To access the efficient inventory system, managers should consider all the situations that have happened in reality. One of these situations is the presence of the defective items in each received lot and the other situation is being the group of customers that do not wait to fulfill their requirements from the vendor and choose another one to get their orders so the proportion of the backordered items becomes lost sales. In this paper we consider both mentioned situations simultaneously to model the inventory system while the proportion of backordering is constant and the imperfect rate follows a uniform distribution, also the particular sampling process is considered that is explained in detail in ''Problem definition''. Our purpose in this paper is to access the optimum value for the total revenue in a year by a particular solution method that is provided in ''Solution method''. After these sections we provide the numerical results in ''Numerical result'' to show the effect of sensitive parameters on the decision variables and the total profit.

This research presents an inventory system with partial backordering and imperfect process with a stochastic numbers of products that are defective per order. This problem is known and there are some customers that do not wait for their orders to be fulfilled so a particular proportion of backordered items become lost sales. This paper considers both mentioned situations simultaneously while the number of defective items follows a uniform distribution and the proportion of backordering is constant. This condition is modelled. The cost function of this inventory model includes order cost, holding cost and two types of shortage costs, one of them is related to backordered items and the other one is related to lost sales. This paper also provides a solution method to obtain optimum values for the decision variables, order quantity and total shortage, then we derive the value of the total cost function according to the optimum values obtained for the decision variables. Finally, some numerical results and diagrams are provided to show how some parameters affect the values of the decision variables and the cost function.