Facility location problems arise for planning and stationing serve centers. The solution of facility location problem can be modeled by a fixed point model. The hard point is that the collection of alternative sites is non-convex set while solving facility location problem between urban inner and outer ring expressway. Commonly used method for solving fixed point model requires the definition set of variables is a convex set. The contribution of this paper is to apply the theorem of existence of the solution of fixed point problem to discuss the facility location problem in this kind of non-convex set and find the numerical result by the pathway tracing method. Numerical example shows that the numerical method by tracing the homotopy pathway with predictor-corrector is an effective method to solving the facility location problem and a global optimal solution can be obtained.
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