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We consider a network in which a set of vehicles is responsible for the pickup and delivery of messages that arrive according to Poisson process with message pickup and delivery locations distributed uniformly at random in a region of bounded area A. The vehicles are required to pickup and deliver the messages so that the average delay is minimized. In this paper, we provide lower bounds on the delay achievable by fully controlled policies, depending on the information constraint in place. We prove that for any policies in which only the source location information is known upon message arrival, the optimal average delay scaling is Θ(λ(n)A/v 2 n). If in addition to source location, destination locations of messages are known to the vehicles, the optimal average delay scaling can be reduced to Θ(λ(n)A/v 2 n 3/2). We note that these scaling bounds are achievable given the service policies we have previously described in .
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... In some recent work researchers have obtained asymptotic performance scalings for certain multi-agent systems. Let us mention [9, 10] on various dynamic routing problems. One can also find in the computer science litterature a large number of papers focusing on the minimum number of agents necessary to perform certain tasks: for instance, the minimum number of pursuers to catch an evader , or the minimum number of guards for an art gallery . ...
In this paper, we consider a simple linear exponential quadratic Gaussian (LEQG) tracking problem for a multi-agent system. We study the dynamical behaviors of the group as we vary the risk-sensitivity parameter, comparing in particular the risk averse case to the LQG case. Then we consider the evolution of the performance per agent as the number of agents in the system increases. We provide some analytical as well as simulation results. In general, more agents are beneficial only if noisy agent dynamics and/or imperfect measurements are considered. The critical value of the risk sensitivity parameter above which the cost becomes infinite increases with the number of agents. In other words, for a fixed positive value of this parameter, there is a minimum number of agents above which the cost remains finite.
In pickup and delivery problems vehicles have to transport loads from origins to destinations without transshipment at intermediate locations. In this paper, we discuss several characteristics that distinguish them from standard vehicle routing problems and present a survey of the problem types and solution methods found in the literature.
In a previous paper , we introduced a new model for stochastic and dynamic vehicle routing called the dynamic traveling repairman problem (DTRP), in which a vehicle traveling at constant velocity in a Euclidean region must service demands whose time of arrival, location and on-site service are stochastic. The objective is to find a policy to service demands over an infinite horizon that minimizes the expected system time (wait plus service) of the demands. We showed that the stability condition did not depend on the geometry of the service region (i.e. size, shape, etc.). In addition, we established bounds on the optimal system time and proposed an optimal policy in light traffic and several policies that have system times within a constant factor of the lower bounds in heavy traffic. We showed that the leading behavior of the optimal system time had a particularly simple form which increases much more rapidly with traffic intensity than the system time in traditional queues (e.g. M/G/1). In this paper, we extend these results in several directions. First, we propose new bounds and policies for the problem of m identical vehicles with unlimited capacity and show that in heavy traffic the system time is reduced by a factor of 1/m2 over the single server case. Policies based on dividing the service region into m equal subregions
We analyze a class of stochastic and dynamic vehicle routing problems in which demands arrive randomly over time and the objective is minimizing waiting time. In our previous work [Oper. Res. 39, No. 4, 601-615 (1991; Zbl 0736.90027) and ibid. 41, No. 1, 60-76 (1993; Zbl 0776.90018)], we analysed this problem for the case of uniformly distributed demand locations and Poisson arrivals. In this paper, using quite different techniques, we are able to extend our results to the more realistic case where demand locations have an arbitrary continuous distribution and arrivals follow only a general renewal process. Further, we improve significantly the best known lower bounds for this class of problems and construct policies that are provably within a small constant factor relative to the optimal solution. We show that the leading behavior of the optimal system time has a particularly simple form that offers important structural insight into the behavior of the system. Moreover, by distinguishing two classes of policies our analysis shows an interesting dependence of the system performance on the demand distribution.
We propose and analyze a generic mathematical model for dynamic, stochastic vehicle routing problems, the dynamic traveling repairman problem (DTRP). The model is motivated by applications in which the objective is to minimize the wait for service in a stochastic and dynamically changing environment. This is a departure from classical vehicle routing problems where one seeks to minimize total travel time in a static, deterministic environment. Potential areas of application include repair, inventory, emergency service and scheduling problems. The DTRP is defined as follows: Demands for service arrive in time according to a Poisson process, are independent and uniformly distributed in a Euclidean service region, and require an independent and identically distributed amount of on-site service by a vehicle. The problem is to find a policy for routing the service vehicle that minimizes the average time demands spent in the system. We propose and analyze several policies for the DTRP. We find a provably optimal policy in light traffic and several policies with system times within a constant factor of the optimal policy in heavy traffic. We also show that the waiting time grows much faster than in traditioal queues as the traffic intensity increases, yet the stability condition does not depend on the system geometry.
We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.
Previous research in the tradeoff between throughput and delay in wireless networks has focused on networks with fixed nodes or nodes with random mobility. In this paper, motivated to study fundamental limits of delay in a wireless network with maximal throughput, we examine delay scaling in networks in which the mobile nodes may control their own motion and schedule the pickup and delivery of messages so as to achieve maximal throughput while minimizing delay. We find that by letting nodes control their mobility, even with very minimal information, results in significant delay reduction.
In this paper a stochastic and dynamic model for the Pick-up and Delivery Problem is developed and analyzed. Demands for service arrive according to a Poisson process in time. The pick-up locations of the demands are independent and uniformly distributed over a service region. A single vehicle must transport the demands from the pick-up to the delivery location. Once a demand has been picked up it can only be dropped off at its desired delivery location. The delivery locations are independent and uniformly distributed over the region, and they are independent of the pick-up locations. The objective is to minimize the expected time in the system for the demands. Unit-capacity vehicle and multiple-capacity vehicle variations are considered. For each variation, bounds on the performance of the routing policies are derived for light and heavy traffic. The policies are analyzed using both analytical methods and simulation.
: Dynamic vehicle routing and dispatching refers to a wide range of problems where information on the problem is revealed to the decision maker concurrently with the determination of the solution. These problems have recently emerged as an active area of research due to recent technological advances that allow real-time information to be quickly obtained and processed. There are probably as many variants of these problems as there are real-world applications. After a brief overview of this broad domain, the paper will then focus on problems motivated by courier services and demand responsive transportation systems. These systems typically evolve within a local service area over a relatively short period of time (typically, a few hours), thus putting stringent response time requirements on the decision maker. They are also characterized by a strong routing component: many tasks can be allocated at once to the same vehicle and these tasks must be appropriately sequenced. The ...
Vrp with pickup and delivery,” in The Vehicle Routing Problem
G. Desaulniers, J. Desrosiers, A. Erdmann, M. Solomon, and F. Soumis, “Vrp with pickup and delivery,” in The Vehicle Routing Problem, P. Toth and D. Vigo, Eds. Mathematics and Applications, 2002, pp. 225–242. SIAM Monographs on Discrete