Elsevier

Operations Research Letters

Volume 47, Issue 6, November 2019, Pages 465-472
Operations Research Letters

Quantile forecasting and data-driven inventory management under nonstationary demand

https://doi.org/10.1016/j.orl.2019.08.008Get rights and content

Abstract

In this paper, a single-step framework for predicting quantiles of time series is presented. Subsequently, we propose that this technique can be adopted as a data-driven approach to determine stock levels in the environment of newsvendor problem and its multi-period extension. Theoretical and empirical findings suggest that our method is effective at modeling both weakly stationary and some nonstationary time series. On both simulated and real-world datasets, the proposed approach outperforms existing statistical methods and yields good newsvendor solutions.

Introduction

In various fields of production/ inventory management, economic, engineering etc., predicting quantiles of a random process provides essential information for decision-making which is ignored by conventional point estimation of the conditional expectation. Moreover, many of these applications emphasize short-term forecasting where time series-based models, considering the internal structure of a process involved over time, are often preferable to explanatory approaches [38], [41]. Accordingly, a time series model is incorporated in this paper. This paper starts with its application in newsvendor problem, one of the most fundamental stochastic inventory models, because it is well-known that the optimal stock level is the so-called critical quantile. In practice, when the distribution of demand is unknown, managers should determine the inventory level based on historical demand observations. This problem essentially boils down to the prediction of the critical quantile of the future demand. Thus, this paper focuses on solving a newsvendor problem, while our results can be easily extended to predict quantiles of a random process in other fields.

Major existing approaches for solving newsvendor problem (quantile prediction) with historical data in the literature can be classified into three groups. The first category holds a strong assumption that all observations can be considered independent and identically distributed (i.i.d.) sample drawn from a real underlying distribution. Next, a standard treatment is to first estimate the distribution and then to replace the distribution with its estimation in the decision-making step (see [15], [24], [44] for a review). As a result, policies derived in such case are very susceptible to the parametric assumption of the demand distribution. Recently, to reduce this limitation, non-parametric methods have also been proposed [4], [6], [8], [16], [23], [29], [30], [32], [34]. Such an assumption of the i.i.d, though facilitates the establishment of asymptotic optimality of those methods, suffers from a major practical limitation that demand in real life changes over time, and that it is generally time-correlated.

The mentioned concern leads to the second stream of approaches that consider time-correlated demand [10]. Most of these papers assume perfect knowledge of the demand evolution function with the innovations to be i.i.d. normal with zero mean and unknown variance [15], [18], [35]. And quantile estimates are generally obtained following a two-step procedure, i.e., mean squared error (MSE) method is employed to estimate the parameters of a predefined model, and subsequently variance of the innovation is estimated from the forecast errors [2], [5], [10], [11], [14], [14], [17]. Besides the fact that the normality of errors is often violated, simple models such as AR(1), considering only linear relationship, is also unrealistic in practice. The choice of an improper evolution model may generate drastic errors in the forecasts.

Inspired by the idea of quantile regression proposed by [26], the third type of approaches perform quantile estimation in a single step and relax the normality assumption on innovations. Introduced in 1978, quantile regression was developed to estimate the quantiles in a linear regression model. Later, it was extended in [27] to the autoregressive case with AR(p) models. Thereafter, researchers also propose different nonlinear quantile autoregressive models [3], [13], [19], [31]. However, most of the aforementioned methods only work with stationary or at least locally stationary time series. To extend the methods to the nonstationary context, more assumptions regarding the form of trend and non-seasonality should be made, as in [20], [33], thus further limit the practicality.

Thus, one of the aims of this paper is to provide a single-step non-parametric solution for quantile forecasting. Then, we argue that it serves as a data-driven approach, which works with time-correlated or even nonstationary demand data to make inventory decisions. Note that for stationarity, there are two common interpretations: the formal mathematical definition of stationary process in mathematics and statistics; the case where the demand distributions do not change over time (i.i.d.) in inventory literature. In this study, its mathematical definition is exploited. To the best of our knowledge, this is the first study in the data-driven inventory management field that deals with a general autoregressive demand process of an unknown form. In addition, we show that the myopic policy remains optimal under autoregressive demand, without requiring the demand process to be statistically increasing as in the previous literature. Moreover, compared with the existing neural network-based methods for quantile prediction by [36] and [43], our method is capable of dealing with nonstationary time series, and it does not require previous quantile values as input, which are not observable in practice. In fact, this is also the first time that a nonstationary time series can be dealt with without prespecified forms of seasonality or trend.

The rest of this paper is organized as follows. In Section 2, the details of our proposed method denoted as DPFNN-Based Quantile Autoregression (DPFNN-QAR) are presented, as well as its application on stationary time series. In Section 3, the application of quantile autoregression as a data-driven approach to solve newsvendor problem is discussed, and its efficiency is illustrated by using case study which is nonstationary. In Section 4, the extension to multi-period inventory control problem is considered, in which excess inventory and backlogged demand can be carried over to the next period. Section 5 contains the concluding remarks and suggests the directions for future investigation.

Section snippets

The time series model

We extend the restrictive linear AR(p) model for time series, and consider a more general autoregressive model to define a process with potentially complicated nonlinear structure: Yt=g(Yt1,,Ytp)+εt,where g() can be any continuous function of unknown form, random innovations {εt} are i.i.d. with mean 0 and unknown variance σ2, but are not necessarily normally distributed. Note that this is a stronger assumption compared with the white noise innovation assumed in the traditional linear

Data-driven newsvendor problem

In this section, the application of DPFNN-based quantile forecasting (DPFNN-QAR) is considered in the field of inventory management, and the newsvendor problem, one of the most fundamental stochastic inventory models, is first dealt with.

Multiperiod safety stock

Now we consider the extension to multi-period newsvendor scenario, where excess inventory will be carried to the next period and unmet demand will be backlogged. We introduce the following new notations:

  • xt: initial inventory at the beginning of period t, negative xt means backlogged demand

  • T: number of periods in the planning time horizon

  • γ: discounting factor to calculate the present value of future costs

Furthermore, x1=0 is given. To facilitate the derivation of closed-form ordering policy, it

Conclusion

The contribution made by this paper is twofold. On the one hand, a novel neural network model (DPFNN-QAR) is proposed to forecast quantiles of a random process. The proposed model exploits the universal approximating capability of neural networks and utilizes the idea of quantile regression. Furthermore, by adding the linear shortcuts in the neural network model, the proposed method is the first, to the best of our knowledge, that captures both stationary and nonstationary time series without

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