Abstract The paper presents the deterministic finite time horizon inventory lot size model, without backlogs and with no lead time, for a single commodity, with some specified markets. The specified markets are represented by the family b( t)= kt r of demand functions where k>0, r>−2 are known parameters and t stands for time, 0< t 0⩽ t⩽ T. The strict positivity of t 0. compared to the restrictive condition t 0=0 which has been already solved, is crucial and implies entirely different analytical techniques. An important special case is the affine function ( r = 1) partly treated already by Donaldson . The problem is to find the optimal schedule of replenishments, i.e., the number and timings of orders. The problem is completely resolved (compared to a recent heuristic by Silver ) and the solution is given in a closed form and is proven to be unique. Numerical examples are provided.