PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM


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1 PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN (Received July 1998; revisions received June 1999, December 1999; acceped January 2000) We sudy a perishable invenory sysem under a fixedcriical number order policy. By using an appropriae ransformaion of he sae vecor, we derive several key samplepah relaions. We obain bounds on he limiing disribuion of he number of oudaes in a period, and we derive families of upper and lower bounds for he longrun number of oudaes per uni ime. Analysis of he bounds on he expeced number of oudaes shows ha a leas one of he new lower bounds is always greaer han or equal o previously published lower bounds, whereas he new upper bounds are someimes lower han and someimes higher han he exising upper bounds. In addiion, using an expeced cos crierion, we compare opimal policies and differen choices of criicalnumber policies. INTRODUCTION In his paper we sudy an invenory sysem for a produc wih a fixed finie lifeime. An unsold uni of invenory becomes unusable (perishes) if i is sill in he sysem when is lifeime ends. Such unis, called oudaes, mus be hrown away and canno be sold. A key performance measure for a perishable invenory sysem is he number of iems ha mus be discarded in such a manner. Indeed, i is precisely he oudaing phenomenon ha differeniaes perishable invenory sysems from nonperishable invenory sysems. From a modeling sandpoin, he very fac ha perishabiliy is explicily included in a formulaion suggess ha he oudaing process has a significan impac on he invenory sysem. The mos frequenly sudied applicaion for perishable invenory models has been he conrol of invenories of blood producs. In his case, he invenory faciliy is a hospial blood bank. The daily demand for blood is a random quaniy ha depends on he number of ransfusions needed each day. When a paien requires a ransfusion, he demand is saisfied from invenory, provided ha he blood is here. Thus, i is imporan, when ordering blood supplies from a regional blood cener, ha a hospial ake ino accoun he key facors ha influence is invenory levels. One such influence is he fac ha blood producs are perishable. Alhough he lifeime of a uni of invenory depends on he ype of blood produc in quesion, a common hread is ha such producs canno be used if hey remain in sock for oo long. The survey paper by Prasacos (1984) provides an overview of he numerous issues involved in blood invenory managemen. Oher examples of perishable invenories include food producs such as mea, milk, and produce. The recursions sudied herein can also be used o model a problem faced by commercial airlines. The airlines have he abiliy o carry packages inside he cargo hold of heir airplanes. However, he amoun of space available in he hold depends upon he number passengers on he fligh. When here are more passengers on a fligh, here is more luggage in he hold, and hence here is less space available for cargo. Thus, he shipping capaciy is a random quaniy. When an airline agrees o ship a parcel, i does no necessarily need o ravel on any paricular fligh. In fac, he airline may agree o ship he package wihin a specified ime window. The lengh of his window may be measured in ineger unis, corresponding o he number of scheduled flighs beween he package s origin and desinaion wihin he ime window. If he package sill has no been shipped by he end of he window, he airline mus pay for i o be sen by some oher means. Relaing his cargo problem o he perishable invenory model, we see ha capaciy is he analog of demand, he number of flighs wihin he ime window is he analog of he produc lifeime, and he packages ha mus be sen by oher means are he analog of he oudaes. For a differen modeling approach and a more deailed descripion of such cargo problems when here is an enire nework of flighs, see Kasilingam (1996). Decision models ypically require an expression for he expeced number of oudaes per uni ime o evaluae ordering policies (under some cos srucure). However, closedform expressions for he expeced number of oudaes have proved o be unobainable, even for relaively simple ordering policies. Thus, many auhors have focused on deriving bounds and approximaions. In ligh of he imporance of he number of oudaes as a performance measure, we will focus primarily on providing improved bounds on boh he expecaion and disribuion of he number of oudaes for a reasonable class of ordering policies. Opimal dynamic ordering policies in perishable invenory sysems are known o be quie difficul o compue, because he sae of he sysem depends no only on he Subjec classificaions: Invenory, perishable: heurisics, performance bounds, sochasic. Area of review: Logisics & Supply Chain Operaions X/01/ $ elecronic ISSN 455 Operaions Research 2001 INFORMS Vol. 49, No. 3, May June 2001, pp
2 456 / Cooper amoun of invenory being held, bu also on he ages of he invenory. For some insighs on opimal policies, refer o Fries (1975), Nahmias (1975), and Nandakumar and Moron (1993). The complexiy of opimal policies, as well as he difficulies involved in compuing hem, has led many auhors o analyze heurisic mehods for conrolling perishable invenories. One such mehod, proposed by Chazan and Gal (1977), Cohen (1976), and Nahmias (1976), is he fixedcriical number order policy, in which orders are placed so ha he oal amoun of invenory is he same a he end of each ime period, regardless of he ages of he invenory. Compuaional sudies by Nahmias (1976, 1977) and Nandakumar and Moron (1993) show ha under a variey of differen assumpions, such fixedcriicalnumber policies can be quie good when compared wih oher mehods, as well as o opimal policies. In addiion, hese policies provide a good baseline agains which oher ypes of policies can be compared. For nonperishable invenory sysems, i is well known ha under fairly general cos srucures, here are opimal policies of his ype (i.e., order so ha invenory is brough up o a fixed number each period). See Poreus (1990) for an overview of such resuls. As is he case wih nonperishable sysems, fixedcriical number policies perform bes for perishable invenory sysems when here are no large fixed perorder coss. This is indeed he case in he sudies menioned above; hey consider a siuaion where ordering coss are a linear funcion of he number of iems ordered. In cases where here are significan charges for placing an order, i may be more appropriae o use an s S ype heurisic (see Nahmias 1978). For more discussion of oher ypes of heurisics and exensive reviews of he perishable invenory lieraure, see Nahmias (1982) or Prasacos (1984). For some of he earlies work in perishable invenory heory, see Van Zyl (1964). The abovecied compuaional sudies have analyzed heurisics for choosing he bes fixedcriical number policy, and have compared he performance of his policy o ha of oher ypes of policies. This paper will have a differen emphasis. Fixedcriical number policies have already been shown o be good in many cases; herefore, we will focus primarily on descripive analyses (alhough we will also presen numerical comparisons among various criicalnumber policies and opimal policies). We sudy he pahwise and limiing properies of he Markov chain induced by such a policy. The complicaions inroduced boh by he mulidimensionaliy of he sae vecor and by he perishabiliy of he invenory render he sae recursions ineresing in heir own righ. The analysis, in urn, yields a number of bounds for various performance measures, some of which improve previously published bounds. The basic building blocks of his sudy are several pahwise relaions saisfied by he oudaing process and he sae vecor of invenories. Wih hese, we obain a new family of bounds on he expeced longrun number of oudaes per ime period when a fixedcriical number order policy is employed. Bounds and approximaions for he expeced longrun number of oudaes per uni ime have been used in newsboyype formulaions o choose he bes criical level (see Nahmias 1976, 1977; Nandakumar and Moron 1993). A leas one of he new lower bounds on he expecaion is igher han lower bounds appearing in he lieraure, whereas he new upper bounds on he expecaion are someimes beer and someimes worse han hose in he lieraure. In addiion, he proof echniques provide insigh ino he inuiion behind some of he exising resuls. Also, we briefly compare opimal policies and various criicalnumber policies, where he choice of criical number is based on differen expressions for he he number of oudaes. We derive bounds on he limiing disribuion of he number of oudaes in a period. Using hese, we compue an upper bound on he variance of he saionary number of oudaes per period. Such bounds could prove useful when rying choose he bes possible fixedcriicalnumber policy subjec o a qualiyofservice consrain. The works cied above have focused on opimaliy crieria involving he expeced value of he difference beween revenues and coss. The res of he paper is organized as follows: 1 inroduces he model and provides a key saespace ransformaion; 2 includes a number of pahwise relaionships saisfied by he oudae process; 3 conains bounds on he disribuion of he age of he oldes and younges invenory in he sysem as well as a discussion of he relaionship beween perishable and nonperishable sysems; 4 presens a family of upper and lower bounds on he expeced longrun number of oudaes per uni ime; 5 conains a family of refined bounds on he limiing disribuion of oudaes, along wih corresponding bounds on he expecaion and variance; 6 compares he new bounds o previously published resuls; 7 describes mehods for choosing he criical number, and compares he resuling choices o opimal policies; and 8 provides a brief summary. 1. THE MODEL Throughou we assume ha he fixedcriical numberorder policy is o order up o a nonnegaive ineger m, so ha a he end of each period here are exacly m unis in invenory. In addiion, we will ake he fixed lifeime of a uni of invenory o be n 2 ime unis. A nonnegaive, inegervalued amoun of demand D arrives a he beginning of each ime uni. Assume ha D is a sequence of independen, idenically disribued random variables, independen of he iniial sae, wih disribuion G. Le G d = 1 G d. We denoe by G k he kfold convoluion of G, and we define G k d = 1 G k d. We will describe he sae of he sysem a he end of ime period by he ndimensional vecor X = X 1 Xn, where X i is he amoun of invenory of age i in he sysem a ime. In each period = 1 2 evens occur in he following order: (1) Demand D arrives and is saisfied from he curren invenory, using oldes invenory firs.
3 Any porion of he demand in excess of m canno be saisfied and is los. (2) The unsold porion of X 1 n (if here is any) perishes. This quaniy is he number of oudaes in period. (3) An order is placed so ha he oal level of invenory in he sysem is brough back o m. The amoun ordered is assumed o arrive immediaely. (4) All invenories age one ime uni. Noe ha his means he jusordered invenory is now considered o be age 1. The vecor of invenories a he end of period is X. Formally, X 1 and D deermine X via he recursion X 1 = m X i (1) i=2 X i = X i 1 1 (D + + X 1) j i= 2 n (2) j=i Here a + = max a 0. This model was proposed by Chazan and Gal (1977), and an equivalen model wih a slighly differen sae space was sudied independenly by Cohen (1976). From he recursions, i follows ha X = X forms a Markov chain wih sae space S = x n + n x i = m where + denoes he nonnegaive inegers. Noe ha S has ( ) m+n 1 n 1 elemens (see, e.g., Chaper 1, Proposiion 6.2 in Ross 1998). For example, when n = 20 and m = 100 (he values assumed in he examples of 6), he cardinaliy of S is approximaely ; his shows why direc calculaion may be impossible and why easily compuable bounds are imporan. For 1 i n, we define he ndimensional vecor m i = m i given by m i i = m and m i j = 0 for j i. m i 1 n These saes, in which all invenory is he same age, will play an imporan role laer. A major quaniy of ineres is he number of oudaes ha occur a ime 1, defined by Z X n 1 D + (3) We assume hroughou ha P D 1 m/n > 0. Under his assumpion, i can be shown ha X has a single absorbing, aperiodic, posiiverecurren class and possibly some ransien saes. Thus, here exiss a unique saionary disribuion, saisfying = (4) where is he ransiion marix of he Markov chain X. We will denoe by X he random variable for which P X = x = lim P X = x = x Furhermore, i follows from (3) ha here exiss a random variable Z, independen of X 0, such ha P Z = z = lim P Z = z = P X n D + = z We also define he cumulaive number of oudaes up o and including ime by W Z s (5) s=1 Cooper / 457 In he subsequen secion, we will sudy he longrun number of oudaes per ime period, lim 1 W In ligh of he previous discussion, he above limi holds wih probabiliy one. Furhermore, = lim E 1 W, and = EZ. By sandard properies of Markov chains, i follows ha if X 0 has disribuion, hen X is saionary; consequenly EZ =, and Z has he same disribuion as Z for all 1. Laer we will provide compuable bounds for, because direc compuaion is no viable in view of he size of S. In addiion o he oudaes, here are a number of oher performance measures of ineres, such as orders, sales, los sales, and he number of iems held in invenory. The srucure of a fixedcriical number policy makes mos of hese easy o compue once here is an expression for he number of oudaes; we shall address his issue in more deail in 8. We end his secion by inroducing a ransformaion of he sae vecor ha will allow us o derive a number of imporan samplepah relaions for he oudae process Z. Consider he ndimensional process Y, where he quaniy Y i represens he oal invenory of age i or greaer when he sae of he sysem is X ; formally, Y i j=i X j i= 1 n (6) The sae space of Y is S = y n + 0 y n y n 1 y 1 = m. In an abuse of noaion, we say ha Y = m i when Y j = m for j i and Y j = 0 for j>i(i.e., when X = m i. Because X converges in disribuion o X, we can also alk abou he limiing random variable Y, defined in he obvious way. In he remainder of he paper we will swich freely beween X and Y, using whichever sae represenaion is more convenien. Lemma 1. The process Y saisfies he recursion Y 1 = m (7) Y i = Y i 1 1 max D Y n 1 + i= 2 n (8) We omi he proof, which requires he raher edious verificaion of several cases. Recalling our assumpions on D, i can be seen from (7) and (8) ha Y is a Markov chain. The relaion (8) is Equaion (7), in Cohen (1976). Using inducion (on j) and he fac ha if a b and b 0 hen a + b + = a b +, we can now obain from (7) and (8) he following key resul, which is needed in many proofs. Lemma 2. The process Y saisfies he relaion Y k = Y k j j s= j+1 max D s Y n s j min k 1 2 k n (9)
4 458 / Cooper The relaionships in Lemma 2 have a simple inerpreaion. The amoun of invenory a ime of age k or greaer is he amoun of invenory a ime j of age k j or greaer, minus wha has lef he sysem since ime j. Invenory leaves he sysem in wo possible ways: I is aken by demand or i oudaes; herefore, he amoun ha leaves he sysem a ime s is he maximum of D s and Ys 1 n. When j = 1, his argumen also yields an inuiive explanaion for Lemma THE OUTDATE PROCESS We begin by describing a fundamenal samplepah relaion saisfied by he oudae process Z. In he ineres of breviy, we presen he following wihou proof. Proposiion 1. The oudae process Z saisfies Z = Y n k+1 k s= k+1 D s 1 s= k+1 Z s + k = 1 min n (10) Observe ha (10) shows ha Z is no a Markov chain. A variaion of (10) appears in Cohen (1976) for he special case in which k = n. In such a case Y n 1 = m when n 0. Proposiion 1 has he following inerpreaion: The amoun of oudaes in period is exacly he amoun of invenory ha was age a leas n k + 1 a ime k and ha has no lef he sysem by ime by eiher oudaing or being purchased by some demand. Noe ha invenory younger han n k + 1 a ime k canno oudae in or before period, since i will be younger han n a he end of period 1 if i is sill in he sysem. Adding W 1 o boh sides of (10) and leing k = min n yields max W 1 Y n +1 0 s=1 D s if 1 n 1 W = max W 1 m+ W n s= n+1 D (11) s if n Nex, we will resae a resul originally appearing in Chazan and Gal (1977), which is a simple consequence of (11). Hereafer, saemens abou random variables such as A B should be aken o mean ha he relaion holds for every possible realizaion of he demand sequence D. Such saemens involving vecors are aken o mean ha he relaion holds componenwise. Inuiively, Corollary 1 saes ha older iniial socks of invenory lead o larger cumulaive amouns of oudaes up o, for every ime. Corollary 1 (Chazan and Gal). Suppose W is he cumulaive oudaing process when he iniial condiion is Y 0, and W is he cumulaive oudaing process when he iniial condiion is Y 0.IfY 0 Y 0, hen W W for 0. The nex resul describes he pahwise behavior of he oudae process when demand in an nperiod inerval is large relaive o he fixed criical number m. In paricular, if he demand in each period always exceeds he raio m/n, hen a ypical uni of invenory ordered in period will always see a leas m unis of demand before i becomes old enough o perish in period + n. Consequenly, afer allowing a sufficien amoun of ime o overcome iniial condiions, he number of oudaes each period will be zero. The following can be proved by inducion on. Proposiion 2. Suppose ha P D 1 < m/n = 0, and le Y 0 have an arbirary disribuion on S. Then Y j m j 1 m n + 1 j min n (12) and Z = 0 for n. This resul can be viewed as he analog of he resul of Chazan and Gal (1977), which gives an exac expression for he expeced number of oudaes in an nperiod span for he special case when P D 1 m/n = 1. In addiion, he argumen used in he proof of he above can, afer minor modificaions, be used o demonsrae rigorously ha P D 1 < m/n < 1 is a sufficien condiion for he exisence of a single aperiodic, posiiverecurren class for he Markov chain X. In he case when P D 1 < m/n = 1, he Markov chain may be reducible or periodic. When P D 1 m/n = 1, Chazan and Gal show ha = m/n ED 1 (13) Observe ha when demand is deerminisically equal o m/n (his is he borderline case covered by boh he resul of Chazan and Gal and Proposiion 2), his expression gives = 0, which agrees wih Proposiion 2. In fac, neiher of hese degenerae cases is paricularly realisic. Any disribuion wih suppor on he enire nonnegaive inegers falls ino neiher of he degenerae caegories. Furhermore, even when he possible values of demand are limied, a sysem operaor would likely choose m so ha we would have posiive probabiliy of demand values boh above and below m/n. Of course, he choice of m depends on wha sors of goals and coss are involved for he operaor. For insance, a cos funcion ha imposes a severe penaly on oudaes will lead o a relaively low choice of m. Likewise, a cos funcion ha places high coss on unsaisfied demand will cause he sysem operaor o choose a large value for m. We conclude his secion wih a simple bound on he disribuion and mean of Z. The following saemen includes he imporan special case when =. Proposiion 3. For n, P Z >z G n m 1 z z = 0 m 1 (14) Hence, m 1 G n m 1 z = E m z=0 + D s ) (15) s=1
5 Proof. For n, Proposiion 1 implies Z = Y 1 n s= n+1 m s= n+1 D s 1 s= n+1 Z s + D s + (16) Then (14) holds for finie, and i holds for = by leing. Since Z is a nonnegaive, inegervalued random variable, = EZ = z=0 P Z >z, and so (15) follows from (14). We noe ha a similar resul has been obained, alhough no specifically peraining o he limiing disribuion of oudaes under a fixedcriical number policy, by Nahmias (1976) (see Lemma 2.1 on p. 1,004 and he discussion surrounding i). In 5, we will presen refinemens of boh (14) and (15). Observe ha he pahwise inequaliy (16) is, in general, igh in he sense ha here are sample pahs for which here is equaliy. Specifically, if X n 1 = m, hen i follows from (1) and (2) ha 1 s= n+1 Z s = 0, giving exacly he righhand side of (16). 3. THE OLDEST AND YOUNGEST INVENTORY In his secion we derive bounds on he seadysae disribuion of he age of he oldes and younges invenory in he sysem. These resuls will be used in subsequen secions in he developmen of bounds on boh he disribuion and he expeced value of he oudaes. Firs we inroduce wo funcions ha will be used hroughou he developmen of he bounds, S 1 n, and S 1 n are given by y = max i y i = m y = max i y i > 0 Inuiively, y represens he age of he younges invenory in he sysem when he sae is y. Likewise, y represens he age of he oldes invenory in he sysem when he sae is y. To simplify fuure noaion, we define 0 = 0 i = G i m 1 n = 1 i = 1 n 1 Proposiion 4. The disribuions of he oldes and younges invenory in he sysem saisfy P Y i i 1 i n 1 (17) P Y i G 0 i 1 2 i n (18) The proof of Proposiion 4 can be found in he appendix. Saemen (17) has he simple inerpreaion ha if, a ime, he cumulaive demand over he las i (wih i n 1) periods has been m or more, hen here is no invenory lef ha was in he sysem a ime i. The bound provided Cooper / 459 in (18) can be undersood as follows. Suppose we have fixed a ime, and suppose ha here was no invenory in he sysem of age greaer han or equal o n i + 2a ime i + 1. Then, here could have been no oudaes in periods i + 2. If, in addiion, demands were zero in periods i + 2, hen no new invenory would have been ordered in periods i + 2 ; hence, here can be no invenory wih age less han i in he sysem a ime. I is eviden from he proof ha (17) and (18) hold for Y when is larger han n. However, we will no require his level of generaliy for our laer resuls. We will now briefly discuss a pahwise relaionship beween perishable and nonperishable invenory sysems relevan o he proof of he proposiion. Consider he process Y on S, which saisfies he recursion Ỹ 1 = m (19) Ỹ i = Ỹ i 1 1 D + i= 2 n (20) The process Ỹ is idenical o Y excep ha in Ỹ, invenory ha reaches he end of is lifeime does no perish. Raher, i coninues o remain a age n unil i is consumed by some demand. Equivalenly, we could allow invenories o coninue aging pas age n; however, hen he process Ỹ would no be defined on S. Noe ha his alernaive descripion of he nonperishable sysem does no change he recursions (19) and (20) for he range i = 1 n; i simply requires us o expand he dimension of he sae space o ge a complee picure of he sysem. In eiher case, he quaniy Ỹ i represens he amoun of invenory of age greaer han or equal o i a ime. For he purposes of a comparison o he perishable sysem, his is he only range in which we are ineresed. Nandakumar and Moron (1993) have used he idea of comparing a perishable sysem wih a nonperishable sysem o obain bounds on opimal order quaniies wihin a Markov decision process framework. A consequence of Lemma 1 and expressions (19) and (20) is he following, which relaes he pahs of he processes Y and Ỹ. Lemma 3 saes ha, if a some ime, he nonperishable sysem has older invenories han does he perishable sysem, hen he nonperishable sysem will coninue o have older invenories a all subsequen imes. Lemma 3. Suppose 0. IfY Ỹ, hen Y +s Ỹ +s for all s 0. Noe ha Lemma 3 would no be valid for X and X, a process corresponding o a nonperishable sysem (and defined in he obvious way). We canno, in general, conclude ha X will dominae X componenwise, regardless of he iniial saes. Combining (19) and (20) and Lemma 3, we see ha inequaliy (33) in he appendix arises by comparing, in a pahwise sense, he process Y and is analog Ỹ in which no invenory oudaes. Specifically, suppose we ake Ỹ i = Y i ; hen, Y i+1 Ỹ i+1 = Ỹ 1 i s= i+1 + D s = m s= i+1 D s +
6 460 / Cooper 4. BOUNDS ON THE EXPECTED NUMBER OF OUTDATES In his secion we derive families of upper and lower bounds on he average oudaes,. The basic mehod used in he derivaion of he upper bounds will be o employ he previous secion s bound, for each possible age, on he seadysae probabiliy ha he oldes iem in he sysem is younger han (or equal o) he age in quesion. We hen consider he process sared from seady sae, and we condiion on he age of he oldes invenory being, say, i. This allows us o obain a pahwise upper bound on he cumulaive oudae process by considering anoher oudae process sared from he sae wih all m unis of invenory of age exacly i. The oudaes for his comparison process can be expressed by a simple formula. A similar mehod is used in he derivaion of he lower bounds. For each i, we use he bound on he seadysae probabiliy ha he age of he younges iem will be greaer han or equal o i. Saring he process from seady sae, we condiion on he age of he younges iem in he sysem and obain a lower bound by comparing he cumulaive oudaes in a pahwise sense wih hose generaed by he process ha sars wih all invenory of age exacly i. As we will see in 6, a simpler varian of his mehod of pahwise comparison is also he key behind previously published lower bounds on. We are now ready o presen he main resuls of his secion. Noe ha in he following heorem, he bounds include only erms involving convoluions of he demand disribuion (and expecaions wih respec o hose convoluions). Furhermore, he highes convoluion needed for compuaion of he bounds is he nfold convoluion. Observe also ha he following heorem conains a family of upper and lower bounds. Theorem 1. For k = 1 n, he longrun number of oudaes per period saisfies k 1 k k 1 k i=n k+1 i=n k+1 E i i 1 E m D ) (21) =1 ( G 0 i 1 G 0 i n i ) + m D ) (22) =1 Proof. We prove (21) firs. Fix k 1 n, and suppose ha Y 0 has he saionary disribuion (equivalenly, X 0 has disribuion ). If he iniial sae Y 0 is m i, hen we can use (11) o show ha { 0 if k<n i + 1 W k = m + D u if k n i + 1 If Y 0 = i, hen Y 0 m i. Therefore, by Corollary 1 and he fac ha he demand sequence is independen of he iniial sae, we have E W k Y 0 = i 1 k E So, E W k = + m D u ) P Y 0 = i E W k Y 0 = i P Y 0 = i 1 k E + m D u ) (23) By (17) we see ha i j=1 P Y 0 = i i for i = 1 n 1, because he sysem is saionary. We can now apply Lemma 4, Par (a) from he appendix o (23) wih + ) c i = 1 k E m D u p i = P ( Y 0 = i ) q i = i o ge E W k = ( ) + ) i i 1 1 k E m D u i=n k+1 ( ) + i i 1 E m D u ) The resul (21) follows afer division by k, since E W k = k when he sysem is saionary. Using similar mehods, we will now prove (22). Fix k and assume he sysem is saionary. If Y 0 = i hen Y 0 m i, and hence + E W k Y 0 = i 1 k E m D u ) So, E W k = P Y 0 = i E W k Y 0 = i P Y 0 = i 1 k E + m D u ) (24) We now apply Lemma 4 Par (b) o (24) wih + ) c i = 1 k E m D u p i = P ( Y 0 = i ) r i = G 0 i 1
7 o ge E W k = ( ) G 0 i 1 G 0 i n i 1 k E + ) m D u i=n k+1 E ( G 0 i 1 G 0 i n i ) + m D u ) Dividing by k complees he proof, because E W k = k since he sysem is saionary. 5. REFINED BOUNDS ON THE OUTDATE DISTRIBUTION In he previous secion, we were able o obain bounds on he longrun average number of oudaes per period by condiioning on he ages of he oldes and younges iems in invenory. The bounds on he saionary probabiliies of hese quaniies, embodied by Proposiion 4, can also be used o provide a refinemen of he bound on he disribuion funcion of Z (14). This refinemen also yields an improvemen on he earlier bound on he expecaion of Z given by (15). For k = n, we see immediaely ha (25) improves (14), because 1 0 G 0 n 1 = 1 G 0 n 1 1. Likewise, i follows ha n. Theorem 2. For k = 1 n, P Z >z ( ) 1 n k G 0 n k+1 k 1 G k m 1 z z = 0 m 1 (25) Hence, k 1 n k G 0 n k+1 k 1 k + E m D ) (26) =1 The proof can be found in he appendix. I is also possible o obain bounds on higher momens of Z by using (25). For insance, i is sraighforward o verify ha E A 2 = a=0 2a + 1 P A>a when A is a nonnegaive, inegervalued random variable. Applying his fac o Z, and hen using (25), we obain E ( ) Z 2 k ( ) 1 n k G 0 n k+1 k 1 m 1 2z + 1 G k m 1 z (27) z=0 We can now combine (27) wih lower bounds on from Theorem 1 o obain an upper bound on he variance of Z. Corollary 2. Le = min 1 n and = max 1 n. Then Var Z 2 (28) Cooper / 461 The resuls of his secion may poenially be useful when evaluaing fixedcriical number policies wihin he conex of a consrained opimizaion problem. Previous work in perishable invenory heory has largely focused on choosing ordering policies so as o minimize expeced coss. In ligh of his, a direcion for fuure work will be o sudy he effec of including probabiliy or variance consrains. The inclusion of such consrains may be more imporan for he cargo problem, where oudaes may be he cause of cusomer dissaisfacion, which is ypically difficul o measure in erms of a cos funcion. 6. COMPARISON OF THE BOUNDS In his secion, we compare he new bounds wih hose exising in he lieraure. Chazan and Gal (1977) obained he resul ha, where = n 1 m E min n 1 n =1 D n 1 m. Afer some rouine algebra, his lower bound can be wrien as + = n 1 E m D (29) =1 To see ha Theorem 1 provides a igher lower bound, noe ha n = 1 ( ) G 0 i 1 n G 0 i n i ( E m =1 + D ) E m + D )) This quaniy is nonnegaive because G 0 n i and m =1 D + m n =1 D + for i = 1 n. The difference will be sricly posiive if G 0 0 and here is an i for which >G 0 n i and E m =1 D + >E m n =1 D +. These condiions are me by any disribuion ha has posiive suppor on 0 1 m. We also noe ha he mehod of proof in Theorem 1 yields and inerpreaion of he lower bound of Chazan and Gal. We sar he process according o he saionary disribuion as in he proof and hen noe ha y m 1 for any iniial sae y. Thus, he cumulaive number of oudaes up o ime n when he process sars in m 1 provides a pahwise lower bound for he cumulaive oudaes up o ime n for any iniial sae. Of course, he cumulaive oudaes up o ime n when Y 0 = m 1 are exacly m n =1 D +. Taking expecaions hen yields he resul. The proof presened by Chazan and Gal is based on a similar idea. In addiion, Chazan and Gal (1977) show ha, where = n 1 m E min D 1 n 1 m. Rearranging, we see ha = n 1 E m nd 1 + =1
8 462 / Cooper Table 1. Numerical comparison of bounds. Disribuion Mean PI PI AllorNone(100,0.975) AllorNone(100,0.950) AllorNone(100,0.925) Geomeric Geomeric Geomeric Poisson Poisson Poisson Uniform(0,5) Uniform(0,10) Uniform(0,15) There are cases for which k < for each k. I can be proven ha one such example occurs when he demand disribuion is given by p if d = 0 P D = d = 1 p if d = m (30) 0 oherwise where 0 <p<1. This siuaion will arise if demand is nearly deerminisic in he sense ha wih a very high probabiliy i is m, bu in some very rare cases i fails o maerialize. In such cases i is reasonable ha a sysem operaor would choose a criical level of m, since his would exacly mee he demand nearly all of he ime. Alernaively, if p is large, hen demand is usually zero, excep in some relaively rare cases, when i is m. If here are significan lossales coss, he operaor may again choose a criical level of m. Noe also ha his case is easy o analyze, because he only possible saes for he process X are m i i = 1 n; his allows explici compuaion of he saionary disribuion. Noe ha sligh perurbaions of G can yield he same qualiaive allornohing behavior bu will desroy he simple srucure ha allows us o wrie down explicily. Observe ha i is no surprising ha he upper bounds from Theorem 1 do well in his case, because he bounds are derived from comparisons wih iniial saes m i. For his demand disribuion, hese are he only possible saes for X. There are many examples for which k > for all k. In ligh of his, i is bes o compue all of he upper bounds and use he smalles of hem. In cases where demand is frequenly smaller han m/n, he bound is quie good. An analysis of Chazan and Gal s clever derivaion of shows why his is rue. The argumen involves runcaing each demand a m/n (i.e., replacing each D by min m/n D ), and hen using he exac formula for given by (13). This exac formula gives an upper bound on he mean number of oudaes for he original sysem, because cumulaive oudaing is pahwise monoone nonincreasing in demand. When demand is no above m/n oo ofen (and no oo far above m/n when i is above m/n), hen his runcaion does no enail much of a loss. Thus, he expression (13) is almos he average number of oudaes for he unruncaed demand sream. Conversely, when demand is frequenly larger han m/n, he runcaion causes a significan change in he demand sream. In hese cases, (13) may no give a good approximaion o he unruncaed sysem. Table 1 gives a numerical comparison of he bounds obained above and hose provided in Chazan and Gal (1977) for a number of differen disribuions wih m = 100 and n = 20. We will use he bes choices of new upper and lower bounds in he comparison. Define = min 1 n 1 n = max 1 n For each disribuion, we have compued he bounds for hree separae mean demand values: 2.5, 5.0, and 7.5. These were chosen o reflec cases when demand is less han, equal o, and greaer han m/n. Observe ha he numerical resuls suppor he inuiive explanaion given above of wha condiions lead o being beer han, and wha condiions lead o being beer han. Allor None a p refers o he disribuion where P D = 0 = p and P D = a = 1 p. We will use percenage improvemen ( PI) and  percenage improvemen ( PI) as a measuremen of he relaive amoun of improvemen in he new bounds. Define ( ) PI = ( ) PI = Noe each percenage improvemen is a number beween 100 and 100. Posiive values mean ha he new bound is igher han he bounds previously available in he lieraure. Larger posiive numbers reflec larger relaive improvemens. Similarly, negaive values indicae ha he previously available bounds are igher han he new bounds. For disribuions ha yield very few oudaes, hese measures may no be paricularly meaningful, because he absolue difference in he bounds will be very small.
9 7. CHOOSING THE CRITICAL NUMBER We nex describe how o use he resuls of he previous secions o choose he criical number m, and compare hese choices of m wih oher choices based on differen expressions for he number of oudaes. In addiion, we compare he resuling policies o finiehorizonexpecedcos opimal policies, i.e., policies ha minimize v E =1 C, where C is he cos incurred in decision epoch and is he end of he planning horizon. We do no consider problems wih discouning or infinie ime horizons here. When making a choice of m, i will be more convenien o aack he equivalen problem of minimizing v/. We consider choices made using wo differen approximaions for he expeced number of oudaes per period: One approximaion will be based on m and m (he parenheical m indicaes he dependence upon m), he oher mehod uses Chazan and Gal s bounds m and m. The primary conclusions of he compuaional sudy described below are ha (1) in mos cases boh ses of bounds lead o he same choice of criical level, and (2) he criical number policy yields an expeced cos very close o ha of he opimal policy. However, here are cases in which he differen bounds yield exremely differen answers. The sudy adds o he mouning body of compuaional evidence ha fixedcriical number ordering policies perform quie well for a wide variey of demand disribuions. Oher published sudies, cied earlier, have also presened similar resuls under a variey of assumpions. We denoe he quaniy ordered in decision epoch by Q, he amoun of los sales in epoch by L, he number of unis held hrough epoch by H, and he oudaes during epoch by Z. There are linear ordering, oudaing, los sales, and holding coss, given by (respecively) c q c z c l, and c h. The cos incurred a decision epoch 1 is given by C c q Q + c z Z + c l L + c h H (31) Noe ha, for reasons o be made eviden shorly, we are making a disincion beween a ime period and decision epoch. For criicalnumber policies, we have ha Q = X 1 1 = Sales + Oudaes = D 1 L 1 + Z 1 L = D m +, and H = m Sales. We have chosen hese definiions of Q and D for he criicalnumber policy so as o make he recursions inroduced in 2 and 3 consisen wih he sequence of evens in oher Markov decision process models; he firs hing o occur in a decision epoch is he ordering/immediae arrival of new invenory; hen demand is realized yielding a realizaion of los sales, whaever invenory remains is charged a holding cos, hen iems oudae. The mehod of assessing oudaing coss is idenical o ha of Fries (1975). To ge a racable opimizaion problem, we subsiue he expressions of he previous paragraph ino (31), and replace he Zs byz m and he Ds by a generic D. Taking expecaions yields Cooper / 463 EC m = c h m + c q c h ED + c l c q + c h E D m + + c q + c z EZ m (32) We consider wo mehods of choosing m. Mehod 1 replaces EZ M by he midpoin of he new bounds, m + m /2. Mehod 2 replaces EZ m by he midpoin of Chazan and Gal s bounds, m + m /2. In each case, we need conduc only a onedimensional search o pick he criical number m ha minimizes EC m. Noe ha we are no considering discouned coss, so we are able o obain direcly a fairly simple cos funcion. Nahmias (1977) proposes a differen mehod (using he midpoin of he Chazan and Gal bounds as a proxy for he expeced oudaing) ha enables one o address discounedcos problems under he addiional complicaing assumpion of random lifeimes. His mehod is also applied by Nandakumar and Moron (1993) o he deerminisiclifeime, infiniehorizon, discounedcos case. We considered problems wih horizon = 1000, lifeime n = 3, and no salvage value a = + 1. Under hese assumpions, we carried ou he above opimizaion procedures for a number of differen choices of he cos parameers. We show resuls for hree differen demand disribuions, each wih mean 5.0: Poisson, geomeric, and allornone(10,0.5) (i.e., P D = 10 = P D = 0 = 0 5). For each of he hree differen disribuions, we used c h = 0, c q = 1 5, and allowed c l o ake values in and c z o ake values in , hereby yielding 20 differen combinaions of cos parameers. In addiion, we sudied a wide array of oher combinaions; however, we do no repor on hose here, because he resuls were no subsanively differen. Tables 2 4 summarize he resuls. In a given cell, he firs line shows he choice of m made by Mehod 1 and is corresponding average cos. Likewise, he second line in each cell shows he choice of m made by Mehod 2 and is corresponding average cos. Each displayed cos is he mean (over 10,000 simulaions) average cos (over = epochs) yielded by simulaion; hus he oal number of decision epochs simulaed for each case was In addiion, he hird row of each cell shows he choice of m ha minimized he mean simulaed oal cos. This m was deermined by simulaing 10,000 demand sequences D =1 and evaluaing he resuling oal coss yielded by differen values for he criical number. The displayed value of m on he hird line is he one ha minimized he mean of he simulaed oal cos (equivalenly, he one ha minimized mean of he simulaed average cos over = 1 ). Also in he hird row is he minimum mean simulaed average cos, corresponding o he minimizing m. All simulaions sared wih an empy sysem. Finally, each cell also shows he expeced cos per uni ime of an opimal policy when he sysem sars empy; his expecaion was obained by recursively compuing he expeced oal cos using sandard Markov decision process echniques and hen dividing by.
10 464 / Cooper Table 2. Criical numbers and coss for allornone demand. c l c z Table 4. Criical numbers and coss for Poisson demand. c z c l Examining Tables 2 4, we see ha in mos cases boh Mehods 1 and 2 picked he m ha minimized coss (among criicalnumber policies) in he simulaions. In addiion, hese choices of m yielded coss ha were generally less han 1% higher han hose of an opimal policy. These resuls are consisen wih hose in Nahmias (1976, 1977) and Nandakumar and Moron (1993). The mos sriking example of he difference beween Mehod 1 and Mehod 2 occurs in he hree cases of he allornone disribuion, where Mehod 1 chooses m = 10 and Mehod 2 chooses m = 0. In he mos exreme case his phenomenon causes a cos increase of abou 16% when using Mehod 2. The reason ha Mehod 2 picks m = 0 can be raced back o he discussion of he compuaion of Table 3. Criical numbers and coss for geomeric demand. c l c z m in he previous secion. The fac ha he only possible posiive values of D are runcaed by m causes Mehod 2 o overesimae he slope of he expeced oudaing cos when viewed as a funcion of m. Table 2 shows idenical coss for he opimal policy and he bes choice of m. This is because he opimal decision rule for he allornone disribuion was a criicalnumber m = 10 rule in every decision epoch excep for (in some cases) = and = 1. In hese epochs close o he end of he horizon, i was someimes opimal o order nohing, even when having oal invenory below 10; however, he difference in oal cos beween he opimal policy and he criicalnumber m = 10 policy was oo small o appear in he able. Inspecion of Table 4 shows he criicalnumber coss racking very close o he opimal numbers. In fac, he opimal policy for he Poisson case prescribed he same order quaniy as did he criicalnumber policy in mos saes and decision epochs. However, he opimal policies were, in general, no criicalnumber policies. Table 3 shows ha for geomeric demand, he criicalnumber policies did no do as well as hey did for Poisson demand. As menioned in he previous secion, he Chazan and Gal upper bound will be igher in cases when demand is relaively low. However, i is no alogeher clear wha causes one approximaion mehod o be beer, in erms of he resuling choice of m, for any paricular problem. By virue of he fac ha he midpoin of he upper and lower bound was used as an approximaion, one could poenially arrive a he bes possible criicalnumber policy by finding upper and lower bounds ha are always equidisan from he rue expeced number of oudaes. If such bounds were obained, hen he midpoin of he bounds would yield an exac expression for he expeced number of oudaes, even if each individual bound were far from he rue value.
11 Nahmias (1977, 1978) used he Chazan and Gal bounds o help derive operaing policies for perishable invenory sysems wih, respecively, random produc lifeimes and fixed order (seup) coss. In he 1977 paper, he found ha policies arising from approximaions uilizing he Chazan and Gal bounds ended o perform beer han hose ha came from mehods firs inroduced in Nahmias (1976). I is also possible o use he bounds m and m o obain approximae cos funcions for he purpose of choosing operaing policies for he morecomplicaed sysems. This could be accomplished by using he mehods employed in he 1977 and 1978 sudies bu wih he new bounds in place of he Chazan and Gal bounds. 8. CONCLUSIONS In his paper, we have sudied a perishable invenory sysem ha uses a fixedcriical number policy, whereby orders are placed each period so ha oal onhand invenory, regardless of age, is reurned o a single fixed number each period. The building blocks of he sudy are a number of samplepah recursions ha relae he dynamics of he oudaing process o hose of a ransformed sae vecor. From hese, we obained families of upper and lower bounds on he expeced number of oudaes per uni ime. In addiion, we derived bounds on he seadysae disribuion of he number of oudaes per uni ime; hese bounds, in urn, led o bounds on higher momens of he number of oudaes. Oher producs of he analysis included comparison resuls wih nonperishable invenory sysems, as well as bounds on he disribuion of he age of he younges and oldes invenory in he sysem. One of he new lower bounds on he expeced number of oudaes was shown o be igher han hose previously published, whereas he upper bounds were someimes beer and someimes worse han hose already available. Numerical resuls suggesed ha he new upper bounds were beer in cases where demand was ofen greaer han m/n, whereas he previously available upper bounds (Chazan and Gal 1977) were beer in cases where demand was ofen less han m/n. The Chazan and Gal bound performed beer in he lower demand siuaions, because i is based on a comparison o an exac expression for he number of oudaes per uni ime in a sysem in which demand is always less han m/n. We conduced a numerical sudy in which we compared, for a variey of demand disribuions and cos parameers, differen choices of criicalnumber policies and opimal policies. We considered choices of he criical number based on separae cos esimaes ha employed eiher he new bounds or he old bounds. Neiher approximaion mehod appeared o be beer across he board. However, in all cases, he performance of he criicalnumber policies was nearly as good as ha of an opimal policy, hereby supporing he asserion ha, in he absence of significan fixedcharge order coss, criicalnumber policies provide a simple and effecive means for managing invenories of a perishable produc. Cooper / 465 APPENDIX Proof (of Proposiion 4). We prove (17) firs. Fix i 1 n 1 and suppose i. From Lemma 2 wih k = i + 1 and j = i, ogeher wih (7), we see ha + m D s (33) Y i+1 s= i+1 Hence, P Y i+1 = 0 P m s= i+1 D s + = 0 = i. Noing ha y i = Y i+1 = 0, wege P Y i i Leing go o infiniy complees he proof of (17). We will demonsrae ha (18) holds when is large enough. The proposiion will hen follow by leing go o. Suppose ha i 1, hen P Y n i+2 i+1 = 0 D s = 0 all s = i + 2 = P ( Y n i+2 i+1 = 0) s= i+2 P D s = 0 (34) = P ( Y i+1 n i + 1 ) G 0 i 1 (35) G 0 i 1 (36) Observe ha Y i = Y i = m for i = 2 n. Thus, i remains only o be shown ha for i 1, Y i n i+2 =m = Y i+1 =0 D s =0 all s = i+2 (37) We see ha { Y n i+2 i+1 = 0} Y n s = 0 all s = i because, by Lemma 2, for s such ha i + 1 s 1, Y n s = Y n+ s i+1 i+1 Y n+ s i+1 i+1 s u= i+2 max D u Y n u 1 + Y n i+2 i+1 Therefore, { Y n i+2 i+1 = 0 D s = 0 all s = i + 2 } = A (38) where we define A { Y n i+2 i+1 = 0 Yn s = 0 all s = i D s = 0 all s = i + 2 } Lemma 2 wih k = i and j = i 1 also implies { { + } Y i =m} = Y 1 i+1 max D s Y n s 1 =m (39) s= i+2 Consequenly, A Y i = m. To prove he opposie inclusion, noe ha Lemma 2 wih k = n j = i 2, and replaced by 1 yields Y n 1 = Y n i+2 1 i+1 s= i+2 max D s Y n s 1 + (40) By (39) i follows ha Y i = m if and only if boh D s = 0 and Ys 1 n = 0 for s = i +2, which in urn, by (40)
12 466 / Cooper implies Y n i+2 i+1 proof. = 0. So, A Y i = m, compleing he Proof (of Theorem 2). Fix i 1 n z 0, m 1, and suppose ha X 0 has he seadysae disribuion. Then P Z >z = P Z >z X i 0 = 0 P Xi 0 = 0 + P Z >z X i 0 0 P Xi 0 0 Applying (1) and (2), we see ha if X0 i = 0, hen Xi+j j = 0 for j = 1 n i. In paricular, Xn i n = 0, so Z = 0. Therefore, we have P Z >z = P Z >z X i 0 0 P Xi 0 0 = P Z >z X i P Xi 0 = 0 P Z >z X i i 1 G 0 i n i (41) To derive his inequaliy, observe ha X i 0 = 0 Y 0 i 1 Y 0 i + 1. Hence, P X i 0 = 0 P Y 0 i 1 Y 0 i + 1. Furhermore, Y 0 i 1 Y 0 i + 1 =,sop Y 0 i 1 Y 0 i + 1 = P Y 0 i 1 + P Y 0 i + 1. Proposiion 4 now gives (41). Proposiion 1 yields P Z >z X i 0 0 = P Y i0 n i + ) D Z >z X i 0 0 =1 =1 + ) P m D >z X i 0 0 =1 + ) = P m D >z =1 The las equaliy follows from he fac ha D 1 D 2 are independen of X 0. Thus, we have + ) P Z >z P m D >z =1 1 i 1 G 0 i n i (42) We now obain (25) by leing k = n i +1 and noing ha if X 0 has disribuion, hen Z k has he same disribuion as Z for k>0. Finally, (26) follows from (25) jus as (15) followed from (14). The following echnical inequaliies, which can be proved by inducion on n, are needed in he proof of Theorem 1. Below, we use he convenion ha 1 i=2 x i = 0. Lemma 4. Suppose n 2 c = c 1 c n p = p 1 p n q = q 1 q n 1, r = r 2 r n, where c 1 c 2 c n. (a) If q i i j=1 p j for i = 1 n 1, hen n 1 p i c i q 1 c 1 + q i q i 1 c i + i=2 (( (b) If r i n j=i p j for i = 2 n, hen ) p j q n 1 )c n j=1 (( ) n 1 p i c i p j r 2 )c 1 + r i r i+1 c i + r n c n j=1 REFERENCES Chazan, D., S. Gal A Markovian model for a perishable produc invenory. Managemen Sci Cohen, M. A Analysis of single criical number ordering policies for perishable invenories. Oper. Res Fries, B Opimal ordering policy for a perishable commodiy wih fixed lifeime. Oper. Res Kasilingam, R. G Air cargo revenue managemen: characerisics and complexiies. Eur. J. Oper. Res Nahmias, S Opimal ordering policies for perishable invenory II. Oper. Res Myopic approximaions for he perishable invenory problem. Managemen Sci On ordering perishable invenory when boh demand and lifeime are random. Managemen Sci The fixed charge perishable invenory problem. Oper. Res Perishable invenory heory: a review. Oper. Res Nandakumar, P., T. E. Moron Near myopic heurisics for he fixedlife perishabiliy problem. Managemen Sci Poreus, E. L Sochasic invenory heory. D. P. Heyman and M. J. Sobel, eds. Sochasic Models. Elsevier Science Publishers, New York, Prasacos, G. P Blood invenory managemen: an overview of heory and pracice. Managemen Sci Ross, S. M A Firs Course in Probabiliy, 5h ed. Prenice Hall, Englewood Cliffs, NJ. Van Zyl, G. J. J Invenory Conrol for Perishable Commodiies. Ph.D. Disseraion, Universiy of Norh Carolina, Chapel Hill, NC. i=2
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