An improved heuristic for the capacitated arc routing problem

The capacitated arc routing problem (CARP) is an important and practical problem in the OR literature. In short, the problem is to identify routes to service (e.g., pickup or deliver) demand located along the edges of a network such that the total cost of the routes is minimized. In general, a single route cannot satisfy the entire demand due to capacity constraints on the vehicles. CARP belongs to the set of NP-hard problems; consequently numerous heuristic and metaheuristic solution approaches have been developed to solve it. In this paper an “ellipse rule” based heuristic is proposed for the CARP. This approach is based on the path-scanning heuristic, one of the mostly used greedy-add heuristics for this problem. The innovation consists basically of selecting edges only inside ellipses when the vehicle is near the end of each route. This new approach was implemented and tested on three standard datasets and the solutions are compared against: (i) the original path-scanning heuristic; (ii) two other path-scanning heuristics and (iii) the three best known metaheuristics. The results indicate that the “ellipse rule” approach lead to improvements over the three path-scanning heuristics, reducing the average distance to the lower bound in the test problems by about 44%.     

Memetic algorithm with route decomposing for periodic capacitated arc routing problem

In this paper, the Periodic Capacitated Arc Routing Problem (PCARP) is investigated. PCARP is an extension of the well-known CARP from a single period to a multi-period horizon. In PCARP, two objectives are to be minimized. One is the number of required vehicles (nv), and the other is the total cost (tc). Due to the multi-period nature, given the same graph or road network, PCARP can have a much larger solution space than the single-period CARP counterpart. Furthermore, PCARP consists of an additional allocation sub-problem (of the days to serve the arcs), which is interdependent with the routing sub-problem. Although some attempts have been made for solving PCARP, more investigations are yet to be done to further improve their performance especially on large-scale problem instances. It has been shown that optimizing nv and tc separately (hierarchically) is a good way of dealing with the two objectives. In this paper, we further improve this strategy and propose a new Route Decomposition (RD) operator thereby. Then, the RD operator is integrated into a Memetic Algorithm (MA) framework for PCARP, in which novel crossover and local search operators are designed accordingly. In addition, to improve the search efficiency, a hybridized initialization is employed to generate an initial population consisting of both heuristic and random individuals. The MA with RD (MARD) was evaluated and compared with the state-of-the-art approaches on two benchmark sets of PCARP instances and a large data set which is based on a real-world road network. The experimental results suggest that MARD outperforms the compared state-of-the-art algorithms, and improves most of the best-known solutions. The advantage of MARD becomes more obvious when the problem size increases. Thus, MARD is particularly effective in solving large-scale PCARP instances. Moreover, the efficacy of the proposed RD operator in MARD has been empirically verified.     

Lower and upper bounds for the mixed capacitated arc routing problem

This paper presents a linear formulation, valid inequalities, and a lower bounding procedure for the mixed capacitated arc routing problem (MCARP). Moreover, three constructive heuristics and a memetic algorithm are described. Lower and upper bounds have been compared on two sets of randomly generated instances. Computational results show that the average gaps between lower and upper bounds are 0.51% and 0.33%, respectively.     

An improved hybrid firefly algorithm for capacitated vehicle routing problem

Firefly algorithm (FA) is a new meta-heuristic which is successfully applied to solve several optimization problems. However, it suffers from a drawback of easily getting stuck at local optima. This paper proposes a new hybrid FA, called CVRP-FA, to solve capacitated vehicle routing problem. In CVRP-FA, FA is integrated with two types of local search and genetic operators to enhance the solution’s quality and accelerate the convergence. The experiments are conducted over 82 benchmark instances. The results demonstrate that CVRP-FA has fast convergence rate and high computational accuracy. It significantly outperforms the other state-of-the-art FA variants in majority of the tested instances.     

Heuristics for the periodic capacitated arc routing problem

The Capacitated Arc Routing Problem (CARP) involves the routing of vehicles to service a set of arcs in a network. This NP-hard problem is extended to a multiperiod horizon, giving a new tactical problem called the Periodic CARP (PCARP). This problem actually occurs in municipal waste collection. Its objective is to assign arcs to periods and to compute the trips in each period, to minimize an overall cost on the horizon. An integer linear program, two insertion heuristics and a two-phase heuristic are developed. These very first algorithms for the PCARP are evaluated on PCARP instances derived from standard CARP benchmarks from literature: the insertion heuristics are very fast but the two-phase method yields better solution costs.     

多车场CARP问题的改进遗传算法求解

带有容量限制的弧路径规划问题来源于城市垃圾回收、街道清扫、邮件投递、校车路线安排和洒水车路线安排等实际问题,多车场CARP问题是具有多个车场的CARP问题。提出了一种先划分区域后进行路径规划的方法来求解多车场CARP问题。该方法先将各服务弧按照离车场距离的远近归并到距离最近的车场,从而转化为单车场CARP问题,然后用改进的遗传算法进行求解;在求解过程中,用模拟退火算法对部分服务弧进行局部调整,使服务弧在一定的范围内在不同的车场之间进行调换,从而避免局部收敛,达到全局优化的效果。以洒水车路线安排为实例,实验结果表明,该算法能有效求解一定规模的多车场CARP问题,为实际应用奠定了基础。     

A two-phase metaheuristic for the cumulative capacitated vehicle routing problem

The cumulative capacitated vehicle routing problem, which aims to minimize the total arrival time at customers, is a relatively new variant of vehicle routing problem. It can be used to model many real-world applications, e.g., the important application arisen from the humanitarian aid after a natural disaster. In this paper, an approach, called two-phase metaheuristic, is proposed to deal with this problem. This algorithm starts from a solution. At each iteration, two interdependent phases use different perturbation and local search operators for solution improvement. The effectiveness of the proposed algorithm is empirically investigated. The comparison results show that the proposed algorithm is promising. Moreover, for nine benchmark instances, the two-phase metaheuristic can find better solutions than those reported in the previous literature.     

A 3.4713-approximation algorithm for the capacitated multicast tree routing problem

Given an underlying communication network represented as an edge-weighted graph G=(V,E), a source node sV, a set of destination nodes DV, and a capacity k which is a positive integer, the capacitated multicast tree routing problem asks for a minimum cost routing scheme for source s to send data to all destination nodes, under the constraint that in each routing tree at most k destination nodes are allowed to receive the data copies. The cost of the routing scheme is the sum of the costs of all individual routing trees therein. Improving on our previous approximation algorithm for the problem, we present a new algorithm which achieves a worst case performance ratio of , where ρ denotes the best known approximation ratio for the Steiner minimum tree problem. Since ρ is about 1.55 at the writing of the paper, the ratio achieved by our new algorithm is less than 3.4713. In comparison, the previously best ratio was .     

Heuristic algorithms for a three-dimensional loading capacitated vehicle routing problem in a carrier

In this paper, we present heuristic algorithms for a three-dimensional loading capacitated vehicle routing problem arising in a real-world situation. In this problem, customers make requests of goods, which are packed in a sortment of boxes. The objective is to find minimum cost delivery routes for a set of identical vehicles that, departing from a depot, visit all customers only once and return to the depot. Apart of the usual 3D container loading constraints which ensure that the boxes are packed completely inside the vehicles and that the boxes do not overlap each other in each vehicle, the problem also takes into account constraints related to the vertical stability of the cargo and multi-drop situations. The algorithms are based on the combination of classical heuristics from both vehicle routing and container loading literatures, as well as two metaheuristic strategies, and their use in more elaborate procedures. Although these approaches cannot assure optimal solutions for the respective problems, they are relatively simple, fast enough to solve real instances, flexible enough to include other practical considerations, and normally assure relatively good solutions in acceptable computational times in practice. The approaches are also sufficiently generic to be embedded with algorithms other than those considered in this study, as well as they can be easily adapted to consider other practical constraints, such as the load bearing strength of the boxes, time windows and pickups and deliveries. Computational tests were performed with these methods considering instances based on the vehicle routing literature and actual customers’ orders, as well as instances based on a real-world situation of a Brazilian carrier. The results show that the heuristics are able to produce relatively good solutions for real instances with hundreds of customers and thousands of boxes.     

The SR-GCWS hybrid algorithm for solving the capacitated vehicle routing problem

The capacitated vehicle routing problem (CVRP) is a well known problem which has long been tackled by researchers for several decades now, not only because of its potential applications but also due to the fact that CVRP can be used to test the efficiency of new algorithms and optimization methods. The objective of our work is to present SR-GCWS, a hybrid algorithm that combines a CVRP classical heuristic with Monte Carlo simulation using state-of-the-art random number generators. The resulting algorithm is tested against some well-known benchmarks. In most cases, our approach is able to compete or even outperform much more complex algorithms, which is especially interesting if we consider that our algorithm does not require any previous parameter fine-tuning or set-up process. Moreover, our algorithm has been able to produce high-quality solutions almost in real-time for most tested instances. Another important feature of the algorithm worth mentioning is that it uses a randomized constructive heuristic, capable of generating hundreds or even thousands of alternative solutions with different properties. These alternative solutions, in turn, can be really useful for decision-makers in order to satisfy their utility functions, which are usually unknown by the modeler. The presented methodology may be a fine framework for the development of similar algorithms for other complex combinatorial problems in the routing arena as well as in some other research fields.     

A hybrid large-neighborhood search algorithm for the cumulative capacitated vehicle routing problem with time-window constraints

This paper introduces a special vehicle routing problem, i.e. the cumulative capacitated vehicle routing problem with time-window constraints (Cum-CVRPTW). The problem can be defined as designing least-cost delivery routes from a depot to a set of geographically-scattered customers, subject to the constraint that each customer has to be served within a time window; accordingly, the objective costs are computed as the sum of arrival times at all the customers. The Cum-CVRPTW finds practical utility in many situations, e.g. the provision of humanitarian aids in the context of natural disasters. The Cum-CVRPTW can be viewed as a combination of two NP-hard problems, i.e. the vehicle routing problem with time windows and the cumulative vehicle routing problem. To effectively address this problem, an effective algorithm is designed, which is based on the frameworks of Large Neighborhood Search Algorithm and hybridizes with Genetic Algorithm. The proposed algorithm adopts a constraint-relaxation scheme to extend the search space, enabling the iterative exploration of both the feasible and infeasible neighborhood solutions of an incumbent solution. Furthermore, some speed-up techniques are designed to reduce the computational complexity. To elucidate its effectiveness, the proposed algorithm is examined on the benchmark instances from the literature. The resultant numerical findings show that the algorithm is able to improve and obtain some best-known solutions found by existing state-of-the-art methods.     

求解车辆路径问题的改进遗传算法

车辆路径问题是一个典型的组合优化类问题,遗传算法是求解此类问题的方法之一。针对遗传算法容易出现“早熟”现象的问题,借鉴免疫算法通过抗体浓度抑制以保持种群多样性的优势以及模拟退火算法的个体选择策略,提出了一种改进的遗传算法,并将其用于解决车辆路径问题。实验验证了算法的有效性以及求解的效率和解的质量。     

Lower bounds for the mixed capacitated arc routing problem

Capacitated arc routing problems (CARP) arise in distribution or collecting problems where activities are performed by vehicles, with limited capacity, and are continuously distributed along some pre-defined links of a network. The CARP is defined either as an undirected problem or as a directed problem depending on whether the required links are undirected or directed. The mixed capacitated arc routing problem (MCARP) models a more realistic scenario since it considers directed as well as undirected required links in the associated network. We present a compact flow based model for the MCARP. Due to its large number of variables and constraints, we have created an aggregated version of the original model. Although this model is no longer valid, we show that it provides the same linear programming bound than the original model. Different sets of valid inequalities are also derived. The quality of the models is tested on benchmark instances with quite promising results.     

Capacitated arc routing problem with deadheading demands

Capacitated arc routing problem (CARP) is the determination of vehicle tours that serve all positive-demand edges (required edge) exactly once without exceeding vehicle capacity while minimizing sum of all tour costs. In CARP, total demand of a tour is calculated by means of all required edges on the tour. In this study, a new CARP variation is introduced, which considers not only required edges but also traversed edges while calculating total demand of the tour. The traversing demand occurs when the traversed edge is either servicing or non-servicing (deadheading). Since the new CARP formulation incurs deadheading edge demands it is called CARP with deadheading demands. An integer linear model is given for the problem which is used to solve small-sized instances, optimally. A constructive heuristic is presented to solve the problem which is a modified version of a well-known CARP heuristic. Furthermore, two post-optimization procedures are presented to improve the solution of the heuristic algorithm. The effectiveness of the proposed methods is shown on test problems, which are obtained by modifying CARP test instances.     

An improved fireworks algorithm for the capacitated vehicle routing problem

The capacitated vehicle routing problem (CVRP), which aims at minimizing travel costs, is a wellknown NP-hard combinatorial optimization. Owing to its hardness, many heuristic search algorithms have been proposed to tackle this problem. This paper explores a recently proposed heuristic algorithm named the fireworks algorithm (FWA), which is a swarm intelligence algorithm. We adopt FWA for the combinatorial CVRP problem with several modifications of the original FWA: it employs a new method to generate "sparks" according to the selection rule, and it uses a new method to determine the explosion amplitude for each firework. The proposed algorithm is compared with several heuristic search methods on some classical benchmark CVRP instances. The experimental results show a promising performance of the proposed method. We also discuss the strengths and weaknesses of our algorithm in contrast to traditional algorithms.     

A decomposition based memetic algorithm for multi-objective vehicle routing problem with time windows

Multi-objective evolutionary algorithm based on decomposition (MOEA/D) provides an excellent algorithmic framework for solving multi-objective optimization problems. It decomposes a target problem into a set of scalar sub-problems and optimizes them simultaneously. Due to its simplicity and outstanding performance, MOEA/D has been widely studied and applied. However, for solving the multi-objective vehicle routing problem with time windows (MO-VRPTW), MOEA/D faces a difficulty that many sub-problems have duplicated best solutions. It is well-known that MO-VRPTW is a challenging problem and has very few Pareto optimal solutions. To address this problem, a novel selection operator is designed in this work to enhance the original MOEA/D for dealing with MO-VRPTW. Moreover, three local search methods are introduced into the enhanced algorithm. Experimental results indicate that the proposed algorithm can obtain highly competitive results on Solomon׳s benchmark problems. Especially for instances with long time windows, the proposed algorithm can obtain more diverse set of non-dominated solutions than the other algorithms. The effectiveness of the proposed selection operator is also demonstrated by further analysis.     

Modified savings heuristics and genetic algorithm for bi-objective vehicle routing problem with forced backhauls

The cost of distribution and logistics accounts for a sizable part of the total operating cost of a company. However, the cost associated with operating vehicles and crews for delivery purposes form an important component of total distribution costs. Small percentage saving in these expenses could result in a large amount of savings over a number of years. Increase in the number of automated teller machines (ATMs) in the bank industry enforced the researchers to concentrate much on the optimization of distribution logistics problem. The process of replenishing money in the ATMs is considered as a scope with bi-objectives such as minimizing total routing cost and minimizing the span of travel tour. Some of the pick-up routes of the problem are forced and it is termed as forced backhauls. This problem is termed as bi-objective vehicle routing problems with forced backhauls (BVFB). We developed three heuristics to solve BVFB. Two heuristics are modified savings heuristics and the third heuristic is based on adapted genetic algorithm (GA). Standard data sets of VRPB of real life cases for BVFB and randomly generated datasets for BVFB are solved using all the three heuristics. The results are compared and found that all the three heuristics are competitive in solving BVFB. GA yields better solution compared to the other two heuristics.     

An improved model for vehicle routing problem with time constraint based on genetic algorithm

A vehicle routing problem (VRP) with time constraint is one of the important problems in distribution and transportation. Thus the generic VRP and its practical extensions are discussed in great detail in the literatures. In the VRP, the service of a customer must start and finish within a given time interval. The objective of this problem is to minimize the cost of servicing the set of customers without being tardy or exceeding the capacity or travel time of the vehicles. In this research we concentrated on developing a GA–TSP model by improving the genetic algorithm (GA) operators and the initial population. For the computational purpose, we developed a GUI (graphic user interface)-type computer program according to the proposed method. The computational results show that the proposed method is very effective on a set of standard test problems and it can be potentially useful in solving the VRPs.     

物流配送路径多目标优化的聚类-改进遗传算法

探讨运输车辆路线安排调度问题的解决方法，提出一种先用优先级综合聚类分析法将客户分类，再用带有控制开关系统的改进遗传算法求解多目标VRP的优化方法。构造了一种随机开关，以此控制遗传算法中的变异运算，增加了群体的多样性，避免了遗传算法中“局部最优现象”的发生。计算机仿真实验证明了该算法的有效性。     

A deterministic tabu search algorithm for the capacitated arc routing problem

The capacitated arc routing problem (CARP) is a difficult optimisation problem in vehicle routing with applications where a service must be provided by a set of vehicles on specified roads. A heuristic algorithm based on tabu search is proposed and tested on various sets of benchmark instances. The computational results show that the proposed algorithm produces high quality results within a reasonable computing time. Some new best solutions are reported for a set of test problems used in the literature.