Mathematical programming for single- and multi-location non-stationary inventory control

Abstract

Stochastic inventory control investigates strategies for managing and regulating inventories under various constraints and conditions to deal with uncertainty in demand. This is a significant field with rich academic literature which has broad practical applications in controlling and enhancing the performance of inventory systems. This thesis focuses on non-stationary stochastic inventory control and the computation of near-optimal inventory policies for single- and two-echelon inventory systems. We investigate the structure of optimal policies and develop effective mathematical programming heuristics for computing near-optimal policy parameters. This thesis makes three contributions to stochastic inventory control.

The first contribution concerns lot-sizing problems controlled under a staticdynamic uncertainty strategy. From a theoretical standpoint, I demonstrate the optimality of the non-stationary (s,Q) form for the single-item single-stocking location non-stationary stochastic lot-sizing problem in a static-dynamic setting; from a practical standpoint, I present a stochastic dynamic programming approach to determine optimal (s,Q)-type policy parameters, and I introduce mixed integer non-linear programming heuristics that leverage piecewise linear approximation of the cost function. The numerical study demonstrates that the proposed solution method efficiently computes near-optimal parameters for a broad class of problem instances.

The second contribution is to develop computationally efficient approaches for computing near-optimal policy parameters for the single-item single-stocking location non-stationary stochastic lot-sizing problem under the static-dynamic uncertainty strategy. I develop an efficient dynamic programming approach that, starting from a relaxed shortest-path formulation, leverages a state space augmentation procedure to resolve infeasibility with respect to the original problem.

Unlike other existing approaches, which address a service-level-oriented formulation, this method is developed under a penalty cost scheme. The approach can find a near-optimal solution to any instance of relevant size in negligible time by implementing simple numerical integrations.

This third contribution addresses the optimisation of the lateral transshipment amongst various locations in the same echelon from an inventory system. Under a proactive transshipment setting, I introduce a hybrid inventory policy for twolocation settings to re-distribute the stock throughout the system. The policy parameters can be determined using a rolling-horizon technique based on a twostage dynamic programming formulation and a mixed integer linear programme.

The numerical analysis shows that the two-stage formulation can well approximate the optimal policy obtained via stochastic dynamic programming and that the rolling-horizon heuristic leads to tight optimality gaps.