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Article Excerpt
1. Introduction
Regression models forecast a variable by estimating its relationship to other variables. It is one of the most widely used statistical tools and is often applied to forecast demand for a product. It is useful when demand depends on independent drivers such as time, seasonal factors, econometric factors, weather, advertising dollars, and promotional events. For example, Glidden Paints, a paint manufacturer, uses linear regression with independent variables including disposable personal income and gross national product to predict demand (Heizer and Render 2005).
In this paper, we consider a single-item periodic review inventory problem with a finite planning horizon of N periods. Demand in each period is modeled by linear regression, and the values of the independent variables are either forecasted or known in advance of the planning horizon. We use a Bayesian formulation to update the regression parameters as new information becomes available. A Bayesian approach to demand modeling is appropriate in an environment of high uncertainty with little historical data (e.g., new product). Such an environment has a high potential for learning and, hence, makes the Bayesian approach useful. To keep the notation simple, we limit our presentation to the case of one independent variable, where demand in each period n is represented by a linear regression model:
[y.sub.n] = [[beta].sub.0] + [[beta].sub.1][x.sub.n] + [[epsilon].sub.n], n = 1, 2, ..., N, (1)
where [y.sub.n] is the demand in period n; [x.sub.n] is the value of the independent variable for period n, exogenously given; [[epsilon].sub.n] is the random error term in period n; and [[beta].sub.0] and [[beta].sub.1] are the unknown regression parameters, namely the "intercept" and "slope," respectively.
We note that all the results in this paper hold when there are multiple independent variables.
The standard Bayesian regression approach assumes a prior distribution on the regression parameters. As demand is observed, the prior distribution is updated to a posterior distribution, which leads to a nonstationary and dependent sequence of demand distributions. Incorporating this Bayesian learning into the periodic review inventory problem results in a dynamic program with a multidimensional state space. In addition to the starting inventory level, the state space includes two key parts. The first part represents a "backward look" due to the Bayesian approach that requires a summary of all the information in the past data. The second part represents a "forward look" due to the regression modeling of demand.
We show that the "backward look" part of the state space can be summarized by a two-dimensional vector, which is updated every period based on demand observations. The "forward look" part of the state space is represented by a vector of future values of the independent variable. We find that a state-dependent base-stock policy is optimal.
Unlike previous work in Bayesian inventory models, we find that the optimal base-stock level is not necessarily monotonic in the demand observations. A monotonicity result holds, however, under certain conditions that involve the future values of the independent variable. We also study the special case of regression forced through the origin. In this case, the optimal base-stock level is monotonic in the demand observations, analogous to previous work in Bayesian inventory models.
Although the optimal policy has a simple structural form, its computation is usually a very complex, if not an intractable, task. To address this computational issue, we propose a tractable procedure based on heuristics that simplify the state space. Our proposed implementation provides near-optimal results.
This paper is organized as follows. In [section]2, we review the related literature. Section 3 introduces the Bayesian linear regression demand structure. Section 4 describes the inventory model and establishes the optimal policy and related structural results. In [section]5, we introduce heuristics that simplify the state space. In [section]6, we study the performance of the heuristics and gain insights to the inventory problem through analytical results and numerical studies. Extensions are discussed in [section]7. In [section]8, we conclude and discuss future directions of this research. All proofs are available in the online technical appendix (provided in the e-companion). (1)
2. Literature Review
Several researchers have studied inventory models where Bayesian updating is used to learn about future demand from past history. The earliest work in Bayesian inventory models was by Dvoretzky et al. (1952). This early work was followed by Scarf (1959), who studied the periodic review inventory problem where demand is a member of the exponential family. Karlin (1960) and Iglehart (1964) extended the analysis to include the case where demand is a member of the range family. More recently, Iyer and Bergen (1997) used Bayesian updating to study the impact of learning on supplier and manufacturer profitability. A number of researchers (Harpaz et al. 1982, Lariviere and Porteus 1999, Ding et al. 2002, Agrawal and Smith 2007, Chen and Plambeck 2008) have used Bayesian updating to study the trade-off between the optimal decision and Bayesian learning when demand observations are censored.
We note that Bayesian inventory models belong to a general class of inventory models that face nonstationary and correlated demand. These models result in a multidimensional state space and a state-dependent optimal policy that can be impossible to compute. Within the Bayesian context, several researchers have proposed approaches to address the complexity of the state space. Azoury and Miller (1984) compare optimal ordering levels of Bayesian and non-Bayesian inventory models. For certain demand distributions, researchers have shown that the state space can be reduced. Scarf (1959) assumes a gamma distribution with an unknown scale parameter and shows that the Bayesian model can be reduced to that of solving another dynamic program with a one-dimensional state space. Azoury (1985) finds the conditions on the demand distribution function that lead to a reduction in the state space of the Bayesian model. She shows that the conditions hold for common distributions such as the gamma, uniform, and Weibull.
Heuristics are common approaches to address the complexity of the state space in an environment of nonstationary and correlated demand. Policies proposed in this environment are often myopic in that they ignore the effect of future periods while minimizing single period costs. Other policies have a myopic structure (mean demand plus safety stock). Lovejoy (1990) studies the performance of a simple myopic policy, and in a Bayesian example where demand is uniformly distributed with an unknown parameter, he finds that the myopic policy provides near-optimal results. Iida and Zipkin (2006) find that the myopic policy performs well under the Martingale model of forecast evolution. Graves (1999), Lee et al. (2000), and Zhang (2004) assume a policy with a myopic structure and characterize its behavior in their specific demand environments, but do not establish the optimality of the policy. Aviv (2003) proposes an adaptive replenishment policy with a simple structure similar to that of the myopic, but with better performance. It has a fixed safety stock, which is appropriate when demand uncertainty is constant.
In this paper, demand is modeled by a normal linear regression model where there is an informative prior probability density function (pdf) on the regression parameters and a Bayesian approach is used to update the pdf on the regression parameters, which subsequently is used to determine the updated demand pdf. This statistical approach is well known, and analyses appear in Jeffreys (1961), Zellner (1971), and Gelman et al. (1995). In our model, we find that the myopic policy can perform poorly. Moreover, a myopic structure with a fixed safety stock would not be appropriate here because demand uncertainty is nonstationary due to the Bayesian updating and the x values. The x values are important in our setting; they can contribute to large changes in the standard deviation from one period to the next. Hence, we introduce a new heuristic that is near optimal and is easy to compute. Also, we outline steps for a manager to identify when it is appropriate to implement the myopic policy versus our new heuristic.
We note that a Bayesian linear regression demand process can also be modeled using a special case of the discrete Kalman filter. A Kalman filter is a technique often used to forecast the state of a process with two recursive equations: (1) an observation equation and (2) a transition equation. The equations are used to update recursively an estimate of the state of the process in a way that minimizes the sum of the squared errors. For the Bayesian linear regression setting, the observation equation is the familiar regression linear model where the regression coefficient vector is the state of the system. The state transition equation is the Bayesian update of the regression coefficient vector.
Aviv (2003) presents a different special case of the Kalman filter to model the demand process in replenishment problems. He develops a time-series model that takes into account the ability of supply chain members to observe different (but possibly correlated) information about demand and integrate these observations into their forecasting and replenishment policies. Whereas in our setting the state vector represents the regression coefficients, in Aviv (2003), the state vector represents demand realizations and possibly information about future demand. We note that the demand predictions in Graves (1999), Chen et al. (2000), Lee et al. (2000), and Zhang (2004) are special cases of the linear state model in Aviv (2003).
To the best of our knowledge, Bayesian linear regression models have not been studied in the context of periodic review inventory models.
3. Bayesian Linear Regression Demand Model
In our periodic review inventory problem, demand in each period is represented by a normal linear regression model as in (1). Let [beta] denote the random vector [([[beta].sub.0], [[beta].sub.1]).sup.T], and let [[~.[chi].sub.n] denote the vector [1, [x.sub.n]]. Then, (1) can be written as
[y.sub.n] = [[~.x].sub.n][beta] + [[epsilon].sub.n], n = 1, 2, ..., N. (2)
We assume that at the beginning of the N-period planning horizon, we have some information about [beta] in the form of a prior distribution. We assume that the random error terms [[epsilon].sub.n], n = 1, 2, ..., N, are identically and independently distributed normal random variables, each with zero mean and common known variance [[sigma].sup.2]. As pointed out by Gelman et al. (1995, p. 261), "prior information is less important for the parameters describing the variance than for the regression coefficients because [[sigma].sup.2] is generally of less substantive interest than [beta]." However, we do consider the case of an unknown [[sigma].sup.2] in [section]7 and show how all of the results of this paper follow.
We interpret the explanatory variables [x.sub.n], n = 1, 2, ..., N, as exogenously given fixed variables, known to the...
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