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Expected inventory order crossovers occur if at the moment of ordering it is expected that orders will not arrive in the sequence they are ordered. Recent research has shown that (a) expected inventory order crossovers will be encountered more frequently in future, and that (b) use of a myopic order-up-to policy based on a stochastic dynamic programming approach leads to improved performance compared to the classical approach. In this paper, we show that the improved policy is still heuristic in nature, as it neglects several control options that are available on the various ordering moments and makes some restrictive assumptions with respect to the separability (i.e., decomposability) of the stochastic dynamic programming problem. We propose further improvements in the policy for situations where a quadratic cost function is appropriate.

Order crossovers occur if orders do not arrive in the same sequence as they were ordered. This phenomenon is caused by differences in the lead-times and review times of the orders. Order crossovers are often neglected in inventory models. Whenever they are taken into account, most inventory models assume that lead-time differences are caused by a stochastic process (Fig. 1). This paper focuses on order crossovers that are the result of dynamic processes instead of stochastic processes. Full-size image (25 K) Fig. 1. Two types of order crossovers. Figure options In dynamic processes, lead-time fluctuations that occur are known in advance and can be anticipated upon. Dynamic lead-time fluctuations may occur due to contract changes, expediting policies, dual-sourcing policies from different geographical areas, transportation mode changes, etc. Riezebos (2006) has argued that these dynamic variations will be encountered more frequently in future. The main reason is that the above-mentioned instruments are increasingly used in modern supply chain management in order to increase flexibility (Robinson et al., 2001; Bradley and Robinson, 2005). However, the use of these instruments may lead to order crossovers. We denote such order crossovers as expected order crossovers, as they can be anticipated upon. Fig. 1 illustrates the difference between both types of order crossovers. This paper studies the consequences of expected order crossovers for a single-item inventory system, with known (variable) ordering moments (discrete-time, periodic review, with variable review periods). It is assumed that the future ordering moments are known in advance, as well as the lead-times at these moments. Order crossovers may occur due to dynamic fluctuations in both lead-times and review periods. This paper assumes that the stochastic variation in these parameters is minor and can therefore be neglected. Demand is stochastic and forecasts for future demand are available and may be updated at the next ordering (i.e. decision) moment. For this problem, Gaalman and Riezebos (2005) showed that the standard order-up-to policy that is available in inventory management systems leads to incorrect decisions in case of expected order crossovers. They derived an improved myopic order-up-to policy based on a stochastic dynamic programming approach. In this paper, we show that the improved policy is still heuristic in nature, as it neglects several control options that are available on the various ordering moments and makes some restrictive assumptions with respect to the separability (i.e., decomposability) of the stochastic dynamic programming problem. We propose further improvements in the policy for situations where a quadratic cost function is appropriate.

The standard myopic ordering policy does not take expected order crossovers into account. Gaalman and Riezebos (2005) developed an improved order-up-to policy, based on a stochastic dynamic programming formulation of the originating problem. This improved myopic order-up-to-rule is shown to be heuristic in nature. It does not completely correct for updated demand forecasts that have become available since the last ordering moment. And it is still myopic in nature, as it assumes that the originating stochastic dynamic programming problem can be reduced to a series of independent optimization problems decisions at each ordering moment. We showed that this assumption does not hold in case of expected order crossovers. We deduced a new generalized ordering policy from a modified stochastic dynamic programming problem. This generalized policy takes into account that one decision can affect (or control) several future inventory positions. The generalized policy is relevant for firms that use multi-sourcing of single-item goods from several geographical locations in the world. These firms might encounter expected order crossovers, but will not be able to manage their inventory position using the models generally available in inventory management modules of ERP systems. Use of the generalized policy will provide benefits in terms of reduced costs and improved service. Future research should be directed towards further insights in this inventory policy, specifically on the issue of generalizing the inventory ordering policy and safety stock determination from within the model.