A Production-Inventory System with Markovian Capacity and Outsourcing Option

Transcription

1 OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp issn X eissn informs doi /opre INFORMS A Producion-Invenory Sysem wih Markovian Capaciy and Ousourcing Opion Jian Yang Deparmen of Indusrial and Manufacuring Engineering, New Jersey Insiue of Technology, Newark, New Jersey 07102, yang@adm.nji.edu Xiangong Qi Deparmen of Indusrial Engineering and Engineering Managemen, Hong Kong Universiy of Science and Technology, Clear Waer Bay, Kowloon, Hong Kong, China, ieemqi@us.hk Yusen Xia Deparmen of Managerial Sciences, Georgia Sae Universiy, Alana, Georgia 30303, ysxia@gsu.edu We sudy he opimal producion-invenory-ousourcing policy for a firm wih Markovian in-house producion capaciy ha faces independen sochasic demand and has he opion o ousource. We find very simple opimal policy forms under fairly reasonable assumpions. In addiion, when he capaciy Markov process is sochasically monoone, he policy parameers decrease in he firm s curren capaciy level under addiional assumpions. All hese resuls exend o he infinie-horizon and undiscouned-cos cases. We analyze comparaive saics and he necessiy of some echnical condiions, and discuss when he ousourcing opion is more valuable. Subjec classificaions: Invenory/producion: uncerainy; dynamic programming/opimal conrol: models; probabiliy: Markov processes. Area of review: Manufacuring, Service, and Supply Chain Operaions. Hisory: Received Sepember 2002; revisions received April 2003, Ocober 2003; acceped January Inroducion Ousourcing has become prevalen in all indusries, boh as a sraegic ool o mainain a firm s brand inegriy while reaining low cos, and as a acical means o hedge agains capaciy and/or demand uncerainies while mainaining he firm s marke share. A he sraegic level, ousourcing, ofen o offshore locaions and of an enire secion of producion, has he benefis of lower operaing cos; quicker ramp-up ime; more ease of adding and dropping programs, services, and markes; and poenially fewer regulaory consrains (Elmaghraby 2000, Quinn and Hilmer 1994). The benefi of using ousourcing (subconracing) as a acical soluion o a firm s daily needs can also be subsanial. A firm usually does no have oal conrol over is demand or even is capaciy, for i is consanly faced wih facors like flucuaing demand sreams, unreliable suppliers, and parially unconrollable in-house elemens such as he work force, machines, power and waer supply, ec. Therefore, he sporadic use of exernal capaciy, when economically feasible, is a good counermeasure o bale he occurrences of delivery delays or los sales. Research on acical ousourcing has mosly been concenraed on he sraegic quesions relaed o price seing, capaciy invesing, and conrac wriing. For insance, using a muliperiod game-heoreical model, Kamien and Li (1990) presened condiions under which acical ousourcing should be carried ou and showed ha i has he effec of producion smoohing. Van Mieghem (1999) used a wo-sage game-heoreical model o analyze ousourcing condiions for hree ypes of conracs beween a firm and is subconracor, wih differen levels of flexibiliy on seing ousourcing coss. The sudy demonsraed he advanage of he ype of conracs wih flexible or negoiable ousourcing coss over he ype of conracs wih pre-fixed ousourcing coss. There also exiss a sizable relaed body of lieraure concerning a reailer s use of a faser and necessarily more expensive secondary supply source o compensae for he sluggishness of he primary supply source. Barankin (1961), Fukuda (1964), Neus (1964), Sehi e al. (2001, 2003), and Wrigh (1968) all derived he opimal policy forms for varians of such problems under he assumpion ha he lead ime of he secondary source is one period less han ha of he primary source. For a problem wih arbirary lead imes, Whiemore and Saunders (1977) found he opimal policy o be of a very complex form. Chiang and Guierrez (1996) reaed a problem where he lead ime of he secondary source is a fracion of ha of he primary source, which in urn is shorer han he invenory review period. Chiang and Guierrez (1998) exended he problem o allow he order from he secondary source o be made more frequenly. Using dynamic programming echniques, hey derived useful properies abou he opimal policies for boh problems. Vlachos and Tagaras (2001) and 328

2 Operaions Research 53(2), pp , 2005 INFORMS 329 Tagaras and Vlachos (2001) considered undiscouned-cos problems where boh he wo lead imes and he review cycle are ineger muliples of a basic period and only one ordering from he secondary source is allowed per review cycle. This paper aims o offer a beer undersanding of acical ousourcing, where a firm is essenially a make-o-sock capaciaed producion sysem and employs an exernal source when is own in-house producion canno saisfy demands in a imely fashion. In our paricular seing, he firm has a Markovian and someimes sochasically monoone capaciy and faces an independen sochasic demand sream. Our goal is o characerize as much as possible he firm s opimal producion and ousourcing policies. The raionale behind he choice of our seing is explained in he following. The uncerainy in demand ofen sems from he uncoordinaed decision making by a vas number of cusomers, while he uncerainy in capaciy comes from a limied number of dependen and independen sources such as raw maerial suppliers, he labor force, machines, and power and waer supply, whose condiions are highly correlaed over ime. Hence, assuming capaciy o be Markovian and demand o be independen is a reasonable choice. Furhermore, here ofen exiss a cerain kind of coninuiy in ime in he condiions of he firm s various inernal componens like he labor force and machines ha ranslaes ino he coninuiy in ime in is overall capaciy. To beer undersand is implicaion, we adop he concep of sochasic monooniciy o rigorously capure his ime-coninuiy propery, which indeed enables us o derive useful and inuiive resuls. For a firm consrained by a fixed capaciy, Federgruen and Zipkin (1986a, 1986b) found he opimal policy o be of he modified base-sock ype, whereas he firm should produce as much as is allowed by he capaciy o a specified base-sock poin. Oher sudies in his vein include Glasserman and Tayur (1994, 1995), Kapuscinski and Tayur (1998), and Tayur (1992). Ciarallo e al. (1994) considered a random, and ye sill independen, capaciy sream. They found he opimal policy o be sill of he base-sock ype. Wang and Gerchak (1996) showed he opimal policy o be of a generalized base-sock ype when boh capaciy and producion yield are random. Some auhors used means oher han he inroducion of random capaciies o model uncerainies on he supply side. Arreola-Risa and DeCroix (1998) found he bes parameers for he s S policy when supply is disruped for periods of random duraions. Gupa (1996) analyzed a coninuous-review invenory sysem adoping he Q r policy whose supply is unavailable for periods of exponenially disribued duraions. Parlar (1997) considered a coninuous-review invenory conrol model wih random lead ime where he supply is alernaely available and unavailable for random duraions. Parlar e al. (1995) showed he opimaliy of he s S policy for an invenory sysem wih seup cos subjec o Markovian supply availabiliy. There are papers whose seings are close o ours. Scheller-Wolf and Tayur (2000) sudied a problem wih wo supply sources (effecively, in-house producion and ousourcing) and Markov-process driven producion bands (lower and upper bounds), coss, and demand levels. They discovered he opimaliy of base-sock-ype policies for he problem under cerain condiions. Bradley (2004, 2005) sudied he same producion-invenory-ousourcing problem in he coninuous-ime seing. He focused on he basesock policy, reaed he exernal source as capaciaed as well, and also considered he issue of in-house capaciy invesmen. In (2005), Bradley proved he opimaliy of he base-sock policy when boh sources form M/M/1 queues, while in (2004) he employed Brownian approximaion o assess he performances of he base-sock policy under more general condiions. For boh cases, he auhor was able o obain a closed-form expression for one of he base levels; and hrough numerical sudies, he showed he benefis brough forh by he ousourcing opion in reducing invenory coss and, more imporanly, in reducing in-house capaciy invesmen. On he oher hand, we only know of wo oher papers ha considered boh he Markovian-syle evoluion and sochasic monooniciy on he supply side. One is Song and Zipkin (1996), whose focal poin was he lead ime. They showed ha he opimal policy reains is convenional form, albei being supply-sae-dependen. The sochasic monooniciy propery also resuls in he monooniciy of he policy parameer under cerain circumsances. The oher is Yang e al. (2004). They sudied he opimal producion and order rejecion policy for a firm facing Markovian capaciy. The policy parameers were found o be monoone as well when he capaciy Markov process is sochasically monoone. In our paper, he firm is consrained by a Markovian in-house producion capaciy, and ye has o face a random demand sream. The firm is allowed o ousource par of is producion o an exernal source, which on op of he uni coss charges a seup cos. The ousourcing seup cos reflecs he exra bookkeeping and oher aciviies unlikely o be avoided when ordering from an exernal source, and i is one of he facors ha ses our paper apar from he aforemenioned recen papers. The oher imporan feaure of our model is is recourse naure. This is caused by our assumpion ha he ime poins a which realizaions of he curren capaciy and demand levels occur separae he ime poins a which decisions of producion and ousourcing are made. Afer making reasonable assumpions abou he producion and ousourcing lead imes and he various cos funcion forms, we are able o derive simple and inuiive properies abou boh ousourcing and producion policies. These properies include: (1) ha he opimal ousourcing policy is always of he s S ype, (2) ha he opimal producion policy is of he modified base-sock ype when cerain oher condiions are saisfied, (3) ha he opimal

3 330 Operaions Research 53(2), pp , 2005 INFORMS policies are of he base-sock ype when here is no ousourcing seup cos, and (4) ha he opimal policy parameers are monoone in he curren capaciy level when he capaciy Markov process is sochasically monoone and here is no ousourcing seup cos. These properies will serve as guidelines for real firms which pracice acical ousourcing. The res of his paper is organized as follows. In 2, we make he necessary echnical assumpions and esablish our mahemaical model for he problem; in 3, we derive main properies abou he opimal ousourcing and producion policies. In 4, we show ha some policy parameers are monoone in he curren capaciy level when he capaciy Markov process is sochasically monoone. We hen exend he resuls o he discree-sae discouned and undiscouned infinie-horizon cases in 5. In 6, we repor our compuaional resuls, which help jusify some of he condiions associaed wih our heoreical resuls and exemplify he benefis of adoping sae-dependen policies and uilizing he ousourcing opions. In 7, we presen our conclusions. 2. Model, Noaion, and Assumpions Throughou, we use he erms posiive, negaive, increasing, decreasing, (quasi-)convex, and K-(quasi-)convex, all in he nonsric sense. We use a posiive ineger o denoe a period and follow he convenion ha period 0 is he erminal period and ha afer period 0, all coss are 0. When he lead ime for producion is 0 and ha for ousourcing is 1, he dynamics of he invenory sysem are very easy o describe: A he very beginning of period, he firm s invenory level is I afer i receives he previous period s ousourcing amoun z +1. Once he firm observes is curren capaciy level, i decides he curren producion level x. Then, he firm observes and saisfies as much as possible he curren demand ha amouns o, wih is unsaisfied porion being backlogged. Finally, he firm decides he curren ousourcing level z. Therefore, for he res of he period, he firm s invenory holding backlogging level is J = I + x. Afer i has jus received he curren ousourcing amoun in he beginning of he nex period, he firm s invenory becomes I 1 = J + z = I + x + z. For he more general case where he lead imes for producion and ousourcing are, respecively, posiive inegers L and L O, he descripion of he sysem dynamics becomes more involved: A he beginning of period, he firm s invenory level is I 0. If LO 1, he firm firs receives he previously ousourced amoun and is invenory level increases o I = I 0 + z +L O; oherwise, I = I 0.IfL 1, he firm hen receives he previously produced amoun and is invenory level increases o I 1 = I + x +L ; oherwise, I 1 = I. Once he firm observes is curren capaciy level, i decides on he curren producion level x. If L = 0, he firm s invenory level immediaely increases o I 2 = I 1 +x ; oherwise, I 2 = I 1. Then, he firm observes and saisfies as much as possible he curren demand ha amouns o, wih is unsaisfied porion being backlogged, so ha is invenory level decreases o J = I 2. Finally, he firm decides on he curren ousourcing level z.ifl O = 0, he firm s invenory level immediaely increases o I 3 = J + z ; oherwise, I 3 = J. For he res of he period, he firm s invenory holding backlogging level is I 3, which will become I 1 0. When L 2, i is quie unnaural o assume ha he currenly known capaciy, updaed once every period, holds he upper limi for a producion run ha lass for L periods. However, his can occur when he firs producion sage, lasing beween 0 and 1 period, poses as he boleneck sage for he enire producion faciliy, or raw maerial availabiliy alone deermines he producion capaciy, or he las L 1 ol periods of he producion run involve merely ransporaion aciviies. Le p 0 be he uni cos of producion, K 0 be he seup cos of ousourcing, q 0 be he uni variable cos of ousourcing, and H I 0 wih H 0 = 0 which is decreasing when I<0 and increasing when I>0 be he cos of holding and backlogging. For sricly negaive s, we assume all coss are 0. Also, le 0 1 be he discoun facor over he span of one period. Suppose ha he firm s capaciy and demand levels in each period are muually independen and are independen of is decisions. Furhermore, suppose is demand levels in differen periods are muually independen as well. On he oher hand, suppose ha he firm s capaciies in differen periods form a Markov process. From now on, and up o 4, we work explicily under he coninuous-sae seing, whose precise definiion is made wih he exisence of cerain coninuous disribuion funcions o be inroduced soon. When disribuions are properly replaced by probabiliy masses, derivaives are replaced by differences, and inegraions are replaced by summaions, we will achieve corresponding resuls for policies and cos funcions under he discree-sae seing, where invenory, capaciy, and demand levels are all confined o ineger values. We op o firs work under he coninuous-sae seing because mos work in producioninvenory conrol are done in his seing. From 5 on, however, we swich back o he discree-sae seing. Now, we le coninuous funcion f C be he probabiliy densiy of capaciy being a he level when he previous period s capaciy is. In a degenerae special case, we will le all capaciies be deerminisic as 1 2.We le coninuous funcion f D be he probabiliy densiy of demand being a level. We define he above densiy funcions on + even hough hey vanish on 0. To guaranee he finieness of he cos funcions o be defined laer, we make he assumpion ha E <+ for every.

4 Operaions Research 53(2), pp , 2005 INFORMS 331 The firm s invenory posiion I a he beginning of period is is invenory level plus he oal on-order producion and ousourcing amoun: I = I 0 + +L =+1 x + +L O =+1 z. The dynamics of he invenory posiion is I 1 = I + x + z. For he wo cases where L O = L and L O = L + 1, we may define G I o be he minimum oal discouned expeced cos ha is conrollable from he momen and I are given. Firs, we suppose ha G 0 0 I 0 = p 0 I 0 (1) This reflecs he fac ha he invenory level I 0 in he end earns he firm p 0 I + 0 in salvage value and coss he firm p 0 I0 in owed producion. Throughou his paper, x + sands for max x 0 and x sands for max x 0. For = 1 2, we have he ieraive relaionship ha, if L O = L, { [ { G I = inf p x + E inf K 1 z > 0 + q z ( + L E [H L I + x + z = L )] + E G 1 1 I + x + z }] } z 0 0 x (2) and if L O = L + 1, { ( G I = inf p x + L E [H L I + x [ + E inf K 1 z > 0 + q z = L )] + E G 1 1 I + x + z ] } z 0 0 x (3) We have L O = L when ousourcing is as fas as in-house producion and L O = L + 1 when ousourcing is one period slower han in-house producion. Due o he exra ransacion and ransporaion aciviies involved in ousourcing, he laer case is more probable. For hese cases, Equaions (2) and (3) are almos he same excep for he invenory-cos erm, and he effec of a sricly posiive L is raher superficial. From now on, we shall concenrae on a special case of he second case where L = 0 and L O = 1. All our resuls can be exended o he wo more general cases wih only sligh modificaions. For oher cases where L O L 1orL O L + 2, we have found ha more variables will be needed o describe he sysem sae, and he corresponding conrol problem is much more complex. Now, when L = 0 and L O = 1, le G I = p I + G I, which can be hough of as he oal expeced cos from he presen o he end, plus he cos of he leas sunk effor needed o achieve he curren invenory level. We inroduce G I because i can ake advanage of he linear producion cos beer han G I can. Now we have G 0 I = 0 (4) and for = 1 2, { G I = q E +inf p q y+e [ H y +inf K 1 w y+ >0 + q p 1 w + E G 1 1 w w y ] } I y I + (5) Tha is, G I =inf G y I y I + (6) G y = p y+e H y +F y (7) F J = q J +inf { K 1 w J >0 + F w w J } (8) F w = q p 1 w+ E G 1 1 w (9) Noe he recourse naure of he above opimizaion problem due o he ime-separae realizaions of capaciy and demand levels wihin one single period. The opimal soluion y I for (6) consiues our opimal producion policy: I ells wha he opimal produce-up-o level should be when he curren capaciy level is and he invenory level righ before producion is I. On he oher hand, he opimal soluion w J for (8) consiues our opimal ousourcing policy: I ells wha he opimal ousource-up-o level should be when he curren capaciy level is and he invenory level righ before making he ousourcing decision is J. Assumpion 1. p, q, and he d I H I s are uniformly bounded by a consan M. Throughou his paper, we use d x f x as a shorhand for he ordinary derivaive df x /dx and use x f x y as a shorhand for he parial derivaive f x y / x y. Wih Assumpion 1, he derivaive of any cos funcion o appear laer can be ieraively found o be bounded by ±10 M/ 1. By he Dominaed Convergence Theorem (see, e.g., Wheeden and Zygmund 1977), we can inerchange he order of differeniaion and expecaion of any of hese cos funcions. 3. The Opimal Policies 3.1. The Opimal Ousourcing Policy Under some mild assumpions, we are able o show ha he opimal ousource-up-o level w J follows he s S ype, where boh parameers are -dependen. To prove his, we need o briefly review he conceps of K-convexiy and K-quasi-convexiy ha are closely relaed o he s S policy.

5 332 Operaions Research 53(2), pp , 2005 INFORMS Definiion 1. For a posiive consan K, f x is said o be K-convex if and only if for any x 1, x 2, and x 3 such ha x 1 <x 2 <x 3, we always have f x 2 x 3 x 2 f x 1 + x 2 x 1 f x 3 + K / x 3 x 1. Definiion 2. For a posiive consan K, f x is said o be K-quasi-convex if and only if for any x 1, x 2, and x 3 such ha x 1 <x 2 <x 3, we never have f x 1 <f x 2 f x 3 + K. A 0-(quasi-)convex funcion is merely a (quasi-)convex funcion. I is very well known and also very easy o prove ha f x is K-quasi-convex if f x is K-convex. Also, for 0 K 1 K 2, f x is K 2 -(quasi-)convex if i is K 1 -(quasi-)convex. The following well-known lemma links K-quasi-convexiy wih he s S policy and shows he preservaion of coninuiy and K-convexiy hrough a cerain minimizaion operaion. Poreus provided proofs for hese resuls (2002) (p. 142, Lemmas 9.11 and 9.12). Lemma 1. Suppose ha f x = inf K 1 y > x + g y y x and g y is coninuous and K-quasi-convex. Then, here will exis wo parameers s and S wih s S (boh allowed o be ± ), such ha for every x, f x = K 1 y x >x + g y x, where y x = S 1 x < s + x 1 x s, and, f x will be coninuous. Also, if g y is K-convex, hen f x will be K-convex as well. Wih he capaciaed producion, he preservaion of K-convexiy and coninuiy hrough he minimizaion operaion in (6) is no previously known. The following lemma shows ha K-convexiy and coninuiy can indeed be preserved. Lemma 2. Suppose ha K and C are wo posiive consans and g y is a K-convex and coninuous funcion. Then, f x = inf g y x y x + C will also be a K-convex and coninuous funcion. Proof. Because x x + C is a compac se and g y is coninuous, we can always find a y x x x + C such ha f x = g y x. Le x 1, x 2, and x 3 be such ha x 1 <x 2 <x 3. Suppose ha y x 1 >y x 3. Then, we mus have x 1 + C>x 3, y x 1 x 3 x 3 + C, y x 3 x 1 x 1 + C, and g y x 1 = g y x 3. Consequenly, we can swap he posiions of y x 1 and y x 3. So, from now on we may always assume ha y x 1 y x 3. Le y 2 = x 3 x 2 y x 1 + x 2 x 1 y x 3 / x 3 x 1. Then, y 2 x 2 x 2 + C jus because y x 1 x 1 x 1 + C and y x 3 x 3 x 3 + C. Hence, f x 2 = g y x 2 g y 2 (10) By he K-convexiy of g y and he posiiviy of K, no maer wheher y x 1 <y x 3 or y x 1 = y x 3, we always have g y 2 y x 3 y 2 g y x 1 + y 2 y x 1 g y x 3 + K / y x 3 y x 1 (11) Combining (10) and (11), and noing he way y 2 is defined, we obain f x 2 x 3 x 2 f x 1 + x 2 x 1 f x 3 + K / x 3 x 1 (12) Tha is, f x is K-convex. Because g y is coninuous, i is uniformly coninuous on any bounded and closed inerval. So, for any >0, we can find a 0 C such ha for any y 1 y 2 0 x+c + saisfying y 1 y 2 <, we would have g y 1 g y 2 <. For any x 0, le f 0 = inf g y y x + x, x + C. Then, we mus have 0 f 0 f x sup g x+ x g y y x x+ x < 0 f 0 f x+ x sup g x+c g y y x+c x+c + x < (13) I herefore follows ha f x+ x f x <. Assumpion 2. For any, H I is convex. Assumpion 2 is a common assumpion used in invenory conrol lieraure, which capures he case of linear holding and backlogging coss. A convex funcion is boh K-convex for any K 0 and coninuous. Assumpion 3. For any, K K 1. Assumpion 3 is needed for he preservaion of K-convexiy and coninuiy from G 1 I o F w. Theorem 1. For = and any, G I is K - convex and coninuous. As a consequence, for = 1 2, here are wo levels s and S (boh allowed o be ± ), wih s S such ha w J = S 1 J < s + J 1 J s solves he opimizaion problem (6). Proof. We prove his heorem by inducion. G 0 I has a value of zero, and herefore is K 0 -convex and coninuous. For = 1 2, suppose ha G 1 I is K 1 - convex and coninuous. Due o (9) and Assumpion 3, F w is coninuous and K 1 -convex, and hence K - convex. By (8) and Lemma 1, he propery for w J is rue, and F J is K -convex and coninuous. By (7) and Assumpion 2, G y is K -convex and coninuous. Finally, G I is K -convex and coninuous due o (6) and Lemma 2. Now define S and s so ha S = sup w F w =inf F w w + s = sup w F w K + F S and w S w J will be of he desired form due o Lemma 1. (14)

6 Operaions Research 53(2), pp , 2005 INFORMS 333 Now, we know ha he opimal ousourcing policy is s S, as long as he producion aciviy is also conduced in an opimal fashion, regardless of wha paricular form he opimal producion policy is in. A sufficien condiion for he producion policy o be of he modified base-sock ype is ha G I is quasi-convex or even convex in I. For wo special cases, we are able o provide such a sufficien condiion Producion Policy for he Deerminisic Capaciy Case In his subsecion, we assume ha capaciies are deerminisic. Then, many previous capaciy-dependen eniies become capaciy independen, and we will drop he corresponding capaciy dependencies in heir noaions accordingly. Previously, many quaniies associaed wih = 0 were no defined. Here, for convenience, we make he following definiions: q 0 = p 0, H 0 I = 0, and F 0 J = q 0 J. Then, according o (4), (6), (7), (8), and (9), we have for = 1 2, ha F J = q J +inf K 1 w J >0 + F w w J (15) F w = q p 1 w+ G 1 w (16) G 1 I =inf G 1 y I y I + 1 (17) G 1 y = p 1 y +E H 1 y 1 +F 1 y 1 (18) Therefore, i is also rue ha F J = q J + inf K 1 w J>0 + q p 1 w + p 1 y + E H 1 y 1 + F 1 y 1 w J w y w + 1 (19) I is also easy o check ha G 0 y = 0. Equaion (19) reflecs he fac ha, wih capaciies being deerminisic, he decision he firm has o make wihin each period is no longer he wo-layered one wih recourse as in he random capaciy case, bu in fac a single-layered one. We may sill preend ha he produce-up-o level is coningen upon he invenory level righ before producion, bu indeed i can be well deermined a he ime when he previous period s ousourcing decision is being made, because here is no uncerainy in he inerim beween he wo menioned ime poins. When he demand disribuion is of a special form and oher reasonable condiions are saisfied, we can show ha he opimal producion policy, when sill being considered as coningen upon he invenory level righ before producion, is of he modified base-sock ype. Le he soluion of (15) be w O J, he soluion of (17) be y 1 O I, and he soluion of (19) be w J y 1 J. Then, we have w J = wo J and y 1 J = yo 1 wo J. Laer, we will always wrie w J in he place of wo J. Noe ha he opimal ousourcing level in period is w J J and he opimal producion level in period 1 is y 1 J w J. From 3.1, we know ha G 1 y and G 1 w are boh K 1 -convex and ha F w and F J are boh K -convex. Define S and s so ha { S =sup w F w =inf F w w + s =sup w F w K + F S and w S (20) Then, we know ha w J = S 1 J < s + J 1 J s (21) The following lemma says ha when ousourcing is more expensive han producion, here will be no ousourcing unless he producion capaciy has been exhaused. We cerainly canno have his resul when he capaciies are random. Lemma 3. If q p 1 > 0, hen he complemenary slackness relaion w J J w J + 1 y 1 J = 0 is rue. Proof. Suppose ha w y is a feasible soluion for (19) ha saisfies boh w>j and y<w+ 1. Le = min w J w+ 1 y. Then, w y is anoher feasible soluion for (19) wih a cos saving of a leas q p 1. Therefore, he opimal soluion w J y 1 J for (19) mus saisfy he desired complemenary slackness relaion. We now suppose ha q p 1 is posiive for all s. By (21), we have w J = S >J when J < s. So, by Lemma 3, we know ha y 1 J = S + 1 when J< s. On he oher hand, his also means ha yo 1 S S = + 1. By (21), we also have w S = S. Therefore, we have y 1 J = S + 1 when J< s or J = S (22) We will call a funcion f x differeniable when i has boh lef and righ derivaives, while hey do no necessarily equal each oher. We say d x f x 0 is wihin a cerain bound if boh of is derivaives are wihin ha bound. The following lemma shows ha d y G y is bounded. Is lower bound will be of paricular use o our laer proof of Theorem 2. Lemma 4. When he F J s, F w s, G I s, and G y s are differeniable, for = 0 1 2, i is always rue ha p q + d I H d y G y p p 1 + d I H +

7 334 Operaions Research 53(2), pp , 2005 INFORMS Proof. We prove he lower bound firs. For = 0, he inequaliy is obvious. Now, we assume ha 1. By (15), we have d J F J q. Then, by (18), we have d y G y p q + E d I H y (23) The desired inequaliy follows immediaely due o he convexiy of H I. We now prove he upper bound. By (15), (16), (17), and (18), we easily obain ha for = 1 2, d y G y p + d I H + + sup d J F J J + d J F J q + sup d w F w w + (24) d w F w q p 1 + sup d I G 1 I I + d I G 1 I sup d y G 1 y y + By combining hese inequaliies wih he fac ha G 0 y = 0, we ge he desired resul. Now, we inroduce he wo conceps ha are essenial o our derivaions leading o he modified base-sock producion policy. Definiion 3. A funcion f x is said o be wih Polya frequency of order 2 (PF 2 ) if for any x 1 <x 2 and y 1 <y 2,i is rue ha f x 1 y 1 f x 2 y 2 f x 1 y 2 f x 2 y 1. Definiion 4. A funcion f x is said o be upward if i changes sign a mos once from o + (or does no change sign a all) as x raverses from o +. A coninuous and piecewise coninuously differeniable funcion f x will be quasi-convex if d x f x is upward. I is known ha if f x is PF 2 and g x is upward and bounded, hen he convoluion r x = + g y f x y dy is upward (see, e.g., Karlin 1968, Theorem 3.1 of Chaper 5, p. 233). Also according o Karlin, a wide variey of commonly seen probabiliy densiies are PF 2. They include such discree disribuions as Poisson and binomial and such coninuous disribuions as uniform, exponenial, and normal. The following heorem saes ha he opimal producion policy is of he modified base-sock ype when he demand disribuion is PF 2 and cerain oher condiions are saisfied. Theorem 2. Under he condiions ha for = 1 2, (a) q p 1 >0 (b) p p 1 +d I H 0 0 (c) p p 1 +d I H p p 1 + d I H f D d + (d) + p 1 max q q 1 d I H 1 f D d 0 (e) f D is PF 2 we have ha for = 0 1 2, G y is coninuous and piecewise coninuously differeniable, d y G y is upward, d y G 0 0, and d y G 0. Consequenly, for = 0 1 2, for some base-sock level B 0, he opimal producion policy y O w is governed in he modifiedbase-sock fashion by y O w = w + 1 w < B + B 1 B w B + w 1 w > B Because his proof is lenghy, i is included in he appendix for he ineresed reader. All condiions imposed on he heorem excep Condiion (d) are easy o undersand. Condiion (a), as menioned before, reasonably requires ha ousourcing be more expensive han producion; Condiion (b) saes ha he backlogging cos more han offses he savings achievable from delayed producion; and Condiion (c) saes ha he holding cos more han offses he savings achievable from advanced producion. Meanwhile, Condiion (e), saing ha he demand disribuion is PF 2, plays a pivoal role in he preservaion of he upwardness of d y G y hrough he aforemenioned resul concerning he convoluion beween an upward funcion and a PF 2 funcion. For a beer undersanding of Condiion (d), suppose ha here are posiive consans h and b for all s ha make H I = h I + + b I. Then, he condiion can be expressed as Pr p p 1 + h p p 1 + h + max q q 1 + b 1 p + b (25) Clearly, Condiion (d) implies he following: Compared o he capaciy level, he demand level is no oo huge; and compared o producion and holding coss, ousourcing and backlogging coss are no oo high. We will discuss he necessiy of Condiions (d) and (e) for he validiy of he base-sock-ype opimal producion policy in In paricular, our numerical analysis showed ha he PF 2 requiremen on he demand disribuion is necessary, and ha Condiion (d) migh have room for improvemen, bu canno be eliminaed Producion Policy for he Zero-Ousourcing Seup-Cos Case In his subsecion, we assume ha here is no ousourcing seup cos. In view of Theorem 1, we have s = S, and he ousourcing policy collapses ino a base-sock policy, while a he same ime, he K -convexiy of F w, F J, G y, and G I collapses ino heir convexiy, and he producion policy is of he modified base-sock ype, where for consan B = sup y yg y <0, we can have y I = max I min I + B. Therefore, we have he following heorem.

8 Operaions Research 53(2), pp , 2005 INFORMS 335 Theorem 3. Suppose ha K = 0 for = 1 2. Then, G I is convex for = As a consequence, for = 1 2, here exiss a level B such ha y I = max I min I + B solves he opimizaion problem (6). The necessiy of he zero-ousourcing seup-cos assumpion for he validiy of he base-sock-ype opimal producion policy will be furher discussed in This resul is hardly surprising given ha wihou he involvemen of any seup cos, all involved cos funcions are now convex. However, even in he curren more resriced seing, our model is differen firs of all from single-source models and somewha from oher dual-source models. The firs poin will be obvious from he fac ha differen uni coss, namely p and q, are sill assumed for producion and ousourcing. The mos imporan differences beween our model and oher dual-source models are: (1) mos dualsource models do no impose capaciy limiaions (Scheller- Wolf and Tayur 2000 and Wrigh 1968 are excepions); (2) almos all dual-source models assume ha a faser source coss more han a slower source wih he implici rade-off being beween speed and cos, while in our model he faser source (in-house producion) needs he help of he more cosly and probably slower source (ousourcing) due exclusively o is finie capaciy; and (3) our model involves opimizaion wih recourse, while almos all dual-source models involve simulaneous mulivariable opimizaion (see, e.g., Chiang and Guierrez 1998; Scheller-Wolf and Tayur 2000; Sehi e al. 2001, 2003; Tagaras and Vlachos 2001). 4. Sochasically Monoone Capaciy 4.1. General Resuls Definiion 5. For wo random variables X and Y,wesay X is sochasically larger han Y, denoed as X s Y, when for every real number z, wehavepr X z Pr Y z ; ha is, when X is more likely o be larger han any given number han Y. We say ha a firm s capaciy Markov process is sochasically monoone when for any,, and any 0, we have + s ; ha is, when a higher curren capaciy level is more prone o lead o a higher capaciy level in he ensuing period. Sochasic monooniciy on he supply side was also considered by Song and Zipkin (1996), whose focal poin was he lead ime. There, he sochasic monooniciy propery resuled in he monooniciy of he policy parameer under cerain circumsances Similar monooniciy was also esablished in he lieraure on demand process, e.g., Song and Zipkin (1993). When wo random variables X and Y saisfy ha X s Y, i is known ha E f X E f Y for any increasing funcion, and in paricular, E X E Y (see, e.g., Ross 1983). So, if he capaciy Markov process is sochasically monoone, hen for any increasing funcion f x,,, and any 0, we will have E f + E f. One discree-sae example for he sochasic monooniciy of he firm s capaciy Markov process comes from he following N -machine example. The firm has N machines, where each machine has wo saes, up and down. In each period, he number of machines ha are up is he firm s capaciy. Suppose ha each machine s sae evolves as a Markov chain, wih he probabiliy of i being up in he nex period given ha i is up in he curren period being p, and he probabiliy of i being down in he nex period given ha i is down in he curren period being q. Tha is, each machine is up and down inermienly, and he up ime follows he geomeric disribuion wih mean 1/ ˆp and he down ime follows he geomeric disribuion wih mean 1/ ˆq, where we have le ˆp = 1 p and ˆq = 1 q. Obviously, he firm s capaciy evolves as a Markov chain as well. The average capaciy E = N ˆq/ ˆp +ˆq because of he balance requiremen pe + ˆq N E = E, and he limiing disribuion of he chain is binomial wih parameers N and ˆq/ ˆp +ˆq. In Yang e al. (2004), i has been proven ha he Markov chain is sochasically monoone if and only if p + q 1. The following heorem says ha under he sochasic monooniciy hypohesis, a higher curren capaciy level leaves he firm beer off for he ime o come. Theorem 4. Suppose ha he firm s capaciy Markov process is sochasically monoone. Then, for = 0 1 2, G I is a decreasing funcion of. Proof. We prove his heorem by inducion. Firs, G 0 I is independen of. Suppose for some = 1 2, G 1 I decreases in. By (9) and he aforemenioned fac regarding condiional expecaions over a monoone funcion, we know F w decreases in. Afer increases, as for he opimizaion problem (8), is objecive funcion decreases while he feasible region is unouched. So F J decreases in. By (7), we know G y decreases in. Afer increases, as for he opimizaion problem (6), is objecive funcion decreases while he feasible region expands. Therefore, G I decreases in No Seup Cos for Ousourcing The following Theorem 5 says ha under he sochasic monooniciy hypohesis and when ousourcing seup coss are zero, a higher curren capaciy level makes i less appealing o have a higher invenory level. As a consequence, he base-sock levels for boh ousourcing and producion decrease in he curren capaciy level. Theorem 5. Suppose ha he firm s capaciy Markov process is sochasically monoone. Then, for = 0 1 2, I G I is an increasing funcion of. As a consequence, for = 1 2, boh S and B are decreasing funcions of.

9 336 Operaions Research 53(2), pp , 2005 INFORMS Proof. We prove his heorem by inducion. Firs, I G 0 I is independen of. Now suppose for some = 1 2 I G 1 I increases in. Then, by (9) and he aforemenioned fac regarding condiional expecaions over a monoone funcion, w F w increases in. According o Theorem 1, S = sup w wf w <0. Consequenly, S decreases in. Because now s = S, we have he following ideniy due o (8): q if J< S J F J = (26) q + w F J if J S By definiion, w F J 0 for J S. This, along wih he monooniciy resul on S, forces J F J o increase in. By (7), we can easily conclude ha y G y increases in. Hence, B as defined in 3.3 decreases in. By (6), we now have he following ideniy: y G I + 0 if I< B I G I = 0 if B I B (27) y G I 0 if I> B The monooniciy of B, B, and yg y ogeher force ha I G I increase in. We will discuss he necessiy of he assumpions of zeroousourcing seup coss and sochasically monoone capaciy Markov process for he monooniciy of he opimal policy parameers in We also noe ha he curren resul is similar in spiri o he one given in Sehi e al. (2003, Theorem 3.3). 5. Exension o he Infinie-Horizon Cases From his secion on, we assume ha he invenory sysem s saes are discree. As menioned in 2, all resuls in he previous wo secions will remain valid once proper replacemens are made. Here, we inheri he discree-sae counerpars of all previous assumpions. Also, we assume ha all given parameers are saionary, and will hereafer suppress he signs associaed wih hem as long as no confusion shall arise. Moreover, we assume ha he capaciy Markov chain is irreducible and he demand is nonzero (here would be nohing o prove if i were zero). For any ineger-variable funcion f x,wele x f x = f x+1 f x. Also, le h 0 = I H 0 and h = I H + be he minimum and maximum uni holding coss, respecively, and b 0 = I H 1 and b = I H be he minimum and maximum uni backlogging coss, respecively. All hese values are finie by Assumpion 1. Our goal is o show ha all he properies enjoyed by he cos funcions and policies are sill reained by he counerpars of he funcions and policies in boh he discouned and undiscouned discree-sae infinie-horizon cases. In he following, we jus presen he mos imporan resuls. All definiions, inermediae echnical lemmas, and proofs are relegaed o he appendix The Discouned Case Assumpion 4. h 0 > 0. Tha is, holding invenory incurs a leas some cos. Assumpion 5. q p. Tha is, ousourcing is more expensive han nex-period producion. Theorem 6 is he pivoal resul in his subsecion. I saes he convergence of G I as ends o +. The resuls following his heorem essenially confirm ha under addiional mild and reasonable assumpions, all our previous resuls are valid when he ime horizon becomes infiniely long. Theorem 6. As ends o +, here exiss a poinwise finie limi, say G I, ofg I. In any region of he I -space wih a bounded I-projecion, he convergence is also uniform. Theorem 7. G I solves (5) where i replaces G I on he lef-hand side and G 1 I on he righ-hand side. The soluions y I = I + x I and w J = J + z J for (5) wih G I on boh sides consiue he opimal producion and ousourcing policies for he saionary infinie-horizon problem. Theorems 8 and 9 confirm ha all properies possessed by he finie-horizon policies are possessed by he infinie-horizon policies as well. Theorem 8. G I is K-convex. Consequenly, here are wo levels s and S wih s S such ha w J = S 1 J < s + J 1 J s. In he firs special case where he firm has a consan capaciy, he condiions corresponding o (a), (b), (c), and (d) in Theorem 2 are saisfied, and he demand disribuion is PF 2, hen y G y is upward, y G 1 0, and y G 0, and consequenly, here exiss a level B 0 1 such ha y I = max I min I + B ; and in he second special case where K = 0, G I is convex and consequenly, s = S and here again exiss a level B such ha y I = max I min I + B. Theorem 9. Suppose ha he firm s capaciy Markov process is sochasically monoone. Then, G I decreases in ; and in he second special case where K = 0, I G I increases in and, consequenly, boh S and B decrease in The Undiscouned Case We are o show ha under addiional mild and reasonable assumpions, all he previous resuls are sill valid when here is no discoun over ime and he crierion is o minimize he expeced ime-average cos. Here, we inheri all previous assumpions wih being replaced by 1. Also, whenever he concep of he curren period is needed, we assume his period o be period 0 (periods afer he curren period will be associaed wih negaive numbers).

10 Operaions Research 53(2), pp , 2005 INFORMS 337 Le be he se of all hisory-dependen policies, prescribing acions x and z for he firm in any period given he evoluionary hisory of he sysem. For a sysem ha, saring from he curren sae 0 I 1, evolves under he Markovian capaciy sream, he independen demand sream, and an arbirary policy in, we use I o denoe he sysem s invenory level in he very beginning of period. The undiscouned-cos problem seeks a policy in ha solves { g = inf lim sup 1/ E [ s=0 p x s + H I 0 s + x s s ] } + K 1 z s > 0 + q z s 0 I 0 (28) Theorem 2.1 in Chaper V of Ross (1983) says ha, as long as G I in (29) is uniformly bounded in he enire I -space, he above g and can be solved by solving he alernaive g + G I = min { p x + E H I + x + E min K 1 z > 0 + q z + E G I + x + z z 0 0 x } (29) where G I bears he inerpreaion of he relaive cosliness of he sae I when he sysem evolves under he opimal policy saring from i. The aforemenioned requiremen is oo srong for our problem here. However, a closer sudy of he proof offered in Ross reveals ha we acually only need lim sup E G I G 0 I 0 = 0 (30) for any 0 I 0 -pair and any 0, where 0 conains he opimal policy. In his secion, we solve (29) firs and hen prove (30) for he hus-found G I. We now add an superscrip o any infinie-horizon eniy associaed wih discoun facor. For he solvabiliy issue, he required g can be chosen as a limi poin of g 0 I 0 = 1 G 0 I 0 as 1 for a fixed 0 I 0 = 0 -pair, and for any and I, he required G I can be chosen as a limi poin of G I = G I G 0 I 0 as 1. Theorem 10. In any sequence n n = 0 1 2, here exiss a subsequence, say i n n = 0 1 2, such ha g i n 0 I 0 converges as n ends o +. Assumpion 6. E > 0. Assumpion 7. E 2 <+. Assumpion 8. There is a posiive consan R such ha E C 2 1 R for any 1 and 2. Theorem 11. In any sequence n n = 0 1 2, here exiss a subsequence, say i n n = 0 1 2, such ha for any and I, G i n I converges as n ends o +. Use an arbirary sequence n n = converging o 1 as he -sequence assumed in Theorem 10, and hen use he resuling -subsequence as he -sequence assumed in Theorem 11. We ulimaely obain a subsequence i n n = 0 1 2, as well converging o 1, such ha g i n 0 I 0 converges, say o g, and ha G i n I for any and I converges, say o G I. The following Theorem 12 corresponds o Theorem 7 in he discouned case. Theorem 12. g and G I, hus-defined, solve Equaion (29). The soluions y I = I + x I and w J = J + z J for (29) would consiue he opimal policy for his undiscouned case had we proved (30). Theorems 13 and 14 confirm ha all properies possessed by he finiehorizon policies are possessed by he undiscouned averagecos crierion policies as well. Theorem 13. G I is K-convex. Consequenly, here are wo levels s and S wih s S such ha w J = S 1 J < s + J 1 J s. In he firs special case where he firm has a consan capaciy, he condiions corresponding o (a), (b), (c), and (d) in Theorem 2 are saisfied, and he demand disribuion is PF 2. Then, y G y is upward, y G 1 0, and y G 0, and consequenly, here exiss a level B 0 1 such ha y I = max I min I + B ; and in he second special case where K = 0, G I is convex and, consequenly, s = S, and here again exiss a level B such ha y I = max I min I + B. Theorem 14. Suppose ha he firm s capaciy Markov process is sochasically monoone. Then, G I decreases in ; and in he second special case where K = 0, I G I increases in and, consequenly, boh S and B decrease in. Afer making some addiional mild assumpions, we can finally prove (30) for he soluion G I we have found for (29). Assumpion 9. b 0 > 0. Tha is, backlogged orders incur a leas some cos. Assumpion 10. For any and posiive I, E C D I I for some sricly posiive consan. Assumpion 10 connoes ha he firm s average capaciy does no exceed he average demand by oo much. This is paricularly rue when is capaciy Markov chain is posiive recurren and has a finie long-run average capaciy. Theorem 15. The ideniy (30) is rue for a properly defined 0, and hence (29) is he opimaliy equaion for our undiscouned-cos case.

11 338 Operaions Research 53(2), pp , 2005 INFORMS Table 1. Theorem 2 assumpions: Only condiion (d) is violaed. y8 I I = = Discussion In his secion, compuaional resuls are presened ha help o jusify some of he condiions associaed wih our earlier resuls, and show he benefis of aking he Markovsyle evoluion of he producion capaciy ino managerial consideraion as well as he advanages of he ousourcing opporuniies Seings for he Compuaional Sudy Because he examples are all of he discree-sae naure, we can resor o he brue-force dynamic programming echnique o calculae he opimal coningency plans. In all our examples, parameers are saionary. Hence, he signs will be suppressed whenever possible. As a defaul, we le p = 10, q = 11, K = 30, = 0 99, and H I = h 0 I + + b 0 I wih defaul values h 0 = 1 and b 0 = 2. For he sochasic producion capaciy, we use wo models: he aforemenioned N -machine model as he defaul one and anoher one ha we name he polarized model. For he N -machine model, our defaul parameers are N = 20, p = 0 9, and q = 0 7. For hus-chosen p and q, he firm s capaciy Markov chain is sochasically monoone, and is long-run expeced capaciy is E = N ˆq/ ˆp +ˆq = / = 15. In he polarized model, he capaciy can only change 1 uni per ransiion. Specifically, suppose ha he maximum capaciy is N. Then, for = 1 2 N 1, P = 1 = 1 /N, P = + 1 = /N; and P = 1 0 = 1, P = N 1 N = 1. We call his model polarized because in i he firm s capaciy ends o say a eiher very high or very low levels. Because of symmery, he longrun expeced capaciy in he model is N/2. Noe ha he N -machine model is reasonably close o realiy, while he polarized model is an exreme case where fuure capaciy levels are srongly dependen on he curren capaciy level. We use wo differen demand disribuions. Our defaul demand disribuion is a modified Poisson disribuion defined as follows. Le 0 and be wo inegers. For i = 0 1 0, le i = e i /i!, he probabiliy densiy a i for a Poisson disribuion wih mean. Le = , and i = i / for i = Our modified Poisson disribuion has P = i = i for i = 0 1 0, and P = i = 0 for all oher is. I is easy o prove ha his disribuion is PF 2. Our defaul parameers for his model are = 14 and 0 = 35. We also have he occasion o use a discree uniform demand disribuion U a b, which is defined on a a+1 b, wih a and b being posiive inegers and a b. For such a disribuion, P = i = 1/ b a + 1 for i = a a + 1 b, and hence E = a+b /2 and Var = b a b a+2 /12. This disribuion is also PF 2. In he following, all our examples are wih he defaul configuraions unless explicily saed oherwise Discussion of Necessary Condiions We demonsrae examples found from he compuaional sudy in which he claims made by our earlier heorems collapse under he relaxaion of cerain assumpions On Theorem 2 s Assumpions. Theorem 2 has five condiions, of which he firs hree are quie naural. Therefore, we only discuss Condiions (d) and (e). In he following examples, Condiions (a) (c) are always saisfied. We firs invesigae he necessiy of Condiion (d). In doing his, we use a PF 2 demand disribuion wih only wo possible values wih P = 7 = P = 8 = 0 5. The consan capaciy level is eiher 2, 3, or 4. Then, Condiion (d) is violaed in ha he lef-hand side of (25) is 1 while is righ-hand side is Under he hree differen consan capaciy levels, Table 1 repors he opimal produce-up-o levels y8 I a ime = 8 and various invenory levels I. In he enry corresponding o = 2 and I = 22, i is opimal o bring he invenory level up o 24 insead of keeping he invenory level unchanged. This implies ha he opimal producion policy is no of he modified base-sock ype. In he able, we have bold produce-up-o levels ha deviae from he supposed modified base-sock policy. Table 2. Theorem 2 assumpions: Only condiion (e) is violaed. y10 I I = =

12 Operaions Research 53(2), pp , 2005 INFORMS 339 Table 3. Theorem 3 assumpion: The ousourcing seup cos is nonzero. y6 I I = = We hen sudy he necessiy of Condiion (e), which insiss ha he demand disribuion be PF 2. Le H I = 2 I I, he consan capaciy level be eiher 5, 6, or 7. Le he demand disribuion be non-pf 2 in ha P = 0 = 0 25, P = 2 = 0 45, P = 20 = 0 3, and P = = 0 for all oher s. This example does no violae Condiion (d) because he lef-hand side of (25) is 0.3, while is righ-hand side is Table 2 repors he opimal produce-up-o levels y10 I. As in Table 1, we have bold produce-up-o levels ha deviae from he supposed modified base-sock policy On Theorem 3 s Assumpion. Theorem 3 saes ha he opimal producion policy is of he modified basesock ype under sochasic capaciy levels when K = 0. To show he necessiy of he condiion K = 0, we use he polarized capaciy model wih N = 20 and he uniform discree demand disribuion U 9 11, and le he ousourcing seup cos K be a is defaul value 30. We find ha he opimal producion policy is no of he modified base-sock ype. Table 3 repors some of he irregulariies in he opimal producion policy a = On Theorem 5 s Assumpions. Theorem 5 says ha he opimal policy parameers decrease wih he curren capaciy level when he ousourcing seup cos is 0 and he capaciy Markov process is sochasically monoone. We firs demonsrae he necessiy of he zero-seup-cos assumpion. We use he defaul modified Poisson demand wih = 3 and he N -machine capaciy model wih N = 6 and p = q = 0 99, and le he ousourcing seup cos K = 50. Our sudy shows ha he parameers of he opimal ousourcing policy may be nonmonoone wih he capaciy. Table 4 shows some of he parameers a ime = 10. Nex, we show he necessiy of he assumpion ha he capaciy Markov process is sochasically monoone. We use a sochasically nonmonoone capaciy Markov chain where P = 3 0 = 1, P = 0 1 = 1, P = 2 2 = P = 3 2 = 1/2, P = 0 3 = P = 1 3 = P = 2 3 = 1/3, and P = n m = 0 for oher m ns. We le he demand disribuion be of he defaul modified Poisson form wih = 1. When he ousourcing seup cos is 0, Table 4. Theorem 5 assumpions: The ousourcing seup cos is nonzero s 10 S he opimal ousourcing policy becomes a base-sock policy wih parameers S and he opimal producion policy is of he modified base-sock ype wih parameers B. We find ha hese parameers are no monoone in a = 20, as shown in Table 5. Noe ha here is no definiion for B when = Value of he Markovian Srucure In he following, we always use he infinie-horizon resuls o conduc our comparisons. Firs, we compare he performance of our model wih ha of he model in which he Markov-syle evoluion of he capaciy is ignored; i.e., he performance of he model in which producion capaciies over differen periods are considered independen. The opimal policy neglecing he Markovian propery of he capaciy can be found by leing P = n +1 = m be he seady-sae probabiliy of he capaciy level being a n regardless of m. For our N -machine model, his probabiliy can be easily compued because we know is seady-sae disribuion is binomial. For any oher model, we can obain he corresponding seady-sae disribuion by solving P = and i = 1, where P is he one-sep ransiion marix for he capaciy evoluion and is he row vecor conaining all seady-sae probabiliies. Each cos erm repored below is he seady-sae average cos a invenory level 0; i.e., i G i 0. In Table 6, we show he percen savings ha our Markovian policy can achieve over he aforemenioned non- Markovian policy when he problem parameers in he defaul seing of N -machine capaciy and modified Poisson demand are se a various N,, p, and q values. The closer p + q is o 2, he less independen he capaciy levels over adjacen periods are, and he more one can predic fuure capaciy levels from he curren capaciy level. In essence, he Markovian policy heeds he predicion, while he non-markovian policy does no. Hence, he percen savings are greaer for p q -pairs wih p + q close o 2. On he oher hand, when p + q is close o 0, he capaciy levels end o oscillae from period o period, and no much acion can be aken o uilize he accurae Table 5. Theorem 5 assumpions: The capaciy Markov process is sochasically nonmonoone S B 20 N/A 2 1 2

13 340 Operaions Research 53(2), pp , 2005 INFORMS Table 6. Comparison wih he non-markovian policy. N p q = (0.1, 0.1) (0.5, 0.5) (0.9, 0.7) (0.9, 0.9) (0.95, 0.95) predicion. Therefore, he percen savings a p = q = 0 9 or 0.95 are, in general, larger han hose a p = q = 0 1. There is also a general rend ha he fewer machines he firm has, he greaer percen savings i can achieve. This is largely because he firm s capaciy level is more volaile when here are fewer machines, and consequenly he informaion abou he curren capaciy level is more valuable for predicing fuure levels. In general, he percen savings are no very impressive under our defaul N -machine capaciy seing. When we use he polarized capaciy model (wih N = 20), we can obain more savings. The savings are more dramaic when he discree uniform demand disribuion is used. Such a demand disribuion also gives us a chance o adjus no only he mean demand level, bu also he demand variance. In Figure 1, we display he percen cos savings under he new seings when he paricular demand disribuion varies. In he figure, he discree uniform demand disribuion U a b is represened by wo parameers, he mean E = E = a + b /2 and he sandard deviaion Sd = Var = b a b a + 2 /12. A larger Sd represens a larger demand variabiliy, and Sd = 0 implies a deerminisic demand. In Figure 1, he percen cos savings decrease in a fairly dramaic way as he demand variance increases. When he demand variance is higher, he benefi of knowing more abou fuure capaciy levels is more likely o be offse by he uncerainy in fuure demand levels. Hence, he degree o which he qualiy of he firm s producion and ousourcing decisions depends on he qualiy of fuure capaciy predicions should decrease as he demand variance increases. Also, he percen cos savings firs increase and hen decrease wih he mean demand level. In general, he higher he demand level, he more relevan he capaciy level is in he decision process. Therefore, he rising mean demand levels make he benefis of knowing more abou fuure capaciy levels more useful. However, when he mean demand level becomes much higher han he mean capaciy level, ousourcing coss will dominae producion coss under boh of our reamens, and hus he percen cos savings from producion will begin o decrease. In Figure 2, we display he percen cos savings under differen uni ousourcing coss q and differen demand disribuions U a b wih N = 8. In Figure 2, he percen cos savings increase wih he ousourcing cos. The firm can use ousourcing o parially offse he imperfecion in is producion decision. However, more cosly ousourcing makes his less possible, and a he same ime makes he qualiy of he producion decision more accounable for he combined qualiy of producion and ousourcing decisions. Thus, knowing more abou fuure capaciy levels becomes more imperaive when he ousourcing cos increases. We can repea he observaion in Figure 2, as in Figure 1, ha he percen cos savings firs increase and hen decrease wih he mean demand level. Figure 1. Comparison wih he policy no using he Markovian propery under differen demand disribuions. Figure 2. Comparison wih he policy no using he Markovian propery under differen ousourcing coss. 6.00% 16.00% Percenage Savings 5.00% 4.00% 3.00% 2.00% 1.00% Sd = 0 Sd = Sd = Sd = 2 Percenage Savings 14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% Demand disribuion U [0,6] U [0,8] U [0,10] U [0,12] U [0,14] U [0,16] 0.00% % Demand Mean E Uni Ousourcing Cos

14 Operaions Research 53(2), pp , 2005 INFORMS 341 Table 7. Comparison wih he policy wihou he ousourcing opion. Varied K parameer: Savings Varied q parameer: Savings Varied b 0 h 0 (2,1) (4,1) (8,1) (2,4) (2,8) parameer: Savings Varied parameer: Savings Drivers of he Ousourcing Opion Value We now compare he policies wih and wihou ousourcing opporuniies. The laer policy can be obained by making ousourcing seup cos K =+. Obviously, he opporuniy o ousource helps he firm o reduce is oal cos. Wha we are ineresed in knowing is he magniude of he cos reducion and wha facors may affec hese magniudes. The percen cos savings due o he ousourcing opporuniy under various parameer changes are lised in Table 7. When ousourcing is more expensive, he firm uses ousourcing less, and he difference beween having and no having he ousourcing opion becomes narrower. In paricular, we have varied he ousourcing seup cos K and he uni ousourcing cos q, respecively, while keeping oher parameers a heir defaul values. The opion of ousourcing is also more valuable when invenory coss are higher. Also, he percen cos savings increase more rapidly wih he uni backlogging cos b 0. Afer all, he ousourcing opporuniy allows orders backlogged due o in-house capaciy slack o be filled quickly. The value of he ousourcing opion increases quickly wih he demand level. When = 16 or 18, which is larger han E = 15, he cos funcion for he wihouousourcing case sill converges due o he help of he discoun facor. However, during he acual execuion of is opimal policy, he firm will have o le he backlogging level pile up oward +, which is no realisic a all. In his sense, when he average demand level is above he average capaciy level, he opion of ousourcing guaranees a feasible sable soluion, which is oherwise unavailable. Finally, we sudy he benefi of using he ousourcing opion under differen mean demand levels and demand variabiliies by employing he discree uniform demand disribuion under differen values of E and Sd. The percen cos savings resuls are lised in Figure 3. These resuls clearly show ha even under he same mean demand level, using he ousourcing opion can achieve more cos savings when demand levels are more volaile, while a he same ime, hey sill show ha he ousourcing opion is more valuable under larger mean demand levels. We may explain he firs new rend by drawing an analogy wih resuls in queueing heory: More volaile demand levels, even under he same mean, are bound o lead o higher backlogging levels wihou he remedy of an ousourcing opion. Hence, Figure 3. Percenage Savings 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% Comparison wih he policy no using he ousourcing opion under differen demand disribuions. E = 14.5 E =14 E = 13.5 E =13 E = Sandard Deviaion (Sd) of Demand Disribuion he difference of he resuls beween having and no having he ousourcing opion widens as demand variance grows. 7. Concluding Remarks We have sudied he operaional problem of how a firm, facing a Markovian sream of available capaciy levels and an independen random sream of demand levels, can opimally carry ou is day-o-day producion and ousourcing aciviies. Under very reasonable cos assumpions, we have shown ha he firm s ousourcing policy should always be of he capaciy-dependen s S ype; and under some more assumpions, we have proved ha he firm s producion policy should be of he capaciy-dependen modified basesock ype. When he capaciy Markov process is sochasically monoone, we have been able o show ha he policy parameers all decrease in he curren capaciy level. We have also exended our resuls o he infinie-horizon and undiscouned-cos versions of he problem. Our compuaional sudy has demonsraed he necessiy of some of our assumpions. The sudy has also clearly shown he benefis of uilizing he Markovian propery of he capaciy and of exercising he ousource opion. More specifically, we can draw a few conclusions from he compuaional sudy. A firm would be beer off if managers were aler o he ime coninuiy of is producion capaciy, especially when (1) he coninuiy is srong enough o render he sochasically independen descripion of he capaciy levels very inaccurae, (2) he mean demand level is high in comparison wih he mean capaciy level, (3) demand forecass ends o be accurae, or (4) he ousourcing opion is expensive. Also, reaining an ousourcing opion is beneficial o he firm, especially when (1) he opion is cheap relaive o in-house producion; (2) invenory holding or, more imporanly, backlogging is expensive; (3) he mean demand level is high compared o he firm s own capaciy level; or (4) i is difficul o make accurae demand forecass.

15 342 Operaions Research 53(2), pp , 2005 INFORMS Appendix Proof of Theorem 2. We prove his heorem by inducion. When = 0, he inducion hypoheses are obviously rue. Now, we suppose he hypoheses are rue for 1 for some 1, and will prove ha i is rue for as well. We will prove he consequence of he inducion hypoheses along he way. Now, G 1 y is coninuous and coninuously differeniable, d y G 1 y is upward, d y G 1 0 0, and d y G Le B 1 = min inf y d yg 1 y > 0 1. The jus-menioned hypoheses dicae ha B 1 is beween 0 and 1. Also by (17), his level B 1 characerizes y 1 O w in he following modified-base-sock fashion: y O 1 w = w w < B B 1 1 B 1 1 w B 1 + w 1 w > B 1 (31) In view of Condiion (a) and (22), we mus have B 1 S + 1 (32) and hence y 1 J = S J < s + J s J< B B 1 1 B 1 1 J B 1 + J 1 J > B 1 (33) Noe ha he opimal producion level is simply governed by y 1 J w J = 1 1 J < B B 1 J 1 B 1 1 J B 1 (34) By (31) and (17), we have G 1 w + 1 when w< B 1 1 G 1 w = G 1 B 1 when B 1 1 w B 1 G 1 w when w> B 1 (35) Also, G 1 w is coninuous and piecewise coninuously differeniable. By combining hese wih (32), (16), and (15), we can ge K q J + q p 1 S + G 1 S + 1 when J< s = q J + q p 1 s + G 1 s + 1 p 1 J + G 1 J + 1 F J = when s J< B 1 1 p 1 J + G 1 B 1 when B 1 1 J B 1 p 1 J + G 1 J when J> B 1 (36) So, F J is coninuous and piecewise coninuously differeniable. By combining hese and (18), we hen have ha G y is coninuous and piecewise coninuously differeniable, and ha d y G y = + R f D y d (37) where { p q +d I H J when J< s p p 1 +d I H J when s R J = + d y G 1 J + 1 J< B 1 1 p p 1 +d I H J when B 1 1 J B 1 p p 1 +d I H J + d y G 1 J when J> B 1 (38) By Condiion (b) and he convexiy of H I, R J 0 when J< s B 1 1 0; by Condiion (b), he convexiy of H I, and he definiion of B 1, R J 0 when s J< B 1 1 0; by Condiions (b) and (c) and he convexiy of H I, R J is upward when B 1 1 J B 1 ; and by Condiion (c), he convexiy of H I, and he definiion of B 1, R J 0 when J> B 1 0. Therefore, R J is upward. Also, Lemma 4 abou he boundedness of d y G y leads o he boundedness of R J. By Condiion (e) and he aforemenioned fac abou he convoluion beween an upward and bounded funcion and a PF 2 funcion, we can conclude ha d y G y is upward. To prove ha d y G 0 0, we only have o noe ha R J 0 for J 0, and ye d y G 0 = + 0 On he oher hand, we have d y G = B R f D d (39) p p 1 +d I H + 1 B 1 B 1 f D + d y G 1 f D d p p 1 +d I H d s + p p 1 +d I H + 1 B 1 + f D d + s + d y G p q +d I H f D d (40) For convenience, we call he above four erms a 1, a 2, a 3, and a 4, respecively. By he definiion of B 1 and he fac ha d y G 1 y is upward, we know ha a 1 +a B 1 0 p p 1 +d I H f D d (41)

16 Operaions Research 53(2), pp , 2005 INFORMS 343 By he lower-bound par of Lemma 4, Condiion (a), and he facs ha d I H 0 and B 1 1,wehave + a 3 + a 4 p p 1 + d I H f D + 1 B 1 d s = + 1 B s d y G f D d p 1 q f D d p 1 max q q 1 d I H 1 f D d (42) Therefore, Condiion (d) will lead o ha d y G 0. Deails for Secion 5 Define funcion L I = 1 E H I s+1 o be he average holding-backlogging cos from period o he end when he iniial invenory level is I and no producion or ousourcing is ever carried ou. Lemma 5. For any, I 0, and 0, when I>0, we have L I + I L I 1 h 0 / h 0 + b E / 1 2 I I. Proof. By he fac ha E < +, for any 0, we have he Chebyshev-ype inequaliy [ ] P s+1 I E /I (43) By he convexiy of H I and posiiviy of I, wehave ( ) ( ) H I + I s+1 H I s+1 ( h 0 I 1 s+1 <I ) b I 1 ( ) s+1 I (44) Combine (43) and (44), and we obain [ ( )] [ ( )] E H I + I s+1 E H I s+1 [ h 0 I P s+1 <I ] b I P [ ] s+1 I h 0 I 1 E /I b I E /I = h 0 h 0 + b E /I I (45) Consequenly, L I + I L I [ ( = (E 1 H I + I )] s+1 ( h 0 [ ( E H I )]) s+1 1 h 0 + b E ) 1 /I I = 1 h 0 / h 0 + b E / 1 2 I I (46) Corollary 1. For any, I 0, and 0, when I>0, we have G I + I G I 1 h 0 / h 0 + b E / 1 2 I + 1 p I. Proof. On any sample pah of he -period problem wih an iniial sae I, we may adop he opimal decisions for he corresponding sample pah of he -period problem wih an iniial sae I + I, where is invenory level is always I higher han ha of he former sae. In his way, he producion and ousourcing coss are he same on boh pahs. Cerainly, he opimal decisions for he former pah can perform even beer. Therefore, G I + I G I 1 p I = G I + I G I + p I [ ( 1 (E 1 H I + I + x s+1 + ) ] [ s+1 E H 1 + z s+1 ( I + ) ]) z s+1 s+1 x s+1 L I + I L I (47) where he x s s and z s s are he opimal producion and ousourcing levels when he sysem sars in he beginning of period a sae I + I ; he second inequaliy uilizes he posiiviy of he abovemenioned levels and he convexiy of H I. We hen have he desired resul by Lemma 5. For convenience, define U y so ha U y =E H y +E inf K 1 w y+ >0 + q w y+ + E G 1 1 w w y (48) By (3), we know ha G I = inf p y I + U y I y I + (49) Corollary 2. For any, y 0, and 0, when y>0, we have U y + y U y 1 h 0 / h 0 + b E / 1 2 y p y.

17 344 Operaions Research 53(2), pp , 2005 INFORMS Proof. On any sample pah of he -period problem saring from posproducion level y, we may adop he opimal decisions for he corresponding sample pah of he -period problem saring from posproducion level y + y. The res follows he argumen in he proof of Corollary 1. Assumpions 4 and 5 will be needed from now on. For any = 1 2, define consans w and ȳ so ha w = h 0 + b E 1 1 q +1 p + +1 h 0 (50) ȳ = h 0 + b E (51) p + h 0 By Assumpions 4 and 5, boh w and ȳ are posiive, w = lim + w = h 0 + b E / 1 1 q + h 0 < +, and ȳ = lim + ȳ = h 0 + b E / 1 1 p + h 0 < +. So, w = sup w = 1 2 <+ and ȳ = sup ȳ = 1 2 <+. The following resul is essenial for he proof of Theorem 7 and is also used in he proof of Theorem 6 laer on. Corollary 3. For any period = 1 2, he firm s opimal ousource-up-o level w J max J w max J w. Proof. By Corollary 1, Relaionship (9), Assumpions 4 and 5, and he definiion of w, F w increases in w when w w. However, he firm always has he opion o ousource exacly o max J w. Thus, we have he desired resul. The following resul is used in he proof of Theorem 6, even hough a slighly weaker resul would suffice. Corollary 4. For any period = 1 2, he firm s opimal produce-up-o level y I max I min I + ȳ max I min I + ȳ. Proof. By Corollary 2, Relaionship (49), Assumpions 4 and 5, and he definiion of ȳ, U y increases in y when y ȳ. However, he firm always has he opion o produce exacly o max I min I + ȳ. Thus, we have he desired resul. Proof of Theorem 6. In a 1 -period problem, we can always adop he firs 1 -period porion of he opimal policy for he -period problem wih he same iniial sae. Then, on corresponding sample pahs, when I 0 for he 1 -period problem is some I, I 1 for he -period problem is also I. However, by (4) and (5), we have G 1 1 I = q E + inf p q y + E H y + inf K 1 w y + >0 + q p w w y I y I + 1 (52) and hence, G 1 I p E + inf 1 p y + E H y I y I + 1 p E + 1 p I (53) where he firs inequaliy is due o Assumpion 5 and he paricular naure of he inner opimizaion, and he second inequaliy is due o he posiiviy of he coss and he paricular naure of he ouer opimizaion. Averaging over all corresponding sample pahs, we obain G I G 1 I + p E p E I 1 = I = I G 1 I + p E p I 1 E = G 1 I p E p I (54) where he firs expecaion should be undersood o have aken ino accoun he behavior of he underlying policy, and he second inequaliy is due o he fac ha he invenory level can decrease no more han he curren demand level afer each period. On he oher hand, in he -period problem, we can always adop he opimal policy for he 1 -period problem for he firs 1 periods, and hen neiher produce nor ousource in Period 1. Therefore, G I G 1 I + 1 E H I 1 = I = I G 1 I + 1 M I + E + max w ȳ (55) where again he firs expecaion has aken ino accoun he behavior of he underlying policy, and he second inequaliy is due o Corollaries 3 and 4, which limi he incremen of he invenory level, and he fac ha he invenory level can decrease no more han he curren demand level in any period, and Assumpion 1. Combining (54) and (55) while aking he fac ha E < + ino consideraion, we obain a posiive consan A such ha G I G 1 I A 1 + I (56) Because 0 1, =0 converges absoluely and he remnan erm + =+1 converges o 0. Therefore, G I = is a Cauchy sequence a every I -pair, and hence, converges o some G I. Also, from he form of he coefficien erm in (56), we know ha he convergence is uniform in any region of he I -space wih a bounded I-projecion. We know ha G I = p I +G I. By Theorem 6, G I is he limi of G I. The following lemma abou G I will be useful in 5.2.

18 Operaions Research 53(2), pp , 2005 INFORMS 345 Lemma 6. For any and I, G I 0. Proof. By Corollaries 3 and 4, in a -period problem saring from I, I 0 is bounded from above by Q 1 + I for some posiive consan Q. Because all oher coss excep he erminal cos are posiive, we have G I Q 1 + I. The resul is rue due o Theorem 6. Proof of Theorem 7. Due o (56) and he fac ha G 0 I = 0, here is a posiive consan C such ha G 1 I C 1 + I. By Theorem 6 and he Dominaed Convergence Theorem, F w converges o F w, which is defined in (9) wih G I on he righhand side; and a any fixed, wih F w F w being bounded by a linear funcion of w, he convergence is uniform in any bounded w-region. Also, i is easy o see ha here is a posiive consan D such ha F w D 1 + w. By Corollary 3, a any J - pair, he infimum in (8) is effecively aken in he uniformly bounded w-region J max J w. Then, F J converges o F J, which is defined by (8) wih F w on he righ-hand side. In urn, a any fixed, wih F J F J being bounded by a linear funcion of J, he above convergence is uniform in any bounded J -region; and F J is bounded by L 1 + J for some posiive L. Then, also due o he fac ha E < +, Assumpion 1, and he Dominaed Convergence Theorem, we know ha G y converges o G y, which is defined by (7) wih F J on he righ-hand side; and ha he convergence is uniform in any bounded y-region. A any I -pair, he infimum in (6) is aken in a uniformly bounded y-region. Hence, we have ha G I converges o he righ-hand side of (6) wih G y replacing G y. Proof of Theorem 8. The K-convexiy of G I is due o he poinwise convergence of he K-convex G I s o i and he fac ha K-convexiy susains poinwise convergence. In he firs special case, he upwardness of y G y is due o he poinwise convergence of upward G y s o G y and he simple fac ha upwardness susains poinwise convergence. In he second special case, he convexiy of G I is due o he poinwise convergence of he convex G I s o i and he fac ha convexiy susains poinwise convergence. The consequen resuls follow similar argumens in Theorems 1, 2, and 3, respecively. Proof of Theorem 9. The firs wo resuls can be proved by employing he relevan resuls in Theorems 4, 5, and 6, and by using he basic fac abou he inerchangeabiliy of monooniciy and convergence. The consequence of he second resul can be achieved by using similar argumens in Theorem 5. Proof of Theorem 10. Saring from sae 0 I 0, he firm can op o always produce nohing and ousource all jus-realized demand in every period. Hence, we have 0 G 0 I 0 K + q E (57) 1 and herefore, g 0 I 0 K + re < +. Because here is always a converging subsequence in a bounded sequence, we have he inended resul. For capaciy levels 1 and 2, define sopping ime C 2 1 so ha C 2 1 = inf = = 2 (58) is he number of fuure periods needed for he capaciy level o reach exacly 2 for he firs ime when he curren capaciy level is 1. For a posiive ineger I, define sopping ime D I so ha { } s D I = inf s = u I (59) is he smalles number of periods needed for he cumulaive demand from he curren period on o exceed level I. Assumpions 6 and 7 will be needed from now on. By a direc applicaion of Wald s Ideniy and he limiing heorem for he excess in he renewal heory (Wolff 1989), and due o Assumpions 6 and 7, we have he exisence of a posiive consan L such ha I/E E D I I/E + E 2 / 2 E 2 + L (60) Therefore, we know ha E D I < +, and herefore D I < + almos surely. Assumpion 8 will be needed from now on. Proof of Theorem 11. Firs, for any wo pairs 1 I 1 and 2 I 2, we are o show ha G 1 I 1 G 2 I 2 is bounded from above by a value ha is independen of. Wihou loss of generaliy, we assume ha I 1 I 2. When he sysem s curren sae is 1 I 1, he firm can op o produce nohing and ousource I 2 I iems in period 0. Then in period 1, he sysem s sae is 1 I 2. From his period on, he firm can op o always produce nohing and ousource all jus-realized demand in every period unil period C 2 1, a which he sysem is in sae 2 I 2. Due o Lemma 6, he posiiviy of all coss, and he fac ha 0 1, collec all coss from period 0 o his period, and we have G 1 I 1 K + q I 2 I 1 + E C 2 1 q E + G 2 I 2 (61) When he sysem s curren sae is 2 I 2, he firm can op o neiher produce nor ousource for he firs D I 2 I 1 1 periods and ousource D I 2 I 1 I 2 I 1 iems for he D I 2 I 1 h period. Therefore, he sysem s sae becomes D I 2 I 1 2 I 1 in period D I 2 I 1. Then, he firm can op o produce nohing and ousource all jus-realized demand in every period for he nex C 1 D I 2 I 1 2 periods.

19 346 Operaions Research 53(2), pp , 2005 INFORMS In period 1 D I 2 I 1 C 1 D I 2 I 1 2 1, he sysem s sae becomes 1 I 1. Again, due o Lemma 6 and he facs ha all coss are posiive and ha 0 1, collec all coss from period 0 o he aforemenioned period, and we will have G 2 I 2 K + q E 1 + E C 1 D I 2 I G 1 I 1 (62) Combining (61), (62), he definiion for G I, he facs ha G I = p I + G I, E < +, and E > 0, and Assumpion 8, we can conclude ha here exiss a posiive consan B, such ha for any and I, G I B 1 + I <+ (63) Using Cauchy s diagonal argumen, we can find a subsequence i n n = such ha for every and I, G i n I converges as n ends o +. For more deail, he reader may refer o he proof of Theorem 11 in Yang e al. (2004). The proof of Theorem 12 can be done in a similar fashion o ha for he laer is done, given ha we have found an upper bound for w J ha is uniform in. The earlier bounds w (he -superscrip is new) found righ before Corollary 3 grow in owards + as ends o 1. Therefore, we need a new bound for a leas hose s ha are close o 1. Lemma 8, which is preceded by he necessary Lemma 7, does exacly his. Lemma 7. Suppose = 1 2, = 3/4 1/ 1, and I = E + 3 E 2 E 2 1/2. Then, for any 1 and any I I, i is rue ha E D I 1 1 /2. Proof. Firs, E 1 u = E and var 1 u = E 2 E 2. So, by a Chebyshev-ype inequaliy and he way I is defined, we have [ ] P 1 u I 1 3 (64) Hence, for I I, [ ] [ ] P D I = P 1 u I P 1 u I 1 3 (65) By he way is defined, for 1 we find [ D I 1 ] E 1 D I 1 = 3 4 (66) Combine (65) and (66), and we obain [ D I 1 E [ ] E D I 1 1 ] 1 D I P D I 3 /4 2/3 = /2 (67) Lemma 8. Suppose ha = 2 b + K / h 0, = 3/4 1/ 1, and I = E + 3 E 2 E 2 1/2. Then, for any 1, any, and any I I 1, we have I G I p q. Consequenly, he firm s opimal ousource-up-o level in he -discouned infinie-horizon problem w J max J I 1. Proof. On any sample pah of he -discouned infiniehorizon problem wih iniial sae I, we may for he firs few periods adop he opimal decisions as hough we are on he corresponding sample pah for he -discouned infinie-horizon problem wih iniial sae I + 1, unil he pre-ousourcing level becomes sricly less han 0, a which poin we may ousource one iem more han wha we would opimally ousource on he corresponding sample pah. Aferwards, we may ac opimally. Le x 0 x 1 and z 0 z 1 be he producion and ousourcing decisions before he pre-ousourcing level reaches below 0. Define sopping ime O I so ha { s O I = inf s = 1 2 I + x 1 u s s 1 } 1 u + z 1 u 1 (68) By he posiiviy of he producion and ousourcing levels, we have O I D I + 1 (69) Because for he problem wih iniial sae I, in he opimal way we can fare a leas as well as wha is allowed by he prescribed acions in he las paragraph, we have I G I p [ O I ( E 1 I H I + x 1 u 1 1 u + ) ] z 1 u K + q E O I 1 [ O I 1 ] h 0 E 1 b + K + q E O I 1 [ D I+1 1 h 0 E ] b 1 + K + q h 0 /2 b + K + q q (70) where he second inequaliy is due o he facs ha all erms bu he las one inside I H are posiive by he definiion of O I and ha H I is convex. The hird inequaliy is due o (69) and he range of, he fourh inequaliy is due o Lemma 7, and he las inequaliy is due o he way is defined.

20 Operaions Research 53(2), pp , 2005 INFORMS 347 From (9), we see ha w F w q p + p q = 1 q 0 when w I 1. Then by (8), we obain he consequen resul. Wih Lemma 8 and he earlier bounds w ha are obviously bounded for 0, we know ha w J is bounded by a consan (independen of ). Proof of Theorem 12. The proof is similar o ha for Theorem 7, and is paricularly close o ha for Theorem 12 in Yang e al. (2004). The only curren difficuly lies in proving he inerchangeabiliy of he limi and infimum relaed wih (8). However, ha has been aken care of by he -independen bound for w J. Thus, we omi he deailed proof here. Proof of Theorem 13. The proof is almos he same as ha for Theorem 8. We omi i here. Proof of Theorem 14. The proof is almos he same as ha for Theorem 9. We omi i here. I is clear ha for all he cos funcions defined for he discouned-cos case, we have heir ilde -clad undiscouned-cos counerpars. When no having o be concerned wih differen s, we can achieve an upper bound for w J for he undiscouned-cos case ha is igher han he uniform bound for he various w J s. Lemma 9. For any and I 0, we have I G I h 0 I + 1 /E b + K + q p. Consequenly, for w U = b h 0 + K E / h 0, he firm s opimal ousource-up-o level w J max J w U. Proof. Oher han he fac ha here we work on wo sample pahs of he undiscouned-cos problem, he proof is idenical o ha for Lemma 8 up o righ before (70). From ha poin on, we have I G I p [ O I ( E I H I + x 1 u 1 1 u + ) ] z 1 u K + q E O I 1 h 0 b K + q E D I h 0 b K + q h 0 I + 1 /E b + K + q (71) Now, from (9), we see ha w F w h 0 w + 1 / E b + K when w 0. Then by (8), we obain he consequen resul. Lemma 10. For any and y 0, we have y U y h 0 y + 1 /E b + K + q. Consequenly, for I U = b h 0 + K + q p E / h 0, he firm s produce-up-o level y I max I min I + I U. Proof. On any sample pah of a problem saring from posproducion level y, we may for he firs few periods adop he opimal decisions as hough we are on he corresponding sample pah for he problem wih posproducion level y + 1, unil he pre-ousourcing level becomes sricly less han 0, a which poin we may ousource one iem more han wha we would opimally ousource on he corresponding sample pah. Aferwards, we may ac opimally. Using he same argumen as in he proof of Lemma 9, we have y U y h 0 y + 1 /E b + K + q (72) Then, by (49), we obain he consequen resul. Assumpion 9 will be needed from now on. For any and posiive I, define sopping ime C D I, which describes he smalles number of periods needed for he average oal capaciy o cach up wih I plus he average oal demand: C D I { ( s } =inf s =1 2 1 u 1 u ) I (73) Assumpion 10 will be needed from now on. The following Lemma 11 can be hough of as he dual of Lemma 9. I provides he lowes pre-ousourcing invenory level a which he firm mus have a sricly posiive ousourcing level, and also provides he firm s lowes ousource-up-o level when ousourcing is invoked. The proof is slighly more involved han ha for Lemma 9, bu is sill similar in spiri. Lemma 11. For any and I 0, we have I G I 1 min b 0 + q p b 0 1 I b 0. Consequenly, for J L = b 0 + q p / b 0 + K 2 b 0 + q p / b 0 / b 0, he firm s opimal ousourcing level z J 1 when J J L, and for w L = b 0 + q p / b 0, he firm s opimal ousource-up-o level w J max J w L + 1 when is opimal ousourcing level z J 1. Proof. On any sample pah of he problem wih iniial sae I, we may for he firs few periods adop he opimal decisions as hough we are on he corresponding sample pah for he problem wih iniial sae I 1, unil eiher he pre-ousourcing level reaches such a negaive level ha he opimal decision for he corresponding sample pah would ousource a sricly posiive level, or when he preproducion level reaches such a negaive level ha he opimal decision for he corresponding sample pah would produce up o a sricly posiive level, a which poin we may ousource or produce one iem less han wha we would opimally ousource or produce on he corresponding sample pah. Aferwards, we may ac opimally. Because he firs-ever sricly posiive ousourcing aciviy a a posiive preousourcing level mus occur afer some sricly