In this paper, we apply the data envelopment analysis technique to propose a minimax model for the fixed cost allocation problem under two assumptions: all decision making units (DMUs) become strong CCR-efficient under common weights after the fixed cost allocation; the maximum relative difference between allocated costs and the corresponding scale sharing costs is minimized. In the pure input case and the pure output case, we deduce the analytical solution of the minimax model; in the general case, we propose an algorithm for the minimax model, which always produces a unique fixed cost allocation solution with good computational efficiency. Compared with other allocation methods based on the similar assumption, the new approach can avoid inappropriate gaps among costs with the help of appropriately allocating costs among DMUs according to their individual inputs and outputs. Numerical analysis shows the validity and superiorities of the new approach.