The newsvendor problemwith unknown distributionU Benzion1, Y Cohen2 and T Shavit3∗1Department of Economics, Ben-Gurion University of the Negev, Israel; 2Department of Management andEconomics, The Open University of Israel, Israel; and 3School of Business, College of Management, Israel
Newsvendor theory assumes that the decision-maker faces a known distribution. But in real-life situations,demand distribution is not always known. In the experimental study which this paper presents, half of theparticipants assuming the newsvendor role were unaware of the underlying demand distribution, while theother half knew the demand distribution. Participants had to decide how many papers to order each day (for 100days). The experimental findings indicate that subjects who know the demand distribution behave differentlyto those who do not. However, interestingly enough, knowing the demand distribution does not necessarilylead the subject closer to the optimal solution or improve profits. It was found that supply surplus at a certainperiod strongly affects the order quantity towards the following period, despite the knowledge of the demanddistribution.Journal of the Operational Research Society (2010) 61, 1022–1031. doi:10.1057/jors.2009.56Published online 20 May 2009
In newsvendor theory, optimal order and expected profit arefunctions of (1) price—the item’s purchase or selling price,and the salvage price; and (2) demand distribution (Nahmias,1994, 2005). Previous literature tends to assume that thedecision-maker faces know demand distribution or estimateddistribution. Schweitzer and Cachon (2000) suggest that theassumption that the decision-maker knows the distribution ofdemand ‘is a reasonable assumption when the decision-makerhas access to a substantial amount of historical data for similarproducts’ (p. 405). Fisher and Raman (1996) discovered thateven though a fashion apparel manufacturer changes styleseach year, the demand distribution for similar styles closelyresembles that of previous years, although admittedly thisobservation refers to a very specific context of application.
While it is possible that with enough historical data peoplemay reasonably approximate the type of distribution andestimate its parameters, this is not always the case in real-life scenarios. There are many real-life instances where thedemand distribution is unknown. This is especially true forthe newsvendor problem, which is a single-period problemthat includes one time occasions, specific holiday accessories,and special projects for which demand cannot be accuratelyprojected. However, demand may be obscured even in otherinventory models. For example, in presenting a new productto a new market, the vendor might face unknown demanddistribution since s/he has no access to historical data for
∗Correspondence: T Shavit, School of Business Administration, Collegeof Management, 7 Rabin Avenue, Rishon-Le’Zion, Israel.E-mail: [email protected]
similar products. Even opening new business premises in anew area is a situation where the demand distribution is notknown in advance.
This paper tests experimentally, the order policy in thenewsvendor problem in two cases: (1) where decision-makersknow the demand distribution, and (2) where decision-makersdo not know the demand distribution. Based on earlier liter-ature, it is assumed that decision-makers in the former cate-gory will order quantities that are closer to the optimal order,than those in the latter category.
In the presented computerized learning experiment, eachparticipant assumed the role of a newsvendor and had todecide how many papers to order each day over 100 days. Theparticipants were paid at the end of the experiment accordingto their gains in a random period. The main finding is thatknowing the demand distribution does not necessarily help indiscovering the optimal solution or in improving the subject’sprofits. In both cases, subjects are asymptotically convertingto a personal order quantity that is different from theoptimal one.
Another important issue is the effect of feedback on thesubjective order. The feedback in this case is the gap betweenthe actual demand realization and the order of the previousday. It was found that this gap plays an important role inthe participants order decisions. When the subject’s orderis greater than the subsequent demand realization (a supplysurplus) s/he has to throw the old newspapers away, absorbinga financial loss. Large losses are an incentive to be morecautious over the next period (ie reducing the order). On theother hand, when the subject orders less than the demand(a demand surplus) s/he sells all the ordered products and asa result faces only positive gain. While subjects may feel that
U Benzion et al—Newsvendor and unknown distribution 1023
they lose the potential gain because they could not supplyall the demand, our results indicate that the effect of supplysurplus and demand surplus are not symmetric. Specifically,the effect of the former is stronger than that of the latter. Thereason for this is that supply surplus is perceived as a realloss, while demand surplus is perceived as a ‘theoretical’ loss.We also show that subjects who do not know the demanddistribution are more affected by the feedback under concernas compared to the subjects who do know the demanddistribution.
The rest of the paper is organized as follows: Section 2is a literature review; Section 3 defines the hypotheses ofour study; Section 4 describes the experimental procedure;Section 5 presents the primary results and provides somepossible explanations; and Section 6 summarizes the conclu-sions.
2. Literature review
The related literature focuses on four aspects of thenewsvendor problem: (1) The expected profit maximizingorder quantity, (2) demand estimation, (3) theoretical ana-lysis, and (4) experimental studies. This review is orderedaccordingly, with a sub-section for each aspect.
2.1. Expected profit maximizing order quantity
The classical newsvendor problem (Whitin, 1955) deals witha single period order of perishable (or degradable) products,which will either be sold, or else will lose part, or all, oftheir value (Polatoglu, 1991). The newsvendor (the decision-maker), facing uncertain demand (D) from a known distribu-tion function F(D) with a probability density function f (D),has to decide on the order quantity Q. The cost of each unitis C and the selling price for the customer is P . Assuming ashortage penalty cost (typically for reputation loss) of R, andsalvage value of S, the model finds the optimal order quantity(Q∗) by maximizing the expected profit: E[�(Q)]. The wellknown formula for maximizing E[�(Q)] is (Khouja, 1999):
F(Q∗) =(P − C + R
P − S + R
)(1)
Another well known format of the same formula is given byNahmias (2005) as follows: Let the marginal cost of shortage(underage) beCu=P−C+R. Let the marginal cost of overagebe C0 = C (or if a salvage value S is returned C0 = C − S).The second well known format of Equation (1) is (Nahmias,2005):
F(Q∗) =(
Cu
CU + C0
)(2)
This result could be used to find optimal order quantitywhen the demand distribution is known. Both Gallego andMoon (1993) and Alfares and Elmorra (2005) deal with thecase where only the mean and the standard deviation areknown, but not the whole distribution. Although they call it
‘distribution free’ model, it is substantially different from anunknown demand distribution model where the mean and thedispersion are unknown.
2.2. Demand estimation
Demand estimation in the presence of censored observationsarising from lost sales is considered by Nahmias (1994) forthe case of the normal distribution and by Silver (1993) fordiscrete distribution with Bayesian updating. Azoury (1985)developed ordering policies for unknown distributions withexact demand observations.
Lovejoy (1990) showed that a simple inventory policybased upon a critical fractile can be optimal or near-optimalin some inventory models with parameter-adaptive demandprocesses. In these, some parameter of the demand distri-bution is not known with certainty, and estimates of theparameter are updated in a statistical fashion as demand isobserved through time. Bounds on the value loss relativeto optimal cost, when using the critical fractile policy, wascalculated directly from the problem data.
Lariviere and Porteus (1999) analyse an empirical Bayesianinventory problem wherein unmet demand is both lost andunobserved. The full Bayesian problem was solved by scalingsolutions to a simple problem that can be examined beforethe product is stocked. The stock level for a given period wasobtained by rescaling the value from the normalized problemusing the best estimate of the size of the market. It wasassumed that the cost parameters were stationary over timeand that the underlying demand process supported indepen-dent and identically distributed demands over time. Resultsindicated that size and precision effects support critical func-tions in stalking information. Findings also showed that whenthe stock out penalty is sufficiently small, the product will beprofitable regardless of the sequence of observed sales. Eppenand Iyer (1997) focus on the problem of buying fashion goodsfor the big book of a catalogue merchandiser. They develop amodel of demand for an individual item. The model uses bothhistorical data and buyer judgement. They build a stochasticdynamic programming (DP) model of the fashion buyingproblem that incorporates the model of demand. The DPmodel was used to derive the structure of the optimal inven-tory control policy. They then develop an updated newsboyheuristic that is intuitively appealing, easily implemented andperforms very well. Similar numerical experiments show thatthe current company practice does not yield consistently goodresults when compared to the optimal solution. Burnetas andSmith (2000) provided a nonparametric approach to learningunknown demand and proved that in the long run, the averageprofit with incomplete information adjusts to the optimal profitwith full information.
2.3. Theoretical analysis
Some of the theoretical papers suggest that known demanddistribution improves the ordering policy. Heching et al
1024 Journal of the Operational Research Society Vol. 61, No. 6
(2002) analyse the sales and price data from a specialtyretailer of women’s apparel. The data set contains 184 stylessold during the Spring 1993 season. A demand model washypothesized, fit to the data and then analysed to obtainestimates of revenues under various pricing policies. Bothfull information and adaptive policies were considered. Theoptimal prices suggested by the models were compared withthe actual prices the revenues generated by various policiesand were estimated. The analysis suggested that if the firmhad made smaller mark-downs earlier in the sales season, itwould have increased its revenues significantly. The resultsalso indicated that model-based pricing schemes can poten-tially increase revenue. Xu and Hopp (2004) studied variousapproaches to demand learning in the context of a one-shotinventory replenishment problem with dynamic pricing. Theystudied full information, non-learning, passive learning andactive learning pricing policies, all of which take the formof a product of a series of indices and a power function ofon-hand inventory. Through computational experiments, theyfound that non-learning pricing policies often perform verypoorly, and that passive learning pricing policies can appro-ximate the optimal active learning pricing policy. Finally,they observed that the values of information and learningare magnified by coordination of pricing with the inventorydecision.
Burnetas and Smith (2000) presented theoretically thecombined problem of pricing and ordering a perishableproduct with demand distribution approximation usingcensored demand observations resulting from lost sales.They developed an adaptive pricing and ordering policy withasymptotic property. According to this policy the averagerealized profit per period adjusts with a probability of one tothe optimal value under complete information on the distri-bution. They claimed that according to their model, when thedemand distribution is assumed to be in a specific parametricclass (normal or uniform), a more efficient ordering policycan be developed by incorporating this additional knowledge.
The specific values of the historical sales levels must beused for estimating the unknown parameters, meaning thatthe decision-maker is using his or others’ knowledge, basedon historical sales, to estimate the unknown parameters.
2.4. Experimental studies
Although the optimal order-size maximizes expected profits,experimental studies of the newsvendor model in recentyears have demonstrated that decision-makers systemati-cally deviate from the optimal order because of differentbehavioural biases. In all of these studies the participantswere informed about the demand distribution. For example,Schweitzer and Cachon (2000), analysing 15 decision orderperiods with known uniform distribution, showed that whenthe marginal profit is larger (smaller) than the cost, parti-cipants tend to order less (more) than the optimal order.Schweitzer and Cachon claimed that there are behavioural
factors that lead to deviation from the optimal order, suchas risk aversion, loss aversion, underestimation of opportu-nity cost or waste aversion. When Bolton and Katok (2004)subsequently extended the Schweitzer and Cachon experi-ment to 100 decision periods, they found that the addi-tional periods enhanced experience and slowly improvednewsvendor performance.
Benzion et al (2008) used 100 decision periods in thenewsvendor problem with both low and high profit margin(as do Schweitzer and Cachon, 2000) and both uniform andnormal distributions. They found that as experience is gath-ered, there is a convergence to a stationary order quantity,which is significantly different from the optimal order. Theyalso found that in the first purchase decision periods, parti-cipants tend to order quantities that are closer to the meandemand than in the last periods. This bias towards the meanexplains the deviation of participants from the optimal order.The effect of time proximity of events on decisions wasalso tested. The findings show that if the current demand isgreater than the order, participants tend to increase their order,and if demand is lower than the order, they tend to reducethem. However, participants learn throughout the experimentto reduce this effect. This is consistent with the findingsof Lurie and Swaminathan (2008), who examined the effectof feedback frequency on performance in four newsvendorexperiments. The feedback was received after making thedecision on the order quantity and included the demand quan-tity, the order quantity, the resultant profit and the cumulativeprofit for the period. Those who made decisions regardingseveral periods received feedback that included the order ineach period, profit in each period and cumulative profit in eachperiod. They found that subjects, who receive frequent feed-back, tend to focus on the most recent data. They concludedthat more frequent feedback slows the learning process.
Bostian et al (2008) evaluated the effect of learning andadaptive behaviour on order decisions in the newsvendorproblem. They used five-period intervals to obtain a long-runperspective on period-by-period losses and gains. The resultsindicate that decision-makers deciding at infrequent intervalsseem to behave like inexperienced decision-makers. The indi-vidual data show that participants respond to recent gains andlosses, but that this response weakens over time. In an experi-mental study of the mutual decisions of wholesalers’ priceand retailers’ purchase in the newsvendor problem, Keser andPaleologo (2004) found that:
The decision heuristic of the retailers is based on an anchor pointcreated by the price-quantity combination in the first periods. Inthe following periods, the retailer will adapt his order quantityto changes in the wholesale price around this anchor point. Hewill decrease his order quantity as the wholesale price increases,and vice versa. (p 17)
However, they did not explain or describe the retailers ‘anchorpoint’.
U Benzion et al—Newsvendor and unknown distribution 1025
3. Hypotheses
In this section we present some background information basedon which we construct our main research hypotheses.
3.1. The effect of knowing the demand distribution
Our main hypothesis is that knowing the demand distribu-tion helps subjects to improve profits and to approach moreclosely the expected profit maximizing order quantity (to becalculated using Equation (2)). We base our hypothesis onthe theoretical approach that suggests that more informationon the demand distribution may improve the order policy (egHeching et al, 2002; Xu and Hopp, 2004; Burnetas and Smith,2000).
H1: Participants’ order quantity and profit.Participants’ order quantity is closer to the optimal ordercalculated by the newsvendor problem model and hencetheir profit is also higher when they know the demanddistribution.
3.2. The adjustment/learning process
Previous experimental studies show an adjustment processof the order quantity that leads towards the expected profitmaximizing quantity, but converges to a value that is still farfrom it (eg Bolton and Katok, 2004; Benzion et al, 2008).In a process covering 100 periods, we expect to find thateach individual adjusts to his own level. We expect differentindividuals to converge to a different order level, and thisorder level is usually different from the optimal order.
H2: Adjustment process.Individuals adjust over time and converge on a specific(personal) order level (the order level deviationsbecoming smaller over time).
As a result the absolute change in the order quantitybetween two consecutive periods is reduced over time.
3.3. The feedback effect
The effect of feedback on decision-making has been tested bymany previous studies (eg Erev and Barron, 2001; Johnsonet al, 2005). The effect of feedback on inventory decision-making and the learning process was tested for differenttasks by Atkins et al (2002), and Diehl and Sterman (1995).Benzion et al (2008), Bostian et al (2008) and Lurie andSwaminathan (2008), found that in the newsvendor problem,subjects respond to recent gains and losses, and tend to focuson the most recent data. Accordingly, we would expect that(1) the order quantity increases when there is a shortage(demand surplus) in the previous period; and (2) the orderquantity decreases when there is a supply surplus of perish-able/degradable products in the previous period (based onSchweitzer and Cachon, 2000; Lurie and Swaminathan, 2008;Benzion et al, 2008).
We examine whether the effects of supply surplus anddemand surplus are symmetric.
A supply surplus creates a real loss since the subjecthas ordered more than required when compared to demandsurplus, where the subject misses out on a potential gain.Since subjects are loss averse (eg Thaler, 1990; Thaler et al,1992), we expect to find that the effect of a supply surplus isstronger than the effect of a demand surplus.
We also examine whether knowing the demand distributionaffects the strength of the feedback effect. We assume thatsubjects who know the distribution base their decision on thedemand distribution (Benzion et al, 2008) and on the previousperiod. However, subjects who do not know the distributionbase their decision mostly on previous periods. The result isthat in an unknown distribution scenario, the effect of theprevious period is stronger.
H3: Feedback effect.
(a) The size of the order increases in cases where there isa demand surplus in a previous period and decreaseswhen there is a supply surplus from the previousperiod.
(b) The effect of a supply surplus is stronger than theeffect of a demand surplus.
(c) When the distribution is unknown, the effect of theprevious period is stronger than when it is known.
4. The experiment
4.1. Experimental design
The experiments were programmed using Visual Basic andExcel and included 121 management students (sophomoresand juniors), who had taken a basic course in statistics.
The students were from the Open University of Israel andHolon Institute of Technology. The experiments took place ina computer laboratory and lasted on average 1 h (not includingpresenting and explaining the instructions). Each subject wasfree to progress at his or her own pace, independently ofthe other participants in the experiment, and with no timelimitation.
Prior to the experiment, participants were divided into eightgroups in order to examine the combinations of two profitlevels, two variance levels (using different distributions) andknown and unknown distributions.
The choice of different control parameter levels, such asprice, cost values and distributions, followed previous exper-imental studies, for example Schweitzer and Cachon (2000)and Bolton and Katok (2004). The groups were assigned thefollowing costs and selling prices: A selling price of 12 NIS(New Israeli Shekels) per unit for all the groups. For the highprofit groups, the cost was 3 NIS per unit (the profit per unitwas therefore 9 NIS). For the low profit groups, the cost was9 NIS per unit (so the profit per unit was 3 NIS).
1026 Journal of the Operational Research Society Vol. 61, No. 6
We chose to test both the uniform and the normal demanddistributions. The use of uniform distribution is common inprevious experimental studies such as Schweitzer and Cachon(2000), Bolton and Katok (2004) and Benzion et al (2008).Wetest also the normal distribution since the demand in real-lifesituations may have this distribution. The normal distributionis also used in theoretical studies such as Nahmias (1994) andin experimental studies such as Benzion et al (2008).
The uniform demand range consisted of 1–300 products.The normal demand distribution had a mean of 150 (�=150)and a SD of 50 (� = 50), ensuring that 99.7% of the demanddistribution was within the range of 1–300 (as in the uniformdistribution).
For the low profit level, the respective values of the optimalorder for the uniform and normal distributions were 75 and116. For the high profit levels, the respective values of theoptimal order for the uniform and normal distributions were225 and 184.
4.2. The experimental procedure
Before the experiment, participants were handed writteninstructions (see Appendix A in Benzion et al (2008)),including examples. For the groups working with a knowndistribution, the demand distribution was given to participantsas follows:
(1) For Uniform distribution, participants were told that eachvalue from 1 to 300 has the same likelihood of beingchosen.
(2) For the Normal distribution, participants were told thatthe order is randomly selected from normal distribution,which they were familiar with. However, in order tobetter explain the normal distribution, participants weregiven a table with demand results of 100 simulateddays. In this way, the participants could use a relativelylarge sample to learn about the nature of the normaldistribution.
(3) The subjects with unknown distributions were told thatthe demand was taken from a distribution, without anyinformation about its nature.
Before starting the experiment, the participants were askedto practice the computer software for 10 periods. In theexperiment, participants run 100 inventory purchase decisionperiods. In each period, participants were informed of thecost and price of the product. Each period was followed by apresentation of the actual demand; the total cost of the order;the total revenue; the demand /supply surplus; the forfeitedprofits due to inventory shortage; and the profit. The datawere presented in a table format.
To provide concrete incentives, at the end of the experimentone of the periods was randomly selected and the participantswere paid proportionally to the profit in the selected periodin cash (the average payment was 20 NIS, or about $5).
5. Results
The results were analysed by calculating the correlationbetween the dependent variable (average order, average profitor average absolute change) and the periods’ number (period).
We also used blocks of 20 periods and compared thesubjects’ behaviour in the first 20 periods block and the last20 periods block. This kind of analysis emphasizes the trendover time, if it exists. The use of period blocks in the analysisof multi-period experiments is very common (see eg Boltonand Katok (2004) and Lurie and Swaminathan (2008)). Thenumber of periods in each block differs from one researchto another. The analysis by blocks also helps us to comparebetween the cases where demand was known versus when itwas unknown.
Table 1 presents, for each group, the correlation betweenthe average order in each period and the periods’ number(period). We also preset the average order over all the periodsand the average order in the first and last blocks of 20 periods.We used a paired t-test to examine the null hypothesis thatthe average order in the first 20 periods block equals theaverage order in the last 20 periods block. We also comparethe average order of the first and last blocks, and of all theperiods, to the optimal order. Under the value of each averageorder we present the p-value of the t-test to examine the nullhypothesis that the average order is not different from theoptimal order.
The average order in all cases was found to be significantlydifferent from the optimal order. In dealing with the knowndistributions, the correlations between the average order andthe period are significant. This means that a consistent trendwas found for the average order in the course of the 100periods. In all these cases, subjects are moving towards theoptimal order according to the newsvendor problem’s theo-retical solution. We also see that the average order in the first20 periods is significantly different from the average orderin the last 20 periods. So, subjects who knew the distribu-tion used their knowledge to improve their order. However,as mentioned in Benzion et al (2008) subjects are biasedto the distribution’s mean when the distribution is knownbecause subjects have a tendency to choose the average(‘Central Tendency Bias’, Helson, 1964; Crawford et al,2000).
When the distribution is unknown, the correlations betweenthe average order and the period are relatively low and mostlynonsignificant. (The correlation is significant only for the caseof the unknown normal distribution with high profit.) Also,no significant difference was found between the first and lastblocks of 20 periods, meaning that subjects did not changethe average order.
Subjects that know the demand distribution, order onaverage the same as the subjects who do not know thedistribution. This is inconsistent with Hypothesis 1. In thecase of normal distribution with low profit we find that (1)subjects who did not know the demand distribution ordered
U Benzion et al—Newsvendor and unknown distribution 1027Ta
ble1
Average
orderin
allperiod
sandin
thefirst
andlast
20-periodblocks
Dem
and
Subjects’
Correlation
First
20-
Last20
-Testing
Average
ofOptim
alpa
ttern,
and
know
ledge
withperiod
period
block
period
block
blocks
diff.
allperiod
sorder
profi
tlevel
ofdeman
daverage
average
pattern
Slop
e(F
value,
sig)
Order
quan
tity
Order
quan
tity
t-testa ,
p-value
Order
quan
tity
Order
quan
tity
Uniform
demand,
Low
profi
tKnown
−0.53
(F=
39.1
,sig
=0.00
)14
8.55
(p
=0.00
)12
2.61
(p
=0.00
)t=
3.22,p
<0.01
141.67
(p
=0.00
)75
Unk
nown
0.04
(F=
0.15
,sig
=0.69
)14
1.24
(p
=0.00
)13
8.74
(p
=0.00
)t=
0.21,p
=0.41
142.69
(p
=0.00
)75
Uniform
demand,
Highprofi
tKnown
0.37
(F=
15.7
,sig
=0.00
)16
5.53
(p
=0.00
)18
9.52
b(p
=0.00
)t=
−3.96,
p<0.01
175.66
(p
=0.00
)22
5Unk
nown
0.08
(F=
0.57
,sig
=0.45
)16
1.76
(p
=0.00
)16
8.09
(p
=0.00
)t=
−0.77,
p=
0.22
164.33
(p
=0.00
)22
5
Normal
demand,
Low
Profi
tKnown
−0.44
(F=
23.6
,sig
=0.00
)15
4.62
b(p
=0.00
)13
9.81
b(p
=0.00
)t=
2.6,
p=
0.01
149.85
b(p
=0.00
)11
6Unk
nown
−0.13
(F=
1.8,sig
=0.18
)13
0.02
(p
=0.00
)12
5.67
(p
=0.03
)t=
1.1,
p=
0.13
128.28
(p
=0.00
)11
6
Normal
demand,
HighProfi
tKnown
0.57
(F=
46.2
,sig
=0.00
)14
7.56
(p
=0.00
)16
7.02
(p
=0.00
)t=
−4.3,p
<0.01
157.36
(p
=0.00
)18
4Unk
nown
0.27
(F=
7.8,sig
=0.00
)15
4.3
(p
=0.00
)16
2.79
(p
=0.00
)t=
1.5,
p=
0.07
160.39
(p
=0.00
)18
4
a Testthenullhypothesis
that
theaverageorderin
thefirst
20-periods
blockequals
theaverageorderin
thelast
20-periods
block.
bp-value
<0.05
forthehypothesis
that
theaverageorderforknow
ndistributio
nequals
theaverageorderfortheunknow
ndistributio
n.
on average closer to the optimal quantity and further awayfrom the mean than did subjects who knew the distribution;and (2) subjects having normal distribution order furtheraway from the mean and closer to the optimum than thosehaving uniform demand distribution. Note that it should beeasier for subjects with normal distribution to estimate themean than those with uniform distribution, since the normaldistribution is more concentrated around the mean havingabout 68% of the observed demand within one standard devi-ation from the mean, while the uniform distribution holdsonly 33% observations in that range.
Table 2 presents, for each group, the correlation betweenthe average profit in each period and the periods’ number. Wealso present the overall average profit for all the periods andin the first and last 20 period blocks.
Table 2 also compares the average profit (in the first and lastblock and in all the periods) to the maximal expected profit.The maximal expected profit is the expected profit (over allpossible demand levels based on Equation (2)) for the optimalorder. Under each average profit we present the p-value of thet-test to examine the null hypothesis that the average profit isnot different from the maximal expected profit.
Over all the treatments the average profit is significantlylower from the maximal expected profit. We do find that theaverage profit in the last 20 periods block is significantlyhigher than the average profit in the first 20-periods block.However, due to high demand variability and nonlinearity oflearning, there were no strong trends (ie correlations betweenthe average profit in each period and the period were close tozero, meaning that the profit is not growing linearly). Overallthere is no basic difference between the average profit ofsubjects with known and with unknown distribution. This isinconsistent with Hypothesis 1.
For low profit with normal distribution, we find higherprofit for the unknown distribution. This is consistent with thefinding that the average order of the subjects with unknowndistribution is closer to the optimal order than for the subjectswith known distribution. In the first block of 20 periods theaverage profit in the unknown distribution (for low profitwith normal distribution group) is positive and in the knowndistribution (for low profit with normal distribution group),negative. This may explain why the subjects in the unknowndistribution are keeping relatively low order throughout theperiods, while the subjects with known distribution, are moti-vated to change their profit from negative to positive and asa result, change their order towards the optimal order along100 periods. This is not the case in normal distribution withhigh profit, since in this case the subjects face losses lessfrequently when compared with the low profit group. Theresults in Tables 1 and 2 suggest that knowing the distributionis not a critical condition for improving the order policy. Thisis inconsistent with the theoretical approach that suggests thatmore information on the demand distribution may improvethe order policy (eg, Burnetas and Smith, 2000; Hechinget al, 2002; Xu and Hopp, 2004).
1028 Journal of the Operational Research Society Vol. 61, No. 6
Table2
Average
profi
tin
allperiod
sandin
thefirst
andlast
20-periodblocks
Dem
and
Subjects’
Correlation
First
20-
Last20
-Testing
Average
ofMaxim
alpa
ttern,
and
know
ledge
withperiod
period
block
period
block
blocks
diff.
allperiod
sexpected
profi
tlevel
ofdeman
daverage
average
Profit
pattern
Slop
e(F
value,
sig’)
Avg.profi
tAvg.profi
tt-testa ,
p-value
Avg.profi
tExpected
profi
t
Uniform
demand,
Low
profi
tKnown
0.01
(F=
0.02
,sig
=0.88
)−2
24.8
(p
=0.00
)−1
23.96c
(p
=0.00
)t=
−2.03,
p=
0.03
−59.55
(p
=0.00
)112.5
Unk
nown
−0.01
(F=
0.00
,sig
=0.95
)−2
43.33
(p
=0.00
)−2
19.63
(p
=0.00
)t=
−0.36,
p=
0.36
−87.21
(p
=0.00
)112.5
Uniform
demand,
Highprofi
tKnown
−0.001
(F=
0.00
,sig
=0.99
)62
1.28
c(p
=0.00
)70
4.72
(p
=0.00
)t=
−3.33,
p<0.01
846.19
(p
=0.00
)10
12.5
Unk
nown
0.05
(F=
0.25
,sig
=0.62
)58
7.25
(p
=0.00
)72
4.00
(p
=0.00
)t=
−6.79,
p<0.00
834.43
(p
=0.00
)10
12.5
Normal
demand,
Low
Profi
tKnown
0.07
(F=
0.45
,sig
=0.50
)−2
4.25
b(p
=0.00
)98
.78c
(p
=0.00
)t=
−3.64,
p<0.01
136.41
b(p
=0.00
)234.4
Unk
nown
−0.005
(F=
0.00
,sig
=0.96
)79
.02
(p
=0.00
)13
6.24
(p
=0.00
)t=
−3.70,
p<0.00
184.14
(p
=0.00
)234.4
Normal
demand,
HighProfi
tKnown
0.02
(F=
0.04
,sig
=0.85
)91
8.8
(p
=0.00
)99
3.99
(p
=0.00
)t=
−5.39,
p<0.01
1069
.95
(p
=0.01
)10
92.9
Unk
nown
0.07
(F=
0.52
,sig
=0.47
)88
3.85
(p
=0.00
)99
0.92
(p
=0.00
)t=
−2.88,
p<0.01
1056
.62
(p
=0.01
)10
92.9
a Testthehypothesis
that
theaverageprofi
tin
thefirst
20-periods
blockequals
theaverageprofi
tin
thelast
20-periods
block.
bp-value
<0.05
forthehypothesis
that
theaverageprofi
tforknow
ndistributio
nequals
theaverageprofi
tfortheunknow
ndistributio
n.c for
p-value
<0.1.
Next, we present in Table 3, the absolute change in orderquantities between each two successive periods. Tracking themagnitude of this change during the 100 decision periods, wecan learn about the convergence to a specific order quantity.Again we present the correlation between the average absolutechange and the period number, for the cases of first block,last block and all periods.
Table 3 shows that the correlations between the averageabsolute change and the period number are negative andsignificant, meaning that subjects reduce their average abso-lute change over 100 periods. In most of the cases, the abso-lute change in the last 20-period block is smaller than thechange in the first 20-period block. These findings are consis-tent with the convergence process suggested in Hypothesis 2.It is interesting to note that in certain price and distribu-tion combinations (low profit and uniform distribution, highprofit with normal distribution), the average absolute changeis significantly lower when the distribution is known (in theother cases there are no significant differences). A possibleexplanation is that subjects with unknown distribution haveto learn about the distribution during the 100 periods and asa result change their order more frequently.
Next, we present a regression analysis regarding the orderquantity change in each period (relative to previous period).The use of regression analysis helps us to examine the feed-back effect and test Hypotheses H3a, H3b and H3c.
The notation used is as follows:
�Qt the change in order quantity, relative to the previousperiod order.
St−1 the supply surplus in the previous period (St−1 =Order − Demand).
SI t−1 Surplus indicator (SI t−1=0: supply surplus, SI t−1=St−1: demand surplus)
K I t−1 Indicator for knowing the demand distribution:
(K I t−1 = 0-known distribution, K I t−1 = St−1-unknowndistribution).
� regression intercept (a constant having the regressionvalue when all the independent variables equal zero)
� regression coefficient for St−1 (St−1 = Order −Demand).
� regression coefficient for SI t−1 (SI t−1 = St−1 =shortage = demand surplus)
� regression coefficient for K I t−1 (K I t−1 = St−1 =unknown demand distribution)
The dependent variable is (�Qt ) and the regression equationis:
�Qt = � + � × St−1 + � × SI t−1 + � × K I t−1 (3)
Table 4 presents the regression results including the coefficientvalues for the various conditions (demand distributions andprofit levels).
AUTHOR COPY
U Benzion et al—Newsvendor and unknown distribution 1029Ta
ble3
Average
absolute
period
toperiod
orderchange
inallperiod
sandin
thefirst
andlast
20-periodblocks
Dem
andpa
ttern,
and
Subjects’
Correlation
First
20-
Last20
-Testing
Average
ofprofi
tlevel
know
ledgeof
withperiod
period
block
period
block
blocks
diff.
allperiod
sdeman
dpa
ttern
average
average
Slop
e(F
value,
sig’)
Absolutechan
geAbsolutechan
get-testa ,
p-value
Absolute
chan
ge
Uniform
demand,
Low
profi
tKnown
−0.57
(F=
46.5
,sig
=0.00
)41
.15
12.99b
t=
−2.71,
p<0.01
22.40b
Unk
nown
−0.19
(F=
3.5,sig
=0.06
)43
.35
35.45
t=
−1.82,
p=
0.04
37.82
Uniform
demand,
Highprofi
tKnown
−0.31
(F=
10.5
,sig
=0.00
)33
.88
21.63
t=
−3.37,
p<0.01
29.82
Unk
nown
−0.23
(F=
5.91
,sig
=0.02
)41
.96
30.12
t=
−1.59,
p=
0.07
35.77
Normal
demand,
Low
profi
tKnown
−0.32
(F=
11.4
,sig
=0.00
)16
.58
12.05
t=
−1.20,
p=
0.12
15.02
Unk
nown
−0.39
(F=
17.2
,sig
=0.00
)19
.13
8.69
t=
−3.61,
p<0.01
15.3
Normal
demand,
Highprofi
tKnown
−0.34
(F=
12.98,sig
=0.00
)14
.81c
9.05
t=
−2.66,
p=
0.01
11.59c
Unk
nown
−0.30
(F=
9.72
,sig
=0.00
)28
.84
14.69
t=
−1.29,
p=
0.1
16.31
a Testthehypothesis
that
theaverageabsolute
change
inthefirst
20-periods
blockequals
theaverageabsolute
change
inthelast
20-periods
block.
bp-value
<0.05
forthehypothesis
that
theaverageabsolute
change
forknow
ndistributio
nequals
theaverageabsolute
change
theunknow
ndistributio
n.c for
p-value
<0.1.
As explained below, Table 4 supports Hypotheses H3a, H3b
and H3c.
• As Hypothesis H3a indicates, the change in order quantityis negatively related to the supply surplus (�< 0). Thismeans that after a period where supply is greater thandemand, subjects tend to reduce their order (compared tothe previous period). Also, after a period of shortage (wheresupply is less than demand), subjects tend to increase theirorder.
• As Hypothesis H3b indicates, the effect of supply surplusis stronger than the effect of demand surplus, since a realloss has a stronger affect than the loss of potential gain.This is reflected by positive �, giving the demand surpluslower adjustment over the one suggested by �. The feed-back has a stronger effect on supply surplus and as aresult, the order changes more dramatically after a supplysurplus.
• As Hypothesis H3c indicates, the effect of the previousperiod is stronger for subjects that do not know the demanddistribution. This is reflected by negative �, giving thesesubjects extra adjustment over that suggested by �. Subjectswho do not know the distribution use the feedback morethan subjects who know the distribution and as a resultchange their order more decisively.
6. Conclusions
This paper compares experimental decision-making in thenewsvendor problem in cases where demand distributionis known, with cases where it is unknown. The findingsare quite surprising since demand information does notimprove the subjects’ profits, nor does it affect the extent towhich the order quantity approaches the theoretical optimalsolution. On the other hand, the results show that subjectsare affected by a surplus of perishable products. More-over, the order levels are more affected by a loss due tounsold products than a loss of potential sales realized bydemand surplus. This is consistent with the loss aversionapproach.
It is evident that proximity in time (eg, previous perioddemand) seems to dominate decisions to such an extentthat there is no significant difference in behaviour betweenthose who know the demand distribution and those who donot know it. Indeed, knowing the distribution may createbehavioural biases such as the ‘central tendency bias’ towardsthe mean.
While the large inventory systems with routine opera-tions may be fully automated, the newsvendor problem is asingle-period problem that is typically determined person-ally by the acquisition manager (see http://oam.ocs.doc.gov/,http://www.hhs.gov/oamp/, accessed 25 January 2009) or theprocurement manager (http://procurement-online.com, acce-ssed 25 January 2009) or the logistics manger (http://www.logisticsmanager.com; accessed 25 January 2009). Even if
AUTHOR COPY
1030 Journal of the Operational Research Society Vol. 61, No. 6
Note: Under each coefficient the significance is in brackets.
some of these managers are well trained (eg, http://www.dau.mil/, accessed 25 January 2009), they do not always knowthe demand distribution, and in many cases the demand isvery difficult to analyse. Therefore, in many cases, they facethe same similar situations where distribution is unknownand they have to decide how many units to order. Companiesare investing large sums in surveys and statistical analysisin order to forecast or better understand demand distribu-tion. As we have shown in this paper, knowing the demanddistribution does not necessarily improve the subject’s deci-sion in terms of expected profit. The orders are expected tobe biased by recent events and the demand mean, so thatthey do not optimize the expected profit for the long run (ifindeed the demand distribution is stationary). While suchbehaviour is sub-optimal for stationary distribution, it isvery efficient for unstable conditions where the distributionchanges significantly with time. This raises a new questionfor future research: do subjects facing a random processalways treat these processes as nonstationary (even if theyare stationary)? Some other directions for future researchmay include examining the purchase decision behaviour in anonstationary demand process, and in a Markovian demandprocess.
Acknowledgements— The authors express their gratitude to the OpenUniversity of Israel for the grant that enabled them to conduct this research.
References
Alfares KH and Elmorra HH (2005). The distribution-free newsboyproblem: Extensions to the shortage penalty case. Int J Prod Econ93–94(8): 465–477.
Atkins PWB, Wood RE and Rutgers PJ (2002). The effects of feedbackformat on dynamic decision-making. Organ Behav Hum Dec 88(2):587–604.
Benzion U, Choen Y, Peled R and Shavit T (2008). Decision-makingand the newsvendor problem: An experimental study. J Opl ResSoc 59: 1281–1287.
Bolton G and Katok E (2004). Learning-by-doing in the newsvendorproblem: A laboratory investigation of the role of experience andfeedback. Manuf Serv Opns Mngt 10(3): 519–520.
Bostian AJA, Holt CA and Smith AM (2008). The newsvendor ‘Pull-to Center’ effect: Adaptive learning in a laboratory experiment.Manuf Serv opns mngt 10(4): 590–608.
Burnetas AN and Smith CE (2000). Adaptive ordering and pricingfor perishable products. Opns Res 48(3): 436–443.
Crawford E, Huttenlocher J and Engebretson PH (2000). Categoryeffects on estimates of stimuli: Perception or reconstruction?Psychol Sci 11(4): 280–284.
Diehl E and Sterman J (1995). Effects of feedback complexity ondynamic decision-making. Organ Behav Hum Dec 62: 198–215.
Eppen GD and Iyer AV (1997). Improved fashion buying withBayesian updates. Opns Res 45(6): 805–819.
Erev I and Barron G (2001). On adaptation, maximization, andreinforcement learning among cognitive strategies. Working paper.Columbia University Business School.
Fisher M and Raman A (1996). Reducing the cost of demanduncertainty through accurate response to early sales. Opns Res44(1): 87–99.
Gallego G and Moon I (1993). The distribution free newsboy problem:Review and extensions. Int J Prod Econ 44(8): 825–834.
Heching A, Gallego G and Van Ryzin G (2002). Mark-down pricing:An empirical analysis of policies and revenue potential at oneapparel retailer. J Revenue Retailing Mngt 1(2): 139–160.
Helson H (1964). Adaptation-Level Theory. Harper & Row: New York.Johnson J, Tellis GJ and Macinnis DJ (2005). Losers, winners and
biased trades. J Cons Res 32(2): 324–329.Keser C and Paleologo G (2004). Experimental investigation of
retailer-supplier contracts: The wholesale price contract. CIRANOWorking paper.
Khouja M (1999). The single-period (news-vendor) problem:Literature review and suggestions for future research. Omega 27:537–553.
Lariviere MA and Porteus EL (1999). Stalking information: Bayesianinventory management with unobsereved lost sales.Mngt Sci 45(3):346–363.
Lovejoy WS (1990). Myopic policies for some inventory models withuncertain demand distributions. Mngt Sci 36(6): 724–738.
Lurie NH and Swaminathan JM (2008). Is timely information alwaysbetter? The effect of feedback frequency on decision making inthe newsvendor problem. Organ Behav Hum Dec 108(March):315–329.
Nahmias S (1994). Demand estimation in lost sales inventory systems.Nav Res Log 41: 739–757.
Nahmias S (2005). Production and Operations Analysis, 5th edn.McGraw-Hill: New York.
Polatoglu LH (1991). Optimal order quantity and pricing decisions insingle-period inventory systems. Int J Prod Econ 23: 175–185.
Schweitzer ME and Cachon GP (2000). Decision bias inthe newsvendor problem with a known demand distribution:Experimental evidence. Mngt Sci 46(3): 404–420.
Silver E (1993). Bayesian updating of an arbitary discrete distributionunder a special case of partial information. Stat Stoch Model 9(4):615–638.
U Benzion et al—Newsvendor and unknown distribution 1031
Thaler RH, Kahneman D and Knetsch JL (1992). The endowmenteffect, loss aversion and status quo bias. In: Thaler RH and RichardH (eds). The Winner’s Curse. Princeton University Press: Princeton,New Jersey,
Whitin TM (1955). Inventory control and price theory. Mngt Sci 2:61–80.
Xu X and Hopp WJ (2004). Dynamic pricing and inventory control:The value of demand learning. Working paper. NorthwesternUniversity, Evanston, Illinois.
Received May 2008;accepted March 2009 after two revisions
