Proceedngs of SKISE Fall Conference, Nov 14-15, 2008 The ( Inventory Polcy n Producton- Inventory Systems Joon-Seok Km Sejong Unversty Seoul, Korea Abstract We examne the effectveness of the conventonal ( model n managng producton-nventory systems wth fnte capacty, stochastc demand, and stochastc order processng tmes. We show that, for systems wth fnte producton capacty, order replenshment lead tmes are hghly senstve to loadng and order quantty. Consequently, the choce of optmal order quantty and optmal reorder pont can vary sgnfcantly from those obtaned under the usual assumpton of a load-ndependent lead tme. More mportantly, we show that for a gven ( polcy the conventonal model can grossly under or over-estmate the actual cost of the polcy. In cases where a setup tme s assocated wth placng a producton order, we show that the optmal ( polcy derved from the conventonal model can, n fact, be nfeasble. Key-words : ( Inventory Polcy, Producton-Inventory Systems, Make-To-Stock ueue 1. Introducton One of the most common methods for managng nventory systems wth stochastc demand s the reorder pont/order quantty polcy, also known as the ( polcy. Under a ( polcy, fnshed goods nventory s contnuously revewed and a new producton order s placed each tme nventory poston (on-hand nventory + outstandng orders backorders) falls to a reorder pont r. Values for and r are, typcally, selected so that nventory costs (orderng costs + holdng costs + backorderng costs) are mnmzed. Polces of the ( type have been wdely studed, wth a rch body of lterature on ( models datng back to the late 50 s (see, for example, Gallher et al. [7] and Hadley and Whttn [8]). For sngle tem nventory systems under the standard assumptons, an optmal polcy can, n fact, be shown to exst wthn the class of ( polces [5]. The applcablty of ( polces has been extended to nventory systems wth multple tems and multple echelons (see for example Atkns and Iyogun [1] and Chen and Zheng [6] and the references theren). In determnng optmal values for and r, t s assumed n most of the exstng lterature that order replenshment lead tmes are not senstve to the loadng of the producton faclty or to the order quantty. In fact, the most common assumpton s a fxed postve lead tme [10]. Ths assumpton s realstc when the nventory system s fully decoupled from the producton system through large nventory holdng at the producton faclty or at subsequent stages of the supply chan (for example, when a local retaler s replenshed from a large regonal warehouse). It may also be realstc when non-producton lead tme s long (for example, when transportaton lead tmes are sgnfcantly longer than manufacturng lead tmes). However, for most ntegrated producton-nventory systems, these assumptons rarely hold. In fact, wth ncreased emphass on lean manufacturng, fnshed goods nventory at factores has become much more tghtly controlled and materal handlng tme between the producton faclty and the fnshedgoods warehouse s often mnmal. Smlar prncples are beng appled to the entre supply chan wth much tghter couplng between retalers, dstrbutors and supplers takng place. Ths 183
The ( Inventory Polcy Producton-Inventory Systems Joon-Seok Km ncreased couplng means that dstrbutors and retalers are often mmedately affected by congeston and delays on the factory floor [4]. In ths paper, we examne the mpact of explctly modelng producton lead tmes on nventory costs, order replenshment lead tmes, and the optmal ( polcy. We contrast results obtaned usng our model wth those obtaned usng the conventonal or standard ( model. We show that, for systems wth fnte producton capacty, order replenshment lead tmes are hghly senstve to loadng and order quantty. Consequently, the choce of optmal order quantty and optmal reorder pont can vary sgnfcantly from those obtaned under the usual assumpton of a load-ndependent lead tme. More mportantly, we show that for a gven ( polcy the conventonal model can grossly under or overestmate the actual cost of the polcy dependng on the value of lead tme used n the standard model. Equally sgnfcant, we show that the order quantty that mnmzes order replenshment lead tme s not necessarly the one that mnmzes total cost. In cases where a setup tme as well as a setup cost, s assocated wth placng a producton order, the optmal ( polcy derved from the standard model can be, n fact, nfeasble. To llustrate our results we restrct our dscusson ntally to the case where demand s Posson dstrbuted snce ths s one of the few cases for whch the optmal soluton to the standard model can be easly obtaned. 2. The Producton-Inventory ( Model In contrast to the standard ( model, we consder an ntegrated producton-nventory system, where nventory s replenshed from a producton faclty wth fnte capacty. Inventory s managed accordng to a ( polcy smlar to the standard polcy. That s, a producton order of sze s placed each tme nventory poston drops to the reorder pont r. As n the standard model, the order s delvered once all unts have been produced (t s possble to extend the model to allow for unt by unt replenshment). As n the standard model, we assume that demand occurs one unt at a tme accordng to a Posson process wth rate λ. Although customer orders arrve ndvdually, a producton order s placed only after customer orders are receved. Therefore, the nter-arrval tme of orders to the producton system s Erlang wth phase. In contrast to the standard model, we let producton capacty be fnte wth a postve processng rate μ. We consder the case where unt processng tmes are ndependent, dentcal and exponentally dstrbuted random varables wth mean 1/ μ. Snce the processng tme of an order of sze s the sum of ndependent, dentcal and exponentally dstrbuted random varables, order processng tme s Erlang wth phase. Thus, the producton faclty can be vewed as an E /E /1 queueng system (.e. a sngle-server queueng system wth Erlang nter-arrval and processng tmes). If we let refer to nventory level, n to the number of producton orders n the producton system (each order s of sze ), z to the number of customer unt demands that have yet to be ordered (an order s placed only when nventory poston reaches the reorder pont), and j to the number of backorders, then t s easy to see that = { + r n z} + and j = {n + z r} +. The number of nventory and the number of backorders are dependent upon r, n and z. For example, f we have =3, r=10, n=1, and z=2, the nventory level s 8 (3+10-1(3)-2=8) and the number of backorders s 0. Notng that z s unformly dstrbuted on {0, 1,, 1} (ths follows from the fact that nventory poston s unformly dstrbuted on {r + 1, r + 2,, r + }[5]), t s straghtforward to show that: 1 + r PI ( ) ( ), = 1, 2,..., +r (1) and 1 + r + j PB ( j) ( ), j = 1, 2,..., (2) where P I (-),P B (-), and P N (-) refer, respectvely, to the steady state probablty dstrbuton of 184
Proceedngs of SKISE Fall Conference, Nov 14-15, 2008 nventory level, number of backorders, and number of orders n the producton faclty (n queue + n servce). For the steady state probabltes to exst, we requre the stablty condton ρ= λ/μ< 1, where ρrefers to the utlzaton of the producton faclty. From 1 and 2, we can obtan average nventory, average number of backorders and average order replenshment lead tme as follows: I = = 1 + r = 1 P I ( ), (3) B = P B( ), and (4) =1 L = P ( N ) / λ. (5) A closed form expresson for the probablty dstrbuton of the number of customers n an E /E /1 system s dffcult to obtan. However, the probabltes can be computed numercally (see for example Neuts [9] for a matrx-geometrc approach). Notng that P N () converges to zero as approaches nfnty, I, B and L can be computed to desred accuracy. Our cost functon can now be wrtten as: C p ( λ + r + r = K + h P ( ) = 1 N + r + + b P ( ) = 1 N We can show that C P ( s convex n r (see Appendx 1). Therefore, for each order sze the optmal reorder pont can be easly obtaned usng standard convex optmzaton. The optmal order quantty can be obtaned by an exhaustve search over the range of feasble order szes (although numercal results suggest that C P ( s jontly convex n and r, showng jont convexty analytcally s dffcult). 3. Comparsons wth the standard ( model In ths paper, the standard ( model refers to a sngle tem contnuous revew nventory model where demand s statonary and occurs one unt at a tme wth rate λ. An order of fxed quantty s placed whenever nventory poston drops to a fxed (6) reorder pont r. Orders are delvered after a fxed postve lead tme L. The other settngs reman same as the producton-nventory ( model. The long run average cost C S ( under the above condtons s gven by the followng (see, for example, Federgruen and Zheng [5]): C s λ ( = K + h ( y ) + b = y + 1 + r y = 1 = 0 ( y) p p (7) where p s the probablty that total demand durng lead tme s. The above expresson s exact when the nventory poston n steady state s unformly dstrbuted on {r+1, r+2,..., r+} and s n-dependent of demand durng lead tme. Ths condton s satsfed when demand s Posson, n whch case p s gven by p = e λl ( λl) /!. Although closed form expressons for the optmal order quantty and optmal reorder pont ( *, r * ) are dffcult to obtan, an effcent algorthm for calculatng * and r * has been proposed by Federgruen and Zheng [5]. Snce the order quantty affects both the dstrbuton of the arrval process to and the processng tme at the producton faclty, t s easy to show that lead tme s senstve to our choce of order quantty (see expresson 5). Also, snce congeston at the producton faclty s affected by both the avalable capacty and demand level, lead tme s affected by the utlzaton of the producton faclty. The behavor of average lead tme, for varyng order quanttes and utlzaton levels, s llustrated n Fgure 1. As we can see, lead tme s ncreasng n both and ρ. Whle the fxed lead tme ncreases n the standard model, the producton-nventory system experences more congeston n ts producton faclty. Therefore, for each fxed lead tme n the standard model, dfferent utlzaton should be used to generate the same average lead tme n the producton-nventory model. Because lead tme s affected by the choce of whch, n turn, affects 185
The ( Inventory Polcy Producton-Inventory Systems Joon-Seok Km our choce of r, the optmal values of and r obtaned respectvely by the standard and the producton-nventory models are generally dfferent. Fgure 2. The effect of lead tme on optmal order quantty (λ = 2, h = 1.0, b = 1.0, K = 10.0) Fgure 1. The effect of order quantty on average lead tme (λ = 2, µ = 4.0, 3.33, 2.857, 2.5, h = 1.0, b = 1.0, K = 10.0) Dependng on the value of lead tme used n the standard model, the dfference n these values can be sgnfcant. Ths s llustrated n Fgures 2 and 3, where values of * and r * for both models are shown for varyng assumptons of fxed lead tme (we use the notaton ( * s, r * s ) and ( * p, r * p ) to dfferentate, respectvely, between the optmal soluton to the standard ( model and the producton-nventory model). It s nterestng to note that generally bgger * s and smaller r * s were chosen n the producton-nventory model when lead tme ncreases. More mportantly, the value of fxed lead tme for whch the two optmal order quanttes are the same does not generally correspond to the actual lead tme experenced by the producton-nventory system. The fact that * and r *, as well as lead tme, are dfferent under the two models, means that average nventory and average number of backorders are also dfferent. In turn, ths means that the estmated costs of the optmal polces under the two models can also be very dfferent. In fact, dependng on the value of lead tme used n the standard model, ths dfference can be qute sgnfcant. Ths s llustrated n Fgure 4. Note that the fxed lead tme does not usually correspond to the actual lead tme n the producton-nventory system. Fgure 3. The effect of lead tme on optmal reorder pont (λ = 2, h = 1.0, b = 1.0, K = 10.0) Fgure 4. The effect of lead tme on optmal cost (λ = 2, h = 1.0, b = 1.0, K = 10.0) More mportantly, the standard ( model can sgnfcantly under-estmate the true costs of mplementng the correspondng optmal polcy (or n fact, any ( polcy). In Fgure 5, we show the dfference between the optmal costs, as estmated by the standard model, and the true costs of 186
Proceedngs of SKISE Fall Conference, Nov 14-15, 2008 mplementng ths polcy as obtaned from the producton nventory model. That s the ( *, r * ) chosen n the standard model were drectly nserted nto the cost functon of the producton-nventory model n order to get the exact cost when the system experences the congeston. Agan, we can see that ths dfference s dependent on the choce of lead tme (n the standard model) and can be qute large. Fgure 5. The Impact of lead tme on estmated and actual costs (λ = 2, µ = 2.5, h = 1.0, b = 1.0, K = 10.0) In addton to nducng errors n estmatng nventory holdng costs, backorderng costs, and orderng costs, the standard model can lead to errors n estmatng other performance metrcs, such as customer order fulfllment tme, fll rate, or the probablty of stock-out. For example, customer order fulfllment tme (the duraton from the tme when a customer places an order to the tme when the order s shpped) s gven, respectvely for the standard and producton-nventory models, by the followng: F s B + r = = λ = 1 = y + 1 ( y) p λ (8) B m + r + m F = = P (9) p λ N m = 1 Snce average order fulfllment tme s lnearly ncreasng n the average number of backorders, order fulfllment s smlarly senstve to the choce of order quantty. 4. The Impact of Setup Tmes In many producton systems, a setup tme s requred pror to the ntaton of an order. A nonzero setup tme ncreases order-processng tme and, therefore, ncreases order lead tme. Snce both average nventory and number of backorders are senstve to lead tme, the ntroducton of setup tme affects total cost. In turn, ths affects the value of the optmal order quantty and the optmal reorder pont. More mportantly, a non-zero setup tme affects the stablty condton of the producton system and places a mnmum requrement on order sze. Ths can be seen by notng that wth non-zero setup tmes the stablty condton s gven by the followng: λs/+λ/µ < 1, (10) whch can be rewrtten as: > λs/(1 λ/µ), (11) where S s average setup tme. The rght-hand sde of (11) represents, for a gven average setup tme, the mnmum feasble order sze (.e., a smaller order sze would result n nfntely long lead tmes). We should note that, although a setup cost s ncluded n the standard ( model, the model does not account for setup tme nor does t account for the relatonshp between order sze and frequency of setups, and ther jont effect on producton capacty. Therefore, the standard model could lead to the choce of an nfeasble order quantty. In order to examne the effect of setup tme on lead tme, optmal order quantty, and optmal reorder pont, we consdered the case where setup tmes are exponentally dstrbuted. In ths case, order processng tme can be represented by a generalzed Erlang dstrbuton wth + 1 phases where phase 1 represents setup tme. A smlar approach to the one descrbed n secton 4 can be used to compute average nventory level, average number of backorders, and average lead tme. These can then be used to determne optmal order quantty and reorder pont. 187
The ( Inventory Polcy Producton-Inventory Systems Joon-Seok Km Fgure 6. The effect of order quantty on average lead tme (λ = 2, µ = 2.5, h = 1.0, b = 1.0, K = 10.0) further ncreases n result n suffcent ncreases n the order processng tme leadng to an overall ncrease n lead tme. Smlar observatons were made by Karmarkar [8] and Benjaafar and Shekhzadeh [2] for make to order systems. In Fgures 7 and 8, we llustrate the mpact of setup tme on the value of the optmal order quantty and reorder pont. As we can see, both quanttes are hghly senstve to setup tme. The effect of setup tme s partcularly pronounced when utlzaton s hgh. We should note that snce these effects are gnored by the standard model, the values of * and r * obtaned from the standard model could be agan very dfferent from those we obtan usng the producton-nventory model. Fgure 7. The effect of setup tme on optmal order quantty (λ = 2, µ = 3.333, 2.8571, 2.5, h = 1.0, b = 1.0, K = 10.0) Fgure 8. The effect of setup tme on optmal reorder pont (λ = 2, µ = 3.333, 2.8571, 2.5, h = 1.0, b = 1.0, K = 10.0) In Fgure 6, we show numercal results that llustrate the effect of order quantty on lead tme for varyng values of setup tme. It s nterestng to note that, n contrast to the case wth zero setup tme, the effect of order quantty on lead tme s not monotonc. Intal ncreases n reduce lead tme by reducng the frequency of setups. However, 5. Concluson In ths paper, we proposed the productonnventory ( models wth fnte capacty, stochastc demand, and stochastc processng tmes and we demonstrated the mportance of modelng the dependency of lead tme on system loadng n producton-nventory systems. Specfcally, we showed that the optmal order quantty and optmal reorder pont could vary sgnfcantly from those obtaned under the usual assumpton of a fxed lead tme. We also showed that the costs estmated usng a fxed lead tme (even f ths fxed lead tme s a good approxmaton of the actual lead tme) could be sgnfcantly dfferent from those actually experenced by the producton-nventory system. For systems where the ntaton of a producton order s preceded by a setup, we found that system stablty s dependent on order sze. Therefore, gnorng ths dependency could lead to system nstablty. These results hghlght the fact that producton and nventory systems cannot be managed (or modeled) separately. It s crtcal when the companes adopt the strateges such as lean manufacturng, where managers should control tghtly both the factory floor and warehouses. Unfortunately, for ( nventory models, the nteracton between producton and nventory has gone largely under-studed. Therefore, there s a 188
Proceedngs of SKISE Fall Conference, Nov 14-15, 2008 great opportunty for extendng the exstng lterature on nventory to nclude ths mportant nteracton. Although we have focused on the ( polcy n ths paper, t s worthwhle to carry out smlar analyss wth respect to other common polces such as perodc revew polces. It s also worthwhle to extend the analyss to systems wth multple echelons. In that case, capturng the nteracton between nventory and producton one hand, and nventory and transportaton on the other, becomes mportant. Appendx 1 Lemma 1: Average nventory cost C P ( s convex n r. Proof: In order to show that C P ( s convex n r, t s suffcent to show that average nventory and average number of backorders are convex n r. Let I( refer to average nventory for gven and r. In order to show that average nventory s convex n r, we need to show that [ I( r + 1) + I( r 1) ] 2 I (. Notng that 1 1 I( = {( + PN (0) + ( + r 1) PN ( ) + K, + r 2 + r 1 + 2PN ( ) + PN ( t s not too dffcult to show that I( r + 1) + I( r 1) 1 + r I( ( ) 0. 2 2 Hence, average nventory s convex n r. Smlarly, we can show that B( r + 1) + B( r 1) 1 + r B( ( ) 0. 2 2 where B ( r ) s the average number of backorders. Thus, B ( r ) s also convex n r, whch completes our proof. References [1] Atkns, D. and P. Iyogun, Perodc versus Can- Order Polces for Coordnated Mult-tem Inventory Systems, Management Scence, 34 (1988), 791-795. [2] Benjaafar, S. and M. Shekhzadeh, "Schedulng Polces, Batch Szes, and Manufacturng Lead Tmes," IIE Transactons, 29(1997), 159-166. [3] Buzacott, J. and G. Shanthkumar, Stochastc Models of Manufacturng Systems, Prentce Hall, Englewood Clffs, NJ, 1992. [4] Benjaafar, S., W.L. Cooper and J.S. Km, Benefts of Poolng n Producton-Inventory Systems, Management Scence, 51(2005), 548-565. [5] Federgruen, A. and Y. S. Zheng, An Effcent Algorthm for Computng an Optmal (r, ) Polcy n Contnuous Revew Stochastc Inventory Systems, Operatons Research, 40 (1992), 808-813. [6] Chen, F. and Y. S. Zheng, Evaluatng Echelon Stock (R, n) Polces n Seral Producton /Inventory Systems wth Stochastc Demand, Management Scence, 40(1994), 1262-1275. [7] Gallher, H. P., P. M. Morse and M. Somond, Dynamcs of Two Classes of Contnuous Revew Inventory Systems, Operatons Research, 7(1957), 362-384. [8] Hadley, G. and T. Whtn, Analyss of Inventory Systems, Prentce-Hall, Englewood Clffs, NJ 1963. [9] Neuts, M. F., Matrx-Geometrc Solutons n Stochastc Models: An Algorthmc Approach, The Johns Hopkns Unversty Press, 1981. [10] Slver, E. A., D. F. Pyke, and R. Peterson, Inventory Management and Producton Plannng and Schedulng, John Wley, New York, NY, 1998. [11] Karmarkar, U. S., Lot Szes, Leadtmes, and In-Process Inventores, Management Scence, 33(1987), 409-418. 189