IPA Derivatives for Make-to-Stock Prouction-Inventory Systems With Backorers Uner the (Rr) Policy Yihong Fan a Benamin Melame b Yao Zhao c Yorai Wari Abstract This paper aresses Infinitesimal Perturbation Analysis (IPA) in the class of Make-to Stock (MTS) prouction-inventory systems with backorers uner the continuous-review ( Rr ) policy where R is the stock-up-to level an r is the reorer point. A system from this class is traitionally moele as a iscrete system with iscrete eman arrivals at the inventory facility an iscrete replenishment orers place at the prouction facility. Here however we map an unerlying iscrete MTS system to a Stochastic Flui Moel (SFM) counterpart in which stochastic flui-flow rate processes with piecewise constant sample paths replace the corresponing traitional iscrete eman arrival an replenishment stochastic processes uner very mil regularity assumptions. The paper then analyzes the SFM counterpart an erives close-form IPA erivative formulas of the time-average inventory level an time-average backorer level with respect to the policy parameters R an r an shows them to be unbiase. The obtaine formulas are comprehensive in the sense that they are compute for any initial inventory state an any time horizon an are simple an fast to compute. These properties hol the promise of utilizing IPA erivatives as an ingreient of offline esign algorithms an online management an control algorithms of the class of systems uner stuy. Keywors an Phrases: Infinitesimal Perturbation Analysis (IPA) IPA erivatives ( Rr ) policy Make-to-Stock Prouction-Inventory System (MTS system) Stochastic Flui Moel (SFM). AMS 2000 Subect Classification: 60G7 90B05 90B30 a Rutgers University Rutgers Business School Newark an New Brunswick Department of Management Science an Information Systems 80 University Avenue Newark NJ 0702. Email: fanyihon@anromea.rutgers.eu b Rutgers University Rutgers Business School Newark an New Brunswick Department of Supply Chain Management an Marketing Sciences 94 Rockafeller R. Piscataway NJ 08854. Email: melame@rbs.rutgers.eu c Rutgers University Rutgers Business School Newark an New Brunswick Department of Supply Chain Management an Marketing Sciences 80 University Avenue Newark NJ 0702. Email: yaozhao@anromea.rutgers.eu Georgia Institute of Technology School of Electrical an Computer Engineering Atlanta GA 30332. Email: wari@ee.gatech.eu
. Introuction In this paper we consier a class of single-stage single-prouct Make-to-Stock (MTS) prouction-inventory systems with backorers uner the continuous-review ( Rr ) policy where R is the stock-up-to level an r is the reorer point ( 0r R). An MTS system consists of a prouction facility couple to an inventory facility: the inventory facility is visite by a stream of emans an the prouction facility replenishes the inventory facility. The system is riven by ranom eman an possibly ranom prouction processes. We assume that the prouction facility has an unlimite supply of raw material so it never starves. The continuous-review ( Rr ) policy aims to maintain the inventory position (inventory on han plus replenishments en route minus backorers) at the inventory facility between r an R but the actual inventory level may initially excee R or it may rop below r or below zero; we mention that in the flui-flow moel of this paper the replenishments-en-route component is null. Uner the ( Rr ) policy with backorers inventory shortfalls are consiere backorere inventory (essentially negative inventory ). This policy utilizes feeback information from the inventory facility to the prouction facility to moulate replenishment. More specifically the prouction facility alternates between two operational states as follows: Busy state: Replenishment from the prouction facility is turne on. This state can be in effect initially or subsequently after the inventory position rops to or below r but oes not reach or excee R. Ile state: Replenishment from the prouction facility is turne off. This state can be in effect initially or subsequently after the inventory position reaches R from below but oes not rop to or below r. The general escription above of an MTS system can be moele using two relate but istinct paraigms: the (traitional) iscrete paraigm an the Stochastic Flui Moel (SFM) paraigm. We mention that MTS systems are usually specifie as iscrete baseline moels since MTS system escriptions are naturally amenable to such setting. If moeling convenience reners an SFM formulation preferable (as will be iscusse later) the iscrete baseline moel nees to be mappe to an SFM counterpart. In the iscrete MTS moel with backorers emans arrive iscretely with some eman size an the inventory facility attempts to satisfy incoming emans on a first come first serve (FCFS) basis. Demans that cannot be immeiately satisfie completely from inventory on han wait in a FCFS buffer at the inventory facility until sufficient replenishment arrives. Whenever the inventory level rops to or below r (possibly resulting in a shortfall) an orer is place at the prouction facility so as to raise the inventory position to R. A replenishment corresponing to each orer arrives at the inventory facility after a ranom lea time. Thus once the inventory position rops to or below R it is maintaine at R thereafter. In contrast an analytical SFM counterpart has no notion of iniviual emans an consequently no notion of iniviual lea times an iniviual replenishments. Rather an SFM substitutes a stochastic process of eman arrival rate for the stochastic process of iscrete eman arrivals an a stochastic process of inventory replenishment rate for the stochastic process of replenishment orer arrivals. The operational states of the prouction facility are moulate by inventory-level hitting times: a busy state is inaugurate when the inventory level hits r from above an an ile state is inaugurate when the inventory level hits R from below. Note that in both a iscrete MTS moel or its SFM - -
counterpart the inventory level may excee R only initially since once it rops to or below R it can never excee R again. A schematic of the SFM is epicte in Figure. Service Rate (t) Prouction Facility Outstaning Orers X(t) Figure. SFM schematic of an MTS prouction-inventory system with backorers uner the ( Rr ) policy The subect matter of this paper is Infinitesimal Perturbation Analysis (IPA) of MTS systems with backorers uner the ( Rr ) policy. IPA is a technique for obtaining sample path erivatives of a ranom variable L() with respect to some parameters of interest. For IPAbase applications to be statistically accurate it is essential that the IPA erivative shoul be unbiase in the sense that the expectation an ifferentiation operators commute i.e. E L( ) E[ L( )] ; otherwise it is sai to be biase. Sufficient conitions for unbiase IPA erivatives are given in the following result. Fact (see Rubinstein an Shapiro (993) Lemma A2 p. 70) L L L is Lipschitz continuous in an the (ranom) Lipschitz constants have An IPA erivative () is unbiase if (a) For each the IPA erivatives () exist w.p. (with probability ) (b) W.p. () finite first moments. Replenishment Rate (t) Inventory Facility Inventory Level I(t) Backorers Level B(t) Deman Arrival Rate (t) IPA erivatives are nonparametric in the sense that they can be compute from observe ata without knowlege of the unerlying probability law. Consequently they can be compute from simulation runs or in real-life systems eploye in the fiel an the values can potentially be use in stochastic optimization. This property hols out the promise of utilizing IPA erivative formulas to provie sensitivity information on system metrics with respect to control parameters of interest an can serve as the theoretical unerpinning for offline esign algorithms an online control algorithms. We point out that SFM formulations often enoy an important avantage over their iscrete counterparts: IPA erivatives in SFM setting ten to be unbiase while their iscrete counterparts ten to be biase; two cases in point of bias are Heielberger et al. (988) for iscrete queueing settings an Fu (994) for iscrete inventory systems uner the ( ss ) R r - 2 -
policy. Comprehensive iscussions of IPA erivatives an their applications can be foun in Fu an Hu (997) an Cassanras an Lafortune (999). A consierable boy of recent work on IPA in queuing context grew out of the observation that IPA erivatives in SFM setting ten to be unbiase. Here transactions (e.g. packets in a telecommunications transmission system) are moele as flui flows so that ranom iscrete arrivals become ranom arrival rates an ranom iscrete services become ranom service rates. Wari et al. (2002) obtaine IPA erivatives in SFM setting for the loss volume an bufferworkloa time average with respect to buffer size a parameter of the arrival rate process an a parameter of the service rate process. Some other representative papers are Liu an Gong (999) Cassanras et al. (2002) Cassanras et al. (2003) Paschaliis et al. (2004) an Sun et al. (2004). The aforementione papers obtaine IPA erivatives in SFM setting for telecommunication networks an showe that the IPA erivatives are unbiase an easily computable. More recently IPA erivatives were obtaine in SFM setting for MTS systems uner a continuous-review base-stock policy (a special case of the ( Rr ) policy where R r). Paschaliis et al. (2004) consiere multi-stage MTS systems with backorers in SFM setting assuming that inventory at each stage is controlle by a continuous-time base-stock policy. The paper compute the right-sie IPA erivatives of the time average inventory level an service level with respect to the base-stock level an then use them to etermine the optimal base-stock level at each stage. One limitation of the aforementione papers (in both queueing an MTS settings) is that they constrain the system to start from a prescribe initial state. In contrast Zhao an Melame (2006 2007) consiere any initial inventory state for MTS systems with backorers or lost-sales uner the base-stock policy an erive sie IPA erivative formulas as neee. Specifically the latter erive IPA erivative formulas for the time average inventory level backorers or lost sales with respect to the base-stock level as well as a parameter of the prouction rate process. Finally these papers also showe that the IPA erivatives are unbiase an easily computable. In this paper we consier an SFM formulation of MTS systems with backorers uner the continuous-review ( Rr ) policy. The choice of an SFM for the system is motivate in this case by the fact that the IPA erivatives of the time-average inventory level an time-average backorer level with respect to the reorer point r are biase in the iscrete moel formulation. More specifically whenever the inventory position is eplete to the reorer point a replenishment orer is place; however if the reorer point is slightly ecrease then no orer is place thereby engenering a singularity in the IPA erivative with respect to r. Using the SFM formulation this paper erives the IPA erivative formulas of the time average inventory level an backorer level with respect to the stock-up-to level R an the reorer point r uner any initial inventory state. Furthermore the IPA erivatives are shown to be unbiase an easily computable. Throughout the paper we use the following notational conventions an terminology. Nx () enotes a neighborhoo of x where x may be vector value. A function fx () is sai to be locally ifferentiable at x if it is ifferentiable in a neighborhoo of x ; it is sai to be locally inepenent of x if it is constant in a neighborhoo of x. The inicator function of set A is enote by A an x max{ x 0}. Finally all computations of erivatives of integrals make use of the generalize Leibniz integral rule in Lemma A. an Corollary A. of Appenix A. - 3 -
The rest of the paper is organize as follows. Section 2 specifies the mapping from iscrete MTS moels with backorers uner the ( Rr ) policy to SFM counterparts. Section 3 escribes the performance metrics an parameters of interest. Section 4 obtains IPA erivative formulas an shows them to be unbiase. Finally Section 5 conclues the paper an Appenix A provies some auxiliary material. 2. Mapping a Discrete MTS Moel to an SFM Counterpart We next procee to map the traitional iscrete MTS moel with backorers uner the ( Rr ) policy into an SFM counterpart. By convention all stochastic processes to follow are assume right-continuous. { I( t): t 0} is the inventory level process where It () is the flui volume of inventory on han at time t. { Bt ( ): t 0} is the backorer level process where Bt () is the flui volume of all backorers at time t. { Wt ( ): t 0} is the extene inventory level process [see Zhao an Melame (2006 2007)] where I ( t) if B( t) 0 W( t) I( t) B( t) B( t) if I( t) 0 Thus Wt () etermines both It () an Bt ()(an vice versa). (2.) { Xt ( ): t 0} is the outstaning orers process where Xt () is the flui volume of outstaning orers at time t. { ( t): t 0} is the eman arrival rate process where () t is the incoming eman arrival rate at time t. { ( t): t 0} is the prouction rate process where () t is the prouction rate (capacity) of the prouction facility at time t. { ( t): t 0} is the replenishment rate process where () t is the replenishment rate of prouct from the prouction facility to the inventory facility at time t. We now procee to exhibit the formal efinitions of all flui-moel components of the MTS system uner stuy. - 4 -
The initial state of the system is given by ( X( 0) W ( 0)) subect to the constraints { W( 0) R} { X( 0) 0} an { W( 0) r} { X( 0) 0}. Furthermore We shall characterize the prouction system state at time t by Xt () namely the state is ile on the event { Xt ( ) 0} an it is busy on { Xt ( ) 0}. Given an initial state ( X( 0) W ( 0)) an the processes { ( t)} an { ( t)} we efine the evolution of the system as follows. While in an ile state the inventory level process is governe by the one-sie stochastic ifferential equation an the moel satisfies t I( t) Wt ( ) ( t) (2.2) t () t 0 (2.3) Bt ( ) Xt ( ) 0. (2.4) While in a busy state the extene inventory level process is governe by the one-sie stochastic ifferential equation an the moel satisfies the conservation equation an the relationships Wt ( ) ( t ) ( t ) ( t ) ( t ) (2.5) t Xt ( ) I( t) Bt ( ) R (2.6) I( t) [ RXt ( )] Bt ( ) [ Xt ( ) R]. (2.7) Let [ 0 T ] be a finite time interval for some prescribe time horizon T. Similarly to Wari et al. (2002) we efine two types of sample-path events: Exogenous events. An exogenous event occurs in sample path at time t if t is a ump time such that ( t) ( t ) or ( t) ( t ). Enogenous events. An enogenous event occurs in sample path at time t if t is an extene-inventory level hitting time such that W( t) R or W( t) r or W( t) 0. 3. Performance Metrics an Parameters We shall be intereste in the following performance metrics in the mappe SFM over a finite interval [ 0 T ]: The time average of flui volume of inventory on-han over the interval [ 0 T ] given by T MI ( T) I( t) t T. (3.) 0 The time average of flui volume of backorers over the interval [ 0 T ] given by T MB ( T) B( t) t T. (3.2) 0-5 -
Let enote a generic parameter of interest from a close an boune omain. We shall routinely write R() r() MI ( T ) MB ( T ) an so on to explicitly isplay the epenence of a ranom variable on its parameters of interest. Thus our goal is to erive formulas for the IPA erivatives MI ( T ) an M ( ) B T in SFM setting using sample path analysis an to show them to be unbiase. The two IPA parameters of interest are: The stock-up-to level of the inventory facility R () The reorer point of the inventory facility r(). (3.3). (3.4) 4. IPA Derivatives Define a sequence of intervals ( P( ) Q( ))... J ( ) subintervals of [ 0 ) T such that ( ) 0 as the orere extremal Xt for all t ( P( ) Q( )) an J() is the [ T ); the term extremal means here that the (ranom) number of such intervals inaugurate in 0 enpoints P () an Q () are obtaine via the inf an sup functions respectively. By convention if any of these enpoints oes not exist then it is set to. We refer to intervals of the form [ P( ) Q( )) as busy intervals since the system is in the busy state for their uration. In a similar vein intervals of the form [ Q ( ) P ( )) will be referee to as ile intervals. Thus from a certain inex an on P () is an inventory hitting time of r while Q () is an inventory hitting time of R. We have the following observation for the hitting times P () an Q (). Observation For any finite time interval [ 0 T ) we have w.p. P( ) Q( ) P( ) Q ( )... P ( ) Q ( ). (a) When they exist 2 2 J( ) J( ) (b) P () 0 on { X( 0) 0} an P () 0 on { X( 0) 0}. P () (c) Q ( t) t W( 0) () r an [ ( t ) ( t )] t R r 0 P() on { X( 0) 0}. () (e) Q () [ ( t ) ( t )] t R W ( 0 ) on { X( 0) 0}. 0 Q () [ ( t ) ( t )] t R r. P () - 6 -
(f) P () () t t Rr. Q () Throughout this chapter we assume the following regularity conitions. Assumption (a) W.p. the eman arrival rate process { ( t)} an the prouction rate process { ( t)} are piecewise-constant an have a finite number of iscontinuities in any finite time interval such that the time points at which the iscontinuities occur are inepenent of the parameters of interest. (b) W.p. ( P( )) a* ( P ( )) ( P ( )) D* ( Q ( )) ( Q ( )) * an ( Q ( )) A* where a * D * *an * inepenent of an. A are positive eterministic constants (c) No multiple sample-path events occur simultaneously w.p.. () The initial inventory state oes not epen on namely w.p. W( 0 ) W( 0) an X( 0 ) X( 0) for all. Assumption implies the following observation. Observation 2 (a) w.p. J( ) J* where J * is a positive eterministic constant inepenent of. (b) There is a neighborhoo NP ( () ) such that for all ( t ) N( P () ) ( t) ( P () ) 0 an ( t) ( t) ( P( ) ) ( P( ) ). (c) There is a neighborhoo NQ ( () ) such that for all ( t ) NQ ( () ) ( t) ( Q () ) an ( t) ( t) ( Q( ) ) ( Q( ) ) 0. All results in the sequel are to be unerstoo as holing w.p.. 4. IPA Derivatives with Respect to R To obtain the IPA erivatives of MI ( T ) an MB ( T ) with respect to the stock-up-to level R we make the following assumptions. Assumption 2 (a) R (). (b) The processes { ( t)} an { ( t)} an the reorer point r are inepenent of the parameter R (). Lemma (a) P () is locally inepenent of. (b) P () an Q () are continuous. - 7 -
Proof To prove part (a) we consier two cases. Case : The system starts in busy state namely on the event { X( 0) 0} whence the result follows from P () 0 by part (b) of Observation. Case 2: The system starts in ile state namely on the event { X( 0) 0} whence P () 0 by part (b) of Observation. The result follows from part (c) of Observation since by part () of Assumption an part (b) of Assumption 2 W () 0 () t an r are all inepenent of. We prove part (b) by inuction letting R() be some given stock-up-to level. Clearly P () is trivially continuous in by part (a) thereby establishing the inuction basis for part (b) of this lemma. Q assume that P () is continuous by the inuction To prove the inuctive step for () hypothesis. Next subtracting the equation in part (e) of Observation with from the same equation with yiels Q ( ) Q ( ) [ ( t) ( t)] t [ ( t) ( t)] t P( ) P( ) Q ( ) P( ) [ ( t) ( t)] t [ ( t) ( t)] t Q ( ) P( ) (4.) Sening 0 in Eq. (4.) an focusing on the rightmost sie above we note that the secon integral tens to zero by the inuction hypothesis an that by part (c) of Observation 2 the integran is positive an constant in some neighborhoos of Q () an Q ( ). Consequently we conclue that lim[ Q( ) ( )] 0 0 Q. Finally to prove the inuctive step for P () assume that Q () is continuous by the inuction hypothesis. Next subtracting the equation in part (f) of Observation with from the same equation with yiels P( ) P( ) ( t) t ( t) t Q ( ) Q ( ) P( ) Q ( ) ( t) t ( t) t P( ) Q ( ) (4.2) Sening 0 in Eq. (4.2) an focusing on the rightmost sie above we note that the secon integral tens to zero by the inuction hypothesis an that by part (b) of Observation 2 the integran is positive an constant in some neighborhoos of P () an P ( ). Consequently we conclue that lim[ P( ) ( )] 0 0 P. - 8 -
Lemma 2 P () an Q () are continuously ifferentiable. Proof We prove the lemma by inuction letting R() be some given stock-up-to level. Clearly P () is continuously ifferentiable in by part (a) of Lemma thereby establishing the inuction basis for this lemma. To prove the inuctive step for Q () assume that P () is continuously ifferentiable by the inuction hypothesis. In view of part (a) of Assumption an Lemma Eq. (4.) can be written w.p. after ivision by a sufficiently small as [ ( Q ( )) ( Q ( ))][ Q ( ) Q ( )] [ ( P ( )) ( P ( ))][ P ( ) P ( )] (4.3) Sening 0 in Eq. (4.3) we note that by the inuction hypothesis its right-han sie converges an the limit is continuous. Furthermore since ( Q( )) ( Q( )) 0 by part (c) of Observation 2 we conclue that Q () is continuously ifferentiable. To prove the inuctive step for P () assume that Q () is continuously ifferentiable by the inuction hypothesis. In view of part (a) of Assumption an Lemma Eq. (4.2) can be written w.p. after ivision by a sufficiently small as ( P ( ))[ P ( ) P ( )] ( Q ( ))[ Q ( ) Q ( )] (4.4) Sening 0 in Eq. (4.4) we note that by the inuction hypothesis its right-han sie converges an the limit is continuous. Furthermore since ( ( )) 0 by part (b) of Observation 2 we conclue that P () is continuously ifferentiable. Lemma 3 an [ ( ) ( )] [ ( ) ( )] [ ( Q ( )) ( Q ( ))] Q ( ) [ ( P ( )) ( P ( ))] P ( ) Q( ) Q( ) P( ) P( ) P - 9 -
P () ( ) ( P ( )) P ( ) ( Q ( )) Q ( ) Q () Proof By Eq. (2.5) parts (a) an (c) of Assumption an part (b) of Assumption 2 we have that { ( t)} an { ( t)} are inepenent of continuous at the hitting times P () an Q () an have a finite number of iscontinuities between successive hitting times. Lemma 3 now follows from Lemma A. Corollary A. an Lemma 2. Lemma 4 Consier a mappe SFM of an MTS system with backorers uner the ( Rr ) policy subect to Assumption an 2. Then for any 0T 0 P () ( Q ( )) Q ( ) (4.5) ( P ( )) an [ ( P ( )) ( P ( ))] P ( ) Q ( ) (4.6) ( Q( )) ( Q( )) Proof We first prove the lemma for. Eq. (4.5) for follows from part (a) of Lemma. To prove Eq. (4.6) for we consier two cases. Case : The system starts in busy state namely on the event { X( 0) 0}. It follows that P () 0 by part (b) of Observation. Next note that by Eq. (2.5) Q( ) [ ( ) ( )] R ( ) W ( 0 ) which on ifferentiation with respect to with the 0 ai of Lemma 3 yiels after some manipulation Q () ( Q( )) ( Q( )) on { X( 0) 0} since W () 0 () t an () t are all inepenent of by part () of Assumption an part (b) of Assumption 2. Eq. (4.6) for now follows from the equation above since ( Q( )) ( Q( )) by part (c) of Observation 2. 0 Case 2: The system starts in ile state namely on the event { X( 0) 0} whence P () 0. Note that by part (e) of Observation Q() P [ ( ) ( )] R ( ) r which on () ifferentiation with respect to with the ai of Lemma 3 an Eq. (4.5) yiels after some manipulation - 0 -
Q () on { X( 0) 0} ( Q( )) ( Q( )) since W () 0 { ( t)} { ( t)} an r are all inepenent of by part () of Assumption an part (b) of Assumption 2. Eq. (4.6) now follows for from the equation above since P() 0 an ( Q( )) ( Q( )) 0 by part (a) of Lemma an part (c) of Observation 2 respectively. Next we prove Eq. (4.5) for. By part (f) of Observation we have P () ( ) R( ) r which on ifferentiation with respect to with the ai of Q () Lemma 3 yiels ( P ( )) P ( ) ( Q ( )) Q ( ) an after some manipulation the above becomes ( Q( )) Q( ) P (). ( P ( )) Eq. (4.5) for now follows from the equation above since ( P ( )) 0 by part (b) of Observation 2. Finally we prove Eq. (4.6) for. By part (e) of Observation we have Q () ( ) ( ) R( ) r which on ifferentiation with respect to with the P () ai of Lemma 3 yiels ( Q ( )) ( Q ( )) Q ( ) ( P ( )) ( P ( )) P ( ) an after some manipulation the above becomes ( P ( )) ( P ( )) P ( ) Q (). ( Q ( )) ( Q ( )) Eq. (4.6) for now follows from the equation above since ( Q( )) ( Q( )) 0 by part (c) of Observation 2. Lemma 5 Consier a mappe SFM of an MTS system with backorers uner the ( Rr ) policy subect to Assumption an 2. Then for any 0T an t [ 0 T] (a) On the event A( ) { 0t Q ( )} It ( ) 0 (4.7) - -
(b) On the events B ( ) { Q ( ) t P ( )} Bt ( ) 0. (4.8) I( t ) ( Q( )) Q( ) (4.9) Bt ( ) 0. (4.0) (c) On the events C ( ) { P( ) t Q ( )} { I( t ) 0} 2 I( t ) [ ( P ( )) ( P ( ))] P ( ) (4.) Bt ( ) 0. (4.2) () On the events D ( ) { P( ) t Q ( )} { Bt ( ) 0} 2 Proof To prove part (a) we consier two cases. It ( ) 0 (4.3) Bt ( ) [ ( P ( )) ( P ( ))] P ( ). (4.4) Case : The system starts in busy state namely on the event { X( 0) 0} so P () 0 by part t (b) of Observation. Next note that by Eq. (2.5) W( t ) W( 0) [ ( ) ( )] on 0 the event A( ) { X( 0) 0}. By part () of Assumption an part (b) of Assumption 2 we have W () 0 { ( t)} an { ( t)} are all inepenent of implying that Wt ( ) is inepenent of ona( ) { X( 0) 0}. It follows from Eq. (2.) that both It ( ) an Bt ( ) are also inepenent of ona( ) { X( 0) 0} from which Eqs. (4.7) an (4.8) follow by Eq. (2.). Case 2: The system starts in ile state namely on the event { X( 0) 0} so P () 0 by part (b) of Observation. Since A( ) { 0t P ( )} { P ( ) t Q ( )} we shall treat each of the constituent events separately. t On the event { 0t P ( )} { X( 0) 0} W( t ) W( 0) ( ) by Eq. (2.2). 0 Since W () 0 an { ( t)} are inepenent of it follows that Wt ( ) is inepenent of on { 0t P( )} { X( 0) 0}. Eqs. (4.7) an (4.8) now follow from Eq. (2.). - 2 -
On the event { P( ) t Q( )} { X( 0) 0} the system is in the busy state so ( ) t W t r [ ( ) ( )] by Eq. (2.5). Since P( ) r { ( t)} an { ( t)} are P( ) all inepenent of by part (a) of Lemma an part (b) of Assumption 2 it follows that. Eqs. (4.7) an (4.8) now Wt ( ) is inepenent of on{ P( ) t Q( )} { X( 0) 0} follow from Eq. (2.). To prove part (b) note that by Eq. (2.2) we have on each event B () t I( t ) R( ) ( ) an Bt ( ) 0 which on ifferentiation with respect to Q () yiel Eqs. (4.9) an (4.0) respectively. To prove parts (c) an () note that by Eq. (2.5) we have on each event C ( ) D( ) t W( t ) r [ ( ) ( )] which on ifferentiation with respect to yiels on C ( ) D( ) P () Wt ( ) [ ( P ( )) ( P ( ))] P ( ). On C() one has I( t ) W( t ) an Bt ( ) 0 while on D () one has It ( ) 0 an Bt ( ) W( t ) both by Eq. (2.) from which parts (c) an () follow. Note that the J() events { P( ) t Q( )} { I( t ) B( t ) 0} nee not be consiere because 2 they have probability zero by part (c) of Assumption. Theorem Consier a mappe SFM of an MTS system with backorers uner the ( Rr ) policy subect to Assumptions an 2. Then for any 0T an t [ 0 T] the IPA erivatives of the inventory time average an the backorer time average with respect to the stock-up-to level parameter R are given by J() M ( T ) ( Q ( )) Q ( ) min{ P ( ) T} Q ( ) an I T J() min{ Q ( ) T} ( ( )) ( ( )) ( ) { I( t ) 0} T P () 2 P P P t J() min{ Q ( ) T} B( ) ( ( )) ( ( )) ( ) T P () { B( t ) 0} 2 M T P P P t (4.5) (4.6) - 3 -
where P () an Q () are given by Eqs. (4.5) an (4.6) respectively. Furthermore the IPA erivatives in Eqs. (4.5) an (4.6) are unbiase for each finite T0an every. Proof Applying the generalize Leibniz integral rule to Eqs. (3.) an (3.2) yiels T T MI ( T ) I( t ) t I( t ) t. T 0 T 0 (4.7) T T MB ( T ) Bt ( ) t Bt ( ) t. T 0 T 0 (4.8) To see that note that the en-points of the interval [ 0 ] times in the interior of [ 0 T ] satisfy by the continuity of { It ( )} an { Bt ( )} T o not epen on while the hitting I( P ( ) ) P ( ) I( P ( ) ) P ( ) B( P ( ) ) P ( ) B( P ( ) ) P ( ) I( Q ( ) ) Q ( ) I( Q ( ) ) Q ( ) BQ ( ( ) ) Q ( ) BQ ( ( ) ) Q ( ) Consequently Eqs. (4.7) an (4.8) follow by rearranging the terms on the right-han sum in Corollary A.. Eqs. (4.5) an (4.6) now follow by substituting the values of the erivatives compute in Lemma 5 into Eqs. (4.7) an (4.8). We next prove that the IPA erivative formulas are unbiase using Fact. The proofs of Eqs. (4.7) an (4.8) establish that Conition (a) of Fact is satisfie for both MI ( T ) an MB( T ) for each finite T 0. By parts (a) an (b) of Assumption an part (a) of Observation 2 Eqs. (4.5) an (4.6) are boune in w.p.. Since any ifferentiable function with such a boune erivative is Lipschitz continuous an its Lipschitz constant trivially has a finite first moment it follows that Conition (b) of Fact hols for both MI ( T ) an MB( T ) for each finite T 0 thereby completing the proof. 4.2 IPA Derivatives with Respect to r To obtain the IPA erivatives of MI ( T ) an MB ( T ) with respect to the reorer point r we make the following assumptions. Assumption 3 (a) r() where. - 4 -
(b) The processes { ( t)} an { ( t)} an the stock-up-to level R are inepenent of the parameter. Lemma 6 The hitting times P () an Q () are continuous. Proof Similar to that of Lemma. Lemma 7 The hitting times P () an Q () are continuously ifferentiable. Proof Similar to that of Lemma 2. Lemma 8 an P () Q () Proof Similar to that of Lemma 3. [ ( ) ( )] [ ( ) ( )] [ ( Q ( )) ( Q ( ))] Q ( ) [ ( P ( )) ( P ( ))] P ( ) Q( ) Q( ) P( ) P( ) ( ) ( P ( )) P ( ) ( Q ( )) Q ( ) Lemma 9 Consier a mappe SFM of an MTS system with backorers uner the ( Rr ) policy subect to Assumption an 3. Then for any 0T an 0 an X( 0) 0 an X( 0) 0 ( P ( )) P () ( Q( )) Q( ) ( P ( )) 0 an X( 0) 0 Q () [ ( P ( )) ( P ( ))] P ( ) otherwise ( Q( )) ( Q( )) (4.9) (4.20) Proof We first prove Eq. (4.9) for. We consier two cases. - 5 -
Case : The system starts in busy state namely on the event { X( 0) 0}. It follows that P () 0 which implies the first line of Eq. (4.9). Case 2: The system starts in ile state namely on the event { X( 0) 0} whence P () 0. P( ) Note that by part (c) of Observation one has ( t ) t W ( 0 ) r ( ) which on 0 ifferentiation with respect to yiels after some manipulation P () on { X( 0) 0} ( P ( )) since W () 0 an { ( t)} are all inepenent of by part () of Assumption an part (b) of Assumption 3 respectively. The secon line of Eq. (4.9) now follows from the equation above since ( P ( )) 0 by part (b) of Assumption. Next we prove Eq. (4.9) for. By part (f) of Observation one has P () ( ) Rr( ) which on ifferentiation with respect to yiels Q () ( P ( )) P ( ) ( Q ( )) Q ( ) an after some manipulation the above becomes ( Q( )) Q( ) P (). ( P ( )) Eq. (4.9) for now follows from the equation above since ( ( )) 0 by part (b) of Assumption. Finally we prove Eq. (4.20). We first consier on the event { X( 0) 0} an note that by part () of Observation Q( ) [ ( ) ( )] R W ( 0 ). Since W () 0 0 { ( t)} { ( t)} an R are all inepenent of by part () of Assumption an part (b) of Assumption 3 the first line of Eq. (4.20) now follows. Otherwise for on the event { X( 0) 0} or for one has by part (c) an (e) of Observation Q () ( ) ( ) R r( ) which on ifferentiation with respect to yiels P () ( Q ( )) ( Q ( )) Q ( ) ( P ( )) ( P ( )) P ( ) an after some manipulation the above becomes ( P ( )) ( P ( )) P ( ) Q (). ( Q( )) ( Q( )) P - 6 -
The secon line of Eq. (4.20) now follows from the equation above since ( Q( )) ( Q( )) 0 by part (b) of Assumption. Lemma 0 Consier a mappe SFM of an MTS system with backorers uner the ( Rr ) policy subect to t [ T] Assumption an 3. Then for any 0T an 0 (a) On the events A( ) { 0t P( )} { X( 0) 0} an 2 2 A ( ) { 0t P ( )} { X( 0) 0} It ( ) 0 (4.2) Bt ( ) 0. (4.22) (b) On the events B ( ) { P ( ) t Q ( )} { X( 0) 0} { I( t ) 0} an J() B2 ( ) { P( ) t Q( )} { I( t ) 0} 2 I( t ) [ ( P ( )) ( P ( ))] P ( ) (4.23) Bt ( ) 0. (4.24) (c) On the events C ( ) { P ( ) t Q ( )} { X( 0) 0} { Bt ( ) 0} an J() C2( ) { P( ) t Q( )} { Bt ( ) 0} 2 It ( ) 0 (4.25) B( t ) [ ( P ( )) ( P ( ))] P ( ). (4.26) () On the events D ( ) { Q ( ) t P2 ( )} { X( 0) 0} Proof To prove part (a) on ( ) Next note that by Eq. (2.2) J() an D2( ) { Q( ) t P ( )} 2 I( t ) ( Q( )) Q( ) (4.27) Bt ( ) 0. (4.28) A note that the system starts in the ile state implying P () 0. W( t ) W( 0) ( ) t on ( ) 0 A. By part () of - 7 -
Assumption an part (b) of Assumption 3 we have that W () 0 an { ( t)} are inepenent of respectively implying that Wt ( ) is inepenent of on A. It follows from Eq. (2.) that both It ( ) an Bt ( ) are also inepenent of on (4.22) follow on A (). () A () from which Eqs. (4.2) an To prove part (a) on A2( ) note that the system starts in busy state implying P () 0. A ( ) { 0 t Q ( )} { X( 0 ) 0 } Next ecompose A2 ( ) A 2 ( ) A2 2( ) where 2 an A ( ) { Q( ) t P( )} { X( 2 2 2 0 ) 0 } separately. t an treat each of the constituent events On A () 2 one has W( t ) W( 0) [ ( ) ( )] by Eq. (2.5). Since W () 0 0 { ( t)} an { ( t)} are all inepenent of by part () of Assumption an part (b) of Assumption 3 it follows that Wt ( ) is inepenent of on A () 2. It further follows from Eq. (2.) that both It ( ) an Bt ( ) are also inepenent of on A () 2 from which Eqs. (4.2) an (4.22) follow on A () 2. t On A () 22 one has W ( t ) R ( ) by Eq. (2.2). Since Q( ) R an Q() { ( t)} are all inepenent of by Eq. (4.20) an part (b) of Assumption 3 it follows that Wt ( ) is inepenent of on A () 22. It further follows from Eq. (2.) that both It ( ) an Bt ( ) are also inepenent of on A () 22 from which Eqs. (4.2) an (4.22) follow on A () 22. B( ) B ( ) C ( ) C ( ) one has To prove parts (b) an (c) note that on the event 2 2 t by Eq. (2.5) W( t ) r( ) [ ( ) ( )] which on ifferentiation with respect P () to yiels on B ( ) B2 ( ) C ( ) C2( ) W( t ) [ ( P ( )) ( P ( ))] P ( ). On B( ) B2( ) one has I( t ) W( t ) an Bt ( ) 0 while on C( ) C2( ) one has It ( ) 0 an Bt ( ) W( t ) both by Eq. (2.) from which parts (b) an (c) follow. Note that the events { P( ) t Q( )} { X( 0) 0} { I( t ) Bt ( ) 0} J() an { P( ) t Q( )} { I( t ) B( t ) 0} nee not be consiere because they 2 have probability zero by part (c) of Assumption. - 8 -
Finally to prove part () note that on the event D( ) D2( ) one has by Eq. (2.2) t I( t ) R ( ) an ( ) 0 Q () yiel Eqs. (4.27) an (4.28) respectively. Bt which on ifferentiation with respect to Theorem 2 Consier a mappe SFM of an MTS system with backorers uner the ( Rr ) policy subect to Assumptions an 3. Then for any 0T an t [ 0 T] the IPA erivatives of the inventory time average an the backorer time average with respect to the reorer point parameter r are given by MI ( T ) ( Q ( )) Q ( ) min{ P2 ( ) T} min{ Q ( ) T} T J() ( Q ( )) Q ( ) min{ P ( ) T} min{ Q T ( ) T} an min{ Q( ) T} { X( 0) 0} { ( ) 0} T M ( T ) [ ( P( )) ( P( ))] P( ) min{ P ( ) T} I t t T 2 J() min{ Q ( ) T} [ ( ( )) ( ( ))] ( ) P () { I( t ) 0} 2 P P P t B min{ Q ( ) T} { X( 0) 0} [ ( ( )) ( ( ))] ( ) T min{ P { ( ) 0} ( ) T} B t J() min{ Q ( ) T} [ ( P ( )) ( P ( ))] P ( ) P () { B( t ) 0} t T 2 P P P t (4.29) (4.30) where P () an Q () are given by Eqs. (4.9) an (4.20) respectively. Furthermore the IPA erivatives in Eqs. (4.29) an (4.30) are unbiase for each finite 0 T an every. Proof. Applying the generalize Leibniz integral rule to Eqs. (3.) an (3.2) an using an argument similar to that in the proof of Theorem yiels T T I 0 0 M ( T ) I( t ) t I( t ) t. T T (4.3) T T MB ( T ) Bt ( ) t Bt ( ) t. T 0 T 0 (4.32) - 9 -
Eqs. (4.29) an (4.30) now follow by substituting the values of the erivatives compute in Lemma 0 into Eqs. (4.3) an (4.32). We next prove that the IPA erivative formulas are unbiase using Fact. The proofs of Eqs. (4.29) an (4.30) establish that Conition (a) of Fact is satisfie for both MI ( T ) an MB( T ) for each finite T 0. By parts (a) an (b) of Assumption an part (a) of Observation 2 Eqs. (4.29) an (4.30) are boune in w.p.. Since any ifferentiable function with such a boune erivative is Lipschitz continuous an its Lipschitz constant trivially has a finite first moment it follows that Conition (b) of Fact hols for both MI ( T ) an MB( T ) for each finite T 0 thereby completing the proof. 5. Conclusion This paper treate an SFM moel of an MTS system with backorers uner the continues-review ( Rr ) policy. In aition to eriving the IPA erivative formulas of the time average inventory level an backorer level with respect to the stock-up-to level R an the reorer point r uner any initial inventory state the IPA erivatives were shown to be unbiase an easily computable. Future research irections inclue both theoretical an practical topics. On the theoretical sie the SFM can be use to obtain IPA erivatives in more general supply chains with ifferent policies or with multiple stages or multiple proucts. On the practical sie it woul be of interest to investigate the efficacy of SFM IPA erivatives as an ingreient in offline esign algorithms an online control algorithms of MTS systems an their extensions. Appenix A This Appenix provies a generalize Leibniz integral rule which extens the classical Leibnitz Integral Rule below. Fact 2 (see Fikhtengolts (965)Volume II pp. 45-46) Let ft ( ) be efine on the rectangle [ AB ] [ ] an let a() an b() be two ifferentiable functions over [ ] satisfying Aa( ) b ( ) B for each [ ]. Let D{( t ): [ ] t [ a( ) b( )]}. Suppose that ft ( ) is continuous on D with a continuously ifferentiable partial erivative ft ( ) on D. Then the function b() F( ) f( t ) t a() an its erivative is given by b() is ifferentiable on [ ] F( ) f( t ) t a() b() f( t ) t f( b( ) ) b( ) f( a( ) ) a( ) a() - 20 -
We nee a slight generalization of the Leibnitz Integral rule for the case where ft ( ) is allowe to be iscontinuous in t at a finite number of points of each interval [ a( ) b( )] incluing its en points. Lemma A. (Generalize Leibnitz Integral Rule) Let ft ( ) be efine on the rectangle [ AB ] [ ] an let a() an b() be two ifferentiable functions over [ ] satisfying Aa( ) b ( ) B for each [ ]. Let D {( t ): [ ] t ( a( ) b( ))} an D2 {( t ): [ ] t [ a( ) b( )]}. Suppose that ft ( ) is continuous on ft ( ) on D with a continuously ifferentiable partial erivative D. Suppose further that there exists a function gt ( ) such that. gt ( ) is continuous on gt ( ) on D 2. 2. For every [ ] Then the function D 2 with a continuously ifferentiable partial erivative f( t ) g( t ) on ( a( ) b( )). b() F( ) f( t ) t a() an its erivative is given by b() is ifferentiable on [ ] F( ) f( t ) t a() b() f( t ) t f( b( ) ) b( ) f( a( ) ) a( ) a() b() g( t ) t g( b( ) ) b( ) g( a( ) ) a( ) a() Proof If the continuity of ft ( ) an ft ( ) hols at the en points a() an b() then the Lemma hols trivially by applying the classical Leibnitz Integral Rule to ft ( ). Otherwise let b() a() an note that gt ( ) satisfies the classical Leibniz Integral Rule. since the integrans iffer in value only in a finite G( ) g( t ) t Clearly F( ) G( ) for every [ ] number of points. Furthermore the assumptions on ft ( ) an gt ( ) imply that g( a( ) ) f( a( ) ) an g( b( ) ) f( b( ) ). The Lemma now follows by applying the classical Leibniz Integral Rule to gt ( ). - 2 -
Corollary A. Let the function ft ( ) be efine in the rectangle [ AB ] [ ] an let ai () i n be ifferentiable functions on [ ] satisfying Aa( ) a 2( ) an ( ) B for every [ ]. For i n let D i {( t ): [ ] t ( ai( ) ai ( ))} an D2 i {( t ): [ ] t [ ai( ) ai ( )]}. Suppose that for each in ft ( ) is continuous on D i with a continuously ifferentiable partial erivative ft ( ) on D i. Suppose further that there exist functions gi( t ) in such that. For each in gi( t ) is continuous on D 2i with a continuously ifferentiable partial erivative gi( t ) on D 2i. 2. For every [ ] f( t ) gi( t ) on each ( ai ( ) ai ( )) in. Then the function a n () F( ) f( t ) t a () is ifferentiable on [ ] an its erivative is given by an() F( ) f( t ) t a() a () n n f( t ) t f( ai ( ) ) ai ( ) f( ai ( ) ) ai ( ) a() i Proof Just ecompose the integral into a ( ) n n ai ( ) F( ) f( t ) t f( t ) t a( ) a ( ) i i an apply Lemma A. to each term on the right. - 22 -
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