Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated. The feasible region is formed as the intersection of two inequality fuzzy systems defined by frank family of t-norms is considered as fuzzy composition. First, the resolution of the feasible solutions set is studied where the two fuzzy inequality systems are defined with max-Frank composition. Second, some related basic and theoretical properties are derived. Then, a necessary and sufficient condition and three other necessary conditions are presented to conceptualize the feasibility of the problem. Subsequently, it is shown that a lower bound is always attainable for the optimal objective value. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. Finally, an algorithm is presented to solve the problem and an example is described to illustrate the algorithm. Additionally, a method is proposed to generate random feasible max-Frank fuzzy relational inequalities. By this method, we can easily generate a feasible test problem and employ our algorithm to it.