Exploring the nonlinear dynamics of the lost-sales order-up-to policy

Abstract

With most inventory theory investigating linear models, the dynamics of nonlinear inventory systems is not well understood. We explore the dynamics of the order-upto policy under lost-sales for the case of i.i.d. normally distributed demand and unit lead times. We consider the ideal minimum mean squared error forecast and two alternative scenarios: partial demand observation and dynamic demand forecasting, providing a broad understanding of the operational performance of lost-sales systems. In each scenario, we obtain analytical expressions for the order, inventory, and satisfied demand distributions. This allows us to quantify the Bullwhip and inventory variance amplification ratios as well as the fill rate and the inventory cover. We show the lost sales nonlinearity induces complex behaviors in inventory systems. Interestingly, lost sales smooth supply chain dynamics, significantly affecting the trade-off between service level and average inventory holding. We also reveal the inventory downsides of demand censoring and the production damages induced by dynamic forecasts. We identify a key parameter, the relative safety margin, that characterizes the performance of lost-sales systems. We finish by offering some prescriptive results for the optimal safety stock and capacity level in both a retail and a manufacturing lost-sales setting.

Assessing the Bullwhip effect in supply chain: trends, gaps, and overlaps

“…Disney et al [89] studied a nonlinear inventory system with OUT policy and lost sales, comparing sales and backlogging to measure the BWE and inventory amplification and recovery. The lost sales model mitigates the BWE, contrasting with a backlogging system under some demand and forecasting conditions.…”

Assessing the Effect of a Novel Production Control Policy on a Two-Product, Failure-Prone Manufacturing/Distribution Scenario

“…On the other hand, the inventory threshold of lost sales is adopted by the manufacturer to limit the backlog quantity. In fact, it is assumed that the cost of backlog is larger than the cost of lost sales when the backlog level exceeds the inventory threshold of lost sales [11,13,14]. Finally, the manufacturing system is connected to a distribution chain consisting of a distributor and final users.…”

“…If the inventory level is not enough to completely fulfill the demand, a stock-out is generated and partial replenishment is used. In addition, the backlog is allowed without exceeding a threshold of lost sales [13,14], defined as −k • μ d p . In fact, it is assumed that, in this case, the cost of backlog is larger than the cost of lost sales and, thus, the manufacturer does not accept to further increase the backlog level (i.e., negative inventory level).…”

The effects of quantity discounts on supply chain performance: Looking through the Bullwhip lens

“…(9) describes the decision making of the retailer around the discount offered by the manufacturer. It is appropriate to note that this equation introduces a nonlinearity in the supply chain model, which has strong implications on the dynamic behaviour of production and distribution systems, as discussed by Nagatani and Helbing, 2004 , Wang et al, 2014 , and Disney et al (2020) .…”