The basic, deterministic, multi-item inventory problem is considered. The production rates (decision variables) are represented by nonnegative functions defined on the interval [0, T], T < ∞. The demands appear at discrete time points. The production cost consists of two parts. The first is a setup cost which depends on the joint production (or joint replenishment) rates. (The minor setup, connected with particular items, is zero.) The second part is described by a linear function of the production rates. The holding cost is linear. An auxiliary allocation problem is considered. Necessary and sufficient conditions for an optimal allocation are given. Under these conditions some qualitative properties of the multi-item joint replenishment inventory problem are obtained and a method of transformation to a one-item problem is given. A construction of the optimal allocation is presented.
