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The Newsvendor under Demand Ambiguity: CombiningData with Moment and Tail InformationSoroush Saghafian, Brian Tomlin

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

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OPERATIONS RESEARCHVol. 64, No. 1, January–February 2016, pp. 167–185ISSN 0030-364X (print) � ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.2015.1454

© 2016 INFORMS

The Newsvendor under Demand Ambiguity: CombiningData with Moment and Tail Information

Soroush SaghafianHarvard Kennedy School, Harvard University, Cambridge, Massachusetts, [email protected]

Brian TomlinTuck 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-drivenapproaches 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 BeliefMaximum Entropy (SOBME), which allows the manager to effectively combine demand observations with distributionalinformation in the form of bounds on the moments or tails. In the proposed approach, the decision maker forms a belief aboutpossible demand distributions, and dynamically updates it over time using the available data and the partial distributionalinformation. We derive a closed-form solution for the updating mechanism, and highlight that it generalizes the traditionalBayesian mechanism with an exponential modifier that accommodates partial distributional information. We prove theproposed approach is (weakly) consistent under some technical regularity conditions and we analytically characterize its rateof convergence. We provide an analytical upper bound for the newsvendor’s cost of ambiguity, i.e., the extra per-period costincurred because of ambiguity, under SOBME, and show that it approaches zero quite quickly. Numerical experimentsdemonstrate that SOBME performs very well. We find that it can be very beneficial to incorporate partial distributionalinformation when deciding stocking quantities, and that information in the form of tighter moment bounds is typically morevaluable than information in the form of tighter ambiguity sets. Moreover, unlike pure data-driven approaches, SOBME isfairly robust to the newsvendor quantile. Our results also show that SOBME quickly detects and responds to hidden changesin the unknown true distribution. We also extend our analysis to consider ambiguity aversion, and develop theoretical andnumerical 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. IntroductionRandomness in demand and/or supply is a major challengein sales and operations planning. A common paradigm inthe operations literature endows the decision maker withfull distributional information about the underlying randomvariable(s). Full distributional information is, however, astrong assumption and is not typically met in practice. Analternative “pure data-driven” paradigm endows the decisionmaker only with past observations/realizations of the randomvariable (the “data”); see, e.g., Azoury (1985), Liyanage andShanthikumar (2005), Levi et al. (2007, 2015), Huh et al.(2011). Reality often lies somewhere in between these twoparadigms: the decision maker typically has some partialdistributional information and some data.

In this paper, we propose and analyze a methodology fordecision making in settings in which the decision maker doesnot know the true distribution but has access to some partialdistributional information and some limited data. This formof ambiguity, a.k.a Knightian uncertainty, about the truedistribution arises in various settings in operations including

new product launches and new process introductions. Insuch settings, firms devote significant efforts to characteriz-ing demand or yield before the new product or process islaunched and then observe a growing time series of realiza-tions, i.e., data, after launch. In new product introductions,forecasting demand and choosing stocking quantities is acommon challenge across a wide array of industries (Kahn2002, AMR 2008), especially “during the commercializationstage (prelaunch preparation and launch) where new productforecasts drive a variety of multifunctional decisions,” (Kahn2002, p. 133). For example, consider the consumer-productscompany Nabisco (now owned by Mondelez International):

Several statistical techniques are used to forecast sales 0 0 0 0However, most of these techniques need a minimum of twoto three years of historical information to build a robustmodel. This presents a big problem when developing fore-casts for newly introduced product 0 0 0 0 At Nabisco, beforea new product is rolled out, [the] planning forecast devel-oped by marketing research 0 0 0 is used for 0 0 0productionplanning 0 0 0 and inventory deployment. However, when the

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mailto:[email protected]

Saghafian and Tomlin: The Newsvendor under Demand Ambiguity168 Operations Research 64(1), pp. 167–185, © 2016 INFORMS

product is rolled out, actual sales can be substantially less ormore than the planned numbers [leading to] lost sales [or]excess inventory. (Amrute 1998, p. 7).

Let us consider the kind of demand-forecast informa-tion that firms generate before and after a new product islaunched. According to surveys by Lynn et al. (1999) andothers, prelaunch forecasts are primarily generated throughmarketing research and judgmental techniques, with the topfour approaches being customer/market research, followedby jury of executive opinion, sales force composite, andlook-like analysis (Kahn 2002). Customer/market researchencompasses a variety of approaches, including surveys ofpotential buyers. The jury of executive opinion approachsolicits and combines judgments from managers represent-ing various functions, whereas the sales force compositeapproach solicits and combines judgments from the salesforce (Aaker et al. 2010) who are typically better aware ofmarket conditions; see, e.g., Saghafian and Chao (2014), andthe references therein. In look-like analysis, the firm identifies“a similar product, sold previously, and extrapolates fromthat product’s prior sales” (Fisher and Raman 2010, p. 85).These judgmental approaches, such as sales force composite,can be used to estimate demand uncertainty by having thesalespeople “provide a sales estimate range in addition to thesingle number representing the expected sales level [or] tohave them estimate the probability of each potential sale.”(Aaker et al. 2010, p. 747). Because of the subjective natureof theses estimation processes, different people will providedifferent forecasts, i.e., different probability distributions. Assuch, these prelaunch techniques enable the firm to generatea set of possible demand distributions and a range for theexpected sales.

After a product has been launched, the firm can use actualdemand observations to update its beliefs about demand.However, market research efforts do not end, at least in partbecause “the rate of new product introduction is fast andfrequent, meaning data series are often short” (Lynn et al.1999, p. 566). Ongoing market research generates forecastinformation through test sales (Fisher and Raman 2010),simulated test markets (Lynn et al. 1999), and surveys of cus-tomer purchase intentions (Aaker et al. 2010). These ongoingpostlaunch market-research efforts provide valuable demandinformation separate from the actual demand observations.

Therefore, in planning stocking quantities for a new orrecently released product, the operations function has twoseparate and evolving sets of information about the unknowndemand distribution. One set is the pre and postlaunch marketresearch including the judgmental approaches discussedabove. The other set is the actual demand observationsover time. This information setting is not captured bythe traditional “full-information” paradigm in operationsmanagement that endows the decision maker with completeknowledge of the demand distribution. Neither it is capturedby the more recent “pure data-driven” approaches that werespecifically designed for settings in which the decision makerhas access to demand observations but no other information.

We develop an approach tailored to intermediate infor-mation settings—such as new product and new processintroductions—in which a decision maker operates in anenvironment of ambiguity characterized by two evolvingsources of information about a relevant random variable(e.g., demand): “exogenous research” and observations (the“data”). Based on the above discussion of pre and postlaunchforecasts, we formulate the exogenous research informationas a set of possible demand distributions as well as potentialupper and/or lower bounds on the moments and/or tailsof the demand distribution. Our proposed approach is asecond-order, maximum-entropy based approach, which weterm Second Order Belief Maximum Entropy (SOBME).It allows the decision maker to effectively combine theabove-mentioned sources of information dynamically overtime to update its beliefs about the possible distributionsgoverning the demand realizations. In our approach, thedecision maker updates its second-order beliefs, i.e., dis-tribution over possible distributions, so as to minimize therelative entropy to its prior belief subject to the most recentlyobserved demand and the current market research informa-tion. In doing so, we follow the core concept of maximumentropy, widely used in estimation and information theory, inwhich (i) beliefs are updated so that the posterior coincideswith the prior as closely as possible, and (ii) only thoseaspects of beliefs for which new evidence was gained areupdated (Jaynes 1957, 1981; Cover and Thomas 2006).

We embed our SOBME approach in a classical operationssetting—the repeated newsvendor—and analytically andnumerically examine its performance. We focus on a repeatednewsvendor setting because that is a common paradigmin the operations literature for exploring approaches thatrelax the assumption of full demand information. However,SOBME is general and could be used in any setting inwhich a decision maker does not know the distribution of anunderlying random variable but has access to data and tosome partial distributional information. Such settings willbecome increasingly common in the era of big data analytics.Our paper provides both theoretical and managerially relevantresults.

On the theoretical side, we derive a closed-form solutionfor the SOBME belief-updating mechanism, and show thatit contains traditional Bayesian updating—both parametricand nonparametric—as a special case. As we will show,SOBME generalizes the Bayesian updating mechanism byincorporating the partial distributional information throughan exponential modifier. We explore the asymptotic behaviorof SOBME and show that it is weakly consistent. That is,under some technical conditions, the updated distributionconverges to the true unknown distribution almost surely inthe weak neighborhoods of it. We also analytically establisha rate of convergence for SOBME. Furthermore, we providea performance guarantee for SOBME by developing anupper bound on the newsvendor’s cost of ambiguity, i.e., theincrease in the expected per-period newsvendor’s cost as

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compared to the full distributional knowledge case. We showthat the cost of ambiguity approaches zero quite quicklywhen SOBME is used. We also explore an ambiguity-averseversion of our repeated newsvendor problem, and developtheoretical results that characterize the optimal newsvendorquantity under ambiguity aversion. In doing so, we establisha generalization of the traditional newsvendor quantile result.We also analytically and numerically examine how SOBMEperforms in this ambiguity-averse setting.

On the managerial side, we numerically establish thatSOBME performs well—as measured by the percentageincrease in cost as compared to full distributional knowledge—even when the number of demand observations is limited.This is of practical importance because, as noted above,firms often do not have access to a long series of demandobservations when planning stocking quantities for recentlyintroduced products. Given the efforts devoted by companiesto generate exogenous research about the demand distribu-tion, it is important to understand whether such research isvaluable, when it is valuable, and what type of research ismore valuable. We numerically explore the value of exoge-nous research by comparing the performance of SOBMEwith that of pure data-driven approaches, such as SampleAverage Approximation (SAA), which rely exclusively onthe demand realizations. We show that there is significantvalue to exogenous research, especially when demand obser-vations are limited or the newsvendor quantile (the desiredservice level) is high. Unlike SAA, the SOBME approachperforms well even at high service levels. We find thatresearch resulting in tighter moment bounds is more valuablethan research resulting in a more compact set of possibledistributions. As discussed above, “like-products” are onesource of exogenous research. Based on a new-productintroduction case motivated by a real-world setting (Allonand Van Mieghem 2011), we show how moment bounds(for demand) can be developed from like-products, andnumerically show that there is significant benefit to momentbound information even if the number of like-products isquite small. We also numerically show that SOBME iseffective at detecting and quickly reacting to changes in theunderlying unknown distribution, an important and practicalbenefit in dynamic environments. We prove that ambiguityaversion leads to lower order quantities. This has importantimplications for practice if individual managers vary in theirambiguity aversion.

The rest of the paper is organized as follows. The lit-erature is reviewed in §2. The model is presented in §3,and the proposed SOBME approach is developed in §4.A performance guarantee for the repeated newsvendor settingis developed in §5. Numerical results are presented in §6.Ambiguity aversion is considered in §7. Section 8 concludesthe paper.

2. Literature ReviewDespite some early and notable exceptions, e.g., Scarf (1958,1959), the operations literature has typically assumed that

the decision maker has full information about the distributionof relevant random variables. A number of approaches havebeen adopted to relax this strong assumption.

Some papers adopt a parametric approach, in whichthe distribution comes from a parametric family but theparameter value is unknown. Bayesian approaches havebeen used in the context of demand uncertainty (Scarf1959, Azoury 1985, Lariviere and Porteus 1999, Wangand Mersereau 2013) and supply uncertainty (Tomlin 2009,Chen et al. 2010). An “operational statistics” approachfor simultaneously estimating an unknown parameter andoptimizing an operational planning problem is introduced inLiyanage and Shanthikumar (2005) and further explored inChu et al. (2008) and Ramamurthy et al. (2012).

The distribution of interest may not come from a para-metric family or the decision maker may not know whichfamily it comes from. With this in mind, a number ofdifferent nonparametric approaches have been proposed andanalyzed in various inventory planning contexts. SampleAverage Approximation (SAA) (see, e.g., Kleywegt et al.2002; Shapiro 2003; Levi et al. 2007, 2015) is a data-drivenapproach that solves a sample-based counterpart to the actualproblem. For the SAA approach, Levi et al. (2007 and 2015)develop bounds on the number of samples required forthe expected cost to come arbitrarily close (with a highconfidence) to the expected cost if the true distributionwas known. Other data-driven, nonparametric approachesinclude the bootstrap method (Bookbinder and Lordahl1989), the concave adaptive value estimation (CAVE) algo-rithm (Godfrey and Powell 2001), and adaptive inventorymethods mainly used for censored demand data (Huh andRusmevichientong 2009, Huh et al. 2011).

An alternative nonparametric approach is a robust onein which certain moments of the distribution are known,and the decision maker maximizes (minimizes) the worst-case expected profit (cost) over all distributions with theknown attribute; see, e.g., Scarf (1959), Gallego and Moon(1993), Popescu (2007). Because such approaches can beconservative, Perakis and Roels (2008) adopt a regret-basedapproach in which the decision maker knows the mean.Zhu et al. (2013) consider a newsvendor setting where thedemand distribution is only specified by its mean and eitherits standard deviation or its support, and develop a methodthat minimizes the ratio of the expected cost to that offull information. Moments, however, might not be knownwith certainty, and Delage and Ye (2010) develop a robustapproach in which the mean and covariance satisfy someknown bounds. Wang et al. (2015) devise a likelihood robustoptimization (LRO) approach, in which the set of possibledistributions are considered to be only those that make theobserved data achieve a certain level of likelihood. It is thenassumed that the decisions are made based on the worst-casedistribution within this set.

A long-established approach when selecting a probabilitydistribution subject to known moments is to choose thedistribution that has the maximum entropy (ME). Despite its

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widespread use in fields such as estimation theory, physics,statistical mechanics, and information theory among others,there are only a few papers in the operations literaturethat have adopted maximum entropy based approaches.Andersson et al. (2013) examine a nonrepeating newsvendormodel in which the decision maker only knows the meanand variance of the demand distribution. They comparean ME approach with robust approaches and find thatthe ME approach performs better on average. Differentfrom Andersson et al. (2013), (i) we examine a repeatednewsvendor setting and so the decision maker has access topast data to dynamically update his beliefs; and (ii) we allowfor (evolving) upper/lower bounds on moments and tailsinstead of assuming a known mean and variance. Maglarasand Eren (2015) apply the ME approach in a revenuemanagement context under a discrete demand assumptionand static linear equalities, and establish its asymptoticalbehavior. Unlike their work, we (i) consider a newsvendorsetting, (ii) permit continuously distributed demand, (iii)allow for dynamically changing inequalities (lower and upperbounds on the moments and tails that may change overtime), and (iv) build distributions over possible distributionsas opposed to the random variable itself, i.e., a second-orderapproach as opposed to a first-order one.

The concept of maximizing entropy when updating proba-bility distributions in response to new information, under thestrong assumption that the first moment is exactly known,has been shown to include Bayesian updating as a specialcase; see Caticha and Giffin (2006) and Giffin and Caticha(2007). We adopt and develop this approach by (i) creat-ing a mechanism that builds a belief/distribution about thepossible distributions of the underlying random parameter(as opposed to directly building a distribution about it); (ii)allowing for partial distributional information in the form ofarbitrary upper and/lower bounds on the moments and tails,which may evolve over time (as opposed to the very specialcase where the first moment is perfectly known and thisinformation is static). Importantly, we also embed it in asequential decision-making setting (a repetitive newsvendor),provide a performance guarantee for it, and analyticallyestablish its asymptotic behavior including its consistencyand rate of convergence. To the best of our knowledge, ourproposed method is among the very first in the operationsliterature that is both data-driven and information-drivenand also fully dynamic. The approach provides a method toeffectively combine dynamically evolving partial informationon moments and tails with data and observations, presentinga widely applicable tool (especially in the current trend ofbusiness analytics) for decision making under distributionalambiguity.

3. The ModelWe consider a repeated newsvendor problem over periods t ∈

´ 2= 811 0 0 0 1 T 9, with T ¶�. Let h > 0 and p > 0 denote theper-unit holding and shortage penalty costs, respectively. Let

L4x5= h6x7+ +p6−x7+, where 6x7+ 2= max8x109. Demandrealizations are generated as i.i.d. random variables froma true distribution that is unknown to the decision maker(DM). Unless otherwise stated (e.g., §6.3), we assume thetrue distribution is stationary. The DM, “he,” as a conventionhereafter, has two types of evolving information about thedemand distribution.

• Data/Observations. At the end of period t = 11 0 0 0 1 T ,the DM knows the realization of demand Di in each of theperiods i = 11 0 0 0 1 t. This information is denoted as Ft . Wedefine F0 = � to reflect the fact that there is no demandrealization prior to period 1.

• Exogenous Research. This information is contained intwo sets.

—Ambiguity Set. As noted in §1 for the example ofnew product introductions, prelaunch marketing researchgenerates a set of possible demand distributions basedon the judgments of various relevant constituents, e.g.,managers, sales people, prospective customers. Supposethe sets of possible demand distributions and associateddensities (assuming they exist) are given by the ambiguitysets D 2= 8F �

D 2 � ∈ì9 and D′ 2= 8f �D 2 � ∈ì9, respectively,

where � is a scalar indicator associated with a distribution,and ì is an arbitrary, typically uncountable, set. Without lossof generality, we assume all the distributions in D have acommon support denoted by ¤, e.g., ¤= 601�5. We assumethat the ambiguity set D contains the true distribution F �∗

D ,where �∗ ∈ì is simply an indicator for the true distribution.

The fact that the indicator � is a scalar is not a limitation.We make no assumption about the shape of the distributionscontained in the ambiguity set. The set may contain anydistribution, and the distributions do not need to come fromthe same family; see, for example, our numerical studyin §6. In addition, families with more than one parameterare allowed. It is simply a matter of associating a scalar �with an instance of the family.

—Moment and Tail Bounds. As noted in §1, pre andpostlaunch marketing research can also provide the DM withadditional demand information in the form of bounds on (i)the expected value of demand, and (ii) tail probabilities. Atperiod t ∈´, we denote the lower and upper bounds on theexpected value of demand by Ɛt4D5 and Ɛ̄t4D5, respectively,with 0 ¶ Ɛt4D5¶ Ɛ̄t4D5.1 We let the tail related informationat period t be of the form Pr4D ¾ u5¶ bt4u5, for somenonincreasing function bt4u52 ¤→ 60117. We denote thismoment and tail information by the set F̂t , and note thatallowing it to change over time is a generalization of the staticcase in which the moment bounds and the tail informationare fixed over the horizon. In this latter case, F̂t = F̂0 forall t ∈´, where F̂0 is the initial information before the firsttime period has occurred. We note that the moment and tailinformation need to be consistent, i.e., should not contradicteach other. Otherwise, there may not be any distributionin the ambiguity set that satisfies them, in which case theproblem we discuss below is infeasible. The consistencybetween the data/observations and the moment/tail bounds

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can be monitored and statistically tested. Our approachcan allow the decision maker to also include endogenouslygenerated moment bounds based on the data/observations;see §6.3.

Because the DM does not know the true demand distribu-tion, his decision in each period will be based on his currentbelief about the distribution. We will introduce and developour approach to belief representation and updating in thenext section. Before doing so, we specify the sequence ofevents in each period t ∈´:

1. The DM determines an order quantity qt ¾ 0 (givenhis current belief about the demand distribution), orders it,and receives it.

2. Demand is realized, with the realized value denotedby � ′

t , and the newsvendor cost L4qt − � ′t5 is incurred.

3. New exogenous research, if any, arrives regardingmoment and tail bounds, and the information set F̂t isspecified. Note that F̂t = F̂t−1 if no new research arrives.

The DM updates his belief about the demand distributionfor period t + 1.

When determining his order quantity at the beginning ofperiod t (step 1), the DM solves2

minqt¾0

ƐD�Ft−11 F̂t−16L4qt −D571 (1)

where ƐD �Ft−11 F̂t−1denotes that the expectation is taken

with respect to the DM’s current belief about the demanddistribution, and that this current belief (however it wasformed) can only avail of the information available at thatpoint in time, i.e., the prior demand realizations Ft−1 andthe most recent moment and tail information set F̂t−1. Werefer to the above optimization problem as the DM’s problemand to its adopted policy as the DM’s policy. The valuesof demand, however, are drawn from a hidden underlyingdistribution. Therefore, the DM’s policy is associated witha true cost that depends on the true hidden underlyingdistribution. This true cost represents the ultimate measureof the policy’s performance.

4. A Belief Updating Approach ThatCombines Data and Moment/TailBounds

In this section, we first sketch the outline of our approachto representing and updating beliefs, and contrast it withother existing approaches (§4.1). Next, in §4.2, we formallydevelop the approach and present the key belief-updatingresult. Finally, in §4.3, we analytically examine the asymp-totic behavior of the approach by investigating its consistencyand convergence rate.

4.1. Outline of the Approach

We adopt a distribution-over-distributions approach. At thebeginning of period t, the DM’s belief about the demanddistribution is represented by a distribution over the set of

possible demand distributions contained in the ambiguity set.In essence, the DM assigns a probability that any particulardistribution in the ambiguity set is the true distribution. Atthe end of period t, having observed the value of period-tdemand, the DM updates his belief about the probabilityof each distribution in the ambiguity set; that is, a newdistribution (over the distributions in the ambiguity set) isformed. The DM adopts the maximum entropy principlewhen creating this new distribution: beliefs are updated sothat (i) the posterior coincides with the prior as closely aspossible, and (ii) only those aspects of beliefs for whichnew evidence was gained are updated. Furthermore, theupdating is done in a manner that ensures conformanceto the current moment and tail bound information. Thesenotions are mathematically formalized in §4.2.

We call this a Second Order Belief Maximum Entropy(SOBME) approach because the DM’s beliefs are representedas a distribution over possible demand distributions ratherthan being codified directly as a demand distribution (a first-order approach). We note that if the DM is ambiguity neutral,i.e., the DM’s period-t objective function is given by (1),then there exists a single first-order demand distributionthat would yield an equivalent period-t objective valueto that given by a distribution-over-distributions approach.However, there is a question as to how such a first-orderdemand distribution would be developed. SOBME providesa useful and practically relevant method to dynamicallycharacterize the first-order distribution as a weighted averageof the distributions in the ambiguity set. From a practicalperspective, our SOBME approach aligns directly withnew-product forecasting judgmental methods (see §1) inwhich different people estimate the probability of each saleslevel. Such methods yield a set of distributions, one fromeach person. SOBME provides a method to dynamicallyupdate the probability associated with each distribution and,as we will see in subsequent sections, it performs verywell. Furthermore, as we will see in §7, a distribution-over-distributions approach is necessary if the DM is ambiguityaverse/loving, and SOBME provides a good approach forsuch settings.

Before we formalize the SOBME approach, we contrast itconceptually with pure data-driven approaches and robustoptimization approaches. In pure data-driven approaches,SAA for example, the DM constructs (in each time period)a new demand distribution based exclusively on the historyof demand observations. SOBME differs from SAA alongtwo important dimensions: (i) it enables the DM to avail ofmoment and tail bound information in addition to the demandobservations, and (ii) the DM’s beliefs are representedas a distribution over distributions rather than as a singledistribution. SOBME is also very different from robustoptimization approaches in which decisions are made onlywith respect to the worst-case distribution in the ambiguityset. In SOBME, the DM assigns weights to all members ofthe ambiguity set. This helps avoid the tendency of robustoptimization methods to make overly conservative decisions.

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Moreover, our approach updates these weights based on newrealizations and evolving moment and tail information.

4.2. Formal Development of theSOBME Approach

At the beginning of period t ∈´, the DM’s belief about thedemand distribution is represented as a density function ft4�5over the set of possible demand distributions contained inthe ambiguity set D 2= 8F �

D 2 � ∈ì9.3 At the end of period t,having observed the period-t demand and knowing the mostrecent moment and tail information, the DM updates hisbelief from ft4�5 to ft+14�5. In what follows, we formallydevelop the approach for updating ft4�5 to ft+14�5.

The updating mechanism (at the end of period t) isbased on the following three inputs. One input is the jointdensity (from the beginning of period t) over the demandand the demand distribution indicator; this is given byft4�1�5= ft4�5f

�D 4�5, where f �

D 4�5 is the demand densityassociated with the indicator �. The second input is theobserved value of the demand in period t, denoted by � ′

t .The third input is the most recent moment and tail boundinformation, i.e., F̂t .

Let f ot+14�1�5 denote some joint density (at the end of

period t) over the demand observation that was realizedin period t and the demand distribution indicator randomvariable. The output of our updating mechanism, definedthrough the optimization program described below, willbe a specific joint density denoted by f ∗o

t+14�1�5. This willdetermine the DM’s updated belief ft+14�5 using ft+14�5=∫

¤ f ∗ot+14�1�5d�, i.e., the DM’s updated belief ft+14�5 is

given by the marginal density of f ∗ot+14�1�5.

The updating mechanism chooses a joint density f ot+14�1�5

[the “posterior”] to minimize the Kullback-Leibler (KL)divergence (a.k.a. relative entropy) between f o

t+14�1�5 andft4�1�5 [the “prior”] while recognizing that the demandobservation in period t took on some specific value, � ′

t , andthat the joint density f o

t+14�1�5 must imply a demand densitythat conforms to the most recent moment and tail boundinformation, i.e., F̂t . The Kullback-Leibler (KL) divergenceis formally defined as:

Definition 1 (Kullback-Leibler Divergence). TheKullback-Leibler (KL) divergence or the relative entropybetween two joint densities f o

t+1 and ft defined on ¤×ì is:

dKL4fot+1�ft5=

∫

¤

∫

ìf ot+14�1�5 log

f ot+14�1�5

ft4�1�5d�d�

= Ɛf ot+1

[

logf ot+14�1�5

ft4�1�5

]

0

This is a general definition for any two densities, butdescribed in terms of our specific notation.4 Observethat dKL4f

ot+1�ft5¾ 0, and dKL4f

ot+1�ft5 is convex in pair

4f ot+11 ft5; see, for example, p. 32 of Cover and Thomas

(2006).

The updating mechanism is given by the following func-tional optimization program, in which the decision variable isthe function f o

t+12 ¤×ì→�+, and is chosen to minimizethe relative entropy (or equivalently maximize the negativerelative entropy).

minf ot+12¤×ì→�+

dKL4fot+1�ft5 (2)

s.t.

Ɛt4D5¶∫

¤

∫

ìf ot+14�1�5�14�5d�d� ¶ Ɛ̄t4D5 (3)

∫

¤

∫

ìf ot+14�1�5�24�1u5d�d� ¶ bt4u5 ∀u ∈¤ (4)

∫

ìf ot+14�1�5d�= �4� − � ′

t5 ∀� ∈¤ (5)

∫

¤

∫

ìf ot+14�1�5d�d� = 11 (6)

where �14�5 2= Ɛ4D ��5=∫

¤ �dF �D 4�5, �24�1u5 2= 1 −

F �D 4u5, and �4 · 5 is the Dirac delta function.5 It should be

noted that constraints (4) and (5) are not single constraints:they each represent an infinite number of constraints.

We first discuss the role of constraints (3) and (4). Note that∫

¤ f ot+14�1�5d� gives the marginal density of the demand

distribution indicator �. Therefore, constraint (3) reflectsthe fact that the DM’s updated belief must result in a meandemand that conforms to the most recent upper and lowermoment bounds, Ɛ̄t4D5 and Ɛt4D5, respectively. Similarly,constraint (4) reflects the fact the DM’s updated belief mustresult in a demand distribution whose tail probability (at anygiven u) is bounded above by the most recent tail bound bt4u5.

Next we discuss the role of constraints (5) and (6). Recallthat the updating optimization program is selecting a jointdensity f o

t+14�1�5 over the demand observation that was real-ized in period t and the demand distribution indicator randomvariable. The marginal density

∫

ìf ot+14�1�5d� in (5) is the

density over the possible demand observations. Of course, therealized value of the observation is already known to be � ′

t .Therefore, any valid marginal density (valid, in the sense thatit reflects what the DM already knows) should place all itsweight on � = � ′

t and no weight on � 6= � ′t . However, it also

needs to be a legitimate density function. These two require-ments are fulfilled by using the Dirac delta function (seeEndnote 5) and setting

∫

ìf ot+14�1�5d�= �4� − � ′

t5 ∀� ∈¤,which is constraint (5). Finally, constraint (6) is a normaliza-tion that ensures f o

t+14�1�5 is a joint density in the sensethat it integrates (over all possible values of both arguments)to 1. We note this constraint is in fact redundant giventhe property that

∫ �

−��4x5dx = 1, but we include (6) for

expositional clarity.This updating program is an infinite-dimensional optimiza-

tion program that has a continuously differentiable convexobjective function and linear constraints. This allows us touse KKT-like conditions along with the variational principleto characterize the global optimum solution, f ∗o

t+14�1�5.Let �̄t and �t be the Lagrangian multipliers for the upper

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bound and lower bound in constraint (3), respectively. Let�t4u5 and �t4�5 represent the multipliers for (the infinitenumber of) constraints (4) and (5), respectively. Defininggt4�5 2= 4�̄t − �t5�14�5+

∫

¤ �t4u5�24�1u5du, we are ableto obtain the following closed-form result.

Theorem 1 (Second Order Joint Belief Updating). Thesolution to the optimization program (2)–(6) is given by:

f ∗ot+14�1�5=

ft4�1�5�4� − � ′t5e

−gt4�5

∫

ìft4�

′t1 �̃5e

−gt4�̃5 d�̃0

Proof. All proofs are contained in Appendix B (availableas supplemental material at http://www.dx.doi.org/10.1287/opre.2015.1454).

Corollary 1 (Second Order Belief Updating). Theupdated density, ft+14�5, over the demand distributionindicator (� ∈ì) is:

ft+14�5=ft4�5ft4�

′t ��5e

−gt4�5

∫

ìft4�̃5ft4�

′t � �̃5e

−gt4�̃5 d�̃1

where ft4�′t ��5= f �

D 4�′t5.

Remark 1 (Bayesian Updating). When the moment andtail bounds, i.e., constraints (3) and (4), are not binding in theupdating optimization program, the updating result simplifiesto ft+14�5= ft4�5ft4�

′t ��5/4

∫

ìft4�̃5ft4�

′t � �̃5 d�̃5. That is,

the updating result reduces to a traditional Bayesian updatingresult. This occurs when (i) there are no moment and tailbounds, or (ii) these bounds are sufficiently loose that theydo not constrain the DM’s updated beliefs, i.e., when they arenot informative given his prior belief. In general, however,the updating is performed using an exponential modifier thatcaptures the effect of the moment and tail bound information.This is an important result with widespread applicationsbeyond the motivating example of this paper. It establishesSOBME as a generalization of Bayesian updating for settingsin which partial distributional information is available inaddition to data/observations.6

Recall that we make no restriction on the family orfamilies of distributions included in the ambiguity set. Fromthis perspective, SOBME can be viewed as a nonparametricapproach to belief representation and updating. Furthermore,we emphasize that the SOBME belief representation andupdating result are not limited to a repeated newsvendorsetting. They apply to any setting in which a DM has partialdistributional information (in the form of an ambiguity setand, possibly, moment and tail bounds) about a randomvariable of interest, and wishes to dynamically combinethis information with an evolving history of observations.Finally, we note that there are alternative formulations to theoptimization program (2)–(6), and SOBME can be derivedfrom these alternatives. For example, one can implicitlyinvoke the Dirac delta function used in constraint (5) anddefine a new distribution f̃t4�5 = ft4�

′t1�5. Introducing

this notation, program (2)–(6) can be rewritten as one thatminimizes the KL divergence between the posterior ft+14�5and the prior f̃t4�5 subject to moment and tail constraints.

In what follows, we apply the SOBME approach in arepeated newsvendor setting. Considering the sequenceof events described in §3, we have the following. At thebeginning of period t, the DM determines his order quantityqt ¾ 0 to minimize the expected newsvendor cost givenhis current belief about the demand distribution, which isrepresented by the density function ft4�5 over the set ofpossible demand distributions contained in the ambiguity setD 2= 8F �

D 2 � ∈ì9. Applying the well-known newsvendorquantile result, the optimal quantity decision under SOBMEis given by:

qSOBMEt 2= inf

{

q ∈¤2 Ɛ�6F�D 4q57¾ p

p+h

}

1 (7)

where Ɛ�6F�D 4q57 =

∫

�∈ìF �D 4q5 ft4�5d�. After ordering

this quantity, the order is received, demand � ′t is realized,

the newsvendor cost L4qSOBMEt − � ′

t5 is incurred, and newexogenous research, if any, arrives regarding moment andtail bounds. At the end of period t, knowing the realizeddemand and the latest moment and tail bounds, the DMupdates his belief from ft4�5 to ft+14�5 using Corollary 1.

For later use, let QSOBMEt denote the period-t order quantity

random variable. QSOBMEt takes on a realized value qSOBME

t ,specified by (7), given any particular sample path of demandrealizations in periods 11 0 0 0 1 t − 1. Let

q�∗

= inf{

q ∈¤2 F �∗

D 4q5¾ p

p+h

}

denote the optimal order quantity if the DM knows the truedistribution. This optimal order quantity is static, becausethe true distribution is stationary. In the next section, weshow that QSOBME

t → q�∗

almost surely as t → �.

4.3. Asymptotic Behavior of SOBME: Consistencyand Convergence Rate

In this section, we study two important asymptotic propertiesof SOBME. The first property is consistency. The secondproperty relates to the rate at which the demand distributionbuilt by SOBME converges to the true distribution. We for-malize the consistency concept below, but loosely speaking,this means that the DM eventually places (almost) all hisweight on the true demand distribution, with the result thatthe DM’s quantity decision converges (with probability one)to the quantity decision under full distributional information:QSOBME

t → q�∗

almost surely as t → �. This requires oneto consider a shrinking neighborhood of �∗, and to showthat the DM will eventually assign a weight of one to thisshrinking neighborhood. The type of consistency dependson how such a shrinking neighborhood is constructed. Weestablish weak consistency, which considers weak neigh-borhoods of �∗. That is, we consider those � ∈ ì that

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are “close” to �∗, where closeness is measured in terms ofan expectation-like distance (see Appendix A for formaldefinitions).

To establish consistency and a rate of convergence, weneed to introduce the notion of coherent information for themoment and tail bounds.

Definition 2 (Coherent Information). The sequenceof information 8F̂t2 t ∈´9 is said to be coherent if (i) thesequence of lower (upper) bounds on the first moment, i.e.,8Ɛt4D52 t ∈´9 (8Ɛ̄t4D52 t ∈´9), is nondecreasing (nonin-creasing), with Ɛt4D5¶ Ɛ4D5¶ Ɛ̄t4D5 for all t ∈´ whereƐ4D5 is the true demand mean, and (ii) for any u ∈¤, thesequence of tail upper bounds 8bt4u52 t ∈´9 is nonincreasingwith bt4u5¾ F �∗

D 4u5.

If the moment and tail bound information is not coherent,then we should not expect an approach that uses such infor-mation to necessarily converge to the true distribution. Forexample, the information might either be “wrong” in someperiod, i.e., the moment and tail bounds may rule out thetrue distribution, or the sequence of bounds might be non-monotonic, e.g., periodic, which could prevent convergenceeven if the information is correct.

Recall that ft4�5 is the density function (at the startof period-t) over the demand distribution indicator �. LetF SOBMEt denote its cumulative distribution, where we use the

superscript SOBME to highlight that it is the result of theSOBME updating method. The following result establishesthe weak consistency of SOBME. We refer the reader toAppendix A for formal definitions of weak consistency andKL neighborhoods used in the following theorem.

Theorem 2 (Consistency of SOBME). Suppose that(a) the sequence of information 8F̂t2 t ∈ ´9 is coherent,and (b) the initial distribution F SOBME

1 is such that∫

K��∗

dF SOBME1 > 0 for every � > 0 (where K�∗

� is a KLneighborhood of �∗). Then:

(i) F SOBMEt is weakly consistent.

(ii) QSOBMEt → q�∗

almost surely as t → �.

In addition to consistency (showing almost-sure conver-gence to the true distribution), it is useful to explore therate/speed of convergence. In a later section, we will examinethis and other questions numerically. Here, we develop analmost-sure analytical bound on the distance between thetrue demand density and the one constructed using SOBME.To this end, for x ∈¤, let f SOBME

D1 t 4x5 2=∫

�∈ìft4�5f

�D 4x5d�

denote the demand density generated by SOBME at the startof period t.

Theorem 3 (Rate of Convergence). Suppose that thesequence of information 8F̂t2 t ∈´9 is coherent and thatregularity conditions 3 and 4 (see Appendix A) hold. Then,there exists constants �1 and �2 such that for large enough t:

dH4f�∗

D �f SOBMED1 t 5¶ 6�16�

2t + 2e−�2t�

2t 771/2 a.s.1

where �t = max8rt1 y1/2t 9, with rt and yt defined in Appendix

A, and dH4f1�f25 is the Hellinger distance (defined inAppendix A) between any two probability densities f1 and f2.

The above result establishes �t = max8rt1 y1/2t 9 as a con-

vergence rate for SOBME. As discussed in Appendix A,this rate of convergence depends on the tightness of theambiguity set (measured through rt) and the precision of theinitial prior (measured through yt).

5. Newsvendor’s Cost of Ambiguity:A Performance Guarantee

Having characterized the asymptotic behavior of SOBME,we next explore the increase in the average cost per period(during a horizon of T <�) that the DM experiences becauseof ambiguity. We refer to this cost as the newsvendor’s costof ambiguity. We will derive an analytical upper bound forthe newsvendor’s cost of ambiguity when SOBME is used.This will enable us to develop a performance guarantee. Tothat end, let

CSOBMET 2=

1T

T∑

t=1

Ɛf �∗

D6L4QSOBME

t −Dt5−L4q�∗

−Dt57 (8)

denote the random average (per period) newsvendor’s cost ofambiguity under SOBME, which depends on the demand real-izations up to the beginning of period T , i.e., the realizationof vector DT = 4Di1 i = 1121 0 0 0 1 T − 15 which is generatedbased on the true distribution. Taking the expectation withrespect to this vector, we denote by

cSOBMET 2= ƐDT

6CSOBMET 71 (9)

the expected average per period newsvendor’s cost of ambi-guity. It should be noted that CSOBME

T is a random variablebut cSOBME

T is a number. In (8), QSOBMEt is a random variable

with a realization that depends on Dt: the expectation in(8) is only with respect to Dt , whereas this randomness isremoved through the outer expectation in (9).

The following results provide upper bounds for CSOBMET

and cSOBMET when the support of all the distributions in the

ambiguity set can be bounded from above. Bounded demandis not a strong assumption in practice because demand isnever infinite.

Theorem 4 (Newsvendor’s Cost of Ambiguity). Supposethe support ¤ is 601 d̄7 for some d̄ ∈ �+. Letting k 2=2√

2d̄max8p1h9, the following hold for any T :

4i5 CSOBMET ¶ k

√

1T

T∑

t=1

dKL4f�∗

D �f SOBMED1 t 51 (10)

almost surely.

4ii5 cSOBMET ¶ k

√

ƐDT

[

1T

T∑

t=1

dKL4f�∗

D �f SOBMED1 t 5

]

0 (11)

Part (i) of the above result shows that the random aver-age per period KL divergence between the true density

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and the one built using SOBME nicely bounds, in thealmost sure sense, the average per period cost of ambiguity.Part (ii) shows a related result for the expected averageper period cost of ambiguity. We next develop a bound forƐDT

641/T 5∑T

t=1 dKL4f�∗

D �f SOBMED1 t 57, the expected average per

period KL divergence between the true density and the onebuilt using SOBME. Because of the result of Theorem 4 part(ii), this will enable us to reach our end goal of providing aperformance guarantee for SOBME.

To bound ƐDT641/T 5

∑Tt=1 dKL4f

�∗

D �f SOBMED1 t 57, we need

some general conditions on (i) the (dynamic) information onmoment and tail bounds (F̂t), (ii) the ambiguity set (D′), (iii)the initial prior used by the DM under SOBME (f SOBME

1 ),and (iv) the convergence behavior around �∗. For (i), we usethe coherent information condition (Definition 2) to ensurethat the DM is not relying on false information. For (ii)–(iv),we use the “smoothness” and “soundness” conditions dis-cussed in Clarke and Barron (1990) for Bayesian updatingmechanisms. We briefly list those conditions here for ourabove-mentioned goal, and refer interested readers to Clarkeand Barron (1990) for more detailed discussions. It shouldbe noted that the “smoothness” condition is related to thebehavior around �∗ and the “soundness” condition is relatedto the speed of weight accumulation around �∗ as more andmore observations are made.

Condition 1 (Smoothness). The density f �D 4�5 and the

divergence dKL4f�∗

D �f �D 5 are both twice continuously differ-

entiable at �∗. Furthermore, (i) there exists an � > 0 suchthat:

Ɛ

[

sup�∈ì2 ��−�∗�<�

∣

∣

∣

∣

¡2

¡�2log f �

D 4�5

∣

∣

∣

∣

2]

<�1

and (ii) I4�5 and f SOBME1 4�5 are both finite, positive, and

continuous at �∗, where I4 · 5 is the Fisher information andf SOBME

1 4 · 5 is the initial prior density.

Condition 2 (Soundness). The ambiguity set is said tobe sound if the convergence of a sequence of � values isequivalent to the weak convergence of the correspondingdemand distributions:

�→�∗⇔ F �

D → F �∗

D 0

The above conditions allow us to provide an effectivebound for the expected value of the average per period KLdivergence under our proposed updating mechanism.

Theorem 5 (Expected Average per Period KL Diver-gence). Suppose the coherent information, smoothness, andsoundness conditions hold. Then:

ƐDT

[

1T

T∑

t=1

dKL4f�∗

D �f SOBMED1 t 5

]

¶ 1T

[

12

logT

2�e+

12

log I4�∗5+ log1

f SOBME1 4�∗5

+ o415]

0

(12)

The above result states that the expected average perperiod KL divergence between the true density andthe one built using SOBME is at worst of the form41/T 56 1

2 log4T /42�e55+ l7 for some constant l that dependson the initial prior and the Fisher information, which areboth also affected by the “tightness” of the ambiguity set, D′.The important implication is that it allows us to provide thefollowing performance guarantee for our proposed SOBMEapproach: based on Theorem 4 part (ii), the expected average(per period) newsvendor’s cost of ambiguity under SOBMEgoes to zero at worst at a fast rate

√logT /T as T → �.

In other words, although a DM facing demand ambiguityinevitably incurs an extra per-period cost compared to a DMwho knows the true demand distribution, the average extraper-period cost approaches zero quite quickly if SOBME isused. The actual rate depends on the quality of the dynamicinformation the DM has on moments and tails; Theorem 5provides an upper bound by considering the worst case, i.e.,no information.

Theorem 6 (Performance Guarantee). If ¤= 601 d̄7 forsome d̄ ∈�+, then under the conditions of Theorem 5:

cSOBMET ¶ k′

√

logTT

+ o41/√T 51 (13)

where k′ = 2d̄max8p1h9, and hence, cSOBMET → 0 as T → �

at a rate of√

logT /T .

Theorem 6 establishes that the DM does not need to havefull distributional information to make suitable stockingdecisions: SOBME enables the DM to deal effectively withambiguity.

6. Numerical ExperimentsAs discussed in §1, companies reduce ambiguity in newproduct introductions by devoting significant market-researchefforts to gather information about the demand distribution.That is, firms do not rely exclusively on demand observationswhen forming beliefs about an unknown demand distribution;they can and do generate additional partial information aboutthe demand distribution. In this paper, such information iscodified by the ambiguity set and moment and tail bounds,i.e., the exogenous research. It is managerially important tounderstand the value of such exogenous research in an oper-ational setting (here, a repeated newsvendor). Furthermore,assuming there is value in the exogenous research, it ishelpful to examine which type of information (ambiguity setor moment bounds) provides more value, and what factors,e.g., newsvendor quantiles, affect the value. Answers to thesequestions—which we provide through numerical studiesbelow—can help managers understand when it is worth theireffort to obtain exogenous information and what form ofinformation is most valuable.

A natural way to explore the value of exogenous researchis to compare our information setting—data and exogenous

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research—to two benchmark settings: (i) full information,and (ii) a data-only information setting. For this secondbenchmark, we need to choose an approach tailored to adata-only setting. Various pure data-driven approaches havebeen developed in the literature as we discussed in §2, amongwhich we consider the Sample Average Approximation (SAA)approach (see, e.g., Kleywegt et al. 2002; Shapiro 2003;Levi et al. 2007, 2015), Scarf’s method (Scarf 1958, Gallegoand Moon 1993), and the recently developed likelihoodrobust optimization (LRO) approach (Wang et al. 2015).

Under SAA, the DM at the beginning of period t+ 1 sets

qSAAt+1 2= inf

{

q ∈¤2 F̂ tD4q5¾

p

p+h

}

1 (14)

where F̂ tD4q5 is the empirical c.d.f. after t observations:

F̂ tD4q5 2= 41/t5

∑tk=1 �8�

k ¶ q9, with �k denoting the demandrealization in period k. In Scarf’s method, the DM minimizesthe worst-case expected cost among all the distributions thathave a mean equal to the sample mean and a variance equalto the sample variance. Using Scarf’s result (Scarf 1958), theoptimal ordering quantity at the beginning of period t+ 1 is

qSCARFt+1 2=mt +

st2

(

√

p

h−

√

h

p

)

1 (15)

where mt and st are the sample average and sample standarddeviation of the observations made in periods 1121 0 0 0 1 t.In the dynamic LRO approach (Wang et al. 2015), the setof possible distributions in each period t contains onlythose that make the observed history of data achieve acertain level of likelihood, and the optimal decisions aremade based on the worst-case distribution within that set. Inour repeated newsvendor setting, we numerically observedthat SAA (weakly) outperformed both Scarf’s method anddynamic LRO, and so we adopt the SAA approach as therepresentative for the data-only setting.7 That is, for brevity,we do not present our numerical results for Scarf’s methodand LRO, and instead focus on the performance of SAAamong the pure data-driven approaches.

6.1. Study Design

In addition to examining the value of exogenous informationby comparing the performances of SOBME and SAA,our numerical study explores the performance of SOBMErelative to a DM who has full distributional information.It also investigates how the performance of SOBME and thevalue of exogenous information are influenced by the truedistribution, the type of exogenous information (ambiguityset and moment bounds), and the newsvendor quantilep/4p+h5.

True Distribution. The true distribution, F �∗

D , in our studyis assumed to have a mean of 15. The true distribution iseither normal (with a coefficient of variation (C.V.) of 0.2)or exponential, with a label “A” denoting the normal caseand “B” denoting the exponential case.

Ambiguity Set. We consider two cases for the ambiguityset D. In Case I, the ambiguity set contains all exponentialdistributions with a mean in 6101205 as well as all normaldistributions with a mean in 6101207 and a C.V. of 0.2. InCase II, the ambiguity set includes all those distributions inCase I plus the mixtures (with equal weights of 0.5) of anyexponential with mean 6101205 and a normal with the samemean (and a C.V. of 0.2). In other words, Case II representsa larger ambiguity set that contains Case I and mixtures ofdistributions in Case I.8

Therefore, the combination of ambiguity sets and thetrue distribution leads to four different cases: I-A, I-B, II-Aand II-B. In all cases, we assume SOBME starts with auniform distribution on the ambiguity set, i.e., it starts withthe highest information entropy.

Moment Bound Information. For each of these four cases,we consider two possible sets of moment bound information,8F̂t2 t ∈ ´9. In one situation, the DM has no momentinformation beyond those dictated by the ambiguity set(referred to as the “No Bounds” situation). In the othersituation (referred to as the “Tight Bounds” situation),the DM has dynamically evolving first moment boundinformation. In this “Tight Bounds” situation, the initiallower and upper bounds are those dictated by the ambiguityset, i.e., Ɛ04D5= 10 and Ɛ̄04D5= 20, but these bounds getprogressively tighter by 1.5 units in each period until theyhit Ɛt4D5= 1405 and Ɛ̄t4D5= 1505, after which they remainconstant. To be conservative when evaluating SOBME, weassume that the DM does not have tail bound information.Such information, if available, would of course improve theperformance.

Newsvendor Quantile. The above combination of twoambiguity sets, two true distributions, and two momentinformation sets leads to eight cases. For each of these cases,as our base-case scenario, we set the newsvendor quantile top/4p+h5= 0075, with h= 1 and p = 3. However, as we willdescribe later, we also explore the impact of the newsvendorquantile by varying its value in 80050100551 0 0 0 100959.

Runs. Each case is run for 100 periods and the averagecost is calculated for the SOBME approach, the full informa-tion case, i.e., when the true distribution is known, and thepure data-driven SAA approach. We purposely choose alimited time horizon because we know that the performanceof both SOBME and SAA will asymptotically approach thatof full information as more and more demand observationsbecome available. In other words, when ample data exists, itis less important which approach is used, as long as it isa consistent approach. As such, we are mainly interestedin how well SOBME performs when data is limited. Thelimited-data setting also better represents the new productintroduction we discussed in §1. The average costs arecalculated by conducting 50 independent replications witheach replication being a sample path generation, i.e., demandrealizations in periods 1 to 100. Under each sample path andin each period, the related expected newsvendor costs are

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calculated directly and based on the closed-form newsvendoroverage plus underage costs of the DM’s ordered quantity.

In each period, the closed-form solution of Theorem 1(and Corollary 1) are used to calculate the updated SOBMEbelief function and then the newsvendor problem is solvedbased on the updated beliefs. The belief-updating probleminvolves calculating the function gt4�5 which requires findingLagrangian multipliers �̄t and �t . Note that �t4u5 canbe set to zero because there is no tail constraint. Thesetwo multipliers are calculated efficiently in each periodt by employing a numerical search method which takesadvantage of the properties of the underlying optimizationproblem.9 A similar numerical search can be used forcalculating �t4u5 when it is nonzero. We note that thebelief-updating optimization problem is independent of theDM’s ordering problem, and so the belief-updating aspectof SOBME can be applied for a wide range of decision-making problems in which there is distributional ambiguity.The computational burden in calculating various integralsinvolved in our method (arising from the continuous first andsecond-order distributional assumptions, i.e., the sets ¤ andì being continuous) is overcome by using high precisionfunction interpolations and advanced numerical methods forcalculating integrals.

6.2. Results

For each of the eight cases in the study design, Figure 1reports the Average Cost Optimality Gap (%) for bothSOBME and SAA, where the optimality gap refers to the per-centage difference in cost as compared to the full informationcase in which the DM knows the true distribution. As can beseen, SOBME performs very well: its optimality gap is lessthan 1% after 20 periods in six of the eight cases. We nextdiscuss how the exogenous research information (momentbounds and ambiguity set) and the true distribution impactthe performance of SOBME relative to full information andto the pure data-driven representative, SAA.

The Effect of Exogenous Information. The impact ofthe moment bound information on SOBME’s performancecan be explored by comparing the SOBME plots in theleft column (No Bounds) with the corresponding SOBMEplots in the right column (Tight Bounds). Moment boundinformation is clearly valuable: SOBME’s optimality gapapproaches zero much more rapidly when tight momentbound information is available. The impact of the ambiguityset on SOBME’s performance can be explored by comparingCase I-A with Case II-A and comparing Case I-B with CaseII-B. We observe that the performance is quite similar inboth comparisons, indicating that SOBME is quite robust tothe ambiguity set. We tested this observation further (forthe case in which the true distribution was normal) by alsoconsidering a setting in which the ambiguity set containedonly normal distributions. Contracting the ambiguity setdid not materially impact SOBME’s performance. The factthat SOBME’s performance is robust to the ambiguity set isindeed an important benefit because it allows the DM to

expand his ambiguity set when he has more doubt about theform of the true distribution.

By comparing the difference between SOBME’s perfor-mance and that of SAA in each subfigure, we see that theDM benefits significantly from exogenous information in theform of the ambiguity set and moment bounds. This indicatesthat there is potentially great value in marketing research andrelated actions that can generate this kind of information. Inaddition, our results suggest that the information containedin the moment bounds is more valuable than those containedin the ambiguity set. Hence, the DM should prefer a tighterset of moment bounds than a tighter ambiguity set: effortsthat result in tighter moment bounds are in general morevaluable than those that result in tighter ambiguity sets.

The Effect of the True Distribution. Comparing the nor-mal cases (I-A and II-A) with the exponential cases (I-Band II-B), we see that SOBME performs better when thetrue distribution is exponential. In the exponential cases,SOBME’s optimality gap is less than 1% after a few periodsunder both “No Bounds” and “Tight Bounds” situations.That SOBME performs better when the true distributionis exponential is due to the fact that the newsvendor costis much more sensitive to distributional mis-specificationwhen the true distribution is normal than when the truedistribution is exponential. To observe this consider thefollowing cases within our numerical setting (h = 1 andp = 3): (i) the true distribution is normal (C.V. = 002), butthe DM believes with probability one that it is exponential(C.V. = 1) (both having a mean of 15), and (ii) the truedistribution is exponential (C.V. = 1) but the DM believeswith probability one it is normal (C.V. = 002) (both having amean of 15). In case (i) the DM orders 20.79 units whereasthe true optimal ordering quantity is 17.03. This results in anoptimality gap of about 55.15% (in terms of the percentageadditional expected cost incurred). In case (ii), the DMorders 17.03 whereas the true optimal ordering quantityis 20.79. This results in an optimality gap of only about2.48% (in terms of the percentage additional expected costincurred). Comparing these two cases shows that a wrongdistributional assumption is less consequential when the truedistribution is exponential than when it is normal. Hence,having a good estimate of the true distribution is moreimportant when the true distribution is normal than when itis exponential. This fact shows why SOBME performs betterwhen the true distribution is exponential than when it isnormal. Interestingly, however, SAA’s performance is slightlybetter when the true distribution is normal than when it isexponential. What accounts for this? There is a force thatcounteracts the robustness under the exponential distribution.The DM sees a more volatile set of observations under anexponential distribution than under the normal distribution.This significantly influences the rate of convergence of SAA.This second driving force is stronger than the first one forSAA. This is not the case for SOBME, because it is not asreliant on observations as SAA, i.e., SOBME is not using anempirical distribution.

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

0

2

4

6

8

0 20 40 60 80 100

t

Avg

. cos

t opt

. gap

(%

) SOBME

SAA

(a) Case I-A-no bounds

0 20 40 60 80 100

t

Avg

. cos

t opt

. gap

(%

)

0

1

2

3

4

5

6

(b) Case I-A-tight bounds

0 20 40 60 80 100

t

Avg

. cos

t opt

. gap

(%

)

0

1

2

3

4

5

6

7

(c) Case I-B-no bounds

0 20 40 60 80 100

t

Avg

. cos

t opt

. gap

(%

)

0

1

2

3

4

5

6

7

(d) Case I-B-tight bounds

0 20 40 60 80 100

t

(e) Case II-A-no bounds

0

2

4

6

8

Avg

. cos

t opt

. gap

(%

)

0

1

2

3

4

5

(g) Case II-B-no bounds

Avg

. cos

t opt

. gap

(%

)

0 20 40 60 80 100

t

0

1

2

3

4

5

6

(h) Case II-B-tight bounds

Avg

. cos

t opt

. gap

(%

)

0 20 40 60 80 100

t

0 20 40 60 80 100

t

(f) Case II-A-tight bounds

Avg

. cos

t opt

. gap

(%

)

0

1

2

3

4

5

6

SOBME

SAA

SOBME

SAASOBME

SAA

SOBME

SAASOBME

SAA

SOBME

SAA

SOBME

SAA

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

0.5 0.6 0.7 0.8 0.9 1.00

10

20

30

40

50

60

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

0

5

10

15

20

(c) Case I-B-no bounds (d) Case I-B-tight bounds

pp + h–

0.5 0.6 0.7 0.8 0.9 1.0

pp + h–

0.5 0.6 0.7 0.8 0.9 1.0

pp + h–

0.5 0.6 0.7 0.8 0.9 1.0

pp + h–

Avg

. cos

t opt

. gap

(%

)A

vg. c

ost o

pt. g

ap (

%)

0

5

10

15

20

Avg

. cos

t opt

. gap

(%

)

0

10

20

30

40

50

60

Avg

. cos

t opt

. gap

(%

)SOBME

SAASOBME

SAA

SOBME

SAASOBME

SAA

The Effect of the Newsvendor Quantile. To explore theimpact of the newsvendor quantile on the performance ofSOBME, we focus on the four Case I studies (exponentialand normal true distributions; No Bounds and Tight Boundsinformation), and vary the newsvendor quantile p/4p+ h5 ∈

80050100551 0 0 0 100959, while fixing h= 1. Figure 2 shows theperformance of SOBME and SAA for each of the four cases.The performance of SAA and SOBME are averaged over 50replications as well as the first 20 periods. We first note thatour study confirms what has already been reported in theliterature for SAA: its performance significantly degradesas the newsvendor quantile increases (see, e.g., Levi et al.2015). In contrast, we observe that SOBME’s performanceis much more robust: its performance degradation is verylow. For instance, in Figure 2(a), although SAA has anoptimality gap of 55% for a newsvendor quantile of 009,SOBME has an optimality gap of only 10%. When momentbound information is available, Figures 2(b) and (d), thedifference in performance between SOBME and SAA is evengreater. Recall that the difference in performance betweenSOBME and SAA measures the value of the exogenousinformation. Thus, our result suggests that this type ofinformation is particularly valuable in settings characterizedby high newsvendor quantiles. Moreover, the robustness ofSOBME to the newsvendor quantile suggests that SOBME is

a very suitable approach if a high service level is required orif inventory cost parameters are subject to estimation error.

In summary, the above study has the following managerialimplications: (i) there is significant value in exogenousinformation (as measured by the comparison of SOBMEand SAA), (ii) market research that leads to tighter momentbounds is more valuable than research that leads to a morecompact ambiguity set, and (iii) it is especially important toavail of exogenous information (through SOBME) whenthe newsvendor quantile is high because pure data-drivenapproaches (SAA) perform much worse at high quantiles.

As mentioned above, the SOBME approach is not limitedto a repeated newsvendor setting. It can be used in anymultiperiod setting in which a DM has partial distributionalinformation (in the form of an ambiguity set and, possibly,moment and tail bounds) about a random variable of interest.A general measure of SOBME’s performance is the qualityof the distribution generated by SOBME, i.e., how closeis it to the true distribution from which the observationsare drawn? We investigate this in two different studies inAppendix C, the second of which uses data from a casebased on a real-world setting (Allon and Van Mieghem 2011)to show how moment bounds can be developed from “like-products.” Among other results, we find that the quality ofthe SOBME-generated distribution is very good, that partial

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information in the form of moment bounds is particularlyvaluable early in the horizon, and that companies can getsignificant benefit from moment bound information even ifthe number of like-products is quite limited. We refer thereader to Appendix C for further details.

6.3. Extensions

6.3.1. Endogenous Bounds. We have assumed to thispoint that the sequence of moment bound information8Ɛt4D51 t ∈´9 and 8Ɛ̄t4D51 t ∈´9 are derived solely fromexogenous research. However, our SOBME framework canalso accommodate dynamic bounds that are endogenouslygenerated from the data observations. For instance, using theCentral Limit Theorem (CLT) at the end of period t, a 1 −�confidence interval for the first moment is given by

It 2= 4mt − z�/2 × st/√t1mt + z�/2 × st/

√t51 (16)

where z�/2 is the inverse of the standard normal c.d.f.evaluated at 1 −�/2, and mt and st are the sample averageand sample standard deviation obtained from observationsmade in periods 1121 0 0 0 1 t. In the absence of exogenousbounds, the DM can use the boundaries of It as momentbounds. If the DM has exogenous bounds given by somesequences 8Ɛt4D51 t ∈´9 and 8Ɛ̄t4D51 t ∈´9, then he canreplace them with the respective boundaries of It in a period twhen It ⊂ 6Ɛt4D51 Ɛ̄t4D57, i.e., when the endogenous boundsare tighter than the exogenous ones. This ensures that themoment bound information remains coherent when theexogenous bound information is coherent.

We investigated the value of endogenous bounds by apply-ing this approach (using a 90% confidence interval) to casesI-A-No Bounds and I-A-Tight Bounds. Figure 3 depictsthe performance of SOBME for the four possible situations

Figure 3. The effect of endogenous bounds on the per-formance of SOBME (No.: no exogenousor endogenous bounds; End.: endogenousbounds only; Exg.: exogenous bounds only;End. + Exg.: both endogenous and exogenousbounds).

0 20 40 60 80 1000

1

2

3

4

t

End. + Exg.

Exg.

End.

No.

Avg

. cos

t opt

. gap

(%

)

(No/Tight Bounds with/without Endogenous Bounds). As canbe seen, endowing the DM with endogenous bounds doesnot materially improve SOBME’s performance. Irrespectiveof the endogenous bounds, SOBME is already availing ofthe data to updates its probabilities across the potentialdistributions in the ambiguity set. Therefore, the data used togenerate the endogenous bounds has also been used (indepen-dently) to appropriately “weight” the possible distributions,and those that conform to the endogenous bounds will havereceived more weight. In other words, SOBME is alreadyusing the observations, i.e.,“the endogenous data,” in aneffective way, and hence using them to create supplementarymoment bounds is not of significant help. We also note thatthe endogenous bounds are typically tight and, therefore,useful, only when they are generated based on ample data,i.e., a large t. However, as discussed earlier, when ampledata exists, any estimation approach that is consistent willgenerate good results and it matters less which one is used.Moreover, it is precisely because they do not have ample datathat firms devote efforts to generating exogenous researchon the demand distribution.

6.3.2. Change Detection. We have assumed to this pointthat the true demand distribution is stationary throughout thehorizon. In practice, however, the distribution may change atcertain points in time because of, for example, changes ineconomic factors, competitor actions, market penetration,and time of year. Some of these changes might be amenableto prediction by managers, and SOBME can accommodatesuch predications through updated moment bounds. Otherchanges might not be predictable and completely hidden tothe DM, and so it is important to test how well approachessuch as SOBME and SAA detect and respond to unpredictedchanges.

We numerically examined the ability of SOBME andSAA to detect and respond to such a hidden/unpredictedchange in the type of demand distribution. In particular, weuse a new setting, setting Case I-C, in which, unknown tothe DM, the true distribution changes from normal (C.V. =002, mean = 15) to exponential (C.V. = 1, mean = 15) inthe middle of the horizon, i.e., period 50. We evaluatethe performance of SOBME (under both “No Bounds”and “Tight Bounds”) and SAA compared to a DM thatknows the true distribution throughout the horizon, i.e.,both before and after the change. The results are presentedin Figure 4. Not surprisingly, both SOBME and SAA areaffected by the change at period 50. The change initiallyaffects SOBME slightly more than SAA because SOBMEwas already getting very close to the true distribution beforethe change whereas SAA was further from it. Also, wenote the spike is lower (and comparable to that for SAA)when SOBME has tight bounds. Very importantly, however,irrespective of the bounds, SOBME recovers much morerapidly than SAA: it quickly recognizes the hidden changeand adjusts to it. This reflects the fact (analytically establishedin the performance guarantee and numerically observed) that

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Figure 4. Comparison of SAA and SOBME for changedetection (Setting Case-I-C) (NB: No Bounds;TB: Tight Bounds).

0 20 40 60 80 1000

1

2

3

4

5

6

t

SOBME (TB)

SOBME (NB)

SAA

Avg

. cos

t opt

. gap

(%

)

SOBME’s cost performance converges quite quickly to thatof full information. This characteristic is particulary desirablein dynamic and nonstationary environments (where theunderlying true distribution is subject to inevitable changes)as it allows the DM to automatically and quickly adjust itsdecisions.

7. Repeated Newsvendor Under AmbiguityAversion

We assumed that the DM was ambiguity neutral in thepreceding sections. In general decision-making settings, aDM may, in fact, be ambiguity averse, ambiguity neutral, orambiguity loving; see, e.g., Klibanoff et al. (2005, 2009).In what follows, we extend our results and demonstratehow the SOBME approach can be also used when the DMis not ambiguity-neutral. We adopt the general frameworkproposed by Klibanoff et al. (2005), hereafter KMM, forsmooth decision making under ambiguity. Tailoring theKMM decision-making framework (see equation (1) ofKMM) to a newsvendor problem, we cast the DM’s quantitydecision at the beginning of period t as

maxqt¾0

Ɛf SOBMEt 4�56ê4Ɛf �

D6−L4qt −D57571 (17)

where ê4 · 5 represents the DM’s ambiguity attitude, andf SOBMEt 4�5 is the second-order belief built using SOBME.

When ê4 · 5 is affine, the DM is ambiguity-neutral, whichis the case considered in the previous sections. However,when ê4 · 5 is concave (convex), the DM is ambiguity-averse(loving).10

In the case of an ambiguity-neutral DM, as discussed in§4.1, there is a type of period-t equivalence between second-order and first-order approaches in the sense that the expectedvalue of the DM’s objective, see (1), depends on the beliefft4�5 and the ambiguity-set densities f �

D 4x5 only through thedemand density f SOBME

D1 t 4x5 2=∫

�∈ìft4�5f

�D 4x5d�. There-

fore, a first-order approach that somehow arrived at an

identical demand density would result in the same period-tobjective. Importantly, this theoretical equivalence holdsonly when the DM is ambiguity neutral. For settings withan ambiguity averse/loving DM, the objective, i.e., (17),depends on ft4�5 and f �

D 4x5 in a manner that cannot becaptured through f SOBME

D1 t 4x5 2=∫

�∈ìft4�5f

�D 4x5d�. There-

fore, it is crucial to adopt a distribution-over-distributionsapproach. SOBME provides a natural and effective approachfor ambiguity averse/loving settings.

In what follows, we first explore some structural propertiesof the decisions made by an ambiguity-averse newsvendor,i.e., properties of the period-t objective (17), when ê4 · 5 isconcave. We then explore the effect of the DM’s ambiguityaversion on the newsvendor ordering quantity. To this end,we consider the negative exponential function ê4x5= −e−�x

with parameter �¾ 0, which is increasing concave and has aconstant coefficient of ambiguity �= −ê′′4x5/ê′4x5. In thissetting, a higher � represents a higher ambiguity aversion.For notational convenience, we use U�4q5= Ɛf �

D6L4q −D57

to denote the random variable that represents the expecteddisutility of ordering q units, with its realization dependingon � ∈ì. The KMM objective (17) is then equivalent to

minqt¾0

Ɛf SOBMEt 4�56e

�U�4qt57= minqt¾0

MU�4qt54�51 (18)

where MU�4qt54�5 is the moment generating function of

U�4qt5 evaluated at �. For technical reasons, we assume that(i) the ambiguity set D is such that this moment generatingfunction exists and is differentiable with respect to q, and(ii) all the demand distributions in D are differentiable andhave densities in D′. Finally, we note that minimizing themoment generating function in (18) is equivalent to minimiz-ing the so-called entropic disutility 41/�5 logMU�4qt5

4�5, andhence we use that form in proofs when more convenient.

Lemma 1 (Convexity). The moment generating functionMU�4q54�5 is convex in q (∀�¾ 0), and hence, (18) is aconvex program.

We let qKMMt 4�5 denote the optimal solution to (18) on a

given sample path. That is, qKMMt 4�5 is the period-t optimal

ordering quantity of a newsvendor (with a coefficient ofambiguity �) given a particular realization of demands upto and including period t− 1. We let QKMM

t 4�5 denote therandom variable associated with qKMM

t 4�5 when lookingover all possible sample paths through period t − 1.

Lemma 2 (Optimal Ambiguity-Averse Ordering Quan-tity). For �> 0, let Z�4q5=U�4q5+ 41/�5 logF �

D 4q5. Theoptimal ordering quantity qKMM

t 4�5 is either zero or thesolution to the implicit equation

MZ�4q54�5

MU�4q54�5=

p

p+h0 (19)

The above lemma proves that the optimal ordering quantity,if positive, sets the ratio of the relevant moment generatingfunctions equal to the well-known newsvendor quantile

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p/4p+h5. This can be viewed as a generalization of theclassical newsvendor quantile result. Using this result, wenow establish how ambiguity aversion influences the optimalordering quantity (in any given period t), and how theoptimal ordering quantity (at any given ambiguity coefficient)behaves as t tends to infinity.

Theorem 7 (Ambiguity Aversion Effect). The optimalordering quantity under ambiguity-aversion satisfies thefollowing:

(i) qKMMt 4�5 is nonincreasing in �.

(ii) lim�→0+ qKMMt 4�5= qSOBME

t (defined by (7)).(iii) lim�→� qKMM

t 4�5= 0.(iv) limt→� QKMM

t 4�5= q�∗ almost surely when the con-ditions of Theorem 2 hold.

A higher level of ambiguity aversion, i.e., a higher �,results in a lower ordering quantity. As � approaches zero,the DM acts similarly to an ambiguity-neutral DM, thatis, his order quantity approaches qSOBME

t defined in (7). As� approaches �, the DM only considers the worst-caseoutcome, i.e., becomes a minimax optimizer, and henceorders nothing so as to avoid any overage cost. Regardlessof the DM’s ambiguity aversion level, as more and moredemand observations are made (t → �), his ordering quantityconverges to that of a DM who completely knows thetrue demand distribution. This highlights the asymptoticalsuitability of SOBME under any ambiguity aversion level.

We numerically explore the performance of SOBMEunder the KMM criterion by considering the “Case I-A-NoBounds” setting introduced in §6.1. Figure 5 presents theaverage cost optimality gap (%) for three levels of ambiguityaversion: �= 00005, �= 00010, �= 00050. The average costoptimality gap (%) is the percentage extra cost of the orderedquantities of a DM who is facing demand ambiguity and usesthe objective (17) compared to a DM who completely knowsthe true demand distribution. For each period shown inFigure 5, the optimality gap is averaged over 20 independentsample path replications, i.e., demand observations overthe horizon. Consistent with Theorem 7(iv), the average

Figure 5. Performance of SOBME under the KMMcriteria (Case I-A-No Bounds).

0 5 10 15 200

5

10

15

20

t

� = 0.005

� = 0.010

� = 0.050

Avg

. cos

t opt

. gap

(%

)

optimality gap goes to zero as t increases. Moreover, as �decreases, the performance under ambiguity aversion tendsto that observed in our previous ambiguity-neutral results.

Although Lemma 2 characterizes the period-t optimalorder quantity in general, a much sharper characterization canbe established in certain cases. In particular, under (i) low butpositive ambiguity aversion, or (ii) a normality assumptionon the disutility distribution, we can derive closed-formsolutions for the ambiguity-averse newsvendor problemby applying the certainty equivalence principle. Usingthis principle drastically simplifies the problem, becausewe only need to consider the first and second momentsof the disability U�4q5 = Ɛf �

D6L4q −D57 as opposed to

its moment generation function. In what follows, we letF SOBMED1 t 4�5 2= Ɛ�1 t6F

�D 4�57=

∫

�∈ìft4�5F

�D 4�5d� denote the

demand distribution built by SOBME (used for decision-making in period t) and we let F SOBME4−15

D1 t 4 · 5 denote itsinverse. First, we consider the case in which the DM has alow but positive level of ambiguity aversion.

Theorem 8 (Low Ambiguity Aversion). Let �4qt5 and�4qt5 denote the mean and standard deviation of U�4qt5(with respect to �), respectively. Then

(i) The optimal ordering quantity under ambiguity-aversion is

qKMMt 4�5=argmin

qt

{

�4qt5+�

2�24qt5+

�

2�24qt5+O4�25

}

0

(20)

(ii) The solution to (20) is

qKMMt 4�5= F

SOBME4−15D1 t

(

p

p+h−��4qKMM

t 5−O4�25

)

1

(21)

where �4qt5 2= Ɛ6F �D 4qt5U

�4qt57− Ɛ64p/4p+h55U�4qt57.

We see that when the DM has a low but positive level ofambiguity aversion, the optimal ordering quantity is a per-turbed version of the traditional critical quantile newsvendorsolution. This perturbation, is approximately (with an errorof order �2 denoted by O4�25) linear in the coefficient ofambiguity, �, with a slope of −�4qKMM

t 5.Next, we consider a case where U�4q5 has an approxi-

mately Normal distribution based on the DM’s second-orderbelief. See Appendix D for an example illustrating thevalidity of this approximation.

Remark 2 (Normal Approximation). Suppose at someperiod t, U�4qt5 is approximately Normal with mean �4qt5and standard deviation �4qt5. Then

(i) The newsvendor problem under ambiguity aversion isequivalent to the mean-variance optimization minqt

�4qt5+

4�/25�24qt5. That is,

qKMMt 4�5' arg min

qt

{

�4qt5+�

2�24qt5

}

0 (22)

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(ii) The solution to (22) is

qKMMt 4�5

'FSOBME4−15D1t

(

p

p+h−�Cov4F �

D 4qKMMt 51U�4qKMM

t 55

)

0 (23)

As in the case of low ambiguity aversion, the optimalordering quantity is again a perturbed version of the tradi-tional critical quantile solution, with the linear perturbationnow having a slope of −Cov4F �

D 4qKMMt 51U�4qKMM

t 55.

8. ConclusionThere are many multiperiod settings in which a decisionmaker does not have full distributional information abouta random variable of interest but may have some partialdistributional information in the form of bounds on momentsor tails. Over time, the decision maker gains access to demandobservations. Techniques which rely exclusively on partialdistributional information ignore the value of observations. Incontrast, pure data-driven techniques avail of observations butignore partial distributional information. With the growingattention to business analytics, it is essential to bridge thisgap by developing approaches that allow decision makers tobenefit from both data and partial distributional information.

In this paper, using the maximum entropy principle, wedeveloped an approach (SOBME) that allows a decisionmaker to effectively use partial distributional informationand data/observations. Moreover, SOBME can accommodatedynamically evolving partial information. We established thatSOBME is a natural generalization of traditional Bayesianupdating; an exponential modifier is introduced to incorporatethe partial distributional information. This is an importantresult, with applications far beyond the newsvendor focus ofthis paper. We proved that SOBME is weakly consistent andcharacterized its rate of convergence. Applying SOBME in arepeated newsvendor setting, we introduced and analyzed thenotion of newsvendor’s cost of ambiguity. We provided ananalytical bound for this cost when the DM uses SOBME.More broadly, our results indicate that information-theoreticalapproaches, such as the one we developed in this paper,provide a natural way for characterizing an upper bound forthe newsvendor’s cost of ambiguity.

Our numerical investigation demonstrated that SOBMEperforms well, not just outperforming a pure data-drivenapproach but often coming close to the performance of fulldistributional information. We also found that unlike a puredata-driven approach such as SAA, SOBME’s performancedoes not significantly degrade as the newsvendor quantileincreases, making it a suitable candidate for settings withhigh service level requirements. By comparing our approachto pure data-driven approaches, we also generated insightsinto the value of partial distribution information. Our resultsindicate that it can be very beneficial to incorporate suchinformation when deciding about the stocking quantities.Also, we found that information in the form of tighter

moment bounds is typically more valuable than informationin the form of tighter ambiguity sets.

Finally, we examined an ambiguity-averse version of ourrepeated newsvendor problem, and showed how SOBMEcan be applied in that setting. We established theoreticalresults that characterize the optimal newsvendor quantityunder ambiguity aversion, and demonstrated that SOBME iswell-suited to an ambiguity-averse setting.

In closing, we emphasize that the SOBME belief represen-tation and updating approach can be applied to any setting inwhich a decision maker has access to observations/data andsome partial distributional information. We hope to applyand test SOBME in various other settings in future work.

Supplemental Material

Supplemental material to this paper is available at http://dx.doi.org/10.1287/opre.2015.1454.

Acknowledgments

The authors are grateful to Retsef Levi for his invaluable suggestions,comments, and discussions which helped to improve this paper.The authors also thank the department editor, Chung Piaw Teo, theassociate editor, and two anonymous referees for their constructivesuggestions that improved the content and exposition of this paper.

Endnotes

1. For clarity, we only consider the first moment, but theapproach and results are readily extended to higher moments.2. We assume that the DM is ambiguity-neutral until §7which considers other ambiguity attitudes.3. At the beginning of period t = 1, the DM chooses aninitial density f14�5 to reflect his starting belief aboutthe probability of each demand distribution and his initialmoment/tail bound information. For instance, a uniformdensity over all � ∈ ì (representing a maximum initialentropy) would be appropriate, if (a) the DM did not believeany particular distribution was more likely to be the trueone, and (b) ì is such that a uniform distribution over all� ∈ì can be defined.4. For the more general class of �-divergence, we referinterested readers to Pardo (2006), Ben-Tal et al. (2013),and the references therein. We also note that dKL4·�·5 is nota metric in the usual sense as it is not symmetric.5. The Dirac delta function, sometimes called the unitimpulse function, is a generalized function (often) definedsuch that �4x5 = � if x = 0 and �4x5 = 0 otherwise,with

∫ �

−��4x5dx = 1 (Kreyszig 1988). It can be defined

more rigorously either as a measure or a distribution.It can be viewed as the limit of the Gaussian function:�4x5= lim�→04e

−x2/2�2/4

√2��55. Importantly, the Dirac

delta function has the property that for any function g4 · 5,∫ +�

−�g4x5�4x− x05dt = g4x05.

6. Giffin and Caticha (2007) also show a related result,but unlike our framework, they only derive it under thevery strong assumptions that the first moment is perfectly

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known and that this is the only information available, i.e.,no tail bounds and no upper and/or lower moment bounds.Furthermore, their work does not examine the asymptoticbehavior (consistency, rate of convergence, etc.) of theupdating mechanism nor does it embed the distribution-formation problem in a decision-making context.7. Our results for SAA, LRO, and Scarf’s method echothose presented in Wang et al. (2015) which explores theperformance of LRO and Scarf’s methods as comparedto SAA for a single-shot newsvendor setting with ampledata. We also observed that SAA has certain computationaladvantages over LRO, which is an important benefit in adynamic multiperiod setting such as ours. Scarf’s approachis closer to SAA in terms of computational complexity, butits performance is dominated by SAA.8. In Case I, we consider ì= 6101307, where � ∈ 6101205indicates an exponential distribution with mean �, and� ∈ 6201307 indicates a normal distribution with a mean�− 10 and a C.V. of 0.2. The construction of ì in Case IIis done in an analogous manner.9. For our numerical experiments, we use a gradient-based numerical method in Mathematica. Other numericalapproaches can also be used.10. The internal expectation of the newsvendor cost L4 · 5in (17) implies a risk-neutral DM. This could be relaxed butwe focus on incorporating ambiguity attitudes and not riskattitudes.

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Soroush Saghafian is an assistant professor at Harvard KennedySchool, Harvard University. His research focuses on the applicationand development of operations research methods in studyingstochastic systems with specific applications in healthcare andoperations management.

Brian Tomlin is a professor at the Tuck School of Business atDartmouth. His research lies in the general area of operations andsupply chain management, with risk being a key focus of much ofhis work.

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