Consider a category of product variants distinguished by some attribute such as color or

Transcription

1 O the Relatioship Betwee Ivetory Costs ad Variety Beefits i Retail Assortmets Garrett va Ryzi Siddharth Mahaja Graduate School of Busiess, Columbia Uiversity, Uris Hall, 3022 Broadway, New York, New York The Fuqua School of Busiess, Duke Uiversity, Box 90120, Durham, North Carolia gjv1@columbiaedu mahaja@dukeedu Cosider a category of product variats distiguished by some attribute such as color or flavor A retailer must costruct a assortmet for the category, ie, select a subset variats to stock ad determie purchase quatities for each offered variat We aalyze this problem usig a multiomial logit model to describe the cosumer choice process ad a ewsboy model to represet the retailer s ivetory cost We show that the optimal assortmet has a simple structure ad provide isights o how various factors affect the optimal level of assortmet variety We also develop a formal defiitio of the level of fashio i a category usig the theory of majorizatio ad examie its implicatios for category profits (Variety; Ivetory; Retailig; Cosumer Choice; Assortmet; Optimizatio; Newsboy; Fashio; Majorizatio; Multiomial Logit) 1 Itroductio ad Literature Review Merchadisig is a fudametal resposibility of retail maagers Basic merchadisig questios iclude: What variats should be stocked withi ay give category of merchadise? How much of each variat should oe buy? What is the trade-off betwee depth (large quatities of each stock keepig uit (SKU)) ad breadth (large umbers of SKUs)? How do factors such as margi ad market size affect the assortmet decisio? Ad should oe maage the buyig decisio for fashio ad basic categories differetly? While product variety has bee studied i the ecoomics ad marketig literature (see Lacaster (1990) for a survey), such assortmet questios typically have ot bee addressed directly I particular, this literature focuses primarily o variety at the Maagemet Sciece 1999 INFORMS Vol 45, No 11, November 1999 pp market level (Shuga (1989)), or variety i terms of product lie structurig (Gree ad Krieger (1985)) I cotrast, assortmet plaig requires detailed decisios o variety ad ivetory levels for colors, flavors, or sizes withi a relatively arrow category of merchadise at the idividual store level Uderstadig assortmet beefits ad costs at this microlevel requires more detailed modelig of both the choice process ad the ivetory cost A exceptio is the early work of Baumol ad Ide (1956) who cosider the joit impact of retail variety o cosumer demad ad store operatig costs The authors use a highly simplified model i which variety is a scalar quatity ad both ivetory costs ad aggregate demad are determiistic Nevertheless, their focus o modelig costbeefit trade-offs i retail variety is very much i the spirit of our work They also cosider several iterestig issues such as withi-store search costs ad customer travel costs, which we do ot model directly /99/4511/1496$ electroic ISSN

2 Broadly speakig, our work addresses the costs ad beefits of product variety as aalyzed from a joit marketig-operatios perspective This topic is receivig icreased research attetio de Groote (1994) cosiders the issue of modelig variety i the cotext of maagig a maufacturig eviromet Kekre ad Sriivasa (1990) discuss how broadeig the product lie affects market share ad productio costs MacDuffie et al (1996) use data from the Iteratioal Motor Vehicle Program to study the effect of product variety o labor productivity ad cosumer perceived product quality, while Fisher ad Itter (1996) coduct a empirical study o the impact of product variety o automobile assembly operatios Yet a distiguishig feature of retail variety decisios, as opposed to productio variety decisios, is that there are few direct costs (set-ups, chage-overs, etc) of variety for a retailer 1 Despite this fact, retailers i practice do ot stock the maximum level of variety possible eg, a typical clothig store is ot stocked with oe piece each of 20,000 differet SKUs Why? First, there are ofte costraits o display space which obviously limit the variety a retailer ca offer But more subtly, the idirect costs of stock-outs ad overstockig impose implicit costs o variety I this paper, we propose ad aalyze a theoretical model which sheds light o such variety trade-offs i retail assortmets We cosider a merchadise category made up of several product variats While the cocept of a category of merchadise is ot well defied i geeral, we have i mid a specialized category of merchadise, cosistig of alterative variats, offered at idetical retail prices ad havig idetical uit costs The assumptio of idetical prices ad costs is restrictive but realistic i certai cases (eg, color/size/flavor variety) More geerally, it represets a stylized model of reality Plausible examples fittig these assumptios iclude differet titles i a assortmet of music CDs or books; apparel, where variats are differet colors or sizes; foods products (ice cream, cereal, soup), where variats are 1 Of course, clearly there are some direct trasactios ad iformatio processig cost that are drive by the umber of SKUs a retailer carries However, i most cases these are relatively small costs compared to the cost of goods ad other operatig expeses differet flavors; or cosumer products (toothpaste, soap, cosumer electroics), where variats correspod to differet brads with similar features ad price poits A retailer must decide which subset of variats should be offered ad how much ivetory of each should be stocked Addig variats to the assortmet icreases the likelihood that cosumers will purchase somethig from the assortmet However, icludig more choice alteratives reduces the volume of demad for each variat idividually This thiig or fragmetig of total demad icreases the variability of demad for each variat, which i tur teds to drive up ivetory costs The mai thrust of this paper is to aalyze this variety trade-off, ad the explore the strategic implicatios for differet categories of merchadise ad differet operatig ad competitive eviromets Demad i our model derives from a stochastic choice process i which idividual purchase decisios are made accordig to a multiomial logit (MNL), radom utility model The supply process the retailer uses is modeled as a ewsboy problem The MNL plays a cetral role i the theory of discrete choice (Aderso et al 1992), ad has also bee successfully used i ecoometric studies to estimate demad for differetiated products (Guadagi ad Little (1983)) Likewise, the ewsboy model is a fudametal stochastic ivetory model, ad has played a cetral role sice the early developmet of ivetory theory (Arrow et al 1958) Therefore, it is importat to uderstad assortmet decisios uder these two classical models Ideed, a mai cotributio of our work is to begi mergig these two importat streams of research Below, we use this model to derive several isights o retail assortmets I doig this, we shall give the model two alterative iterpretatios The first is that it represets short-ru variety ad ivetory decisios for a specialized merchadise category withi a store (eg a buyer s decisio for the assortmet of me s dress shirts for the fall seaso) I this case, we are iterested i short-ru costs ad profits Short-ru costs would typically iclude both direct costs ad Maagemet Sciece/Vol 45, No 11, November

3 2 We are assumig here that there are approximately costat returs to scale i a store s fixed costs i this case also variable ad/or short-ru opportuity costs for the shelf space required to stock each uit Alteratively, we will also view the model as a stylized represetatio of the log-ru, strategic assortmet decisios of a etire specialty store, i which the assortmet is cosidered to be the etire rage of products carried by the store I this case, the uiform margi assumptio ca be cosidered simply as a rough approximatio of the typical margis withi a give specialty retail segmet For such strategic aalysis, oe must cosider profit ad cost over the log ru For example, uit cost should iclude the (appropriately amitorized) fixed cost of store space, because store size ca be icreased i the log ru (eg, by leasig more sellig space) 2 I this way, the model ca provide isights o strategic issues of breadth of assortmet ad store size The results i 4 o characterizig a category of merchadise are particularly useful to gai isights about which store formats are best suited to which categories of merchadise Fially, we ote that i a recet paper, Smith ad Agrawal (1996) cosider a problem that is very similar to ours However, the work differs i some importat respects First, rather tha usig the MNL, they propose their ow model of demad substitutio They also cosider other cost compoets ad impose a variety of costraits The resultig model is therefore more realistic for decisio support purposes; however, it is aalytically itractable, ad the authors must resort to umerical solutio of a oliear, iteger program I cotrast, our primary aim is to obtai theoretical isights rather tha to provide a decisio support tool As a result, we use a parsimoious model that builds o established choice ad ivetory models To keep the model simple, we do ot iclude ay direct costs or costraits o variety decisios, though direct variety cost i particular would ot be hard to icorporate Ideed, we believe it is a iterestig observatio that, eve i the absece of ay direct variety costs or costraits, there still exists a sigificat trade-off betwee ivetory costs ad variety beefits Our work, i this sese, provides a theoretical complemet to the work of Smith ad Agrawal (1996), ad there are iterestig extesios alog both lies of research The remaider of the paper is orgaized as follows: I 2, we formulate several versios of our assortmet model for a sigle merchadise category I 3, we fid a simple characterizatio of the optimal subset of variats I 32, we ivestigate how the scale of the busiess, gross profit margis ad the attractiveess of outside alteratives affect the optimal level of variety offered I 4, we compare categories similar i cost structure ad volume of busiess, but havig differet profiles of customer preferece (utility) This leads aturally to a defiitio of a fashioable merchadise category based o the theory of majorizatio (Marshall ad Olki 1979) The implicatios of our results for store operatig strategies are discussed i 42 2 Model Formulatio 21 Variats, Prices, ad Costs The set of possible variats is deoted N {1, 2,, }, ad we let S N deote the subset of variats stocked by the store Each variat is offered at a idetical retail price, p, ad has a idetical uit cost, c As metioed, we ca allow differet prices ad costs, provided the ratio p/c is the same for all variats However, for clarity of expositio we will retai the assumptio of idetical prices ad costs We assume that p is exogeously determied Edogeizig the price decisio would be a desirable extesio, but it complicates the problem cosiderably Ideed, eve joit optimizatio of ivetory ad price for a sigle variat is complex (see Hempeius (1970), Karli ad Carr (1962), Mills (1959), ad Whiti (1955)) Yet there are certaily realistic cases i which a retailer s pricig flexibility is quite limited, i particular whe price competitio is very high ad/or maufacturers exert cotrol over retail prices (maufacturer suggested retail price (MSRP)) 22 The Customer Choice Process Each potetial customer cosiders the subset S of variats offered by the store A customer may choose 1498 Maagemet Sciece/Vol 45, No 11, November 1999

4 to purchase a variat i S or she may choose ot to purchase at all This choice decisio is modeled usig the multiomial logit (MNL) model (see Luce (1959) ad Luce ad Suppes (1965)) For a recet moograph o the MNL, see Aderso et al (1992) The MNL is a geeralizatio of the classical model of utility maximizig cosumers, ad is widely used See for example Guadagi ad Little (1983) for applicatio i marketig or Be-Akiva ad Lerma (1985) for applicatio i estimatig travel demad Each idividual i the populatio associates a utility U j with the variats j S I additio, there is a o purchase optio, deoted j 0, with associated utility U 0 A idividual chooses the variat with the highest utility amog the set of available choices, {U j : j S {0}} Note that icreasig the choice set S makes it more likely that a give cosumer chooses to purchase, because a cosumer chooses to purchase oly if max {U j } U 0 O the other had, the variats are choice alteratives i the sese that if the set S is elarged by addig a variat i, a customer will switch from i to j if U i U j Thus, the selectio of the set S affects the choice decisio of each cosumer ad hece the total demad for each variat The key differece betwee the MNL ad classical utility maximizatio is that the utility U j is assumed to be uobservable, so that a idividual s choice is ucertai Moreover, utility values ca vary from idividual to idividual due to heterogeeity of preferece amog customers (the value of a size small shirt depeds o the purchaser), ad these differeces amog customers may also be uobservable a priori 3 I the MNL, these characteristics are modeled as radom compoets i the utility values The utility U j is decomposed ito two parts: oe part, deoted u j, represets the omial (expected) utility; the other part, deoted j, is a zero-mea radom variable represetig the differece betwee a idividual s actual utility, U j, ad the omial utility, u j Thus, U j u j j We ote that while the actual utility U j 3 The iterpretatio of radom utility as beig uobservable heterogeeity i a populatio of cosumers was formalized by Maski (1977); Luce (1959) ad Luce ad Suppes (1965) cosidered the radomess as beig primarily due to a idividual s iheret icosistecy i makig choice decisios assiged to variat j differs from idividual to idividual, the omial utility u j for variats is the same for all idividuals The radom compoet, j, is modeled as a Gumbel (extreme value of Type I, see, for example, Evas et al 1993) radom variable with distributio P( j x) exp( e ((x/ ) ) ) with mea zero ad variace 2 2 /6 ( is Euler s costat, 05772) Here is a positive costat, with a higher value of correspodig to a higher degree of heterogeeity amog the populatio The assumptio of zero mea is without loss of geerality, while the Gumbel distributio is used primarily because it is closed uder maximizatio (Gumbel 1958) The assumptio of the Gumbel distributio i the MNL model, while restrictive, leads to a simple form of the choice probabilities as show below i Equatio (1) Give the choices S ad the o-purchase optio, a cosumer chooses the optio with the largest utility Let q j P(U j max{u i : i S {0}}) deote the probability that variat j has the maximum utility A stadard result of the MNL yields (see Aderso et al 1992, p 39): q j where, for j 0,,, wedefie v j i S v i v 0 (1) v j e uj/ (2) We refer to the quatities v j as prefereces Note that v j is icreasig i the omial utility u j, so a high value of v j correspods to variats with higher expected utility Also, as metioed above, (1) reflects the fact that variats are choice alteratives, i the sese that the choice probabilities deped o S For example, if we add a variat l S to S, the deomiator i (1) icreases by v l ; hece, the probability that a customer selects oe of the origial variats decreases However, the overall probability that a customer selects somethig from the set S, q j, icreases Below, we work directly with the value v j ad let v deote the vector (v 0, v 1,, v ) The MNL possesses a somewhat restrictive property kow as the idepedece from irrelevat alteratives (IIA) property Roughly, IIA states that the ratio of choice probabilities for alteratives i ad j is idepe- Maagemet Sciece/Vol 45, No 11, November

5 det of the choice set S cotaiig the alteratives The IIA property is ot realistic if the choice set cotais alteratives that ca be grouped such that alteratives withi a group are more similar tha alteratives outside the group, because addig a ew alterative reduces the probability of choosig similar alteratives more tha dissimilar oes Therefore, the MNL model should be restricted to choice sets cotaiig alteratives that are, i some sese, equally dissimilar (eg, differet colors or differet sizes, but ot differet color-size combiatios) Despite these well-kow limitatios, however, the MNL model remais widely used ad therefore serves as a useful startig poit to uderstad ivetory uder cosumer choice We ext impose two assumptios cocerig the customer choice process, which we call static choice assumptios (These same static choice assumptios are made i the work of Smith ad Agrawal 1996) Assumptio (A1) Customers choose based oly o kowledge of the set S, ad they have o kowledge of the ivetory status of the variats i S Assumptio (A2) If a customer selects a variat i S ad the store does ot have it i stock, the customer does ot udertake a secod choice, ad the sale is lost Note that as a result of Assumptios A1 ad A2, variats are ot substitutes i the sese that a customer will dyamically substitute oe variat for aother if their first choice is out of stock Rather, it is oly a customer s iitial choice that is iflueced by the set of alteratives that are offered While these are perhaps ot the most realistic assumptios, they provide a reasoable startig poit ad, importatly, greatly simplify a otherwise quite complex demad process I particular, assumig that a customer s choice is made from stock o had itroduces a complicated depedecy betwee customer choice decisios ad ivetory status (We address a problem i which customers choose dyamically based o the o-had stock i Mahaja ad va Ryzi (1998)) However, there are retail eviromets where Assumptios A1 ad A2 may reasoably approximate certai types of customer behavior For example, A1 certaily holds for customers of a catalog retailer who do ot kow the ivetory status of items prior to orderig I a store settig, if customers choose based o ispectio of floor models, the status of the o-had ivetory is also ot readily discerable As aother justificatio of A1, Smith ad Agrawal (1996) argue that i may retail settigs opportuity costs of shortages are quite high, ad hece optimal ivetory decisios hedge strogly agaist stock outs (see 24) I such situatios, stock outs are relatively rare evets, ad hece most customers are ideed choosig from the full assortmet 4 As for Assumptio A2, it may be plausible if oe views customers as relatively uiformed prior to visitig the store Upo ispectig the choice set S (eg, the store s floor models), they lear about the variats offered ad idetify oe that they like best If the store the turs out to be out of stock, the ewly-iformed customer elects to go elsewhere to obtai his or her preferred variat rather tha settlig for aother alterative Thus, the very act of ispectig S chages the iformatio a customer has about his or her possible choices, makig the first ad secod choice fudametally differet Aother settig i which Assumptios A1 ad A2 are plausible approximatios is whe oe views the customer s iitial choice as a store-visit decisio That is, S is viewed as a store s strategic variety decisio (eg, the collectio of brads they routiely offer), ad customers are viewed as makig a choice whether or ot to visit a store (eg, select a variat i S) based oly o their kowledge of S (I this case, the o-purchase utility should be iterpreted as the maximum utility amog all other available alteratives, icludig visitig other stores or ot shoppig) Sice customers are ot physically i the store whe they make their store-visit decisio, they are uaware of the curret ivetory status of idividual brads (A1) Upo visitig the store, they lear if their preferred brad is i stock; if it is ot, they go elsewhere or decide ot to purchase at all (A2) Of course, each of these explaatios, i tur, im- 4 However, it is ot clear, a priori, whether or ot these rare stock-out evets truly have a egligible effect o ivetory ad variety decisios 1500 Maagemet Sciece/Vol 45, No 11, November 1999

6 plies other behavioral assumptios that may be equally upalatable For example, the uiformedcustomer explaatio assumes strog preferece oce a favorite variat is idetified The store-visit explaatio assumes o learig about log-ru ivetory availability, ad so forth Nevertheless, Assumptios A1 ad A2 are partially defesible, ad, as we show below, they lead to ituitively satisfyig isights o retail assortmets 23 Two Models of the Aggregate Demad The above descriptio details how a idividual customer makes his or her decisio Aggregate demad, however, is determied by the collectio of decisios made by all customers We cosider two models of aggregate demad I the idepedet populatio model, aggregate demad is the result of a series of idepedet choices from a heterogeeous populatio of cosumers I the tred-followig populatio model, aggregate demad is the result of a series of depedet choices from a homogeeous populatio Each model is appropriate i differet retail settigs, as we discuss below (See Marvel ad Peck 1995 for a similar pair of demad models) 231 The Idepedet Populatio Model I the idepedet populatio model, each customer assigs utilities to the variats i the subset S offered by the store based o idepedet samples of the MNL model described above The utility U ij that customer i assigs to variat j is give by U ij u j ij where the { ij ; i 1} are iid radom variables Hece, the differet customers make iid choices amog the variats offered by the retailer accordig to the MNL probabilities q j defied i (1) Sice customers are heterogeeous ad idepedet, the observatio of oe customer s choice reveals o additioal iformatio about the choice of subsequet customers The mea umber of customers makig choice decisios per uit time is Let the umber of customers selectig variat j per uit time be deoted Y j, ad ote that by Assumptios A1 ad A2, EY j q j We assume Y j is ormally distributed with a stadard deviatio that is a power fuctio of the mea; that is, the stadard deviatio is ( q j ), where 0 ad 0 1 The assumptio that 0 ad 0 1 implies that the coefficiet of variatio is decreasig i If the coefficiet of variatio is costat, the assortmet profit is simply proportioal to Wedoot cosider the case where the coefficiet of variatio is icreasig i volume; however, this does ot seem to be a very atural case We assume that 1 q j q j, so that the probability of havig egative demad is small A atural special case of this model is whe the total volume of customers visitig the store is Poisso with mea The the total demad for variat j, Y j, is also Poisso with mea q j, because the MNL results i a thiig of the aggregate Poisso demad I this case, a ormal approximatio to the Poisso distributio yields 1 ad 1 2 This approximatio is justified if EY j 1 The idepedet purchase model is useful for basic product categories, i which aggregate cosumer preferece is relatively stable ad the primary ucertaity is over idividual prefereces for color, size, or flavor that iheretly vary from oe customer to the ext 232 The Tred-Followig Populatio Model The tred-followig populatio model assumes a opposite extreme A fixed umber of customers make choice decisios Each customer has idetical valuatios of the utilities for the variats, ad this uiform set of utilities is determied by a sigle sample of the MNL model So, { ij 1} As a result, each customer makes the same choice Hece, oce the outcome of oe customer s choice is observed, the decisio of subsequet customers is perfectly predictable However, these commo utilities are ot observable to the retailer prior to makig the assortmet decisio, ad the retailer therefore has a icetive to hedge agaist this ucertaity by stockig more tha oe variat The demad for variat j, deoted Y j, uder this model is (a scaled) Beroulli radom variable with probability mass fuctio, P Y j y q j y, 1 q j y 0, 0 otherwise This models a extreme case of tred-followig behavior For example, it is appropriate as a stylized model of color or style variety for tredy apparel (3) Maagemet Sciece/Vol 45, No 11, November

7 Next, we aalyze ivetory cost uder both these models of aggregate demad 24 The Supply Processes We cosider the supply model We do ot cosider costraits o the umber of uits stocked Such costraits may affect the structure of the optimal assortmet i the short ru ad are a worthy topic of research O the other had, over a log time horizo such costraits ca be relaxed at a cost, either by reallocatig store space amog categories or by expadig the sellig space of the store This is the perspective we take below The supply model is aalyzed uder both the idepedet ad tred-followig demad models Both cases yield qualitatively similar profit fuctios but provide differet isights ito the assortmet decisio 241 Supply Cost Uder the Idepedet Populatio Model We assume without loss of geerality that the sales seaso lasts oe uit of time The umber of customers selectig variat j, Y j, is ormally distributed with mea q j ad stadard deviatio ( q j ) as described above For each uit ot sold, we assume the loss to the retailer is the cost c For each uit short, the opportuity cost to the retailer is the loss i margi, p c We assume that there is o salvage value, though this assumptio ca easily be relaxed deote the umber of uits of variat j Let x j stocked by the store ad x x 1,,x The expected profit made by the retailer o the jth variat is the E[ p mi { x j, Y j } cx j ], so the maximum expected profit give S ad v, deoted I (S, v), is I S, v max x 0 E p mi x j, Y j cx j (4) The optimal profit depeds o S ad v through their effect o the choice probabilities q j, which i tur determie the demad Y j Uder the assumptio of a ormal distributio, the optimal procuremet level, deoted x* j, is give by where x* j q j z q j, (5) z 1 1 c p, (6) ad ( z) deotes the cdf of a stadard ormal z2 /2 radom variable (We let ( z) (1/ 2 )e deote the stadard ormal desity fuctio) Defie x* ( x* 1,, x* ) to be the optimal vector of stockig levels, where we adopt the covetio that x* j 0, j S Substitutig this expressio for x* j, i (4) we fid I S, v p c q j p e z 2 /2 q j, (7) 2 where we have used the fact that for a stadard ormal radom variable Z, E(Z z) ( z) z(1 ( z)) 242 Supply Cost Uder the Tred-Followig Populatio Model I this case, a costat umber of customers visit the store ad each makes idetical utility valuatios The valuatios are ukow to the retailer at the time the assortmet decisios are made, ad the distributio of demad for variat j, Y j, is give by (3), as described above The expected profit from variat j give x j is agai E[ p mi{ x j, Y j } cx j ] Oe ca easily determie that the optimal expected profit, deoted T (S, v), is T S, v pq j c, (8) usig the fact that the optimal procuremet level is x* j 0ifpq j c ad x* j if pq j c Note that either demad model ca also be iterpreted as the sigleperiod cost i a periodic review system with lost sales as i the work of Smith ad Agrawal (1996) 3 The Optimal Assortmet Problem With these differet profit fuctios i had, we ca formulate the optimal assortmet selectio problem The problem admits a two level hierarchy; amely, at the lower level the retailer selects the optimal stockig levels give S ad v, yieldig the profits (7) or (8) At a higher level, the retailer chooses the best set of variats S by solvig max S N (S, v), where the appropriate profit fuctio is used Let S* deote a 1502 Maagemet Sciece/Vol 45, No 11, November 1999

8 optimal solutio to this problem The the correspodig pair (S*, x*) defies a optimal assortmet for the store 31 The Structure of the Optimal Assortmet Without loss of geerality, let the variats be ordered accordig to decreasig values of v j, so that v 1 v 2 v To see that the optimal subset S* is ot trivial, cosider the case where 8 ad v 1 v 2 10 while v 8 1 It turs out that the set {1, 8} ca be better tha the set {1, 2} That is, it may be better to pair a popular variat with a very upopular variat rather tha offerig two popular variats Oe or the other could domiate depedig o the cost structure ad the vector of prefereces of the variats, v To see why, ote that the variat that gets paired with Variat 1 will add some icremetal sales O the other had, its presece will divert demad from Variat 1 I the extreme case, itroducig a secod variat provides a egligible icrease i overall sales ad oly results i splittig total demad betwee the two variats The resultig icrease i variability requires more total ivetory ad results i lower expected profits Ideed, from example like this oe ca show that the most profitable assortmet of k variats eed ot cosist of the most popular k variats Despite such couterituitive examples, the optimal set S* turs out to be structurally quite simple To show this, however, we eed the followig lemma (proof i Appedix): Lemma 1 Let 0 v 1 ad defie the followig fuctios: f v j v 0 (9) g I p c v j p e z 2 /2 2 1 v j v j v 0 (10) g T pv j cf p cf (11) The the fuctios h I ( ) g I ( )/f( ) ad h T ( ) g T ( )/f( ), are both quasi-covex i The fuctio h I ( ) (similarly for h T ( )) represets the profit associated with addig a variat with preferece to the existig set S Whe 0, the fuctio reduces to I (S, v) ad whe v j for j S, it equals I (S { j}, v) Because the fuctio is quasicovex, it follows that if we wat to maximize h I ( ) o the closed iterval [0, max {v j : j S}], the maximum is achieved at the ed poits of the iterval So the profit is maximized by either ot addig ay more variats or by addig the variat with highest preferece amog those ot icluded i S The same holds for the tred followig demad case This property allows us to prove our mai result o the structure of the optimal assortmet Theorem 1 Let A i {1,, i} for 1 i The for each of the assortmet problems defied above, there exists a S* {A 1,,A } that maximizes store profits Proof The proof is by costructio ad is the same for both cases of the assortmet problem Let S* be a optimal subset with cardiality m, ad let the objects i S* be deoted v* j with v* 1 v* 2 v* m 0 If S* A m the theorem holds trivially If S* A m, the there exists a v j S* such that v j v* m However, from the quasi covexity of the fuctios h( ) i Lemma 1, it must also be true that we ca either remove v* m or exchage it for v j v* m without decreasig profits Redefie S* to be this ew set ad repeat the procedure Evetually, oe arrives at a optimal set S* {A 1,, A } I words, the optimal assortmet ca be restricted to oe of possible types We simply choose the best i variats, where 1 i To gai some ituitio about the result, recosider our earlier compariso of the sets {1, 2} ad {1, 8} We ca ow say the followig: If {1, 8} is ideed better tha {1, 2}, the it must be true that {1} is the optimal subset for the merchadise category That is, the oly reaso that pairig a popular item with a very upopular item is preferred is because it is i some sese closer to offerig oly the popular variat More geerally, while the most profitable assortmet of k variats eed ot cosist of the most popular k vari- Maagemet Sciece/Vol 45, No 11, November

9 ats, the optimal assortmet does cosists of the most popular k variats for some value of k 32 Implicatios The simple structure of the optimal assortmet makes it easy to defie the level of variety a store offers That is, more variety correspods directly to a higher idex i amog the possible subsets {A 1,,A } This raises iterestig questios about what affects the level of variety offered i a optimal assortmet The aswers to these questios are resolved by the followig theorem (proof i Appedix): Theorem 2 For all i 1, a) ( A i 1, v) ( A i, v) for sufficietly high sellig price p b) ( A i 1, v) ( A i, v) for sufficietly low opurchase preferece v 0 c) ( A i 1, v) ( A i, v) (idepedet populatio cases oly) for sufficietly high store volume Part (a) states that high margis create a icetive to stock higher levels of variety This is ituitive, sice as margis icrease the risk of lost sales domiates the risk of overstockig Therefore, a wider variety is offered to miimize the likelihood of customers ot purchasig The result suggests that as margis rise i a category, existig retailers will have a icetive to broade the rage of merchadise they carry 5 Part (b) cosiders the effect of o-purchase utility o variety A high o-purchase utility could correspod to a product category that is somewhat frivolous (eg, souveirs, toys, or jewelry), i which ot purchasig is a commo outcome Alteratively, a high o-purchase utility ca represet the existece of may attractive outside alteratives, icludig other stores i close proximity carryig similar merchadise I either case, as the o-purchase utility declies, the prospect of losig a purchase to a exteral optio decreases while the threat of withi-assortmet caibalizatio icreases Therefore, it is i the retailer s iterest to 5 We ote that Shuga (1989) shows that it is possible for lower levels of variety to be optimal for higher priced categories whe demad is highly price sesitive The reaso is that the reduced purchase volume from high prices ca be eough to offset the positive effect of higher margis Part (a) assumes store volume ad purchase probabilities are fixed decrease the breadth of the assortmet For example, this result predicts that stores i less competitive retail eviromets will ted to offer lower variety tha similar stores i more competitive retail eviromets Part (c) implies that, i the idepedet demad case, as the volume of busiess icreases, high variety becomes icreasigly more profitable, ad a store will carry all variats for a sufficietly high volume That is, there are scale ecoomies i offerig variety ad, as traffic grows, a store ot oly stocks more uits of each variat, but also teds to stock more variats The reaso higher variety becomes more desirable i the idepedet populatio case is due to the risk poolig iheret i a large umber of idepedet purchase decisios As the volume of purchase decisios goes up, the relative amout of overstockig ad uderstockig error goes dow ad the cost of havig fragmeted purchase decisios become relatively smaller Hece, more variety becomes profitable Super stores are plausible examples of this sort of scale effect Ideed, the results suggest a reaso why the super store format is ecoomically viable There may be a atural, positive feedback i this format; the large variety attracts a high volume of traffic which i tur allows the super store to profitably offer a large variety of merchadise But this scale ecoomy oly exists for the idepedet purchase case I the tred-followig case, (8) shows that profit is directly proportioal to That is, volume has o effect o the relative profitability of differet levels of assortmet variety Hece, there are o scale ecoomies; a large store simply places bigger bets ad eds up makig proportioately bigger losses While this is clearly a stylized model, the geeral coclusios appear cosistet with retailig practices For example, cosider the categories of merchadise offered by the super stores They ted to be those that experiece highly fragmeted but largely idepedet demad, eg Borders (books, magazies), Home Depot (hardware, home remodelig), Staples (office supplies), ad Toys R Us (childre s toys) I cotrast, small stores, idepedet boutique clothig stores for example, ted to thrive with merchadise 1504 Maagemet Sciece/Vol 45, No 11, November 1999

10 categories that have strog tred-followig customers Here, store scale is ot a impedimet to profitability Ideed, i the tred-followig case, competitive advatage derives from havig itimate kowledge of local markets ad the buyig flexibility to respod to that iformatio This is precisely the advatage small, idepedet stores have 4 Defiig Fashio Usig Majorizatio Orderig Thus far, our aalysis has bee restricted to a sigle category We ext cosider comparisos amog differet merchadise categories We show below that the eveess of the preferece vector v provides a atural measure of fashio ad this otio ca be made precise usig the theory of majorizatio (Marshall ad Olki 1979) 41 Characterizig a Fashioable Merchadise Category For a vector x, let [i] deote a permutatio of the idices {1, 2,, } satisfyig x [1] x [2] x [] We the have the followig defiitio of the partial order based o majorizatio: Defiitio 1 For x, y, x is said to be majorized by y, x y (y majorizes x), if i 1 x [i] i 1 y [i], ad for k k all k 1,, 1, i 1 x [i] i 1 y [i] Ituitively, a oegative vector y that majorizes x teds to have more of its mass cocetrated i a few compoets I our problem, majorizatio provides a appropriate measure for the degree of fragmetatio i cosumer preferece withi a give category of merchadise Ideed, we propose the followig defiitio: Defiitio 2 A merchadise category v is said to be more fashioable tha w if (v 1,,v ) (w 1,,w ) To compare two categories v ad w usig Defiitio 2, we must make the assumptio that they have the same umber of variats, ad that i 1 v i i 1 w i However, these assumptios are ot restrictive Ideed, oe ca add a arbitrary umber of variats j with preferece values v j 0 without alterig the problem Also, oe ca scale all values of v or w (ad the o-purchase preferece, v 0 or w 0 ) by a arbitrary multiplier without affectig the resultig choice probabilities ad thus esure that i 1 v i i 1 w i, so the secod assumptio is ot restrictive either Fially, we emphasize that majorizatio oly produces a partial orderig o vectors, so eve after these adjustmets it is etirely possible that two vectors v ad w caot be ordered accordig to Defiitio 2 Defiitio 2 says that, for fashio categories, prefereces are more evely spread out across variats Alteratively, i the tred-followig demad case, the degree of fragmetatio of the retailer s prior iformatio of cosumer prefereces for variats is higher for the fashio category tha for the basic category Note also that oe ca iterpret v ad w either as two categories at the same poit i time or oe category observed at two differet poits i time I the latter case, oe ca the meaigfully talk of a category becomig more fashioable over time Pashigia s (1988) study of bed sheets (white vs facy) is a example of such a category He shows that the fractio of sales of white sheets has declied over time, while the fractio of facy sheets, which icludes all other colors ad patters, has rise Such behavior correspods, roughly, to a icrease i fashio over time accordig to our defiitio Note that it is commo to thik of tredy ad fashioable as beig syoymous However, i our termiology, fashio correspods to fragmetatio i preferece (or prior iformatio o preferece), while tredy correspods to correlated purchase behavior amog cosumers We require the followig stadard result (see Marshall ad Olki 1979): Lemma 2 If g: 3 is a covex fuctio ad x y the, i 1 g( x i ) i 1 g( y i ) We ext show that the optimal profit obtaied from two merchadise categories is itimately related to the majorizatio orderig Theorem 3 Cosider two merchadise categories, v ad w, that have idetical cost structures ( p ad c), demad volumes,, ad equal o purchase utilities (v 0 w 0 ) Defie r ad l such that ( A r, v) max S N (S, v) ad ( A l, w) max S N (S, w), where ca be Maagemet Sciece/Vol 45, No 11, November

11 either of the assortmet profit fuctios, (7) or (8) The if v w, ( A r, v) ( A l, w) Proof We first prove the result for the idepedet demad fuctio (7) ad the for the fuctio (8) The proofs proceed as follows: First, we fid r such that A r is the optimal subset for the fashio category v We the costruct a feasible solutio set for the basic category w that yields at least as much profit asthe optimal profit for the fashio category v From the defiitio of majorizatio, we ca fid a t r such that, t 1 r j 1 w j w t j 1 v j, where 0 1 Defie the covex fuctio, g x p c x p e z 2 /2 x, L 2 L r where L j 1 v j v 0,0 1, ad let w be the r-dimesioal vector such that w (w 1, w 2,, r r w t 1, w t, 0,, 0) Note that j 1 w j j 1 v j Sice (v 1,, v r ) w, usig Lemma 2, it follows that r j 1 r g v j g w j (12) j 1 Now sice g(0) 0, the right had side of (12) is simply h I ( w t ), where, h I is the quasicovex fuctio defied i Lemma 1 with subset of variats S A t 1 Usig (12) ad the quasicovexity of h I,it follows that r I A r, v j 1 r g v j g w j j 1 max I A t 1, w, I A t, w I A l, w This completes the proof for the idepedet populatio case For the tred-followig populatio case (profit fuctio (8)), we defie the covex fuctio g T ( x) ( /L)(px cl) r, where L j 1 v j v 0, as defied above Replacig g by g T, ad h I by h T i the argumet above, we coclude that T ( A r, v) T ( A l, w) I cotrast to the previous assumptios o v ad w, the coditios equal ad v 0 w 0 are restrictive Essetially, these two assumptios serve to equalize the relative demad ( market potetial ) of each category This follows sice if we have two assortmets, S ad T with i S v i i T w i (equally attractive) ad they face the same mea umber of customers, the the mea umber of customers who decide to purchase is also the same That is, ( i S v i )/( i S v i v 0 ) ( i T w i )/( i T w i w 0 ) This ormalizatio is ecessary to isolate the effect of fashio o profits because demad volume strogly affects a category s profits Oe should therefore view Theorem 3 as primarily a theoretical compariso The profits of two categories are affected, i geeral, by a combiatio of volume, gross margi ad fashio effects I a real-world compariso of two categories, v 0 w 0 is equivalet to assumig that the outside choice alteratives for each category are equally attractive The assumptio of equal may approximate a case where two categories are stocked at the same store ad is viewed as a measure of store traffic Accordig to the above theorem, if oe merchadise category is more fashioable tha aother, the, all other thigs beig equal, the optimal profit of the fashio category will be lower tha that of the more basic category The ituitive reaso for this is that the risk of ivetory overage ad uderage is higher due to the higher fragmetatio of cosumer purchase decisios i the fashio category Thus, eve uder optimal variety ad stockig decisios, the fashio category is less profitable at a give price However, oe might questio whether basic categories, uder similar demad volumes, could reasoably be expected to be more profitable i the log ru Ideed, oe would expect the market to evetually compesate retailers for the added costs of fashio categories by allowig higher market prices That is, retailers recogizig the higher profitability of a basic category would have a icetive to add the category to their store The resultig icrease i the umber of retailers offerig this category would, i tur, ted to icrease price competitio util the basic category becomes less profitable ad further ew etrats are discouraged A simple approximatio of this effect is obtaied by freeig up the price variable ad askig the questio: At what prices do two categories have equal profits? 1506 Maagemet Sciece/Vol 45, No 11, November 1999

12 The followig corollary shows that these equalizig prices are ordered if the categories are ordered accordig to our defiitio of fashio (proof i Appedix): Corollary 1 Assume v w, all other parameters except price, p, are idetical ad that prices are adjusted to satisfy the equilibrium profit coditio, max S N S, v max S N S, w e, where is either of the assortmet profit fuctios (7) or (8) ad e is a arbitrary, oegative equilibrium profit level Let the equilibrium prices be deoted p v ad p w, respectively, for categories v ad w The p v p w Clearly, the equilibrium profit hypothesis behid this result represets a highly simplistic view of retail markets I reality, factors such as locatio, assortmet, store image ad customer service eable a retail store to differetiate itself from its competitors I additio, categories may ot have the same costs for space, fixtures, sigage, etc Also, customers may shop stores to purchase a basket of products from differet categories based o a shoppig list (see, for example, Bell et al (1997)), ad stores may differetiate themselves based o the total basket they offer Despite these myriad limitatios, Corollary 1 does provide a simple ad ituitively appealig explaatio of the fact (see Pashigia (1988) for empirical evidece) that fashio goods ted to have higher margis tha basic goods I short, Corollary 1 simply suggests that higher gross margis may serve to compesate retailers for the icreased ivetory risks iduced by the highly fragmeted purchase choices of the fashio category (Lazear (1986) proposes a alterative theory for this effect based o differeces i the variability of reservatio prices for each category) 42 Implicatios The equilibrium price results of Corollary 1 ad the scale ecoomy results of Theorem 2 have iterestig implicatios for a retailer s operatig strategy Corollary 1 suggests that equilibrium margis ted to be higher for fashio categories High margis, i tur, may justify chages i the way goods are supplied to stores I particular, they may justify usig fast ad potetially expesive logistics processes (eg, air freight) to repleish stocks of popular variats i seaso rather tha stockig to forecasts While this may be a viable strategy to maage a fashio category with depedet (tred-followig) purchase behavior, Theorem 2 suggests that there are scale ecoomies to offerig variety i the idepedet (o-tred-followig) populatio case As a result, fast repleishmet may ot be a viable logistics strategy for this type of category because competig retailers ca mitigate fashio risks usig large-scale store formats (or cetralized warehouses) without resortig to expesive logistics optios Dvorak ad Paassche (1996), writig i McKisey Quarterly, describe several retailer s logistical strategies that are cosistet with these coclusios The authors describe the operatios of oe (aoymous) fast-to-market high fashio retailer as follows: High fashio is a high-risk busiess [Most] retailers usually air-ship oly those items that have uexpectedly ru out But this retailer air-ships all items that have sold better tha expected durig tests Noe of this speed is cheap, but the expese is more tha covered by higher sales ad fewer mark-dows For a retailer of casual cotto clothes, which are moderately priced fashio items ad correspod more closely to the idepedet (o-tred-followig) case, they describe a very differet strategy: What is importat is to make sure stores are always stocked with the right color, size ad desig Productio lead times are log [ad] price costraits also rule out shippig goods by air The thrust of the logistics strategy is therefore to achieve ot speed, but a smooth, seamless trasitio from oe wave of goods to the ext To cope with variatios i demad for specific colors, sizes, or desigs a secod strad of the retailer s strategy itroduces flexibility Regioal warehouses, located close eough to stores to allow cost-effective, frequet repleishmet, provide buffer stocks ready to fill gaps o the store s shelves This descriptio is cosistet with the scale sesitive characteristics of the idepedet populatio case Appedix Proof of Lemma 1 We use the followig result from Magasaria (1969): The fuctio g /f is quasicovex o X if (i) g is covex ad f 0 for all v X ad (ii) f is liear o X The fuctio f defied i (9) is liear i It is easy to show that the fuctios g I ad g T defied by (10) ad (11), respectively, are Maagemet Sciece/Vol 45, No 11, November

13 covex i Thus, the fuctios h of the lemma satisfy the above coditios Proof of Theorem 2 From (1), we have v j q j v j v 0 Defie q j t j, whe S A i ad q j w j, whe S A i 1 The from (7), i the idepedet demad case, I ( A i 1, v) I ( A i, v) if ad oly if, i 1 i p w j t c j 1 j 1 j p e z 2 /2 i 1 i w j t j 2 j 1 j 1 Because i 1 i j 1 w j j 1 t j, this yields 1 c p 2 1 e z 2 /2 j 1 i 1 w i j j 1 i 1 i j 1 w j j 1 t j t j (13) Similarly, for the tred-followig demad case, we have by (8) that T ( A i 1, v) T ( A i, v) implies i 1 j 1 pw j c j 1 i pt j c (14) To prove part (a), we ote that by (6), z 1 (1 (c/p)) is a icreasig fuctio of p So the left had side of (13), is a icreasig fuctio of p, while the right had side is idepedet of p Thus, for sufficietly high p, A i 1 is better tha A i For (14), ote that for sufficietly high p, the iequality becomes p( i 1 i j 1 w j j 1 t j ) c, which is satisfied for large eough p sice i 1 i j 1 w j j 1 t j This completes the proof of part (a) For provig part (b), ote that q j 3 v j / v j as v 0 3 0, ad hece i 1 i j 1 w j 3 j 1 t j As a result, the deomiator of the right had side of both (13) teds to zero, while oe ca easily show the umerators ted to a positive costat, which establishes the result Reversig the iequality i (14) ad usig the fact that i 1 j 1 w j 3 i j 1 t j as v the claim holds i the tred-followig case as well Part (c) is true because the left had side of both (13) is icreasig i Proof of Corollary 1 We eed to show that (S, v) is a icreasig fuctio of p Hece, ( A r, v) max S A1 A (S, v), is also a icreasig fuctio of p Combiig this property with Theorem 3, it the follows that the equilibrium prices satisfy the stated coditio For the idepedet demad case, differetiatig I (S, v) with respect to p, we obtai the followig coditio for I (S, v) tobea icreasig fuctio of p, 1 1 c 1 p c 1 p c 2 1 q j q j 1 p z 2 /2 e where prime deotes the derivative with respect to p Uder our assumptio that 1 q j q j, the left had side is greater tha 1, while the right had side is less tha 1, sice [ 1 (1 (c/p))] 0, so the above coditio is always satisfied For the tred followig demad case, it is easily see that T (S, v) is a icreasig fuctio of p Refereces Aderso, S P, A de Palma, J F Thisse 1992 Discrete Choice Theory of Product Differetiatio The MIT Press, Cambridge, MA Arrow, K J, S Karli, H Scarf, eds 1958 Studies i the Mathematical Theory of Ivetory ad Productio Staford Uiversity Press, CA Baumol, W J, E A Ide 1956 Variety i retailig Maagemet Sci Bell, D R, T Ho, C Tag 1997 Determiig where to shop: Fixed ad variable costs of shoppig Workig Paper, The Aderso School, UCLA, Los Ageles, CA Be-Akiva, M, S Lerma 1985 Discrete Choice Aalysis: Theory ad Applicatio to Predict Travel Demad The MIT Press, Cambridge, MA de Groote, X 1994 Flexibility ad marketig/maufacturig coordiatio Iterat J Productio Ecoom Dvorak, R E, F va Paassche 1996 Retail logistics: Oe size does t fit all McKisey Quart Evas, M, N Hastigs, B Peacock 1993 Statistical Distributios Joh Wiley, New York Fisher, M L, C D Itter 1996 The impact of product variety o automobile assembly operatios Workig Paper, Operatios ad Iformatio Maagemet, The Wharto School, Philadelphia, PA Gree, P, A Krieger 1985 Models ad heuristics for product lie selectio Marketig Sci 4, 1 19 Guadagi, P M, J D C Little 1983 A logit model of brad choice calibrated o scaer data Marketig Sci Gumbel, E J 1958 Statistics of Extremes Columbia Uiversity Press, New York Hempeius, A L 1970 Moopoly with Radom Demad Rotterdam Uiversity Press, Rotterdam, The Netherlads Karli, S, C R Carr 1962 Prices ad optimal ivetory policies Arrow, K J, S Karli, ad H Scarf eds Studies i Applied Probability ad Maagemet Sciece Staford Uiversity Press, Staford, Califoria Kekre, S, K Sriivasa 1990 Broader product lie: A ecessity to achieve success? Maagemet Sci Lacaster, K 1990 The ecoomics of product variety: A survey Marketig Sci Lariviere, M, E L Porteus 1996 The ice cream stockig problem Preseted i Sessio SC06, INFORMS, Washigto DC, May 5, 1996 Lazear, E P 1986 Retail pricig ad clearace sales Amer Ecoom Rev Luce, R 1959 Idividual Choice Behaviour: A Theoretical Aalysis Wiley, New York, P Suppes 1965 Preferece, utility ad subjective probability 1508 Maagemet Sciece/Vol 45, No 11, November 1999

14 R Luce, R Bush, ad E Galater, eds Hadbook of Mathematical Psychology Vol 3, Wiley, New York MacDuffie, J P, K Sethurama, M L Fisher 1996 Product variety ad maufacturig performace: Evidece from the iteratioal automotive assembly plat study Maagemet Sci Mahaja, S, G va Ryzi 1998 Stockig retail assortmets uder dyamic cosumer substitutio To appear, Oper Res Magasaria, O L 1969 Noliear Programmig McGraw-Hill, New York Maski, C 1977 The structure of radom utility models Theory ad Decisios Marshall, A W, I Olki 1979 Iequalities: Theory of Majorizatio ad its Applicatio Academic Press, New York Marvel, H P, J Peck 1995 Demad ucertaity ad returs policies Iterat Ecoom Rev Mills, E S 1959 Ucertaity ad price theory Quart J Ecoom Pashigia, P B 1988 Demad ucertaity ad sales: A study of fashio ad markdow pricig Amer Ecoom Rev Shuga, S M 1989 Product assortmet i triopoly Maagemet Sci Smith, S A, N Agrawal 1996 Maagemet of multi-item retail ivetory systems with demad substitutio Workig Paper, Decisio ad Iformatio Scieces, Sata Clara Uiversity, Sata Clara, CA Whiti, T M 1955 Ivetory cotrol ad price theory Maagemet Sci Accepted by Awi Federgrue; received October 1, 1997 This paper has bee with the authors 5 moths ad 10 days for 3 revisios Maagemet Sciece/Vol 45, No 11, November