Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds


 Kellie Megan McBride
 2 years ago
 Views:
Transcription
1 OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp issn X eissn informs doi /opre INFORMS Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Xiangwen Lu Cisco Sysems, 210 Wes Tasman Drive, San Jose, California 95134, JingSheng Song Fuqua School of Business, Duke Universiy, Durham, Norh Carolina 27708, Amelia Regan Compuer ScienceSysems, School of Informaion and Compuer Science, Universiy of California, Irvine, California 92697, We consider a finiehorizon, periodicreview invenory model wih demand forecasing updaes following he maringale model of forecas evoluion (MMFE). The opimal policy is a saedependen basesock policy, which, however, is compuaionally inracable o obain. We develop racable bounds on he opimal basesock levels and use hem o devise a general class of heurisic soluions. Through his analysis, we idenify a necessary and sufficien condiion for he myopic policy o be opimal. Finally, o assess he effeciveness of he heurisic policies, we develop upper bounds on heir value loss relaive o opimal cos. These soluion bounds and cos error bounds also work for general dynamic invenory models wih nonsaionary and auocorrelaed demands. Numerical resuls are presened o illusrae he resuls. Subjec classificaions: invenory, forecasing, MMFE, approximaion, error bounds. Area of review: Manufacuring, Service, and Supply Chain Operaions. Hisory: Received June 2003; revisions received Augus 2004, May 2005; acceped November Inroducion Demand forecasing is essenial for invenory planning, especially when he demand environmen is highly dynamic and he procuremen lead imes are long. How o adjus invenory planning decisions according o demand forecasing updaes is of grea ineres o managers and for decades has araced many researchers. I is well known ha he opimal invenory policy for any dynamic forecasing invenory model is very complex, so boh in pracice and in he research lieraure considerable aenion has been given o much simpler myopic policies. For example, several auhors have eiher proposed using myopic policy as invenory policy (e.g., Graves 1999 and Aviv 2003) or esablished sufficien condiions under which a myopic policy can be opimal in specific demandforecasing models (e.g., Johnson and Thompson 1975, Miller 1986). Also, due o racabiliy, some recen works in he supply chain managemen lieraure employ he myopic policy in a demandforecasing environmen o gain various insighs such as he value of informaion sharing (e.g., Lee e al. 2000), collaboraive planning, forecasing and replenishmen (e.g., Aviv 2001, 2002), and quanifying bullwhip effec (e.g., Chen e al. 2000). However, some basic quesions remain. In paricular, are he insighs gained from he myopic policy sill valid in a sysem under an opimal policy? This is equivalen o asking how good he myopic policy is in general dynamic demandforecasing invenory models. Also, if he myopic policy is no good enough, are here any simple adjusmens ha can improve he performance significanly? More generally, how can we evaluae he performance of a myopic policy or any oher heurisic policy in erms of heir value loss relaive o opimal policy? These are he quesions we aim o address in his paper. We consider a singleiem, periodicreview invenory sysem wih demand forecasing updaes. The demand can be imecorrelaed and nonsaionary over ime or follow any demandforecasing model. For simpliciy, we assume ha he forecas evoluion follows he maringale model of forecas evoluion (MMFE), developed independenly by Graves e al. (1986) and Heah and Jackson (1994). The MMFE is quie sraighforward, general, and flexible. I can represen nonsaionary and imecorrelaed demands. I can also accommodae judgmenal forecass as well as commonly used ime series models such as he auoregressive moving average (ARMA) model. Oher feaures of he invenory model are sandard, such as full backlogging; a consan replenishmen lead ime; and linear ordering, invenoryholding, and backorderpenaly coss. Several auhors have adoped he MMFE o sudy producioninvenory planning issues. For insance, Güllü (1996) uses a woperiod MMFE o assess he value gained from using a dynamic demandforecasing model. Graves e al. (1998) address how o adjus he maerial requiremen schedule when he safeysock plans are modified from 1079
2 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1080 Operaions Research 54(6), pp , 2006 INFORMS period o period due o he modificaion of he demand forecass. Tokay and Wein (2001) focus on one ype of forecascorreced invenory policy in a sysem wih finie capaciy and obain closedform approximaions. Gallego and Özer (2001) consider a model of advance demand informaion and characerize he form of opimal policy. Their demand model can be viewed as a special case of he MMFE. Among he sudies using MMFE, Iida and Zipkin (2006) is mos closely relaed o ours. They show ha a demandforecas dependen basesock policy is opimal and develop bounds on he opimal basesock levels. They furher develop a piecewiselinear approximaion of he cos funcions and a simulaionbased echnique o solve he problem approximaely. Finally, hey esablish condiions under which he myopic policy is opimal. In his paper, we make wo major conribuions o he lieraure. The firs is he developmen of easierocompue bounds on opimal basesock levels, using a differen approach from ha of Iida and Zipkin (2006). Our approach also allows us o provide a necessary and sufficien condiion for he myopic policy o be opimal and gain deeper insighs. The second conribuion is he developmen of error bounds on he value loss of any heurisic policy relaive o he opimal cos, a subjec no sudied by Iida and Zipkin. These coserror bounds can also be used o evaluae heurisic policies in any dynamic demandforecasing invenory sysems. Our main idea for developing bounds on he opimal basesock levels is hrough a samplepah approach o approximae he firsorder condiion funcion in he dynamicprogram formulaion. We develop an explici expression of he firsorder condiion o see clearly he radeoff beween he marginal cos in he curren period and he marginal cos for he fuure periods. We use boh he probabiliy of oversock for a number of periods (i.e., no orders would be placed for a number of periods because he preorder invenory level is higher han he opimal basesock level in hese periods) and he magniudes of hese oversocks o esimae he marginal fuure cos. The noion of obaining bounds on he opimal basesock levels by esimaing he marginal fuure cos in dynamic invenory models is no new. See Moron (1978) and Moron and Penico (1995) for models wih nonsaionary and independen demands. These works use he probabiliy of oversock for a number of periods o develop lower bounds on he opimal basesock levels. By employing more informaion han he probabiliy of oversock, we obain significanly igher lower bounds; see, e.g., Example 7. Anoher approach o obaining bounds on he opimal basesock levels in dynamic invenory models is o allow disposal of sock earlier han he end of he horizon. This approach ransforms he original problem ino a shorer planning horizon problem. Solving he shorer planning horizon problem wih he upper or lower bound (on he marginal fuure cos) being erminal cos leads o he upper or lower bound on he opimal basesock levels. Using lower bound zero and upper bound he maximum salvage cos in he erminal period, Moron (1978) and Iida and Zipkin (2006) develop bounds on he opimal basesock levels for he independen demand model and MMFE, respecively. In fac, our approach is general enough o rea hese mehods as special cases; see Examples 5 and 6. However, solving a shorer planning horizon problem opimally is sill compuaionally challenging even for small problem sizes under MMFE. In he lieraure of dynamic invenory models, a commonly used mehod o evaluae he performance of a heurisic policy is o esimae he gap beween he lower and upper bounds on he opimal basesock levels. Moron (1978) and Iida and Zipkin (2006) show ha, under cerain condiions, he gap beween he upper and lower bounds by solving a shorer planning horizon (say k + 1 period) problem (wih zero or maximum salvage value as he erminal cos) goes o zero as k goes o infiniy. However, for he MMFE examined in his research, i is no pracical o solve a k + 1 period problem opimally even for small k. This moivaes us o explore an alernaive approach o evaluaing he effeciveness of heurisic policies. Our approach is a samplepah worscase approach. We develop upper bounds on he cos difference beween any heurisic policy and he opimal policy. We also develop lower bounds on he opimal cos. This leads o upper bounds on he cos error of any heurisic relaive o he opimal policy. To our knowledge, Lovejoy s (1990, 1992) are he only previous aemps in he dynamic invenory lieraure o esablish cos error bounds on subopimal policies. While Lovejoy focused on myopic and sopped myopic policies, we derive cos error bounds for any heurisic policy. Our echniques are also differen from his. When applied o myopic policies, our bounds are significanly igher. (Since he compleion of our sudy, Levi e al. (2004) have examined a similar invenory model o ours and show ha he cos of a dualbalancing policy is wihin 200% of he cos of opimal policy.) Several oher ypes of forecasing models have been sudied in he lieraure. One is a Bayesian model of updaing demand disribuion from pas hisory; see, e.g., Scarf (1959, 1960), Azoury (1985), and Lovejoy (1990). The second is a ime series approach he demand process is he ARMA process or he ARIMA (inegraed ARMA) process; see, e.g., Johnson and Thompson (1975), Miller (1986), Reyman (1989), and Graves (1999). The hird approach is o model he demand as a Markovmodulaed sochasic process; see, e.g., Lovejoy (1992), Song and Zipkin (1993), and Treharne and Sox (2002). For furher discussion of he lieraure, see Iida and Zipkin (2006). The res of his paper is organized as follows. Secion 2 inroduces he basic noaion and he model formulaion. Secion 3 discusses he firsorder condiion and he myopic policy. Secion 4 presens he soluion bounds, while 5
3 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1081 develops he cos error bounds of any heurisic policy relaive o opimal policy. Finally, 6 presens numerical examples, and 7 concludes he paper. 2. Model and Formulaion We consider a T period periodicreview invenory sysem wih sochasic demand and zero replenishmen lead ime. (The exension o sysems wih a fixed consan lead ime can be done by following he sandard argumen.) Le D be he acual demand in period. The demand process D = 1 2 T can be nonsaionary and correlaed over ime. A any period, we generae forecass of he demand for all fuure periods in he horizon. A he beginning of each period, an ordering decision is made based on he invenory saus and he demand forecas. Then, he placed orders arrive. During he period, demand is realized and fulfilled as much as possible. Unsaisfied demand is fully backlogged. A he end of he period, invenoryholding and backorderpenaly coss are charged, and demand forecass are updaed. There are linear coss for ordering, invenory holding, and backlogging, respecively. We use ĉ, ĥ, and ˆb o represen he uni ordering, invenory holding, and backorderpenaly coss in period, 1 T, respecively. We also assume ha here is a salvage value ĉ T +1 a he end of period T. The objecive is o minimize he oal expeced cos. To updae he demand forecass, we can eiher follow a sandard forecasing ool such as a imeseries model, or use oher echniques such as exper judgmen, or do boh. Le D +i be he forecas made a he end of period for he demand in period + i, i = 0 T. Because forecass are made afer he curren demand informaion is revealed, D = D. Le D = demand forecas vecor made a he end of period = D +1 D T, where D 0 is he iniial forecas vecor. We consider wo ypes of forecas updaes: addiive and muliplicaive. For addiive updaes, define e +i = D +i D 1 +i as he forecas updae made a he end of period for demand in period + i. Denoe Var e = 2 and le e = demand forecas updae vecor made a he end of period = e e +1 e T. We assume ha he forecass are unbiased, i.e., E e s = 0 s. We also assume ha he forecas updaes e, = 1 2 T are independen over ime. The forecas updaes wihin a period, however, are no necessarily independen because hey migh rely on he same or relaed informaion. For muliplicaive updaes, similarly define e +i = D +i /D 1 +i. Here, E e +i = 1. Again, le e be he forecas updae vecor made a he end of period. Weassume ha boh D +i and e +i are posiive and he forecas updaes are independen over ime. As before, he forecas updaes wihin a period are no necessarily independen. For exposiion simpliciy, we mainly focus on he addiive model hroughou he paper (wih he excepion of 6.2). However, all he resuls hold rue for he muliplicaive model. We furher assume ha he forecas updaes have a coninuous disribuion, and hus he oneperiod cos has a unique minimum poin. The forecas updaes in differen periods can have differen disribuions. This model is broader han he original MMFE (e.g., Heah and Jackson 1994), which assumes he mulivariae normal disribuion. We now formulae he problem as a dynamic program. Le I (respecively, I ) be he invenory level a he beginning of period afer (respecively, before) ordering. The sae of he sysem a he beginning of period is I D 1. Because he lead ime is zero, he sysem dynamics are I +1 = I D = I D 1 e D +i = D 1 +i + e +i for 0 i T Le C y D 1 be he expeced holding and backorder coss charged o period, given ha I = y and he laes forecas for demand in period is D 1. Then, C y D 1 = ĥ E y D 1 + e + + ˆb E D 1 + e y + where x + = max x 0 and D 1 + e = D. Le V x D 1 be he opimal oal expeced coss from period hrough T, given ha I = x and he laes forecas for he fuure demands is D 1. We have he following recursive funcional equaions: V T +1 x D T = ĉ T +1 x V x D 1 = min ĉ y x + C y D 1 y x + E V +1 y D 1 + e D Nex, we make a ransformaion o simplify he exposiion. Se h = ĥ + ĉ ĉ +1 and b = ˆb ĉ ĉ +1. Le C y D 1 =h E y D 1 +e + +b E D 1 +e y + = ĉ ĉ +1 y + C y D 1 V x D 1 =ĉ x + V x D 1 We obain V T +1 x D T = 0 (1) V x D 1 = min y D 1 y x 1 T (2) where G y D 1 C y D 1 + E V +1 y D 1 + e D (3)
4 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1082 Operaions Research 54(6), pp , 2006 INFORMS From now on, we call he ransformed funcions C y D 1 and V x D 1 he oneperiod expeced cos and he opimal oal expeced cos from period o period T, respecively. I can be shown ha G y D 1 is convex in y. Le s D 1 be is minimizer. Then, he basesock policy wih ime and saedependen basesock levels s D 1 is opimal (e.g., Iida and Zipkin 2006). However, he mulidimensional Equaions (1) and (2) are exremely difficul o compue, and so is s D 1. In he res of his paper, we develop racable approximaions of s D 1 and provide error bounds on he cos of using he approximae policies. For simpliciy, we someimes suppress he argumen in he basesock levels. For example, we wrie s +i wih he undersanding of s +i D +i 1. Also, le D +i represen he cumulaive demand in periods 1 hrough + i 1, i.e., i 1 i 1 D + i = D +j = D +j +j j=0 j=0 Using his noaion, (3) can be rewrien as G y D 1 C y D 1 + E V +1 y D + 1 D Throughou his paper, for any funcion y D, wihou any confusion, we use y D o denoe he parial derivaive / y y D. In addiion, for any real numbers u and v, we denoe u v = max u v and u v = min u v. 3. The FirsOrder Condiion and Myopic Policies 3.1. The FirsOrder Condiion: A SamplePah View Because s D 1 is he soluion of he firsorder condiion G D 1 = 0, o develop approximaions of s D 1, we begin wih examining he consiuions of G D 1. Assume ha he invenory level afer ordering in period is y, i.e., I = y. By he dominaed convergence heorem, i is sraighforward o show ha E V +1 y D + 1 D = E V +1 y D + 1 D. Therefore, G y D 1 = C y D 1 + E V +1 y D + 1 D (4) Noe ha V +1 x D = G +1 x s+1 D D, which implies 0 x s V +1 x D +1 D = (5) G +1 x D >0 x>s+1 D So, V +1 = 0 if he opimal basesock level in period + 1, s+1 D, is reachable (Veino 1965), i.e., he preorder invenory level, I+1 = y D + 1, is no greaer han he opimal basesock level s+1 D. Therefore, he decision y a ime has no effec on he fuure cos if s+1 D is reachable. In oher words, y affecs (increases) he cos of periods + 1 and beyond only if he opimal basesock level in period + 1isno reachable, in which case V +1 is posiive. Similarly, for any fuure period + i, V +i is nonzero (posiive) only if none of s+1 s +2 s +i is reachable, i.e., I+j = y D +j >s +j, j = 1 2 i. This means ha he evens A +j y D 1 happen for all j = 1 2 i, where A +j y D 1 = y D + j >s +j D +j 1 (6) Wihou confusion, we suppress y D 1, he iniial condiion a ime, in his noaion mos of he ime. Le I A be he indicaor funcion of A and I A c = 1 I A. Noe ha he decision y a ime affecs period + k and beyond only if A +i happens for all i = 1 k. This implies E I A +1 A +i V +i y D + i D +i 1 > 0 and E I A +1 A +i 1 A c +i V +i y D +i D +i 1 =0 (7) The following proposiion provides a decomposiion of G. Proposiion 1. For any ime, G y D 1 = C y D 1 + y D 1 (8) where y D 1 = E V +1 y D + 1 D E I A +1 A +i T = i=1 Moreover, y D 1 0. C +i y D + i D +i 1 +i (9) Proof. According o (4), we need o work only on E V +1 y D + 1 D : E V +1 y D + 1 D = E I A +1 V +1 y D + 1 D + E I A c +1 V +1 y D + 1 D = E I A +1 V +1 y D + 1 D = E I A +1 C +1 y D + 1 D +1 + E I A +1 V +2 y D + 2 D +1
5 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1083 = E I A +1 C +1 y D + 1 D +1 + E I A +1 A +2 V +2 y D + 2 D +1 + E I A +1 A +2 c V +2 y D + 2 D +1 = E I A +1 C +1 y D + 1 D +1 + E I A +1 A +2 V +2 y D + 2 D +1 T = = E I A +1 A +i C +i y D +i D +i 1 +i i=1 = y D 1 The second and he fifh equaliies are due o (7). Finally, y D 1 0 follows from (5). From Proposiion 1, he opimal invenory decision is a radeoff beween he marginal cos in he curren period C y D 1 and he marginal fuure cos y D 1. While he marginal curren cos can be posiive or negaive, from (5) he marginal fuure cos is always nonnegaive. Applying (8), he firsorder condiion for opimaliy is G s D 1 D 1 = C s D 1 D 1 + s D 1 D 1 = 0 (10) Noe ha C s D 1 D 1 = b + h P D 1 + e s D 1 b = b + h F s D 1 D 1 b where F is he cumulaive disribuion funcion of forecas error erm e. Treaing s D 1 D 1 as a known consan, from (10) we can express he opimal basesock level as ( s D 1 = D 1 + F 1 b s D ) 1 D 1 (11) b + h Because s D 1 D 1 depends on he enire forecas evoluion, i is very difficul o obain Myopic Policies One common approach o deal wih he difficuly of obaining s D 1 D 1 is o ignore i and use only he firs erm, C y D 1, o approximae G y D 1. This resuls in he socalled myopic policy a basesock policy wih he basesock level solving C y D 1 = 0. Le s m D 1 be he myopic basesock level a ime given ha he forecas vecor a he beginning of is D 1. Then, ( ) s m D 1 = D 1 + F 1 b (12) b + h If F is a normal disribuion and we le be he sandard normal disribuion funcion (remember ha is he sandard deviaion of he forecasing error erm e ), hen ( ) s m D 1 = D b (13) b + h Here, 1 b / b + h is ermed he safey facor. The myopic policy simply uses a lower bound zero o approximae he erm s D 1 D 1 0 and hus is an upper bound on s D 1, i.e., s D 1 s m D 1 Remark. In he case of a consan lead ime L, (12) becomes ( ) L s m D 1 = D 1 +i + F L+1 1 b (14) b + h i=0 where F L+1 is he cumulaive disribuion funcion of L j=i e +i +j. L i=0 The simple expressions of (12) and (13) furher explain why he myopic policy is popular and when i migh cause subopimaliy. A any ime, he policy parameer is he sum of wo erms: he laes forecas of he curren period demand D 1 and a safey sock ha depends only on he disribuion of forecas error e. Noe ha in mos forecasing models, i is reasonable o have =. If, furher, he cos parameers are saionary, hen he safey sock is a consan. Under hese seings, using he myopic policy, we would sock in he firs period he consan safey sock plus he forecased demand for he firs period, D 0 1. Then, in he subsequen periods, we need only o adjus he order quaniy according o he realized demand in he previous period and he laes demand forecas for he curren period. More specifically, if we have ordered in period 1, so he posorder invenory posiion in ha period I 1 = s 1 m and he preorder invenory posiion I = s 1 m D 1 1, hen q = s m I = D 1 D D 1 1 = D 1 + e 1 1 is he order quaniy in period if his quaniy is nonnegaive. If q is nonnegaive for all sample pahs for all, which means ha he myopic basesock level in each period is reachable, hen he cos in each period is minimized, so he myopic policy is opimal. I urns ou ha q 0is boh necessary and sufficien for he myopic policy o be opimal. In general, we have he following. Proposiion 2. The myopic policy is opimal if and only if P s m D 1 D + 1 >s m +1 D = 0 for all (15) If boh he cos parameers and he forecas updae process are saionary, (15) is equivalen o q = D 1 + e for all (16) Proof. The sufficien condiion can be shown easily by inducion; we omi he deails here. To show he necessary condiion, suppose ha P s m D >s+1 m >0. This implies ha E I s m D >s+1 m C +1 sm D D +1 > 0.
6 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1084 Operaions Research 54(6), pp , 2006 INFORMS Because V +2 0, we have G sm D 1 >0, which means ha he myopic policy canno be opimal a conradicion. Now assume saionary coss and forecas updaes. Condiion (16) requires ha he demand forecas for he curren period (period ) be large enough o offse he negaive deviaion of he forecas error in he previous period. I is conceivable ha his condiion can be me easily if he demand process has a nondecreasing rend. This is consisen wih he general undersanding of when he myopic policy is expeced o be opimal. I is ineresing o see ha even when demand has a decreasing rend, he myopic policy can sill be opimal. Indeed, Iida and Zipkin (2006) offers such an example (see heir independen, nonsaionary demand example), provided he following condiion holds: For every sample pah of forecas updaes, demands are nonnegaive, i.e., D = D 1 + e 0 for all (17) Because e 1 1 and e have he same disribuion and e and D 1 are uncorrelaed, (17) implies (16). Alhough he demand may have a decreasing rend, (17) ensures ha he forecas errors will be bounded by he lowes possible demand. In oher words, if he demand in one period is low, he forecas errors in all periods have o be sufficienly small. Noe ha because D 1 can be correlaed wih e 1 1, (16) does no necessarily imply (17). Therefore, Iida and Zipkin s condiion is sufficien bu no necessary. We now illusrae his by an example adaped from Güllü (1997). In his example, demands in wo consecuive periods are negaively correlaed. If he demand in he curren period is high (respecively, low), hen he demand in he nex period is expeced o be lower (respecively, higher). This ype of demand paern is common when cerain markeing effors are in place. For example, he demand righ afer a promoion period is expeced o be lower because of forward buying during he promoion period. Example 3. Le be he mean demand, and se D 0 = e = N 0 2 e +1 = e s = 0 for all s + 2 D = D 0 + e 1 + e for all Here, N 0 2 is he normal random variable N 0 2 runcaed a 25 and 25. (Obviously, given he large runcaion limis, N 0 2 is basically N 0 2.) Assume ha = 0 9, = 100, and = 40. Because e 1 = 0 9e 1 1 and , we have q = D 1 +e 1 1 = +e 1 +e 1 1 = Hence, he myopic policy is opimal. However, D 1 + e can be negaive, so (17) is no saisfied. When would myopic policies behave poorly? From he necessary and sufficien condiion (16) we see ha, for saionary coss and forecas updaes, he myopic policy is subopimal if here exis sample pahs in which q < 0 for some. This could happen if here is a sudden drop in demand or here is a decreasing demand rend so ha D 1 is oo small o offse he negaive deviaion of e 1 1. A similar observaion is made by Song and Zipkin (1996) using a Markov modulaed sochasic process o model demand facing obsolescence. 4. Bounds on Opimal BaseSock Levels Recall ha he myopic policy simply approximaes he nonnegaive erm s D 1 D 1 by zero and hus is an upper bound on he opimal basesock level s D 1. In his secion, we develop a general class of upper (lower) bounds on s D 1 D 1 (hese imply bounds on G ), which, according o (11), yield lower (upper) bounds on he opimal basesock level s D 1. Observe from he expression of (9) ha wha makes difficul o compue is is dependence on he opimal basesock levels from period + 1 on hrough he evens A +j in (6), j = 1 i. Our idea for approximaion is o replace he opimal basesock level s+j D +j 1 in A +j wih simplerocompue bounds. We presen a procedure o consruc he bounds recursively. We laer show hrough examples ha our procedure includes many exising bounds in he lieraure as special cases Consrucion of he Bounds Se T u D T 1 = T l D T 1 = 0 and st u D T 1 = st l D T 1 = st m D T 1 (he myopic basesock level in period T ). For any ime < T, suppose ha we have obained upper and lower bounds on he opimal basesock level in period + i, s+i u D +i 1 and s+i l D +i 1, respecively, for all i = 1 T. (For example, we may have s+i u D +i 1 = s+i m D +i 1 and s+i l D +i 1 = 0.) Assume ha I = y, and define A u +i y D 1 = y D + i > s u +i D +i 1 A l +i y D 1 = y D + i > s l +i D +i 1 A m +i y D 1 = y D + i > s m +i D +i 1 T l y D 1 = E I A u +1 Au +i u y D 1 i=1 C +i y D + i D +i 1 +i = E I A u +1 Am +1 C +1 y D + 1 D +1 E I A l +1 Al +i 1 Am +i T + i=2 C +j y D + i D +i 1 +i
7 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1085 S l D 1 = zero poin of C y D 1 + u y D 1 S u D 1 = zero poin of C y D 1 + l y D 1 Noe ha l y D 1, u y D 1, S l D 1, and S u D 1 all depend on he known bounds s+i u D +i 1 and s+i l D +i 1, i = 1 T, implicily. If all he evens A m +1 y D 1 A m +i y D 1 happen, for example, he invenory level before ordering is greaer han he myopic basesock level in all periods +1 hrough +i given ha he invenory level afer ordering in period is y. Therefore, no order will be placed in hese periods under he myopic policy. Similar discussion applies o A u +i y D 1 and A l +i y D 1. As we shall show below, using he righ combinaion of A u +i y D 1, A l +i y D 1, and A m +i y D 1 o replace A +i y D 1 in y D 1 can resul in bounds on y D 1. We have Theorem 4. Theorem 4 (Bounds on Opimal BaseSock Levels). For any given, = 1 T, under he above assumpion and consrucion, we have l y D 1 y D 1 u y D 1 (18) This implies ha G y D 1 is bounded below and above by C y D 1 + l y D 1 and C y D 1 + u y D 1, respecively. Moreover, hese boundingfuncions are increasingfuncions of y, so heir zero poins S u D 1 and S l D 1 are unique and have he exac expression as in (11) wih replaced by l and u, respecively. Finally, S l D 1 s D 1 S u D 1 Proof. I is sufficien o show ha (18) holds; oher pars are sraighforward. Noe ha when = T, from he erminal condiion (1), all he inequaliies in (18) become equaliies. Now, consider <T. Firs, noe ha from he consrucion, A u +i A +i A l +i and Am +i A +i, implying I A u +i I A +i I A l +i and I A m +i I A +i for all i = 1 T. For every sample pah of D + 1, wehave C +1 y D + 1 D +1 <0 for s +1 D <y D + 1 <s m +1 D So, E I A +1 A u +1 Am +1 C +1 y D + 1 D Therefore, E I A +1 C +1 y D + 1 D +1 = E I A +1 A u +1 Am +1 C +1 y D + 1 D +1 + E I A u +1 Am +1 C +1 y D + 1 D +1 E I A u +1 Am +1 C +1 y D + 1 D +1 (19) Similarly, E I A +1 A +i C +i y D + i D +i 1 +i = E I A +1 A +i 1 A m +i C +i y D + i D +i 1 +i + E I A +1 A +i 1 A +i A m +i C +i y D + i D +i 1 +i E I A +1 A +i 1 A m +i C +i y D + i D +i 1 +i E I A l +1 Al +i 1 Am +i C +i y D + i D +i 1 +i i = 2 T (20) The firs inequaliy is due o he fac ha I A +i A m +i C +i y D + i D +i 1 +i 0. The second inequaliy is from I A m +i C +i y D + i D +i 1 +i 0. Combining (9), (19), and (20), we obain y D 1 u y D 1. On he oher hand, recall ha V +1 y D + 1 D is nonnegaive, so I A +i A u +i V +i y D + i D +i 1 0 (21) Then, E I A +1 V +1 y D + 1 D 1 = E I A u +1 V +1 y D + 1 D + E I A +1 A u +1 V +1 y D + 1 D E I A u +1 V +1 y D + 1 D = E I A u +1 G +1 y D + 1 D = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 A +2 V +2 y D + 2 D +1 = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 V +2 y D + 2 D +1 + E I A u +1 A +2 A u +2 V +2 y D + 2 D +1 E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 V +2 y D + 2 D +1 = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 G +2 y D + 2 D +1 = E I A u +1 C +1 y D + 1 D +1 + E I A u +1 Au +2 C +2 y D + 2 D +1 + E I A u +1 Au +2 A +3 V +3 y D + 3 D +2 T E I A u +1 Au +i C +i y D +i D +i 1 +i i=1 = l y D 1 (22) All he inequaliies are due o (21). Noe ha he second and fifh equaliies are due o he fac ha when A u +i happens, we have y D + i >s+i, so no order will be
8 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1086 Operaions Research 54(6), pp , 2006 INFORMS placed in period + i under he opimal policy. The hird and sixh equaliies are due o (7). From (9) and (22), we obain y D 1 l y D 1. To summarize, from Theorem 4, for any, based on any upper and lower bounds on he opimal basesock levels from period +1 unil T, we can consruc lower and upper bounds on he opimal basesock level in period. In his way, we develop a general class of bounds on he opimal basesock levels Connecions wih Exising Bounds Clearly, he only difference beween hese new approximaions and he myopic policy is he adjusmens in he fracile ha deermines he magniude of he safey sock. The myopic is a special case of our resul in which we se l y D 1 = 0. Below we provide several oher examples o illusrae he connecion of he bounds in Theorem 4 wih he exising bounds in he lieraure. The soluion o he shorerhorizon, k + 1 period problem wih zero erminal cos is referred o as he kperiod ahead soluion. I has been shown in he lieraure ha his soluion is an upper bound on opimal basesock level for he nonsaionary independen demand model (Moron 1978) and for he MMFE (Iida and Zipkin 2006). In Example 5, we show ha his resul can be viewed as a special case of our procedure. For simpliciy, we illusrae only he case of k = 1. The idea for general k is similar. Example 5 (OnePeriod Ahead Policy). Seing s+1 u = s+1 m and s+2 u = yields Au +2 = y D + 2 > s+2 u D +1 = y D + 2 > =. Therefore, E I A u +1 Au +i C +j y D + i D +i 1 +i = 0 for all i 2 and l y D = E I Am +1 C +1 y D + 1 D +1. According o Theorem 4, he soluion o C y D 1 + E I A m +1 C +1 y D + 1 D +1 = 0 is an upper bound on s D 1. In he following, we show ha our procedure generalizes and ighens some lower bounds in he lieraure. For insance, Example 6(b) has been shown by Moron (1978) and Iida and Zipkin (2006) for independen and MMFE demand models, respecively. Example 6(c) provides igher lower bounds on he opimal basesock level han ha of Example 6(b). Example 6. Le s+i D k +i 1 be he opimal soluion o a k + 1 period problem (from period unil period + k) for period + i wih erminal cos of T i=k+1 h +i. Then, (a) s+i D k +i 1 solves he following equaions recursively: C +i y D +i 1 +i + k +i y D +i 1 = 0 (23) where k y D 1 = E [ I ( y D + 1 >s k ) +1 C +1 y D + 1 D +1 ] [ + +E I ( y D + 1 >s k ) +1 I ( y D +k+1 >s k ) +k+1 ( T h +i )] (24) i=k+1 (b) s+i D k +i 1 s+i D +i 1, i = 0 k. (c) Replacing T i=k+1 h +i in (24) wih T i=k+1 h +ip y D + i >s+i m D +i 1, solving (23) leads o a igher lower bound for s D 1. Proof. By he same logic in developing Proposiion 1, we obain (a). Because he proof of (c) is similar o ha of (b), below we show only (b). We use inducion. Because C +k y D +k 1 +k + k +k y D +k 1 C +k y D +k 1 +k + +k y D +k 1 we know ha s k +k D +k 1 s+k D +k 1. Suppose ha s+j D k +j 1 s+j D +j 1 holds for i j k. We only need o show s+i 1 D k +i 2 s+i 1 D +i 2. For simpliciy, we show he case of i =1. Applying a similar proof of (22) o he k + 1 period problem under consideraion, we know ha replacing s+j D k +j 1 1 j k wih is upper bound s+j D +j 1 leads o a lower bound on k y D 1. Tha is, C y D 1 + k y D 1 k 1 C y D 1 + E I A +1 A +i i=1 C +i y D + i D +i 1 +i + E I ( y D + k + 1 >s k ) ( T )] +k+1 h +i i=k+1 C y D 1 + y D 1 [ I A +1 A +k Therefore s k D 1 s D 1 holds because s k D 1 solves C y D 1 + k y D 1 = 0 and s D 1 solves C y D 1 + y D 1 = 0. Nex, we illusrae ha our procedure generalizes and ighens oher lower bounds presened in he lieraure ha are obained hrough avenues differen from solving shorerhorizon problems.
9 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1087 Example 7. Se s+1 u D = s+1 m D. According o Theorem 4, we have y D 1 u 1 y D 1, where u 1 y D 1 = E I A m +1 C +1 y D + 1 D +1 Because E I A l +1 Al +j 1 I Am +j T + j=2 C +j y D + j D +j 1 +j C +j y D + j D +j 1 +j = b +j + h +j P D +j 1 +j + e +j +j y D + j b +j h +j (25) we furher have u 1 y D 1 u 2 y D 1 u 3 y D 1, where u 2 u 3 T y D 1 =h +1 P A m +1 + h +j P A l +1 Al +j 1 Am +j j=2 T y D 1 =h +1 P A l +1 + h +j P A l +1 Al +j 1 Al +j According o Theorem 4, he soluion o each of he following hree equaions: j=2 C y D 1 + u i y D 1 = 0 i= (26) is a lower bound on he opimal basesock level. Furhermore, because u i y D 1 increases in y, wehave C y D 1 + u i y D 1 C y D 1 + u i s m D 1 D 1 y s m D 1 Suppose ha S l i D 1 solves C y D 1 + u i s m D 1 D 1 = 0. Then, S l i D 1 is newsvendor soluion, which is easier o compue bu is a looser lower bound (han he soluion o (26)): s D 1 S l 1 D 1 S l 2 D 1 S l 3 D 1 where S l i D 1 = D 1 + ( b u i s m D 1 D 1 b + h ) i = (27) The lower bound S l 3 D 1 is essenially he lower bound developed in Moron (1978) and Moron and Penico (1995). They developed he lower bound on he opimal basesock level for he case of independen demand invenory models and heir resul can be exended o he demandforecasing invenory models. In addiion o he probabiliy of oversock hey employ, we use more informaion such as he magniude of oversock, y D +j s+j D +j 1 given I = y, o esimae he marginal fuure cos. Observe from (25) ha h +j P A l +1 Al +j 1 Al +j can be much greaer han E I A l +1 Al +j 1 Am +j C +j, j = 1 2 T. So, u 1 y D 1 can be much smaller han u 3 y D 1, which could lead o a significanly igher lower bound on opimal basesock levels. Indeed, for a woperiod problem, one of our lower bounds, he soluion o (26) for he case of i = 1, is he exac opimal soluion, while heir lower bound S l 3 D 1 canno be opimal. In fac, none of he bounds in (27) can be opimal. Now we presen some easieroimplemen bounds, which are used in he numerical sudies in 6. Example 8. Se s u +1 D = s m +1 D, s u +2 D +1 = s m +2 D +1, and s u +3 D +2 =. We obain A u +3 = y D + 3 >su +3 D +1 = y D + 3 > = which implies E I A u +1 Au +j C +j y D + j D +j 1 +j = 0 j 3 According o Theorem 4, S u 1 D 1, he soluion o C y D 1 +E I A m +1 C +1 y D +1 D +1 +E I A m +1 Am +2 C +2 y D +2 D =0 (28) is an upper bound for s D 1. Example 9. Seing s u +1 D = s m +1 D and s l +i D +i 1 =, 2 i T, wehave A l +i = y D + i > sl +i D +i 1 = y D + i > = Therefore, E I A l +1 Al +i 1 Am +i C +i y D + i D +i 1 +i =E I A l +1 Am +i C +i y D +i D +i 1 +i j 3 Thus, S l 4 D 1, he soluion o C y D 1 + E I A m +1 C +1 y D + 1 D +1 E I A l +1 Am +j C +j y D +j D +j 1 +j =0 T + j=2 (29) is a lower bound on s D 1 according o Theorem 4.
10 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1088 Operaions Research 54(6), pp , 2006 INFORMS 5. CosError Bounds In his secion, we consider how o esimae he value loss of a heurisic relaive o opimal cos. Le V H x D 1 be he oal expeced cos of a given heurisic policy H in periods hrough T, assuming ha he preorder invenory level in period is x and he forecas vecor made a he end of period 1isD 1. The cos error of H relaive o he opimal cos is defined by err = V 1 H x D 0 V 1 x D 0 100% V 1 x D 0 Our approach is o develop an upper bound for V H x D 1 V x D 1 and a lower bound for V x D 1 because V x D 1 is usually compuaionally impossible o obain. Recall ha s m is an upper bound for he opimal basesock level; herefore, we assume ha s H D 1 s m D Upper Bound on V H x D 1 V x D 1 Take any period, and assume ha I = x. There are wo possible siuaions ha require differen reamens: s H D 1 s D 1 and s H D 1 s D 1. We sudy hem separaely and presen he resuls in Lemmas 10 and 11. Based on hese resuls, we develop upper bounds on V H x D 1 V x D 1, which is presened in Theorem 12. Lemma 10. If s H D 1 s D 1, we have x D 1 V x D 1 1 D 1 + E +1 x sh D 1 D D V +1 x s H D 1 D D (30) where 1 D 1 = C s H D 1 D 1 C s u D 1 s m D 1 D 1 (31) Proof. If s H D 1 s D 1, hen C s H D 1 D 1 C s D 1 D 1. Furhermore, because s m D 1 minimizes C D 1,wehave C x s H D 1 D 1 C x s D 1 D 1 1 D 1 (32) Noe ha x D 1 = C s H D 1 x D 1 + E +1 sh D 1 x D D and V x D 1 = C s D 1 x D 1 + E V +1 s D 1 x D D We obain x D 1 V x D 1 = C x s H D 1 D 1 C x s D 1 D 1 + E +1 x sh D 1 D D E V +1 x s D 1 D D 1 D 1 + E +1 x sh D 1 D D E V +1 x s D 1 D D 1 D 1 + E +1 x sh D 1 D D V +1 x s H D 1 D D Here, he firs inequaliy is due o (32). The second inequaliy is because x s H D 1 D x s D 1 D and V +1 z D increases in z. Lemma 11. If s H D 1 >s D 1, hen x D 1 V x D 1 2 D 1 + E +1 x sh D 1 D D V +1 x s H D 1 D D (33) and x D 1 V x D 1 3 D 1 + max E +1 s D 1 D D V +1 s D 1 D D E +1 x D D V +1 x D D (34) Here, 2 D 1 = 1 C s H D 1 D 1 + u s H D 1 D 1 1 = s H D 1 s l D 1 + T 3 D 1 = E I A h +1 Ah +k 1 Am +k 2C +k sh D 1 k=1 D + k D +k 1 +k A h +i = sh D 1 D + i > s H +i D +i 1 2 = s H D 1 s l D 1 s H D 1 D s H +1 D + (35) Proof. We firs show ha (33) holds. We have he following key observaion: x D 1 V x D 1 = s H D 1 x D 1 G s D 1 x D 1 = G x s H D 1 D 1 G x s D 1 D 1 + x s H D 1 D 1 G x s H D 1 D 1
11 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1089 = G x s H D 1 D 1 G x s D 1 D 1 + C x s H D 1 D 1 C x s H D 1 D 1 + E +1 x sh D 1 D D E V +1 x s H D 1 D D = G x s H D 1 D 1 G x s D 1 D 1 + E +1 x sh D 1 D D E V +1 x s H D 1 D D (36) Noe ha G y D 1 s D 1 y s H D 1 0. We also have G x s H D 1 D 1 G x s D 1 D 1 = x s H D 1 x s D 1 G y D 1 y= x s D 1 x s H D 1 s H D 1 s D 1 + G y D 1 y=s H = s H D 1 s D 1 + C sh D 1 D 1 + s H D 1 D 1 1 C sh D 1 D 1 + u sh D 1 D 1 = 2 D 1 (37) The second inequaliy is due o (18). Combining (36) and (37) yields (33). We nex show ha (34) holds. If x s H D 1 s D 1, wehave x D 1 V x D 1 = C x D 1 C x D 1 + E +1 x D D E V +1 x D D = E +1 x D D E V +1 x D D (38) If x<s H D 1,wehave x D 1 = 0 D 1 (39) Because s m D 1 s H D 1, we also have C s H D 1 D 1 C s D 1 D 1 0 (40) Therefore, x D 1 V x D 1 = 0 D 1 V x D 1 0 D 1 V 0 D 1 = C s H D 1 D 1 C s D 1 D 1 + E +1 sh D 1 D D E V +1 s D 1 D D (41) E +1 sh D 1 D D E V +1 s D 1 D D (42) =E +1 sh D 1 D D E +1 s D 1 D D + E +1 s D 1 D D E V +1 s D 1 D D (43) The firs equaliy is due o (39). The firs inequaliy is due o V x D 1 increases in x, and he second inequaliy is due o (40). We furher show he following in he appendix: E +1 sh D 1 D D E +1 s D 1 D D 3 D 1 (44) From (43) and (44), if x<s H D 1, we have ha (34) holds. On he oher hand, if x s H D 1, from (38), (34) also holds. This complees he proof. Based on Lemmas 10 and 11 and noing ha n D 1, n = 1 2 3, do no depend on he iniial invenory x, we develop upper bounds on V H x D 1 V x D 1 as follows. Theorem 12 (Maximum Gap beween he Coss of Any Heurisic Policy and he Opimal Policy). For any given heurisic policy H and any, x D 1 V x D 1 1 D 1 n D 1 [ T ] + E 1 k D k 1 n k D k 1 n= 2 3 (45) k=+1 Proof. Suppose ha n = 2. For = T, because we assume ha st H sm T = s T, we do no need o consider sh T >s T. For he case st H s T, from Lemma 10, we have ha (45) holds. Suppose ha (45) holds for V+i H x D V +i x D, i = 1 T, for some. Also, recall ha he upper bound in (45) is independen of x. In he following, we show ha (45) holds for period, he proof will hen be compleed by inducion. No maer wheher s H D 1 is less han s D 1 or no, from (30) and (33), we have x D 1 V x D 1 1 D 1 2 D 1 + E +1 x sh D D V +1 x s H D D By inducion, (45) holds for n = 2. Similarly, based on (30) and (34), (45) also holds for n = 3. Remarks. (a) The maximum value loss of using he heurisic policy H is given by he righhand side of (45) wih = 1. (b) The righhand side of (45) does no depend on he preorder invenory level x, so wha we have is a worscase analysis. In oher words, he righhand side of (45) is an absolue upper bound on he cos difference beween he heurisic and he opimal policy. We can modify 1 D 1 n D 1 o develop a recursive upper bound for V1 H x D 1 V 1 x D 1, which depends on x. We illusrae his by modifying 1 D 1 2 D 1 ino
12 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1090 Operaions Research 54(6), pp , 2006 INFORMS 1 D 1 and 2 D 1 only. The oher is much more complex. 1 D 1 = C s H D 1 x D 1 C s u D 1 s m D 1 x D 1 2 D 1 = 2 D 1 = 1 C sh D 1 D 1 + u sh D 1 D 1 1 = s H D 1 x s l D 1 x + (c) In he numerical sudy, we use he following fac: T 3 D 1 E I A h +1 Am +k 2 k=1 C +k sh D 1 D + k D +k 1 +k (46) Taking any upper bound on he opimal basesock level ha is no greaer han he myopic basesock level (such as he myopic soluion) as a heurisic policy, we do no need o consider he case of s H D 1 s D 1. This yields he following corollary. Corollary 13. The maximum gap beween he coss of he opimal policy and any upperbound policy s u D 1, = 1 2 T, ha saisfies s u D 1 s m D 1 for all is 1 x D 0 V 1 x D 0 T 3 1 D 0 + E 3 D 1 where 3 1 D D 1 T = =2 T E 3 D 1 (47) =2 E s u D 1 D s u +1 D + I A m +k k=1 C +k su D 1 D + k D +k 1 +k (48) ( T h +k )E s u D 1 D s u +1 D + (49) k=1 In paricular, he coserror bounds for he myopic policy is given by replacing s u D 1 wih s m D 1 in (47). Noe ha, compared wih (46), he bound (48) is much easier o evaluae bu can be much looser (see he examples in 6). Using a differen approach, Lovejoy (1992) developed an upper bound for he cos difference beween he opimal policy and he myopic policy. Because s m D 1 s D 1, insead of developing 3, Lovejoy esimaes he cos of disposing of he oversock s m D s+1 m +, which yields an upper bound on he exra cos due o he decision made in period. For example, according o Lovejoy (1992), he disposal cos can be se a T j=+1 h j. Thus, Lovejoy s bound is o replace 3 D 1 wih T k=1 h +k E s u D 1 D s+1 u D +. From (49), he error bound developed in his paper can be much igher, as illusraed in he numerical examples in Lower Bound on V x D 1 The derivaion of a lower bound on V x D 1 is quie sraighforward. We assume ha an upper bound on opimal basesock level s u D 1 is known (recall ha s m D 1 is a special case of s u D 1 ). We have he following. Proposiion 14. V x D 1 C x s u D 1 s m D 1 D 1 E C +j s u +j D +j 1 T + j=1 s m +j D +j 1 D +j 1 +j (50) Proof. Because s+j D +j 1 s+j u D +j 1 s+j m D +j 1 s+j m D +j 1 for all j, wehave V x D 1 C x s D 1 D 1 E C +j s +j D +j 1 D +j 1 +j T + j=1 C x s u D 1 s m D 1 D 1 E C +j s u +j D +j 1 T + j=1 s m +j D +j 1 D +j 1 +j This lower bound is similar o hose in Lovejoy (1990, 1992) when s u D 1 is chosen o be s m D Numerical Sudy 6.1. Resul Illusraion In his subsecion, we presen a numerical sudy o illusrae he resuls developed in he previous secions. We also compare our coserror bounds wih hose developed in Lovejoy (1990, 1992). We use an AR(1) demand forecas model. Tha is, D = + D for all < 1 where E D =, he coefficien of correlaion of demands in wo successive periods is, and are i.i.d. N 0 2 random variables. A any ime period, afer D is revealed, we generae a new forecas for he demand in period + 1as D +1 = +1 + D
13 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1091 Using he new forecas for period + 1, we obain a new demand forecas for period + 2as D +2 = +2 + D = D Similarly, we obain he new forecas for period + i as D +i = +i + i D Therefore, we have 0 i T D = D +1 D T e = e e T = T Noe ha = e = D D 1 is he onesep forecasing error. The AR(1) model has been adoped by several auhors in he recen supply chain managemen lieraure o sudy he value of informaion sharing and collaboraive forecasing (see, e.g., Aviv 2001 and Lee e al. 2000). Due o racabiliy, hese auhors focus on myopic policies. A naural quesion is: Can he findings of hese sudies also apply o a sysem under an opimal policy? This is equivalen o asking wheher he myopic policy is sufficienly good for sysems wih he AR(1) demand model. Previous research has esablished cerain sufficien condiions under which he myopic policy is opimal when he demand follows AR(1) (see, e.g., Johnson and Thompson 1975 and Iida and Zipkin 2006). To shed ligh on he above issues, our numerical sudy focuses on parameers which do no saisfy hese sufficien condiions. More specifically, he ime horizon T = 10 and here is a replenishmen lead ime L = 3. Two ses of cos parameers were chosen. One se is nonsaionary: h 2j+1 = 1, b 2j+1 = 19, h 2j = 9, b 2j = 11 j = 0 4. The oher se is saionary: h i = 1, b i = 20, i 1. Iniial Demand. We choose several values of D 0 : D 0 0 = 0 + p 0, where 0 = / 1 2 1/2, p For any fixed p, D 0 = + p 0 and D 0 = D 0 1 D 0 2 D 0 T. We se 0 / 1 = 0 3 or Wih hese parameers, he demand can demonsrae a wide range of variabiliy. For example, suppose ha 0 / 1 = 0 3, +1 = 5, 1 = 100, = 0 9, and p = 3. We have D 1 N and D 2 N D 13 N Demand Trend. We firs consider consan over ime: = 100 for all 1. We hen consider wih a decreasing rend: +i = +i 1 5, 1 = 100. We es differen values of Heurisics Evaluaed. We evaluae he performances of hree policies: he myopic policy s m, he upper bound policy S u 1 given in (28), and he heurisic policy S H defined by S H D 1 = S u 1 D S l 4 D (51) Here, S l 4 D 1 is given by (29). The parameer is chosen o minimize max 1 D 1 3 D 1 in period (i.e., o minimize he upper bound on he relaive cos error). Also, we use (31) o compue 1 D 1 and we use (35) and (46) o compue 3 D 1. To illusrae he heurisic S H D 1 under consideraion, le us consider he firsperiod problem for he case of decreasing demand rend, nonsaionary cos parameers, and 0 / 1 = 0 3. Suppose ha D 0 = = Wehaves1 m D 0 = , S u 1 1 D 0 = , and S l 4 1 D 0 = We furher have S1 H D 0 = wih he parameer = 0 68 (which means ha he heurisic is closer o S u 1 1 D 0 ). In evaluaing he heurisic S1 H D 0, we have 2 = 1 32, 1 1 D 0 = 2 43, 3 1 D 0 = 2 44, and max 1 1 D D 0 = Upper Bounds on Relaive Cos Error. Tables 1 4 show err, he upper bound on relaive cos error of he approximae policies, where err is defined by err = V 1 H x D 0 V 1 x D 0 100% V 1 x D 0 err = RHS of 45 RHS of % where RHS sands for righhand side. MLovejoy, Measy, and Myopic are upper bounds on relaive cos error for myopic policy when we apply Lovejoy s mehod defined in (49), he easierocompue upper bound defined in (48), and a general mehod defined in (47) o evaluae he myopic policy, respecively. Because Lovejoy s mehod is no for evaluaing a general heurisic policy, our mehod is used o evaluae S H D 1 given in (51). Specifically, we use (31), (35), and (46) o compue he upper bound on V1 H x D 0 V 1 x D 0 as defined in 45 for n = 3 o evaluae he heurisic which is referred o as S H D 1. Tables 1 and 3 repor some special cases, while Tables 2 and 4 provide he averages. As menioned before, we esed all cases of and p For he cases we do no repor, he resuls are usually beer, i.e., he upper bounds on he relaive cos errors are smaller for he same value of. Sofware Used. We used Malab version 6.5 for he numerical sudy. We employed wo rouines for inegraion: he single inegraion funcion quad( ) and double inegraion funcion dblquad( ). We also used he sandard normal probabiliy densiy and cumulaive disribuion funcions: normpdf( ) and normcdf( ). We se he maximum error o be for precision. We observe ha for nonsaionary cos parameers, he myopic policy can be far from opimal; in one case he value loss is as high as 45%. On he oher hand, wih saionary cos parameers, he myopic policy is very close o opimal. The maximum value loss is around 2%.
14 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1092 Operaions Research 54(6), pp , 2006 INFORMS Table 1. Upper bound on relaive cos error for nonsaionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H p 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend = The heurisic policy S H is very close o opimal wheher or no he cos parameers are saionary. The maximum value loss is % for nonsaionary coss and % for saionary coss. This also indicaes ha our upper bound on he cos difference beween he heurisic and opimal policy is very igh. This observaion can be furher verified by comparing he upper bound on he relaive cos error on he myopic policy beween our mehod and Lovejoy s (1990, 1992) mehod. The examples show ha our cos error bounds are usually 20% o 1% of Lovejoy s (1992) bounds. The reason is, when he oversock (by ordering up o he myopic basesock level) happens, he exra cos due o he oversock is usually much smaller han he oal cos for holding he iems from he curren period unil he end of he planning horizon he basis for Lovejoy s esimaion. The performance of S u 1 is also very close o opimal; is maximum value loss is 2.26% in all he cases examined. Thus, his policy is recommended o be used if he myopic policy fails perform well. Noe ha he woperiodahead policy is a slighly igher upper bound on he opimal basesock level han S u 1.So he good performance of S u 1 implies good performance of he woperiodahead policy. This is consisen wih he finding by Treharne and Sox (2002), who show ha he woperiodahead policy is very close o opimal for he Markov modulaed demand model. I is ineresing o observe ha, as decreases, he performances of all hree policies end o be beer (closer o opimal). We find his difficul o explain. For example, on one hand, if <0 and he demand in period is lower han, he expeced demand in period +1 will be higher han +1, so boh he probabiliy and he magniude of oversock decrease, which favors he myopic policy. On he oher hand, if he demand in period is higher han, he expeced demand in period + 1 will be lower han +1, which seems o be agains he myopic policy. Table 2. Upper bound on relaive cos error for nonsaionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend =
15 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1093 Table 3. Upper bound on relaive cos error for saionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H p 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend = The following example, however, sheds some ligh on why he myopic policy may be near opimal for near 1. Assume ha cos parameers are saionary and = 100 for all 1. Then, he safey sock, ss, for he myopic policy is he same across differen periods due o (14). More specifically, he forecas error of he lead ime demand is I has a normal disribuion wih mean zero and sandard deviaion where is he sandard deviaion of. Thus, he myopic policy has safey sock ss = ( ) b 1 b + h In paricular, s m = D ss s m +1 = D ss Now, consider he case = 0 9 and 0 / 1 = 0 3. Because D 1 1 has a normal disribuion, by direc compuaion, he probabiliy of oversock is max 3 p 3 P s m D s+1 m D 1 1 = p Var D Wih such a low (close o zero) probabiliy of oversock under he myopic policy, he myopic level is likely o be always reachable and hus likely o be opimal Comparison wih Iida and Zipkin (2006) In his subsecion, we compare our approximaions wih hose in Iida and Zipkin (2006). For his purpose, we follow heir choice of demand paern and cos parameers. More specifically, demand follows a muliplicaive model. The experimens include wo paerns of iniial forecass: rends and cycles. A firs, he iniial forecas for period 1 is 250. We hen adjus ha value and se he iniial forecass for periods 2 o 16 as follows: Trends: Linear rends wih hree slopes: +10, 0, and 25, numbered 1 hrough 3 (represened by rend in he able). Cycles: Four ypes of cycle: none, 50 sin, 50 cos, and 50 cos ; numbered 1 hrough 4 (represened by C in Tables 5 and 6). Table 4. Upper bound on relaive cos error: Average case for saionary cos parameers. MEasy MLovejoy Myopic S H MEasy MLovejoy Myopic S H 0 / 1 = 0 3 Trend = 0 0 / 1 = 0 3 Trend = / 1 = 0 25 Trend = 0 0 / 1 = 0 25 Trend =
16 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1094 Operaions Research 54(6), pp , 2006 INFORMS These paerns can produce nonposiive iniial forecass (he rend erm plus he cycle erm); such values are rese o one. The muliplicaive forecas updaes have mulidimensional lognormal disribuions. Le s = log e s. The s hen are joinnormally disribued wih covariance marix (52) and mean vecor Oher parameers are h = 2 and b = 10. The planning horizon T = We also examine he case of demand uncerainy being revealed early. To represen early resoluion of uncerainy, a new se of problems is consruced by reversing he order of he diagonal elemens in he covariance marix (52) as done in Iida and Zipkin. Wih hese daa, we compare our heurisic soluion S H D 1 defined in (51) wih Iida and Zipkin s (2006, 4.3) soluions obained by using heir wo approximaion echniques o solve he dynamic program. The resuls are presened in Tables 5 and 6, respecively. Here, 1 and 2 are drawn from Tables 1 and 2 of an earlier version of heir paper, daed December 20, 2004, and err is he relaive cos error of our heurisic policy given by (51). Noe ha 1 measures he error of funcional approximaions and 2 measures he sampling error in Iida and Zipkin. In oher words, 1 is he upper bound on he relaive cos error while 2 is he compuaional error. See ha earlier version of Iida and Zipkin for more deail. For he problems esed, our heurisic policy is near opimal; is maximum value loss is less han 1% in all he cases examined. When he demand uncerainy is revealed early, our policy appears even beer. The reason is ha when he demand uncerainy is revealed earlier, he safey sock is reduced and hus he possibiliy of oversock is reduced. As a consequence, he gap beween he upper and lower bounds on he opimal basesock level can be very small (or even zero), rendering our heurisic o be near opimal or even opimal. Ou of he oal 120 cases repored in Tables 5 and 6, in 109 cases (more han 90%) our mehod is beer han Iida and Zipkin s (judging from he magniudes of he errors). Our mehod is usually beer when (1) he planning horizon is shorer, or (2) he demand paern increases or says he same, or (3) he demand uncerainy reveals early. Even in oher cases, such as when he demand paern decreases, our mehod can sill be beer. 7. Conclusions We have examined a singleiem, periodicreview invenory sysem wih demandforecas updaes following he Maringale model of forecas evoluion (MMFE). The opimal policy is a saedependen basesock policy ha is compuaionally inracable o obain. Using a samplepah approach, we developed a general class of racable bounds on he opimal basesock levels, which generalized and improved he exising bounds in he lieraure. We hen used hese bounds o consruc nearopimal policies. Our numerical examples showed ha our heurisics ouperform he myopic policy significanly. The samplepah approach also allowed us o idenify a necessary and sufficien condiion for he myopic policy o be opimal, which sharpens our inuiion on his policy. Furhermore, our samplepah Table 5. Comparison wih Iida and Zipkin s (2006) mehod: Base case. Trend C T = 2 T = 4 T = 8 T = 12 T = 16 C T = 2 T = 4 T = 8 T = 12 T = err err err err err err
17 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1095 Table 6. Comparison wih Iida and Zipkin s (2006) mehod: Early resoluion of demand uncerainy. Trend C T = 2 T = 4 T = 8 T = 12 T = 16 C T = 2 T = 4 T = 8 T = 12 T = err err err err err err approach enabled us o perform worscase analysis and derive upper bounds on he value loss of any heurisic policy (including he myopic policy). These appear o be he firs se of coserror bounds on he performance of any heurisic policies in dynamic invenory models. Numerical examples demonsraed ha our error bounds improve he exising error bounds in he lieraure for evaluaing he performance of myopic policies. Finally, boh he soluion bounds and he coserror bounds developed in his paper can be easily adaped o general dynamic invenory models wih nonsaionary or auocorrelaed demands. Appendix Proof of (44). Define Ĩ +i = invenory level afer ordering in period + i following he heurisic policy under consideraion given I +1 = s H D 1 D, Î +i =invenory level afer ordering in period + i following he heurisic policy under consideraion given I +1 = s D 1 D. We have Ĩ +i Î +i 0 i= 1 2 T (53) Ĩ +1 Î +1 s H D 1 D s H +1 D s D 1 D s H +1 D s H D 1 s D 1 s H D 1 D s H +1 D + = 2 (54) The reason is, considering period + 1, if he heurisic orders for boh I +1 = sh D 1 D and I +1 = s D 1 D, hen Ĩ +1 Î +1 = 0; if he heurisic places an order for only one siuaion, because s H D 1 D s D 1 D, we have Ĩ +1 Î +1 = s H D 1 D s+1 H D s H D 1 s D 1 ; if he heurisic does no place an order for boh siuaions, we have Ĩ +1 Î +1 = s H D 1 s D 1. We focus on developing an upper bound for E V+1 H s H D 1 D D E V+1 H s D 1 D D by developing an upper bound for C +i Ĩ +i C +i Î +i, i = 1 T. Le i 0 = min j s H D 1 D + j s H +j D +j 1 Then, period +i 0 is he firs period afer when he heurisic will place an order given I +1 = sh D 1 D. Thus, we have Ĩ +i0 = s H +i 0 D +i0 1. Because Î +i0 s H +i 0 D +i0 1 = Ĩ +i0, from (53), we have Ĩ +i0 = Î +i0. Therefore, C +j Ĩ +j C +j Î +j = 0 j = i 0 T (55) The following evens happen for all k<i 0 : A h +k = sh D 1 D + k > s H +k D +k 1 In any of hese periods, say period + k k < i 0,ifĨ +k s m +k D +k 1, because Î +k Ĩ +k s m +k D +k 1, and one period expeced cos C +k z decreases in z when z< s m +k D +k 1, wehave C +k Ĩ +k C +k Î +k 0 (56) If Ĩ +k >s m +k D +k 1 + 2, because Ĩ +k Î +k 2 (from (54)) and C +k z increases in z when z>s m +k D +k 1, we have C +k Ĩ +k C +k Î +k C +k Ĩ +k C +k Ĩ +k 2 (57)
18 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds 1096 Operaions Research 54(6), pp , 2006 INFORMS If s m +k D +k Ĩ +k > s m +k D +k 1, because C +k Î +k C +k s m +k D +k 1, wehave C +k Ĩ +k C +k Î +k C +k Ĩ +k C +k s m +k D +k 1 (58) In summary, if Ĩ +j >s H +j D +j 1, j = 1 k, and Ĩ +k > s m +k D +k 1, from (57) and (58), we have C +k Ĩ +k C +k Î +k C +k Ĩ +k C +k Ĩ +k 2 s m +k D +k 1 (59) Considering all hese possibiliies, we have E C +k Ĩ +k C +k Î +k = E I A h +1 Ah +k 1 Ah +k c C +k Ĩ +k C +k Î +k + E I A h +1 Ah +k 1 Ah +k C +k Ĩ +k C +k Î +k = E I A h +1 Ah +k 1 Ah +k C +k Ĩ +k C +k Î +k = E I A h +1 Ah +k 1 Am +k C +k Ĩ +k C +k Î +k + E I A h +1 Ah +k 1 Ah +k Am +k C +k Ĩ +k C +k Î +k E I A h +1 Ah +k 1 Am +k C +k Ĩ +k C +k Î +k E I A h +1 Ah +k 1 Am +k C +k Ĩ +k C +k Ĩ +k 2 s m +k D +k 1 = E I A h +1 Ah +k 1 Am +k C +k s H D 1 D + k D +k 1 +k C +k s H D 1 D + k 2 s m +k D +k 1 D +k 1 +k = E I A h +1 Ah +k 1 Am +k C +k D +k 1 +k 2 s H D 1 D + k s m +k D +k 1 s H D 1 D +k 2 s m +k D +k 1 s H D 1 D +k E I A h +1 Ah +k 1 Am +k 2 C +k sh D 1 D + k D +k 1 +k The second equaliy is due o (55), i.e., when A h +1 A h +k 1 Ah +k c happens, Ĩ +k = Î +k. The firs inequaliy is due o (56), and he second inequaliy is due o (59). The las equaliy is due o he medium heory, and he las inequaliy is due o C +k z D +k 1 +k increases in z and C +k z D +k 1 +k 0 when z s+k m D +k 1. Finally, we have E +1 sh D 1 D D E = E [ T k=1 { C +k Ĩ +k C +k Î +k +1 s D 1 D D }] T E I A h +1 Ah +k 1 Am +k 2C +k sh D 1 k=1 D + k D +k 1 +k = 3 D 1 Acknowledgmens The auhors hank he associae edior and wo anonymous referees for many helpful suggesions ha improved he paper s exposiion. This research was suppored in par by NSF grans DMI and DMI and he Naional Naural Science Foundaion of China award no References Aviv, Y The effec of collaboraive forecasing on supply chain performance. Managemen Sci. 47(10) Aviv, Y Gaining benefis from join forecasing and replenishmen processes: The case of auocorrelaed demand. Manufacuring Service Oper. Managemen 4(1) Aviv, Y A imeseries framework for supplychain invenory managemen. Oper. Res. 51(2) Azoury, K. S Bayes soluion o dynamic invenory models under unknown demand disribuion. Managemen Sci. 31(9) Chen, F., Z. Drezner, J. K. Ryan, D. SimchiLevi Quanifying he bullwhip effec in a simple supply chain: The impac of forecasing, lead imes, and informaion. Managemen Sci. 46(3) Gallego, G., O. Özer Inegraing replenishmen decision wih advance demand informaion. Managemen Sci. 47(10) Graves, S. C A singleiem invenory model for a nonsaionary demand process. Manufacuring Service Oper. Managemen 1(1) Graves, S. C., D. B. Kleer, W. B. Hezel A dynamic model for requiremens planning wih applicaion o supply chain opimizaion. Oper. Res. 46(3) S35 S49. Graves, S. C., S. Meal, Y. Dasu, Y. Qing Twosage producion planning in a dynamic environmen. S. Axsaer, C. Schneeweiss, E. Silver, eds. MuliSage Producion Planning and Conrol. Lecure Noes in Economics and Mahemaical Sysems. SpringerVerlag, Berlin, Güllü, R On he value of informaion in dynamic producion/invenory problems. Naval Res. Logis. 43(2) Güllü, R A woechelon allocaion model and he value of informaion under correlaed forecass and demands. Eur. J. Oper. Res. 99(2) Heah, D. C., P. L. Jackson Modeling he evoluion of demand forecass wih applicaion o safey sock analysis in producion/disribuion sysems. IIE Trans. 26(3) Iida, T., P. Zipkin Approximae soluions of a dynamic forecasinginvenory model. Manufacuring Service Oper. Managemen 8(4) Johnson, G., H. Thompson Opimaliy of myopic invenory policies for cerain dependen demand process. Managemen Sci Lee, H. L., K. C. So, C. S. Tang The value of informaion sharing in a wo level supply chain. Managemen Sci. 46(5) Levi, R., M. Pal, R. Roundy, D. Shmoys Approximaion algorihms for sochasic invenory conrol. Working paper, Cornell Universiy, Ihaca, NY. Lovejoy, W. S Sopped myopic policies in some invenory models wih uncerain demand disribuions. Managemen Sci. 36(6) Lovejoy, W. S Sopped myopic policies in some invenory models wih generalized demand processes. Managemen Sci. 38(5) Miller, B. L Scarf s sae reducion mehod, flexibiliy, and a dependen demand invenory model. Managemen Sci. 34(1) Moron, T. E The nonsaionary infinie horizon invenory problem. Managemen Sci. 24(14)
19 Lu, Song, and Regan: Invenory Planning wih Forecas Updaes: Approximae Soluions and Cos Error Bounds Operaions Research 54(6), pp , 2006 INFORMS 1097 Moron, T. E., D. W. Penico The finie horizon nonsaionary sochasic invenory problem: Nearmyopic bounds, heurisics, esing. Managemen Sci. 41(2) Reyman, G Sae reducion in a dependen demand invenory model given by a ime series. Eur. J. Oper. Res. 41(2) Scarf, H Bayes soluions of he saisical invenory problem. Ann. Mah. Sais. 30(2) Scarf, H Some remarks on Bayes soluion o he invenory problem. Naval Res. Logis. Quar. 7(4) Song, J. S., P. Zipkin Invenory conrol in a flucuaing demand environmen. Oper. Res. 41(2) Song, J. S., P. Zipkin Managing invenory wih he prospec of obsolescence. Oper. Res. 44(1) Tokay, L. B., L. M. Wein Analysis of a forecasingproducioninvenory sysem wih saionary demand. Managemen Sci. 47(9) Treharne, J. T., C. R. Sox Adapive invenory conrol for nonsaionary demand and parial informaion. Managemen Sci. 48(5) Veino, A. F., Jr Opimal policy for a muliproduc, dynamic, nonsaionary invenory problem. Managemen Sci. 12(3)
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationForecasting, Ordering and Stock Holding for Erratic Demand
ISF 2002 23 rd o 26 h June 2002 Forecasing, Ordering and Sock Holding for Erraic Demand Andrew Eaves Lancaser Universiy / Andalus Soluions Limied Inroducion Erraic and slowmoving demand Demand classificaion
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationMACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR
MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationAs widely accepted performance measures in supply chain management practice, frequencybased service
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 6, No., Winer 2004, pp. 53 72 issn 523464 eissn 5265498 04 060 0053 informs doi 0.287/msom.030.0029 2004 INFORMS On Measuring Supplier Performance Under
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces imeseries smoohing forecasing mehods. Various models are discussed,
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationAnalysis of Tailored BaseSurge Policies in Dual Sourcing Inventory Systems
Analysis of Tailored BaseSurge Policies in Dual Sourcing Invenory Sysems Ganesh Janakiraman, 1 Sridhar Seshadri, 2, Anshul Sheopuri. 3 Absrac We sudy a model of a firm managing is invenory of a single
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationResearch on Inventory Sharing and Pricing Strategy of Multichannel Retailer with Channel Preference in Internet Environment
Vol. 7, No. 6 (04), pp. 365374 hp://dx.doi.org/0.457/ijhi.04.7.6.3 Research on Invenory Sharing and Pricing Sraegy of Mulichannel Reailer wih Channel Preference in Inerne Environmen Hanzong Li College
More informationINTRODUCTION TO FORECASTING
INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren
More informationRevisions to Nonfarm Payroll Employment: 1964 to 2011
Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationForecasting. Including an Introduction to Forecasting using the SAP R/3 System
Forecasing Including an Inroducion o Forecasing using he SAP R/3 Sysem by James D. Blocher Vincen A. Maber Ashok K. Soni Munirpallam A. Venkaaramanan Indiana Universiy Kelley School of Business February
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101115. Macroeconomericians
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationII.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationThe effect of demand distributions on the performance of inventory policies
DOI 10.2195/LJ_Ref_Kuhn_en_200907 The effec of demand disribuions on he performance of invenory policies SONJA KUHNT & WIEBKE SIEBEN FAKULTÄT STATISTIK TECHNISCHE UNIVERSITÄT DORTMUND 44221 DORTMUND Invenory
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationChapter 5. Aggregate Planning
Chaper 5 Aggregae Planning Supply Chain Planning Marix procuremen producion disribuion sales longerm Sraegic Nework Planning miderm shorerm Maerial Requiremens Planning Maser Planning Producion Planning
More informationpolicies are investigated through the entire product life cycle of a remanufacturable product. Benefiting from the MDP analysis, the optimal or
ABSTRACT AHISKA, SEMRA SEBNEM. Invenory Opimizaion in a One Produc Recoverable Manufacuring Sysem. (Under he direcion of Dr. Russell E. King and Dr. Thom J. Hodgson.) Environmenal regulaions or he necessiy
More informationPrice elasticity of demand for crude oil: estimates for 23 countries
Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationTime Series Analysis Using SAS R Part I The Augmented DickeyFuller (ADF) Test
ABSTRACT Time Series Analysis Using SAS R Par I The Augmened DickeyFuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationSupplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect RiskTaking?
Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec RiskTaking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationInternational Journal of Supply and Operations Management
Inernaional Journal of Supply and Operaions Managemen IJSOM May 05, Volume, Issue, pp 5547 ISSNPrin: 859 ISSNOnline: 855 wwwijsomcom An EPQ Model wih Increasing Demand and Demand Dependen Producion
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationHotel Room Demand Forecasting via Observed Reservation Information
Proceedings of he Asia Pacific Indusrial Engineering & Managemen Sysems Conference 0 V. Kachivichyanuul, H.T. Luong, and R. Piaaso Eds. Hoel Room Demand Forecasing via Observed Reservaion Informaion aragain
More informationIssues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d
These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationChapter 8 Student Lecture Notes 81
Chaper Suden Lecure Noes  Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop
More informationSPEC model selection algorithm for ARCH models: an options pricing evaluation framework
Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,
More informationInformation Acquisition for Capacity Planning via Pricing and Advance Selling: When to Stop and Act?
OPERATIONS RESEARCH Vol. 58, No. 5, Sepember Ocober 2010, pp. 1328 1349 issn 0030364X eissn 15265463 10 5805 1328 informs doi 10.1287/opre.1100.0798 2010 INFORMS Informaion Acquisiion for Capaciy Planning
More informationA ProductionInventory System with Markovian Capacity and Outsourcing Option
OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030364X eissn 15265463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A ProducionInvenory Sysem wih Markovian Capaciy
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationA OneSector Neoclassical Growth Model with Endogenous Retirement. By Kiminori Matsuyama. Final Manuscript. Abstract
A OneSecor Neoclassical Growh Model wih Endogenous Reiremen By Kiminori Masuyama Final Manuscrip Absrac This paper exends Diamond s OG model by allowing he agens o make he reiremen decision. Earning a
More informationTowards Optimal Capacity Segmentation with Hybrid Cloud Pricing
Towards Opimal Capaciy Segmenaion wih Hybrid Cloud Pricing Wei Wang, Baochun Li, and Ben Liang Deparmen of Elecrical and Compuer Engineering Universiy of Torono Absrac Cloud resources are usually priced
More informationImproving timeliness of industrial shortterm statistics using time series analysis
Improving imeliness of indusrial shorerm saisics using ime series analysis Discussion paper 04005 Frank Aelen The views expressed in his paper are hose of he auhors and do no necessarily reflec he policies
More informationON THURSTONE'S MODEL FOR PAIRED COMPARISONS AND RANKING DATA
ON THUSTONE'S MODEL FO PAIED COMPAISONS AND ANKING DATA Alber MaydeuOlivares Dep. of Psychology. Universiy of Barcelona. Paseo Valle de Hebrón, 171. 08035 Barcelona (Spain). Summary. We invesigae by means
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationLEASING VERSUSBUYING
LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss
More informationNikkei Stock Average Volatility Index Realtime Version Index Guidebook
Nikkei Sock Average Volailiy Index Realime Version Index Guidebook Nikkei Inc. Wih he modificaion of he mehodology of he Nikkei Sock Average Volailiy Index as Nikkei Inc. (Nikkei) sars calculaing and
More informationDynamic programming models and algorithms for the mutual fund cash balance problem
Submied o Managemen Science manuscrip Dynamic programming models and algorihms for he muual fund cash balance problem Juliana Nascimeno Deparmen of Operaions Research and Financial Engineering, Princeon
More informationTowards Optimal Capacity Segmentation with Hybrid Cloud Pricing
Towards Opimal Capaciy Segmenaion wih Hybrid Cloud Pricing Wei Wang, Baochun Li, and Ben Liang Deparmen of Elecrical and Compuer Engineering Universiy of Torono Torono, ON M5S 3G4, Canada weiwang@eecg.orono.edu,
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationMarket Liquidity and the Impacts of the Computerized Trading System: Evidence from the Stock Exchange of Thailand
36 Invesmen Managemen and Financial Innovaions, 4/4 Marke Liquidiy and he Impacs of he Compuerized Trading Sysem: Evidence from he Sock Exchange of Thailand Sorasar Sukcharoensin 1, Pariyada Srisopisawa,
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationPermutations and Combinations
Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes  ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he kvalue for he middle erm, divide
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationAn Optimal Strategy of Natural Hedging for. a General Portfolio of Insurance Companies
An Opimal Sraegy of Naural Hedging for a General Porfolio of Insurance Companies HongChih Huang 1 ChouWen Wang 2 DeChuan Hong 3 ABSTRACT Wih he improvemen of medical and hygienic echniques, life insurers
More information