In this paper, we study a supervisory control problem under a constraint on memory size as well as a constraint described by a class of quantitative linear temporal logic, which enables us to consider how well the specification is satisfied. Our control objective is to design a finite-memory supervisor such that the satisfaction value of the specification formula on the controlled system is larger than or equal to a given threshold. We adopt a Safraless synthesis methodology and reduce the problem to solving a safety game parameterized with the memory size of a supervisor. On the safety game, we compute a winning strategy by leveraging a ranking function. Our definition of the ranking function relies on the product automaton, which captures both the behavior of the controlled plant and that of an automaton transformed from the specification and the threshold.