Nonparametric Predictive Inference for Inventory Decisions

Abstract

In inventory theory, many scenarios have been studied with the aim of determining an optimal order strategy, typically with the aim to maximise expected profit. Traditionally, a stochastic model with a known probability distribution for random demand is assumed. In this thesis, an alternative approach to inventory problems is presented, with the aim of basing the order strategy on information in the form of previously observed demands, adding only quite minimal further assumptions. Nonparametric Predictive Inference (NPI) is used to predict a future demand given observations of past demands. NPI makes only a few modelling assumptions, which is achieved by quantifying uncertainty through lower and upper probabilities.

As the first use of NPI in inventory theory, the basic scenario of inventory for a single period is considered. We present NPI lower and upper probabilities for the event that the random profit achieved for one future period is non-negative, which can be used to determine an optimal inventory level. As second optimality criterion, we consider the NPI lower and upper expected profits for the next period. We also consider optimisation of a weighted average of the NPI lower and upper probabilities and expected profits.

We also develop the NPI method for two-period inventory problems, in which we choose to maximise expected profit as the optimal criterion for determining optimal inventory levels. We derive the optimal inventory level for the two-period model with a single order. We presume that we are filling the inventory for both periods at the same time. Therefore, the future demand will be a combination of the future demand for the first period and the future demand for the second period. We also derive the optimal inventory levels for both periods in the two-period independent demands model. First, we determine the optimal inventory level for the second period, assuming there is a remaining stock (or shortage) from the first period, and with that optimal strategy for the second period, we then optimise over the first period.

Attention is also given to the situation of the two-period model with dependent demands. The NPI bootstrap (NPI-B) method is applied to deal with this model and the complexities in some of the inventory models. We study different strategies for the inventory levels to determine which one of those is optimal based on maximising the average profit.

The NPI method and the classical method are compared through simulations. Several cases are studied, some where the assumptions underlying the classical method are fully correct, so the classical method performs better; for a large number of observations, there is a tendency for the NPI to be close to the classical method. In the other cases where the assumed model is not well aligned with reality, the NPI method performs better.