Estimation of Jointly Normally Distributed Demand for Cross-Selling Items in Inventory Systems with Lost Sales

Abstract

Demand estimation is often confronted with incomplete information of censored demand because of lost sales. Many estimators have been proposed to deal with lost sales when estimating the parameters of demand distribution. This study introduces the cross-selling effect into estimations, where two items are cross-sold because of the positive externality in a newsvendor-type inventory system. We propose an approach to estimate the parameters of a jointly normally distributed demand for two cross-selling items based on an iterative framework considering lost sales. Computational results based on more than two million numerical examples show that our estimator achieves high precision. Compared with the point estimations without lost sales, all the relative errors of the estimations of demand expectation, standard deviation, and correlation coefficient are no larger than 2% on average if the sample size is no smaller than 800. In particular, for demand expectation, the error is smaller than 1% if the comprehensive censoring level is no larger than four standard deviations (implying a 2σ-level of safety stock for each item), even if the sample size decreases to 50. This implies that the demand estimator should be competent in modern inventory systems that are rich in data.