Robust Inventory Management: An Optimal Control Approach

Abstract

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) 2 L 2 [0, T]. The demand rate function d(t) is unknown and is only assumed to belong to an uncertainty set ⌦ ⇤by the strong law of large numbers for stochastic processes limWe 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 ⌧ 2 [0, T], the past demand on [0, ⌧) is observable. In the first dynamic model, we ignore this information, and define a variant of ⌦ that is well formed for the remaining planning horizon [⌧, T]. In the second model, we define a variant of ⌦ for [⌧, T] that utilizes the past demand information, though we make a simplifying technical assumption about the consistency of the demand on [0, ⌧) and ⌦. 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 2 [0, T], are encouraging.

Comparison of different approaches to multistage lot sizing with uncertain demand

“…The framework is applied to static decisions and linear decision rules. Influenced both by robust optimization and control theory, Wagner (2018) frame demand uncertainty through stochastic processes and develops closed forms of optimal ordering rate functions for the static and dynamic cases. Under seasonality trends, reoptimization capabilities pay off in significantly better objective values.…”

Feedback-Based Deterministic Optimization Is a Robust Approach for Supply Chain Management under Demand Uncertainty

“…Solyalı et al 20 provided a tractable moving horizon heuristic for the nonzero fixed order cost robust inventory management problem, which is computationally intractable for large network instances due to the presence of binary variables. Wagner 21 defined an inventory management problem at the intersection of robust optimization and optimal control theory and analyzed the basic offline model and online variants based on rolling horizon optimization.…”

Robust optimization approaches for purchase planning with supplier selection under lead time uncertainty

“…Ben-Tal, Golany, Nemirovski, & Vial (2005) provide a tractable reformulation of the adjustable robust optimization to deal with the inventory management problem in a two-echelon supply chain with flexible commitments contracts. Wagner (2018) gives a closed-form solution for the infinite dimension version of the problem. See & Sim (2010) provide a second-order conic program for truncated linear replenishment policies (that correspond to an S policy), and they characterize the uncertain demand by covariance and directional deviation in addition to the classical mean and standard deviation.…”