Production, Manufacturing and Logistics Characterizing order processes of continuous review ðs; SÞ and ðr ; nQ Þ policies Ann M. Noblesse a,⇑ , Robert N. Boute a,b , Marc R. Lambrecht a , Benny Van Houdt c a Research Center for Operations Management, University KU Leuven, Belgium b Technology & Operations Management Area, Vlerick Business School, Belgium c Department of Mathematics and Computer Science, University of Antwerp, Belgium article info Article history: Received 5 March 2013 Accepted 25 January 2014 Available online 4 February 2014 Keywords: Inventory Batching Bullwhip effect Stochastic processes abstract We provide a novel approach to characterize the order process of continuous review ðs; SÞ and ðr; nQ Þ inventory policies, and study the impact of the batching parameter (the value of Q or S  s) on the vari- ability in the order process. First, we characterize the distribution of the time between orders, as well as the distribution of order sizes. We find that the coefficient of variation (cv) of the time between orders is smaller than the cv of the time between demands. The size of the orders can exhibit either variance amplification or dampening, compared to the demand sizes, depending on the demand size distribution and the value of the batching parameter. This may motivate a supplier to adjust his imposed fixed order cost to influence the batching size. Second, we look at the compound order process, defined by the num- ber of units ordered during an arbitrary interval. The compound order process always exhibits variance amplification compared to the compound demand, which increases linearly in the batching parameter for large values of Q or S  s; for small values, the variance amplification is fluctuating. We point out that the time interval, during which the number of units ordered/demanded is observed, also impacts the level of variance amplification, and we show to what extent larger time intervals (resulting in more aggregation of the data) lead to lower values of variance amplification. Both perspectives (looking at time between orders and order quantities, or observing the compound order process) provide useful information for the upstream supplier. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction In the early fifties, Arrow, Harris, and Marschak (1951) intro- duced the ðs; SÞ inventory policy as a way to balance the cost of inventory with the cost of placing orders, hence exploiting the economies of scale when ordering. In an ðs; SÞ inventory policy, a lower stock level s and an upper stock level S are established: no order is placed until inventories fall to s or below, whereupon an order is placed to restore the inventory position to the level S. This means that orders are placed with a batch size, that is always lar- ger than or equal to the value of S  s. The more the inventory level falls below s (which depends on the stochastic demand size ob- served prior to the order is placed), the more the order quantity will exceed S  s; this is the stochastic overshoot. Several authors showed, in different settings, that an ðs; SÞ policy is optimal when a fixed order cost is present (Iglehart, 1963; ?; Scarf, 1960; Veinott, 1966). Today, the ðs; SÞ inventory policy is still of main importance to inventory theory and ordering policies and is incorporated in business software of many companies all over the world (Caplin & Leahy, 2010). When materials flow in fixed batch sizes, such as full truckloads or containers, an ðr; nQ Þ inventory policy can be adopted, which is comparable, but slightly different to the ðs; SÞ policy (Federgruen & Zheng, 1992; Li & Sridharan, 2008). In an ðr; nQ Þ inventory policy, if the inventory position reaches the order point r, an order is placed equal to the smallest multiple of Q that raises the inventory posi- tion above r. In the supply chain literature it is widely accepted that batching of orders creates a bullwhip effect, indicating that the variability in the order process is larger than the variability in demand (see e.g., Burbidge, 1961; Lee, Padmanabhan, & Whang, 1997; Potter & Disney, 2006). This distorted information can lead to serious inefficiencies upstream in the supply chain (Lee et al., 1997). It is argued that larger batch sizes create more bullwhip (e.g., Burbidge, 1961), and hence batching is to be avoided. However, Potter and http://dx.doi.org/10.1016/j.ejor.2014.01.058 0377-2217/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Tel.: +32 16326899. E-mail addresses: ann.noblesse@kuleuven.be (A.M. Noblesse), robert.boute@ kuleuven.be (R.N. Boute), marc.lambrecht@kuleuven.be (M.R. Lambrecht), benny. vanhoudt@ua.ac.be (B. Van Houdt). European Journal of Operational Research 236 (2014) 534–547 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor