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An minimization problem with a linear objective function subject to fuzzy relation equations using max-product composition has been considered by Loetamonphong and Fang. They first reduced the problem by exploring the special structure of the problem and then proposed a branch-and-bound method to solve this 0-1 integer programming problem. In this paper, we provide a necessary condition for an optimal solution of the minimization problems in terms of one maximum solution derived from the fuzzy relation equations. This necessary condition enables us to derive efficient procedures for solving such optimization problems. Numerical examples are provided to illustrate our procedures.