Using an iterative procedure of maximum likelihood estimations to solve the newsvendor problem with censored demand

This paper proposes a new data-driven solution approach for solving a newsvendor problem, where the parameters of the demand distribution are unknown and only sales (censored demand) can be observed. The procedure can be applied to different demand distributions. Compared to the previous parametric literature our approach allows the value at which demand is censored to vary, and we design an iterative solution procedure where the newsvendor updates their order size when new sales data is observed. The core of the procedure is an estimation phase where the newsvendor finds an optimal order size, using a novel maximum likelihood approach, which explicitly incorporates censored data. Moreover, the maximum likelihood part of the procedure is not specific to the newsvendor problem, and can therefore be used to solve other inventory management problems in future research or practice. In this paper, we explore numerically both the negative binomial distribution and the Poisson distribution, and we show that our log-likelihood function is concave for the Poisson distribution. In our comprehensive numerical experiments, we show that the procedure generally arrives at the optimal order size in short sales seasons with 25 to 100 periods. Moreover, by the 100th period the 25% and 75% quantiles of our experimental data are close to the optimal order size. We also introduce and discuss the regret of the algorithm and compare the algorithm to algorithms designed to minimize regret.