  This article presents a framework and develops a formulation to solve a path
planning problem for multiple heterogeneous Unmanned Vehicles (UVs) with
uncertain service times for each vehicle--target pair. The vehicles incur a
penalty proportional to the duration of their total service time in excess of a
preset constant. The vehicles differ in their motion constraints and are
located at distinct depots at the start of the mission. The vehicles may also
be equipped with disparate sensors. The objective is to find a tour for each
vehicle that starts and ends at its respective depot such that every target is
visited and serviced by some vehicle while minimizing the sum of the total
travel distance and the expected penalty incurred by all the vehicles. We
formulate the problem as a two-stage stochastic program with recourse, present
the theoretical properties of the formulation and advantages of using such a
formulation, as opposed to a deterministic expected value formulation, to solve
the problem. Extensive numerical simulations also corroborate the effectiveness
of the proposed approach.
