In the present investigation a vertex covering center location model has been developed and modeled in stochastic environment. The problem has been dealt after relaxing the restrictions of the network and modeled as a multi-objective location problem where the existing demand points are not known but are instead random variables specified by probability distributions. Both bivariate uniform and bi-variate exponential distributions have been considered to design the models. Compromise solution methods for the different probability distributions have been developed. It has been proved that the solutions obtained by these procedures will give global optimal solutions. Recti-linear distance norm has been considered as the distance measure as it is more appropriate in different realistic situations.