MAKE-TO-STOCK SYSTEMS WITH BACKORDERS: IPA GRADIENTS. Yao Zhao Benjamin Melamed

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1 Proceeings of the 24 Winter Simulation Conference R. Ingalls, M. Rossetti, J. Smith, B.Peters, es. MAKE-TO-STOCK SYSTEMS WITH BACKORDERS: IPA GRADIENTS Yao Zhao Benjamin Melame Department of Management Science an Information Systems Rutgers Business School Newark an New Brunswick Rutgers University Piscataway, NJ 8854, U.S.A. ABSTRACT We consier single-stage, single-prouct Make-to-Stock systems with ranom eman an ranom service (prouction) rate, where eman shortages at the inventory facility are backorere. The Make-to-Stock system is moele as a stochastic flui moel (SFM), in which the traitional iscrete arrival, service an eparture stochastic processes are replace by corresponing stochastic flui-flow rate processes. IPA (Infinitesimal Perturbation Analysis) graients of various performance metrics are erive with respect to parameters of interest (here, base-stock level an prouction rate), an are showe to be unbiase an easy to compute. The (ranom) IPA graients are obtaine via sample path analysis uner very mil assumptions, an are inherently nonparametric in the sense that no specific probability law nee be postulate. The formulas erive can be use in simulation, as well as in real-life systems. 1 INTRODUCTION The objective of this paper is to erive IPA graients (erivatives) of the ranom variables that moel performance metrics of interest in Make-to-Stock systems, an to show them to be unbiase. Specifically, let L(θ) be a ranom variable, parameterize by a generic real-value parameter θ from a close an boune set. The IPA graient of L(θ) with respect to θ is the ranom variable L (θ) L(θ), provie it exists almost surely. Furthermore, L(θ) is sai to be unbiase, if the expectation an ifferentiation operators commute, namely, E[ L(θ)] E[L(θ)]; otherwise, it is sai to be biase. Sufficient conitions for unbiase IPA erivatives are given in the following lemma. Lemma 1 (see Rubinstein an Shapiro (1993), Lemma A2, p. 7) An IPA erivative L (θ) is unbiase, if (a) (b) For each θ, the IPA erivatives L (θ) exist with probability 1 (w.p.1). W.p.1, L(θ) is Lipschitz continuous in, an the (ranom) Lipschitz constants, K(θ), have finite first moments. Comprehensive iscussions of IPA erivatives an their applications can be foun, for example, in Glasserman (1991) an Ho an Cao (1991). Most papers on stochastic prouction-inventory systems (an Make-to-Stock systems in particular) postulate specific probability laws that govern the unerlying stochastic processes (e.g., Poisson eman arrivals an exponential service times). For simple systems, such as the one-stage Maketo-Stock variety, close-form formulas of key performance metrics (e.g., statistics of inventory levels an lost sales or backorers), have been erive as functions of control parameters. For example, Zipkin (1986) an Karmarkar (1987) obtain the optimal control of such systems with respect to batch sizes an re-orer points by stanar optimization techniques. For more complex Make-to-Stock systems, such as the multi-stage serial variety, close-form formulas are not available. Buzacott, Price, an Shanthikumar (1991) carrie out sample path analysis for a 2-stage prouction system, governe by the continuous-time base-stock policy. Diffusion moels an eterministic flui moels have been propose in orer to mitigate the analytical an computational complexity of performance evaluation an optimal control. For example, Wein (1992) use a iffusion process to moel a multi-prouct, single-server Make-to-Stock system, while Veatch (22) iscusse iffusion an flui-flow moels of serial Make-to-Stock systems. Note, however, that iffusion moels require a heavy traffic conition to be vali approximations (Wein 1992). In a similar vein, while eterministic flui-flow moels provie valuable insights into the control rules of such systems, eterministic

2 moeling may well result in substantial numerical errors (Veatch 22). Simulation has been wiely use to stuy the performance of complex prouction systems. Uner perioicreview policies, system performance is evaluate only at specific review times. For example, Glasserman an Tayur (1995) consiere a class of prouction systems uner perioic-review moifie base-stock policy, an estimate via simulation its performance metrics an expecte IPA erivatives. Fu (1994) erive IPA erivatives for prouction systems uner perioic-review with the (s, S) policy. In contrast, iscrete-event simulation can track system performance continuously. For instance, Caramanis an Liberopoulos (1992) evelope a numerical technique base-on iscreteevent simulation an IPA to esign a near optimal flow controller for failure-prone manufacturing systems. All in all, most papers on stochastic prouction-inventory systems postulate a specific unerlying probability law, an focus on off-line control an optimization algorithms. The stochastic flui moel (SFM) improves on the aforementione moels in that it moels flui-flow queueing systems subject to ranomness. Like in traitional queueing systems, an SFM consists of a buffer an a server. However, the operation of the system iffers from its traitional counterpart in that the workloa moves like (continuous) flui rather than (iscrete) jobs. More specifically, a flui stream is injecte into a buffer accoring to some stochastic arrival-rate process, an is ischarge by a server accoring to some stochastic service-rate process. Finally, we point out that, ceteris paribus, SFM systems enjoy important avantages over their iscrete counterparts. First, IPA graients in SFM setting are unbiase, while their counterparts in iscrete queueing systems are by an large biase (Heielberger et al. 1988). Secon, IPA graients in SFM setting are nonparametric in the sense that no probability law nee be postulate, so that they may be compute both in simulation an real-life systems. Consequently, IPA graients, erive in SFM setting, can provie important information an insights into their iscrete counterparts, by applying graient formulas obtaine in SFM setting to traitional queueing systems. Motivate by the consierations above, Wari et al. (22) erive IPA graients for the loss volume an bufferworkloa time average, for simple queues in SFM setting; each of these metrics was ifferentiate with respect to buffer size, a parameter of the arrival rate process an a parameter of the service rate process. The paper showe the IPA graients to be unbiase, easily computable an nonparametric. Most recently, Paschaliis et al. (24) treate multi-stage prouction-inventory systems with continuoustime base-stock policy in SFM setting, an compute IPA graients of the time average inventory level an service level with respect to base-stock levels, an use them to etermine optimal base-stock levels at each stage. In this paper, we erive the IPA graients of the time average inventory level an backorers for the class of single-prouct, single-stage Make-to-Stock systems. Our proof methoology iffers markely from Paschaliis et al. (24), an we further erive new IPA formulas for the time average inventory level an backorers with respect to a prouction rate parameter. Our ultimate goal is to use the graient information for control an optimization of supply chains, which will be the subject of further research. Throughout the paper, we use the following notational conventions an terminology. The inicator function of set A is enote by 1 A, an we enote x + max{x,}. A function f(x) 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 rest of the paper is organize as follows. Section 2 presents the prouction-inventory moel uner stuy. Section 3 erives IPA graient formulas for the MTS systems with backorers. Finally, Section 4 summarizes the results an iscusses future work. 2 MAKE-TO-STOCK SYSTEMS WITH BACKORDERS Consier the traitional single-stage, single-prouct Maketo-Stock system (MTS system, for short), consisting of a prouction facility an an inventory facility. The two facilities interact: the latter sens orers to the former, while the former prouces stock to replenish the latter. The prouction facility is comprise of a queue that houses a prouction server (a single machine, a group of machines or a prouction line), precee by an infinite buffer that hols outstaning prouction orers. We assume that the prouction facility has an unlimite supply of raw material, so it never starves. The inventory facility satisfies incoming emans on a first come first serve (FCFS) basis, an is controlle by a continuous-time base-stock policy with some base-stock level S>. More specifically, the inventory an prouction facilities are couple: the inventory facility places orers as iscrete jobs in the prouction facility s buffer, while the prouction facility restocks the inventory facility. The eman process consists of an interarrival-time process of emans an their ranom magnitues. Demans arrive at the inventory facility an are satisfie from inventory on han (if available). Otherwise, the shortage is backorere an the eman waits in a FCFS buffer at the inventory facility until the prouction facility replenishes the inventory facility with the shortage amount. We now procee to map the traitional iscrete MTS system into an SFM version, shown in Figure 1. Here, the level-relate stochastic processes are flui volumes, where I(t) is the volume of inventory on-han at time t, B(t) is the volume of backorers at time t, an X(t) is the volume

3 of outstaning orers at time t. In a similar vein, Trafficrelate stochastic processes moel ranom flow rates, where α(t) is the rate of incoming emans at time t. µ(t) is the prouction rate at time t, λ(t) is the rate of incoming outstaning orers at time t, an ρ(t) is the traffic rate of prouct replenishment at time t. Similarly to Wari et al. (22), we efine two types of events along a sample path. We say that an exogenous event occurs whenever a jump occurs in the sample path of {α(t)} or {µ(t)}, an that an enogenous event occurs whenever a time interval is inaugurate, in which X(t) orx(t) S. L I an L B are ranom variables for each T. However their epenence on the sample path an on T is suppresse to simplify the notation. Let θ enote a generic parameter of interest with a close an boune omain,. We write S(θ), µ(θ, t), L I (θ), L B (θ) an so on, to explicitly isplay the epenence of a performance ranom variable on its parameter of interest. Our objective is to erive close-form formulas for the IPA graients L I (θ) L I (θ) an L B (θ) L B(θ) in SFM setting, using sample path analysis, an to show them to be unbiase. The interval [,T] can be partitione into two types of alternating perios: 1. Surplus perios are perios uring which B(θ,t). 2. Shortage perios are perios uring which B(θ,t) >. Figure 1: The Queue-Inventory System with Backorers We impose the following mil regularity conitions (cf. Wari et al. (22)). Assumption 1 (a) The processes {α(t)} an {µ(t)} have rightcontinuous sample paths that are piecewise continuously-ifferentiable, w.p.1. (b) Each of the processes, {α(t)} an {µ(t)}, has a finite number of iscontinuities in any finite time interval, w.p.1. (c) No multiple events occur simultaneously, w.p.1. Let [,T] be a finite time interval. In this paper, we will be intereste in the following performance metrics: the time average of the flui volume of inventory on-han over the interval [,T], given by L I 1 T I(t)t, (1) an the time average of the flui volume of backorers over the interval [,T], given by L B 1 T B(t)t. (2) The parameters of interest are the base-stock level of the inventory facility, S, an a parameter of the prouction rate process {µ(t)} (to be efine later). Observe that the metrics For each θ, let I j (θ), j 1,...,, be the successive surplus perios in [,T], an let B k (θ), k 1,...,K(θ), be the successive shortage perios in [,T], so that [,T] I j (θ) K(θ) B k (θ), (3) where K(θ) 1. We can now rewrite Eqs. (1) - (2) as L I (θ) L B (θ) 1 T 1 T I(θ,t)t 1 T B(θ,t)t 1 K(θ) T I j (θ) B k (θ) I(θ,t)t, (4) B(θ,t)t. (5) We next procee to erive IPA graients for the MTS system, assuming throughout the following initial conitions: I() S (full inventory), B() (no backorers), an X() (empty outstaning orer buffer). 3 IPA DERIVATIVES In the SFM version of the MTS moel with backorers, {λ(t)} is given by an {ρ(t)} is given by ρ(t) λ(t) α(t), t, (6) { µ(t), if X(t) > min{µ(t), λ(t)}, if X(t) (7)

4 This moel further satisfies the conservation relation I(t) B(t) + X(t) S, (8) where the inventory volume process is given by I(t) [S X(t)] +, (9) the backorer volume process is given by B(t) [X(t) S] +, (1) an the ynamics of the volume process of outstaning orers, {X(t)}, in the prouction facility are governe by the one-sie stochastic ifferential equation, X(t) t + {, X(t) an α(t) µ(t), α(t) µ(t), otherwise. (11) Observe that the stochastic erivative above oes not epen on S, but only on α(t) an µ(t). 3.1 IPA Graients with Respect to the Base-Stock Level In this section we treat the IPA erivatives, L I (θ) an L B (θ), where θ is the base-stock level, S(θ) θ, θ. The following assumptions are mae throughout this section. Assumption 2 (a) (b) The processes {α(t)} an {µ(t)} are inepenent of the parameter θ. The ranom erivatives L I (θ) an L B (θ) exist w.p.1. Note that this assumption alreay follows from Assumption 1 in the special case that {α(t)} an {µ(t)} have piecewise-constant sample paths. Notice that the time points at which {I(θ,t)} reaches S or are generally functions of θ. However, the time points at which {I(θ,t)} ceases to be full (equivalently, {X(t)} ceases to be zero) are locally inepenent of θ, because they correspon to a jump or a change of sign in {α(t) µ(t)}, an this ifference process is inepenent of θ by (a) of Assumption 2. Let I j (θ) [, H j (θ)), j 1,...,, enote the j-th surplus perio in [,T], where is its start point an H j (θ) is its en point (note the epenence on the base-stock level, θ, except for the initial G 1 ). We use the convention that H (θ) T when I(θ,T) > (i.e., when a surplus perio is still in progress at time T ). A generic sample path is epicte in Figure 2. By Assumption 1 an 2, the sample paths of an H j (θ)) are locally ifferentiable functions of θ, w.p.1.; furthermore, the number of surplus perios,, is locally inepenent of θ, w.p.1. Figure 2: A Generic Sample Path of an MTS System with Backorers Proposition 1 L I (θ) 1 T 1 T Proof. From Eq. (4), L I (θ) 1 T For every θ, [H j (θ) ] 1 {I(θ,t)>} t. (12) I(θ,t)t. Since is locally inepenent of θ, ifferentiating the equation above with respect to θ yiels L I (θ) 1 T Next, for each j 1,...,, I(θ,t)t. (13) I(θ,t)t I(θ,) + I(θ,H j (θ)) H j (θ) + I (θ, t) t. (14) For j 1, G 1 (θ)/, since G 1 is inepenent of θ, while for j 2,...,, I(θ,) by efinition. Similarly, I(θ,H j (θ)) for j<an for j with H (θ) < T by efinition. For j with H (θ) T one has H (θ)/, since in this case, H (θ) is locally inepenent of θ. Hence, the first two

5 terms on the right-han sie of Eq. (14) vanish for all j 1,...,. Furthermore, I(θ,t) S(θ) X(t) uring the surplus perios by Eq. (9), an X(t) is inepenent of θ S(θ) by Eq. (11) an the initial conition X(), I (θ, t) whence 1 over (, H j (θ)). It follows that I (θ, t) t [H j (θ) ]. (15) Eq. (12) now follows from Eq. (15) in view of Eqs. (13) - (15). We point out that Proposition 1 agrees with the result obtaine in Paschaliis et al. (24) by other methos. Next, observe that B k (θ) [H k (θ), G k+1 (θ)), k 1,...,K(θ), enotes the k-th shortage perio in [,T], where H k (θ) is its start point an G k+1 (θ) is its en point (note the epenence on the base-stock level θ). If I(θ,T) >, then K(θ) 1, while if B(θ,T ) >, then K(θ). We use the convention that G K(θ)+1 (θ) T when B(θ,T ) > (i.e., when a shortage perio is still in progress at time T ). By Assumption 1 an 2, the realizations of H k (θ) an G k+1 (θ) are locally ifferentiable functions of θ, an the number of shortage perios, K(θ), is locally inepenent of θ, w.p.1. Proposition 2 For every θ, L B (θ) 1 T 1 T Proof. From Eq. (8), K(θ) B(θ,t) I(θ,t) S(θ) + X(t), [G k+1 (θ) H k (θ)] 1 {B(θ,t)>} t. (16) t [,T]. Substituting the formula above into Eq. (5), one has L B (θ) 1 T [I(θ,t) S(θ) + X(t)] t L I (θ) S(θ) + 1 T X(t)t. (17) Since S(θ) θ an {X(t)} oes not epen on θ, ifferentiating Eq. (17) with respect to θ yiels, L B (θ) L I (θ) 1. Eq. (16) now follows by substituting Eq. (12) into the above, since by Eq. (3), [,T] can be partitione into surplus an shortage perios. To show that the IPA graients are unbiase, consier any θ,θ + θ. Since {X(t)} is inepenent of θ an S(θ) θ, we have from Eq.(9), I(θ + θ, t) I(θ,t) [S(θ + θ) X(t)] + [S(θ) X(t)] + θ, (18) an from Eq. (1), B(θ + θ, t) B(θ,t) [X(t) S(θ + θ)] + [X(t) S(θ)] + θ. (19) Proposition 3 Uner Assumption 1 an 2, the IPA erivatives L I (θ) an L B (θ) are unbiase. Proof. To show that L I (θ) an L B (θ) are unbiase, we use Lemma 1. First, Conition (a) of Lemma 1 is satisfie by part (b) of Assumption 2 for both L I (θ) an L B (θ). Next, by Eq. (18), L I (θ + θ) L I (θ) 1 T [I(θ + θ, t) I(θ,t)] t 1 T an by Eq. (19), I(θ + θ, t) I(θ,t) t θ, (2) L B (θ + θ) L B (θ) 1 T [B(θ + θ, t) B(θ,t)] t 1 T B(θ + θ, t) B(θ,t) t θ. (21) Eqs. (2) an (21) establish that Conition (b) of Lemma 1 hols for both L I (θ) an L B (θ). The proof of the proposition is complete. 3.2 IPA Graients with Respect to the Prouction Rate In this section we treat the IPA erivatives, L I (θ) an L B (θ), where θ is a parameter of the prouction rate, µ(θ, t), such that for all θ an all t [,T], µ(θ, t) µ (θ, t) 1. (22) This functional form correspons to a linear relationship between θ an the prouction rate. More picturesquely, θ can be viewe as a knob whose turning" scales the prouction rate. The following assumptions are mae throughout this section.

6 Assumption 3 Proposition 4 For every θ, (a) (b) The process {α(t)} an the base-stock level, S, are inepenent of the parameter θ. The ranom erivatives L I (θ) an L ζ (θ) exist, w.p.1. The surplus perios, I j (θ) [, H j (θ)), generally epen on θ for all j 1,...,, except for the initial G 1. Let [U j,m (θ), V j,m (θ)), m 1,...,M j (θ), be the subintervals in I j (θ) uring which I(t) S, with the notational conventions V 1, (θ) G 1 U 1,1, V j, (θ) for j>1, an U j,mj (θ)+1(θ) H j (θ), for all j 1,...,. We let H (θ) T if I(θ,T) >. Assumption 1 an 3 imply that all U j,m (θ) an V j,m (θ) are locally ifferentiable functions with respect to θ. For all t [V j,m (θ), U j,m+1 (θ)) in surplus perio, I j (θ), let F j,m (θ) be the (common) most recent time point at which the on-han inventory was S. Note that the F j,m (θ) are well-efine by the assumption I() S (though they may fall in the current or preceing surplus perios). Assumption 1 an 3 imply that each F j,m (θ) is a locally continuouslyifferentiable function of θ. From the efinitions above it follows that for all 1 j an 1 m M j (θ), Vj,m (θ) U j,m (θ) I (θ, t) t, (23) while Eq. (9) implies for all 1 j an 1 m M j (θ), I(θ,t) S X(θ,t) >, t (V j,m (θ), U j,m+1 (θ)). (24) Furthermore, Eq. (11) implies for all 1 j an 1 m M j (θ), X(θ, t) t + α(t) µ(θ, t), t (F j,m (θ), U j,m+1 (θ)), (25) whence, L I (θ) 1 M 1 (θ) [U 1,m+1 (θ) V 1,m (θ)] 2 2T + 1 2T m1 {[U j,1 (θ) F j,1 (θ)] 2 [ F j,1 (θ)] 2 j2 + M j (θ) [U j,m+1 (θ) V j,m (θ)] 2 }. (27) m1 Proof. From Eqs. (4), we can write L I (θ) 1 T I(θ,t)t, (28) since is locally inepenent of θ. Differentiating each term in Eq. (28) with respect to θ yiels, I(θ,t)t I(θ,) + I(θ,H j (θ)) H j (θ) Hj (θ) I (θ, t) + t I (θ, t) t, j 1,...,, (29) an the proof alreay appears in Proposition 1. Furthermore, from Eq.(23), we can rewrite Eq. (29) for each j 1,..., as I(θ,t)t M j (θ) m Uj,m+1 (θ) V j,m (θ) I (θ, t) t, (3) since the M j (θ) is locally inepenent of θ. Next, ifferentiate Eq. (24), an substitute Eq. (25) into the result. In view of Eq. (26) we can now euce for every t (V j,m (θ), U j,m+1 (θ)) the representation, X(θ,t) X(θ,F j,m (θ)) + F j,m (θ) [α(τ) µ(θ, τ)] τ [α(τ) µ(θ, τ)] τ, t (F j,m (θ), U j,m+1 (θ)), F j,m (θ) (26) since X(θ,F j,m (θ)). I (θ, t) X(θ, t) [α(τ) µ(θ, τ)] τ F j,m (θ) [α(f j,m + (θ)) µ(θ, F j,m + (θ))] F j,m(θ) + τ t F j,m (θ). (31) F j,m (θ)

7 To see that, note that only two cases are possible for α(t) µ(θ, t): (i) either it jumps at t F j,m (θ) (implying F j,m (θ)/, since the jumps of both {α(t)} an {µ(t)} are locally inepenent of θ), or (ii) it crosses zero continuously at t F j,m (θ) (implying α(f j,m + (θ)) µ(θ, F j,m + (θ)) α(f j,m(θ)) µ(θ, F j,m (θ)) ). Either way, the first term on the thir right-han equality of (31) vanishes, an the rest follows from Eq. (22). Next, substituting Eq. (31) into Eq. (3), we can now write for j 1,...,, I (θ, t) M j (θ) m t M j (θ) m Uj,m+1 (θ) V j,m (θ) { 1 2 [U 2 j,m+1 (θ) V 2 j,m (θ)] [t F j,m (θ)] t F j,m (θ)[u j,m+1 (θ) V j,m (θ)]}. (32) Eq. (27) now follows by substituting Eq. (32) into Eq. (29), an then substituting Eq. (29) into Eq. (28), an noting the ientities V 1, U 1,1 G 1 (inepenent of θ), an F j,m (θ) V j,m (θ) for all j 1,..., an m 2,...,M j (θ). We next erive L B (θ). Forθ, the shortage perios, B k (θ) [H k (θ), G k+1 (θ)), generally epen on θ for all k 1,...,K(θ). By Assumption 1 an 3, its en points are locally ifferentiable functions with respect to θ. For each shortage perio, B k (θ), let F k (θ) be the most recent time point at which the on-han inventory was S. The existence of the F k (θ) follows from the initial conition I() S. In view of Assumption 1 an 3, F k (θ) is a locally continuouslyifferentiable function of θ. Observe that F k (θ) may or may not belong to the immeiately preceing surplus perio. Eq.(1) implies for all 1 k K(θ), B(θ,t) X(θ,t) S >, t (H k (θ), G k+1 (θ)). (33) Furthermore, Eq. (11) implies for all 1 k K(θ), X(θ, t) t + α(t) µ(θ, t), t ( F k (θ), G k+1 (θ)), (34) whence, since X(θ, F k (θ)), X(θ,t) X(θ, F k (θ)) + F k (θ) F k (θ) [α(τ) µ(θ, τ)] τ [α(τ) µ(θ, τ)] τ, t ( F k (θ), G k+1 (θ)). (35) Proposition 5 L B (θ) 1 2T K(θ) For every θ, {[G k+1 (θ) F k (θ)] 2 [H k (θ) F k (θ)] 2 }. (36) Proof. From Eq. (5) we obtain by ifferentiation with respect to θ, L B (θ) 1 T K(θ) H k (θ) B(θ,t)t, (37) since K(θ) is locally inepenent of θ as a consequence of Assumption 1 an 3. Differentiating each term in Eq. (37) with respect to θ yiels, B(θ,t)t B(θ,H k (θ)) H k(θ) H k (θ) + B(θ,G k+1 (θ)) G k+1(θ) B(θ, t) + t H k (θ) B(θ, t) t, k 1,...,K(θ). (38) H k (θ) To see that, note that if B(θ,T ), then by assumption, B(θ,H k (θ)) B(θ,G k+1 (θ)) for all k 1,...,K(θ), while if B(θ,T ) >, then G K(θ)+1(θ) because G K(θ)+1 (θ) T is locally inepenent of θ. Next, ifferentiate Eq. (33), an substitute Eq. (34) into the result. In view of Eq. (35), we can now euce for every t (H k (θ), G k+1 (θ)) the representation, B(θ, t) X(θ, t) [α(τ) µ(θ, τ)] τ F k (θ) [α( F k + (θ)) µ(θ, F k + (θ))] F k (θ) F k (θ) τ F k (θ) t. (39) To see that, note that only two cases are possible for [α(t) µ(θ, t)]: (i) either it jumps at t F k (θ) (implying F k (θ)/, since the jumps of both {α(t)} an {µ(t)} are locally inepenent of θ), or (ii) it crosses zero continuously at t F k (θ) (implying α( F + k (θ)) µ(θ, F + k (θ)) α( F k (θ)) µ(θ, F k (θ)) ). Either way, the first term on the thir right-han equality of (39) vanishes, an the rest follows from Eq. (22).

8 Next, substituting Eq. (39) into Eq. (38), we can now write for k 1,...,K(θ), H k (θ) K(θ) B(θ,t)t H k (θ) { F k (θ)[g k+1 (θ) H k (θ)] [ F k (θ) t] t 1 2 [G2 k+1 (θ) H k 2 (θ)]}. (4) Eq. (36) now follows by substituting Eq. (4) into Eq. (37). Proposition 6 Uner Assumption 1 an 3, the IPA erivatives L I (θ) an L B (θ) are unbiase. Proof. Conition (a) of Lemma 1 is satisfie by part (b) of Assumption 3, so it remains to prove Conition (b) of Lemma 1. Observe that Eq. (11) is a special case (for an unlimite buffer capacity) of Eq. (2.1) in Wari an Melame (21). Since our initial conition is the same as that of Wari an Melame (21), Proposition 3.2 (ibi.) implies that for any θ,θ + θ, X(θ + θ, t) X(θ,t) θ τ θ t. (41) From (41), we reaily obtain by appeal to Eq. (9), I(θ + θ, t) I(θ,t) [S X(θ + θ, t)] + [S X(θ,t)] + max{ X(θ + θ, τ) X(θ,τ) : τ t} θ t, (42) an similarly, Finally, by Eq. (42), B(θ + θ, t) B(θ,t) θ t. (43) L I (θ + θ) L I (θ) 1 T [I(θ + θ, t) I(θ,t)] t 1 T 1 T an by Eq. (43), I(θ + θ, t) I(θ,t) t θ t t T θ, (44) 2 L B (θ + θ) L B (θ) T θ. (45) 2 4 CONCLUSIONS We have erive IPA graient formulas for the MTS system with backorers in SFM setting for the time average inventory level an backorers level with respect to the base-stock level an a parameter of the prouction rate process. The IPA graients erive are unbiase, nonparametric an their formulas are easy to compute an intuitive. The methoology employe in this paper hols out the promise of generalizations an extensions. First, in orer to implement IPA-base control schemes, the formulas nee to be generalize to arbitrary initial conitions. This is so, because the system may potentially be in any state when a control action is applie, at which point the IPA graient computation nees to be restarte from that (general) state. Secon, control schemes can be evise an reaily implemente, base on both system state an IPA graient information. Finally, our methoology can be extene to MTS systems with lost sales an beyon. Future extensions an generalizations will be reporte elsewhere. ACKNOWLEDGMENTS This research was one uner the auspices of the Rutgers Center of Supply Chain Management at the Rutgers Business School - Newark an New Brunswick. REFERENCES Buzacott, J., S. Price, an J. Shanthikumar Service level in multi-stage MRP an base-stock controlle prouction systems. In New Directions for Operations Research in Manufacturing, es. G. Fanel, T. Gullege, an A. Hones, Springer, Berlin, Germany. Caramanis, M., an G. Liberopoulos Perturbation analysis for the esign of flexible manufacturing system flow controllers. Operations Research 4: Fu, M Sample Path Derivatives for (s,s) Inventory Systems. Operations Research 42: Glasserman P Graient estimation via perturbation analysis. Kluwer Acaemic Publishers, Boston, Massachusetts. Glasserman P., an S. Tayur Sensitivity analysis for base-stock levels in multiechelon prouction-inventory systems. Management Science 41: Heielberger, H., X.R. Cao, M. Zazanis, an R. Suri Convergence properties of infinitesimal analysis estimates. Management Science 34: Ho, Y. C., an X.R. Cao Perturbation Analysis of Discrete Event Simulation. Kluwer Acaemic Publishers, Boston, Massachusetts. Karmarkar, U Lot sizes, lea-times an in-process inventories. Management Science 33:

9 Paschaliis, I., Y. Liu, C.G. Cassanras, an C. Panayiotou. 24. Inventory Control for Supply Chains with Service Level Constraints: A Synergy between Large Deviations an Perturbation Analysis. Annals of Operations Research 126: Rubinstein, R.Y., an A. Shapiro Discrete Event Systems: Sensitivity Analysis an Stochastic Optimization by the Score Function Metho. John Wiley an Sons, New York, New York. Veatch, M.H. 22. Using flui solutions in ynamic scheuling. In Analysis an moeling of manufacturing systems (S. B. Gershwin, Y. Dallery, C. T. Papaopoulos, J. M. Smith, es.) , Kluwer, New York. Wari, Y., an B. Melame. 21. Variational bouns an sensitivity analysis of traffic processes in continuous flow moels. Journal of Discrete Event Dynamic Systems 11: Wari, Y., B. Melame, C.G. Cassanras, an C.G. Panayiotou. 22. On-line IPA graient estimators in stochastic continuous flui moels. Journal of Optimization Theory an Applications 115(2): Wein, L. M Dynamic scheuling of a multiclass Make-to-Stock queue. Operations Research 4: Zipkin, P Moels for esign an control of stochastic, multi-item batch prouction systems. Operations Research 34: visual moeling environments. Melame was aware an AT&T Fellow in 1988, an an IEEE Fellow in He became an IFIP WG7.3 member in 1997, an was electe to Beta Gamma Sigma in His aress is <melame@rbs.rutgers.eu> an his web page is < AUTHOR BIOGRAPHIES YAO ZHAO is an assistant professor at the Rutgers Business School, Department of Management Science an Information Systems. In 22 he receive a Ph.D. egree from the Department of Inustrial Engineering an Management Science at Northwestern University. He is a member of INFORMS. His aress is <yaozhao@anromea.rutgers.eu> an his web page is < BENJAMIN MELAMED is a Professor at the Rutgers Business School, Department of Management Science an Information Systems, an a member of RUTCOR (Rutgers Center for Operations Research) an the Rutgers Center for Supply Chain Management. Before joining Rutgers University, Melame taught at the Department of Inustrial Engineering an Management Science at Northwestern University, an was an AT&T Fellow at Bell Laboratories, following which he serve at NEC as Deputy Director, Hea of the Performance Analysis Department, an NEC Fellow. Melame s research interests inclue system moeling an analysis (especially telecommunications systems an traffic moeling), simulation, stochastic processes an