Inventory Policy Implications of On-Line Customer Purchase Behavior

Transcription

1 Invenoy Polcy Implcaons of On-Lne Cusome Puchase Behavo Hay Goenevel Smon School of Busness Unvesy of Rochese Rochese, NY 4627 Panab Majumde Fuqua School of Busness Duke Unvesy Duham, NC Absac In hs pape we wll examne some mplcaons of onlne daa fo a classcal opeaons managemen model, vs. he Economc Ode Quany model. Cusome wang behavo on ndvdual odes (whch occu dung sockous) foms he bass fo evaluang he poenal backodes. The poenal aacon of educng nvenoy holdng coss mus be balanced wh he loss due o los sales. We clealy delneae he condons unde whch s pofable o sock ou evey odeng cycle, and he condons unde whch he adonal economc ode quany model sll holds. In ode o allow paccal applcaon of he model, we develop a numbe of dffeen appoaches o he poblem of esmang he backode funcon fom avalable on-lne ansacon daa.. Inoducon The emegence of web based onlne ealng has changed he manne n whch cusome behavo can be acked and pofled. A ecen acle poflng Yahoo pesens he poenal and pfalls n ulzng hs daa fo new busness models [0]. New ools ae avalable ha allow one o age pomoons and poduc offengs o cusomes pedsposed o puchasng hem as ndcaed by cusomes obseved pas behavo. The avalably of exensve cusome puchase behavo daa s changng he way n whch dffeen funconal aeas can fomulae opmal accs and saeges. Ths has pacula elevance fo makeng. Fo nsance; Acve Buyes Gude helps vsos selec models of vaous consume eleconc ems (see hp:// They evaluae he cusome's pefeences fo abues by conducng a faco analyss usng dynamcally geneaed flash cad compasons. Smlaly, n ealng, some ses ack he onlne bowsng behavo and have a hsoy of ems ha a pacula cusome fnds neesng. Ths daa s used o sugges smla ems fo puchase. Amazon (hp:// s a popula eale wh exemely sophscaed cusome managemen. Wha Amazon.com has done s nven and mplemen a model fo neacng wh mllons of cusomes, one a a me (see []). Cusomes love Amazon no because offes he lowes pces-- doesn'--bu because he expeence has been cafed so caefully ha mos of us acually enjoy (see [6]). Thee ae mplcaons fo nvenoy managemen as well. As an example, hs daa s also beng used o sugges alenae ems n case a pacula em s ou of sock, o o povde an updaed esmae fo he wang me ll new sock aves. Ths allows he cusome o make nfomed choces abou subsuon and/o backodeng. Ths behavo s now vsble o he eale, and can fom he bass fo bee busness decsons. The challenge s no meely o sugges eplacemen ems o povde wang mes, bu o use opeaonal polces and managemen saeges o bee suppo hs neacon beween he fm and he cusome. In hs pape we wll examne some mplcaons of onlne daa fo a classcal opeaons managemen model, vs. he Economc Ode Quany model. Cusome wang behavo on ndvdual odes (whch occu dung sockous) foms he bass fo evaluang he poenal backodes. The poenal aacon of educng nvenoy holdng coss mus be balanced wh he loss due o los sales. We clealy delneae he condons unde whch s pofable o sock ou evey odeng cycle, and he condons unde whch he adonal economc ode quany model sll holds. In ode o allow paccal applcaon of he model, we develop a numbe of dffeen appoaches o he poblem of esmang he backode funcon fom avalable on-lne ansacon daa. 2. Leaue Suvey Thee s a sgnfcan body of ecen leaue on onlne ealng. The nceasng populay of he web has led o an exploson of eseach elaed o eal subsuon behavo (see [2] [3] [8] [4] [2]). These papes llusae he effec of dealed cusome behavo daa on he Opeaons Managemen leaue. The EOQ fomula was nally deved n [], and he elaed leaue s volumnous, see e.g. [9] [23]. The EOQ model wh backodes has been dscussed by [9] as well. They consde a lnea backode funcon, and deve a polcy based on a backode cos pe un backodeed pe me un. A numbe of auhos have exended he EOQ model by consdeng backodes. Backode models whee only a facon of he demand s backodeed when a sockou occus ae examned n [5] [6] [20] [22]. Ou model easly handles hese knds of mxues beween backodes

2 (pen-up demand) and los sales. Ou model does no have explc coss of no sasfyng demand fom on-hand sock. Insead, some of he demand dung he sockou peod uns no los sales, whch educes evenue fo he eale. Backodes have been he subjec of exensve eseach n he conex of sochasc models. An ode pon ode quany model wh a mxue of backodes and los sales s nvesgaed n [8]. Los sales wll occu when backodes exceed a cean heshold level. Geneal backode coss n sochasc nvenoy models ae dscussed n [5]. The cos sucue of backodes has been exended by some auhos. Invenoy polces when he backode coss have fxed and popoonal componens ae examned n [4]. Anohe appoach o modelng he effec of sock-ous can be found n [7], whee s assumed ha he demand ae s nfluenced by backodes. Howeve, o he bes of ou knowledge, he appoach o modelng he effec of sockous pesened n ou model s new, as s he chaacezaon of condons unde whch peodc avalably s an opmal polcy. Thee s also exensve eseach on peshable nvenoes, when he sock a hand may decay, o become obsolee. One epesenave pape [3] negaes he sockng decson wh backodeng. Whle we do no have peshable nvenoes, we have peshable backodes, n he sense ha backodes gow less han popoonaely wh he elapsed sockou me. Fnally, n [2] a ecen pape uses an exponenal pen-up demand funcon n he conex of sochasc lead mes. Lee focuses on algohms o oban he opmal sockou peod whou devng any sucual esuls. Ths s he only pape we have found so fa whch examnes a non-lnea pen-up demand funcon. In conas, ou pape pesens he sucual esuls fo any pen-up demand funcon, as well as he paamec condons unde whch hee dffeen polces ae opmal: he EOQ sockng polcy, he peodc sockng polcy and he no-nvenoy polcy. Many EOQ nvenoy models focus on nvenoy as a cos, as he ognal wok [] dd. The decson make conols a cos cene, and aemps o mnmze coss whle delveng a cean sevce level. Ofen, hs leads o suaons whee he opmal decson s unpofable, as aleady obseved n [22]. In conas, ou objecve s o maxmze pofs, whch allows us o choose no o opeae f s no pofable. A dffeen way o hnk of he peodc avalably polces n hs pape s as a bdge beween classcal supply fom nvenoy polces (whee a eale aemps o fll demand fom on-hand sock) and sockless ealng polces (whee he eale acs puely as an ode ake who passes he demand on o hs supple). Indeed, boh exemes ae specal cases of ou geneal model. We use faconal pogammng o solve he dscouned cash flow opmzaon. A good noducon o non-lnea faconal pogammng can be found n [7]. Ou algohm s smla, bu specfcally cafed fo ou needs. 3. The Invenoy Model We make all he sandad assumpons of he Economc Ode Quany (EOQ) model, bu wh some addonal assumpons egadng wha happens when he eale uns ou of sock (whch s no allowed n he EOQ model). The eale expeences a consan deemnsc demand of D uns of poduc pe un me fo a gven poduc. The eale can place an ode a any me fo any quany of poduc desed, and he delvey wll be made afe a consan lead me. Thee s a fxed ode cos of F pe ode placed, a puchasng cos of c pe un puchased, non-fnancal nvenoy holdng cos of h pe un kep n nvenoy pe un me, and snce we wll use connuous me dscounng, a dscoun ae of pe me un. The eale sells ems a a pce of p pe un sold. The eale seeks o maxmze oal dscouned pof. Conay o he sandad EOQ assumpons, we assume ha when he eale uns ou of sock, some of he demand s backodeed and some of he demand s los. In pacula, f he eale has been ou of sock fo me uns, he oal pen-up o backodeed demand equals D (). The funcon D () s assumed o sasfy he followng popees: D (0) = 0, D () s concave and non-deceasng, and D () D () ( sd ) fo all > s 0. Of couse f D () s dffeenable, hese condons can be saed as 0 D ( ) D, and D () s non-nceasng. I wll be convenen o use a scaled veson of he pen-up demand funcon, so we defne k () = D ()/ D. Ths funcon wll heefoe sasfy he followng assumpons: k(0) = 0, k ( ) s non-deceasng and concave. Defne he subdffeenal k () of k a as he se of all values x such ha ks () k () + xs ( ) fo all s 0. In pacula he assumpons mply ha k s almos eveywhee dffeenable, and ha he subdffeenal k always exss and s deceasng n he obvous sense,.e., f < s, hen k () k ( s) fo evey k () k() and k () s k() s. Usng sandad dynamc pogammng agumens, s easy o see ha a cyclc polcy s opmal. Fo convenence, we wll assume ha he eale sas wh no sock, and wh no backodes. The eale mus hen decde, fs, how long o allow backodes o buld up befoe akng delvey (), and second, how long (x) o keep he em n sock afe ha. Ehe of hese me peods can be zeo: f = 0, he classcal EOQ suaon esuls, f x = 0, he eale follows a sockless polcy,.e., he eale s bascally jus an ode ake. Hence dung a sngle cycle he followng evens occu. A me 0 (he sa of he cycle), he eale uns ou of sock. Pen-up demand accumulaes dung he nex me uns. A ha momen, an ode of Q uns of poduc s eceved. The pen-up demand D () s sasfed as soon as he delvey of poduc s made a me, so he esulng

3 nvenoy s Q D(). Noe ha doesn make sense o no sasfy all he pen-up demand a me : he eale could jus ode D () Q moe uns and sell hem mmedaely a me. Ths would lead o a posve cash flow of ( p c)( D( ) Q ) a me. If hs pen-up demand s no sasfed a me, some of may un no los sales, and he ne pesen value of he emande s educed because of dscounng. Dung he neval (, + x), he nvenoy of Q D() s dawn down a a ae of D by he egula demand, and a me + x he eale uns ou of sock agan. Hence x = ( Q D() )/ D= Q/ D k() () o Q= D( x+ k() ). (2) Fnally, o avod pahologcal cases, we assume p > c 0, F > 0, h > The Opmal Invenoy Polcy We fs deve an expesson fo he ne pesen value assocaed wh a polcy ( x, ). We assume ha a me 0 he eale uns ou of sock. Then dung he neval (0, ) hee ae no cash flows. A me, he eale puchases D( x+ k() ) uns of poduc a dscouned cos e ( cd( x+ k( ) ) + F ). A he same me, he eale sells Dk() uns wh a dscouned evenue of e pdk(). Dung he neval (, + x ) he eale sells D uns of poduc pe me peod, whch leads o oal dscouned evenue of + x y = D ( x pd e dy p e e ). Dung hs me neval nvenoy deceases fom Dx o 0, whch means ha he dscouned holdng cos ove hs peod equals + x y ( ) ( x D e hd + x y e dy = h e x + ). Defne α = D( p c ), β = ( D/ )( p+ h/ ), and = ( D/ )( c+ h/ ). Then he dscouned value of all cash flows dung he fs cycle can be expessed as x e ( αk() F x+ β( e )). Fnally he oal dscouned value of all cash flows fo an nfne hozon s he sngle cycle value mulpled by ( x) /( e + ), whch equals x αk () + β( e ) x F π( x, ) =. (3) x e e I s no had o show ha n maxmzng (3) we can confne ouselves o he egon Ψ = {( x, ) : x 0, 0,( x, ) (0,0)} Hence we wan o fnd G = sup { π(,):(,) x x Ψ } (4) Defne he funcon x f( x,, G) = αk( ) + β( e ) x F x Ge ( e ), hen we can we G = sup { G: f( x,, G) = 0, ( x, ) Ψ }. (5) We wll explo he fac ha f s (jonly) concave n x and o develop an effcen algohm fo solvng poblem (4). Befoe we can chaaceze he opmal polcy, we need some lemmas. Lemma G < β. Poof: Fs, noe ha fo all 0 we have αk ( ) α= ( β ) ( β )( e ), and fo all x 0 we have ( x x β e ) x ( β )( e ), hence fo evey ( x, ) Ψwe have x ( β )( e e ) F π( x, ) < β. x e e Lemma 2 Le 0 < G < β. The poblem zg = max { f( x,, G) : x 0, 0} (6) has a unque soluon ( xg, G) gven by xg = ln( β G) ln, (7) 0 f αk (0) G, G = he unque value fo whch (8) αk () = Ge ohewse. (If k s no connuously dffeenable, hen k () should be nepeed as a suably chosen subdffeenal of k a ). Poof: Noe ha f s sepaable n x and. I s hus saghfowad o show ha f s jonly concave n x and, and ha (7) and (8) ae he fs ode condons fo a global maxmum. Lemma 3 Le 0 < G < β. Then sgn( z ) = sgn( G G). G Poof: We consde he obvous hee cases sepaaely. G xg Case : z G > 0. Defne ε = zg /( e e ) > 0, hen x f( xg, G, G+ ε) = f( xg, G, G) ε( e e ) = 0, and * hence G π( xg, G) = G+ ε> G. Case 2: z G = 0. Then f( xg, G, G ) = 0, and snce f(0,0, G) = F < 0, hs mples ( xg, G) 0, so ( xg, G) Ψ and G π( xg, G) = G. Fuhemoe, fo abay ( x, ) Ψ we know f( x,, G) 0, and snce f s scly deceasng n G on Ψ, hs mples by (5) ha G G. We conclude ha n hs case G = G. Case 3: z G < 0. We need o show G < G. If G 0, we ae done, snce he lemma assumes G > 0. So assume ε= π ( x, ) > 0 fo some ( x, ) Ψ. We wll nex show ha Φ = {( x, ) Ψ: π( x, ) ε} s compac. Noe Φ = {( x, ) Ψ: f( xε,, ) 0}, and snce f (0,0, ε ) = F < 0, Φ = {( x, ) : f( x,, ε) 0, x 0, 0}. Snce f s connuous n x and, follows ha Φ s closed. Usng ha f s sepaable and concave and ha f ends o when ehe x o ends o, s no had o show ha Φ s bounded. Hence

4 G = sup{ π( x, ):( x, ) Ψ} = sup{ π( x, ):( x, ) Φ} = π( x, ) fo some ( x, ) Ψ. Bu snce f( x,, G) z G < 0, hs mples G < G. The hee cases ogehe mply he lemma. Lemma 4 Defne k( ) = lm k( ), hen αk( ) + β (+ ln( β/ )) > F G > 0. (9) Poof: Noe f( x,,0) αk( ) + β (+ ln( β/ )) F fo evey ( x, ) Ψ, and ha he bound s gh (hs s easly shown usng he same appoach as he poof of lemma 2). Hence f he condon on he lef n (9) s ue, hee exss a ( x, ) Ψ such ha f( x,,0) > 0. Bu hs mples G π( x, ) > 0. If on he ohe hand he condon on he lef n (9) s false, hen f( x,,0) 0 fo evey ( x, ) Ψ, and hs mples G 0. Theoem If αk( ) + β (+ ln( β/ )) > F, hen he opmal polcy ( x, ) and G sasfy β G x = ln ( ), (0) ( β ) e k ( ) = G f ( β ) k (0) > G, () = 0 ohewse, β αk ( ) Ge + β F ln G = 0. (2) ( ) If αk( ) + β (+ ln( β/ )) F, hen G = 0 (.e., s opmal neve o puchase he em). Poof: Lemmas 2 and 3 mply ha he values x, and G sasfy equaons (0), () and x x αk ( ) Ge β( e ) Ge + + x F= 0. Subsung (0) no hs las equaon gves (2). The las saemen follows fom lemma 4 and he fac ha π( x, ) as fo any fxed x. Theoem gves exac opmaly condons. An mmedae consequence s Coollay Assume αk( ) + β (+ ln( β/ )) > F, and le G sasfy he equaon β G G+ ln ( ) = β F, (3) hen = 0 f and only f k (0) G. α Coollay gves he pecse condons unde whch he classcal EOQ polcy (of no plannng o un ou of sock) s opmal. Theoem nspes he followng vey effcen algohm o denfy he opmal polcy and s cos. To smplfy noaon, defne ˆ(, ) ( ln β G f G = f,, G ) β Ge β G = αk () + ( ln ) F, Algohm A: IF lm ˆ f(,0) 0 : = ; G : = 0 THEN ELSEIF f ˆ(0,( β ) k (0)) < 0 THEN le be he unque soluon o he equaon f ˆ(,( β ) e k ()) = 0 (4) (f k s no dffeenable, hen k () s a suably chosen subgaden of k a ); G : = ( β ) e k ( ) ELSE : = 0; le G be he soluon o he equaon (n G) f ˆ(0, G ) = 0 ( 5) END IF x : = ln β G To pove he coecness of he algohm, we need he followng echncal lemma. Lemma 5 The funcon () ˆ h = f(,( β ) e k ()) s nceasng n, and by appopae choces of k () assumes all values beween f ˆ(0,( β ) k (0)) and lm ˆ f(,0) 0 as vaes fom 0 o. Poof: Noe ha we can we h () = h(, k ()) + h3( h2(, k'()), (6) whee e h( ) = α( k() k'() ) F, ( ) h2 (, k ()) = β ( β ) e k (), h3( x) = ( x ln x). Noe ha h 2 (, k ()) s nceasng n, and can assume all values beween ( β ( β ) k (0))/ and β/ by choosng appopae values fo he subgaden k () of k a. Noe also ha h ( ) 3 x s nceasng n x fo x. We conclude ha he second em of (6) s nceasng n and can assume all values beween h3( h2(0, k'(0)) and h ( β/ ). 3 Nex, we wll show ha h (, k ()) s nceasng n as well. Choose s> 0. Noe ha he concavy of k mples ha ks () k () ( s k ) () s and k () s k () fo evey choce of subgadens of k a s and. Hence h(, s k ()) s h(, k ()) α s e e = ks ( ) + k () s k () k () s e e ( s ) k () s + k () s ( ) ( s ) e ( e ( ) )( s ) k ( s) s = ( s )( e ) k ( s) 0, x whee he nex o las nequaly follows snce e x fo all x. Hence h (, k ()) s nceasng n and assumes all values beween F (nclusve) and

5 αk( ) F (he lae value possbly excluded). Theoem 2 Algohm A coecly solves poblem (4). Poof: The IF condon of he algohm s smply he condon unde whch he eale can make a posve NPV on he em (see he las pa of heoem ), so = ; G = 0 f s sasfed. Defne G = ( β ) k (0) hen by lemma 2, = 0 G and x = (ln( β G ) ln ) / G, and he ELSEIF condon of he algohm s equvalen o z < 0 G whch n un s equvalen o G < G by lemma 3. Hence f he ELSEIF condon s sasfed, G = ( β ) e k ( ) by () and subsung hs no (2) gves (4). If s no sasfed, = 0 by () and subsung hs no (2) gves (5). Fnally, lemma 5 guaanees ha (4) has a unque soluon (, k ( )) wheneve needs o be solved, and s easy o show ha he same holds fo (5). A few commens ae n ode. Fs, noe ha he opmaly of planned sockous (he ELSEIF es n he algohm) depends only on k (0), he facon of demand ha s no mmedaely los a he momen he eale fs uns ou of sock. The condon s always sasfed when k (0) =, so some sockous wll always be opmal n hs case. On he ohe hand, when k (0) <, he opmaly of havng some sockous depends on he value of he fxed ode coss F, and fo small enough values of F wll be opmal o avod sockous alogehe. When k (0) = 0, he IF and ELSEIF ess n he algohm acually concde (noe ha k (0) = 0 mples k () 0 snce k s nomalzed, non-deceasng and concave), and hence ehe he eale doesn sell he em a all, o he wans o avod sockous alogehe. The decson on whehe o sell he em a all (he IF es n he algohm) depends only on k( ), he maxmum (nomalzed) facon of demand ha s no los when he eale has no sock fo a vey long me. Hence he specfc fom of he funcon k plays a vey lmed ole n makng hese basc decsons. Of couse n ode o deve an opmal sockng polcy n he case ha planned sockous ae opmal, moe nfomaon abou he pen-up demand funcon s necessay. Noe ha equaon (5) can be solved effcenly by usng Newon s mehod o fnd he unque oo y of he equaon y ln y= + F/ (a convenen sang pon /2 s y= + (2 F / ), he soluon o he appoxmae 2 equaon obaned fom ln y y + ( y ) ), and 2 hen calculang G = β y. Equaon (4) can of couse easly be solved usng bsecon. When k s dffeenable, he condons unde whch he LHS of (4) s concave nvolve he hd devave of k, so a smple mplemenaon of Newon s mehod s no ecommended. When k s pece-wse lnea, he LHS of (4) s convex n beween beakpons of k (), whle s concave n k () a he beakpons. Hence once he beakpon o segmen whch conans has been denfed, a couple of Newon-Raphson seps wll quckly yeld he soluon wh gea accuacy. 5. Esmang he backode funcon In hs secon we un o he ssue of mplemenng he model n pacce. In pacula, one needs o esmae he backode funcon. We wll skech a numbe of dffeen appoaches o hs esmaon poblem ha could be useful n vaous ccumsances. Assume ha a web eale uses a web se ha povdes poenal cusomes wh nfomaon on how long wll be unl he eale can shp he poduc. Cusomes hen ehe place an ode fo one un of he poduc (whch wll be shpped as soon as becomes avalable fo shppng), o hey ex whou odeng. The eale collecs he followng nfomaon elaed o a pacula poduc of nees. Ove a peod of M days, he eale esponded o n nques ha he shppng delay would be days, and hese nques esuled ns sales, = 0,,..., N. Noe ha even when he poduc s avalable fo shpmen, some poenal cusomes sll don place an ode. Fom hs daa, we wsh o esmae he undelyng demand ae D and he (scaled) pen-up demand funcon k (). We assume ha nques ae geneaed accodng o some saonay pocess wh a ae of µ pe day. Le he andom vaable S j denoe he numbe of uns sold esulng fom he j-h nquy ha occus when he shppng delay s days. We assume ha { Sj : j=,2,...} ae d fandom vaables wh mean τ. If esmaes τˆ fo τ ( = 0,..., N) ae avalable, one can esmae he scaled pen-up demand funcon by lnea nepolaon on hese esmaes: k( ) = τˆ, = 0,,2,..., N. ˆ τ 0 = The saghfowad appoach o esmang he aveage numbe of uns sold pe nquy as a funcon of he numbe of days unl shpmen s of couse nave τ ˆ = s / n. The poblem wh hs appoach s ha he esulng scaled pen-up demand funcon need no be concave, snce s que possble ha n he avalable daa s / n < sj / njfo some j>. Of couse s coune-nuve (o say he leas) ha a longe shppng delay would lead o a hghe expeced numbe of uns sold pe nquy, so we wll eque ha ou esmaes sasfy 0 τˆ ˆ ˆ N τn... τ0. In he emande of hs secon we dscuss seveal ways o handle hs esmaon poblem. 5. Paamee Esmaon fom a Specfc Funconal Fom The exponenal pen-up demand funcon s gven by D D () = ( e ). θ

6 Ths fom ases f we assume ha pen-up demand sasfes he dffeenal equaon D () = θd D() wh sang condon D (0) = 0. Hee s he ae a whch pen-up demand dsspaes (e.g. because s sasfed by a compeo o because subsues ae acqued by consumes), and θ s he facon of demand ha s no mmedaely los when he eale uns ou of sock. So hee s exponenal decay of pen-up demand, compaable o adoacve decay. I s no had o show ha he exponenal pen-up demand funcon sasfes ou assumpons f > 0 and 0 θ. To be complee, we have k ( ) = θ( e ) /, k( ) = θ/, k ( ) = θe and k (0) = θ. A smple appoach o esmang values fo and θ s o mnmze N 2 n 0 s θ k ( e ) s 0 n, = = he squaed sum of eos beween he values of k calculaed usng he naïve esmaes τˆ nave and he values of k calculaed wh he specfed funcon. An alenave specfc funconal fom s he logahmc funcon gven by θd D () = ln( + ). I s easy o vefy ha hs mples k () = ( θ/ )ln( + ), k( ) =, k ( ) = θ/( + ) and k (0) = θ. 5.2 The Cusome Uly Appoach Defne he aveage esdual uly R= U p, whee U s he uly ha he aveage cusome deves fom he poduc f s mmedaely avalable, and p s he puchase pce. In addon, a cusome has a dsuly fo wang fo he em, so le bx ( ) be he backode dsuly, f he cusome has o wa x me uns. Then a cusome decdes o puchase he em f and only f R b( x) + ε > 0, whee ε s a cusome specfc andom vaable wh mean 0, and x s he me he cusome has o wa fo he em. Fo specfc funconal foms of bx ( ) one can use pob o log egesson analyss o esmae P{ R b( x) + ε > 0}, and he nomalzed pen-up demand funcon s calculaed as k () = P ε > bx ( ) Rdx 0 ( ) Clealy, snce P( ε > bx ( ) R) s non-nceasng n x and canno exceed he value of, he esulng funcon k () sasfes he sandad assumpons of secon 2. If he funcon bx ( ) s assumed o be lnea, hen pob esmaon leads o k () = [ Φ ( ν) Φ ( ν)], k () = Φ( ν), k (0) = Φ ( ν), k( ) = Φ ( ν ), whee Φ denoes he sandad nomal cumulave dsbuon funcon, and Φ ( y) = Φ ( y) y( Φ( y)) s he famla sandad nomal lnea loss funcon (see e.g. [23, page 458]), and ν and ae paamees obaned fom he pob egesson. Clealy, he esmaed paamee needs o be posve fo hs appoach o make sense. If he funcon bx ( ) s assumed o be lnea, hen log esmaon leads o ν ν k () = [ln( + e) ln( + e ] ν k () = /( + e ) ν k( ) = ln( + e ) whee ν and ae paamees obaned fom he log egesson and agan we need o be posve. The uly appoach has as addonal advanages ha on can coec fo pce changes (and possbly addonal envonmenal facos) n he esmaon, as well as say somehng abou he mpac of pce changes (and possbly addonal envonmenal facos) on opmal polcy, pof, ec. 5.3 Pecewse Lnea Appoxmaon Usng Isoonc Regesson We can use a consaned maxmum-lkelhood appoach o oban esmaes τˆ fo he values τ ha sasfy he consans 0 τˆ ˆ ˆ N τn... τ0. Ths amouns o fndng he soonc egesson of he pons (, s / n) wh weghs n (see [9, p. 32]). The esul s a pece-wse lnea appoxmaon wh beakpons 0 = 0 < < < n, and slopes θ0 > θ > > θ n 0. If we defne j () = max{: } fo any 0, hen j () k () = ( 0 ) ( ), + θ = j k () = θ j() f j(), k () = [ θ, θ ] f =. 6. Implcaons j() j() j() Fo on-lne eales, hs s an deal oppouny o examne he nvenoy managemen polces n he conex of cusome behavo. Mos ses aleady have he echnology o collec exemely dealed daa abou he eal-me shoppng behavo of vsos. The polcy algohm and esmaon echnques pesened hee lend hemselves o auomaon que easly, n he same manne ha smple EOQ and sockng level calculaons ae sandad ounes n many exsng enepse sofwae soluons. They can be used decly o change he sockng polcy as well as paamees on an auomac bass, o moe ealscally, be used o sugges changes o a decson-make. In case of sockous, many savvy eales ls smla ems o pomoe subsuon, and also povde he esockng dae o allow he cusome o wa. Thus, subsuon and wang by cusomes allows hem o aac a lage make (o a a lowe cos) han f hey had a fxed assomen of socked ems. Hence hee s a gowng end o allow cusomes o make he own subsuon/wa/leave decsons. Whaeve be he (aguable) benefs of hs polcy n

7 he bcks wold of ealng, hee ae many moe benefs fo he clcks wold of on-lne ealng. In many cases cusomes ae condoned o a shppng delay, and may be wllng o wa fo a sho me n ode o ge he fs choce. The on-lne wold povdes a wealh of ackng nfomaon abou cusome pefeences, sockou and subsuon behavo. In addon, s possble o ensue ha when socks ave, all backlogs ae nsananeously cleaed. To hese oganzaons an undesandng of cusome wang behavo s ccal n ode o mplemen pofable nemen sockng polces - clealy a song agumen fo sysemac analyss of sockou behavo (as well as subsuon behavo) n he on-lne ealng ndusy. 6. Enepse Resouce Plannng (ERP) Sofwae Exsng ERP sofwae oday has pe-pogammed ounes ha allow calculaon of nvenoy paamees, e.g. safey sock (usng pevous ode daa), eode pons, and ode quanes. We have ncluded he algohms ha ae mplemenable as addonal ounes o fnd ou he opmal sockng polcy fo ems of nees, as well as an algohm o calculae maxmum lkelhood esmaos (n lnea unnng me). Ths allows neesed fms o decly mplemen he polces ha we descbe hee. Jus as mos sockng level calculaon pocedues ae used no o decly change values bu o sugges changes o a decson make, we envsage ha he addonal algohms pesened hee can be used o sugges changes n polcy o he decson make. Ths allows fo a degee of flexbly ha s cuenly no avalable n sockng polcy decsons. 6.2 Ohe Consdeaons The peodc avalably polces ha we have descbed have some ohe mplcaons as well. Snce we decease he aveage nvenoy holdng me n such polces (as compaed o a pue EOQ polcy), hs has an mpac on accounng measues, lke Reun on Equy. Fo onlne fms hs may be an mpoan consdeaon f he pefomance, ncenves and even exsence depends upon common accounng measues of pofably. Snce each ode s lage hen he pue EOQ polcy ode, hee may be quany dscouns o shppng economes ha kck n f a peodc avalably polcy s adoped. Whle hs s no explcly modeled n hs pape, s plausble ha quany dscouns o shppng economes makes such a polcy even moe aacve. Whle we have looked a he nvenoy polcy ha can capalze on he nceased vsbly of cusome behavo, s equally mpoan o have busness pocesses ha can suppo. Fo nsance, s necessay o have logscs funcons ha can negae well wh a egula acvy of fllng backodes. Onlne busness models have o be suppoed by all funconal aeas n andem. I s ceanly no enough o meely apply a peodc avalably nvenoy polcy a he waehouse. 7. Concluson Ou model s jus a sa howeve. In he fuue, sophscaed on-lne eales wll use he cusome daa o model cusome behavo n he face of sockous, pce changes, avalable subsues n he eale's collecon, and possbly compeve acons. Hence hee wll be an nceasng need fo models ha negae he adonal emphass on eplenshmen decsons wh ohe aspecs of logscs sysems. In hs pape we sa fom a dffeen pespecve on sockous and he cos of backodes. We have exended he EOQ model n an nuve manne whch shows he specfc condons unde whch he EOQ model s opmal, and condons unde whch EOQ s no opmal (o even pofable) bu peodc avalably s opmal and pofable. Ths epesens a smple bu poweful exenson of hs classc nvenoy model. Poenal exensons o hs eseach nclude consdeng many ems o be socked a one locaon, he ssue of subsuon beween mulple ems, and he case of compeon beween eales usng dffeen sockng polces. Each of hese occus n pacce, and s elevan o boh eseach and ndusy. Thee ae mpoan applcaons o he feld of on-lne eal nvenoy, and hee ae mplcaons fo all nvenoy sysems, ncludng supply chans. In any suaon whee a sockou causes a paal non-lnea loss of sales we asse ha s mpoan o asses he pofably of peodc avalably. Refeences [] Alsop, Sewa. I m Beng on Amazon.com, Foune Magazne, Apl 30, 200, 48. [2] Anupnd, Rav; Dada, Maqbool & Gupa, Sachn. Esmaon of Consume Demand wh Sock-Ou Based Subsuon: An Applcaon o Vendng Machne Poducs, Makeng Scence, 998, 7, [3] Bassok, Yehuda; Anupnd, Rav & Akella, Ram. Sngle-Peod Mulpoduc Invenoy Models wh Subsuon, Opeaons Reseach, 999, 47, [4] Cenkaya, S. & Pala, M. Opmal Myopc Polcy fo a sochasc nvenoy poblem wh fxed and popoonal backode coss, EJOR, 998, 0, [5] Chen, Fanguo & Zheng, Yu-Sheng. Invenoy models wh geneal backode coss, EJOR, 993, 65, [6] Colvn, Geoffey. Shakng Hands On The Web, Foune Magazne, May 4, 200, 54. [7] Dnkelbach, W. On Non-Lnea Faconal Pogammng, Managemen Scence, 967, 3, [8] Ens, Rcado & Kouvels, Panagos. The Effecs of Sellng Packaged Goods on Invenoy Decsons, Managemen Scence, 999, 45, [9] Hadley, G. & Whn, T. M. Analyss of Invenoy Sysems,

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