In this paper we study a dynamic vehicle routing problem in which there are multiple vehicles and multiple classes of demands. Demands of each class arrive in the environment randomly over time and require a random amount of on-site service that is characteristic of the class. To service a demand, one of the vehicles must travel to the demand location and remain there for the required on-site service time. The quality of service provided to each class is given by the expected delay between the arrival of a demand in the class, and that demand's service completion. The goal is to design a routing policy for the service vehicles which minimizes a convex combination of the delays for each class. First, we provide a lower bound on the achievable values of the convex combination of delays. Then, we propose a novel routing policy and analyze its performance under heavy load conditions (i.e., when the fraction of time the service vehicles spend performing on-site service approaches one). The policy performs within a constant factor of the lower bound (and thus the optimal), where the constant depends only on the number of classes, and is independent of the number of vehicles, the arrival rates of demands, the on-site service times, and the convex combination coefficients.
Deep Dive into Dynamic Multi-Vehicle Routing with Multiple Classes of Demands.
In this paper we study a dynamic vehicle routing problem in which there are multiple vehicles and multiple classes of demands. Demands of each class arrive in the environment randomly over time and require a random amount of on-site service that is characteristic of the class. To service a demand, one of the vehicles must travel to the demand location and remain there for the required on-site service time. The quality of service provided to each class is given by the expected delay between the arrival of a demand in the class, and that demand’s service completion. The goal is to design a routing policy for the service vehicles which minimizes a convex combination of the delays for each class. First, we provide a lower bound on the achievable values of the convex combination of delays. Then, we propose a novel routing policy and analyze its performance under heavy load conditions (i.e., when the fraction of time the service vehicles spend performing on-site service approaches one). The
Consider a bounded environment E in the plane which contains n service vehicles. Demands for service arrive in E sequentially over time and each demand is a member of one of m classes. Upon arrival, a demand assumes a location in E, and requires a class dependent amount of on-site service time. To service a demand, one of the n vehicles must travel to the demand location and perform the on-site service. If we specify a policy by which the vehicles serve demands, then the expected delay for demands of class α, denoted D α , is the expected amount of time between a demands arrival and its service completion. Then, given coefficients c 1 , . . . , c m > 0, the goal is to find the vehicle routing policy that minimizes
By increasing the coefficients for certain classes, a higher priority level can be given to their demands. This problem, which we call dynamic vehicle routing with priority classes, has important applications in areas such as UAV surveillance, where targets are given different priority levels based on their urgency or potential importance.
In classical queuing theory (i.e., queuing systems in which the demands are not spatially distributed), the problem of priority queues has received much attention, [1]. In [2] the authors characterize the region of delays that are realizable by a single server. This analysis is performed under the assumption that the customer (demand) interarrival times and service times are distributed exponentially. In [3] the achievable delays are studied in more a general setting known as queuing networks.
If service demands are spatially distributed, then providing service becomes a problem in dynamic vehicle routing (DVR). One of the first DVR problems was the dynamic traveling repairperson problem (DTRP) [4,5]. The DTRP is the single class version of the dynamic vehicle routing with priority classes problem studied in this paper. In [4,5], the authors study the expected delay of demands and propose optimal policies in both heavy load (i.e., when the fraction of time the service vehicles spend performing on-site service approaches one), and in light load (i.e., when the fraction of time the service vehicles spends performing on-site service approaches zero). In [7], and [8], decentralized policies are developed for the DTRP. Spatial queuing problems have also been studied in the context of urban operations research [9], where approximations are used to cast the problems in the traditional queuing framework. In our previous paper [10], we introduced and studied dynamic vehicle routing with priority classes, for the case of two classes and one vehicle. For this case we derived a lower bound on the achievable delay values and proposed the Randomized Priority policy, which performed within a constant factor of the lower bound, for all convex combination coefficients.
The contributions of this paper are as follows. We extend the dynamic vehicle routing with priority classes problem to n service vehicles and m classes of demands. The extension of our previous analysis to multiple classes of demands is very nontrivial. We derive a new lower bound on the achievable values of the convex combination of delays, and propose a new policy in which each class of demands is served separately from the others. We show that the policy performs with a constant factor of 2m 2 of the optimal. Thus, the constant factor is independent of the number of vehicles, the arrival rates of demands, the on-site service times, and the convex combination coefficients. We also comment on the source of the gap between the upper and lower bounds.
The paper is organized as follows. In Section 2 we give some asymptotic properties of the traveling salesperson tour. In Section 2.2 we formalize the problem and in Section 3 we derive a lower bound, and in Section 4 we introduce and analyze the Separate Queues policy. Finally, in Section 5 we present simulation results.
In this section we summarize the asymptotic properties of the Euclidean traveling salesperson tour, and formalize dynamic vehicle routing with priority classes.
Given a set Q of N points in R 2 , the Euclidean traveling salesperson problem (TSP) is to find the minimum-length tour of Q (i.e., the shortest closed path through all points). Let TSP(Q) denote the minimum length of a tour through all the points in Q. Assume that the locations of the N points are random variables independently and identically distributed, uniformly in a compact set E with area |E|; in [11] it is shown that there exists a constant β TSP such that, almost surely, lim
The constant β TSP has been estimated numerically as β TSP ≈ 0.7120 ± 0.0002, [12]. The bound in equation ( 1) holds for all compact sets E, and the shape of E only affects the convergence rate to the limit. In [9], the authors note that if E is “fairly compact [square] and fairly convex”, then equation ( 1) provides an adequate estimate of the optimal TSP tour length for values of N as low as 15.
Conside
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