The concept of \(\varvec{k}\) -level opacity is used to determine the level of opacity in a system. If the tolerance level \(\varvec{k}\) is higher, then the degree of opacity for any secret information will also be higher. However, if more non-secret information are not distinguished from secret ones, then the cost of achieving a control strategy will also be higher. In this paper, an optimization model that minimizes the discount total choosing cost while maintaining \(\varvec{k}\) -level opacity and releasing all the secret string is presented. If the secret language is \(\varvec{k}\) -level opaque w.r.t. its closure, the closure or the infimal controllable super-language of the closure is the optimal solution. If the secret language is not \(\varvec{k}\) -level opaque w.r.t. its closure, then a Multi-Stage Optimization Problem with Multi-Selection at each Stage (MSOP-MSS) is proposed. By analyzing the MSOP-MSS, an algorithm for handling the situation was proposed and used to obtain control strategy for the optimization model. The control strategy has been proven to be optimal. Finally, a complete algorithm, computational complexity analysis, and application examples are described.
