The Newsvendor Under Demand Ambiguity: Combining Data with Moment and Tail Information

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Soroush Saghafian, Brian Tomlin

To cite this article: Soroush Saghafian, Brian Tomlin (2016) The Newsvendor under Demand Ambiguity: Combining Data with Moment and Tail Information. Operations Research 64(1):167-185. http://dx.doi.org/10.1287/opre.2015.1454

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INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org OPERATIONS RESEARCH Vol. 64, No. 1, January–February 2016, pp. 167–185 ISSN 0030-364X (print) ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.2015.1454 © 2016 INFORMS

The Newsvendor under Demand Ambiguity: Combining Data with Moment and Tail Information

Soroush Saghaﬁan Harvard Kennedy School, Harvard University, Cambridge, Massachusetts, Soroush_Saghaﬁ[email protected] Brian Tomlin Tuck School of Business at Dartmouth, Hanover, New Hampshire, [email protected]

Operations managers do not typically have full information about the demand distribution. Recognizing this, data-driven approaches have been proposed in which the manager has no information beyond the evolving history of demand observations. In practice, managers often have some partial information about the demand distribution in addition to demand observations. We consider a repeated newsvendor setting, and propose a maximum-entropy based technique, termed Second Order Belief Maximum Entropy (SOBME), which allows the manager to effectively combine demand observations with distributional information in the form of bounds on the moments or tails. In the proposed approach, the decision maker forms a belief about possible demand distributions, and dynamically updates it over time using the available data and the partial distributional information. We derive a closed-form solution for the updating mechanism, and highlight that it generalizes the traditional Bayesian mechanism with an exponential modifier that accommodates partial distributional information. We prove the proposed approach is (weakly) consistent under some technical regularity conditions and we analytically characterize its rate of convergence. We provide an analytical upper bound for the newsvendor’s cost of ambiguity, i.e., the extra per-period cost incurred because of ambiguity, under SOBME, and show that it approaches zero quite quickly. Numerical experiments demonstrate that SOBME performs very well. We find that it can be very beneficial to incorporate partial distributional information when deciding stocking quantities, and that information in the form of tighter moment bounds is typically more valuable than information in the form of tighter ambiguity sets. Moreover, unlike pure data-driven approaches, SOBME is fairly robust to the newsvendor quantile. Our results also show that SOBME quickly detects and responds to hidden changes in the unknown true distribution. We also extend our analysis to consider ambiguity aversion, and develop theoretical and numerical results for the ambiguity-averse, repeated newsvendor setting. Keywords: data-driven newsvendor; maximum entropy; distribution ambiguity; moment information; ambiguity aversion. Subject classifications: inventory/production: applications, uncertainty: stochastic; probability: entropy. Area of review: Operations and Supply Chains. History: Received June 2014; revisions received November 2014, April 2015, October 2015; accepted October 2015. Published online in Articles in Advance January 11, 2016.

1. Introduction new product launches and new process introductions. In Randomness in demand and/or supply is a major challenge such settings, ﬁrms devote signiﬁcant efforts to characteriz- in sales and operations planning. A common paradigm in ing demand or yield before the new product or process is the operations literature endows the decision maker with launched and then observe a growing time series of realiza- full distributional information about the underlying random tions, i.e., data, after launch. In new product introductions, variable(s). Full distributional information is, however, a forecasting demand and choosing stocking quantities is a strong assumption and is not typically met in practice. An common challenge across a wide array of industries (Kahn alternative “pure data-driven” paradigm endows the decision 2002, AMR 2008), especially “during the commercialization maker only with past observations/realizations of the random stage (prelaunch preparation and launch) where new product variable (the “data”); see, e.g., Azoury (1985), Liyanage and forecasts drive a variety of multifunctional decisions,” (Kahn Shanthikumar (2005), Levi et al. (2007, 2015), Huh et al. 2002, p. 133). For example, consider the consumer-products (2011). Reality often lies somewhere in between these two company Nabisco (now owned by Mondelez International):

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. paradigms: the decision maker typically has some partial Several statistical techniques are used to forecast sales 0 0 0 0 distributional information and some data. However, most of these techniques need a minimum of two In this paper, we propose and analyze a methodology for to three years of historical information to build a robust decision making in settings in which the decision maker does model. This presents a big problem when developing fore- not know the true distribution but has access to some partial casts for newly introduced product 0 0 0 0 At Nabisco, before distributional information and some limited data. This form a new product is rolled out, [the] planning forecast devel- of ambiguity, a.k.a Knightian uncertainty, about the true oped by marketing research 0 0 0 is used for 0 0 0 production distribution arises in various settings in operations including planning 0 0 0 and inventory deployment. However, when the

product is rolled out, actual sales can be substantially less or We develop an approach tailored to intermediate infor- more than the planned numbers [leading to] lost sales [or] mation settings—such as new product and new process excess inventory. (Amrute 1998, p. 7). introductions—in which a decision maker operates in an Let us consider the kind of demand-forecast informa- environment of ambiguity characterized by two evolving tion that firms generate before and after a new product is sources of information about a relevant random variable launched. According to surveys by Lynn et al. (1999) and (e.g., demand): “exogenous research” and observations (the others, prelaunch forecasts are primarily generated through “data”). Based on the above discussion of pre and postlaunch marketing research and judgmental techniques, with the top forecasts, we formulate the exogenous research information four approaches being customer/market research, followed as a set of possible demand distributions as well as potential by jury of executive opinion, sales force composite, and upper and/or lower bounds on the moments and/or tails look-like analysis (Kahn 2002). Customer/market research of the demand distribution. Our proposed approach is a encompasses a variety of approaches, including surveys of second-order, maximum-entropy based approach, which we potential buyers. The jury of executive opinion approach term Second Order Belief Maximum Entropy (SOBME). solicits and combines judgments from managers represent- It allows the decision maker to effectively combine the ing various functions, whereas the sales force composite above-mentioned sources of information dynamically over approach solicits and combines judgments from the sales time to update its beliefs about the possible distributions force (Aaker et al. 2010) who are typically better aware of governing the demand realizations. In our approach, the market conditions; see, e.g., Saghafian and Chao (2014), and decision maker updates its second-order beliefs, i.e., dis- the references therein. In look-like analysis, the firm identifies tribution over possible distributions, so as to minimize the “a similar product, sold previously, and extrapolates from relative entropy to its prior belief subject to the most recently that product’s prior sales” (Fisher and Raman 2010, p. 85). observed demand and the current market research informa- These judgmental approaches, such as sales force composite, tion. In doing so, we follow the core concept of maximum can be used to estimate demand uncertainty by having the entropy, widely used in estimation and information theory, in salespeople “provide a sales estimate range in addition to the which (i) beliefs are updated so that the posterior coincides single number representing the expected sales level [or] to with the prior as closely as possible, and (ii) only those have them estimate the probability of each potential sale.” aspects of beliefs for which new evidence was gained are (Aaker et al. 2010, p. 747). Because of the subjective nature updated (Jaynes 1957, 1981; Cover and Thomas 2006). of theses estimation processes, different people will provide We embed our SOBME approach in a classical operations different forecasts, i.e., different probability distributions. As setting—the repeated newsvendor—and analytically and such, these prelaunch techniques enable the firm to generate numerically examine its performance. We focus on a repeated a set of possible demand distributions and a range for the newsvendor setting because that is a common paradigm expected sales. in the operations literature for exploring approaches that After a product has been launched, the firm can use actual relax the assumption of full demand information. However, demand observations to update its beliefs about demand. SOBME is general and could be used in any setting in However, market research efforts do not end, at least in part which a decision maker does not know the distribution of an because “the rate of new product introduction is fast and underlying random variable but has access to data and to frequent, meaning data series are often short” (Lynn et al. some partial distributional information. Such settings will 1999, p. 566). Ongoing market research generates forecast information through test sales (Fisher and Raman 2010), become increasingly common in the era of big data analytics. simulated test markets (Lynn et al. 1999), and surveys of cus- Our paper provides both theoretical and managerially relevant tomer purchase intentions (Aaker et al. 2010). These ongoing results. postlaunch market-research efforts provide valuable demand On the theoretical side, we derive a closed-form solution information separate from the actual demand observations. for the SOBME belief-updating mechanism, and show that Therefore, in planning stocking quantities for a new or it contains traditional Bayesian updating—both parametric recently released product, the operations function has two and nonparametric—as a special case. As we will show, separate and evolving sets of information about the unknown SOBME generalizes the Bayesian updating mechanism by demand distribution. One set is the pre and postlaunch market incorporating the partial distributional information through research including the judgmental approaches discussed an exponential modifier. We explore the asymptotic behavior

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. above. The other set is the actual demand observations of SOBME and show that it is weakly consistent. That is, over time. This information setting is not captured by under some technical conditions, the updated distribution the traditional “full-information” paradigm in operations converges to the true unknown distribution almost surely in management that endows the decision maker with complete the weak neighborhoods of it. We also analytically establish knowledge of the demand distribution. Neither it is captured a rate of convergence for SOBME. Furthermore, we provide by the more recent “pure data-driven” approaches that were a performance guarantee for SOBME by developing an speciﬁcally designed for settings in which the decision maker upper bound on the newsvendor’s cost of ambiguity, i.e., the has access to demand observations but no other information. increase in the expected per-period newsvendor’s cost as Saghaﬁan and Tomlin: The Newsvendor under Demand Ambiguity Operations Research 64(1), pp. 167–185, © 2016 INFORMS 169

compared to the full distributional knowledge case. We show the decision maker has full information about the distribution that the cost of ambiguity approaches zero quite quickly of relevant random variables. A number of approaches have when SOBME is used. We also explore an ambiguity-averse been adopted to relax this strong assumption. version of our repeated newsvendor problem, and develop Some papers adopt a parametric approach, in which theoretical results that characterize the optimal newsvendor the distribution comes from a parametric family but the quantity under ambiguity aversion. In doing so, we establish parameter value is unknown. Bayesian approaches have a generalization of the traditional newsvendor quantile result. been used in the context of demand uncertainty (Scarf We also analytically and numerically examine how SOBME 1959, Azoury 1985, Lariviere and Porteus 1999, Wang performs in this ambiguity-averse setting. and Mersereau 2013) and supply uncertainty (Tomlin 2009, On the managerial side, we numerically establish that Chen et al. 2010). An “operational statistics” approach SOBME performs well—as measured by the percentage for simultaneously estimating an unknown parameter and increase in cost as compared to full distributional knowledge— optimizing an operational planning problem is introduced in even when the number of demand observations is limited. Liyanage and Shanthikumar (2005) and further explored in This is of practical importance because, as noted above, Chu et al. (2008) and Ramamurthy et al. (2012). firms often do not have access to a long series of demand The distribution of interest may not come from a para- observations when planning stocking quantities for recently metric family or the decision maker may not know which introduced products. Given the efforts devoted by companies family it comes from. With this in mind, a number of to generate exogenous research about the demand distribu- different nonparametric approaches have been proposed and tion, it is important to understand whether such research is analyzed in various inventory planning contexts. Sample valuable, when it is valuable, and what type of research is Average Approximation (SAA) (see, e.g., Kleywegt et al. more valuable. We numerically explore the value of exoge- 2002; Shapiro 2003; Levi et al. 2007, 2015) is a data-driven nous research by comparing the performance of SOBME approach that solves a sample-based counterpart to the actual with that of pure data-driven approaches, such as Sample problem. For the SAA approach, Levi et al. (2007 and 2015) Average Approximation (SAA), which rely exclusively on develop bounds on the number of samples required for the demand realizations. We show that there is significant the expected cost to come arbitrarily close (with a high value to exogenous research, especially when demand obser- confidence) to the expected cost if the true distribution vations are limited or the newsvendor quantile (the desired was known. Other data-driven, nonparametric approaches service level) is high. Unlike SAA, the SOBME approach include the bootstrap method (Bookbinder and Lordahl performs well even at high service levels. We find that 1989), the concave adaptive value estimation (CAVE) algo- research resulting in tighter moment bounds is more valuable rithm (Godfrey and Powell 2001), and adaptive inventory than research resulting in a more compact set of possible methods mainly used for censored demand data (Huh and distributions. As discussed above, “like-products” are one Rusmevichientong 2009, Huh et al. 2011). source of exogenous research. Based on a new-product An alternative nonparametric approach is a robust one introduction case motivated by a real-world setting (Allon in which certain moments of the distribution are known, and Van Mieghem 2011), we show how moment bounds and the decision maker maximizes (minimizes) the worst- (for demand) can be developed from like-products, and case expected profit (cost) over all distributions with the numerically show that there is significant benefit to moment known attribute; see, e.g., Scarf (1959), Gallego and Moon bound information even if the number of like-products is (1993), Popescu (2007). Because such approaches can be quite small. We also numerically show that SOBME is conservative, Perakis and Roels (2008) adopt a regret-based effective at detecting and quickly reacting to changes in the approach in which the decision maker knows the mean. underlying unknown distribution, an important and practical Zhu et al. (2013) consider a newsvendor setting where the benefit in dynamic environments. We prove that ambiguity demand distribution is only specified by its mean and either aversion leads to lower order quantities. This has important its standard deviation or its support, and develop a method implications for practice if individual managers vary in their that minimizes the ratio of the expected cost to that of ambiguity aversion. full information. Moments, however, might not be known The rest of the paper is organized as follows. The lit- with certainty, and Delage and Ye (2010) develop a robust erature is reviewed in §2. The model is presented in §3, approach in which the mean and covariance satisfy some and the proposed SOBME approach is developed in §4. known bounds. Wang et al. (2015) devise a likelihood robust A performance guarantee for the repeated newsvendor setting Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. optimization (LRO) approach, in which the set of possible is developed in §5. Numerical results are presented in §6. distributions are considered to be only those that make the Ambiguity aversion is considered in §7. Section8 concludes observed data achieve a certain level of likelihood. It is then the paper. assumed that the decisions are made based on the worst-case distribution within this set. 2. Literature Review A long-established approach when selecting a probability Despite some early and notable exceptions, e.g., Scarf (1958, distribution subject to known moments is to choose the 1959), the operations literature has typically assumed that distribution that has the maximum entropy (ME). Despite its Saghafian and Tomlin: The Newsvendor under Demand Ambiguity 170 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

widespread use in fields such as estimation theory, physics, L4x5 = h6x7+ + p6−x7+, where 6x7+ 2= max8x1 09. Demand statistical mechanics, and information theory among others, realizations are generated as i.i.d. random variables from there are only a few papers in the operations literature a true distribution that is unknown to the decision maker that have adopted maximum entropy based approaches. (DM). Unless otherwise stated (e.g., §6.3), we assume the Andersson et al. (2013) examine a nonrepeating newsvendor true distribution is stationary. The DM, “he,” as a convention model in which the decision maker only knows the mean hereafter, has two types of evolving information about the and variance of the demand distribution. They compare demand distribution. an ME approach with robust approaches and find that • Data/Observations. At the end of period t = 11 0 0 0 1 T , the ME approach performs better on average. Different the DM knows the realization of demand Di in each of the from Andersson et al. (2013), (i) we examine a repeated periods i = 11 0 0 0 1 t. This information is denoted as Ft. We newsvendor setting and so the decision maker has access to define F0 = to reflect the fact that there is no demand past data to dynamically update his beliefs; and (ii) we allow realization prior to period 1. for (evolving) upper/lower bounds on moments and tails • Exogenous Research. This information is contained in instead of assuming a known mean and variance. Maglaras two sets. and Eren (2015) apply the ME approach in a revenue —Ambiguity Set. As noted in §1 for the example of management context under a discrete demand assumption new product introductions, prelaunch marketing research and static linear equalities, and establish its asymptotical generates a set of possible demand distributions based behavior. Unlike their work, we (i) consider a newsvendor on the judgments of various relevant constituents, e.g., setting, (ii) permit continuously distributed demand, (iii) managers, sales people, prospective customers. Suppose allow for dynamically changing inequalities (lower and upper the sets of possible demand distributions and associated bounds on the moments and tails that may change over densities (assuming they exist) are given by the ambiguity 0 time), and (iv) build distributions over possible distributions sets D 2= 8FD 2 ∈ ì9 and D 2= 8fD 2 ∈ ì9, respectively, as opposed to the random variable itself, i.e., a second-order where is a scalar indicator associated with a distribution, approach as opposed to a first-order one. and ì is an arbitrary, typically uncountable, set. Without loss The concept of maximizing entropy when updating proba- of generality, we assume all the distributions in D have a bility distributions in response to new information, under the common support denoted by ¤, e.g., ¤ = 601 5. We assume ∗ strong assumption that the first moment is exactly known, that the ambiguity set D contains the true distribution FD , has been shown to include Bayesian updating as a special where ∗ ∈ ì is simply an indicator for the true distribution. case; see Caticha and Giffin (2006) and Giffin and Caticha The fact that the indicator is a scalar is not a limitation. (2007). We adopt and develop this approach by (i) creat- We make no assumption about the shape of the distributions ing a mechanism that builds a belief/distribution about the contained in the ambiguity set. The set may contain any possible distributions of the underlying random parameter distribution, and the distributions do not need to come from (as opposed to directly building a distribution about it); (ii) the same family; see, for example, our numerical study allowing for partial distributional information in the form of in §6. In addition, families with more than one parameter arbitrary upper and/lower bounds on the moments and tails, are allowed. It is simply a matter of associating a scalar which may evolve over time (as opposed to the very special with an instance of the family. case where the first moment is perfectly known and this —Moment and Tail Bounds. As noted in §1, pre and information is static). Importantly, we also embed it in a postlaunch marketing research can also provide the DM with sequential decision-making setting (a repetitive newsvendor), additional demand information in the form of bounds on (i) provide a performance guarantee for it, and analytically the expected value of demand, and (ii) tail probabilities. At establish its asymptotic behavior including its consistency period t ∈ ´, we denote the lower and upper bounds on the ¯ and rate of convergence. To the best of our knowledge, our expected value of demand by Ɛt4D5 and Ɛt4D5, respectively, ¯ 1 proposed method is among the very first in the operations with 0 ¶ Ɛt4D5 ¶ Ɛt4D5. We let the tail related information literature that is both data-driven and information-driven at period t be of the form Pr4D ¾ u5 ¶ bt4u5, for some and also fully dynamic. The approach provides a method to nonincreasing function bt4u52 ¤ → 601 17. We denote this ˆ effectively combine dynamically evolving partial information moment and tail information by the set Ft, and note that on moments and tails with data and observations, presenting allowing it to change over time is a generalization of the static a widely applicable tool (especially in the current trend of case in which the moment bounds and the tail information

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. ˆ ˆ business analytics) for decision making under distributional are fixed over the horizon. In this latter case, Ft = F0 for ˆ ambiguity. all t ∈ ´, where F0 is the initial information before the first time period has occurred. We note that the moment and tail information need to be consistent, i.e., should not contradict 3. The Model each other. Otherwise, there may not be any distribution We consider a repeated newsvendor problem over periods t ∈ in the ambiguity set that satisfies them, in which case the ´ 2= 811 0 0 0 1 T 9, with T ¶ . Let h > 0 and p > 0 denote the problem we discuss below is infeasible. The consistency per-unit holding and shortage penalty costs, respectively. Let between the data/observations and the moment/tail bounds Saghafian and Tomlin: The Newsvendor under Demand Ambiguity Operations Research 64(1), pp. 167–185, © 2016 INFORMS 171

can be monitored and statistically tested. Our approach possible demand distributions contained in the ambiguity set. can allow the decision maker to also include endogenously In essence, the DM assigns a probability that any particular generated moment bounds based on the data/observations; distribution in the ambiguity set is the true distribution. At see §6.3. the end of period t, having observed the value of period-t Because the DM does not know the true demand distribu- demand, the DM updates his belief about the probability tion, his decision in each period will be based on his current of each distribution in the ambiguity set; that is, a new belief about the distribution. We will introduce and develop distribution (over the distributions in the ambiguity set) is our approach to belief representation and updating in the formed. The DM adopts the maximum entropy principle next section. Before doing so, we specify the sequence of when creating this new distribution: beliefs are updated so events in each period t ∈ ´: that (i) the posterior coincides with the prior as closely as 1. The DM determines an order quantity qt ¾ 0 (given possible, and (ii) only those aspects of beliefs for which his current belief about the demand distribution), orders it, new evidence was gained are updated. Furthermore, the and receives it. updating is done in a manner that ensures conformance 2. Demand is realized, with the realized value denoted to the current moment and tail bound information. These 0 0 by t, and the newsvendor cost L4qt − t5 is incurred. notions are mathematically formalized in §4.2. 3. New exogenous research, if any, arrives regarding We call this a Second Order Belief Maximum Entropy ˆ moment and tail bounds, and the information set Ft is (SOBME) approach because the DM’s beliefs are represented ˆ ˆ specified. Note that Ft = Ft−1 if no new research arrives. as a distribution over possible demand distributions rather The DM updates his belief about the demand distribution than being codified directly as a demand distribution (a first- for period t + 1. order approach). We note that if the DM is ambiguity neutral, When determining his order quantity at the beginning of i.e., the DM’s period-t objective function is given by (1), period t (step 1), the DM solves2 then there exists a single first-order demand distribution that would yield an equivalent period-t objective value

min ˆ 6 4q − D571 (1) to that given by a distribution-over-distributions approach. ƐDFt−11 Ft−1 L t qt ¾0 However, there is a question as to how such a first-order demand distribution would be developed. SOBME provides where Ɛ ˆ denotes that the expectation is taken D Ft−11 Ft−1 a useful and practically relevant method to dynamically with respect to the DM’s current belief about the demand characterize the first-order distribution as a weighted average distribution, and that this current belief (however it was of the distributions in the ambiguity set. From a practical formed) can only avail of the information available at that perspective, our SOBME approach aligns directly with point in time, i.e., the prior demand realizations F and t−1 new-product forecasting judgmental methods (see §1) in the most recent moment and tail information set Fˆ . We t−1 which different people estimate the probability of each sales refer to the above optimization problem as the DM’s problem level. Such methods yield a set of distributions, one from and to its adopted policy as the DM’s policy. The values each person. SOBME provides a method to dynamically of demand, however, are drawn from a hidden underlying update the probability associated with each distribution and, distribution. Therefore, the DM’s policy is associated with as we will see in subsequent sections, it performs very a true cost that depends on the true hidden underlying well. Furthermore, as we will see in §7, a distribution-over- distribution. This true cost represents the ultimate measure distributions approach is necessary if the DM is ambiguity of the policy’s performance. averse/loving, and SOBME provides a good approach for such settings. 4. A Belief Updating Approach That Before we formalize the SOBME approach, we contrast it Combines Data and Moment/Tail conceptually with pure data-driven approaches and robust Bounds optimization approaches. In pure data-driven approaches, SAA for example, the DM constructs (in each time period) In this section, we first sketch the outline of our approach a new demand distribution based exclusively on the history to representing and updating beliefs, and contrast it with of demand observations. SOBME differs from SAA along other existing approaches (§4.1). Next, in §4.2, we formally two important dimensions: (i) it enables the DM to avail of develop the approach and present the key belief-updating moment and tail bound information in addition to the demand result. Finally, in §4.3, we analytically examine the asymp- Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. observations, and (ii) the DM’s beliefs are represented totic behavior of the approach by investigating its consistency as a distribution over distributions rather than as a single and convergence rate. distribution. SOBME is also very different from robust optimization approaches in which decisions are made only 4.1. Outline of the Approach with respect to the worst-case distribution in the ambiguity We adopt a distribution-over-distributions approach. At the set. In SOBME, the DM assigns weights to all members of beginning of period t, the DM’s belief about the demand the ambiguity set. This helps avoid the tendency of robust distribution is represented by a distribution over the set of optimization methods to make overly conservative decisions. Saghafian and Tomlin: The Newsvendor under Demand Ambiguity 172 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

Moreover, our approach updates these weights based on new The updating mechanism is given by the following func- realizations and evolving moment and tail information. tional optimization program, in which the decision variable is o the function ft+12 ¤ × ì → +, and is chosen to minimize 4.2. Formal Development of the the relative entropy (or equivalently maximize the negative SOBME Approach relative entropy). At the beginning of period t ∈ ´, the DM’s belief about the o min dKL4ft+1ft5 (2) f o 2 ×ì→ demand distribution is represented as a density function ft45 t+1 ¤ + over the set of possible demand distributions contained in s.t. the ambiguity set 2= 8F 2 ∈ ì9.3 At the end of period t, Z Z D D o ¯ having observed the period-t demand and knowing the most Ɛt4D5 ¶ ft+141 5145 d d ¶ Ɛt4D5 (3) ¤ ì recent moment and tail information, the DM updates his Z Z f o 41 5 41 u5 d d b 4u5 ∀ u ∈ (4) belief from ft45 to ft+145. In what follows, we formally t+1 2 ¶ t ¤ ¤ ì develop the approach for updating ft45 to ft+145. Z o 0 The updating mechanism (at the end of period t) is ft+141 5 d = 4 − t5 ∀ ∈ ¤ (5) based on the following three inputs. One input is the joint ì Z Z density (from the beginning of period t) over the demand o ft+141 5 d d = 11 (6) and the demand distribution indicator; this is given by ¤ ì f 41 5 = f 45f 45 f 45 t t D , where D is the demand density where 45 2= 4D 5 = R dF 45, 41 u5 2= 1 − 1 Ɛ ¤ D 2 associated with the indicator . The second input is the F 4u5, and 4 · 5 is the Dirac delta function.5 It should be 0 D observed value of the demand in period t, denoted by t. noted that constraints (4) and (5) are not single constraints: The third input is the most recent moment and tail bound they each represent an infinite number of constraints. ˆ information, i.e., Ft. We first discuss the role of constraints (3) and (4). Note that o R o Let ft+141 5 denote some joint density (at the end of f 41 5 d gives the marginal density of the demand ¤ t+1 period t) over the demand observation that was realized distribution indicator . Therefore, constraint (3) reflects in period t and the demand distribution indicator random the fact that the DM’s updated belief must result in a mean variable. The output of our updating mechanism, defined demand that conforms to the most recent upper and lower through the optimization program described below, will moment bounds, ¯ 4D5 and 4D5, respectively. Similarly, ∗o Ɛt Ɛt be a specific joint density denoted by ft+141 5. This will constraint (4) reflects the fact the DM’s updated belief must determine the DM’s updated belief ft+145 using ft+145 = result in a demand distribution whose tail probability (at any R f ∗o 41 5 d, i.e., the DM’s updated belief f 45 is ¤ t+1 t+1 given u) is bounded above by the most recent tail bound bt4u5. ∗o given by the marginal density of ft+141 5. Next we discuss the role of constraints (5) and (6). Recall o The updating mechanism chooses a joint density ft+141 5 that the updating optimization program is selecting a joint [the “posterior”] to minimize the Kullback-Leibler (KL) o density ft+ 41 5 over the demand observation that was real- o 1 divergence (a.k.a. relative entropy) between ft+141 5 and ized in period t and the demand distribution indicator random f 41 5 [the “prior”] while recognizing that the demand R o t variable. The marginal density ft+141 5 d in (5) is the 0 ì observation in period t took on some specific value, t, and density over the possible demand observations. Of course, the that the joint density f o 41 5 must imply a demand density 0 t+1 realized value of the observation is already known to be t. that conforms to the most recent moment and tail bound Therefore, any valid marginal density (valid, in the sense that ˆ information, i.e., Ft. The Kullback-Leibler (KL) divergence it reflects what the DM already knows) should place all its is formally defined as: 0 0 weight on = t and no weight on 6= t. However, it also needs to be a legitimate density function. These two require- Definition 1 (Kullback-Leibler Divergence). The ments are fulfilled by using the Dirac delta function (see Kullback-Leibler (KL) divergence or the relative entropy Endnote 5) and setting R f o 41 5 d = 4 − 05 ∀ ∈ , between two joint densities f o and f defined on × ì is: ì t+1 t ¤ t+1 t ¤ which is constraint (5). Finally, constraint (6) is a normaliza- o o tion that ensures ft+141 5 is a joint density in the sense o Z Z o ft+141 5 dKL4ft+1ft5 = ft+141 5 log d d that it integrates (over all possible values of both arguments) ¤ ì ft41 5 to 1. We note this constraint is in fact redundant given o R Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. f 41 5 t+1 the property that − 4x5 dx = 1, but we include (6) for = Ɛf o log 0 t+1 expositional clarity. ft41 5 This updating program is an infinite-dimensional optimiza- This is a general definition for any two densities, but tion program that has a continuously differentiable convex described in terms of our specific notation.4 Observe objective function and linear constraints. This allows us to o o that dKL4ft+1ft5 ¾ 0, and dKL4ft+1ft5 is convex in pair use KKT-like conditions along with the variational principle o ∗o 4ft+11 ft5; see, for example, p. 32 of Cover and Thomas to characterize the global optimum solution, ft+141 5. ¯ (2006). Let t and t be the Lagrangian multipliers for the upper Saghafian and Tomlin: The Newsvendor under Demand Ambiguity Operations Research 64(1), pp. 167–185, © 2016 INFORMS 173

bound and lower bound in constraint (3), respectively. Let this notation, program (2)–(6) can be rewritten as one that

t4u5 and t45 represent the multipliers for (the inﬁnite minimizes the KL divergence between the posterior ft+145 ˜ number of) constraints (4) and (5), respectively. Deﬁning and the prior ft45 subject to moment and tail constraints. g 45 2= 4¯ − 5 45 + R 4u5 41 u5 du, we are able In what follows, we apply the SOBME approach in a t t t 1 ¤ t 2 to obtain the following closed-form result. repeated newsvendor setting. Considering the sequence of events described in §3, we have the following. At the Theorem 1 (Second Order Joint Belief Updating). The beginning of period t, the DM determines his order quantity solution to the optimization program (2)–(6) is given by: qt ¾ 0 to minimize the expected newsvendor cost given

0 −gt 45 his current belief about the demand distribution, which is ∗o ft41 54 − t5e ft+141 5 = R 0 represented by the density function ft45 over the set of f 401 5e˜ −gt 45˜ d˜ ì t t possible demand distributions contained in the ambiguity set 2= 8F 2 ∈ ì9. Applying the well-known newsvendor Proof. All proofs are contained in Appendix B (available D D quantile result, the optimal quantity decision under SOBME as supplemental material at http://www.dx.doi.org/10.1287/ is given by: opre.2015.1454). Corollary 1 (Second Order Belief Updating). The SOBME p qt 2= inf q ∈ ¤2 Ɛ6FD 4q57 ¾ 1 (7) updated density, ft+145, over the demand distribution p + h indicator ( ∈ ì) is: R where Ɛ6FD 4q57 = ∈ì FD 4q5 ft45 d. After ordering 0 −gt 45 0 ft45ft4t 5e this quantity, the order is received, demand t is realized, ft+145 = R 1 SOBME 0 0 −gt 45˜ the newsvendor cost L4q − 5 is incurred, and new ì ft45f˜ t4t 5e ˜ d˜ t t exogenous research, if any, arrives regarding moment and 0 0 where ft4t 5 = fD 4t5. tail bounds. At the end of period t, knowing the realized demand and the latest moment and tail bounds, the DM Remark 1 (Bayesian Updating). When the moment and updates his belief from ft45 to ft+145 using Corollary1. tail bounds, i.e., constraints (3) and (4), are not binding in the SOBME For later use, let Qt denote the period-t order quantity updating optimization program, the updating result simplifies random variable. QSOBME takes on a realized value qSOBME, to f 45 = f 45f 40 5/4R f 4˜ 5f 40 ˜ 5 d˜ 5. That is, t t t+1 t t t ì t t t specified by (7), given any particular sample path of demand the updating result reduces to a traditional Bayesian updating realizations in periods 11 0 0 0 1 t − 1. Let result. This occurs when (i) there are no moment and tail bounds, or (ii) these bounds are sufficiently loose that they ∗ ∗ p q = inf q ∈ ¤2 F 4q5 do not constrain the DM’s updated beliefs, i.e., when they are D ¾ p + h not informative given his prior belief. In general, however, the updating is performed using an exponential modifier that denote the optimal order quantity if the DM knows the true captures the effect of the moment and tail bound information. distribution. This optimal order quantity is static, because This is an important result with widespread applications the true distribution is stationary. In the next section, we SOBME ∗ beyond the motivating example of this paper. It establishes show that Qt → q almost surely as t → . SOBME as a generalization of Bayesian updating for settings in which partial distributional information is available in 4.3. Asymptotic Behavior of SOBME: Consistency addition to data/observations.6 and Convergence Rate Recall that we make no restriction on the family or In this section, we study two important asymptotic properties families of distributions included in the ambiguity set. From of SOBME. The first property is consistency. The second this perspective, SOBME can be viewed as a nonparametric property relates to the rate at which the demand distribution approach to belief representation and updating. Furthermore, built by SOBME converges to the true distribution. We for- we emphasize that the SOBME belief representation and malize the consistency concept below, but loosely speaking, updating result are not limited to a repeated newsvendor this means that the DM eventually places (almost) all his setting. They apply to any setting in which a DM has partial weight on the true demand distribution, with the result that distributional information (in the form of an ambiguity set the DM’s quantity decision converges (with probability one)

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. and, possibly, moment and tail bounds) about a random to the quantity decision under full distributional information: SOBME ∗ variable of interest, and wishes to dynamically combine Qt → q almost surely as t → . This requires one this information with an evolving history of observations. to consider a shrinking neighborhood of ∗, and to show Finally, we note that there are alternative formulations to the that the DM will eventually assign a weight of one to this optimization program (2)–(6), and SOBME can be derived shrinking neighborhood. The type of consistency depends from these alternatives. For example, one can implicitly on how such a shrinking neighborhood is constructed. We invoke the Dirac delta function used in constraint (5) and establish weak consistency, which considers weak neigh- ˜ 0 ∗ deﬁne a new distribution ft45 = ft4t1 5. Introducing borhoods of . That is, we consider those ∈ ì that Saghaﬁan and Tomlin: The Newsvendor under Demand Ambiguity 174 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

∗ 1/2 are “close” to , where closeness is measured in terms of The above result establishes t = max8rt1 yt 9 as a con- an expectation-like distance (see Appendix A for formal vergence rate for SOBME. As discussed in Appendix A, definitions). this rate of convergence depends on the tightness of the To establish consistency and a rate of convergence, we ambiguity set (measured through rt) and the precision of the need to introduce the notion of coherent information for the initial prior (measured through yt). moment and tail bounds. Definition 2 (Coherent Information). The sequence 5. Newsvendor’s Cost of Ambiguity: ˆ of information 8Ft2 t ∈ ´9 is said to be coherent if (i) the A Performance Guarantee sequence of lower (upper) bounds on the first moment, i.e., ¯ Having characterized the asymptotic behavior of SOBME, 8Ɛt4D52 t ∈ ´9 (8Ɛt4D52 t ∈ ´9), is nondecreasing (nonin- ¯ we next explore the increase in the average cost per period creasing), with Ɛt4D5 ¶ Ɛ4D5 ¶ Ɛt4D5 for all t ∈ ´ where (during a horizon of T < ) that the DM experiences because Ɛ4D5 is the true demand mean, and (ii) for any u ∈ ¤, the of ambiguity. We refer to this cost as the newsvendor’s cost sequence of tail upper bounds 8bt4u52 t ∈ ´9 is nonincreasing ∗ of ambiguity. We will derive an analytical upper bound for with bt4u5 ¾ FD 4u5. the newsvendor’s cost of ambiguity when SOBME is used. If the moment and tail bound information is not coherent, This will enable us to develop a performance guarantee. To then we should not expect an approach that uses such infor- that end, let mation to necessarily converge to the true distribution. For example, the information might either be “wrong” in some T 1 ∗ SOBME X ∗ SOBME period, i.e., the moment and tail bounds may rule out the C 2= Ɛf 6L4Q −Dt5−L4q −Dt57 (8) T T D t true distribution, or the sequence of bounds might be non- t=1 monotonic, e.g., periodic, which could prevent convergence denote the random average (per period) newsvendor’s cost of even if the information is correct. ambiguity under SOBME, which depends on the demand real- Recall that f 45 is the density function (at the start t izations up to the beginning of period T , i.e., the realization of period-t) over the demand distribution indicator . Let SOBME of vector DT = 4Di1 i = 11 21 0 0 0 1 T − 15 which is generated Ft denote its cumulative distribution, where we use the superscript SOBME to highlight that it is the result of the based on the true distribution. Taking the expectation with SOBME updating method. The following result establishes respect to this vector, we denote by the weak consistency of SOBME. We refer the reader to cSOBME 2= 6CSOBME71 (9) Appendix A for formal definitions of weak consistency and T ƐDT T KL neighborhoods used in the following theorem. the expected average per period newsvendor’s cost of ambi- Theorem 2 (Consistency of SOBME). Suppose that guity. It should be noted that CSOBME is a random variable ˆ T (a) the sequence of information 8Ft2 t ∈ ´9 is coherent, but cSOBME is a number. In (8), QSOBME is a random variable SOBME T t and (b) the initial distribution F1 is such that with a realization that depends on D : the expectation in R SOBME ∗ t dF1 > 0 for every > 0 (where K is a KL K∗ (8) is only with respect to D , whereas this randomness is ∗ t neighborhood of ). Then: removed through the outer expectation in (9). SOBME (i) Ft is weakly consistent. The following results provide upper bounds for CSOBME SOBME ∗ T (ii) Qt → q almost surely as t → . SOBME and cT when the support of all the distributions in the In addition to consistency (showing almost-sure conver- ambiguity set can be bounded from above. Bounded demand gence to the true distribution), it is useful to explore the is not a strong assumption in practice because demand is rate/speed of convergence. In a later section, we will examine never infinite. this and other questions numerically. Here, we develop an Theorem 4 (Newsvendor’s Cost of Ambiguity). Suppose almost-sure analytical bound on the distance between the the support is 601 d¯7 for some d¯ ∈ . Letting k 2= true demand density and the one constructed using SOBME. √ ¤ + SOBME R 2 2d¯max8p1 h9, the following hold for any T : To this end, for x ∈ ¤, let fD1 t 4x5 2= ∈ì ft45fD 4x5 d denote the demand density generated by SOBME at the start s 1 T of period t. 4i5 CSOBME k X d 4f ∗ f SOBME51 (10) T ¶ T KL D D1 t Theorem 3 (Rate of Convergence). Suppose that the t=1 Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. sequence of information 8Fˆ 2 t ∈ ´9 is coherent and that t almost surely. regularity conditions 3 and 4 (see Appendix A) hold. Then, there exists constants and such that for large enough t: s 1 2 T ∗ 2 SOBME 1 X ∗ SOBME SOBME 2 −2t t 1/2 4ii5 c k d 4f f 5 0 (11) dH 4fD fD1 t 5 ¶ 616 t + 2e 77 a.s.1 T ¶ ƐDT KL D D1 t T t=1 1/2 where t = max8rt1 yt 9, with rt and yt defined in Appendix A, and dH 4f1f25 is the Hellinger distance (defined in Part (i) of the above result shows that the random aver- Appendix A) between any two probability densities f1 and f2. age per period KL divergence between the true density Saghafian and Tomlin: The Newsvendor under Demand Ambiguity Operations Research 64(1), pp. 167–185, © 2016 INFORMS 175

and the one built using SOBME nicely bounds, in the The above result states that the expected average per almost sure sense, the average per period cost of ambiguity. period KL divergence between the true density and Part (ii) shows a related result for the expected average the one built using SOBME is at worst of the form 1 per period cost of ambiguity. We next develop a bound for 41/T 56 2 log4T /42e55 + l7 for some constant l that depends 641/T 5 PT d 4f ∗ f SOBME57, the expected average per ƐDT t=1 KL D D1 t on the initial prior and the Fisher information, which are period KL divergence between the true density and the one both also affected by the “tightness” of the ambiguity set, D0. built using SOBME. Because of the result of Theorem4 part The important implication is that it allows us to provide the (ii), this will enable us to reach our end goal of providing a following performance guarantee for our proposed SOBME performance guarantee for SOBME. approach: based on Theorem4 part (ii), the expected average ∗ To bound Ɛ 641/T 5 PT d 4f f SOBME57, we need (per period) newsvendor’s cost of ambiguity under SOBME DT t=1 KL D D1 t √ some general conditions on (i) the (dynamic) information on goes to zero at worst at a fast rate log T /T as T → . ˆ 0 moment and tail bounds (Ft), (ii) the ambiguity set (D ), (iii) In other words, although a DM facing demand ambiguity SOBME the initial prior used by the DM under SOBME (f1 ), inevitably incurs an extra per-period cost compared to a DM and (iv) the convergence behavior around ∗. For (i), we use who knows the true demand distribution, the average extra the coherent information condition (Deﬁnition2) to ensure per-period cost approaches zero quite quickly if SOBME is that the DM is not relying on false information. For (ii)–(iv), used. The actual rate depends on the quality of the dynamic we use the “smoothness” and “soundness” conditions dis- information the DM has on moments and tails; Theorem5 cussed in Clarke and Barron (1990) for Bayesian updating provides an upper bound by considering the worst case, i.e., mechanisms. We brieﬂy list those conditions here for our no information. above-mentioned goal, and refer interested readers to Clarke ¯ and Barron (1990) for more detailed discussions. It should Theorem 6 (Performance Guarantee). If ¤ = 601 d7 for ¯ be noted that the “smoothness” condition is related to the some d ∈ +, then under the conditions of Theorem5: behavior around ∗ and the “soundness” condition is related r to the speed of weight accumulation around ∗ as more and log T √ cSOBME k0 + o41/ T 51 (13) more observations are made. T ¶ T Condition 1 (Smoothness). The density fD 45 and the 0 SOBME ∗ where k = 2d¯max8p1 h9, and hence, c → 0 as T → divergence d 4f f 5 are both twice continuously differ- √ T KL D D at a rate of log T /T . entiable at ∗. Furthermore, (i) there exists an > 0 such that: Theorem6 establishes that the DM does not need to have full distributional information to make suitable stocking ¡2 2 sup log f 45 < 1 decisions: SOBME enables the DM to deal effectively with Ɛ 2 D ∈ì2 −∗< ¡ ambiguity.

SOBME and (ii) I45 and f1 45 are both finite, positive, and continuous at ∗, where I4 · 5 is the Fisher information and 6. Numerical Experiments SOBME f1 4 · 5 is the initial prior density. As discussed in §1, companies reduce ambiguity in new Condition 2 (Soundness). The ambiguity set is said to product introductions by devoting significant market-research be sound if the convergence of a sequence of values is efforts to gather information about the demand distribution. equivalent to the weak convergence of the corresponding That is, firms do not rely exclusively on demand observations demand distributions: when forming beliefs about an unknown demand distribution; they can and do generate additional partial information about ∗ ∗ → ⇔ FD → FD 0 the demand distribution. In this paper, such information is codified by the ambiguity set and moment and tail bounds, The above conditions allow us to provide an effective i.e., the exogenous research. It is managerially important to bound for the expected value of the average per period KL understand the value of such exogenous research in an oper- divergence under our proposed updating mechanism. ational setting (here, a repeated newsvendor). Furthermore, Theorem 5 (Expected Average per Period KL Diver- assuming there is value in the exogenous research, it is gence). Suppose the coherent information, smoothness, and helpful to examine which type of information (ambiguity set

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. soundness conditions hold. Then: or moment bounds) provides more value, and what factors, e.g., newsvendor quantiles, affect the value. Answers to these T 1 ∗ X d 4f f SOBME5 questions—which we provide through numerical studies ƐDT KL D D1 t T t=1 below—can help managers understand when it is worth their 1 1 T 1 1 effort to obtain exogenous information and what form of log + log I4∗5 + log + o415 0 information is most valuable. ¶ T 2 2e 2 f SOBME4∗5 1 A natural way to explore the value of exogenous research (12) is to compare our information setting—data and exogenous Saghaﬁan and Tomlin: The Newsvendor under Demand Ambiguity 176 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

research—to two benchmark settings: (i) full information, Ambiguity Set. We consider two cases for the ambiguity and (ii) a data-only information setting. For this second set D. In Case I, the ambiguity set contains all exponential benchmark, we need to choose an approach tailored to a distributions with a mean in 6101 205 as well as all normal data-only setting. Various pure data-driven approaches have distributions with a mean in 6101 207 and a C.V. of 0.2. In been developed in the literature as we discussed in §2, among Case II, the ambiguity set includes all those distributions in which we consider the Sample Average Approximation (SAA) Case I plus the mixtures (with equal weights of 0.5) of any approach (see, e.g., Kleywegt et al. 2002; Shapiro 2003; exponential with mean 6101 205 and a normal with the same Levi et al. 2007, 2015), Scarf’s method (Scarf 1958, Gallego mean (and a C.V. of 0.2). In other words, Case II represents and Moon 1993), and the recently developed likelihood a larger ambiguity set that contains Case I and mixtures of robust optimization (LRO) approach (Wang et al. 2015). distributions in Case I.8 Under SAA, the DM at the beginning of period t + 1 sets Therefore, the combination of ambiguity sets and the true distribution leads to four different cases: I-A, I-B, II-A p SAA ˆ t and II-B. In all cases, we assume SOBME starts with a qt+1 2= inf q ∈ ¤2 FD4q5 ¾ 1 (14) p + h uniform distribution on the ambiguity set, i.e., it starts with the highest information entropy. where Fˆ t 4q5 is the empirical c.d.f. after t observations: D Moment Bound Information. For each of these four cases, Fˆ t 4q5 2= 41/t5 Pt 8k q9, with k denoting the demand D k=1 ¶ we consider two possible sets of moment bound information, realization in period k. In Scarf’s method, the DM minimizes 8Fˆ 2 t ∈ ´9. In one situation, the DM has no moment the worst-case expected cost among all the distributions that t information beyond those dictated by the ambiguity set have a mean equal to the sample mean and a variance equal (referred to as the “No Bounds” situation). In the other to the sample variance. Using Scarf’s result (Scarf 1958), the situation (referred to as the “Tight Bounds” situation), optimal ordering quantity at the beginning of period t + 1 is the DM has dynamically evolving ﬁrst moment bound r r information. In this “Tight Bounds” situation, the initial SCARF st p h qt+1 2= mt + − 1 (15) lower and upper bounds are those dictated by the ambiguity 2 h p ¯ set, i.e., Ɛ04D5 = 10 and Ɛ04D5 = 20, but these bounds get where m and s are the sample average and sample standard progressively tighter by 1.5 units in each period until they t t ¯ deviation of the observations made in periods 11 21 0 0 0 1 t. hit Ɛt4D5 = 1405 and Ɛt4D5 = 1505, after which they remain In the dynamic LRO approach (Wang et al. 2015), the set constant. To be conservative when evaluating SOBME, we of possible distributions in each period t contains only assume that the DM does not have tail bound information. those that make the observed history of data achieve a Such information, if available, would of course improve the certain level of likelihood, and the optimal decisions are performance. made based on the worst-case distribution within that set. In Newsvendor Quantile. The above combination of two our repeated newsvendor setting, we numerically observed ambiguity sets, two true distributions, and two moment that SAA (weakly) outperformed both Scarf’s method and information sets leads to eight cases. For each of these cases, dynamic LRO, and so we adopt the SAA approach as the as our base-case scenario, we set the newsvendor quantile to representative for the data-only setting.7 That is, for brevity, p/4p +h5 = 0075, with h = 1 and p = 3. However, as we will we do not present our numerical results for Scarf’s method describe later, we also explore the impact of the newsvendor and LRO, and instead focus on the performance of SAA quantile by varying its value in 800501 00551 0 0 0 1 00959. among the pure data-driven approaches. Runs. Each case is run for 100 periods and the average cost is calculated for the SOBME approach, the full informa- 6.1. Study Design tion case, i.e., when the true distribution is known, and the In addition to examining the value of exogenous information pure data-driven SAA approach. We purposely choose a by comparing the performances of SOBME and SAA, limited time horizon because we know that the performance our numerical study explores the performance of SOBME of both SOBME and SAA will asymptotically approach that relative to a DM who has full distributional information. of full information as more and more demand observations It also investigates how the performance of SOBME and the become available. In other words, when ample data exists, it value of exogenous information are inﬂuenced by the true is less important which approach is used, as long as it is

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. distribution, the type of exogenous information (ambiguity a consistent approach. As such, we are mainly interested set and moment bounds), and the newsvendor quantile in how well SOBME performs when data is limited. The p/4p + h5. limited-data setting also better represents the new product ∗ True Distribution. The true distribution, FD , in our study introduction we discussed in §1. The average costs are is assumed to have a mean of 15. The true distribution is calculated by conducting 50 independent replications with either normal (with a coefﬁcient of variation (C.V.) of 0.2) each replication being a sample path generation, i.e., demand or exponential, with a label “A” denoting the normal case realizations in periods 1 to 100. Under each sample path and and “B” denoting the exponential case. in each period, the related expected newsvendor costs are Saghaﬁan and Tomlin: The Newsvendor under Demand Ambiguity Operations Research 64(1), pp. 167–185, © 2016 INFORMS 177

calculated directly and based on the closed-form newsvendor expand his ambiguity set when he has more doubt about the overage plus underage costs of the DM’s ordered quantity. form of the true distribution. In each period, the closed-form solution of Theorem1 By comparing the difference between SOBME’s perfor- (and Corollary1) are used to calculate the updated SOBME mance and that of SAA in each subfigure, we see that the belief function and then the newsvendor problem is solved DM benefits significantly from exogenous information in the based on the updated beliefs. The belief-updating problem form of the ambiguity set and moment bounds. This indicates involves calculating the function gt45 which requires finding that there is potentially great value in marketing research and ¯ Lagrangian multipliers t and t. Note that t4u5 can related actions that can generate this kind of information. In be set to zero because there is no tail constraint. These addition, our results suggest that the information contained two multipliers are calculated efficiently in each period in the moment bounds is more valuable than those contained t by employing a numerical search method which takes in the ambiguity set. Hence, the DM should prefer a tighter advantage of the properties of the underlying optimization set of moment bounds than a tighter ambiguity set: efforts problem.9 A similar numerical search can be used for that result in tighter moment bounds are in general more calculating t4u5 when it is nonzero. We note that the valuable than those that result in tighter ambiguity sets. belief-updating optimization problem is independent of the The Effect of the True Distribution. Comparing the nor- DM’s ordering problem, and so the belief-updating aspect mal cases (I-A and II-A) with the exponential cases (I-B of SOBME can be applied for a wide range of decision- and II-B), we see that SOBME performs better when the making problems in which there is distributional ambiguity. true distribution is exponential. In the exponential cases, The computational burden in calculating various integrals SOBME’s optimality gap is less than 1% after a few periods involved in our method (arising from the continuous first and under both “No Bounds” and “Tight Bounds” situations. second-order distributional assumptions, i.e., the sets ¤ and That SOBME performs better when the true distribution ì being continuous) is overcome by using high precision is exponential is due to the fact that the newsvendor cost function interpolations and advanced numerical methods for is much more sensitive to distributional mis-specification calculating integrals. when the true distribution is normal than when the true distribution is exponential. To observe this consider the 6.2. Results following cases within our numerical setting (h = 1 and For each of the eight cases in the study design, Figure1 p = 3): (i) the true distribution is normal (C.V. = 002), but reports the Average Cost Optimality Gap (%) for both the DM believes with probability one that it is exponential SOBME and SAA, where the optimality gap refers to the per- (C.V. = 1) (both having a mean of 15), and (ii) the true centage difference in cost as compared to the full information distribution is exponential (C.V. = 1) but the DM believes case in which the DM knows the true distribution. As can be with probability one it is normal (C.V. = 002) (both having a seen, SOBME performs very well: its optimality gap is less mean of 15). In case (i) the DM orders 20.79 units whereas than 1% after 20 periods in six of the eight cases. We next the true optimal ordering quantity is 17.03. This results in an discuss how the exogenous research information (moment optimality gap of about 55.15% (in terms of the percentage bounds and ambiguity set) and the true distribution impact additional expected cost incurred). In case (ii), the DM the performance of SOBME relative to full information and orders 17.03 whereas the true optimal ordering quantity to the pure data-driven representative, SAA. is 20.79. This results in an optimality gap of only about The Effect of Exogenous Information. The impact of 2.48% (in terms of the percentage additional expected cost the moment bound information on SOBME’s performance incurred). Comparing these two cases shows that a wrong can be explored by comparing the SOBME plots in the distributional assumption is less consequential when the true left column (No Bounds) with the corresponding SOBME distribution is exponential than when it is normal. Hence, plots in the right column (Tight Bounds). Moment bound having a good estimate of the true distribution is more information is clearly valuable: SOBME’s optimality gap important when the true distribution is normal than when it approaches zero much more rapidly when tight moment is exponential. This fact shows why SOBME performs better bound information is available. The impact of the ambiguity when the true distribution is exponential than when it is set on SOBME’s performance can be explored by comparing normal. Interestingly, however, SAA’s performance is slightly Case I-A with Case II-A and comparing Case I-B with Case better when the true distribution is normal than when it is II-B. We observe that the performance is quite similar in exponential. What accounts for this? There is a force that

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. both comparisons, indicating that SOBME is quite robust to counteracts the robustness under the exponential distribution. the ambiguity set. We tested this observation further (for The DM sees a more volatile set of observations under an the case in which the true distribution was normal) by also exponential distribution than under the normal distribution. considering a setting in which the ambiguity set contained This significantly influences the rate of convergence of SAA. only normal distributions. Contracting the ambiguity set This second driving force is stronger than the first one for did not materially impact SOBME’s performance. The fact SAA. This is not the case for SOBME, because it is not as that SOBME’s performance is robust to the ambiguity set is reliant on observations as SAA, i.e., SOBME is not using an indeed an important benefit because it allows the DM to empirical distribution. Saghafian and Tomlin: The Newsvendor under Demand Ambiguity 178 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

Figure 1. Comparison of SAA and SOBME under various cases.

(a) Case I-A-no bounds (b) Case I-A-tight bounds

8 SAA 6 SAA SOBME SOBME 5 6 4

4 3

2 2 Avg. cost opt. gap (%) Avg. cost opt. gap (%) 1

0 0 0 20406080100 0 20406080100 t t

4 4

3 3

2 2 Avg. cost opt. gap (%) Avg. cost opt. gap (%) 1 1

0 0 0 20406080100 0 20406080100 t t

(e) Case II-A-no bounds (f) Case II-A-tight bounds 8 SAA 6 SAA SOBME SOBME 6 5

4 4 3

2 2 Avg. cost opt. gap (%) Avg. cost opt. gap (%) 1

0 0 0 20406080100 0 20406080100 t t

(g) Case II-B-no bounds (h) Case II-B-tight bounds

6 SAA SAA 5 SOBME 5 SOBME 4 4

2 2

Avg. cost opt. gap (%) 1 Avg. cost opt. gap (%) 1

Figure 2. Sensitivity analyses on the newsvendor quantile p/4p + h5.

(a) Case I-A-no bounds (b) Case I-A-tight bounds 60 SAA 60 SAA 50 SOBME SOBME 50 40 40 30 30 20 20 Avg. cost opt. gap (%) 10 Avg. cost opt. gap (%) 10

0 0 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 p p – – p + h p + h

10 10

5 5 Avg. cost opt. gap (%) Avg. cost opt. gap (%)

0 0 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 p p – – p + h p + h

The Effect of the Newsvendor Quantile. To explore the a very suitable approach if a high service level is required or impact of the newsvendor quantile on the performance of if inventory cost parameters are subject to estimation error. SOBME, we focus on the four Case I studies (exponential In summary, the above study has the following managerial and normal true distributions; No Bounds and Tight Bounds implications: (i) there is significant value in exogenous information), and vary the newsvendor quantile p/4p + h5 ∈ information (as measured by the comparison of SOBME 800501 00551 0 0 0 1 00959, while fixing h = 1. Figure2 shows the and SAA), (ii) market research that leads to tighter moment performance of SOBME and SAA for each of the four cases. bounds is more valuable than research that leads to a more The performance of SAA and SOBME are averaged over 50 compact ambiguity set, and (iii) it is especially important to replications as well as the first 20 periods. We first note that avail of exogenous information (through SOBME) when our study confirms what has already been reported in the the newsvendor quantile is high because pure data-driven literature for SAA: its performance significantly degrades approaches (SAA) perform much worse at high quantiles. as the newsvendor quantile increases (see, e.g., Levi et al. As mentioned above, the SOBME approach is not limited 2015). In contrast, we observe that SOBME’s performance to a repeated newsvendor setting. It can be used in any is much more robust: its performance degradation is very multiperiod setting in which a DM has partial distributional low. For instance, in Figure2(a), although SAA has an information (in the form of an ambiguity set and, possibly, optimality gap of 55% for a newsvendor quantile of 009, moment and tail bounds) about a random variable of interest. SOBME has an optimality gap of only 10%. When moment A general measure of SOBME’s performance is the quality

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. bound information is available, Figures2(b) and (d), the of the distribution generated by SOBME, i.e., how close difference in performance between SOBME and SAA is even is it to the true distribution from which the observations greater. Recall that the difference in performance between are drawn? We investigate this in two different studies in SOBME and SAA measures the value of the exogenous Appendix C, the second of which uses data from a case information. Thus, our result suggests that this type of based on a real-world setting (Allon and Van Mieghem 2011) information is particularly valuable in settings characterized to show how moment bounds can be developed from “like- by high newsvendor quantiles. Moreover, the robustness of products.” Among other results, we ﬁnd that the quality of SOBME to the newsvendor quantile suggests that SOBME is the SOBME-generated distribution is very good, that partial Saghaﬁan and Tomlin: The Newsvendor under Demand Ambiguity 180 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

information in the form of moment bounds is particularly (No/Tight Bounds with/without Endogenous Bounds). As can valuable early in the horizon, and that companies can get be seen, endowing the DM with endogenous bounds does significant benefit from moment bound information even if not materially improve SOBME’s performance. Irrespective the number of like-products is quite limited. We refer the of the endogenous bounds, SOBME is already availing of reader to Appendix C for further details. the data to updates its probabilities across the potential distributions in the ambiguity set. Therefore, the data used to 6.3. Extensions generate the endogenous bounds has also been used (indepen- dently) to appropriately “weight” the possible distributions, 6.3.1. Endogenous Bounds. We have assumed to this and those that conform to the endogenous bounds will have point that the sequence of moment bound information received more weight. In other words, SOBME is already ¯ 8Ɛt4D51 t ∈ ´9 and 8Ɛt4D51 t ∈ ´9 are derived solely from using the observations, i.e.,“the endogenous data,” in an exogenous research. However, our SOBME framework can effective way, and hence using them to create supplementary also accommodate dynamic bounds that are endogenously moment bounds is not of significant help. We also note that generated from the data observations. For instance, using the the endogenous bounds are typically tight and, therefore, Central Limit Theorem (CLT) at the end of period t, a 1 − useful, only when they are generated based on ample data, confidence interval for the first moment is given by i.e., a large t. However, as discussed earlier, when ample √ √ data exists, any estimation approach that is consistent will It 2= 4mt − z/2 × st/ t1 mt + z/2 × st/ t51 (16) generate good results and it matters less which one is used. Moreover, it is precisely because they do not have ample data where z/2 is the inverse of the standard normal c.d.f. that firms devote efforts to generating exogenous research evaluated at 1 − /2, and mt and st are the sample average on the demand distribution. and sample standard deviation obtained from observations made in periods 11 21 0 0 0 1 t. In the absence of exogenous 6.3.2. Change Detection. We have assumed to this point

bounds, the DM can use the boundaries of It as moment that the true demand distribution is stationary throughout the bounds. If the DM has exogenous bounds given by some horizon. In practice, however, the distribution may change at ¯ sequences 8Ɛt4D51 t ∈ ´9 and 8Ɛt4D51 t ∈ ´9, then he can certain points in time because of, for example, changes in replace them with the respective boundaries of It in a period t economic factors, competitor actions, market penetration, ¯ when It ⊂ 6Ɛt4D51 Ɛt4D57, i.e., when the endogenous bounds and time of year. Some of these changes might be amenable are tighter than the exogenous ones. This ensures that the to prediction by managers, and SOBME can accommodate moment bound information remains coherent when the such predications through updated moment bounds. Other exogenous bound information is coherent. changes might not be predictable and completely hidden to We investigated the value of endogenous bounds by apply- the DM, and so it is important to test how well approaches ing this approach (using a 90% conﬁdence interval) to cases such as SOBME and SAA detect and respond to unpredicted I-A-No Bounds and I-A-Tight Bounds. Figure3 depicts changes. the performance of SOBME for the four possible situations We numerically examined the ability of SOBME and SAA to detect and respond to such a hidden/unpredicted Figure 3. The effect of endogenous bounds on the per- change in the type of demand distribution. In particular, we formance of SOBME (No.: no exogenous use a new setting, setting Case I-C, in which, unknown to or endogenous bounds; End.: endogenous the DM, the true distribution changes from normal (C.V. = bounds only; Exg.: exogenous bounds only; 002, mean = 15) to exponential (C.V. = 1, mean = 15) in End. + Exg.: both endogenous and exogenous the middle of the horizon, i.e., period 50. We evaluate bounds). the performance of SOBME (under both “No Bounds” and “Tight Bounds”) and SAA compared to a DM that knows the true distribution throughout the horizon, i.e., 4 No. both before and after the change. The results are presented End. in Figure4. Not surprisingly, both SOBME and SAA are 3 Exg. affected by the change at period 50. The change initially End. + Exg. affects SOBME slightly more than SAA because SOBME

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. was already getting very close to the true distribution before 2 the change whereas SAA was further from it. Also, we note the spike is lower (and comparable to that for SAA)

Avg. cost opt. gap (%) 1 when SOBME has tight bounds. Very importantly, however, irrespective of the bounds, SOBME recovers much more rapidly than SAA: it quickly recognizes the hidden change 0 0 20 40 60 80 100 and adjusts to it. This reﬂects the fact (analytically established t in the performance guarantee and numerically observed) that Saghaﬁan and Tomlin: The Newsvendor under Demand Ambiguity Operations Research 64(1), pp. 167–185, © 2016 INFORMS 181

Figure 4. Comparison of SAA and SOBME for change identical demand density would result in the same period-t detection (Setting Case-I-C) (NB: No Bounds; objective. Importantly, this theoretical equivalence holds TB: Tight Bounds). only when the DM is ambiguity neutral. For settings with an ambiguity averse/loving DM, the objective, i.e., (17), 6 depends on ft45 and fD 4x5 in a manner that cannot be SAA SOBME R captured through fD1 t 4x5 2= ∈ì ft45fD 4x5 d. There- 5 SOBME (NB) fore, it is crucial to adopt a distribution-over-distributions SOBME (TB) approach. SOBME provides a natural and effective approach 4 for ambiguity averse/loving settings. 3 In what follows, we ﬁrst explore some structural properties of the decisions made by an ambiguity-averse newsvendor, 2 i.e., properties of the period-t objective (17), when ê4 · 5 is

Avg. cost opt. gap (%) concave. We then explore the effect of the DM’s ambiguity 1 aversion on the newsvendor ordering quantity. To this end, −x 0 we consider the negative exponential function ê4x5 = −e 0 20 40 60 80 100 with parameter ¾ 0, which is increasing concave and has a t constant coefﬁcient of ambiguity = −ê004x5/ê04x5. In this setting, a higher represents a higher ambiguity aversion. SOBME’s cost performance converges quite quickly to that For notational convenience, we use U 4q5 = 6 4q − D57 ƐfD L of full information. This characteristic is particulary desirable to denote the random variable that represents the expected in dynamic and nonstationary environments (where the disutility of ordering q units, with its realization depending underlying true distribution is subject to inevitable changes) on ∈ ì. The KMM objective (17) is then equivalent to as it allows the DM to automatically and quickly adjust its U 4qt 5 min SOBME 6e 7 = min 451 (18) decisions. Ɛft 45 MU 4qt 5 qt ¾0 qt ¾0

where 45 is the moment generating function of 7. Repeated Newsvendor Under Ambiguity MU 4qt 5 Aversion U 4qt5 evaluated at . For technical reasons, we assume that (i) the ambiguity set D is such that this moment generating We assumed that the DM was ambiguity neutral in the function exists and is differentiable with respect to q, and preceding sections. In general decision-making settings, a (ii) all the demand distributions in D are differentiable and DM may, in fact, be ambiguity averse, ambiguity neutral, or 0 have densities in D . Finally, we note that minimizing the ambiguity loving; see, e.g., Klibanoff et al. (2005, 2009). moment generating function in (18) is equivalent to minimiz- In what follows, we extend our results and demonstrate ing the so-called entropic disutility 41/5 log 45, and MU 4qt 5 how the SOBME approach can be also used when the DM hence we use that form in proofs when more convenient. is not ambiguity-neutral. We adopt the general framework proposed by Klibanoff et al. (2005), hereafter KMM, for Lemma 1 (Convexity). The moment generating function smooth decision making under ambiguity. Tailoring the MU 4q545 is convex in q (∀ ¾ 0), and hence, (18) is a KMM decision-making framework (see equation (1) of convex program. KMM) to a newsvendor problem, we cast the DM’s quantity KMM We let qt 45 denote the optimal solution to (18) on a decision at the beginning of period t as KMM given sample path. That is, qt 45 is the period-t optimal ordering quantity of a newsvendor (with a coefﬁcient of max SOBME 6ê4 6− 4q − D57571 (17) Ɛft 45 ƐfD L t ambiguity ) given a particular realization of demands up qt ¾0 KMM to and including period t − 1. We let Qt 45 denote the KMM where ê4 · 5 represents the DM’s ambiguity attitude, and random variable associated with qt 45 when looking SOBME t − ft 45 is the second-order belief built using SOBME. over all possible sample paths through period 1. When ê4 · 5 is afﬁne, the DM is ambiguity-neutral, which Lemma 2 (Optimal Ambiguity-Averse Ordering Quan- is the case considered in the previous sections. However, tity). For > 0, let Z 4q5 = U 4q5 + 41/5 log FD 4q5. The when ê4 · 5 is concave (convex), the DM is ambiguity-averse optimal ordering quantity qKMM45 is either zero or the 10 t Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. (loving). solution to the implicit equation In the case of an ambiguity-neutral DM, as discussed in

§4.1, there is a type of period-t equivalence between second- MZ4q545 p order and first-order approaches in the sense that the expected = 0 (19) MU 4q545 p + h value of the DM’s objective, see (1), depends on the belief ft45 and the ambiguity-set densities fD 4x5 only through the The above lemma proves that the optimal ordering quantity, SOBME R demand density fD1 t 4x5 2= ∈ì ft45fD 4x5 d. There- if positive, sets the ratio of the relevant moment generating fore, a first-order approach that somehow arrived at an functions equal to the well-known newsvendor quantile Saghafian and Tomlin: The Newsvendor under Demand Ambiguity 182 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

p/4p + h5. This can be viewed as a generalization of the optimality gap goes to zero as t increases. Moreover, as classical newsvendor quantile result. Using this result, we decreases, the performance under ambiguity aversion tends now establish how ambiguity aversion influences the optimal to that observed in our previous ambiguity-neutral results. ordering quantity (in any given period t), and how the Although Lemma2 characterizes the period- t optimal optimal ordering quantity (at any given ambiguity coefficient) order quantity in general, a much sharper characterization can behaves as t tends to infinity. be established in certain cases. In particular, under (i) low but positive ambiguity aversion, or (ii) a normality assumption Theorem 7 (Ambiguity Aversion Effect). The optimal on the disutility distribution, we can derive closed-form ordering quantity under ambiguity-aversion satisfies the solutions for the ambiguity-averse newsvendor problem following: KMM by applying the certainty equivalence principle. Using (i) qt 45 is nonincreasing in . KMM SOBME this principle drastically simplifies the problem, because (ii) lim→0+ qt 45 = qt (defined by (7)). KMM we only need to consider the first and second moments (iii) lim→ qt 45 = 0. of the disability U 4q5 = Ɛf 6L4q − D57 as opposed to QKMM45 = q∗ almost surely when the con- D (iv) limt→ t its moment generation function. In what follows, we let ditions of Theorem2 hold. SOBME R FD1 t 45 2= Ɛ1 t6FD 457 = ∈ì ft45FD 45 d denote the A higher level of ambiguity aversion, i.e., a higher , demand distribution built by SOBME (used for decision- SOBME4−15 results in a lower ordering quantity. As approaches zero, making in period t) and we let FD1 t 4 · 5 denote its the DM acts similarly to an ambiguity-neutral DM, that inverse. First, we consider the case in which the DM has a SOBME low but positive level of ambiguity aversion. is, his order quantity approaches qt defined in (7). As approaches , the DM only considers the worst-case Theorem 8 (Low Ambiguity Aversion). Let 4qt5 and outcome, i.e., becomes a minimax optimizer, and hence 4qt5 denote the mean and standard deviation of U 4qt5 orders nothing so as to avoid any overage cost. Regardless (with respect to ), respectively. Then of the DM’s ambiguity aversion level, as more and more (i) The optimal ordering quantity under ambiguity- demand observations are made (t → ), his ordering quantity aversion is converges to that of a DM who completely knows the true demand distribution. This highlights the asymptotical KMM 2 2 2 qt 45=argmin 4qt5+ 4qt5+ 4qt5+O4 5 0 suitability of SOBME under any ambiguity aversion level. qt 2 2 We numerically explore the performance of SOBME (20) under the KMM criterion by considering the “Case I-A-No Bounds” setting introduced in §6.1. Figure5 presents the (ii) The solution to (20) is average cost optimality gap (%) for three levels of ambiguity aversion: = 00005, = 00010, = 00050. The average cost KMM SOBME4−15 p KMM 2 q 45 = FD1 t − 4q 5 − O4 5 1 optimality gap (%) is the percentage extra cost of the ordered t p + h t quantities of a DM who is facing demand ambiguity and uses (21) the objective (17) compared to a DM who completely knows the true demand distribution. For each period shown in where 4qt5 2= Ɛ6FD 4qt5U 4qt57 − Ɛ64p/4p + h55U 4qt57. Figure5, the optimality gap is averaged over 20 independent We see that when the DM has a low but positive level of sample path replications, i.e., demand observations over ambiguity aversion, the optimal ordering quantity is a per- the horizon. Consistent with Theorem7(iv), the average turbed version of the traditional critical quantile newsvendor solution. This perturbation, is approximately (with an error Figure 5. Performance of SOBME under the KMM of order 2 denoted by O425) linear in the coefficient of KMM criteria (Case I-A-No Bounds). ambiguity, , with a slope of −4qt 5. Next, we consider a case where U 4q5 has an approximately Normal distribution based on the DM’s second-order = 0.050 20 = 0.010 belief. See Appendix D for an example illustrating the = 0.005 validity of this approximation.

15 Remark 2 (Normal Approximation). Suppose at some Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. period t, U 4qt5 is approximately Normal with mean 4qt5 10 and standard deviation 4qt5. Then (i) The newsvendor problem under ambiguity aversion is

Avg. cost opt. gap (%) equivalent to the mean-variance optimization min 4q 5 + 5 qt t 2 4/25 4qt5. That is,

(ii) The solution to (22) is moment bounds is typically more valuable than information in the form of tighter ambiguity sets. KMM qt 45 Finally, we examined an ambiguity-averse version of our repeated newsvendor problem, and showed how SOBME SOBME4−15 p 'F −Cov4F 4qKMM51U 4qKMM55 0 (23) can be applied in that setting. We established theoretical D1t p+h D t t results that characterize the optimal newsvendor quantity As in the case of low ambiguity aversion, the optimal under ambiguity aversion, and demonstrated that SOBME is ordering quantity is again a perturbed version of the tradi- well-suited to an ambiguity-averse setting. tional critical quantile solution, with the linear perturbation In closing, we emphasize that the SOBME belief represen- KMM KMM tation and updating approach can be applied to any setting in now having a slope of −Cov4FD 4qt 51 U 4qt 55. which a decision maker has access to observations/data and 8. Conclusion some partial distributional information. We hope to apply and test SOBME in various other settings in future work. There are many multiperiod settings in which a decision maker does not have full distributional information about Supplemental Material a random variable of interest but may have some partial Supplemental material to this paper is available at http://dx.doi distributional information in the form of bounds on moments .org/10.1287/opre.2015.1454. or tails. Over time, the decision maker gains access to demand observations. Techniques which rely exclusively on partial distributional information ignore the value of observations. In Acknowledgments contrast, pure data-driven techniques avail of observations but The authors are grateful to Retsef Levi for his invaluable suggestions, ignore partial distributional information. With the growing comments, and discussions which helped to improve this paper. attention to business analytics, it is essential to bridge this The authors also thank the department editor, Chung Piaw Teo, the gap by developing approaches that allow decision makers to associate editor, and two anonymous referees for their constructive suggestions that improved the content and exposition of this paper. benefit from both data and partial distributional information. In this paper, using the maximum entropy principle, we developed an approach (SOBME) that allows a decision Endnotes maker to effectively use partial distributional information 1. For clarity, we only consider the first moment, but the and data/observations. Moreover, SOBME can accommodate approach and results are readily extended to higher moments. dynamically evolving partial information. We established that 2. We assume that the DM is ambiguity-neutral until §7 SOBME is a natural generalization of traditional Bayesian which considers other ambiguity attitudes. updating; an exponential modifier is introduced to incorporate 3. At the beginning of period t = 1, the DM chooses an

the partial distributional information. This is an important initial density f145 to reﬂect his starting belief about result, with applications far beyond the newsvendor focus of the probability of each demand distribution and his initial this paper. We proved that SOBME is weakly consistent and moment/tail bound information. For instance, a uniform characterized its rate of convergence. Applying SOBME in a density over all ∈ ì (representing a maximum initial repeated newsvendor setting, we introduced and analyzed the entropy) would be appropriate, if (a) the DM did not believe notion of newsvendor’s cost of ambiguity. We provided an any particular distribution was more likely to be the true analytical bound for this cost when the DM uses SOBME. one, and (b) ì is such that a uniform distribution over all More broadly, our results indicate that information-theoretical ∈ ì can be deﬁned. approaches, such as the one we developed in this paper, 4. For the more general class of -divergence, we refer provide a natural way for characterizing an upper bound for interested readers to Pardo (2006), Ben-Tal et al. (2013),

the newsvendor’s cost of ambiguity. and the references therein. We also note that dKL4··5 is not Our numerical investigation demonstrated that SOBME a metric in the usual sense as it is not symmetric. performs well, not just outperforming a pure data-driven 5. The Dirac delta function, sometimes called the unit approach but often coming close to the performance of full impulse function, is a generalized function (often) deﬁned distributional information. We also found that unlike a pure such that 4x5 = if x = 0 and 4x5 = 0 otherwise, R data-driven approach such as SAA, SOBME’s performance with − 4x5 dx = 1 (Kreyszig 1988). It can be deﬁned

Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. does not significantly degrade as the newsvendor quantile more rigorously either as a measure or a distribution. increases, making it a suitable candidate for settings with It can be viewed as the√ limit of the Gaussian function: −x2/22 high service level requirements. By comparing our approach 4x5 = lim→04e /4 255. Importantly, the Dirac to pure data-driven approaches, we also generated insights delta function has the property that for any function g4 · 5, R + into the value of partial distribution information. Our results − g4x54x − x05 dt = g4x05. indicate that it can be very beneficial to incorporate such 6. Giffin and Caticha (2007) also show a related result, information when deciding about the stocking quantities. but unlike our framework, they only derive it under the Also, we found that information in the form of tighter very strong assumptions that the first moment is perfectly Saghafian and Tomlin: The Newsvendor under Demand Ambiguity 184 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

known and that this is the only information available, i.e., Delage E, Ye Y (2010) Distributionally robust optimization under moment no tail bounds and no upper and/or lower moment bounds. uncertainty with application to data-driven problems. Oper. Res. 58(3):595–612. Furthermore, their work does not examine the asymptotic Diaconis P, Freedman D (1986) On the consistency of Bayes estimates. behavior (consistency, rate of convergence, etc.) of the Ann. Stat. 14(1):1–26. updating mechanism nor does it embed the distribution- Fisher M, Raman A (2010) The New Science of Retailing: How Analytics are Transforming the Supply Chain and Improving Performance formation problem in a decision-making context. (Harvard Bus. Press, Boston). 7. Our results for SAA, LRO, and Scarf’s method echo Gallego G, Moon I (1993) The distribution free newsboy problem: Review those presented in Wang et al. (2015) which explores the and extensions. J. Oper. Res. Soc. 44(8):825–834. Giffin A, Caticha A (2007) Updating probabilities with data and moments. performance of LRO and Scarf’s methods as compared Knuth KH, Caticha A, Center JL, Giffin A, Rodriguez CC, eds. Proc. to SAA for a single-shot newsvendor setting with ample 27th Internat. Workshop on Bayesian Inference and Maximum Entropy data. We also observed that SAA has certain computational Methods in Sci. Engrg. (AIP, Melville, NY), 74–84. Godfrey GA, Powell WB (2001) An adaptive, distribution-free algorithm advantages over LRO, which is an important benefit in a for the newsvendor problem with censored demands, with applications dynamic multiperiod setting such as ours. Scarf’s approach to inventory and distribution. Management Sci. 47(8):1101–1112. is closer to SAA in terms of computational complexity, but Huh WT, Rusmevichientong P (2009) A nonparametric asymptotic analy- its performance is dominated by SAA. sis of inventory planning with censored demand. Math. Oper. Res. 34(1):103–123. 8. In Case I, we consider ì = 6101 307, where ∈ 6101 205 Huh WT, Levi R, Rusmevichientong P, Orlin JP (2011) Adaptive data- indicates an exponential distribution with mean , and driven inventory control with censored demand based on kaplan-meier ∈ 6201 307 indicates a normal distribution with a mean estimator. Oper. Res. 59(4):929–941. Jaynes ET (1957) Information theory and statistical mechanics. Physical − 10 and a C.V. of 0.2. The construction of ì in Case II Rev. 106(4):620. is done in an analogous manner. Jaynes ET (1981) On the rationale of maximum-entropy methods. IEEE 9. For our numerical experiments, we use a gradient- Proc. 70(9):939–952. Kahn KB (2002) An exploratory investigation of new product forecasting based numerical method in Mathematica. Other numerical practices. J. Prod. Innov. Management 19(2):133–143. approaches can also be used. Kleywegt AJ, Shapiro A, Homem-De-Mello T (2002) The sample average 10. The internal expectation of the newsvendor cost 4 · 5 approximation method for stochastic discrete optimization. SIAM J. L Opt. 12(2):479–502. in (17) implies a risk-neutral DM. This could be relaxed but Klibanoff P, Marinacci M, Mukerji S (2005) A smooth model of decision we focus on incorporating ambiguity attitudes and not risk making under ambiguity. Econometrica 73(6):1849–1892. attitudes. Klibanoff P, Marinacci M, Mukerji S (2009) Recursive smooth ambiguity preferences. J. Econom. Theo. 144(3):930–976. Kreyszig E (1988) Advanced Engineering Mathematics (Wiley, New York). Lariviere MA, Porteus EL (1999) Stalking information: Bayesian inventory References management with unobserved lost sales. Management Sci. 45(3): 346–363. Aaker DA, Kumar V, Dayand GS, Leone RP (2010) Marketing Research Levi R, Perakis G, Uichanco J (2015) The data-driven newsvendor problem: (John Wiley & Sons, Hoboken, NJ). New bounds and insights. Oper. Res. 63(6):1294–1306. Allon G, Van Mieghem JA (2011) The mexico china dual sourcing strategy Levi R, Roundy RO, Shmoys DB (2007) Provably near-optimal sampling- simulation. Kellogg Case Number 1-212-500. based policies for stochastic inventory control models. Math. Oper. AMR (2008) Innovation in consumer products: New products from concept Res. 32(4):821–839. to launch, AMR Research 2008 Study. Liyanage LH, Shanthikumar JG (2005) A practical inventory control policy Amrute S (1998) Forecasting new products with limited history: Nabisco’s using operational statistics. Oper. Res. Let. 33(4):341–348. experience. J. Bus. Forcast. Meth. Syst. 17(3):7–11. Luenberger DG (1994) Optimization by Vector Space Methods, 1969 (John Andersson J, Jörnsten K, Nonås SL, Sandal L, Ubøe J (2013) A maximum Wiley & Sons, New York). entropy approach to the newsvendor problem with partial information. Lynn GS, Schnaars SP, Skov RB (1999) A survey of new product forecasting Eur. J. Oper. Res. 228(1):190–200. practices in industrial high technology and low technology businesses. Azoury KS (1985) Bayes solution to dynamic inventory models under Indust. Market. Management 28(6):565–571. unknown demand distribution. Management Sci. 31(9):1150–1160. Maglaras C, Eren S (2015) A maximum entropy joint demand estimation and Barron A, Schervish MJ, Wasserman L (1999) The consistency of posterior capacity control policy. Production Oper. Management 24(3):438–450. distributions in nonparametric problems. Ann. Stat. 27(2):536–561. Pardo L (2006) Statistical Inference Based on Divergence Measures Ben-Tal A, den Hertog D, De Waegenaere A, Melenberg B, Rennen G (Chapman and Hall/CRC, Boca Raton, FL). Perakis G, Roels G (2008) Regret in the newsvendor model with partial (2013) Robust solutions of optimization problems affected by uncertain information. Oper. Res. 56(1):188–203. Management Sci. probabilities. 59(2):341–357. Popescu I (2007) Robust mean-covariance solutions for stochastic optimiza- Bookbinder JH, Lordahl AE (1989) Estimation of inventory reorder level tion. Oper. Res. 55(1):98–112. using the bootstrap statistical procedure. IIE Trans. 21(4):302–312. Ramamurthy V, Shanthikumar G, Shen Z-J (2012) Inventory policy with Boyd SP, Vandenberghe L (2004) Convex Optimization (Cambridge Univer- parametric demand: Operational statistics, linear correction, and sity Press, Cambridge, Cambridge, UK). regression. Production Oper. Management 21(2):291–308. Caticha A, Giffin A (2006) Updating probabilities. Mohammad-Djafari A, Saghafian S, Chao X (2014) The impact of operational decisions on the ed. Proc. 26th Internat. Workshop on Bayesian Inference and Maximum optimal design of salesforce incentives. Naval Res. Log. 61(4):320–340. Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved. Entropy Methods in Sci. Engrg. (AIP, Melville, NY), 31–42. Scarf H (1958) A min-max solution of an inventory problem. Studies in the Chen M, Xia Y, Wang X (2010) Managing supply uncertainties through Mathematical Theory of Inventory and Production (Stanford University bayesian information update. IEEE Trans. Automation Sci. Eng. Press, Redwood City, CA) 201–209. 7(1):24–36. Scarf H (1959) Bayes solution to the statistical inventory problem. Ann. Chu LY, Shanthikumar JG, Shen Z-J (2008) Solving operational statistics Math. Stat. 30(2):490–508. via a bayesian analysis. Oper. Res. Let. 36(1):110–116. Schwartz L (1965) On Bayes procedures. Z. Wahrsch. Verw. Gebiete Clarke B, Barron A (1990) Information theoritic asymptotics of Bayes 4(1):10–26. methods. IEEE Trans. Info. Theo. 36(3):453–471. Shapiro A (2003) Monte Carlo sampling methods. Ruszczynski A, Shapiro A, Cover TM, Thomas JA (2006) Elements of Information Theory (Wiley- eds. Stochastic Programming, Handbooks in Operations Research and Interscience, Hoboken, NJ). Management Science, Vol. 10 (Elsevier, Amsterdam, the Netherlands). Saghafian and Tomlin: The Newsvendor under Demand Ambiguity Operations Research 64(1), pp. 167–185, © 2016 INFORMS 185

Shen X, Wasserman L (2001) Rates of convergence of posterior distributions. Soroush Saghaﬁan is an assistant professor at Harvard Kennedy Ann. Stat. 29(3):687–714. School, Harvard University. His research focuses on the application Tomlin B (2009) Impact of supply learning when suppliers are unreliable. and development of operations research methods in studying Manufacturing Service Oper. Management 11(2):192–209. Wang Z, Mersereau AJ (2013) Bayesian inventory management with stochastic systems with speciﬁc applications in healthcare and potential change-points in demand. Working paper, University of North operations management. Carolina, Chapel Hill, NC. Brian Tomlin is a professor at the Tuck School of Business at Wang Z, Glynn PW, Ye Y (2015) Likelihood robust optimization for Dartmouth. His research lies in the general area of operations and data-driven problems. Comput. Man. Sci. Forthcoming. Zhu Z, Zhang J, Ye Y (2013) Newsvendor optimization with limited supply chain management, with risk being a key focus of much of distribution information. Opt. Meth. Soft. 28(3):640–667. his work. Downloaded from informs.org by [128.103.193.212] on 12 February 2016, at 06:17 . For personal use only, all rights reserved.