a numerical problem. The consideration of multi-factors in a multi-graph with priorities gives another scope of research. JEL Codes: C44 Received: 18/07/2024. Accepted: 29/09/2024. Published: 13/10/2024.

The Chinese Postman Problem and Simulated Annealing applied to urban forest inventory

Routing is the process of defining paths between geographically distinct points, and it has been applied in decision-making processes for services that include displacement, with potential application in the planning of urban forest inventories. Linear Programming is an operations research technique often applied to solve complex problems, such as those related to routing. Thus, the objective of this study was to verify whether the application of a routing model via Integer Linear Programming (ILP) increases the efficiency of an urban forest inventory using real data. First, we compared the empirical division used in the inventory with the application of the ILP method for routing. We also tested an approximate method (Simulated Annealing, SA) to solve the routing problem and reduce the processing time. Finally, we simulated the consequences of applying ILP in different scenarios. The results showed that the ILP improved the inventory efficiency by 6.36%, and the use of SA reduced the processing time by 43,200 times for the most complex scenario, resulting in a solution that was 1.87% of the ILP response value. Our findings show the potential application of ILP in the planning of urban forest inventories, with gains directly proportional to the number of lots in the inventory.

Uncertain Chinese postman problem with budget constraint: a robust optimization approach

Uncertain Chinese postman problem with budget constraint: a robust optimization approach

Approximation algorithms for solving the vertex-traversing-constrained mixed Chinese postman problem

Approximation algorithms for solving the vertex-traversing-constrained mixed Chinese postman problem

Uncertain Maximum Benefit about Chinese Postman Problem

This research deals with the generalization of profit maximization of the Chinese postman with several vehicles under uncertainty, and then it is solved using the Lagrangian algorithm. Four different cases of the model are discussed. The first mode performs Chinese postman routing and assumes that the origin and destination are the same for each vehicle for each cycle and is provided by the user, which is the case of a typical postman. The second takes into account the assumption that the origin and destination of each vehicle should not be the same and allows users to choose the possible origin or destination of the vehicles. The third case takes the origin of each vehicle as input and generates a solution based on finding the best destination for each vehicle. The last parameter, which is very general, allows the optimization model to choose possibly different locations for the origin and destination of the vehicle during each cycle. Modeling these issues in the real world requires the use of uncertainty theory. Liu expressed the theory of uncertainty. Here, is a model of the Chinese postman’s multiple profit maximization problem. So far, there is no such uncertainty analysis of these models. After stating the uncertainty problem of the maximum profit problem of the Chinese postman, we will state the problem using the Lagrangian algorithm. We then compare the results.

Ant Colony Optimization for solving Directed Chinese Postman Problem

The Chinese Postman Problem (CPP) is a well-known optimization problem involving determining the shortest route, modeling the system as an undirected graph, for delivering mail, ensuring all roads are traversed while returning to the post office. The Directed Chinese Postman Problem (DCPP) extends the Chinese Postman Problem (CPP), where the underlying graph representing the system incorporates exclusively directed edges. Similarly to CPP, this problem has plenty of applications in route optimization, interactive system analysis, and circuit design problems. However, due to the added constraint (directionality of edges), DCPP results are more challenging to solve. Although methods to solve it in literature are proposed, typically by using minimum-cost-flow algorithms, the meta-heuristics approaches proposed to deal with it are very limited. In this paper, we propose an innovative meta-heuristic approach to solve DCPP by using an ant colony optimization (ACO) algorithm, i.e., an algorithm that simulates in a simplified way the behavior of some species of ants to solve optimization problems. The efficiency of our ant colony optimization for solving the Directed Chinese Postman Problem (ACO-DCPP) is measured by comparing the ACO outcomes with the results obtained by a recursive algorithm that explores all the possible solutions. Results show that ACO-DCPP is stable and gets the global optimum frequently by using an extremely limited number of solutions explored.

Critical-edge based tabu search algorithm for solving large-scale multi-vehicle Chinese postman problem

The min–max multi-vehicle Chinese postman problem is an NP-hard problem, which is widely used in path planning problems based on road network graphs, such as urban road structure probing planning, urban road underground cavity detection planning, high-voltage line inspection planning, and so on. With the rapid increase in the number of nodes and connections of road network graph, the solution time and path equilibrium constraints pose new challenges to the problem solving. In this paper, we propose a critical-edge tabu search algorithm, CTA-kroutes, for solving the min–max multi-vehicle postman problem for large-scale road networks. First, the initial solution with balanced path lengths is obtained by segmenting the Eulerian paths; second, the critical edges are moved in the initial solution to construct the neighborhood solution, and the tabu search algorithm is used to find the optimal solution iteratively; and lastly, the solution optimization algorithm is used at the end of each iteration to de-duplicate and optimally reconstruct the current search result. Experiments show that the CTA-kroutes algorithm can effectively improve the equalization of multi-vehicle paths and its applicability to large-scale road networks.

Windy Postman Problem Under Uncertainty

The postman problem is one of the optimization problems that has many applications in today’s world. With the progress of science, such optimization problems are used more. The postman’s problem is to find a route that has the minimum cost. The postman starts the route from one point and returns to the same point and must travel all the routes. If the round-trip costs are symmetric, in other words, the round-trip cost is the same, the problem is called the Chinese postman problem, and if the costs are asymmetric, that is, the round-trip cost is different from the return cost, the problem is called the windy postman problem (WPP). Here we describe two models of the WPP, which include the WPP and the rural WPP. The model for these problems in the real world needs to apply uncertainty theory. Liu expressed the theory of uncertainty. Here is a model of the windy postman uncertainty problem and a model of the windy rural postman uncertainty problem. So far, there has been no such uncertainty analysis on these models. To solve these problems, we use the Lagrangian heuristic method after turning it into a deterministic model. To solve these problems, after turning the problem into a deterministic problem, the heuristic Lagrangian method was used, and for the wind postman problem, we stated the corresponding Lagrangian algorithm and for the rural wind postman, the Lagrangian algorithm. Examples of these are problems and uncertainties. We expressed them and finally solved them by direct and Lagrangian methods.

Ant colony optimization for Chinese postman problem

This paper aims to solve the Chinese Postman Problem (CPP) using an Ant Colony Optimization (ACO) algorithm. In graph theory, the CPP looks for the shortest closed path that visits every edge of a connected undirected graph. This problem has many applications, including route optimization, interactive system analysis, and flow design. Although numerous algorithms aimed at solving CPP are present in the literature, very few meta-heuristic algorithms are proposed, and no ACO applications have been proposed to solve them. This paper tries to fill this gap by presenting an ACO algorithm that solves CPP (ACO-CPP). To prove its consistency and effectiveness, ACO-CPP is compared with a Genetic Algorithm (GA) and a recursive algorithm throughout three experiments: (1) recursive-ACO-GA comparisons over randomly generated graphs for the attainment of the global optimum; (2) ACO-GA statistical comparisons over specifically generated graphs; (3) recursive-ACO-GA comparisons by changing ACO hyperparameters over randomly generated graphs for the attainment of the global optimum. The experiments prove that the ACO-CPP algorithm is efficient and exhibits a consistency similar to GA when the number of possible solutions to explore is relatively low. However, when that number greatly exceeds those explored, ACO outperforms GA. This suggests that ACO is more suitable for solving problems with a CPP structure.

The biobjective minimum‐cost perfect matching problem and Chinese postman problem

AbstractIn this paper, we address the biobjective versions of the perfect matching problem (PMP) and the Chinese postman problem (CPP). Both problems are solved by means of integer formulations or separating blossom inequalities, exploiting the PMP relationship with the CPP. In both cases, we first find the set of supported nondominated solutions and then we use them to obtain the nonsupported ones. The set of supported nondominated solutions are obtained solving scalarized integer formulations. To obtain the sets of nonsupported solutions, we resort to solving lexicographic problems based on adding additional linear constraints to the original problems. For this reason, we also characterize the combinatorial structure of the PMP vertices with one or two additional constraints. We also investigate when it is possible to use the PMP to solve CPP in the biobjective case. We report computational experiments comparing the different approaches and formulations based on different types of graphs with up to 700 nodes.

Quantum computing for a profusion of postman problem variants

In this paper we study the viability of solving the Chinese Postman Problem, a graph routing optimization problem, and many of its variants on a quantum annealing device. Routing problem variants considered include graph type, directionally varying weights, number of parties involved in routing, among others. We put emphasis on the explanation of how to convert such problems into quadratic unconstrained binary optimization (QUBO) problems. QUBO is one of two equivalent natural paradigms for quantum annealing devices, the other being the Ising Model. We also expand upon a previously discovered algorithm for solving the Chinese Postman Problem on a closed undirected graph to decrease the number of constraints and variables used in the problem. Optimal annealing parameter settings and constraint weight values are discussed based on results from implementation on the D-Wave 2000Q and Advantage. Results from classical, purely quantum, and hybrid algorithms are compared.

The new approaches for solving hierarchical Chinese postman problem with stochastic travel times

The hierarchical Chinese postman problem (HCPP) aims to find the shortest tour or tours by passing through the arcs classified according to precedence relationship. HCPP, which has a wide application area in real-life problems such as shovel snow and routing patrol vehicles where precedence relations are important, belongs to the NP-hard problem class. In real-life problems, travel time between the two locations in city traffic varies due to reasons such as traffic jam, weather conditions, etc. Therefore, travel times are uncertain. In this study, HCPP was handled with the chance-constrained stochastic programming approach, and a new type of problem, the hierarchical Chinese postman problem with stochastic travel times, was introduced. Due to the NP-hard nature of the problem, the developed mathematical model with stochastic parameter values cannot find proper solutions in large-size problems within the appropriate time interval. Therefore, two new solution approaches, a heuristic method based on the Greedy Search algorithm and a meta-heuristic method based on ant colony optimization were proposed in this study. These new algorithms were tested on modified benchmark instances and randomly generated problem instances with 817 edges. The performance of algorithms was compared in terms of solution quality and computational time.

Algorithm for optimal path planning of a robotic lawnmower

Horticultural lawns include areas of land that are covered with sown grass in urban areas and require periodic mowing in order to best preserve their aesthetic and functional purpose. This activity may require significant involvement of human and material resources, especially in large public areas such as sports fields, parks and picnic areas. Thanks to the progress of computer technology, this tiresome routine work is gradually being taken over by robotic mowers that can bring significant savings in human and material resources. In this paper, the problem of robotic lawnmowers related to path planning in the working environment is presented. Commercial solutions to date are mostly based on choosing a random straight line path until the mower reaches a lawn boundary or obstacle, after which it turns in the opposite direction and continues mowing in an arbitrary direction. This paper presents a path planning method based on the Chinese postman's algorithm, which enables the planning of the most optimal path that the mower will take so that it mows the entire lawn in the most economical way.

Approximation algorithms for solving the heterogeneous Chinese postman problem

Approximation algorithms for solving the heterogeneous Chinese postman problem

Optimization of Snowplow Routes for Real-World Conditions

During the winter season, snowplowing has a significant effect on road users as it is critical to winter road maintenance and operations. The main goal of this study is to generate optimal routes for snowplowing trucks for efficient road maintenance. In addition to the conventional problem of reducing travel time and distance, this study also incorporates actual operational constraints, such as minimum maintenance standards and driver safety, to improve the overall efficiency of operations. To achieve the objectives, we first implemented the Chinese Postman Problem (CPP) to create Euler circuits from the initial routes and then identified the shortest paths by applying Dijkstra’s algorithm. Then, the Tabu search algorithm was chosen as a metaheuristic algorithm for the optimization process that finds near-optimal solutions by considering operational constraints for snowplow routes. Unsafe turning conditions and minimum maintenance standards were taken into account in the objective function defined for the optimization process. In simulations, the route obtained by our approach was compared to one with the application of CPP only in terms of travel distance, time, turning conditions, and road maintenance priority.

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.

An improved heuristic algorithm for the maximum benefit Chinese postman problem

The maximum benefit Chinese postman problem (MBCPP) is a practical generalization of the Chinese postman problem. A distinctive feature of MBCPP is that the postman can traverse each edge arbitrary times and obtains a benefit for each traversal of an edge, which depends on the number of times that the edge has been traversed. (Pearn and Wang, Omega 31 (2003) 269–273) discussed MBCPP under the assumption that the benefits of each edge is a non-increasing function of the number of traversals. They showed that MBCPP under this assumption is NP-hard, and proposed a heuristic algorithm which applies the minimum spanning tree and the minimal-cost T-join algorithms. (Corberán, Plana, Rodríguez-Chía and Sanchis, Math. Program. 141 (2013) 21–48) presented an integer programming formulation and a branch-and-cut algorithm for MBCPP without the assumption on the benefits. This is based on the idea of integrating the benefits of each edge into two benefits, each representing that the edge is traversed an odd or even number of times. In this paper, by applying the idea of Corberán et al., we improve the heuristic algorithm of Pearn and Wang. Our algorithm applies to the general case with no assumption on the benefits, and can perform better even if the benefits are non-increasing. We then analyze the efficiency of our heuristic algorithm in theory and in practice, and prove that it finds the optimal solution when the benefits satisfy a certain property.

A distributed message passing algorithm for the capacitated directed Chinese postman problem

Message passing is a class of extremely powerful distributed iterative algorithms based on probabilistic graphic model, in which little computation performed per iteration on minimal data structure. As a prototypical message-passing algorithm, belief propagation (BP) algorithm has wide applications in various fields of coding theory, machine learning and combinatorial optimization. Due to its distributed and iterative nature, BP algorithm can run effectively and fast on large data networks. In this paper, we study the behavior of Min-Sum BP algorithm for the capacitated directed Chinese postman problem (CDCP). We derive the iterative process of message passing for solving CDCP. As the main result, for any weighted digraph G of size n, if the weight on each edge is nonnegative integral, then our algorithm converges to the optimal solution of CDCP after O(w∗n2) iterations, provided that CDCP has a unique optimal solution, where w∗=max{we:e∈E(G)}.

Smart Parking Management and Planning Based on Chinese Postman Problem

This paper discusses the practicability of the charging for the video collection vehicle in the city of Jinan in China. Firstly, according to the collection vehicle and the charge standard, we have constructed a price loss function for the single parking space and used the method of Chinese postman problem to analyse the parking spaces in the whole area, then obtained the best route by Fleury algorithm which is compared with manual charging at last.

On the periodic hierarchical Chinese postman problem

This paper presents a mathematical formulation and a heuristic approach for a new variant of the Hierarchical Chinese Postman Problem (HCPP). Indeed, we introduce the concept of periodicity, and we define and solve, for the first time, the Periodic-HCPP, denoted as P-HCPP. Given that the resulting integer programming model makes use of a big number of binary variables and given the extended time horizon considered, 30 days in our case, the problem is characterized by a high level of complexity. However, our developed heuristic is able to solve instances having up to 40 nodes, 520 arcs and 5 hierarchies, whereas a general-purpose solver like Gurobi was not able to provide solutions for instances having more than 10 nodes. While the collected results are very encouraging, we provide at the end of this paper a set of possible future extensions of this work.