It sounds surprising for me because all problems that I know to be solvable in psedopolynomial time can be solved in polynomial time within a constant factor. Can you formulate this problem?

Thanks for your attention to my questions. As a follow-up message, I report that I now have the answer to my earlier questions. It is not surprising that there are optimization problems that are neither in APX nor strongly NP-hard. In fact, there is no close connection between the separation boundary of strongly/weakly NP-hardness and that of approximability/in-approximability. For those who would like to have more details, please get in touch via individual communications.