Capacitated Arc Routing Problem Research Articles

Explaining Genetic Programming-Evolved Routing Policies for Uncertain Capacitated Arc Routing Problems

Genetic programming has been successfully used to evolve routing policies that can make real-time routing decisions for uncertain arc routing problems. Although the evolved routing policies are highly effective, they are typically very large and complex, and hard to be understood and trusted by real users. Existing studies have attempted to improve the interpretability by developing new genetic programming approaches to evolve both effective and interpretable (e.g., with smaller program size) routing policies. However, they still have limitations due to the trade-off between effectiveness and interpretability. To address this issue, we propose a new post-hoc explanation approach to explaining the effective but complex routing policies evolved by genetic programming. The new approach includes a local ranking explanation and a global explanation module. The local ranking explanation uses particle swarm optimisation to learn an interpretable linear model that accurately explains the local behaviour of the routing policy for each decision situation. Then, the global explanation module uses a clustering technique to summarise the local explanations into a global explanation. The experimental results and case studies on the benchmark datasets show that the proposed method can obtain accurate and understandable explanations of the routing policies evolved for uncertain arc routing problems. Our explanation approach is not restricted to uncertain arc routing, but has a great potential to be generalised to other optimisation and machine learning problems such as learning classifier systems and reinforcement learning.

A Multi-Objective Genetic Programming Algorithm With α Dominance and Archive for Uncertain Capacitated Arc Routing Problem

The uncertain capacitated arc routing problem is an important combinatorial optimization problem with many applications in the real world. Genetic programming hyper heuristic has been successfully used to automatically evolve routing policies, which can make real-time routing decisions for uncertain capacitated arc routing problems. It is desired to evolve routing policies that are both effective and small/simple to be easily understood. The effectiveness and size are two potentially conflicting objectives. A further challenge is the objective selection bias issue, i.e., it is much more likely to obtain small but ineffective routing policies than the effective ones that are typically large. In this paper, we propose a new multi-objective genetic programming algorithm to evolve effective and small routing policies. The new algorithm employs the α dominance strategy with a newly proposed α adaptation scheme to address the objective selection bias issue. In addition, it contains a new archive strategy to prevent the loss of promising individuals due to the rotation of training instances. The experimental results showed that the newly proposed algorithm can evolve significantly better routing policies than the current state-of-the-art algorithms for uncertain capacitated arc routing problem in terms of both effectiveness and size. We have also analysed the evolved routing policies to show better interpretability.

Learning heuristics for arc routing problems

Solid waste collection and street sweeping can each be formulated as a capacitated arc routing problem (CARP) in which the streets of a city require service by a fleet of vehicles at minimum total cost. Heuristic and meta-heuristic methods have been used to solve CARP because the problem is known to be NP-Hard. In this paper we introduce Splice, a novel machine learning framework that learns heuristics to solve routing problems, such as CARP, defined on the edges (rather than vertices) of a non-euclidean graph representation of the problem. Splice uses a message passing graph neural network to generate a graph embedding, and deep Q-learning feed-forward layers to learn heuristics to construct a giant tour that results in minimum total cost after applying the splitting procedure, iterating until all tasks are serviced. Compared to the two AI-driven approaches to CARP, Splice has a simpler network architecture and does not require a supervised learning method for training. The total costs of the solutions found by Splice are from 11-30% lower than those found by a path scanning heuristic and a memetic meta-heuristic algorithm. When trained on small instances, the heuristics learned by Splice generalize well, i.e., can be used to solve larger instances without retraining. Due to the route-first split-second form of heuristic learned, the Splice framework may be applied to solve other variants of routing problems by adapting the splitting procedure.

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.

Using an exact bi-objective decoder in a memetic algorithm for arc-routing (and other decoder-expressible) problems

We address the bi-objective Capacitated Arc Routing Problem (CARP) by considering two levels of solution interpretation: implicit and explicit solutions. An algorithm that translates implicit solutions into explicit solutions is called a decoder. In this work, the decoder takes as input a permutation of the required edges and generates a Pareto frontier of CARP solutions. While bi-objective CARP was our main focus and starting point, we could also use the proposed framework to solve a bi-objective version of the traveling salesman problem by plugging-in a different decoder. Recall that bi-objective CARP asks to service (the demands of) a set of required edges using a fleet of vehicles of limited capacity so as to minimize: (i) the total travelled distance and (ii) the length of the longest route. Any permutation s of the required edges constitutes an implicit CARP solution. The decoder constructs all non-dominated explicit solutions that service the edges in the order indicated by s, i.e., the decoder is an exact algorithm that returns the optimal Pareto frontier subject to the service order s. To achieve competitive CARP results it is also important to reinforce the decoder using a local search operator that acts on explicit routes (and that may change the service order s). For nine instances, the resulting algorithm was even able to find a new total-cost upper bound, improving upon the best solutions reported in the (considerably larger) mono-objective CARP literature. This shows that (some of) the proposed ideas can also be useful in single objective optimization: the second objective can be seen as a guide for the mono-objective search process.

An improved multi-objective framework for the Rich arc routing problem

This article introduces a Rich Arc Routing Problem (RARP). Required arcs are partitioned into given clusters, and each cluster shall be serviced for prescribed times during the planning horizon. There is a time limit that restricts the visits that can be performed in a single day. Each vehicle has to stop at one of intermediate facilities before the end of the specific hours per day. Accordingly, several days may be necessary to complete all tasks, and paths traveled by different vehicles on the same day shall not overlap. The RARP consists of finding a set of routes with (i) the minimum total distance, (ii) the minimum number of vehicles, (iii) the minimum pure working days, and (iv) the temporal service time interval consistency of each cluster. Recently, a Local-Ideal-Points based Autonomous Space Decomposition (LIP-ASD) algorithm has been demonstrated to be a competitive multi-objective framework. However, randomly generating an initial population in each decomposed space decreases the convergence speed of LIP-ASD. In addition, retaining only one individual preserved in each sub-space limits population diversity. In response to these issues, an Improved LIP-ASD (I-LIP-ASD) for RARP is developed. The two improvements of the proposed algorithm are as follows: (i) the initial population in the current sub-space is generated by combining existing nondominated individuals, and (ii) all nondominated solutions will be output from the evolved population in each decomposed sub-space. Finally, we conduct a series of experiments over converted capacitated arc routing problem instances, as well as analyze them under a real-world case to evaluate and demonstrate the effectiveness of the proposed solution algorithm.

Solving the time capacitated arc routing problem under fuzzy and stochastic travel and service times

AbstractStochastic, as well as fuzzy uncertainty, can be found in most real‐world systems. Considering both types of uncertainties simultaneously makes optimization problems incredibly challenging. In this paper we propose a fuzzy simheuristic to solve the Time Capacitated Arc Routing Problem (TCARP) when the nature of the travel time can either be deterministic, stochastic or fuzzy. The main goal is to find a solution (vehicle routes) that minimizes the total time spent in servicing the required arcs. However, due to uncertainty, other characteristics of the solution are also considered. In particular, we illustrate how reliability concepts can enrich the probabilistic information given to decision‐makers. In order to solve the aforementioned optimization problem, we extend the concept of simheuristic framework so it can also include fuzzy elements. Hence, both stochastic and fuzzy uncertainty are simultaneously incorporated into the CARP. In order to test our approach, classical CARP instances have been adapted and extended so that customers' demands become either stochastic or fuzzy. The experimental results show the effectiveness of the proposed approach when compared with more traditional ones. In particular, our fuzzy simheuristic is capable of generating new best‐known solutions for the stochastic versions of some instances belonging to the tegl, tcarp, val, and rural benchmarks.

Learning Heuristics With Different Representations for Stochastic Routing.

Uncertainty is ubiquitous in real-world routing applications. The automated design of the routing policy by hyperheuristic methods is an effective technique to handle the uncertainty and to achieve online routing for dynamic or stochastic routing problems. Currently, the tree representation routing policy evolved by genetic programming is commonly adopted because of the remarkable flexibility. However, numeric representations have never been used. Considering the practicability of the numeric representations and the capability of the numeric optimization methods, in this article, we investigate two numeric representations on a representative stochastic routing problem and uncertain capacitated arc routing problem. Specifically, a linear representation and an artificial neural-network (ANN) representation are implemented and compared with the tree representation to reveal the potential of the numeric representations and the characteristics of their optimization. Experimental results show that the tree representation is the best choice, but on a majority of the test instances, the numeric representations, especially the ANN representation, can provide competitive performance. Further analyses also show that training a good ANN representation policy requires more training data than the tree representation. Finally, a guideline of representation selection is given.

Knowledge Transfer Genetic Programming With Auxiliary Population for Solving Uncertain Capacitated Arc Routing Problem

The uncertain capacitated arc routing problem (UCARP) is an NP-hard combinatorial optimization problem with a wide range of applications in logistics domains. Genetic programming (GP) hyper-heuristic has been successfully applied to evolve routing policies to effectively handle the uncertain environment in this problem. The real world usually encounters different but related instances due to events, such as season change and vehicle breakdowns, and it is desirable to transfer knowledge gained from solving one instance to help solve another related one. However, the solutions found by the GP process can lack diversity, and the existing methods use the transferred knowledge mainly during initialization. Thus, they cannot sufficiently handle the change from the source to the target instance. To address this issue, we develop a novel knowledge transfer GP with an auxiliary population. In addition to the main population for the target instance, we initialize an auxiliary population using the transferred knowledge and evolve it alongside the main population. We develop a novel scheme to carefully exchange the knowledge between the two populations, and a surrogate model to evaluate the auxiliary population efficiently. The experimental results confirm that the proposed method performed significantly better than the state-of-the-art GP approaches for a wide range of uncertain arc routing instances, in terms of both final performance and convergence speed.

A parameterized lower bounding method for the open capacitated arc routing problem

Consider an undirected graph with demands scattered over the edges and a homogeneous fleet of vehicles to service the demands. In the open capacitated arc routing problem (OCARP) the objective is to find a set of routes that collectively service all demands with the minimum cost. Each vehicle has limited capacity and it can start and finish the route at any node. The OCARP is NP-hard, and its applications include meter reading and cutting path determination problems. State-of-the-art solution methods developed for the OCARP are heuristics, which show good tradeoffs between solution quality and processing time, but do not provide optimality certificates of the obtained solutions. This work focuses on a lower bounding method for the OCARP which can be used to better assess the quality of heuristic solutions. We propose the Relaxed Flow method (RF(k)) which involves the resolution of a mixed integer linear formulation where all vehicles' capacities are modeled as flows on an augmented graph. A parameter k controls the model tightness and RF(k) is shown to be at least as tight as the well-known Belenguer and Benavent's formulation for any k⩾0. To strengthen the model, capacity cuts were included in RF(k) by means of a branch-and-cut framework. Extensive computational experiments conducted on a set of benchmark instances revealed that our method outperformed previous methods. Computational experiments also demonstrated the importance of the parameterization technique to obtain good results. The previously known lower bounds were improved substantially and optimality certificates were attained in 78.9% of the instances. As far as we know this is the first parameterized lower bounding method proposed for an arc routing problem, and we argue it can be generalized to other variants of arc routing problems and general routing problems.

Hybridised ant colony optimisation for the multi-depot multi-compartment capacitated arc routing problem

Hybridised ant colony optimisation for the multi-depot multi-compartment capacitated arc routing problem

A Novel Generalized Metaheuristic Framework for Dynamic Capacitated Arc Routing Problems

The capacitated arc routing problem (CARP) is a challenging combinatorial optimisation problem abstracted from many real-world applications, such as waste collection, road gritting and mail delivery. However, few studies considered dynamic changes during the vehicles' service, which can cause the original schedule infeasible or obsolete. The few existing studies are limited by the dynamic scenarios considered, and by overly complicated algorithms that are unable to benefit from the wealth of contributions provided by the existing CARP literature. In this paper, we first provide a mathematical formulation of dynamic CARP (DCARP) and design a simulation system that is able to consider dynamic events while a routing solution is already partially executed. We then propose a novel framework which can benefit from existing static CARP optimisation algorithms so that they could be used to handle DCARP instances. The framework is very flexible. In response to a dynamic event, it can use either a simple restart strategy or a sequence transfer strategy that benefits from past optimisation experience. Empirical studies have been conducted on a wide range of DCARP instances to evaluate our proposed framework. The results show that the proposed framework significantly improves over state-of-the-art dynamic optimisation algorithms.

Lagrangian relaxation-based decomposition approaches for the capacitated arc routing problem in the state-space-time network

ABSTRACT We study the capacitated arc routing problem in the state-space-time network. A multi-commodity network flow model is formulated in the state-space-time network. Two Lagrangian relaxation-based decomposition methods are developed to tackle this problem. By dualizing the coupling constraints to the objective function, the Lagrangian relaxed problem (LR) can be decomposed into several routing subproblems in the state-space-time network. Then we extend quadratic penalty terms to the LR and obtain an augmented Lagrangian relaxed problem (ALR). By applying the alternating direction method of multipliers and linearization techniques, we decompose the ALR into a sequence of routing subproblems. We solve these subproblems of LR and ALR by a time-dependent dynamic programming algorithm. Feasible solutions are produced according to the result of the LR or ALR. The Lagrangian multipliers are updated by the subgradient optimization method. We implement the proposed methods in three networks, showing that the ALR-based method has better performance.

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.

Study on path planning of mechanized harvesting of ratoon rice in the first season based on the capacitated arc routing problem model.

The stubble crushing caused by the harvester during the first season of ratoon rice harvesting will directly affect the grain yield of the ratoon season. In this work, a harvester path planning method for quadrilateral fields to address the harvester driving path problem of the first season of ratoon rice mechanized harvesting is proposed. This research first analyzes the operational characteristics and requirements of ratoon rice first-season mechanized harvesting, and then models the mechanized harvesting process of ratoon rice in the first season as a capacitated arc routing problem (CARP) considering the fact that the harvester cannot complete the full-coverage harvesting operation at one time due to the limitation of grain bin volume. The genetic algorithm (GA) with strong global search capability is used to solve it, and the selection and variation links of the algorithm are improved. The path planning method proposed in this article can dynamically find the optimal harvester travel route according to the specific conditions of the field and the parameters of harvester implements. The simulation test shows that the CARP method performs better in terms of harvesting path length and crushed area compared to the conventional rectangular detour and foldback reciprocating harvesting paths. The degree of optimization of this method is influenced by various factors such as the width of the cutting table, the turning radius of the harvester, and the size of the grain bin capacity. This research provides a more efficient and flexible path planning method to improve the efficiency of ratoon rice first-season mechanized harvesting operations and optimize the harvester's operating path, which can well meet the operational requirements.

Route Stability in the Uncertain Capacitated Arc Routing Problem

Power line inspections in a microgrid can be modeled as the uncertain capacitated arc routing problem, which is a classic combinatorial optimization problem. As an evolutionary computation method, genetic programming is used as a hyper-heuristic method to automatically evolve routing policies that can make real-time decisions in an uncertain environment. Most existing research on genetic programming hyper-heuristic for the uncertain capacitated arc routing problem only focuses on optimizing the total cost of solutions. As a result, the actual routes directed by the routing policies evolved by genetic programming hyper-heuristic are usually not stable, i.e., the routes have large fluctuations in different uncertain environments. However, for marketing or considering the drivers’ and customers’ perspectives, the routes should not be changed too often or too much. Addressing this problem, this study first proposes a method to estimate the similarity between two routes and then extends it for evaluating the stability of the routes in uncertain environments. A novel genetic programming hyper-heuristic, which considers two objectives, i.e., the solution quality (total cost) and the stability of routes, was designed. Experimental studies demonstrate that the proposed genetic programming is hyper-heuristic with stability in consideration and can obtain more stable solutions than the traditional algorithm, without deteriorating the total cost. The approach provided in this study can be easily extended to solving other combinatorial optimization problems in the microgrid.

Genetic Programming With Knowledge Transfer and Guided Search for Uncertain Capacitated Arc Routing Problem

The uncertain capacitated arc routing problem has many real-world applications in logistics domains. Genetic programming (GP) is a promising approach to training routing policies to make real-time decisions and handle uncertain events effectively. In the real world, there are various problem domains and no single routing policy can work effectively in all of them. Instead of training in isolation, we can leverage the relatedness between the problems and transfer knowledge from previously solved source problems to solve the target problem. The existing transfer methods are not effective enough due to the loss of diversity during the knowledge transfer. To increase the diversity of the transferred knowledge, in this article, we propose a novel GP method that removes phenotypic duplicates from the source individuals to initialize the target individuals. Furthermore, assuming that the transferred knowledge used in initialization already includes all the important knowledge explored for the source problem, it is more effective to explore new regions that have not been explored for the source problem. Therefore, we propose novel genetic operators that prohibit the search from revisiting the source individuals when solving the target problem. To speed up the revisit check, we propose to adapt a powerful hashing method for routing policies that greatly improves the efficiency of the genetic operators. Our experimental results show that the proposed method significantly outperforms the existing GP approaches with knowledge transfer in terms of both initial and final solution quality.

Arc routing with trip-balancing and attractiveness measures — A waste collection case study

This work focuses on a household (or door to door) waste collection problem in the Portuguese municipality of Seixal, which is modelled as a generalisation of a mixed capacitated arc routing problem (MCARP). The MCARP is known to be NP-hard. The proposed methodology uses: i) a GIS (geographic information system), for input/output and to reduce problem dimensions; ii) a matheuristic that iteratively solves a new hybrid model; and iii) two versions of a two-phase matheuristic. The latter pursues the generation of connected and compacted trips. During the first phase, called initial assignment, some links requiring service are assigned to vehicle services. In the second phase, the hybrid model finishes the assignment and generates a feasible set of trips. The quality of the generated solutions is assessed through the total time, as well as by some attractiveness measures. These measures evaluate the adequacy of the solutions for the real case-study, a crucial aspect for trips that need to be accepted by practitioners. With this purpose, a new attractiveness measure that introduces space dependent penalisation of overlaps, named weighted hull overlap, is also proposed. Computational results with 18 Seixal instances, with 265–1223 nodes and 492–2254 links, point to the good performance of the proposed methodology.

An Artificial Bee Colony Algorithm for Static and Dynamic Capacitated Arc Routing Problems

The Capacitated Arc Routing Problem (CARP) is a combinatorial optimization problem, which requires the identification of such route plans on a given graph to a number of vehicles that generates the least total cost. The Dynamic CARP (DCARP) is a variation of the CARP that considers dynamic changes in the problem. The Artificial Bee Colony (ABC) algorithm is an evolutionary optimization algorithm that was proven to be able to provide better performance than many other evolutionary algorithms, but it was not used for the CARP before. For this reason, in this study, an ABC algorithm for the CARP (CARP-ABC) was developed along with a new move operator for the CARP, the sub-route plan operator. The CARP-ABC algorithm was tested both as a CARP and a DCARP solver, then its performance was compared with other existing algorithms. The results showed that it excels in finding a relatively good quality solution in a short amount of time, which makes it a competitive solution. The efficiency of the sub-route plan operator was also tested and the results showed that it is more likely to find better solutions than other operators.

Multiobjective Multidepot Capacitated Arc Routing Optimization Based on Hybrid Algorithm

The multidepot capacitated arc routing problem (CARP) is investigated with the hybrid optimization algorithm of the Dijkstra algorithm and genetic algorithm. The complex multidepot CARP is transformed into multiple single depot CARP by systematic clustering analysis. After completing the system clustering, the Dijkstra algorithm is used to adjust the boundary arc locally and merge it to a reasonable depot, while in the genetic algorithm, the structure of the chromosome is reset to use the path as the way of real coding, and the elite selection is used to decode to obtain the optimal path optimization scheme. Finally, Lanzhou road network data as experimental data, through Matlab to achieve the practicability of the algorithm in sprinkler applications. The results show that the improved genetic algorithm can successfully solve the multi-segment CARP with a certain road network scale, ensuring the correctness and feasibility of the algorithm. In addition, the efficiency of the algorithm in the later iteration is basically controlled at about 0.5 seconds, indicating that the efficiency of the algorithm is worth identifying.