Optimal Policies and Approximations for A Serial. Multi-echelon Inventory System with. Time-correlated Demand

Transcription

1 Optimal Policies and Approximations for A Serial Multi-echelon Inventory System with Time-correlated Demand Lingxiu Dong John M. Olin School of Business Washington University St. Louis, MO dong@olin.wustl.edu Hau L. Lee Graduate School of Business Stanford University Stanford, CA haulee@leland.stanford.edu August 2000 Revision: November 2001 Subject classification: Inventory/production: approximations, multi-echelon, stochastic.

2 Abstract Since Clark and Scarf s pioneering work, most advances in multi-echelon inventory systems have been based on demand processes that are time-independent. This paper revisits the serial multi-echelon inventory system of Clark and Scarf, and develops three key results. First, we provide a simple lower bound approximation to the optimal echelon inventory levels and an upper bound to the total system cost for the basic Clark and Scarf s model. Second, we show that the structure of the optimal stocking policy of Clark and Scarf holds under time-correlated demand processes using Martingale model of forecast evolution. Third, we extend the approximation to the time-correlated demand process, and study in particular for an auto-regressive demand model the impact of leadtimes and auto-correlation on the performance of the serial inventory system.

3 1 Introduction In a seminal paper, Clark and Scarf (1960) study a serial multi-echelon inventory system where demand occurs at the lowest echelon, and show that, when there is no fixed ordering cost in the serial system, the optimal inventory control of the overall system follows a base-stock policy for every echelon. The base-stock levels at each echelon can be solved from a series of single location inventory problems with appropriately defined penalty functions for not having enough inventory to bring the downstream site to its target base-stock level. Extensions of the Clark and Scarf approach have been developed for other production/inventory systems. Schmidt and Nahmias (1985) use the Clark and Scarf decomposition technique to characterize the optimal policy for an assembly system with two components. Rosling (1989) shows, under appropriate assumptions, that a general assembly system is equivalent to a serial system and thus Clark and Scarf s results apply. Axäter and Rosling (1993) establish the condition under which the echelon and installation based policies are equivalent. Chen (2000) generalizes the Clark and Scarf result to the serial systems and assembly systems where the material flow from one level to another has to be in batches (e.g., truck loads). Iida(2000) studies error bounds of near myopic policies in a multi-echelon system. Lee and Whang (1999) show that a performance measurement scheme based on Clark and Scarf s penalty function exists that enables a decentralized serial inventory system to achieve the optimal performance of a centralized system.

4 Although the optimal inventory policy follows a simple form of a base-stock inventory policy, the computation of the optimal base-stock levels can be quite complicated due to the complexity of the induced penalty cost functions. Federgruen and Zipkin (1984) extend Clark and Scarf s results to the infinite horizon case, and show that the computation can be much simplified if one minimizes discounted cost or long-term average cost. For a two-echelon system with Normal demand distribution, they provide a closed-form equation for solving the optimal inventory levels and a closed-form expression for the corresponding optimal cost. Similar calculations can be carried over to a system with more than two echelons, but the complexity of the computation increases considerably. Moreover, even with a closed-form expression, the analysis of the system inventory performance is still quite tedious and often has to rely on numerical methods. Gallego and Zipkin (1999) develop several heuristic methods and conduct numerical studies on system performance sensitivity to the stock positioning under different assumptions of inventory holding costs. Shang and Song (2001) extend the restriction-decomposition heuristic in Gallego and Zipkin to the echelon-stock setting under linear holding cost and give newsvendor type upper and lower bounds to the optimal echelon policies. Most of the research work on multi-echelon inventory systems has been based on stationary demand processes that are independent over time. Although empirically, time-correlated demands are commonly observed (e.g. see Erkip et al., 1990 and Lee et al., 1999), there has been only limited work with time-correlated demand processes. Chen and Zheng (1994) construct a simple proof of Clark and Scarf s 2

5 resultsinaninfinite-horizon case by finding a lower bound on the system-wide cost and constructing a feasible policy to achieve such a lower bound. Using this lowerbounding approach, Chen and Song (2001) relax the independent demand assumption in the classic model and consider a demand process whose demand distribution is determined by the state of an underlying finite-state Markov chain. They show that echelon base-stock policies are still optimal when order-up-to levels are adjusted based on the state of the underlying Markov chain, and provide an algorithm to compute the optimal base-stock levels. For time-series based demand processes, the martingale model of forecast evolution (MMFE) developed by Graves et al. (1986, 1998) and Heath and Jackson (1994) offers a powerful descriptive framework that incorporates both past demands and other influential factors to characterize the forecast processes. Readers are referred to Heath and Jackson for an excellent discussion of motivation of MMFE, to Toktay and Wein (2001) for a review on researches (Güllü, 1996) built upon it. A common demand process used in the literature is the order-one auto-regressive, or AR(1), process with positive auto-correlation (see, for example, Johnson and Thompson, 1975; Lovejoy, 1990&1992; Fotopoulos and Wang, 1988; Reyman, 1989; Scarf, 1959&1960; Miller, 1986; Kahn, 1987; Erkip et al., 1990; Lee et al., 1997; and Lee et al., 1999). In fact AR(1) process with minimum mean-square error forecast is a special case of MMFE. More complex time series demand processes have also been modeled more recently, such as the random walk model of Graves (1999) and Lee and Whang (1998). 3

6 This paper develops three key results for the Clark and Scarf model. First, we develop a simple approximation of the optimal echelon inventory stocking levels that is easy to compute for serial multi-echelon systems with more than two echelons. This approximation provides a good lower bound to the optimal stocking levels for the system. This approximation applies to any convex inventory holding and penalty cost functions, while in the linear function case, the lower bound is same as that of Shang and Song (2001). Second, we show that Clark and Scarf s results can be carried over to the time-correlated demand process using MMFE, but with greater computation complexity compared to its counterpart in the independent demand environment. Finally, we extend the approximation in the independent demand case to the timecorrelated demand process and show that the approximation again provides a lower bound to the optimal stocking levels. The approximation is used in particular for the AR(1) case to explore the role of the cross-time demand correlation in affecting the performance of the system. The remainder of this paper is organized as follows: in section 2 we introduce the serial inventory system; we then present the approximation of optimal inventory levels under an i.i.d. demand process in section 3; in section 4 we establish Clark and Scarf s results under time-correlated demand processes, present the approximation, and illustrate the impacts of the echelon leadtimes and demand auto-correlation on the system performance; in section 5 we show through numerical examples that the approximations in sections 3 and 4 are very accurate and are useful analytical tools of unveiling the fundamental relationship between system parameters and the inventory 4

7 cost; we conclude our findings in section 6. 2 The Serial Multi-echelon System Consider an M-echelon serial periodic-review inventory system where customer demand arises from echelon 1 and orders are placed by echelon 1 to echelon 2, by echelon 2toechelon3,...,andechelonM is the highest echelon that orders from an external supplier with infinite supply. The event sequence is as follows: (a) at the beginning of a period, shipments due in this period arrive, followed by demand occurrence; (b) demand is satisfied from on-hand inventory with complete backlogging; (c) at the end of the period, inventory holding and shortage costs are charged against stocks and/or shortages followed by ordering and shipment decisions. The shipment leadtime from echelon m +1to echelon m is T m, i.e., an order placed by echelon m at the end of period t will arrive at echelon m at the beginning of period t + T m +1.Suchanevent sequence, different from that of Clark and Scarf but the same as that of Federgruen and Zipkin (1984) and Lee and Whang (1999), makes the exposition easier. The results of this paper hold for other sequences of events with appropriate adjustments. Assume that there are no ordering setup costs at all sites. The following notation will be used: ξ : demand in a period, assumed to be i.i.d. in section 3; u m : on-hand inventory level at site m at the end of a period, just before ordering decisions are made; 5

8 w m : vector of amounts due to arrive from site m+1 to site m in future periods (the dimension of the vector equals T m,andtheith element, w i m, is the amount due to arrive in i periods); w 0 m : vector w m without the first element; e m : column vector of ones with dimension of T m ; x m : echelon inventory position at level m at the end of a period, just before ordering decisions are made; x m = v m + w m e m ; v m : echelon inventory at level m at the end of a period, just before ordering decisions are made, i.e., v m = x m 1 + u m ; y m : echelon inventory position at level m at the end of a period, after ordering decisions are made; c m : unit shipment or processing cost from site m +1to site m; L m (v m ):holding and shortage cost in a period at level m, given that the echelon inventory at the end of a period is v m ; α : discount factor per period; v : column vector (v m ) M m=1 ; x : column vector (x m ) M m=1 ; y : column vector (y m ) M m=1 ; 6

9 W : acollectionofcolumnvectorsgivenby(w m ) M m=1 ; W 0 : a collection of column vectors given by (w 0 m) M m=1 ; (W 0, z) :a collection of column vectors given by any vector of dimension M. ³ w 0 m z m M m=1,wherez =(z m ) M m=1,is Let C (v, W) be the system minimum expected discounted cost, given initial echelon inventory levels v and shipments W. The overall inventory control problem can be stated as: C (v, W) (1) = min y:x m y m v m+1 nx M m=1 [c m (y m x m )+L m (v m )] + αe C v + w 1 ξ, (W 0, y x) o, where v M+1 =. Clark and Scarf show that the optimal inventory policy for the overall system follows an echelon base-stock policy at each echelon. Let S m be the optimal orderup-to level at echelon m. Hence, at the end of each period, echelon m places an order to its upstream echelon to bring its echelon inventory position up to Sm.The actual amount that gets shipped depends on the on-hand inventory at the upstream site. Clark and Scarf s results state that we can decompose (1) into a series of singleechelon inventory control problems with the proper penalty cost function Γ m added to the original holding and shortage cost expression, with Sm being the respective solution for echelon m. We state this result explicitly as follows: C (v, W) = X M f m (v m, w m ), (2) m=1 7

10 where f m (v m, w m ) = min cm (y x m )+L m (v m )+Γ m (v m )+αe f m vm + wm 1 ξ, (w y:x m m,y 0 x m ) ª. y Define γ (j), a random variable, as the sum of demands in j periods, where γ (0) = 0. Hence γ (1) = ξ. Since the decision made at the end of period t will not affect the system cost until period t + T m +1,weconsolidatethestatespaceandrewritethe function f m (v m, w m ): f m (v m, w m ) (3) = L m (v m )+Γ m (v m )+αe L m vm + wm 1 ξ + Γ m vm + wm 1 ξ +α 2 E L m vm + wm 1 + wm 2 γ (2) + Γ m vm + wm 1 + wm 2 γ (2) α T m E [L m (v m + w m e m γ (T m )) + Γ m (v m + w m e m γ (T m ))] +g m (x m ), where g m (x m ) (4) = min y:x m y {c m (y x m )+α T m+1 E [L m (y γ (T m +1))+Γ m (y γ (T m + 1))] +αe [g m (y ξ)]}. We interpret the function f m (v m, w m ) as the adjusted (through the penalty function Γ m (v m )) discounted expected cost for echelon m, given the current state of v m and w m.form =1, the penalty function is defined as Γ 1 (v 1 )=0for all values of v 1.For 8

11 m 1, the penalty function is defined as: Γ m+1 (v m+1 ) (5) c m (v m+1 Sm)+α Tm+1 E [L m (v m+1 γ (T m +1))+Γ m (v m+1 γ (T m + 1))] = α T m+1 E [L m (S m γ (T m +1))+Γ m (S m γ (T m + 1))] +αe [g m (v m+1 ξ) g m (S m ξ)] if v m+1 S m; 0 otherwise. For computation purpose, the cost function L m ( ) has to be explicitly specified. We follow the linear cost specification used by Federgruen and Zipkin (1984) and Lee and Whang (1999). Let H m = per-period unit holding cost for inventory at site m, or in transit to site m 1. π = per-period unit backorder cost at site 1. h m = H m H m+1, m =1,..., M, whereh M+1 =0. h m is referred to as the echelon inventory holding cost at level m. We will specify the holding and shortage costs associated with echelons as: L m (v m )=h m v m, m>1; (6) L 1 (v 1 )=h 1 v 1 +(π + H 1 ) v 1 ; (7) where x =( x) + and x + =max(0,x). 9

12 Lee and Whang (1999) show that such an echelon cost definition fully allocates the total system cost to all sites, i.e., H 1 v πv1 + X M H m (u m + w m 1 e m 1 )= X M L m (v m ). m=2 m=1 3 An Approximation of the IID Demand Model Federgruen and Zipkin develop a closed-form solution for a two-echelon system when inventory holding and shortage cost is linear and the end demand follows a Normal distribution. However, generalizing their results to serial multi-echelon systems with more than two echelons involves calculating multi-variate Normal distribution functions and is computationally challenging. Since the difficulty of calculating optimal stocking level Sm is caused by the complexity of the penalty function Γ m ( ), weproceed to develop an approximation for Γ m ( ), denoted by bγ m ( ). Define: bγ 1 ( ) =0; and, for m 1, bγ m+1 (v m+1 ) (8) ³ h i = c m v m+1 bs m + α Tm+1 E L m (v m+1 γ (T m +1))+bΓ m (v m+1 γ (T m +1)) α T m+1 E +αe h ³ ³ i L m S bm γ (T m +1) + bγ m S bm γ (T m +1) h ³ i bg m (v m+1 ξ) bg bsm m ξ, 10

13 where bs m is the solution of bg m (x m ) (9) h i = min {c m (y x m )+α Tm+1 E L m (y γ (T m +1))+bΓ m (y γ (T m +1)) y:x m y +αe [bg m (y ξ)]}. Hence, we have the following approximation for f m (v m, w m ): bf m (v m, w m ) (10) = L m (v m )+bγ m (v m )+αe hl m vm + w 1m ξ + bγ m vm + w 1m ξ i +α 2 E hl m vm + w 1m + w 2m γ (2) + Γ b m vm + w 1m + w 2m γ (2) i α Tm E +bg m (x m ). h i L m (v m + w m e m γ (T m )) + bγ m (v m + w m e m γ (T m )) Functions bγ m, bg m,andf b m together constitute an approximate formulation of the inventory control of the multi-echelon system. Although the approximation is still inductive, i.e., for echelon m+1, the definition of bγ m+1 requires the approximate lower echelon target stocking level, S b m, whose calculation involves Γ b m, the computation is greatly simplified. Using this approximate penalty function, echelon m+1 is penalized even when it can fulfill the order from echelon m, i.e., echelon m +1is penalized for overstocking! Theorem 1 states an intuitive result: the stocking level bs m derived from the approximate inventory control system is a lower bound to the optimal stocking level Sm for each echelon m, m =1,...,M. The proofs of Theorem 1 and others are given in the Appendix. 11

14 Theorem 1 Let bs m be the solution of (9), then bs m S m, m =1,..., M. It turns out that S b m can be computed easily for the specific costfunctions L m ( ) of (6) and (7), and an approximate cost function can be derived in closedform. Define: τ (i, j) =(T i +1)+(T i+1 +1)+... +(T j +1)for i j, and0 otherwise. Φ ( ; t), Φ ( ; t) : the cumulative distribution function (CDF) and the complementary CDF of demand in t periods, respectively. Θ (x) =xφ (x)+φ (x), whereφ ( ) and φ ( ) are the CDF and the probability density function (PDF) of the standard Normal distribution, respectively. µ (t), σ (t) :mean and standard deviation of demand in t periods. Theorem 2 For m =1, 2,...,M, bs m satisfies: ³ Φ bsm ; τ (1,m) = K m, where ³X m K m = i=1 α τ(1,i) (1 α) c i + α Ti+1 h i / (π + H 1 ). Theorem 2 thus gives us a very simple, non-inductive way to calculate bs m. Let ba α m be the cost per period at echelon m by following the order-up-to bs m policy, calculated with the approximate induced penalty cost function bγ m and zero initial inventory. Let ba α be the corresponding system cost per period, i.e., ba α = P M m=1 b A α m. 12

15 Lemma 1 h i ba α 1 = (1 α) c 1 bs 1 + αc 1 µ + α T1+1 h 1 S b1 µ (T 1 +1) Z ³ +α T1+1 (π + H 1 ) t bs 1 dφ (t; T 1 +1); t= bs 1 for m>1, h i ba α m = (1 α) c msm b + αc m µ + α Tm+1 h bsm m µ (T m +1) h ³ i +α Tm+1 E Γ bm S bm γ (T m +1). Lemma 2 bγ m (y) = X m 1 ³y bs m 1 i=1 ατ(i+1,m 1) (1 α) c i + α Ti+1 h i α τ(1,m 1) (π + H 1 ) Z y t= b S m 1 Φ (t; τ (1,m 1)) dt. Furthermore, if the end demand per period is Normal with mean µ and variance σ 2, then bγ m (y) = ³y S m 1 b X m 1 i=1 ατ(i+1,m 1) (1 α) c i + α Ti+1 h i +α τ(1,m 1) σ (τ (1,m 1)) (π + H 1 ) h ³ ³ Θ (ι (y)) Θ ι S bm 1 i, where ι (y) = [y µ (τ (1,m 1))] /σ (τ (1,m 1)). Suppose the end demand per period is Normal with mean µ and variance σ 2. Let ba m betheapproximateaveragecostperperiod,i.e.,α 1, forechelonm when the cost function are given by (8), (9), and (10) and when the stocking levels bs ms 0 are 13

16 followed. Let ba bethecorrespondingaveragecostperperiodforthewholesystem, i.e., b A = P M m=1 b A m. Lemma 3 With Normal demands, ba 1 = c 1 µ + h 1 Φ 1 (1 K 1 ) σ (T 1 +1)+(π + H 1 ) σ (T 1 +1)Θ Φ 1 (1 K 1 ) ; for m>1, ba m = c m µ + h m µ (τ (1,m 1)) +σ (τ (1,m)) (H 1 H m+1 ) Φ 1 (1 K m )+(π + H 1 ) Θ Φ 1 (1 K m ) σ (τ (1,m 1)) (H 1 H m ) Φ 1 (1 K m 1 )+(π + H 1 ) Θ Φ 1 (1 K m 1 ). n o M The telescoping structure of series bam m=1 leads to: Theorem 3 With Normal demands, ba = µ X M m + X M mµ (τ (1,m 1)) + σ (τ (1,M)) (π + H 1 ) φ Φ 1 (1 K M ) m=1 m=1 = µ X M m + X M 1 m+1µ (T m +1)+σ (τ (1,M)) (π + H 1 ) φ Φ 1 (1 K M ). m=1 m=1 Corollary 1 ba/ T m > 0 and ba/ T m ba/ T m+1. Theorem 3 provides a closed-form expression of the approximate system cost. Thesystemcostcanbeviewedasconsistingofthreeparts:(1)theaverageshipping/processing cost; (2) the average inventory holding cost; and (3) the safety stock cost caused by the demand randomness within the total system-wide leadtime τ (1,M). Corollary 1 illustrates the role that leadtimes play in affecting the system 14

17 cost. In general, longer leadtime leads to higher cost. Since holding inventory is more expensive at the downstream sites, reducing lower-echelon leadtimes would have a bigger impact to the system cost than reducing the higher-echelon leadtimes. We are now ready to examine the true cost performance of following the stocking levels of bs m in the multi-echelon system. Let A α m denote the echelon-m cost per period of following the order-up-to S b m policy, calculated with the true h i penalty cost Γ m.notethata α m =(1 α) c m bs m +αc m µ+α Tm+1 h m S bm µ (T m +1) h ³ i + α Tm+1 E Γ m S bm γ (T m +1). Denote A α as the corresponding true systemwide per-period cost for policy bs m. For a two-echelon system, we were able to derive the cost dominance relationship between ba α and A α : Corollary 2 For a two-echelon system:(1) b A α 2 A α 2 ;(2) b A α A α. For the two-echelon case with α 1, i.e., the average cost case, we see that ba provides an upper bound for the true cost of implementing the approximate policy bs m. This is helpful, since the average cost ba is easily computable, as shown in Theorem 3. In section 5, we will show through numerical examples that both bs m and ba α are very accurate approximations of the optimal stocking levels and the optimal system cost. 4 TheTime-correlatedDemandModel When demands are correlated across time periods, the optimal control of the inventory system will be based on the demand forecasts which are revised from period to period 15

18 as information accumulated over time. Let D t be the end-customer demand in period t. Let D t,t+i be the forecast made in period t fordemandinperiodt + i. Hence we can define a vector of demand forecasts made in period t for future periods, D t =(D t,t,d t,t+1,...), withd t,t = D t as the demand realized in period t. Then ² t,t+i = D t 1,t+i D t,t+i, represents the forecast update made for period t + i from period t 1 to t, i = 0, 1,..., and² t =(² t,t+i ) i=0 represents the forecast update vector in period t. We adopt the MMFE developed by Heath and Jackson (1994), which treats forecasts as the conditional expectation of the future demands on the current information set and assumes that the demand random variables form a martingale relative to the corresponding information set. We make the following assumptions explicitly: (1) ² t vectors are i.i.d. multi-variate Normal random vectors with mean 0; (2) there are only finite number of random variables in ² t that are linearly independent, i.e., there exists a set of finite number of random variables in ² t such that each ² t,t+i can be written as a linear combination of the random variables in the set. Assumption (1), essential to MMFE, requires the demand process to be stationary and forecasts to be unbiased. Assumption (2) implies that only finite number of uncertainty factors of the information set affect the forecasts. MMFE is indeed a powerful model of forecast process in that it captures both historical demands and potential information available through other sources in one 16

19 framework (see Aviv, 2001a, for an exploration of MMFE). In fact, an ARMA (autoregressive and moving average) process with minimum mean-square error forecast is a special case of MMFE. To illustrate, we take a look at an AR(1) demand process D t = d + ρd t 1 + ε t, with d>0, 1 < ρ < 1, andε t being i.i.d. Normal with mean zero and variance σ 2 (σ d). Here ρ is the auto-correlation coefficient of demand in two consecutive periods. In this case, D t,t+i = d P i 1 j=0 ρj + ρ i D t, ² t,t+i = ρ i ε t, and clearly ² t,t+i is linearly dependent on ε t. Theorem 4 For a time-correlated demand process that satisfies assumptions (1) and (2), a state-dependent base-stock policy {S (D)} is optimal for a single-echelon inventory system, where the state D is the demand forecast made before making the ordering decision. Let C (v, W, D 0 ) be the minimum expected discounted cost for the serial inventory system, given the forecast D 0 made in the current period, the initial inventory levels v and shipments W. The overall inventory control problem can be stated as C (v, W, D 0 ) (11) = min { X M [c m (y m x m )+L m (v m )] y:x m y m v m+1 m=1 +αe D0 C v + w 1 D 1, (W 0, y x), D 1 }, where v M+1 =. The expectation in (11) is taken over the next period s forecast D 1, but conditional on the demand forecast D 0 in the current period. In the rest 17

20 of this section, we will drop D 0 in the conditional expectation for ease of exposition whenever it is unambiguous. Define D (i, j) as a random variable representing demand in j periods that starts from period i, i.e., the cumulative demand in time interval [i, i + j 1]. Similar to the i.i.d. demand case, we define the adjusted discounted expected cost functions, f m (v m, w m, D 0 ) and g m (x m, D 0 ), and the corresponding penalty cost function Γ m (v m,d 0 ) for echelon m, given that the current period forecast is D 0. f m (v m, w m, D 0 ) (12) = min y:x m y {c m (y x m )+L m (v m )+Γ m (v m,d 0 ) +αe f m vm + w 1 m D 1, (w 0 m,y x m ), D 1 } = L m (v m )+Γ m (v m,d 0 ) +αe L m vm + w 1 m D 1 + Γm vm + w 1 m D 1,D α Tm E [L m (x m D (1,T m )) + Γ m (x m D (1,T m ),D Tm )] + g m (x m, D 0 ), g m (x m, D 0 ) (13) = min y:x m y {c m (y x m ) +α T m+1 E [L m (y D (1,T m +1))+Γ m (y D (1,T m +1),D Tm +1)] +αe [g m (y D 1, D 1 )]}. 18

21 For m =1, Γ 1 (, ) =0;form 1, Γ m+1 (v m+1,d 0 ) c m [v m+1 S m (D 0 )] (14) = +α T m+1 E [L m (v m+1 D (1,T m +1))+Γ m (v m+1 D (1,T m +1),D Tm +1)] α Tm+1 E [L m (S m (D 0 ) D (1,T m +1))+Γ m (S m (D 0 ) D (1,T m +1),D Tm+1)] +αe [g m (v m+1 D 1, D 1 ) g m (S m (D 0 ) D 1, D 1 )] if v m+1 S m (D 0 ) ; 0 otherwise; where Sm (D 0 ) is the solution of (13). Under the assumptions (1) and (2), the dynamic programming formulation of the inventory problem differs from the i.i.d. demand case in that more dimensions are added to the state space, namely the forecasts for future periods, and that the conditional expectation is used. The following theorem shows that the key result of Clark and Scarf for i.i.d. demand system, i.e., the property of echelon decomposition for the optimal policy using the induced echelon penalty cost, remains true in the time-correlated demand setting. Theorem 5 C (v, W, D 0 )= P M m=1 f m (v m, w m, D 0 ). 19

22 We again define the approximate penalty cost function bγ m, similar to the previous section. Define: b Γ 1 (,D 0 )=0;and,form 1, bγ m+1 (v m+1,d 0 ) = c m hv m+1 S b i m (D 0 ) +α Tm+1 E α T m+1 E +αe h i L m (v m+1 D (1,T m +1))+bΓ m (v m+1 D (1,T m +1),D Tm +1) (15) h ³ ³ i L m S bm (D 0 ) D (1,T m +1) + bγ m S bm (D 0 ) D (1,T m +1),D Tm +1 hbg m (v m+1 D 1, D 1 ) bg m ³ bsm (D 0 ) D 1, D 1 i, where b S m (D 0 ) is the solution to: bg m (x m, D 0 ) (16) h = min {c m (y x m )+α Tm+1 E L m (y D (1,T m +1))+ b i Γ m (y D (1,T m +1),D Tm +1) x m y +αe [bg m (y D 1, D 1 )]}. Also, bf m (v m, w m, D 0 ) = L m (v m )+ Γ b m (v m,d 0 )+αe hl m vm + w 1m D 1 + Γm b i vm + w 1m D 1,D 1 +α 2 E hl m vm + w 1m + w 2m D (1, 2) + bγ i m vm + w 1m + w 2m D (1, 2),D α T m E h i L m (v m + w m e m D (1,T m )) + bγ m (v m + w m e m D (1,T m ),D Tm ) +bg m (x m, D 0 ). When demand is time-correlated, it is possible that the physical inventory level at a site is higher than the target stocking level, such that the target inventory level 20

23 may not always be realizable. Indeed, it arises in inventory problems in other settings as well. For example, when demands are Normal, stock levels can exceed the target even when demands are i.i.d., since it is theoretically possible that a negative demand can occur. Another example is the classic paper by Eppen and Schrage (1981) where the optimal allocation of stocks among multiple sites may not be feasible due to stock imbalance. The traditional approach to this problem is to assume that the probability of such occurrence is negligible. In fact, when demand follows an AR(1) process with ρ > 0, the probability of excess inventory (overshoot) is smaller than the standard i.i.d. demand case (see Lemma 1 in Lee et al.,1999, Aviv, 2001b) and the myopic policy is quite an accurate approximation to the optimal policy. Another approach is to make the costless return assumption (see Lee et al., 1997). Specifically, at echelon m<m, if the physical echelon inventory level is higher than the target echelon inventory level, then the excessive inventory will not be charged as echelon m inventory, and echelon m gets a refund at the original purchasing price which is paid by an outside source (for example, a bank); however, the excessive inventory does not leave the system physically and is still available as inventory for the upper echelon, m +1, for future replenishment to echelon m; atechelonm, excessive inventory is returned to the outside supplier with a refund at the original purchasing cost. Under this assumption the myopic policy is indeed optimal. We will use such an assumption in the remainder of the analysis. Again the approximate inventory levels, which can be computed easily, are lower bounds of the optimal stocking levels. 21

24 Theorem 6 bs m (D) S m (D), m =1,..., M. We now study the AR(1) demand process. Since in any given period t, the forecasts of future periods are functions only of demand realization D t,weneedonly D t to represent forecasts made in period t. Given the current period s realized demand D 0,letΦ 0 ( ; t) and Φ 0 ( ; t) be the CDF and the complementary CDF of demand in the subsequent t periods whose mean and standard deviation are given by µ 0 (t) and σ (t) respectively. Theorem 7 For m =1, 2,...,M, b S m (D 0 ) satisfies: ³ Φ 0 S bm (D 0 );τ (1,m) = K m, where ³X m K m = i=1 α τ(1,i) (1 α) c i + α Ti+1 h i / (π + H 1 ). Hence bs m (D 0 )=µ 0 (τ (1,m)) + Φ 1 (1 K m ) σ (τ (1,m)). Let b A α m (D 0 ) be the cost per period at echelon m by following the order-upto bs m (D 0 ) policy and using bγ m as the penalty cost function. Let ba α (D 0 ) be the corresponding system-wide cost per period, i.e., b A α (D 0 )= P M m=1 b A α m (D 0 ). Lemma 4 h i ba α 1 (D 0 ) = (1 α) c 1S1 b (D 0 )+αc 1 µ 0 (1) + α T1+1 h bs1 1 (D 0 ) µ 0 (T 1 +1) Z h i +α T1+1 (π + H 1 ) t bs 1 (D 0 ) dφ 0 (t; T 1 +1); t= S b 1 (D 0 ) 22

25 for m>1, h i ba α m (D 0 ) = (1 α) c m bs m (D 0 )+αc m µ 0 (1) + α Tm+1 h m S bm (D 0 ) µ 0 (T m +1) h ³ i +α Tm+1 E bγm bsm (D 0 ) D (1,T m +1),D Tm+1. Lemma 5 bγ m (y, D 0 ) = ³ y S b X m 1 m 1 (D 0 ) i=1 ατ(i+1,m 1) (1 α) c i + α Ti+1 h i h ³ ³ +α τ(1,m 1) (π + H 1 ) σ (τ (1,m 1)) Θ (ι (y, D 0 )) Θ ι S bm 1 (D 0 ),D 0 i, where ι (x, D 0 )= [x µ 0 (τ (1,m 1))] /σ (τ (1,m 1)). As the discount factor α 1, the average system-wide cost per period b A (D 0 ) has the following closed-form expression: Theorem 8 ba (D 0 ) = µ 0 (1) X M m + X M m [µ 0 (τ (1,m)) µ 0 (T m +1)] m=1 m=1 +σ (τ (1,M)) φ Φ 1 (1 K M ) = µ 0 (1) X M m + X M me µ Tm +1 (τ (1,m 1)) m=1 m=1 +σ (τ (1,M)) (π + H 1 ) φ Φ 1 (1 K M ). where µ Tm+1 (τ (1,m 1)) represents the mean demand in the subsequent τ (1,m 1) periods starting from period T m +1. Corollary 3 ba (D 0 ) is nondecreasing in T 1,T 2,...,T M and ba (D 0 ) / T m ba (D 0 ) / T m+1. 23

26 The above lemmas and theorems share similar structures as those in the i.i.d. and Normal demand case. Indeed, when ρ is set to zero, Theorem 8 is the same as Theorem 3. We now further assume that demands across periods are positively correlated, i.e., ρ > 0. Erkip et al. (1990) and Lee et al. (1999) both report that positively correlated demands were commonly observed in industry. Theorem 9 If ρ > 0, then: (1) S b m (D 0 ) is nondecreasing in ρ and T 1,T 2,...,T m ; (2) ba (D 0 ) is nondecreasing in ρ and T 1,T 2,...,T M ; (3) S b m (D 0 ) / ρ is nondecreasing in T 1,T 2,..., T m and A b (D 0 ) / ρ is nondecreasing in T 1,T 2,...,T M. Hence, as expected, both the approximate stocking levels and the approximate average cost are higher with higher auto-correlated demands, or longer leadtimes, ceteris paribus. Moreover, the sensitivity of stocking levels and cost to the autocorrelation of demand is greater when leadtimes are long. This seems to imply that the interaction effect of demand auto-correlation and leadtime is significant. Similarly, let A α m (D 0 ) be the actual echelon-m cost using bs m (D 0 ) as the target stocking level, i.e., Γ m is used to calculate the penalty cost. Then: h i A α m (D 0 ) = (1 α) c msm b (D 0 )+αc m µ 0 (1) + α Tm+1 h bsm m (D 0 ) µ 0 (T m +1) +α Tm+1 E h ³ i Γ m S bm (D 0 ) D (1,T m +1),D Tm+1. 24

27 The corresponding system cost is A α (D 0 )= P M m=1 Aα m (D 0 ). For a two-echelon system, we have the following cost dominance relationship. Corollary 4 For a two-echelon system: (1) ba α 2 (D 0 ) A α 2 (D 0 ); (2) ba α (D 0 ) A α (D 0 ). 5 Numerical Examples In this section, we first present examples to show that the approximations developed in sections 3 and 4 are accurate for both the i.i.d. and AR(1) demand processes. We then illustrate how demand auto-correlation of the AR(1) process affects the decisions and performance of the system. In the i.i.d. demand scenario, the base case has the following parameters: the demand per period is Normal with mean µ =100and standard deviation σ =20; replenishment leadtimes, T 1 =2, T 2 =1; transportation or processing costs, c 1 =10, c 2 =5; inventory holding costs, H 1 =4, H 2 =2; site 1 shortage cost, π =10;perperiod discount factor, α =0.9. The true optimal stocking levels and system cost are calculated by the method given by Federgruen and Zipkin (1984) for a two-echelon system. Figure 1 shows the approximate and optimal inventory stocking levels under varying σ/µ. We observe, as stated in Theorem 1, that the approximate stocking level, bs 2, is a lower bound of the optimal stocking level, S2.Moreover,thedifference between the two levels is very small (within 3% of the optimal stocking level S2). 25

28 Figure 2 shows the comparison of three system costs: (1) ba α is the system cost using the approximate optimal stocking levels S, b calculated with the approximate penalty cost function bγ; (2)A α is the system cost using bs, but calculated with the penalty cost function Γ; (3)A α is the system cost using optimal stocking levels S,calculated with the true penalty cost Γ. As shown in Corollary 2, we observe that ba α A α. Moreover, the three cost curves are very close (within 1.01% of the optimal system cost A α ). Hence, Figure 2 shows that both ba α and A α are very good approximations of A α. Comparisons based on other parameter ranges, such as c 2 /c 1, H 1 /c 1, H 2 /c 2, π/c 1,andT 2 /T 1 (a total of 61 examples, see Table 1), all show the similar accuracy of the approximation. Figures 1 and 2 about here. To verify the accuracy of this approximation for systems with more than two echelons, we also run comparison for a three-echelon system. The three-echelon system parameters are set as: Normal distribution with mean µ =100and standard deviation σ =20; replenishment leadtimes, T 1 =2, T 2 =1, T 3 =1; transportation or processing costs, c 1 =10, c 2 =5, c 3 =2; inventory holding costs, H 1 =4, H 2 =2, H 3 =1; site 1 shortage cost π =10, per-period discount factor, α =1. Figures 3 and 4 show that the gap between our approximation and the optimal result is very small. In fact, comparisons based on other parameter ranges (a total of 59 examples, see Table 2) also shows similar accuracy (gaps are within 2.55% of the optimal stocking level S3 and 0.36% of the optimal system cost A, respectively). Hence, this approximation 26

29 is reasonably accurate for a three-echelon inventory system as well. Figures 3 and 4 about here. For the AR(1) demand process, the base case is the same as that of the i.i.d. case, with α =1and the AR(1) process specified as d =100, ρ =0.1, D 0 =10, and σ =20. We extended the method of Federgruen and Zipkin (1984) for the i.i.d. case to the AR(1) case to calculate the optimal stocking levels and system costs (see Dong, 1999, for details). Figure 5 shows the approximate and optimal stocking levels under varying σ/d. Again, the approximate stocking level bs 2 (D 0 ) is a lower bound of optimal S2 (D 0 ),andtheirdifference is very small (within 2.20% of A (D 0 )). Similarly, Figure 6 confirms that ba (D 0 ) A (D 0 ). The cost curves are very close to one another, with gaps that are within 1.12%. A total of 71 examples (see Table 3) based on different parameter changes also show similar degree of accuracy. Hence, for both the i.i.d. and AR(1) demand processes, the approximation not only provides an effective and simple computation for the target stocking levels, also generates an accurate representation of the actual system cost by following the approximate stocking levels that is close to optimal. Figures 5 and 6 about here. We further explore the effectiveness of the simple closed-form expression of the approximate model. Figures 7 and 8 show the optimal inventory levels and system costs under varying auto-correlation coefficient ρ and replenishment leadtime T 1. As expected, both the optimal inventory levels and system cost increase as ρ or T 1 27

30 increases. In addition, the cost increments due to leadtime increase, is greater when ρ is higher. Leadtime and demand auto-correlation are thus shown to have significant interaction effects. Leadtime reduction has greater impact on system cost when the demand auto-correlation is high. Such observations match the results of Theorem 9. Hence, the analytical results and the associated qualitative insights that we draw from the approximate model are shown to be valid under the true cost model, based on this set of examples. Figures 7 and 8 about here. 6 Conclusion In this paper, we revisit the Clark and Scarf s model and develop a simple approximation of the induced penalty cost function. This approximation leads to a lower bound on the optimal stocking levels and an upper bound on the average system cost. The approximation is then extended to the time-correlated demand process with MMFE, where Clark and Scarf s decomposition result is shown to hold. In particular, under the AR(1) process the approximation provides a simple, easy to compute closed-form expression for the stocking levels and the average system cost. The closed-form expression from the approximation allows us to investigate how the underlying demand process affects the performance of the inventory system. As noted earlier, an AR(1) model may be a better representation of the underlying demand process in many high-tech and consumer goods industries, and it is indeed used in recent supply chain 28

31 management research studies. Our study of the AR(1) demand process shows that both the system cost and target inventory stocking levels increase as the demand auto-correlation coefficient increases. Hence, in a highly time-correlated demand environment, simplifying the demand assumption to i.i.d. would understate the actual system cost and result in target inventory levels that are too low. We have also shown that the impact of leadtime reduction is greater when the auto-correlation coefficient ρ, is higher. Hence it is more worthwhile to invest in leadtime reduction in a highly time-correlated demand environment. The approximation developed in this paper offers a benchmark for further study of the forecast/information sharing in supply chains. An important avenue of future research is on fine tuned MMFE, which offers great flexibility in modeling information and collaboration in multi-echelon systems. Acknowledgments: The authors gratefully acknowledge the helpful comments of the editor and two anonymous referees. The authors also thank Paul Zipkin, Jing- Sheng Song, Yossi Aviv for insightful discussions. Useful feedback was provided by participants at the 2001 Multi-echelon Inventory Systems Conference at Berkeley. Appendix Proof of Theorem 1. First observe that, in (4), de [g m (y ξ)] /dy = c m. Hence, the first derivative of the minimand on the R.H.S. of (4) w.r.t. y is given by (1 α)c m + α T m+1 (d/dy) E [L m (y γ (T m +1))+Γ m (y γ (T m + 1))]. (A.1) 29

32 Similarly, the first derivative of the minimand on the R.H.S. of (9) w.r.t. y is given by (1 α)c m + α Tm+1 (d/dy) E h i L m (y γ (T m +1))+bΓ m (y γ (T m +1)). (A.2) To show S b m Sm,itsuffices to show that: h i (d/dy) E Γ bm (y Z) (d/dy) E [Γ m (y Z)] for any random variable Z. We show this inequality by induction. When m =1, bγ 1 = Γ 1, and so equality holds. Assume that the inequality holds up to m and S b m Sm.LetF ( ) be the CDF of Z. First, note that, when bs m S m <y z, bγ m+1 (y z) Γ m+1 (y z) = bγ m+1 (y z) is nondecreasing in y. Second, for S m y z, from(5),wehave: (d/dy) Γ m+1 (y) =(1 α) c m +α T m+1 (d/dy) E [L m (y γ (T m +1))+Γ m (y γ (T m + 1))] and from (8), we have (d/dy) b Γ m+1 (y) =(1 α) c m +α T m+1 (d/dy) E h L m (y γ (T m +1))+ Γ b i m (y γ (T m +1)). Hence, we have: = = h i (d/dy) E Γ bm+1 (y Z) Γ m+1 (y Z) Z h i (d/dy) Γ bm+1 (y z) Γ m+1 (y z) df (z) z y S m Z + Z z>y S m z y S m Z + 0. z>y S m h i (d/dy) Γ bm+1 (y z) Γ m+1 (y z) df (z) h i dbγ m+1 (y z) /dy df (z) h i α Tm+1 (d/dy) E bγm (y z γ (T m +1)) Γ m (y z γ (T m +1)) df (z) 30

33 h ³ i Assume that E (d/dy) Γ bm+1 (y Z) Γ m+1 (y Z) <. The interchange of differentiation and integration is justified by the Lebeague Dominance Convergence Theorem. Proof of Lemma 1. Let b A α m (x m ) be the cost per period at echelon m by following the order-up-to bs m policy with initial inventory of x m. Clearly ba α m (x m ) = (1 α) bg m (x m ).Define eg m ( ) as: eg m (x m ) = bg m (x m )+c m x m = (1 α) c m b Sm + αc m µ + α Tm+1 E +αe h ³ i eg m S bm ξ. h ³ L bsm m γ (T m +1) + Γ b ³ i bsm m γ (T m +1) And ba α m (x m )=(1 α)[eg m (x m ) c m x m ]. Note that eg m (x m ) is actually independent of x m,hence ba α m = b A α m (0) = (1 α) eg m (0) = (1 α) c m bs m + αc m µ + α Tm+1 E h ³ ³ i L m S bm γ (T m +1) + bγ m S bm γ (T m +1). By definition of L m ( ), the desired result follows. Lemma A.1 For all m>1 and for a random variable Z, we have: h i (d/dy) E Γ bm (y Z) = (1 α) X m 1 i=1 ατ(i+1,m 1) c i + X m 1 d i=1 ατ(i,m 1) dy E [L i (y Z γ (τ (i, m 1)))]. 31

34 ProofofLemmaA.1. The proof is by induction. We refer the reader to details in Dong (1999). ProofofTheorem2. For m =1, the desired result is obtained by observing that, setting (A.2) to zero yields (1 α) c 1 + α T1+1 (d/dy) E [L 1 (y γ (T 1 + 1))] = (1 α) c 1 + α T 1+1 h 1 (π + H 1 ) Φ (y; T 1 +1) = 0. For m 2, usinglemmaa.1,(a.2)becomes h i (1 α) c m + α Tm+1 h m + α Tm+1 (d/dy) E bγm (y γ (T m +1)) m 1 X = (1 α) c m + α Tm+1 h m + α Tm+1 {(1 α) α τ(i+1,m 1) c i X α τ(i,m 1) d dy E [L i (y γ (T m +1) γ (τ (i, m 1)))]} m 1 + i=1 = (1 α) c m + α Tm+1 h m +(1 α) X m 1 + X m 1 d i=1 ατ(i,m) dy E [L i (y γ (τ (i, m)))] i=1 i=1 ατ(i+1,m) c i = (1 α) X m i=1 ατ(i+1,m) c i + α Tm+1 h m + X m 1 +α τ(1,m) h 1 (π + H 1 ) Φ (y; τ (1,m)) i=2 ατ(i,m) h i = (1 α) X m i=1 ατ(i+1,m) c i + X m i=1 ατ(i,m) h i α τ(1,m) (π + H 1 ) Φ (y; τ (1,m)) Setting (A.2) to zero immediately yields the desired result. ProofofLemma2. Using Lemma A.1 by setting Z to be zero and integrating 32

35 both sides from bs m 1 to y gives: bγ m (y) bγ m ³ b S m 1 = = = Z y m 1 X m 1 X {(1 α) α τ(i+1,m 1) c i + α τ(i,m 1) d t= bs m 1 dt E [L i (t γ (τ (i, m 1)))]}dt i=1 i=1 ³y S m 1 b (1 α) X m 1 i=1 ατ(i+1,m 1) c i + X Z m 1 y i=1 ατ(i,m 1) h i dt α τ(1,m 1) (π + H 1 ) t= S b m 1 X m 1 ³y bs m 1 i=1 ατ(i+1,m 1) (1 α) c i + α Ti+1 h i α τ(1,m 1) (π + H 1 ) Z y t= b S m 1 Φ (t; τ (1,m 1)) dt. Z y t= b S m 1 Φ (t; τ (1,m 1)) dt The result follows from noting that bγ m ³ b S m 1 =0. For Normal demand, note that Θ 0 (x) =Φ (x), sothat = Z y Φ (t; τ (1,m 1)) dt t= S b m 1 µ (t µ (τ (1,m 1))) Φ t= bs m 1 σ (τ (1,m 1)) h Θ (ι (y)) Θ Z y = σ (τ (1,m 1)) dt ³ ³ ι bsm 1 i. Here we state two lemmas that are useful for calculations involving Normal demands. The proofs of these two lemmas are quite tedious and are given in Dong (1999). Lemma A.2 = Z (1/Σ) t= Σ 2 b Σ 2 + b 2 φ ( (a t) /b) Φ ( (a t) /b) φ ((t µ) /Σ) dt µ µ a + µ a µ µ a Φ. Σ2 + b 2 b Σ2 + b 2 33

36 Lemma A.3 = Z (1/Σ) φ ( (a t) /b) φ ((t µ) /Σ) dt t= ³b/ 2 ³ b 2 + Σ 1/ Ã! (a µ)2 2π exp. 2(b 2 + Σ 2 ) Lemma A.4 h ³ i E Γ bm S bm γ (T m +1) = ³ X m 1 S bm µ (T m +1) bs m 1 i=1 ατ(i+1,m 1) (1 α) c i + α Ti+1 h i " Z bsm # γ(t m +1) α τ(1,m 1) (π + H 1 ) E Φ (t; τ (1,m 1)) dt. t= S b m 1 IftheenddemandperperiodisNormalwithmeanµ and variance σ 2,then = h ³ i E Γ bm S bm γ (T m +1) ( m 1 X α ) τ(i+1,m 1) (1 α) c i + α Ti+1 h i i=1 Φ 1 (1 K m ) σ (τ (1,m)) Φ 1 (1 K m 1 ) σ (τ (1,m 1)) ª +α τ(1,m 1) (π + H 1 ) Θ Φ 1 (1 K m ) σ (τ (1,m)) Θ Φ 1 (1 K m 1 ) σ (τ (1,m 1)) ª. Proof of Lemma A.4. The first part is a consequence of Lemma 2. For Normal demand, we can write bs m = µ (τ (1,m)) + Φ 1 (1 K m ) σ (τ (1,m)), 34

37 for m =1,...,M. Hence " Z bsm # γ(t m +1) E Φ (t; τ (1,m 1)) dt t= bs m 1 = σ (τ (1,m 1)) { Z t= ³ h i Θ S bm t µ (τ (1,m 1)) /σ (τ (1,m 1)) dφ (t; T m +1) Θ Φ 1 (1 K m 1 ) }. Let a = bs m µ (τ (1,m 1)), b = σ (τ (1,m 1)), eµ = µ (T m +1), Σ = σ (T m +1), then Z t= Z = (1/Σ) = (1/Σ) ³ ³ Θ bsm t µ (τ (1,m 1)) t= Z t= Z Θ ( (a t) /b) φ ((t eµ) /Σ) dt /σ (τ (1,m 1)) dφ (t; T m +1) ( (a t) /b) Φ ( (a t) /b) φ ((t eµ) /Σ) dt +(1/Σ) φ ( (a t) /b) φ ((t eµ) /Σ) dt t= µ Σ 2 eµ a = b Σ 2 + b φ + eµ a µ eµ a Φ 2 Σ2 + b 2 b Σ2 + b 2 + ³b/ 2 ³ b 2 + Σ 1/ ( ) (a eµ)2 2π exp 1/2 (by Lemmas A.2 & A.3) b 2 + Σ 2 = σ 2 (T m +1)/ (σ (τ (1,m 1)) σ (τ (1,m))) φ Φ 1 (1 K m ) (σ (τ (1,m)) /σ (τ (1,m 1))) Φ 1 (1 K m ) Φ Φ 1 (1 K m ) +σ (τ (1,m 1)) /σ (τ (1,m)) φ Φ 1 (1 K m ) = (σ (τ (1,m)) /σ (τ (1,m 1))) Θ Φ 1 (1 K m ). The desired result follows. ProofofLemma3. Theresultfollowsfromtakingthelimitof ba α m as α 1. See Dong (1999) for the algebraic details. 35

38 Proof of Theorem 3. n b A m o M m=2 is a telescope series. Using Lemma 3, we get ba = µ X M c m + X M m=1 m=1 µ (τ (1,m 1)) +σ (τ (1,M)) H 1 Φ 1 (1 K M )+(π + H 1 ) Θ Φ 1 (1 K M ). Note that H 1 Φ 1 (1 K M )+(π + H 1 ) Θ Φ 1 (1 K M ) = H 1 Φ 1 (1 K M ) +(π + H 1 ) {φ Φ 1 (1 K M ) Φ 1 (1 K M ) Φ Φ 1 (1 K M ) } = (π + H 1 ) φ Φ 1 (1 K M ), since Φ ( Φ 1 (1 K M )) = K M = H 1 / (π + H 1 ) as α 1, and the desired result follows. The alternative expression can be obtained by using h m = H m H m+1, H M+1 =0. Proof of Corollary 1. ba/ T m = H m σ(τ(1,M)) (π + H 1) φ (Φ 1 (1 K M )) 0 for m =1,..., M 1; ba/ T M = 1 2σ(τ(1,M)) (π + H 1) φ (Φ 1 (1 K M )) 0. ba/ T m b A/ T m+1,sinceh 1 H 2... H M. Proof of Corollary 2. (1) h ³ ³ i ba α 2 A α 2 = α T2+1 E Γ b2 S b2 γ (T 2 +1) Γ 2 S b2 γ (T 2 +1) 0, since b Γ 2 ( ) Γ 2 ( ) on <. (2) follows for the fact that b A α 1 = A α 1. For the AR(1) case, we will make use of the property that D n = d X n 1 i=0 ρi + ρ n D 0 + X n i=1 ρn i ε i, 36

39 µ n = d X n 1 i=0 ρi + ρ n D 0, (A.3) σ 2 n = σ 2 1 ρ 2n / 1 ρ 2, D (1,n)=d/ (1 ρ) X n X n 1 ρ i + D 0 = d X n X i 1 i=1 i=1 j=1 ρi j ε j i=1 ρi + X n i=1 i=1 j=0 ρj + D 0 X n i=1 ρi +1/ (1 ρ) X n µ 0 (n) =d X n i=1 X i 1 σ 2 0 (n) =σ 2 X n j=0 ρj + D 0 X n i=1 ρi, i=1 ³X i 1 j=0 ρj 2. X i 1 ρ i ε i, Notice that the variances of D n and D (1,n) are independent of historical demand D 0, hence we can drop subscript 0 from the variance notation. ProofofTheorem4. We first study the finite horizon case. With zero leadtime, the dynamic programming formulation of the problem is f n (x n, D n )=min x n y {c (y x n)+l n (x n )+αe [f n+1 (y D n+1, D n+1 )]}, where L n (x) is convex in x for all n and f N+1 (, ) =0. Note that the expectation is taken over a finite number of multi-variate random variables, no anomaly should be encountered. Let V be the set of functions v (, ) that are convex w.r.t. the first variable; π =... ( appears N times), where is the set of base-stock policies, and for δ, δ (x, D) =S (D) if x<s (D), δ (x, D) =x otherwise. Thefollowingthreestepsaresufficient to show that a base-stock policy is optimal for period n and for every n, and the optimal base-stock level is a function of demand in period n. 37

40 Itcanbeeasilyshownthat,iff n+1 V,thenE [f n+1 (y D n+1, D n+1 )] is convex in y. If f n+1 V, then there exists δ that is optimal for period n. To see this, suppose f n+1 V. Now, we have cy n + L n (x n )+αe [f n+1 (y n D n+1, D n+1 )] being convex in y n. Let Sn be a minimizer of cy n + L n (x n )+αe [f n+1 (y n D n+1, D n+1 )], whichis clearly a function of D n. Let δ (x n, D n )=S n (D n ) if x n <S n (D n ), δ (x n, D n )=x n otherwise. Hence, following δ (x n, D n ) is optimal for period n. If f n+1 V,thenf n V. This follows from: = f n (x n, D n ) c (Sn (D n ) x n )+L n (x n )+αe [f n+1 (Sn (D n ) D n+1, D n+1 )] if x n <Sn (D n ) ; L n (x n )+αe [f n+1 (x n D n+1, D n+1 )] otherwise, and that f n (x n, D n ) is convex and has continuous derivative with respect to x n. The nonzero leadtime case can be extended as in standard inventory literature. Finally, we can extend the result to the infinite horizon case using Proposition 1.6 and Proposition 1.7 in Section 3.1 of Bertsekas (1995), such that the limit of Sn(D) converges to an optimal stationary policy S (D) as n. Proof of Theorem 6. Similar to that of Theorem 1. See Dong (1999) for details. Lemma A.5 For all m>1 and for a random variable D (1,T), we have: h i (d/dy) E Γ bm (y D (1,T),D T ) = (1 α) X m 1 i=1 ατ(i+1,m 1) c i + X m 1 d i=1 ατ(i,m 1) dy E [L i (y D (1,T + τ (i, m 1)))]. 38

41 Proof of Lemma A.5. The proof is similar to that of Lemma A.1. See Dong (1999) for details. Proof of Lemmas 4 and 5. Similar to that of Lemmas 1 and 2. See Dong (1999) for details. Lemma A.6 E h b Γ m ³ b S m (D 0 ) D (1,T m +1),D Tm+1 i = nx m 1 i=1 ατ(i+1,m 1) (1 α) c i + α T i+1 h i o Φ 1 (1 K m ) σ (τ (1,m)) Φ 1 (1 K m 1 ) σ (τ (1,m 1)) ª +α τ(1,m 1) (π + H 1 ) σ (τ (1,m)) Θ Φ 1 (1 K m ) σ (τ (1,m 1)) Θ Φ 1 (1 K m 1 ) ª. Proof of Lemma A.6. The proof is similar to that of Lemma A.4. See Dong (1999) for details. ProofofTheorem8. The first equality is similar to the proof of Theorem 2. For the second equality, we note that µ 0 (τ (1,m)) µ 0 (T m +1) (A.4) = d X τ(1,m) i=1 = d X τ(1,m) i=t m+1+1 X i 1 X τ(1,m) j=0 ρj + D 0 i=1 X i 1 ρ i d X T m+1 j=0 ρj + D 0 X τ(1,m) i=t m+1+1 ρi, i=1 X i 1 X Tm+1 j=0 ρj D 0 i=1 ρ i 39

42 and E µ Tm +1 (τ (1,m 1)) = E d X τ(1,m 1) X i 1 i=1 j=0 ρj + D Tm +1 = d X τ(1,m 1) X i 1 ³ i=1 j=0 ρj + d X T m = d X τ(1,m 1) i=1 = d X τ(1,m 1) i=1 = d X τ(1,m) i=t m+1+1 ³X i 1 X i+tm j=0 X i 1 j=0 ρj + X T m j=0 ρj+i ρ j + D 0 X τ(1,m) X τ(1,m 1) i=1 ρ i X τ(1,m 1) j=0 ρj + ρ Tm+1 D 0 i=1 i=t m +1+1 ρi j=0 ρj + D 0 X τ(1,m) i=t m+1+1 ρi. ρ i X τ(1,m 1) + D 0 ρ i+tm+1 i=1 (A.5) Proof of Corollary 3. Since σ (τ (1,M)) is nondecreasing in T 1,...,T M and σ (τ (1,M)) / T m = σ (τ (1,M)) / T m+1, we only need to check the term P M m=1 h m E[µ Tm +1 (τ (1,m 1))]. Notethat X M h me µ Tm+1 (τ (1,m 1)) m=1 = X M h me d X τ(1,m 1) X i 1 m=1 i=1 j=0 ρj + D Tm +1 X τ(1,m 1) i=1 ρ i which is nondecreasing in T 1,...,T M. To show ba (D 0 ) / T m ba (D 0 ) / T m+1,we examine: ³X M me µ Tm +1 (τ (1,m 1)) / T i m=1 = h i E µ Ti +1 (τ (1,i 1)) ³X M / T i + me µ Tm+1 (τ (1,m 1)) / T i m=i+1 40