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     

The maximum reliability location problem and α-reliablep-center problem: Derivatives of the probabilistic location set covering problem

In the last several years, the modeling of emergency vehicle location has focussed on the temporal availability of the vehicles. Vehicles are not available for service when they are engaged in earlier calls. To incorporate this dynamic aspect into facility location decisions, models have been developed which provide additional levels of coverage. In this paper, two new models are derived from the probabilistic location set covering problem. These models allow the examination of the relationships between the number of facilities being located, the reliability that a vehicle will be available, and a coverage standard. In addition, these models incorporate sectoral specific estimates of the availability of the vehicles. Solution of these models reveals that the use of sectoral estimates leads to facility locations which are distributed to a greater spatial extent over the region to be serviced.     

Probabilistic Formulation of the Emergency Service Location Problem

The problem of locating emergency service facilities is studied under the assumption that the locations of incidents (accidents, fires, or customers) are random variables. The probability distribution for rectilinear travel time between a new facility location and the random location of the incident P _i is developed for the case of P _i being uniformly distributed over a rectangular region. The location problem is considered in a discrete space. A deterministic formulation is obtained and recognized to be a set cover problem. Probabilistic variation of the central facility location problem is also presented.An example and some computational experience are provided to emphasize the impact of the probabilistic formulation on the location decision.     

An analysis of p-median location problem: Effects of backup service level and demand assignment policy

Any solution to facility location problems will consider determining the best suitable locations with respect to certain criteria. Among different types of location problems, involving emergency service system (ESSs) are one of the most widely studied in the literature, and solutions to these problems will mostly aim to minimize the mean response time to demands. In practice, however, a demand may not be served from its nearest facility if that facility is engaged in serving other demands. This makes it a requirement to assign backup services so as to improve response time and service quality. The level of backup service is a key, strategic-level planning factor, and must be taken into consideration carefully. Moreover, in emergency service operations conducted in congested demand regions, demand assignment policy is another important factor that affects the system performance. Models failing to adopt sufficient levels of backup service and realistic demand assignment policies may significantly deteriorate the system performance.Considering the classic p-median problem (pMP) location model, this paper investigates the effects of backup service level, demand assignment policy, demand density, and number of facilities and their locations on the solution performance in terms of multiple metrics. For this purpose, we adopt a combined optimization and simulation approach. We will first modify the classic pMP to account for distances to backup services. Next, we employ a discrete event simulation to evaluate the performance of location schemes obtained from the deterministic mathematical model. Our results provide insights for decision-makers while planning ESS operations.     

带退货和软时间窗的多仓库选址-路径问题研究

选址-路径问题(location routing problems, LRP)是集成物流网络研究中的难题,也是任何一个大型物流配送企业必须面对的管理决策问题。本文在仓库容量约束和车辆容量约束的基础上,结合送取货一体化的配送模式和客户服务时间要求,建立了带退货和软时间窗的多仓库选址-路径(MDLRP)数学模型。针对MDLRP问题求解的复杂性,引入局部搜索算法和重组策略,设计了自适应混合遗传算法,对模型进行整体求解。最后进行数值实验,表明本文提出的模型和改进算法具有实用性和优越性,可为选址和车辆运输决策提供重要参考依据。     

A genetic algorithm for service level based vehicle scheduling

In many practical applications, vehicle scheduling problems involve more complex evaluation criteria than simple distance or travel time minimization. Scheduling to minimize delays between the accumulation and delivery of correspondence represents a class of vehicle scheduling problems, where: the evaluation of candidate solutions is costly, there are no efficient schemes for evaluation of partial solutions or perturbations to existing solutions, and dimensionality is limiting even for problems with relatively few locations. Several features of genetic algorithms (GA's) suggest that they may have advantages relative to alternative heuristic solution algorithms for such problems. These include ease of implementation through efficient coding of solution alternatives, simultaneous emphasis on global as well as local search, and the use of randomization in the search process. In addition, a GA may realize advantages usually associated with interactive methods by replicating the positive attributes of existing solutions in the search process, without explicitly defining or measuring these attributes. This study investigates these potential advantages through application of a GA to a service level based vehicle scheduling problem. The procedure is demonstrated for a vehicle scheduling problem with 15 locations where the objective is to minimize the time between the accumulation of correspondence at each location and delivery to destination locations. The results suggest that genetic algorithms can be effective for finding good quality scheduling solutions with only limited search of the solution space.     

The rollon–rolloff waste collection vehicle routing problem with time windows

This study introduces a rollon–rolloff waste collection vehicle routing problem involving large containers that accumulate huge amounts of garbage at construction sites and shopping districts. In this problem, tractors move one container at a time between customer locations, a depot, disposal facilities, and container storage yards. The complicated constraints discussed in this study arise from having multiple disposal facilities, multiple container storage yards, seven service types of customer demands, different time windows for customer demands and facilities, various types and sizes of containers, and the lunch break of tractor drivers. In addition, real-world issues, such as changing service types, multiple demands at a customer’s location, and tractors with different work schedules, are dealt with. This study proposes a large neighborhood search based iterative heuristic approach consisting of several algorithms for the problem. The effectiveness of the proposed methods is demonstrated by computational experiments using benchmark data, some instances of which are derived from real-world problems.     

A new mixed integer linear model for a rich vehicle routing problem with docking constraints

In this paper we address a rich vehicle routing problem that arises in real-life applications. Among other aspects we consider time windows, simultaneous delivery and pick-up at customer locations and multiple use of vehicles. To guarantee a coordinated material flow at the depot, we include the timed allocation of vehicles to loading bays at which the loading and unloading activities can occur. The resulting vehicle routing problem is formulated as a two-index vehicle-flow model which integrates the routing under real-life conditions and the assignment of vehicles to loading bays at the depot. We use CPLEX 11.0 to solve medium-sized instances that are derived from the extended Solomon test set. The selective implementation of preprocessing techniques and cutting planes improves the solver performance significantly.     

Optimal location of dwell points in a single loop AGV system with time restrictions on vehicle availability

Since the workload of a manufacturing system changes over time, the material handling equipment used in the facility will be idle at certain time intervals to avoid system overload. In this context, a relevant control problem in operating an automated guided vehicle (AGV) system is where to locate idle vehicles. These locations, called dwell points, establish the response times for AVG requests. In this article, a dynamic programming algorithm to solve idle vehicle positioning problems in unidirectional single loop systems is developed to minimize the maximum response time considering restrictions on vehicle time available to travel and load/unload requests. This polynomial time algorithm finds optimal dwell points when all requests from a given pick-up station are handled by a single AGV. The proposed algorithm is used to study the change in maximum response time as a function of the number of vehicles in the system.     

基于时间敏感需求及随机完工期的承诺交货期决策

在新的经济形势和高新技术指引下,产品更新换代的速度加快,基于时间的竞争成为供应链竞争的焦点.在制造商为核心的供应链中,产品需求除了与价格有关外,与承诺交货期也有一定关联,且部分客户愿意为快速交货而支付更高价格.当需求与时间及价格具有敏感性且实际完工期服从一定的随机分布时,建立利润最大化及服务水平约束的承诺交货期决策模型,并对模型进行讨论及优化分析.通过算例验证了模型的有效性,通过参数敏感性分析得出的结论是:当客户服务水平达到一定阈值时,最优承诺交货期将发生改变;价格及交货期敏感系数不影响承诺交货期,但影响产品需求及最终利润;最优承诺交货期与单位提早完工成本是反向变动的关系而与单位延迟完工成本是正向变动的关系;随着完工期均值及标准差的不断增大,最优承诺交货期呈上升趋势,利润、市场需求及价格不断下降.     

A branch and cut algorithm for the location-routing problem with simultaneous pickup and delivery

This paper addresses a location-routing problem with simultaneous pickup and delivery (LRPSPD) which is a general case of the location-routing problem. The LRPSPD is defined as finding locations of the depots and designing vehicle routes in such a way that pickup and delivery demands of each customer must be performed with same vehicle and the overall cost is minimized. We propose an effective branch-and-cut algorithm for solving the LRPSPD. The proposed algorithm implements several valid inequalities adapted from the literature for the problem and a local search based on simulated annealing algorithm to obtain upper bounds. Computational results, for a large number of instances derived from the literature, show that some instances with up to 88 customers and 8 potential depots can be solved in a reasonable computation time.     

The use of simple calibrations of individual locations in making transshipment decisions in a multi-location inventory network

Demands occur at each location in a network of stock-holding retail outlets. Should a location run out of stock between successive replenishments, then subsequent demands may be met either by transshipping from another location in the network or by an emergency supply from a central depot. We deploy an approximate stochastic dynamic programming approach to develop a class of interpretable and implementable heuristics for making transshipment decisions (whether and from where to transship) which make use of simple calibrations of the candidate locations. The calibration for a location depends upon its current stock, the time to its next replenishment and the identity of the location needing stock. A numerical investigation shows strong performance of the proposed policies in comparison with standard industry practice (complete pooling, no pooling) and a recently proposed heuristic. It points to the possibility of substantial cost savings over current practice.     

A Simple and Effective Component Procurement Policy for Stochastic Assembly Systems

We examine the component procurement problem in a single-item, make-to-stock assembly system. The suppliers are uncapacitated and have independent but non-identically distributed stochastic delivery lead times. Assembly is instantaneous, product demand follows a Poisson process and unsatisfied demand is backordered. The objective is to minimize the sum of steady-state holding and backorder costs over a pre-specified class of replenishment policies. To keep the analysis tractable, we impose a synchronization assumption that no mixing occurs between sets of component orders. Combining existing results from queueing theory with original results concerning distributions that are closed under maximization and translation, we derive a simple approximate solution to the problem when lead time variances are identical. In simulations, our derived policy is within 2% of optimal and significantly outperforms policies that ignore either component dependence or lead time stochasticity. It is also quite robust with respect to various model assumptions, except the synchronization one.     

The stochastic generalised assignment problem with Bernoulli demands

This paper deals with the Stochastic Generalised Assignment problem. It presents several models for the special case when demands are independent and Bernoulli distributed. Each model designs an assignment structure before the demands are known. Two policies are considered to handle infeasibilities in particular instances of the demands vector. Model performances are compared under both policies.     

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.     

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.     

Dynamic demand fulfillment in spare parts networks with multiple customer classes

We study real-time demand fulfillment for networks consisting of multiple local warehouses, where spare parts of expensive technical systems are kept on stock for customers with different service contracts. Each service contract specifies a maximum response time in case of a failure and hourly penalty costs for contract violations. Part requests can be fulfilled from multiple local warehouses via a regular delivery, or from an external source with ample capacity via an expensive emergency delivery. The objective is to minimize delivery cost and penalty cost by smartly allocating items from the available network stock to arriving part requests. We propose a dynamic allocation rule that belongs to the class of one-step lookahead policies. To approximate the optimal relative cost, we develop an iterative calculation scheme that estimates the expected total cost over an infinite time horizon, assuming that future demands are fulfilled according to a simple static allocation rule. In a series of numerical experiments, we compare our dynamic allocation rule with the optimal allocation rule, and a simple but widely used static allocation rule. We show that the dynamic allocation rule has a small optimality gap and that it achieves an average cost reduction of 7.9% compared to the static allocation rule on a large test bed containing problem instances of real-life size.     

Comparison of policies in dynamic routing problems

We consider a company that has to satisfy customers' pick-up requests arriving over time every day. The overall objective of the company is to serve as many requests as possible at a minimum operational cost. When organizing its business the company has to fix some features of the service that may affect both service quality and operational costs. Some of these features concern the time a request is taken into account to plan its service, the associated deadline and the way requests are managed when the system is overloaded. In this paper we analyse several policies that can be implemented by the management of a carrier company in a multi-period context. For example, a company might reject all the requests that cannot be feasibly scheduled or accept all the requests and rely on a backup service in order to serve requests that are difficult to handle. Another interesting issue considered in this paper is the impact of collaborative service where two or more carrier companies, with their own customers, decide to share customers in order to optimize the overall costs. We set up a general framework to allow comparison of alternative service policies. Extensive computational results evaluating the number of lost requests and the distance travelled provide interesting insights.     

Planning and approximation models for delivery route based services with price-sensitive demands

Classical vehicle routing problems typically do not consider the impact of delivery price on the demand for delivery services. Existing models seek the minimum sum of tour lengths in order to serve the demands of a given set of customers. This paper proposes approximation models to estimate the impacts of price on delivery services when demand for delivery service is price dependent. Such models can serve as useful tools in the planning phase for delivery service providers and can assist in understanding the economics of delivery services. These models seek to maximize profit from delivery service, where price determines demand for deliveries as well as the total revenue generated by satisfying demand. We consider a variant of the model in which each customer’s delivery volume is price sensitive, as well as the case in which customer delivery volumes are fixed, but the total number of customers who select the delivery service provider is price sensitive. A third model variant allows the delivery service provider to select a subset of delivery requests at the offered price in order to maximize profit.     

A tabu search heuristic for a routing problem arising in servicing of offshore oil and gas platforms

This paper introduces a pickup and delivery problem encountered in servicing of offshore oil and gas platforms in the Norwegian Sea. A single vessel must perform pickups and deliveries at several offshore platforms. All delivery demands originate at a supply base and all pickup demands are also destined to the base. The vessel capacity may never be exceeded along its route. In addition, the amount of space available for loading and unloading operations is limited at each platform. The problem, called the Single Vehicle Pickup and Delivery Problem with Capacitated Customers consists of designing a least cost vehicle (vessel) route starting and ending at the depot (base), visiting each customer (platform), and such that there is always sufficient capacity in the vehicle and at the customer location to perform the pickup and delivery operations. This paper describes several construction heuristics as well as a tabu search algorithm. Computational results are presented.