Robust Inventory Management: An Optimal Control Approach

We formulate and solve static and dynamic models of inventory management that lie at the intersection of robust optimization and optimal control theory. Our objective is to minimize cumulative ordering, holding, and shortage costs over a horizon [0, T], where the variable is a nonnegative ordering rate function q(t) is an element of L-2[0, T]. The demand rate function d(t) is unknown and is only assumed to belong to an uncertainty set Omega={d(t)is an element of L-2[0,T]:mu(a) <= (1/T) integral(t)(0) d(t)dt <= mu(b),a <= d(t) <= b,for all t is an element of[0,T]}; this set is motivated by the strong law of large numbers for stochastic processes lim(T ->infinity)(1/T)integral(T)(0)d(t)dt=mu, where mu is the mean drift. We analyze a static model, where the ordering rate function must be fully specified at time zero, and three dynamic variants, where re-optimizations are allowed during the planning horizon [0, T] at prespecified review epochs. In the dynamic models, at review epoch tau is an element of [0, T], the past demand on [0, tau) is observable. In the first dynamic model, we ignore this information, and define a variant of Omega that is well formed for the remaining planning horizon [tau, T]. In the second model, we define a variant of Omega for [tau, T] that utilizes the past demand information, though we make a simplifying technical assumption about the consistency of the demand on [0, tau) and Omega. In the third dynamic model, we remove this assumption, and we remedy the arising complications using the Hilbert Projection Theorem. In all cases we derive optimal closed-form ordering rate functions that equal either the bounds a or b, or weighted averages of these bounds (sb + ha)/(s + h) or (sa + hb)/(s + h), where s and h are the shortage and holding costs, respectively. The strategies differ by when these four ordering rates are applied, which is determined by an uncertainty-set-dependent partition of the remaining planning horizon. Computational experiments, focused on studying the dynamic variants, supplement the analytical results, and demonstrate that (1) the three variants exhibit comparable performance under well-behaved stochastic demand and (2) the third variant has a significant advantage when demand is seasonal, especially when the review frequency is appropriately selected. Finally, computational comparisons with the omniscient strategy q(t) = d(t), for all t is an element of [0, T], are encouraging.