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In the literature, one of the minimal solutions is an optimal solution of solving a linear objective function subject to fuzzy relational equations with the max-Archimedean composition. Since the objective function is nonlinear so that this characteristic can't be employed again to the optimization problem with a linear fractional objective function. In this paper, according to the characteristics of feasible domain of fuzzy relational equations with max-Archimedean t-norm composition, some theoretical results are presented for exploring such an optimization problem. These results can be employed to cut down the feasible domain first. Hence, the work of computing an optimal solution can be simplified. Then the simplified problem can be converted into traditional linear fractional programming problems and a simple procedure is proposed for optimizing such a problem.