Use of Uncertain Additional Information in Newsvendor Models 1st Sergey Tarima 2nd Zhanna Zenkova Institute for Health and Equity Institute of Applied Mathematics and Computer Science Medical College of Wisconsin Tomsk State University Wauwatosa, U.S.A. Tomsk, Russia [email protected] [email protected] Abstract—The newsvendor problem is a popular inventory Liyanagea and Shanthikumar [4] suggested to use direct management problem in supply chain management and logis- profit maximization which simultaneously incorporates both tics. Solutions to the newsvendor problem determine optimal parameter estimation and expected profit maximization. inventory levels. This model is typically fully determined by a purchase and sale prices and a distribution of random market Quantile estimation methods range from maximum likeli- demand. From a statistical point of view, this problem is often hood estimators to more robust methods minimizing specific considered as a quantile estimation of a critical fractile which risk functions: Koenker’s quantile regression [5] minimizes maximizes anticipated profit. The distribution of demand is a the sum of absolute deviations of residuals; sum of signs of random variable and is often estimated on historic data. In an residuals minimization is suggested in [6]. All of these meth- ideal situation, when the probability distribution of the demand is known, one can determine the quantile of a critical fractile ods directly applicable to solving the newsvendor problem as minimizing a particular loss function. Since maximum likelihood well. estimation is asymptotically efficient, under certain regularity In a situation, when a family of probability distributions assumptions, the maximum likelihood estimators are used for the of the demand is known, a maximum likelihood estimation quantile estimation problem. Then, the Cramer-Rao lower bound can be applied to quantile estimation as maximum likelihood determines the lowest possible asymptotic variance. Can one find a quantile estimate with a smaller variance then the Cramer-Rao estimators (MLEs) are asymptotically efficient under certain lower bound? If a relevant additional information is available regularity assumptions. Then, asymptotically, MLEs reach the then the answer is yes. Additional information may be available Cramer-Rao lower bound for variance, and no other estimator in different forms. This manuscript considers minimum variance has asymptotically smaller variance. Can a quantile estimator and minimum mean squared error estimation for incorporating with a smaller variance than the Cramer-Rao lower bound be additional information for estimating optimal inventory levels. By a more precise assessment of optimal inventory levels, we found? If a relevant additional information is available then maximize expected profit. the answer is yes. Index Terms—Newsvendor model, additional estimation, quan- Many of the above mentioned statistical methods to solving tile estimation, minimum variance, minimum mean squared error the newsvender problem lead to estimators regular enough to have two finite moments. If the two moments exist, then additional information can be combined together with external information (for example, an averaged sales from another I. INTRODUCTION store with similar characteristics) known with a degree of uncertainty (for example, a standard error of this average is Newsvendor model is a popular inventory management known as well) [7]. This approach assumed that the additional model. This model depends on two simple quantities, the information is unbiased, meaning that averaged sales for arXiv:2010.09549v1 [stat.AP] 19 Oct 2020 purchase price of a single unit of a product c and the its price both stores are about the same. A similar assumption was when it is sold, p. The overall profit is fully defined by the made in [8]. It is possible that additional information can critical fractile (= (p − c)=p). If F is the distribution of the be biased, then minimum mean squared error (MSE) can be random demand, D, then the optimal amount of product to considered instead [9], [10]. The use of additional information order is Q = F −1 ((p − c)=p), where F −1 is an inverse of F . known up to a few distinct values is considered in [11]–[14]; This solution secures the highest expected profit. the minimum MSE criterion was also used in these papers. The distribution of the random variable D is often estimated Zenkova and Krainova [15] considered the use of a known using historic data. Hayes [1] considered exponential and quantile for estimating expectations. The net premium using Gaussian models to minimize Expected Total Operating Cost a known quantile for voluntary health insurance was used as in the newsvendor problem. A Bayesian approach was used an illustrative application. to improve the estimation. Similarly, Bayesian methodology This manuscript considers minimum variance and minimum was used to solve inventory problems in [2] and [3]. In more MSE estimation for incorporating additional information. Sec- recent literature Bayesian frameworks is also very popular. tion III presents methodology for combining empirical data (historical sales data directly available for data analysis) and applies to all unbiased or approximately unbiased estimators external information available in form of means and standard with two finite moments. errors. Sections II and IV illustrate the use of this new Given the importance of Product A, it is very difficult to statistical approach to the newsvendor problem for quantile obtain historic sales data from similar retailers. At the same estimation. time, additional data on Product B is much simpler to get from other retailers. Since the information on Product B is viewed II. ILLUSTRATIVE EXAMPLE as less important by other retailers, they freely share their sales Table I reports an artificial dataset with 36 weeks sales data. data over a cup of coffee. Product A is sold at 860 dollars per unit, Product B is sold Consider the following additional information. An owner of at 490 dollars per unit. The retailer pays 660 dollars/unit for a similar retailer store bragged that his store sold 30 thousand product A and 370 dollars/unit for product B. units of Product B in past five years (additional information 1), Product A whereas an owner of another similar store said that his sales of 6576 4263 5340 3697 3535 2651 Product B are higher than 100 units every other week within 2541 2351 3611 3867 4257 6204 the same five year period (additional information 2). Can we 6666 4364 5441 3727 3495 2755 2399 2452 3621 3961 4291 6264 use these two pieces of seemingly irrelevant information to 6600 4333 5391 3732 3662 2498 improve estimation accuracy of the optimal inventory levels 2576 2402 3588 3900 4220 6214 for Product A? The answer is yes, and we will return to this Product B 215 142 155 97 101 83 illustrative example in Section IV. 104 96 102 101 130 215 223 134 157 99 99 87 III. METHODOLOGY 100 97 98 104 131 202 211 139 150 100 105 82 −1 103 98 100 102 127 219 Let θ be the parameter of interest, θ = F ((p − c)=p) TABLE I for the newsvendor problem. The estimator of θ based on THIRTY SIX WEEK SALES HISTORY (IN NUMBERS OF UNITS SOLD) FOR historical data, θ^, is assumed to be an unbiased estimator of PRODUCTS A AND B θ, so that E(θ^) = θ. The θ^ is a normal quantile estimated on historical sales data in Section II (θ^ = 3118:14). In addition to The retailer is mainly interested in Product A as it is θ^, another estimator η~ is available as additional information. associated with high sales and is highly important for the re- This quantity estimates η not θ, and η is a different and tailer’s success. Then, the critical fractile ratio for Product A is possibly multi-dimensional parameter. Specifically, in Section 23:26%(= (860 − 660)=860). The solution to the newsvendor II, η = (η1; η2), where η1 is the mean weakly sales of Product −1 problem is Q = F (0:2326). Relying on previous experience B and η2 is the median weakly sales of Product B. The the retailer is confident that market demand for the two additional information described in Section II can be converted products, A and B, can be described by normal distributions, into a two-dimensional estimate (~η = (115:3846; 100)). The and the demands are likely to be correlated. Seasonal variation number 115:3846 is obtained as a ratio 30; 000=260, because is so small that historic weekly data can be assumed to be there are 260 weeks within a five year period. The first independent. The only complication is that there exists a additional information sets the mean weakly sales of Product difficult to predict clustering, see Figure 1, but this problme B at 115:3846 units, and the second additional information is not in the focus of this manuscript. Normal distributions sets the median sales at 100 units/month. 2 depend on two unknown parameters: µ and σ . Further, we Further, we use “hat” to denote estimators based on empir- 2 will use subscripts A and B to differentiate between µ and σ ical (historical) data and “tilde” for the quantities determined of Products A and B when needed. by additional information. Using the data in Table I, the mean Using Table I, the MLEs of the unknown parameters of weakly sales = 128 (η^1 = 128) and weakly median sales = the normal model are µ^A = 4095:694 (sample mean) and 103:2 (η^2). 2 σ^A = 1791703 (sample variance). Note, the sample variance is To combine additional information with empirical data, we 2 slightly different from the MLE of σA, but the sample variance consider a class of linear combinations 2 is an unbiased estimator of σA and will be used instead. Then, using the 0:2326-level normal quantile, Q = 3118:14 θΛ = θ^ + Λ (^η − η~) (1) (in Product A units).