The 1-median and 1-antimedian problems with continuous probabilistic demand weights

We discuss the 1-median and 1-antimedian problems with probabilistic demand. It is assumed that demand weights of users generated at nodes of the network are independent continuous random variables. In the 1-median problem with probabilistic demand the objective is to find a location of a desirable facility on a network that maximizes the probability that the weighted sum distance does not exceed some pre-determined value T. In the 1-antimedian problem the objective is to locate an undesirable facility and thus we maximize the probability that the total weighted distance is at least T. We discuss how to solve the problems under arbitrary distributions and for small and large networks.