The myopic Order-Up-To policy with a proportional feedback controller

Abstract

We develop a discrete control theory model of a myopic Order-Up-To (OUT) policy reacting to a stochastic demand pattern with Auto Regressive and Moving Average (ARMA) components. We show that the bullwhip effect arises with such a policy despite the fact that it is optimal when the ordering cost is linear. We then derive a set of z-transform transfer functions of a modified OUT policy that allows us to avoid the bullwhip problem by incorporating a proportional controller into the inventory position feedback loop. With this technique, the order variation can always be reduced to the same level as the demand variation. However, bullwhip-effect avoidance always comes at the cost of holding extra inventory. When the ordering cost is piece-wise linear and increasing, we compare the total cost per period under the two types of control policies: with and without bullwhip-effect reduction. Numerical examples reveal that the cost saving can be substantial if the order variance is reduced by using the proportional controller.

Current economic downturn and supply chain: the significance of demand and inventory smoothing

“…From a managerial viewpoint, the advocated smoothing of demand and inventory converts in a highly beneficial reduction of holding costs for all the levels of the supply chain, and in a higher stability of the supply chain in terms of orders and inventory levels. In traditional structures, smoothing replenishment rules are able to reduce bullwhip by 40% and realise economic savings of nearly 20% (Chen and Disney 2007).…”

“…It causes a small perturbation in orders at a downstream stage to have a large effect on the variation of an order or production rate at an upstream stage (Shin et al 2010). Bullwhip is the cause of a range of unnecessary costs in supply chain such as excessive inventory investments throughout the supply chain to cope with the increased demand variability; reduced customer service due to the inertia of the production/distribution system; lost revenues due to shortages; reduced productivity of capital investment; increased investment in capacity, inefficient use of transport capacity; and increased missed production schedules (Holweg et al 2005;Chen and Disney 2007). Examples of industries include telecommunications manufacturing, computer components manufacturing, grocery, retail, automotive industry, electronics industry, furniture industry, food, apparel, and so on.…”

“…Average inventory (Equation (15)) is the mean of a tier's inventory values over the interval T. The metric is commonly used in production-distribution system analysis in order to provide concise information on inventory investment, see for example holding cost modelled as linearly dependent from stock levels in Cachon and Fisher (2000), Disney and Grubbström (2004), Chen and Disney (2007) and Reichhart et al (2008), Dellino et al (2010).…”

“…The amount of the gaps to recover is regulated by the decision parameters known as proportional controller. This class of order-up-to (OUT) policy has come to researchers and practitioners' attention for its noticeable bullwhip dampening properties (Towill 1982;Mason-Jones et al 1997;Towill 2002, 2003;Boute et al 2007;Chen and Disney 2007;Strozzi et al 2007;Chen and Lee 2009;Zhou et al 2010), as it can limit the tiers' overreaction/underreaction for changes in the demand (Cannella et al 2011). Essentially, a smoothing replenishment policy is able to solve the detrimental consequence of the adoption of the classical OUT, as it is well recognised that this policy may lead to the bullwhip effect (Disney and Towill 2003a;Wei et al 2013).…”

An Integrated Inventory-Transportation System with Periodic Pick-Ups and Leveled Replenishment

“…The share to be replenished is regulated by proportional controllers. While downstream processes benefit, smoothing replenishment comes at the cost of a deteriorated service level (Chen and Disney 2007). This strongly limits the applicability for the material collection problem (MCP), where high service levels are required to facilitate production.…”