Time-Varying Demand Inventory Models

Inventory problems are typically separated into two classes of demands, deterministic and stochastic. In the realm of deterministic demands, the dominating assumption is that of uniform or constant demands. The EOQ formula (Harris, 1915) is the typical and much used prototype of the class of methods developed on the uniform demand assumption. The other extreme of discrete deterministic known demands can be solved by dynamic programming methods (Wagner and Whitten, 1958). The area that has had little attention is the more-realistic assumption of known but varying demand such as that which would be obtained from curve fitting historical demand data. This is the demand assumption considered in this analysis. The main basis of this work is the finite production model under the assumption of time varying known demand. We will illustrate how the time varying demand under instantaneous replenishment is readily solved via the proposed procedure. The general solution procedure is to fit the yearly time varying demands via a polynomial to any needed power. The function values, integrals of demand and expected inventory levels can then be solved efficiently and exactly based on this polynomial form and its integrals. Using Leibniz rule for the derivatives of integrals, the problems considered yield concise recursive relationships which can be solved numerically using most equation solvers for the optimal reorder points in time.