An approximation algorithm for the pickup and delivery vehicle routing problem on trees

This paper presents an approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot where there are two types of demands, pickup demand and delivery demand. Customers are located on nodes of the tree, and each customer has a positive demand of pickup and/or delivery.Demands of customers are served by a fleet of identical vehicles with unit capacity. Each vehicle can serve pickup and delivery demands. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. In each tour, a vehicle begins at the depot with certain amount of goods for delivery, visits a subset of the customers in order to deliver and pick up goods and returns to the depot. At any time during the tour, a vehicle must always satisfy the capacity constraint, i.e., at any time the sum of goods to be delivered and that of goods that have been picked up is not allowed to exceed the vehicle capacity. We propose a 2-approximation algorithm for the problem.     

An interactive GRAMPS algorithm for the heterogeneous fixed fleet vehicle routing problem with and without backhauls

In this article, a visual interactive approach based on a new greedy randomised adaptive memory programming search (GRAMPS) algorithm is proposed to solve the heterogeneous fixed fleet vehicle routing problem (HFFVRP) and a new extension of the HFFVRP, which is called heterogeneous fixed fleet vehicle routing problem with backhauls (HFFVRPB). This problem involves two different sets of customers. Backhaul customers are pickup points and linehaul customers are delivery points that are to be serviced from a single depot by a heterogeneous fixed fleet of vehicles, each of which is restricted in the capacity it can carry, with different variable travelling costs.     

Integer linear programming models for a cement delivery problem

We consider a cement delivery problem with an heterogeneous fleet of vehicles and several depots. The demands of the customers are typically larger than the capacity of the vehicles which means that most customers are visited several times. This is a split delivery vehicle routing problem with additional constraints. We first propose a two phase solution method that assigns deliveries to the vehicles, and then builds vehicle routes. Both subproblems are formulated as integer linear programming problems. We then show how to combine the two phases in a single integer linear program. Experiments on real life instances are performed to compare the performance of the two solution methods.     

A goal programming approach to vehicle routing problems with soft time windows

The classical vehicle routing problem involves designing a set of routes for a fleet of vehicles based at one central depot that is required to serve a number of geographically dispersed customers, while minimizing the total travel distance or the total distribution cost. Each route originates and terminates at the central depot and customers demands are known. In many practical distribution problems, besides a hard time window associated with each customer, defining a time interval in which the customer should be served, managers establish multiple objectives to be considered, like avoiding underutilization of labor and vehicle capacity, while meeting the preferences of customers regarding the time of the day in which they would like to be served (soft time windows). This work investigates the use of goal programming to model these problems. To solve the model, an enumeration-followed-by-optimization approach is proposed which first computes feasible routes and then selects the set of best ones. Computational results show that this approach is adequate for medium-sized delivery problems.     

A memetic algorithm for the Multi Trip Vehicle Routing Problem

We consider the Multi Trip Vehicle Routing Problem, in which a set of geographically scattered customers have to be served by a fleet of vehicles. Each vehicle can perform several trips during the working day. The objective is to minimize the total travel time while respecting temporal and capacity constraints.     

考虑客户满意度的同时收发车辆路径问题

当客户要求车辆一次性完成发送以及收集货物的任务时, 只需考虑车辆的路径安排即可.但若客户进一步提出在时间窗内完成的话,就必须考虑客户的等待时间--客户的满意度的衡量标准,等待时间越短满意度越高.因此问题的目标为最小化车辆路径总长度、最小化所有客户等待时间之和.本文通过加权转变为单目标函数,由最邻近法及最廉价插入法得到初始解后经过禁忌搜索算法可得到改进算法,解并通过实例对不同权参数的情况进行了比较.     

Service network design models for two-tier city logistics

This paper focuses on two-tier city logistics systems for advanced management of urban freight activities and, in particular, on the first layer of such systems where freight is moved from distribution centers on the outskirts of the city to satellite platforms by urban vehicles, from where it will be distributed to customers by a different fleet of dedicated vehicles. We address the issue of planning the services of this first tier system, that is, select services, their routes and schedules, and determine the itineraries of the customer-demand flows through these facilities and services. We propose a general scheduled service network design modelling framework that captures the fundamental concepts related to the definition of urban-vehicle tactical plans within a two-tier distribution network. We examine several operational assumptions regarding the management of the urban-vehicle fleet and the flexibility associated with the delivery of goods, and show how the proposed modelling framework can evolve to represent an increasing level of detail. A discussion of algorithmic perspectives completes the paper.     

A greedy look-ahead heuristic for the vehicle routing problem with time windows

In this paper we consider the problem of physically distributing finished goods from a central facility to geographically dispersed customers, which pose daily demands for items produced in the facility and act as sales points for consumers. The management of the facility is responsible for satisfying all demand, and promises deliveries to the customers within fixed time intervals that represent the earliest and latest times during the day that a delivery can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns such as vehicle capacity, delivery time intervals and all relevant costs. The model, which is a case of the vehicle routing problem with time windows, is solved using a new heuristic technique. Our solution method, which is based upon Atkinson's greedy look-ahead heuristic, enhances traditional vehicle routing approaches, and provides surprisingly good performance results with respect to a set of standard test problems from the literature. The approach is used to determine the vehicle fleet size and the daily route of each vehicle in an industrial example from the food industry. This actual problem, with approximately two thousand customers, is presented and solved by our heuristic, using an interface to a Geographical Information System to determine inter-customer and depot–customer distances. The results indicate that the method is well suited for determining the required number of vehicles and the delivery schedules on a daily basis, in real life applications.     

A column generation algorithm for the vehicle routing problem with soft time windows

The Vehicle Routing Problem with Time Windows consists of computing a minimum cost set of routes for a fleet of vehicles of limited capacity visiting a given set of customers with known demand, with the additional constraint that each customer must be visited in a specified time window. We consider the case in which time window constraints are relaxed into “soft” constraints, that is penalty terms are added to the solution cost whenever a vehicle serves a customer outside of his time window. We present a branch-and-price algorithm which is the first exact optimization algorithm for this problem.     

An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

The vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem. It arises when each vehicle performs several routes during the workday due to strict time limits on route duration (e.g., when perishable goods are transported). The routes are defined over customers with a revenue, a demand and a time window. Given a fixed-size fleet of vehicles, it might not be possible to serve all customers. Thus, the customers must be chosen based on their associated revenue minus the traveling cost to reach them. We introduce a branch-and-price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation, using column generation. The pricing subproblems are elementary shortest path problems with resource constraints. Computational results are reported on euclidean problems derived from well-known benchmark instances for the vehicle routing problem with time windows.     

A multi-level composite heuristic for the multi-depot vehicle fleet mix problem

The problem of simultaneously allocating customers to depots, finding the delivery routes and determining the vehicle fleet composition is addressed. A multi-level composite heuristic is proposed and two reduction tests are designed to enhance its efficiency. The proposed heuristic is tested on benchmark problems involving up to 360 customers, 2 to 9 depots and 5 different vehicle capacities. When tested on the special case, the multi-depot vehicle routing, our heuristic yields solutions almost as good as those found by the best known heuristics but using only 5 to 10% of their computing time. Encouraging results were also obtained for the case where the vehicles have different capacities.     

A worst-case analysis for the split delivery vehicle routing problem with minimum delivery amounts

In the vehicle routing problem (VRP), a fleet of vehicles must service the demands of customers in a least-cost way. In the split delivery vehicle routing problem (SDVRP), multiple vehicles can service the same customer by splitting the deliveries. By allowing split deliveries, savings in travel costs of up to 50 % are possible, and this bound is tight. Recently, a variant of the SDVRP, the split delivery vehicle routing problem with minimum delivery amounts (SDVRP-MDA), has been introduced. In the SDVRP-MDA, split deliveries are allowed only if at least a minimum fraction of a customer’s demand is delivered by each visiting vehicle. We perform a worst-case analysis on the SDVRP-MDA to determine tight bounds on the maximum possible savings.     

Simple Heuristics for the Vehicle Routeing Problem with Soft Time Windows

We describe three simple heuristics for the vehicle routeing problem with customer time windows that can be violated by paying appropriate penalties. The customer demands are known, and a homogeneous fleet of vehicles stationed at a single depot is considered. The penalty payable to a customer is assumed to be a linear function of the amount of time window violation. Upper limits are imposed on both the penalty payable and the waiting time allowed at any customer. At each customer in a route, the PC-based heuristics simultaneously determine the actual time to begin service, and the next customer to serve. To achieve this, each heuristic uses different measures to compare the cost of waiting and penalty payable, with the benefit obtained by leaving immediately for the next customer. Computational results on a set of benchmark problems show that the procedure is efficient and enables significant reduction in the number of vehicles required and/or the total route distances while controlling both customer penalties and waiting times.     

Minimising average passenger waiting time in personal rapid transit systems

Personal Rapid Transit (PRT) is an emerging urban transport mode. A PRT system operates much like a conventional hackney taxi system, except that the vehicles are driven by computer (no human driver) between stations in a dedicated network of guideways. The world’s first two PRT systems began operating in 2010 and 2011. In both PRT and taxi systems, passengers request immediate service; they do not book ahead. Perfect information about future requests is therefore not available, but statistical information about future requests is available from historical data. If the system does not use this statistical information to position empty vehicles in anticipation of future requests, long passenger waiting times result, which makes the system less attractive to passengers, but using it gives rise to a difficult stochastic optimisation problem. This paper develops three lower bounds on achievable mean passenger waiting time, one based on queuing theory, one based on the static problem, in which it is assumed that perfect information is available, and one based on a Markov Decision Process model. An evaluation of these lower bounds, together with a practical heuristic developed previously, in simulation shows that these lower bounds can often be nearly attained, particularly when the fleet size is large. The results also show that low waiting times and high utilisation can be simultaneously obtained when the fleet size is large, which suggests important economies of scale.     

Probabilistic analysis of a capactiated vehicle routing problem—I

A fleet of vehicles located at a service center must serve the demands of a set of customers. The amount delivered by each vehicle cannot exceed its capacity and a customer’s demand may not be split over more than one vehicle. In our model, customers locations, as well as their demands are independent identically distributed. Simchi-Levi and Bramel [10] determined the asymptotic value of the optimal solution in this model. We prove here a sharp rate of convergence to the asymptotic value     

Efficient Heuristics for the Heterogeneous Fleet Multitrip VRP with Application to a Large-Scale Real Case

The basic Vehicle Routing Problem (VRP) consists of computing a set of trips of minimum total cost, to deliver fixed amounts of goods to customers with a fleet of identical vehicles. Few papers address the case with several types of vehicles (heterogeneous fleet). Most of them assume an unlimited number of vehicles of each type, to dimension the fleet from a strategic point of view. This paper tackles the more realistic tactical or operational case, with a fixed number of vehicles of each type, and the optional possibility for each vehicle to perform several trips. It describes several heuristics, including a very efficient one that progressively merges small starting trips, while ensuring that they can be performed by the fleet. This heuristic seeks to minimize the number of required vehicles as a secondary objective. It outperforms classical VRP heuristics, can easily handle various constraints, and gives very good initial solutions for a tabu search method. The real case of a French manufacturer of furniture with 775 destination stores is presented.     

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

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

Waiting time in a multi-server cutoff-priority gueue, and its application to an urban ambulance service

We consider a priority queue in steady state with N servers, two classes of customers, and a cutoff service discipline. Low priority arrivals are "cut off" (refused immediate service) and placed in a queue whenever N1 or more servers are busy, in order to keep N-N1 servers free for high priority arrivals. A Poisson arrival process for each class, and a common exponential service rate, are assumed. Two models are considered: one where high priority customers queue for service and one where they are lost if all servers are busy at an arrival epoch. Results are obtained for the probability of n servers busy, the expected low priority waiting time, and (in the case where high priority customers do not queue) the complete low priority waiting time distribution. The results are applied to determine the number of ambulances required in an urban fleet which serves both emergency calls and low priority patients transfers.     

带时间窗偏好的同时配集货且需求可拆分车辆路径问题

针对带时间窗偏好的同时配集货且需求可拆分车辆路径问题,最小化派遣成本、理货成本、时间窗惩罚成本以及油耗成本之和,建立数学模型。设计混合遗传变邻域搜索算法求解问题,在算法中引入时空距离的理念,首先用最近邻插入法和Logistic映射方程生成初始种群;然后利用变邻域搜索算法的深度搜索能力优化算法;提出自适应搜索策略,平衡种群进化所需的广度和深度;设计拆分准则,为各客户设置不同的拆分服务量;提出确定车辆最优出发时间的时差推移法,减少车辆在客户处的等待时间;最后通过多组算例验证本文模型和算法的有效性。     

The open vehicle routing problem with time windows

In this paper, we consider the open vehicle routing problem with time windows (OVRPTW). The OVRPTW seeks to find a set of non-depot returning vehicle routes, for a fleet of capacitated vehicles, to satisfy customers’ requirements, within fixed time intervals that represent the earliest and latest times during the day that customers’ service can take place. We formulate a comprehensive mathematical model to capture all aspects of the problem, and incorporate in the model all critical practical concerns. The model is solved using a greedy look-ahead route construction heuristic algorithm, which utilizes time windows related information via composite customer selection and route-insertion criteria. These criteria exploit the interrelationships between customers, introduced by time windows, that dictate the sequence in which vehicles must visit customers. Computational results on a set of benchmark problems from the literature provide very good results and indicate the applicability of the methodology in real-life routing applications.