A non-parametric adaptive algorithm for the censored newsvendor problem

Abstract

This article studies the problem of determining stocking quantities in a periodic-review inventory model when the demand distribution is unknown. Moreover, lost sales are unobservable in the system and hence inventory decisions are to be made solely based on sales data. Both the non-perishable and perishable inventory problems are addressed. Using an online convex optimization procedure, a non-parametric adaptive algorithm that produces inventory policy in each period that depends on the entire history of stocking decisions and sales observations. With the help of a convex quadratic underestimator of the cost function, it is established that the T-period average expected cost of the inventory policy converges to the optimal newsvendor cost at the rate of O(log T/T) for demands whose expected cost functions satisfy an α-exp-concavity property. It is shown that, when the demand distribution is continuous, this property holds the probability density function over the decision set is bounded away from zero. For other continuous distributions, a “shifted” version of the density function is constructed to show an ε-consistency property of the algorithm so that the gap between the T-period average expected cost of the proposed policy and the optimal newsvendor cost is of the order O(log T/T) + ε (for a given small ε > 0). Simulation results show that the proposed algorithm performs consistently better than two existing algorithms that are closely related to the proposed algorithms.

Keywords:

• Inventory
• unknown demand
• lost sales
• censoring
• adaptive policies

Additional information

Notes on contributors

Arnab Bisi

Arnab Bisi is an Assistant Professor of Decision Sciences and Information Systems at the Johns Hopkins Carey Business School, Baltimore, Maryland. His research and teaching interests include supply chain management, inventory management, operational risks, Six Sigma quality management, stochastic processes, and business statistics. He received a Ph.D. in Mathematics and Statistics from Hong Kong University of Science and Technology, an M.Stat. degree from the Indian Statistical Institute, and a B.Sc. with honors in Statistics from the University of Calcutta.

Karanjit Kalsi

Karanjit Kalsi is a Senior Engineer in the Electricity Infrastructure and Buildings Division at Pacific Northwest National Laboratory in Richland, Washington. His research interests are in control theory for energy systems, power system modeling, smart grid control, and optimization. He received a Ph.D. in Electrical and Computer Engineering from Purdue University and an M.Eng. in Electronics and Electrical Engineering from the University of Sheffield.

Golnaz Abdollahian

Golnaz Abdollahian is a Postdoctoral Fellow in the Department of Electrical and Computer Engineering and Neuroscience Research Institute at the University of California, Santa Barbara. Her research interests are in the areas of neuroscience, bioimage analysis, and video and image content analysis. She received a Ph.D. in Electrical and Computer Engineering from Purdue University and a B.Sc. degree with honors in Electrical Engineering from Sharif University of Technology.