Most of the offline businesses do not run continuously for the entire day, therefore, the concept of idle time is inevitable. Natural idle time is referred to as the closing time when no demand is being fulfilled by the seller, the unproductive time when labor is not getting utilized but could be. In view of this, this paper investigates replenishment policies for the retailer who runs his business for only a part of a day and closes his shop for the remaining time. The demand gets fulfilled only during the opening time period while no customer is entertained during the closing part of the day. The retailer also faces issues of imperfect quality items in the lot received from his supplier. Thus, he carries out a rigorous inspection process of the entire lot so as to fulfill the demand with perfect quality items only. Under the above-stated circumstances, it is difficult to avoid shortages in the model. Thus, the model assumes fully backlogged shortages and is solved under a profit maximizing framework. The model is exemplified to understand its behavior. Further, to gain some managerial insights and to check the robustness of the model parameters, detailed sensitivity analysis is performed.