This paper considers two types of imperfect advance demand information. The advance demand information is given each day, and the actual arrival time is stochastic. In a known demand lead time model, the arrival date of actual demand, which is called demand lead time, is informed, and it is fixed. In unknown demand lead time model, the demand information does not include the demand lead time. In both cases, the actual demand may not occur with a given probability, and urgent demand may appear each day, which requires a product in the same day. The ordering, lost sale, holding and return costs are incurred. In this paper, these two cases are compared under the same situation except demand lead time information. To derive the optimal policy on order and return, each of the models is formulated as a Markov decision process. The optimal ordering policy in each day is derived, which minimizes the total expected cost over a finite horizon. The experimental results show that the total expected cost under the optimal policy in the case of the known lead time is smaller than that in the unknown lead time case, but the difference becomes small when the holding cost is small or the fraction of urgent demand is large.