The Data-Driven Newsvendor Problem: New Bounds and Insights

The Data-Driven Newsvendor Problem: New Bounds and Insights Retsef Levi [email protected]

Georgia Perakis [email protected]

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Joline Uichanco [email protected]

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April 2010

Abstract We consider the well-known newsvendor model, but under the assumption that the underlying demand distribution although exists, is not known, and is not given as part of the input. Instead the only information available is a set of independent random samples that are drawn from the true demand distribution. This model is more realistic to capture practical scenarios. Sample Average Approximation (SAA) is one of the most common non-parametric approaches to solve the above problem; the original objective function is replaced by an average based on independent random samples that are drawn from the true demand distribution. There has been previous work to develop analytical bounds on the number of samples required for the SAA-based solution to be near-optimal (with respect to the true objective). However, they fail to reveal under what properties of the distribution is the SAA approach most attractive. Moreover, computational experiments show that these bounds are far too conservative than needed. We develop a novel analysis that suggests that if a distribution has a large weighted mean spread, then the SAA solution is near-optimal with relatively few samples. We prove a uniform lower bound for the weighted mean spread of any logconcave distribution, which results in a sampling bound for the log-concave family. The bounds we develop are much tighter than previous bounds and match the empirical evidence. We also provide computational comparison between the SAA-based approach and other data-driven approaches.

∗ †

Retsef Levi and Georgia Perakis are with the Sloan School of Management, Massachusetts Institute of Technology Contact author; Joline Uichanco is with the Operations Research Center, Massachusetts Institute of Technology

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Extended Abstract In the classical newsvendor model, a retailer plans to sell a product over a single period to satisfy a stochastic demand with a known distribution. She needs to commit to a stocking quantity before observing the actual demand at the end of the sales period. The retailer incurs an underage cost for each unit of unsatisfied demand, and an overage cost for each unsold unit of product at the end of the period. The goal of the retailer is to choose a quantity to minimize the expected cost. The optimal order quantity is a well-specified quantile of the distribution (sometimes called the newsvendor quantile) which balances the trade-off between the costs of over- and under-ordering [13]. In this classical version, the retailer is assumed to have perfect information about the demand distribution. However, this assumption is unrealistic in many practical scenarios as there is usually partial information about the demand. There exists a large body of literature on models and heuristics that can be applied when only limited demand information is known. These include (but is not limited to) Bayesian learning [11, 6], operational statistics [9], bootstrap statistical procedures [3], sample average approximation [5, 7] and robust optimization [12, 4, 2, 10]. These different approaches strike a different balance between three seemingly contradictory aspects: (i) the assumptions about what is known on the underlying distributions, (ii) the robustness of the resulting solutions, and (iii) the conservativeness of the solution. The focus of this work is on a nonparametric data-driven approach. In this approach, the assumption is that the only information available is a set of samples drawn independently from the true demand distribution, but the true distribution is unknown to the decision-maker. In particular, there are no parametric assumptions on the underlying demand distributions. Under this setting, one of the most common algorithmic approaches is sample average approximation (SAA) [5]. In this approach, one attempts to minimize the average cost over the samples instead of the true expected cost that cannot be computed. A natural benchmark to evaluate the quality of the SAA solution is to compare the expected cost it incurs to the optimal cost that can be obtained if one has full knowledge of the underlying distribution. The ratio of the difference of between the two costs and the true optimal expected cost is what we will refer to as the relative error of the SAA solution. It has been observed empirically that for many stochastic problems the SAA solution typically has small errors even if the number of samples is relatively small [8]. This suggests that SAA is a viable approach even in the presence of limited data. A recent paper by Levi, Roundy and Shmoys [7] establishes analytic bounds on the number of samples required to guarantee that the SAA solution has at most a prespecified relative error with some prespecified confidence. We call these sampling bounds. The sampling bound in [7] is distribution-free and only depends on the error, confidence level and the overage and underage cost parameters. In particular, it does not depend on any property or statistic of the underlying distribution. However, there are several weaknesses in their analysis. Firstly, because their bounds are general, they fail to reveal under what types or properties of demand distributions is the SAA approach most attractive (i.e., likely to have small errors using relatively few samples). For example, when the newsvendor quantile is 0.9, their analysis suggests the same bound regardless of the underlying distribution that can be, say, uniform or normal. Yet intuitively, to accurately estimate the tail of a normal distribution requires mores samples. Secondly, empirical experiments that we conducted suggest that their sampling bound 2

is very conservative. For instance, with only 100 samples drawn from a uniform distribution, the error of the SAA solution (which only uses the samples and does not rely on the fact that they were generated by a uniform distribution) is at most 2% in 82% of repeated experiments (i.e., with confidence level of 82%). Yet to match this same error and confidence, the sampling bound by Levi, Roundy and Shmoys [7] suggests that one needs more than one million samples! In this work, we address these two issues and provide a new analysis that: (i) highlights several important properties of the underlying distribution that make a data-driven framework attractive, and (ii) obtains analytical sampling bounds that match empirical performance. The new bounds that we obtain provide important insights of what properties of the underlying distribution make it “easier” to solve the SAA and get an accurate solution. Tight and informative sampling bounds. We develop a novel analysis that suggests that the number of samples needed to guarantee at most a certain error and confidence level depends on a single parameter of the underlying distribution that we call the weighted mean spread around the newsvendor quantile. The weighted mean spread at a point is the product of two terms: (1) the absolute mean spread, defined to be the expected demand conditioned on the demand being greater than the point minus the expected demand conditioned on it being less than the point, and (2) the value of the probability density at that point. Our analysis shows that when the sample size is fixed, the SAA approach has smaller errors when the distribution has a larger weighted mean spread. If a distribution has a large absolute mean spread at the optimal quantile, then this implies that the expected cost function is flatter around the optimum, and the errors are small even when there is a large difference between the SAA solution and the true optimal solution. On the other hand, if the value of the density at the optimal quantile is large, then less samples are needed to accurately estimate the optimal quantile. We have observed that in many distributions the absolute mean spread and the density value exhibit an inverse relationship. Therefore, a large weighted mean spread corresponds to a distribution for which this inverse relationship is somewhat balanced. We demonstrate through empirical experiments that the sample size predicted by our method has a tight relationship with the target error and confidence level. Since the weighted mean spread is a first moment type of information, it can be estimated from data easily. This implies that from only estimating a single distribution parameter, it is possible to make strong predictions about the accuracy of the SAA solution. Analytical sampling bound for log-concave distributions. We derive a uniform lower bound for the weighted mean spread of any log-concave distribution. We do this through solving the optimization problem of minimizing the weighted mean spread over all distributions subject to the distribution being log-concave. We find that the distribution that solves this problem is an exponential-type. As a consequence, we manage to establish a uniform bound on the number of samples required in the case the underlying demand distribution is log-concave. These bounds are significantly tighter than the bounds in [7]. Moreover, this sampling bound does not require estimating the weighted mean spread. This result has wide applicability because many of the common distributions assumed in inventory management belong to this class. Examples of log-concave distributions include a normal, uniform, exponential, logistic, chi-square, chi, beta and gamma distributions. Moreover, a given set of samples can be verified to be log-concave through a simple nonparametric testing procedures proposed by An [1]. The method-

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ology we propose can be used to derive similar sampling bounds for other distribution families, which we believe is an interesting future direction. Empirical Experiments. Finally, we conduct an extensive computational study comparing the performance of the SAA heuristic against another heuristic that constructs policies using samples from the true distribution. The two methods we compare are: (1) SAA approach, and (2) Best Fit approach which uses the distribution-fitting software EasyFit to find the distribution that best describes the samples from its database of more than 50 distributions. The comparison is made based on the average errors each method incurs. We investigate how the sample size, the coefficient of variation, and nonstandard distributions affects the magnitude of the errors. We find that in most cases, the errors of the SAA method are on par or dominate those of the Best Fit approach. However, when the samples are drawn from a nonstandard distribution (e.g., mixed normals), the Best Fit method results in huge errors, whereas the SAA approach stays robust.

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