321 research outputs found
Approximation Algorithms for the A Priori TravelingRepairman
We consider the a priori traveling repairman problem, which is a stochastic
version of the classic traveling repairman problem (also called the traveling
deliveryman or minimum latency problem). Given a metric with a root
, the traveling repairman problem (TRP) involves finding a tour
originating from that minimizes the sum of arrival-times at all vertices.
In its a priori version, we are also given independent probabilities of each
vertex being active. We want to find a master tour originating from
and visiting all vertices. The objective is to minimize the expected sum of
arrival-times at all active vertices, when is shortcut over the inactive
vertices. We obtain the first constant-factor approximation algorithm for a
priori TRP under non-uniform probabilities. Previously, such a result was only
known for uniform probabilities
The dynamic traveling repairman problem
Includes bibliographical references (p. 30-32).Partially supported by the National Science Foundation. ECS-8717970Dimitris Bertsimas, Garrett van Ryzin
Probabilistic bounds on the Traveling Salesman Problem and the Traveling Repairman Problem
The traveling salesman problem (-TSP) seeks a tour of minimal length
that visits a subset of points. The traveling repairman problem (TRP)
seeks a complete tour with minimal latency. This paper provides constant-factor
probabilistic approximations of both problems. We first show that the optimal
length of the -TSP path grows at a rate of
. The proof
provides a constant-factor approximation scheme, which solves a TSP in a
high-concentration zone -- leveraging large deviations of local concentrations.
Then, we show that the optimal TRP latency grows at a rate of . This result extends the classical Beardwood-Halton-Hammersley theorem to
the TRP. Again, the proof provides a constant-factor approximation scheme,
which visits zones by decreasing order of probability density. We discuss
practical implications of this result in the design of transportation and
logistics systems. Finally, we propose dedicated notions of fairness --
randomized population-based fairness for the -TSP and geographical fairness
for the TRP -- and give algorithms to balance efficiency and fairness
A stochastic and dynamic vehicle routing problem in the Euclidean plane
"February 1990."Includes bibliographical references (p. 29-31).Research supported by the National Science Foundation. DDM-9014751 Research supported by a grant from Draper Laboratory.Dimitris J. Bertsimas, Garrett van Ryzin
Stochastic Dynamic Vehicle Routing in the Euclidean Plane: The Multiple-Server, Capacitated Vehicle Case
In a previous paper [12], 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 subregion
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