Estimation of quantiles of non-stationary demand distributions

ABSTRACT

Many problems involve the use of quantiles of the probability distributions of the problem's parameters. A well-known example is the newsvendor problem, where the optimal order quantity equals a quantile of the demand distribution function. In real-life situations, however, the demand distribution is usually unknown and has to be estimated from past data. In these cases, quantile prediction is a complicated task, given that (i) the number of available samples is usually small and (ii) the demand distribution is not necessarily stationary. In some cases the distribution type can be meaningfully presumed, whereas the parameters of the distribution remain unknown.

 This article suggests a new method for estimating a quantile at a future time period. The method attaches weights to the available samples based on their chronological order and then, similar to the sample quantile method, it sets the estimator at the sample that reaches the desired quantile value. The method looks for the weights that minimize the expected absolute error of the estimator. A method for determining optimal weights in both stationary and non-stationary settings of the problem is developed. The applicability of the method is illustrated by solving a problem that has limited information regarding the distribution parameters and stationarity.

KEYWORDS:

• Demand forecasting
• probability distribution quantiles
• inventory management

Additional information

Notes on contributors

Hadar Amrani

Hadar Amrani is currently a Ph.D. student at Tel Aviv University, Israel, in the Department of Industrial Engineering. He received M.E. and B.Sc. degrees in Industrial Engineering, both from the Technion–Israel Institute of Technology. Having filled several positions in the field of maintenance, planning, and management, he is currently interested in applications of operations research in this field.

Eugene Khmelnitsky

Eugene Khmelnitsky is an Associate Professor at Tel Aviv University, Israel, in the Department of Industrial Engineering. He studied Applied Mathematics at the Institute of Physics and Technology, Moscow, Russia, and received a Ph.D. degree in Industrial Engineering from the University of Tel Aviv. His research interests are in the field of dynamical systems, optimal control, and analytical and numerical methods for production planning and scheduling problems.