Inventory Control Using a L\'evy Process for Evaluating Total Costs under Intermittent Demand

Abstract

Products with intermittent demand are characterized by a high risk of sales losses and obsolescence due to the sporadic occurrence of demand events. Generally, both point forecasting and probabilistic forecasting approaches are applied to intermittent demand. In particular, probabilistic forecasting, which models demand as a stochastic process, is capable of capturing uncertainty. An example of such modeling is the use of L\'evy processes, which possess independent increments and accommodate discontinuous changes (jumps). However, to the best of our knowledge, in inventory control using L\'evy processes, no studies have investigated how the order quantity and reorder point affect the total cost. One major difficulty has been the mathematical formulation of inventory replenishment triggered at reorder points. To address this challenge, the present study formulates a reorder-point policy by modeling cumulative demand as a drifted Poisson process and introducing a stopping time to represent the timing at which the reorder point is reached. Furthermore, the validity of the proposed method is verified by comparing the total cost with that obtained from a case where an ARIMA model is combined with a reorder-point policy. As a main result, while the total cost under ARIMA-based forecasting increases linearly over time, the L\'evy process-based formulation provides an analytical expression for the total cost, revealing that random demand fluctuations cause the expected total cost to grow at a rate faster than linear.