Title: Leadtime Management in a Periodic-Review Inventory System: A State-Dependent Base-Stock Policy

Transcription

1 Elsevier Ediorial Sysem(m) for European Journal of Operaional Research Manuscrip Draf Manuscrip Number: Tile: Leadime Managemen in a Periodic-Review Invenory Sysem: A Sae-Dependen Base-Sock Policy Aricle Type: Regular Paper Secion/Caegory: Invenory Keywords: Invenory, leadime managemen, sae-dependen base-sock policy Corresponding Auhor: Dr. Jun-Yeon Lee, PhD Corresponding Auhor's Insiuion: Universiy of Houson-Vicoria Firs Auhor: Jun-Yeon Lee, PhD Order of Auhors: Jun-Yeon Lee, PhD; Leroy B. Schwarz, PhD Manuscrip Region of Origin: Absrac: We examine a single-iem, periodic-review invenory sysem wih sochasic leadimes, in which a replenishmen order is delivered immediaely or one period laer, depending probabilisically on cosly effor. The objecive is o deermine an invenory policy and an effor-choice sraegy ha minimize he expeced oal coss. Our analyical and compuaional analysis suggess ha (i) a sae-dependen base-sock policy is opimal, (ii) he opimal effor sraegy is such ha he marginal cos of effor is equal o he value of immediae delivery, and (iii) he cos impac of leadime reducion can be very large. We also provide some couner-inuiive resuls, compared wih he radiional muli-period newsvendor model.

2 Tex Only Including Absrac/Tex + Figs + Tables Leadime Managemen in a Periodic-Review Invenory Sysem: A Sae-Dependen Base-Sock Policy Jun-Yeon Lee a,, Leroy B. Schwarz b a School of Business Adminisraion Universiy of Houson-Vicoria Vicoria, TX 7791, USA b Kranner Graduae School of Managemen Purdue Universiy Wes Lafayee, IN 4796, USA 8/23/26 ABSTRACT We examine a single-iem, periodic-review invenory sysem wih sochasic leadimes, in which a replenishmen order is delivered immediaely or one period laer, depending probabilisically on cosly effor. The objecive is o deermine an invenory policy and an effor-choice sraegy ha minimize he expeced oal coss. Our analyical and compuaional analysis suggess ha (i) a sae-dependen base-sock policy is opimal, (ii) he opimal effor sraegy is such ha he marginal cos of effor is equal o he value of immediae delivery, and (iii) he cos impac of leadime reducion can be very large. We also provide some couner-inuiive resuls, compared wih he radiional muli-period newsvendor model. Key Words: invenory, leadime managemen, sae-dependen base-sock policy Corresponding auhor. Mailing address: 63 Lornmead Dr, Houson, TX 7724, USA; Tel: ; leej@uhv.edu

3 1. INTRODUCTION In his paper we examine a single-iem, periodic-review invenory sysem wih sochasic leadimes, in which a replenishmen order is delivered eiher immediaely or one period laer, depending probabilisically on cosly effor. The effor migh ake he form of expediing, he use of beer equipmen, and so on. The oher assumpions are sandard: Unme demands are backordered, ordering coss are linear, and one-period expeced invenory-holding and shorage cos is a convex funcion of iniial invenory. The objecive is o deermine an invenory policy and an effor-choice sraegy ha minimize he expeced oal coss over T ime periods. Characerisically, replenishmen leadimes have boh a fixed and variable componens. The lengh of he fixed componen ypically depends on process-design choices (e.g., he exen of auomaion, choice of common carrier) and environmenal facors (e.g., he naure of he produc, he geographic disance from he supplier). The lengh of he variable componen also depends on choices, in paricular, how each realized leadime can be decreased or increased by choices made and acions aken during each replenishmen cycle. Finally, each realized replenishmen leadime has random variaion, some of which is manageable; and some variaion is sricly random. In his paper we focus on he variable componens: choices made and acions aken during each replenishmen cycle. These choices and acions are represened implicily by effor, wherein beer choices and acions (i.e., more effor) yield shorer leadimes. We analyze he single-period ( T = 1), muli-period ( T < ), and infinie-horizon (T = ) problem. In he single-period problem, despie he non-unimodaliy of he cos funcion, we prove he opimaliy of a sae-dependen base-sock policy: There exiss a criical number such ha i is opimal o place an order if and only if iniial invenory is less han he criical number; and he opimal order-up-o level may depend on iniial invenory. For he muli-period problem, we provide simple bounds for he opimal order-up-o levels. In all of he problems, we show ha he opimal effor sraegy is such ha he marginal cos of effor is equal o he value of immediae delivery. Our compuaional resuls indicae ha a sae-dependen base-sock policy is also opimal for he muli-period problem. They also sugges ha he cos impac of leadime reducion can be very large. In our compuaional sudy he oal coss were reduced 24.7 % on average, compared wih no effor case. We also discovered ha, unlike in he radiional muliperiod newsvendor model, given he same iniial invenory, he opimal order-up-o levels may increase as he end of he horizon approaches. In he infinie-horizon problem, by solving he 1

4 problem using a policy-ieraion procedure, we were able o compue no only he opimal policy bu also he end-of-horizon invenory-valuaion funcion ha induces a saionary opimal policy in he finie-horizon problem. However, we found ha, unlike in he radiional model, he valuaion funcion is non-linear. Mos of he leadime-managemen relaed lieraure has used coninuous-review invenory models o examine one-ime effor in leadime reducion and is impac on invenory coss (e.g., Hariga and Ben-Daya, 1999; Ray e al., 24; Yang e al., 25; and Lee and Schwarz, 26). Only a few papers have sudied leadime managemen in periodic-review invenory scenarios (Ouyang and Chuang, 1998, 2; Chuang e al., 24). In hese models, however, he same level of effor is exered for every replenishmen order regardless of invenory level, and an approximae funcion of he oal expeced annual cos is minimized. A relaed sream of research is on invenory models wih wo delivery modes (e.g., Fukuda, 1964; Chiang and Guierrez, 1996, 1998; and Tagaras and Vlachos, 21), in which he faser delivery mode is available a a higher cos han he slower mode. Each delivery mode has a deerminisic leadime. Our model is differen from he above models in ha i examines he problem of managing sochasic leadimes in a periodic-review invenory model, where effors for leadime reducion are chosen dynamically for each replenishmen order depending on invenory level. Our work is also relaed o he lieraure on sae-dependen (i.e., modified) base-sock policies. Two of he earlies references are Karlin (1958) and Karlin and Scarf (1958). Karlin (1958) shows ha if he ordering cos is convex, hen he opimal base-sock level is a funcion of iniial invenory, since he oal coss, including he convex ordering coss, are reduced by smoohing order quaniies beween ime periods. Karlin and Scarf (1958) show ha if excess demand is los and here is a ime lag in delivery, hen he opimal base-sock level should be modified by iniial invenory, since he cos for fuure ime periods depends joinly on he curren order and iniial invenory. Federgruen and Zipkin (1986a, b) prove ha a modified base-sock policy is opimal for an invenory sysem wih limied producion capaciy, where i is opimal o order up o a arge base-sock level or as close o i as possible. Henig and Gerchak (199) and Anupindi and Akella (1993) show he opimaliy of a sae-dependen base-sock policy when yield is random and when orders are allocaed beween wo suppliers wih differen supply uncerainies, respecively. Our model is similar o some of hese models in a sense ha he 2

5 opimal base-sock level is modified by iniial invenory. However, he reason for he modificaion is differen: In our model, if invenory is low, hen we wan an order o be received sooner and hence exer a higher effor. Bu as he probabiliy of shor leadime increases (hrough he increased effor), we would choose o order eiher less or more. The res of his paper is organized as follows: In secion 2 we presen he basic model. In secion 3 and 4 we analyze he single-period and he muli-period problem, respecively. In secion 5 we provide compuaional resuls for he muli-period problem. In secion 6 he infiniehorizon problem is analyzed. Finally, in secion 7 we summarize he main resuls. 2. THE BASIC MODEL Consider a single-iem, sochasic, periodic-review invenory sysem over T ime periods. Demands D, = 1,2,..., T, for he iem in each period are independen and idenically disribued. Unme demands in every period excep T are backordered; in period T hey are los. A he end of period T lefover invenory is worhless. The acquisiion cos is $ c per uni. There is no fixed order cos. Le A( x ) denoe he one-period expeced invenory-holding and shorage cos given iniial invenory x. We assume ha A( x ) is a convex funcion of x. Le s be is minimizer; i.e., As ( ) Ax ( ) for all x. A replenishmen order will eiher be delivered immediaely or one period laer, depending probabilisically on cosly effor. The effor exered for an order in period deermines he probabiliy p of immediae delivery, and, hence, he probabiliy (1 p ) of delivery nex period. Thus, he choice of effor can be modeled as choosing p. Assume ha we can choose any p, p p p, a cos of ( ) W p, where p and p 1 represen he smalles and he larges effor ha we can choose, respecively. We assume ( ) ; ( ) ; ( ) ; ( ) ; lim ( ) W p = W p > W p = W p > W p =. (1) Tha is, no cos is incurred if no effor is exered; he cos of effor increases as he effor increases; he marginal cos of effor is zero a no effor, bu increases as he effor increases; and i is exremely cosly o choose p p p = p. There is no carryover of effor from period o period. The objecive is o deermine an invenory policy and an effor-choice sraegy ha minimize he expeced oal coss over he T ime periods. In he infinie-horizon problem, we 3

6 minimize he expeced long-run average oal coss per period. 3. THE SINGLE-PERIOD PROBLEM In he single-period ( T = 1) problem, given iniial invenory x, we have o choose an order-up-o level s and an effor level p o: s x; p p p ( ) ( ) ( ) ( ) ( ) Minimize c s x + W p + pa s + 1 p A x (2) Noe ha if we choose s and p, hen wih probabiliy p he expeced invenory-holding and shorage cos will be As ( ), and wih probabiliy ( 1 p) i will be ( ) A x. I is easy o show ha he opimal p for any given s and x, denoed p( s, x ), saisfies: + where [ ω] max { ω, } ( (, )) = ( ) ( ) + W p s x A x A s (3) =. Equaion (3) means ha p should be chosen such ha he marginal cos of he effor, p, is equal o he value of immediae delivery; i.e., he difference beween he oneperiod invenory coss when he replenishmen order is delivered immediaely and one period laer. Now we have o find an opimal order-up-o level s ( ) If y( s, ( ) ( ) ( ) x o: ( ) ( ) ( ) ( ( )) ( ) Minimize y s, x c s x + W p s, x + p s, x A s + 1 p s, x A x (4) s x were a unimodal funcion of s for any given x, hen i would be easy o show ha a sae-dependen base-sock policy is opimal. However, y( s, may no be unimodal in s. See Figure 1 for an example of he shape of y( s, x ). Noneheless, he following proposiion saes ha a sae-dependen base-sock policy is opimal: There exiss a criical number such ha i is opimal o place an order if and only if iniial invenory is less han he criical number; and he opimal order-up-o level may depend on iniial invenory. Proposiion 1. Assume ha demand is a coninuous random variable and A( ) is a coninuous, differeniable, and sricly convex funcion. (a) For any x and s > x, if y( s, s (b) For any x and s > x, if y( s, s >, hen y( s, hen ( s, s > for all x < x < s. ϒ, x<, where 4

7 ϒ( s, y( x, y( s, (c) If x is he value such ha s ( = x and s ( x x x. Proof. (a) I is sraighforward o show ha ha for any x and s (, ) (, ) ( ) > for all x x y s x s = c+ p s x A s. > x, <, hen ( ) s x = x for all Consider any x, x< x < s. If A ( s), hen clearly y( s s c p( s A ( s) Suppose A ( s) <. Then, since A( A( x ) A( s) from he (sric) convexiy of ( ) implies ha p( s, x ) p( s, and hus y( s, x ) s = c+ p( s, x ) A ( s) c+ p( s, A ( s) >. (b) I is sraighforward o show ha for any x and s > x,, = +, >. ϒ ( s, x = c+ p( s, A (. Since A ( < A ( s) by he sric convexiy of A( ), ϒ ( s, x = c+ p( s, A ( < c+ p( s, A ( s) = y( s, s. (c) Suppose ha s ( x ) > x for some x > x. If we le s s ( x ), hen i mus be y( s This implies ha y( s ( s ϒ, x< for all x x A, his, s =., s for all x x from (a). This, in urn, implies ha ϒ, =,, since from (b). Bu, noe ha ( s yxx ( ) ysx ( ) s ( = x; and ha ϒ ( sx, ) = yxx (, ) ysx (, ) since s = s (. This gives a conradicion. ( ) Noe ha Proposiion 1(a) and 1(b) show ha saionary poins, e.g., P xysx, (, ) ( ) Figure 1, move upward relaive o P x, y( x, x in xs, ˆ ˆ as x increases, which means ha once a saionary poin becomes higher han P x, hen i always says above P x as x increases. Proposiion 1(c) saes ha if we define x as he smalles value of iniial invenory such ha s ( = x, hen ( ) s x > x if and only if x< x. 4. THE MULTI-PERIOD PROBLEM In he muli-period ( T < ) problem, if we define TC ( ) from period o T given iniial invenory x, hen we have o solve: x as he minimum expeced oal coss 5

8 { + 1 } ( ) ( ) ( ) ( ) ( ) ( ) ( ) TC x = min c s x + W p + pa s + 1 p A x + E TC s D, = 1,2,..., T (5) s x, p p p where TC ( T + 1. As in he single-period ( 1 T = ) problem, he opimal p in any period for any given s and x, denoed p ( s, x ), saisfies equaion (3); i.e., (, ) Consequenly, problem (5) becomes { + 1 } ( ) = min (, ) + ( ) s x ( ) = ( ) ( ) W p s x A x A s. TC x y s x E TC s D (6) where y( s, x ) is he single-period expeced oal coss, given by (4). Define Noe ha, alhough y( s, (, ) (, ) + ( ) J s x y s x E TC s D + 1 (7) is no unimodal, we were able o prove he opimaliy of a sae dependen base-sock policy in he single-period problem. However, for he muli-period problem, i is difficul o characerize is opimal policy, since he addiion of he minimum expeced fuure coss (i.e., he second erm of J (, ) s x ) makes he problem much harder. Insead, we shall provide simple bounds for he opimal order-up-o levels, which can be used for numerical analysis. In his secion and aferwards we assume ha demand and invenory can ake only on ineger values Define J ( s, as he marginal change of he expeced oal coss as he order-up-o level is increased from s o s + 1, i.e., (, ) ( 1, ) (, ) (, ) ( 1 ) ( ) J s x J s+ x J s x = y s x + E TC+ 1 s+ D TC+ 1 s D (8) where Lemma 1. For any x and s (, ) ( 1, ) (, ) y s x y s+ x y s x (9) x, ( 1, ) ( 1 ) ( ) (, ) (, ) ( 1) ( ) c+ p s+ x A s+ A s y s x c+ p s x A s+ A s (1) Proof. For any x and s x, (, ) ( 1, ) (, ) = c+ W ( p( s+ 1, ) W ( p( s, ) + p( s+ 1, A( s+ 1 ) p( s, A( s) p( s+ 1, p( s, A( y s x = y s+ x y s x + 6

9 ( + 1, ) ( (, )) ( (, )) ( + 1, ) (, ) W ps x W psx W psx p s x psx, Since ( ) (, ) ( 1, ) ( 1) ( ) ( 1, ) (, ) (, ) y s x c+ p s+ x A s+ A s { ( ) ( ) ( ) } + ps + x psx W p sx Ax As ( ) ( ) ( ) If A( A( s), hen W p( s, = A x A s. If A( A( s) ( ) < ( ) ( + 1), we have (, ) ( 1, ) A x A s A s Therefore, <, his implies ha p s x = p s+ x = p. Hence, in any case, ( ) ( ) ( ) { ( ) ( ) ( ) } p s+ 1, x psx, W p sx, Ax As =. (, ) ( 1, ) ( 1) ( ) y s x c+ p s+ x A s+ A s. Similarly, i can be shown ha y( s, c+ p( s, A( s+ 1) A( s) The bounds of y( s,. s < s. Since provided by Lemma 1 depend on he value of iniial invenory, x. To obain x-free lower and upper bounds of y( s,, we define y( s) and y( s) as follows: where ( ) ( ) ( ) ( ) c+ p A s+ 1 A s, if s< s y( s) sup { c+ p( s, A( s+ 1) A( s) } = x s c+ p A s+ 1 A s, if s s ( ) ( ) ( ) ( ) c+ p A s+ 1 A s, if s< s y( s) inf { c+ p( s+ 1, A( s 1) A( s) } x s + = c+ p A s+ 1 A s, if s s s is he minimizer of A( ). (11) (12) If we define ( ) ( ) ( 1 ) ( ) J s y s + E TC+ 1 s+ D TC+ 1 s D (13) ( ) ( ) ( 1 ) ( ) J s y s + E TC+ 1 s+ D TC+ 1 s D (14) hen i is easy o see ha Define J ( s) J( s, J ( s) l as he larges ineger value of s such ha J ( s ) smalles ineger value of s such ha J ( s) says ha he opimal order-up-o level, s ( ). (15) 1 < for all s s ; and u as he for all s s. Then, he following proposiion x, in period is bounded by l and u. 7

10 Proposiion 4. For any period 1, 2,..., T and ( ) s x = x if x u. Proof. Noe ha J ( s J ( s) Hence, ( ) = and any given value of x, ( ), < for all s l l s x u for x u < and ( ) s x = x for x u. Now, o provide a bound for he bounds ( l, u ) Lemma 2. TC ( x 1) TC ( M ( l s x u if x< u, < and J ( s) J ( s, { } + for any x and, where for all s u, we use he following lemma:. ( ) M x Proof. For any x and, since ( ) ( ) ( ) ( ) c+ 1 p A x+ 1 A x, if x< s c+ ( 1 p) A( x+ 1 ) A(, if x s (, ) ( ( + 1 ), ) J s x x J s x x ( ) ( ) ( ) ( ) ( ( ) ) ( ( 1, ) 1) ( ( 1, ) ) ( 1 ( 1) ) ( 1) ( ) TC x+ 1 TC x = J s x+ 1, x+ 1 J s x, x J s x+ x+ J s x+ x M = c+ p x+ A x+ A x (. (16) Lemma 2 means ha he marginal value of a uni of iniial invenory in any period is a mos he negaive of M ( x ) given by (16). This resul is very inuiive because he marginal value of a uni of iniial invenory should be is acquisiion cos plus is conribuion o he expeced invenory-holding and shorage cos muliplied by he probabiliy of an order being delayed. If we define s as he smalles ineger value of s such ha ( ) ( ) ( ) c+ p A s+ 1 A s + E M s D (17) and s as he larges ineger value of s such ha y( s ) provides a lower and upper bound for he bounds (, ) 1 <, hen he following proposiion { } l u. Proposiion 5. For any period = 1, 2,..., T, s l u s. Proof. Noe ha u s if and only if J ( s) for all s s. Bu, for any s s, 8

11 Hence, u ( ) ( 1) ( ) + 1( 1 ) + 1( ) c+ p A( s+ 1) A( s) + E M ( s D) J s = c+ p A s+ A s + E TC s+ D TC s D s for all. To prove s l for all, noe ha s l if and only if J ( s) < for all s < s. We shall use an inducion o prove J ( s) < for all s < s. For period T, noe ha s l s = T. For any x s <, since ( ) s x s x+ 1, T ( ) ( ) ( ) ( ) ( ( ) ) ( ( ), 1 ) ( ( ), ) ( 1 T ( )) ( 1) ( ) TC x + 1 TC x = J s x + 1, x + 1 J s x, x T T T T T T Now, suppose ha for period + 1, (i) s l + 1 For any s Hence, < s, J s x x+ J s x x T T T T = c+ p x A x+ A x c and (ii) ( 1) ( ) TC x + TC x c for all x< s ( ) ( 1) ( ) + 1( 1 ) + 1( ) p A( s+ 1) A( s) < J s = c+ p A s+ A s + E TC s+ D TC s D s l. I can be also shown ha TC ( x + 1) TC ( c for all x< s. This complees he proof. 5. COMPUTATIONAL STUDY We conduced a compuaional sudy for he muli-period problem in order o (i) compare he oal coss under leadime managemen wih no effor case o assess he cos impac of leadime reducion and (ii) examine he opimaliy of a sae-dependen base-sock policy in he muliperiod problem. In our compuaional sudy we esed problems wih T = 4 periods and a runcaed Poisson demand disribuion, i.e., d e ( ) Pr( ) λ λ f d D= d = d! Θ, d =,1,..., D, where D d e λ λ Θ=. d! We assumed a linear invenory-holding cos h per uni per monh and a linear shorage cos π per uni per monh; i.e., ( ) = ( ) + π ( ) + + A x he x D E D x d = 9

12 The value of h was fixed a $1 per uni per monh, and he value of π was se equal o $99 per uni per monh in he base case, wih alernaive values of 66, 82, 11, 142, and 199. The cos-of-effor funcion was: 1 1 W ( p) = κ ( p p) p p p p Noe ha he above cos funcion saisfies all he assumpions in (1). Tha is, i is an increasing, convex funcion of p wih W ( p ) =, W ( p) =, and lim ( ) W p p p =. Is chosen parameer values were κ = 1 and p =.5 in he base case and, alernaively, κ = 2, 3, 5, 7, 1 and p =.2,.3,.4,.6,.7. The value of p was se o 1. The funcions and parameers used are summarized in Tables 1 and 2. In order o assess he cos impac of leadime reducion, we compued he minimum expeced oal coss under leadime managemen and under no effor, and measured he percenage cos reducion. We define: where TC ( NE ) TC IMPACT % = 1 TC ( NE) TC and TC( NE ) are he expeced oal coss over T = 4 periods wih zero iniial invenory under opimal leadime managemen and under no effor, respecively. Using he funcions and parameers described above, we observed ha he cos impac of leadime reducion can be very large. IMPACT% averaged 24.7 % (range: %). See Table 2. Alhough we were no able o prove ha a sae-dependen base-sock policy is opimal in he muli-period model, in our compuaional sudy he opimal policy observed was always such a policy. Tha is, here exised a se of criical numbers, { x }, such ha i is opimal o place an order in period (i.e., s ( x ) > x ) if and only if iniial invenory of he period is less han he criical number (i.e., x <. Furher, he opimal order-up-o level depended on iniial invenory. See Figure 2 for he shape of he opimal order-up-o level as a funcion of iniial invenory. In paricular, for any given parameerizaion, we observed ha he opimal order-up-o level is monoonic wih respec o iniial invenory, approaching s from above or below, as iniial invenory decreases. We inerpre his as follows: As iniial invenory decreases, we exer higher levels of effor, hereby increasing he probabiliy of immediae delivery. As he 1

13 probabiliy of immediae delivery increases, he opimal order-up-o level becomes closer o s, which is he invenory level ha minimizes he expeced invenory-holding and shorage cos for he curren period. We also examined he ime paern of he opimal order-up-o levels across ime periods. I is well known ha in he radiional muli-period newsvendor model, he opimal order-up-o level is non-increasing as he end of horizon approaches because he marginal value of lefovers is non-increasing. However, in our model, his was no always he case. The following is a couner example ha we found: Non-Monooniciy Example: Demand disribuion: Pr{ D = } = Pr{ D = 5} =.1, { } Pr D = 25 =.98 Invenory-holding and backorder penaly coss: A( =.2( x 25) 2 Oher parameer values: c = 4.5, p=.5, k = 1, T = 4 The reason for his non-monooniciy is ha he level of effor in a period affecs he marginal value of iniial invenory in ha period. In paricular, he value of a given level of iniial invenory decreases as he effor increases (since he probabiliy of immediae delivery increases). Hence, depending on he parameers of he business scenario, in our model, he opimal orderup-o level can be observed o increase as he end of he horizon approaches. 6. THE INFINITE-HORIZON PROBLEM The infinie-horizon (T = ) problem is he same as he muli-period (T < ) problem, excep ha we minimize he expeced long-run average oal coss per period over an infinie horizon. The purpose of examining his problem is o (i) compue and observe he opimal policy and (ii) examine under wha end-of-horizon condiion he non-saionary opimal policy becomes saionary in he finie-horizon problem. As expeced, i was observed ha a sae-dependen base-sock policy is opimal for he infinie-horizon problem. For he second purpose of examinaion, noe ha in he radiional finie-horizon newsvendor model wih fixed leadime, he opimal policy becomes saionary if a paricular end-of-horizon invenory-valuaion funcion is added o he model. This invenoryvaluaion funcion is linear, and equal o c x, where c is he acquisiion cos per uni and x is he invenory a he end of he horizon. By solving he infinie-horizon problem of our model, we 11

14 were able o compue he corresponding end-of-horizon invenory-valuaion funcion, denoed v( x ), ha induces a saionary opimal policy in he finie-horizon problem. However, as migh be expeced, he end-of-horizon invenory-valuaion funcion, v( x ), is no linear. See Figure 3. Inuiively, his is because he marginal value of a uni of iniial invenory in he infinie-horizon problem is deermined by is acquisiion cos plus is marginal conribuion o he one-period expeced invenory-holding and shorage cos muliplied by he probabiliy of an order being delayed. In wha follows we briefly explain how o solve he infinie-horizon problem by using a policy-ieraion procedure. Consider he following opimaliy equaion: { } ( ) ( ) ( ) g+ v x = min y s, x + E v s D, x Λ (18) s x where g represens he long-run average oal coss per period and Λ is he space of invenory levels in seady sae. v( x ) represens he value of iniial invenory x. If here exiss a soluion for v ( ) and g in he opimaliy equaion, (18), hen he value of s ha minimizes he righ- hand side of (18), denoed s ( (, is he opimal invenory policy. The opimal effor sraegy p is obained from he equaion (3). Furher, if here exiss a soluion for he opimaliy equaion, (18), hen a saionary policy is opimal (Ross,197). In our compuaional sudy, as in he previous secion, we assumed ha demands in each period can ake on a finie number of values, i.e., D =,1,2..., D. Then, he opimal order-up-o level s ( mus be a mos 2D, since he maximum demand during he curren and he nex period is 2D. Also, noe ha s (, if we assume π p > c (i.e., if he expeced shorage cos ha can be avoided by a uni is greaer han is acquisiion cos). Hence, we only have o search { x } s D for s ( max, 2 { D,...,2D}, x 2D; and, he space of invenory levels in seady sae is Λ=. Given his, he opimaliy equaion, (18), can be solved in a finie number of ieraions by a ypical policy-ieraion procedure (Howard, 196). 7. CONCLUSION We have examined he problem of managing sochasic replenishmen leadimes in a simple, periodic-review invenory sysem, where a replenishmen order can be delivered immediaely or 12

15 one period laer, depending probabilisically on cosly effor. We have examined he singleperiod ( T = 1), muli-period ( T < ), and infinie-horizon (T = ) problem. We have provided several ineresing resuls. In paricular, our analyical and compuaional analysis suggess ha (i) a sae-dependen base-sock policy is opimal and (ii) he cos impac of leadime reducion can be very large. 13

16 REFERENCES Anupindi R, Akella R. Diversificaion under supply uncerainy. Managemen Science 1993; 39(8); Chiang C, Guierrez GJ. A periodic review invenory sysem wih wo supply modes. European Journal of Operaional Research 1996;94; Chiang C, Guierrez GJ. Opimal conrol policies for a periodic review invenory sysem wih emergency orders. Naval Research Logisics 1998;45; Chuang B, Ouyang L, Chuang K. A noe on periodic review invenory model wih conrollable seup cos and lead ime. Compuers & Operaions Research 24;31; Federgruen A, Zipkin P. An invenory model wih limied producion capaciy and uncerain demand I. The average cos crierion. Mahemaics of Operaions Research 1986a; 11(2); Federgruen A, Zipkin P. An invenory model wih limied producion capaciy and uncerain demand I. The discouned cos crierion. Mahemaics of Operaions Research 1986b; 11(2); Fukuda Y. Opimal policies for he invenory problem wih negoiable leadime. Manageimen Science 1964;1(4); Hariga M, Ben-Daya M. Some sochasic invenory models wih deerminisic variable lead ime. European Journal of Operaional Research 1999;113; Henig M, Gerchak Y. The srucure of periodic review policies in he presence of random yield. Operaions Research 199;38(4); Howard RA. Dynamic Programming and Markov Processes. Published joinly by he Technology Press of he MIT and John Wiley & Sons, Inc., New York; 196. Karlin S. Opimal invenory policy for he Arrow-Harris-Marschak dynamic model. In: Arrow K, Karlin S, Scarf H (Eds), Sudies in he mahemaical heory of invenory and producion, Sanford Universiy Press: Sanford, CA; Karlin S, Scarf H. Invenory models of he Arrow-Harris-Marschak ype wih ime lag. In: Arrow K, Karlin S, Scarf H (Eds), Sudies in he mahemaical heory of invenory and producion, Sanford Universiy Press: Sanford, CA; Lee J, Schwarz LB. Leadime reducion in a (Q, r) invenory sysem: An agency perspecive. Acceped for publicaion in Inernaional Journal of Producion Economics;

17 Ouyang L, Chuang B. A minimax disribuion free procedure for periodic review invenory model involving variable lead ime. Inernaional Journal of Informaion and Managemen Sciences 1998;9(4); Ouyang L, Chuang B. A periodic review invenory model involving variable lead ime wih a service level consrain. Inernaional Journal of Sysems Science 2;31(1); Ray S, Gerchak Y, Jewkes EM. The effeciveness of invesmen in lead ime reducion for a make-o-sock produc. IIE Transacion 24;36; Ross SM. Applied Probabiliy Models wih Opimizaion Applicaions. Holden-Day, San Francisco; 197. Tagaras G, Vlachos D. A periodic review invenory sysem wih emergency replenishmens. Managemen Science 21;47(3); Yang G, Ronald RJ, Chu P. Invenory models wih variable lead ime and presen value. European Journal of Operaional Research 25;164(2);

18 Table 1. Funcions and Parameers Funcions Parameers Values Invenory coss h 1 and demand π 66, 82, 99, 11, 142, 199 c 3 λ 5 D k 1, 2, 3, 5, 7, 1 = p p p p p.2,.3,.4,.5,.6,.7 p 1 ( ) ( ) W p k p p - The values wih represen he base case. Table 2. Numerical Resuls on IMPACT% TC(NE) TC IMPACT% BASE CASE π p κ Average

19 Figure 1. Non-Unimodaliy of he Single-Period Expeced Toal Cos, y( s, x ) y( s, x ) P x P x,ˆ s x ŝ s Example: ( 6) ( ) ( ) 2 2 Γ d d f ( d) = 1, d =,1,...,1 1Γ 3 Γ ; ( ) = ( ) ( ) + ; W ( p) ( p ) A x E x D E D x 1 = p Figure 2. Shape of he opimal order-up-o level as a funcion of iniial invenory s ( s st ( x ) s x 17

20 Figure 3. End-of-horizon Invenory-valuaion Funcion, v( x ) v( x 18