This paper focuses on a competitive facility location problem between leader and follower on a network with demands whose weights are given uncertainly and vaguely. By representing them as fuzzy random variables, the optimal location problem can be formulated as a fuzzy random programming problem for finding Stackelberg equilibrium. For solving the problem, it is reformulated as the problem to find the optimal solutions maximizing a degree of necessity under some chance constraint for the leader. Theorems for its complexity are shown based upon the characteristics of the facility location.