This paper addresses the problem of locating a service facility on an undirected network having unreliable edges so that some equitable performance level of network service to the demand nodes is provided, termed as the reliable 1–center problem. Equivalently, a point is sought on a network that maximizes service availability to the most critical demand node, i.e., the node associated with the least service availability. Two sub–problems of the reliable 1–center problem are considered based on the form of the objective function: (a) the reli–minmax problem in which the maximum expected number of unsuccessful responses to demand requests over all nodes is minimized, and (b) the reli–maxmin problem in which the minimum expected number of successful responses to demand requests over all nodes is maximized. The objective functions of these two sub–problems are based on the calculation of 2–terminal reliabilities, which is#P–Complete for general networks. By modeling the operational probability of the edges as an exponential function of the physical distance, we present linear time algorithms that solve both sub–problems when applied on weighted or unweighted tree networks.
