Probabilistic Analysis of Euclidean Capacitated Vehicle Routing

We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of n unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most k customers, where k is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when k is either o(?n) or ?(?n), and they asked whether the ITP algorithm was "also effective in the intermediate range". In this work, we show that when k = ?n, the ITP algorithm is at best a (1+c?)-approximation for some positive constant c?. On the other hand, the approximation ratio of the ITP algorithm was known to be at most 0.995+? due to Bompadre, Dror, and Orlin, where ? is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to 0.915+?. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest

The dynamic nearest neighbor policy for the multi-vehicle pick-up and delivery problem

In this paper, a dynamic nearest neighbor (DNN) policy is proposed for operating a fleet of vehicles to serve customers, who place calls in a Euclidean service area according to a Poisson process. Each vehicle serves one customer at a time, who has a distinct origin and destination independently and uniformly distributed within the service area. The new DNN policy is a refined version of the nearest neighbor (NN) policy that is well known to perform sub-optimally when the frequency of customer requests is high. The DNN policy maintains geographically closest customer-to-vehicle assignments, due to its ability to divert/re-assign vehicles that may be already en-route to pick up other customers, when another vehicle becomes available or a new customer call arrives. Two other pertinent issues addressed include: the pro-active deployment of the vehicles by anticipating in which regions of the service area future calls are more likely to arise; and, imposition of limits to avoid prohibitively long customer wait times. The paper also presents accurate approximations for all the policies compared. Extensive simulations, some of which are included herein, clearly show the DNN policy to be tangibly superior to the first-comefirst-served (FCFS) and NN policies

Applying revenue management to agent-based transportation planning

We consider a multi-company, less-than-truckload, dynamic VRP based on the concept of multi-agent systems. We focus on the intelligence of one vehicle agent and especially on its bidding strategy. We address the problem how to price loads that are offered in real-time such that available capacity is used in the most profitable way taking into account possible future revenues. We develop methods to price loads dynamically based on revenue management concepts.\ud We consider a one leg problem, i.e., a vehicle travels from i to j and can wait at most τ time units in which it can get additional loads from i to j. We develop a DP to price loads given a certain amount of remaining capacity and an expected number of auctions in the time-to-go. Because a DP might be impractical if parameters change frequently and bids has to be determined in real-time, we derived two approximations to speed up calculations. The performance of these approximations are compared with the performance of the DP. Besides we introduce a new measure to calculate the average vehicle utilisation in consolidated shipments. This measure can be calculated based on a limited amount of data and gives an indication of the efficiency of schedules and the performance of vehicles

Opportunity costs calculation in agent-based vehicle routing and scheduling

In this paper we consider a real-time, dynamic pickup and delivery problem with timewindows where orders should be assigned to one of a set of competing transportation companies. Our approach decomposes the problem into a multi-agent structure where vehicle agents are responsible for the routing and scheduling decisions and the assignment of orders to vehicles is done by using a second-price auction. Therefore the system performance will be heavily dependent on the pricing strategy of the vehicle agents. We propose a pricing strategy for vehicle agents based on dynamic programming where not only the direct cost of a job insertion is taken into account, but also its impact on future opportunities. We also propose a waiting strategy based on the same opportunity valuation. Simulation is used to evaluate the benefit of pricing opportunities compared to simple pricing strategies in different market settings. Numerical results show that the proposed approach provides high quality solutions, in terms of profits, capacity utilization and delivery reliability

A Computational Study of Genetic Crossover Operators for Multi-Objective Vehicle Routing Problem with Soft Time Windows

The article describes an investigation of the effectiveness of genetic algorithms for multi-objective combinatorial optimization (MOCO) by presenting an application for the vehicle routing problem with soft time windows. The work is motivated by the question, if and how the problem structure influences the effectiveness of different configurations of the genetic algorithm. Computational results are presented for different classes of vehicle routing problems, varying in their coverage with time windows, time window size, distribution and number of customers. The results are compared with a simple, but effective local search approach for multi-objective combinatorial optimization problems