Prcng Overage and Underage Penaltes for Inventory wth Contnuous Replenshment and Compound Renewal emand va Martngale Methods RAF -Jun-3 - comments welcome, do not cte or dstrbute wthout permsson Junmn Sh ept. of Manageral Scences Robnson College of Busness Georga State Unversty 35 Broad St. Atlanta, GA 333 jmsh@gsu.edu Benjamn Melamed ept. of Supply Chan Management and Maretng Scences Rutgers Busness School -- Newar and New Brunswc 94 Rocafeller Rd. Pscataway, NJ 8554 melamed@busness.rutgers.edu Mchael N. Katehas ept. of Management Scence and Informaton Systems Rutgers Busness School -- Newar and New Brunswc Washngton Par Newar, NJ 7 mn@andromeda.rutgers.edu Abstract hs paper addresses the nventory penalty prcng problem under rs-neutral valuaton prncple. In partcular, we consder an nventory drven by a constant replenshment rate and compound renewal demand stream (d demand nter-arrval tme and d demand quantty), and subject to overage penaltes and underage penaltes. Our soluton approach treats the nventory penaltes of overage and underage as a seres of Amercan optons and constructs auxlary martngale processes n term of nventory process. We provde a necessary and suffcent martngale condton for general compound renewal demands. For the case wth compound Posson demands, explct expressons of penalty functons for the underage and overage are obaned. Keywords and Phrases: Inventory penalty, martngale, overage, underage, rs-neutral prcng.
. INROUCION Consder an nventory wth constant replenshment rate and compound renewal demand stream, and subject to an overage penalty and/or an underage penalty, both assessed aganst the frm. An overage penalty s assessed whenever the nventory level up-crosses a prescrbed target level (or capacty level), whle an underage penalty s assessed whenever a demand arrval encounters a stocout where any porton of the demand cannot be satsfed from on-hand nventory. We defne the perpetual nventory penalty varable as the dscounted total penalty over an nfnte tme horzon. he standard valuaton problem n such nventory system s the computaton of the expected perpetual nventory penalty. he problem we address n ths paper s the nverse valuaton problem of devsng proper dynamc penalty functons for assessng overage and/or underage penaltes, such that the expected perpetual nventory penalty equals a prescrbed value (nterpreted as the present value of the penalty budget gven to an nventory manager). We menton that underage penaltes present a specal problem. Whereas overage mpacts the nventory owner alone, underage mpacts both the frm (e.g., lost sales, lost goodwll) and the customer (e.g., reduced servce levels wth potental busness losses ncurred by customers). Whle t s reasonable to assess overage penaltes n terms of nventory holdng costs, t s much more challengng to assess the frm s loss of goodwll and the costs of long-term and short-term busness loss sustaned by the customer. Smlarly, t s also hard to valuate the full extent of overage penalty because the suppler may sustan addtonal losses stemmng from the enforcement of supply contracts (e.g., returned tems). he latter are typcally gnored n supply chan studes [Zpn ()]. Most nventory optmzaton models capture overage and underage penaltes as mportant cost components, but they rarely address the practcal problem of valuatng them usng real-lfe metrcs. he tradtonal approach s to assume a gven fxed stocout cost that s ncurred as a tangble loss by each stocout [Ernst and Powell (995)]. However, ntangble losses such as goodwll are not explctly captured [Walter and Grabner (975)]. o the best of our nowledge, the lterature on the valuaton nventory penaltes s relatvely sparse. [Schwartz (966), (97)] ponted out that the effect of loss of goodwll s characterzed by the fact that a dsapponted customer reacts n the future to change hs purchasng habts. hus the nature of the effect s that subsequent demand s perturbed, a phenomenon qute dfferent from havng an mmedate penalty cost mposed, whch was defned as perturbed demand. [Schwartz (966), (97)] analyzed some propertes of perturbed demand. In addton, some numercal studes have been carred out recently to evaluate nventory penaltes. For example, n order to estmate the value of lost-sales opportuntes, [Ofenbah (8)] analyzes hstorcal sales data to assess the underlyng lost demand. More recently, there have been some studes that meld fnance models and nventory management, n whch nventory s treated as a fnancal asset and an nventory polcy as a contngent clam, and well-nown fnancal models are appled to nventory management problems. For example, [Stowe and Su (997)] vews nventory as an opton on future sales and shows how some basc nventory problems can be solved usng prcng models. Our soluton approach to the nverse problem s motvated by the rs-neutral valuaton framewor for contngent clams, and specfcally by the specal case of Amercan optons. he prncple of rs-neutral valuaton states that the opton s rs-neutral f ts expected value, dscounted to tme, equals ts present value at tme. It s nown that f a contngent clam s rs neutral, then the stochastc process of ts dscounted values over tme forms an exponental martngale process over an adjusted probablty space [Hull ()]. - -
In ths paper, we shall consder an nventory penalty under the rs-neutral prncple, and treat the nventory penaltes of overage and underage as a seres of Amercan optons (the stoppng tmes at whch penaltes are ncurred are analogous to Amercan opton exercse tmes). o ths end, we shall construct auxlary exponental martngale processes, smlarly to fnancal valuaton models, and derve a functonal equaton for the expected dscounted costs. he auxlary martngale process wll then be used to derve an equaton n terms of the Laplace transform of the pdf functon of stoppng tmes at whch the penaltes are ncurred and an approprate penalty functon. Fnally, we wll use that equaton wth a gven present value of the total penaltes over an nfnte tme horzon he man contrbutons of ths paper are lsted below: () For general demand process, we create an exponental martngale process and provde a necessary and suffcent condton for ts martngale property. () For underage penalty, we derve an explct expresson and show that the penalty value s postvely and exponentally determned by the ntal nventory level and the underage sze as well at a constant rate. (3) For overage penalty, we derve an explct expresson and show that the overage penalty value s negatvely and exponentally determned by the ntal nventory level, but postvely and exponentaly determned by the capacty level at a constant rate. hroughout the rest of ths paper, we use the followng notatonal conventons and termnology. + Let R denote the set of real numbers. For any real number x R, x max{, x }. he ndcator functon { A } s f A s true, otherwse. For a random varable X, ts pdf (probablty densty functon) s denoted by fx ( x), cdf (cumulatve dstrbuton functon) by ( x) and the complementary cdf by F ( x) F ( x). For any a stochastc process FX { ( ): } Xt t over a probablty space (,, P) by the random varables { Xt ( ): t s} defned by X X F, let F X () s denote the -algebra generated. he Laplace transform of a real functon f( x) s zx fz () e fxdx (). If real functons fx ( ) and gx ( ) are defned on [, ), then the convoluton functon of fx ( ) and gx ( ) s gven by ( n ) () u f g ( u) f( u x) g( x) dx. Let f s denote the n-th fold convoluton of tself. Also, we assume the contnuous compound nterest rate, r s a gven constant. Hence, the present value of one unt cash flow at tme t s rt e. he rest of ths paper s organzed as follows. Secton ntroduces assessment functons and relevant bacground of martngale methodology. Secton 3 formulates the nventory model wth general demand arrvals. Secton 4, 5 and 6 treat the case wth compound Posson demands. Specfcally, Secton 5 nvestgates the underage penaly, whle Secton 6 studes the overage penalty. Fnally, Secton 8 concludes ths paper. - -
. ASSESSMEN FUNCIONS In ths study, we shall be nterested n assessng a dscounted nventory cost functon n a fnancally reasonable manner. One abstracted assessment problem of nterest can be formulated as follows. Let { Yt ( ): t } be a real-valued stochastc process (e.g., the evoluton of a stoc ( L) prce, nventory level, etc.), and let { : } be a sequence of httng tme of level L by ( L) the process { Yt ( )} at whch some assessments, W, tae place as functon of the level L and ( L) the state, Y( ), where the assessment may be random or determnstc. Examples nclude exercsng an Amercan opton, ncurrng an underage or overage penalty n an nventory system, ( L) etc. enote the pdf functon of condtoned on Y () by f ( L) ( t y), and ts Y ( ) condtonal Laplace transform by f ( L) ( s y). hen, the rs-neutral value of the acton at Y ( ) tme condtonal on the ntal state s ( L) r ( ) (, ) L CLy E e W ( ) Y y. (.) We shall mae an effort to derve the expected dscounted value of the assessment occurred at the ( L) frst httng tme, ( L) r ( L) C ( L, y) E e W Y( ) y, (.) snce the rs-neutral value of the acton gven by Eq. (.) can be obtaned recursvely va condtonng on the state at frst httng tme. A classcal example of the generc standard valuaton problem above s the expected dscounted defct at run tme n a classcal nsurance model [Gerber and Shu (998)]. Specfcally, let Yt () be the surplus of an nsurance frm at tme t, and Y() y the ntal surplus. he frm wll encounter banruptcy at the run tme when ts surplus falls below zero for the frst tme. he L penalty at run W ( ) s a functon of three varables n general: the surplus mmedately before run, the defct at run, and the tme of run. ( L) In the more specal case that W w ( L ) s determnstc, Eq. (.3) reduces to C L y w L e Y y w L f r y ( L) r (, ) ( ) E ( ) ( ) ( L ) ( ). y A classcal example of the generc standard valuaton problem above s a perpetual Amercan call opton. Specfcally, let K be the exercse prce, and Yt () the stoc prce at tme t. Wthout loss of generalty, assume that Y() < K, so that exercsng the opton at tme t can be - 3 -
+ excluded. hen, the payoff functon s wl ( ) [ L K] L K. We menton that when the rs-free nterest rate s constant, rs-neutral valuaton provdes a well-defned and unambguous valuaton methodology for the Amercan opton under consderaton [Hull ()]. Recall that the standard valuaton problem under the rs-neutral prncpals s to prce the acton exercse of Eq. (.) wth a gven payoff functon wl ( ), whle the nverse valuaton problem s to deduce w( L), gven C( L). In nventory budget-plannng context, the latter can be formulated as follows: an nventory manager s gven a prescrbed perpetual nventory penalty budget that he/she s prepared to allocate to underage penaltes and overage penaltes, as those relate to servce level metrcs of the nventory system. A rs-neutral nventory manager s then faced wth the (nverse) problem of dervng the approprate penalty functons to assess per underage/overage event occurrence, subject to the prncple of rs-neutral valuaton (that s, the constrant that the expected perpetual nventory penalty equals to the prescrbed budget). As shown n Eq. (.), the functonal relatonshp between C( L) and wl ( ) s specfed n terms of f ( L ) ( r y ), and consequently, ths Laplace transform needs to be derved. o ths end, Y ( ) we shall use an auxlary martngale sequence { Mt ( ): t } to defne the requste sequence of penalty functons (see secton 4). Recall that a martngale process { Mt ( ): t } satsfes the regularty condton and the martngale property E Mt () <, t (.4) E Mt () Ms (), s t Ms (), s t. (.5) We shall mae use of the Optonal Stoppng heorem (also called the Optonal Samplng heorem). One verson of the theorem s gven below. heorem [Karr (993)] Let { Mt ( )} be a martngale and let be a stoppng tme of { Mt ( )}, such that (a) E < (b) { Mt ( )} s bounded or unformly ntegrable. hen, E M( ) M( ) M( ). Fnally, we shall mae use of the followng defnton efnton Let { Xt ( ), t } be a stochastc process and let {, } be a sequence of httng tme determned by { Xt ( ), t }. Let { X, } be an embedded sequence, where X X ( ). hen { Xt ( )} has an embedded martngale { X } f t satsfes for any j, E Xj X( t), t X. (.6) - 4 -
he motvaton for defnng auxlary martngale processes as a vehcle for solvng for f ( L)( r) stems from the martngale methodology appled n the context of fnancal dervatves valuaton. In partcular, the evoluton of an underlyng securty prce process, { St ( )}, s often modeled as a geometrc Brownan moton, that s, the process { Xt ( )}, where Xt () log St (), s a shfted Brownan moton, whch s routnely used n the analyss fnancal dervatves (e.g., Amercan opton), due to ts contnuty and ndependent-ncrement property. In a rs-neutral settng, the dscounted-prce process of fnancal dervatves over tme s martngale of the form [Luenberger (998)] rt g( X( t)) Mt () e +, (.7) gxt ( ( )) where gx ( ) s a real-valued functon, and e s the prce of the fnancal dervatves. By the martngale property, we have E [ Mt () M() ] M(). If { Mt ( )} s a bounded process, then ( L) ( L) ( L) gx ( ( )) r + g( X( )) r + g( log L) g( log L) e Ee Ee f ( L) ( r) e. (.8) where the frst equalty s due to the Optonal Stoppng theorem appled to the stoppng tme ( ) L and the second equalty follows from the dentty S( ) L. Eq. (.8) readly mples gx ( ( )) g( logl) f ( L) ( r) e ( ) L, (.9) ( L) whch shows the explct connecton between the Laplace transform of and the martngale of Eq. (.7), and wll be used to solve for the former [cf Eq. 5.7 n Gerber and Shu (998)]. In ths paper, we use the above methodology to dentfy the penalty functon for overage and underage penaltes, subject to the prncple of rs-neutral valuaton, by usng martngale technques smlar to those n [Gerber and Shu (998)], n whch a standard nsurance model was studed. o ths end, the nventory-level process { It ( ): t } wll play a role analogous to Xt () logst (), and the httng tme ( L) wll map to httng tmes of certan nventory levels, whch ncur underage and overage penaltes. In our case, we select gx ( ) real constant, so that the auxlary process { Mt ( ), t }, gven by cx, where c s a rt ci() t Mt () e +, (.) s a martngale. Smlar auxlary martngales are dscussed n [Gerber and Shu (998)]. 3. INVENORY MOEL FORMULAIONS In ths paper we consder a contnuous-revew nventory system wth the lost-sales stocout rule. Replenshment occurs at a constant (determnstc) rate >. he demand stream s denoted - 5 -
by {( A, ) : }, where the arrval tme processes { A } and the correspondng demand process { } are mutually ndependent. In partcular, the arrval tmes { A : } follow a renewal process wth arrval rate, and by conventon A. Let { : } be the nterarrval renewal sequence of demands, where the A A are d wth common densty f () t satsfyng f A (). Let further { N ( t): t } be the countng process of demand arrvals, where N () max{ : } A t n A n t s the number of demand arrvals n the nterval (, t ] wth densty f ( n) P{ A t < A } A N () t n n+ A P{ A t} P { A t} (3.) n *( n) *( n+) n+ f ( t) f ( t) he correspondng ndvdual demands { : } are d wth common cumulatve dstrbuton functon (cdf) F ( x ) and probablty densty functon (pdf) f ( x ), and by conventon. Let the cumulatve demand up to tme t be t and s, NA () () t Gt. hus, for *( n) *( n+) ( n) fgt () () s f () () () n t f t f s. (3.) We shall consder a basc producton-nventory system { It ( ): t }, gven by NA () () () t It I + t. (3.3) Note that a negatve nventory level means that bacorderng s n effect. Next, consder the auxlary process rt ci() t Mt () e + where the values of varable c are selected so as to ensure that each { M M ( A )} s an embedded martngale process. Lemma he process { Mt ( )} has an embedded martngale { M } f and only f c satsfes f ( r c ) f ( c ). (3.4) Proof. See Appendx. We next nvestgate the roots of Eq. (3.4). o ths end, we defne and L ( c ) log f ( r c ) (3.5) - 6 -
and note that Eq. (3.4) s equvalent to he followng result provdes ey propertes of these functons. L () c log f () c, (3.6) L ( c ) L ( c ). (3.7) Lemma (a) L ( c ) s strctly decreasng and strctly concave n c. (b) L () c s strctly decreasng and strctly convex n c. Proof. See Appendx. he followng Proposton provdes a ey property for the roots of Eq. (3.4). Proposton Eq. (3.4) has two real roots, c and c. Proof. Follows mmedately from the Lemma and the fact that L () < L () by Eqs. (3.5) and (3.6). Fgure depcts the functonal form of L ( c ) and L () c, as well as the locaton of the roots, c and c. Fgure. he Functonal form of L ( c ) and L () c and the two roots of L ( c ) L ( c ) - 7 -
4. COMPOUN POISSON EMAN ARRIVALS In ths secton and the rest of ths paper, we consder the case where demand arrvals follow a Posson process wth arrval rate >, that s the densty functon of nterarrval tme satsfes f () t e t. hen Eq. (3.4) smplfes to r c + [ f ( c)]. (4.) Eq. (4.) s well nown n the context of nsurance models, where t s referred to as Lundberg s fundamental equaton [Gerber and Shu (998)]. Accordngly, we shall refer to Eq.(3.4) as the renewal extenson of Lundberg s fundamental equaton. he followng example provdes the two roots for the case wth Posson nter-arrval tme and Exponental demands. Example (Posson arrval and Exponental demands) Consder the case wth exponental demand sze of parameter > Hence, f ( c ) + c. hen Eq. (4.) can be rewrtten as or equvalently, r c + [ ], + c c + ( r ) cr. x, that s f ( x) e. Hence, two roots are obtaned as r + ( r ) 4 r c, c r + + ( r ) 4 r. Wth two roots c and c, we defne the two processes, and () M t e + (4.) () rt c I t () M t e +. (4.3) () rt c I t - 8 -
Proposton If the demand arrvals follow a Posson process and c and c satsfy Eq.(4.), then { M ( t )} and { M ( t )} are both contnuous martngales n t. Proof. he proof s completed by followng the memoryless property of Posson process and replacng A and A j wth t and s, respectvely, n the proof of Lemma. By Proposton, both { M ( t )} and { M ( t )} are martngale processes f c and c satsfy Eq.(4.), or Eq.(3.4) n more general. However, the martngale property does not hold always whle a stoppng tme (e.g, underage or overage stoppng tme) s consdered. Fortunately, n our cases, the martngale property s stll holdng f we assocate { M ( t )} and { M ( t )} wth underage and overage stoppng tme, respectvely. In the followng, we shall assocate { M ( t )} wth underage n secton 5, whle assocate { M ( t )} wth overage n secton 6. 5. UNERAGE PENALY FUNCION In ths secton, we study the basc producton-nventory systems under the lost sales, where unmet demand wll be lost subject to an underage (or lost-sale) penalty. An underage penalty s assessed whenever the nventory level hts (at httng tmes). We are nterested n dervng the explct functon for the underage penalty. o ths end, we shall tae the advantage of the aforementoned process { M ( t )}, by whch we derve some cost propertes pertanng to the system. Under lost sale operaton, all the shortage f any wll be lost, whch s nterpreted as lost sales. he lost-sales sze s denoted by LA ( ) [ IA ( )] +. (5.) he nventory process under lost sales, { It ( ): t }, s gven by the followng equatons It NA() t u + t LA ( ) () [ ], (5.) where the sumaton term represents the cumulatve amount of demands that have been satsfed. hen, the stoppng tmes of underage n the process defned above are denoted by ( ) ( ) ( ) A nf{ > : L( A) > }. where whle by conventon, the lost-sale sze LA ( ) s gven by Eq. (5.), and we use supscrpt to denote the httng nventory level for lost-sale occurrence. Fgure llustrates a sample path of the nventory level process. - 9 -
Fgure. A sample path of the nventory level under lost sales he followng theorem provdes the martngale property of { M ( t )}. heorem he process { M ( t )} defned by Eq. (4.) s a martngale process over Proof. By Proposton, we have { M ( t )} s martngale. Furthermore, f ( ),. ( ) <, we have M by Eq. (4.) snce ra + c I( A). Applyng the optonal samplng theorem, the proof s completed. Wthout loss of genralty, we denote the underage penalty for the th underage by ( ) ( ) ( ( ), ( )) w ( ) I L as a functon of the nventory level rght before underage and the correspondng lost-sales sze. It s of our nterest to derve an explct expresson for the underage penalty. o ths end, we shall frst study the expected dscounted penalty for the frst underage n subsecton 5., and then extend to the sequental underage over nfnte tme horzon n subsecton 5., and fnally derve for the attendant penaly values. 5. he Frst Underage In ths subsecton, our man effort s to derve the expected dscounted cost for the frst underage. o ths end, we shall reconsder the basc nventory process gven by Eq. (5.3). Fgure 3 llustrates a sample path of the nventory process tll the stoppng tme of frst underage. - -
Fgure 3. A sample path of the nventory level In lght of heorem, one mmedately has ( ) M() E M( ) I() u, whch can be further wrtten as ( ) ( ) cu r c I( ) e E e + I ( ) u. (5.4) Example (Exponental demands) Consder the case n whch the demand arrvals follow a Posson dstrbuton and the demand sze has exponental dstrbuton wth pdf functon f ( x) e, where >. wo roots to Eq. (3.4) for ths case have been obtaned n Eaxample. We then have the followng result. Proposton 3 If the demand sze s Exponental wth parameter >, then for any u, the followng s true ( ) r + c c u E e I( ) u e. (5.5) x Proof. See Appendx. In partcular, f u n Eq. (5.5), one has r c E[ e I( ) ] + ( ). (5.6) More specfcally, f the penalty functon s a functon of the underage sze denoted by ( ) ( ) w ( L( )), then the expected dscounted penalty for the frst lost sale can be wrtten as - -
( ) ( ) ( ) r ( ) ( ) ( ) r E[ e w L( ) I( ) u] w ( y) f ( ) ( ) y dye e I u + c c u ( ) e E w ( ) (5.7) As show n Eq. (5.7), the expected dscounted penalty of the frst underage s exponentally decreasng n the ntal nvenory level u. he ncreasng rate s determnted unqally by the root value c. 5. he Underage Sequence In ths subsecton, we shall study the underage sequence over the tme. For >, we have ( ) ( ) ( ) ( ) r r r[ ] E e I( ) u E e I( ) u E e I( ) u ( ) ( ) r r E e I( ) u E e I( ). (5.8) + c ( ) r E e I( ) u where the last equalty hold by Eq. (5.6). A combnaton of Eq. (5.8) and Eq. (5.5) yelds ( ) r c + c u E e I( ) u e. (5.9) We then have the followng result. heorem 3 If the prescrbed perpetual underage penalty budget s p U, the underage penalty ( w ) ( y) are ( ) ( ) dentcal over such that w ( y) w ( y) as a functon of the underage sze, then Proof. By Eq. (5.8), one has ( ) c u c y w ( y) pu e e. (5.) + c c u E E (5.) ( ) r ( ) ( ) e w ( Y) I( ) u e w ( ) c - -
By the prncple of rs-neutral valuaton, the rght hand sde of Eq. (5.) equals to s the amount the manager would pay for underage n all at tme, then p U, whch Note that E ( ) ( ) w w ( ) c c u E w ( ) p e U + c ( ) ( ). he equaton above yelds ( ) c c u w ( ) p e (5.) U ( + c ) Fnally, tang nverse Laplace transform of Eq. (5.), we complete the proof. heorem 3 mples that the penalty value s postvely exponentally determned by the ntal nventory level and the underage sze as well at degree of c. 6. OVERAGE PENALY FUNCIONS In ths secton, we consder producton-nventory systems under the Base-stoc rule, where the nventory has an upper threshold level (or refered to as a capacty level), S >, such that an overage penalty s assessed whenever the nventory level hts S (at httng tmes). Replenshment s governed by the base-stoc polcy as follows: t proceeds at a rate > whle the nventory level s below the S, and s suspended, otherwse. All the shortage of the demand s bacordered. hen, the process of the nventory wth threshold level, { It ( ): t }, s gven by the followng equatons For u It u t NA t Iz S dz () () + { ( ) < }. (6.) < S, we defne a sequence of stoppng tmes when nventory hts the capacty levels, nf { t > : I( t) S}, (6.) where, by convenence, f. Note that for each, snce { It ()} s jump free upward, we have ( ) S I( ) S. (6.3) - 3 -
Fgure 4. A sample path of the nventory level wth upper threshold level S In the followng, we are nterested derve the penalty functon for overage. Let ( S ) ( ) + denote the frst demand arrvng tme after the -th overage occurence. hen the correspondng replenshment dle tme s. (6.4) By the memoryless property of exponental dstrbuton, we have that { } are d, and follow an exponental dstrbuton wth parameter. Whence, for,,3,, t holds that ( S ) r E e. (6.5) + r Let the overage penalty for the -th overage be parameterzed by the target level and denoted by w. hen the expected dscounted penalty for overage sequence s w r r e I u w e I u E () E (). (6.6) In the followng, we frst study the expected dscounted penalty for the frst overage, then extend to the expected dscounted penalty for the overage sequence over an nfnte tme horzon. 6. he Frst Overage Shown n Fgure 5 s a sample path of the nventory level process untl the frst overage occurrence. - 4 -
Fgure 5. A sample path of the nventory process tll the frst overage occurrence Consderng { M ( t )}, we have the followng theorem. heorem 4 he process { M ( t )} s a martngale over,. Proof. By Proposton, we have { M ( t )} s martngale. Furthermore, f <, we have c u M () t e unformly n t by Eq. (4.3). Applyng the optonal samplng theorem, the proof s completed. By heorem 4 we mmedately have M () EM () t I() u. Specfcally, t readly follows f we set t, ( c u r + c I ( ) ) S cs ( r e E e I ) u e e I ( ) u E. (6.7) where the second equalty follows from Eq. (6.3). It mples ( S ) r c [ us] E e I() u e (6.8) hen the expected dscount penalty for the frst overage s ( S ) ( S ) r r E w e I( ) u w E e I( ) u, (6.9) where the second equalty holds by Eq. (6.8) c u S [ ] w e We shall menton that Eqs. (6.7) - (6.9) reman vald even f u or u S. he condton for the ntal nventory level s u S. - 5 -
6. he Overage Sequence For >, we have r r - r[ ] - E e I() u E e I( ) u E e I( ) u r - r[ W + ] E e I( ) u E E e I( ) r - rw ( S ) r E e I( ) u E e E E e I( ) ( S ) c [ xs] r - f ( x) e dx e I( ) u + r E ( S ) c S r - e f ( c ) E e I( ) u + r (6.) Combnng Eq. (6.) and Eq. (6.8) yelds, r ( S ) - c [ u S ] e I u E () e, (6.) where c S e f ( c r ) +. (6.) Note that < < snce < <, + r Substtutng Eq. (6.) nto Eq., we have < < and < f ( c ) <. e c S ( S ) E w r c [ u S] - e I u e w (). (6.3) In partcular, f the overage penalty s dental over, then we have the followng result. heorem 5 If the prescrbed perpetual overage penalty budget s p, and the overage penalty s dental over O and denoted by w where Proof. Settng w c S e f ( c r ). + w, then the overage penalty for overage level S s w n Eq. (6.3), one has ( ) c S u w p e [ ] O. (6.4) - 6 -
r E w e I u e w () c [ us] e w ( S ) c [ us] -. (6.5) By the prncple of rs-neutral valuaton, the most rght hand sde of Eq. (6.5) equals to p O, whch s the amount the manager would pay for overage penaltes n all at tme. herefore p O e he result readly follows by smple algobra. c [ us] w (6.6) heorem 5 mples that the overage penalty value s negatvely exponentally determned by the ntal nventory level u, but postvely exponentaly determned by S at degree of c. In c S partcular, f I() u S, then w p O e f ( c ) + r whch s exponentally ncreasng n S. 7. CONCLUING REMARKS As an nterface between fnance and nventory management, a major goal of ths nvestgaton s to formulate the penalty functons and derve closed-form expressons for overage and underage penaltes under rs-neutral valuaton prncple n a quanttve and methodologcal effort. In partcular, we consder an nventory drven by a constant replenshment rate and compound renewal demand stream, and subject to overage penaltes and underage penaltes. Our soluton approach treats the nventory penaltes of overage and underage as a seres of Amercan optons. o prce such optons, we construct auxlary exponental martngale processes n term of nventory process. he auxlary martngale process wll then be used to acheve some ntermedate results that help derve the expected dscounted penalty costs. Under rs-neutral valuaton prncple, we derve the penalty functon wth a prescrbed perpetual nventory (overage or underage) penalty budget. In partcular, for general compound renewal demands, we obtan the necessary and suffcent martngale condton. We then prove that there are two roots to ths equaton, one s postve whle the other s negatve. Accordngly, we assocate the martngale constructed based on the negatve root wth the underage dscusson, whle the martngale constructed based on the postve root wth the overage. For the case wth compound Posson demands, explct expressons for the underage penalty and overage penalty are obaned. - 7 -
References []. Gerber, H.U. and Shu, E.S.W. (998) On the tme value of run. North Amercan Actuaral Journal, Vol., 48-78. []. Zpn, P. () Foundatons of Inventory Management. Boston: McGraw Hll, 5-6. [3]. Ernst, R. and Powell, S.G. (995) Optmal nventory polces under servce-senstve demand. European Journal of Operatonal Research, 87-, 36-37. [4]. Walter, C. K. and Grabner J. R., (975) Stocout Cost Models: Emprcal ests n a Retal Stuaton. he Journal of Maretng, 39-3, 56-6. [5]. Schwartz, B. L. (966) A new approach to Stocout penaltes. Management Scence, vol., B538-B544. [6]. Schwartz, B. L. (97), Optmal Inventory Polcy n Perturbed emand Models. Management Scence, Vol. 6, B59-B58. [7]. Stowe, J.. and Su, e, (997) A Contngent-Clams Approach to the Stocng ecson. Fnancal Management, Vol. 6, No. 4, 4-58. [8]. Ofenbah, I. (8) Estmaton of lost sales revenue opportunty due to out of stoc nventory, Paper presented at INFORMS, 9 Octubor, 8, NY, USA. [9]. Hull, J. C. () Optons, Futures, and Other ervatve Securtes. Prentce-Hall5 th Edton. []. Karr, A. F. (993) Probablty, Sprnger. []. Ross, S. M. (996) Stochastc Process. Wley Seres n Probablty and Mathematcal Statstcs, nd Edton. []. Luenberger,. G. (998) Investment Scence. Oxford Unversty Press. - 8 -
APPENIX Proof of Lemma Substtutng Eq. (3.3) nto Eq. (.) and notng that It () and Mt () determne each other for each t, we can rewrte the left hand sde of Eq. (.6) for any j as raj cia ( j) Mj M( t), t A + E E e M( t), t A ra+ cia ( ) r[ Aj A] + cia [ ( j) IA ( )] E e e I( t), t A r [ Aj A] + c( [ Aj A] [ S( Aj) S( A)]) M E e I( A ) j ( rc)( Aj A) c + M E e IA ( ) j j ( rc) c M + e + E E e (8.) For the embedded martngale condton to hold, t s necessary and suffcent that the rght-most hand sde of Eq. (8.) equal M, whch s equvalent to the condton f ( r c) f ( c). f r c f c j j ( rc) + c + E e E e j j + + j j ( ) ( ) j f ( r c) f ( c) Eq. (3.4) now readly follows, snce f ( r c) f ( c) s non-negatve. Proof of Lemma o prove part (a), we show that the dervatves of L ( c ) satsfy f ( r c) L ( c) c < f ( r c) (8.) f ( r c) f ( ) ( ) r c f r c c c L ( c) <. (8.3) [ f ( r c)] Eq. (8.) follows from the facts - 9 -
( r c ) f ( r c) E e >, (8.4) ( r c ) t ( r c ) f ( r c) te f ( t) dt e > c E (8.5) where Eq. (8.4) s an mmedate consequence of the Laplace transform and Eq. (8.5) follows by the Lebnz ntegral rule. o prove Eq.(8.3), note that the denomnator s postve, so t remans to show that the numerator s postve too. fferentatng Eq. (8.5) wth the ad of the Lebnz ntegral rule yelds f ( r c ) t ( r c ) ( ) ( ) r c t e f t dt e c E.(8.6) Substtutng Eqs. (8.4), (8.5) and (8.6) nto the numerator of Eq. (8.3), the latter becomes e ( r c) ( r c ) ( ) r c e e E E E (8.7) Fnally, applyng the Cauchy-Schwarz nequalty [Karr (993)] to the product of expectatons n the frst term of Eq. (8.7) results n the nequalty, e ( rc) ( r c ) ( r c ) e e E E > E, (8.8) where the strct nequalty follows from the fact that the random varables on the left-hand sde above are not proportonal to each other. Eq. (8.8) establshes that the numerator of Eq. (8.3) s postve, thereby completng the proof of part (a). o prove part (b), we apply an argument smlar to that for part (a). Accordngly, we show that the dervatves of L () c satsfy f () c L () c < f () c f() c f () c f () c f () c L () c > Eq. (8.9) follows from the facts [ f ( c)] (8.9) (8.) c () cx f c E e e f() x dx > (8.) cx c f () c ( x) e f () x dx E e <. (8.) where Eq. (8.) s an mmedate consequence of the Laplace transform and Eq. (8.) follows by the Lebnz ntegral rule. - -
o prove Eq. (8.), note that the denomnator s postve, so t remans to show that the numerator s postve too. fferentatng Eq. (8.) wth the ad of the Lebnz ntegral rule yelds, cx c f () () c x e f x dx E e >. (8.3) Substtutng Eqs. (8.), (8.) and (8.3) nto the numerator of Eq. (8.), the latter becomes c c c E e e [ e ] E E (8.4) Fnally, applyng the Cauchy-Schwarz nequalty to the frst term of Eq. (8.4) results n the nequalty, c c E [ c e Ee > E e ] (8.5) where the strct nequalty follows from the fact that the random varables on the left-hand sde above are not proportonal to each other. Eq. (8.5) establshes that the numerator of Eq. (8.) s postve, thereby completng the proof of part (b). Proof of Proposton 3 For ease of exposton, let (,, ) and ( ) fxyt u denote the jont densty functon of ( ( ) I ), where x, y, t are non-negatve real numbers. efne rt fxy (, u) e fxyt (,, udt ), and the margnal densty functons are denoted respectvely by fx ( u) fxy (, udy ) and fy ( u) fxy (, udx ). herefore, one has the followng relatonshp ( ) r rt E e I( ) u e f( x, y, t u) dydxdt rt e f( x u) dx By the defnton of condtonal probablty, one further has f ( x + y) fxy (, u) fx ( u). F ( x), L ( ) ( ) (8.6) Note that, by the memorylessness property of exponental dstrbuton the second term on the rght hand sde above can be smplfed as f ( x + y) f ( y). F ( x) Hence, - -
fxy (, u) fx ( uf ) ( y) (8.7) hen, ( ) ( ) r c I( ) + rt c y e I() u E e f( x,y,t u) dxdydt cy e f( x,y u) dxdy cy e f( x u) f ( y) dxdy c y ( ) r e f ( y) dy E e I( ) u ( ) r E e I( ) u + c cu where the fourth equaton holds by Eq. (8.6). By Eq. (5.4), the above equaton equals e, whch therefore completes the proof. - -