In 1991, D. J. Bertsimas and G. van Ryzin introduced and analyzed a model for stochastic and dynamic vehicle routing in which a single, uncapacitated vehicle traveling at a constant velocity in a Euclidian 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. This paper extends our analysis in several directions. First, we analyze the problem of m identical vehicles wtih unlimited capacity and show that that in heavy traffic the system time is reduced by a factor of 1/m2 over the single-server case. One of these policies improves by a factor of two on the est known system time for the single-server case. We then consider the case in which each vehicle can serve at most q customers before returning to a depot. We show that the stability condition in this case depends strongly on the geometry of the region. Several policies that have system times within a constant factor fo the optimum in heavy traffic are proposed. Finally, we discuss extensions to mixed travel cost and system time objectives.
Bertsimas, D. J., and Garrett van Ryzin. "Stochastic and dynamic vehicle routing in the Euclidean plane with multiple capacitated vehicles." Operations Research 41 (1993): 60-76.
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