  This paper introduces two objective functions for computing the expected cost
in the Stochastic Collection and Replenishment (SCAR) scenario. In the SCAR
scenario, multiple user agents have a limited supply of a resource that they
either use or collect, depending on the scenario. To enable persistent
autonomy, dedicated replenishment agents travel to the user agents and
replenish or collect their supply of the resource, thus allowing them to
operate indefinitely in the field. Of the two objective functions, one uses a
Monte Carlo method, while the other uses a significantly faster analytical
method. Approximations to multiplication, division and inversion of Gaussian
distributed variables are used to facilitate propagation of probability
distributions in the analytical method when Gaussian distributed parameters are
used. The analytical objective function is shown to have greater than 99%
comparison accuracy when compared with the Monte Carlo objective function while
achieving speed gains of several orders of magnitude.
