Lost-Sales Problems with Stochastic Lead Times: Convexity Results for Base-Stock Policies

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1 OPERATIONS RESEARCH Vol. 52, No. 5, Septeber October 2004, pp issn X eissn infors doi /opre INFORMS TECHNICAL NOTE Lost-Sales Probles with Stochastic Lead Ties: Convexity Results for Base-Stock Policies Ganesh Janakiraan IOMS-OM Group, Stern School of Business, New York University, 44 West 4th Street, New York, New York , Robin O. Roundy School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, We consider a single-location inventory syste with periodic review and stochastic deand. It places replenishent orders to raise the inventory position that is, inventory on hand plus inventory in transit to exactly S at the beginning of every period. The lead tie associated with each of these orders is rando. However, the lead-tie process is such that these orders do not cross. Deand that cannot be et with inventory available on hand is lost peranently. We state and prove soe saple-path properties of lost sales, inventory on hand at the end of a period, and inventory position at the end of a period as functions of S. The ain result is the convexity of the expected discounted su of holding and lost-sales costs as a function of S. This result justifies the use of coon search procedures or linear prograing ethods to deterine optial base-stock levels for inventory systes with lost sales and stochastic lead ties. It should be noted that the class of base-stock policies is suboptial for such systes, and we are priarily interested in the because of their widespread use. Subject classifications: inventory/production: periodic review; lost sales; base-stock policies; convexity; saple-path properties. Area of review: Manufacturing, Service, and Supply Chain Operations. History: Received January 2001; revisions received May 2002, April 2003; accepted Septeber Introduction We consider a single-location inventory syste with periodic review and stochastic deand. It places replenishent orders to raise the inventory position, that is, inventory on hand plus inventory in transit, to exactly S at the beginning of every period. The lead tie associated with each of these orders is rando. However, the lead-tie process is such that these orders do not cross. Deand that cannot be et with inventory available on hand is lost peranently. First, we establish soe eleentary saple-path properties of lost sales and end-of-period inventories as functions of S. Subsequently, we show that a discounted su of lost sales over a finite horizon is convex in S for every saple path of deands and lead ties. We provide a counterexaple to show that this result is not true when orders cross. We also establish the convexity result for the discounted su of end-of-period inventory position for every saple path of deands and lead ties. Consequently, for a cost odel with holding costs that are linear in the inventory position at the end of the period and lost-sales costs that are linear in the lost sales incurred, the discounted-cost function is easily seen to be convex. This cost odel is appropriate when capital costs are large copared to other inventory holding costs, and payent due dates are based on order placeent dates rather than order arrival dates. We provide an exaple to show that if holding costs are charged on the inventory on hand at the end of each period, then the discounted-cost function need not be convex along every saple path. With an additional assuption about lead ties (see Kaplan 1970), we show that the expected discounted su of end-of-period inventories on hand over a finite horizon is convex in S. With this assuption, we establish the convexity of the expected discounted su of holding and lost-sales costs for cost odels that include two holding ters the first one proportional to the inventory position and the second one proportional to on-hand inventory. This result was first established by Downs et al. (2001) for the case of deterinistic lead ties when holding costs are charged only on the inventory on hand. This result justifies the use of coon search procedures or linear prograing ethods to deterine optial base-stock levels for inventory systes with lost sales and stochastic lead ties. We assue that there are no fixed ordering costs in this environent. Though order-up-to-s policies are optial for systes without fixed costs when excess deand is backordered (for exaple, see Erhardt 1984 for such a result with backorders and stochastic lead ties), these policies 795

2 796 Operations Research 52(5), pp , 2004 INFORMS are known to be suboptial for lost-sales probles with positive lead ties. However, they are very coonly used in practice in lost-sales environents, and this serves to otivate our work. 2. Motivation and Literature Survey Karlin and Scarf (1958) considered an inventory odel with lost sales and a lead tie of one period. They assued deands with positive continuous density functions, a fixed lead tie of one period, linear and proportional purchase and lost-sales costs, and convex, increasing holding costs. They prove that base-stock policies are not optial for these systes and also establish soe eleentary properties of the optial ordering policy. They also state that though order-up-to-s policies are not optial for this situation, there are exaples of their use in both ilitary and industrial areas in lost-sales probles. Their paper also has a brief discussion on finding the optial S when deand is exponentially distributed. With the added assuption that the holding costs are linear, Morton (1969) extended the result of Karlin and Scarf (1958) on the properties of the optial order quantity to the case where the lead tie is constant and equal to soe nonnegative and constant integer. He also developed tight upper and lower bounds on the optial ordering policy and used these bounds in heuristics. Morton (1971) proposed yopic policies as approxiate solutions to this proble as well as a larger class of proportional cost probles, and presented coputational results. Nahias (1979) studied the periodic-review lost-sales proble with set-up costs, rando lead ties without order crossing and partial backordering, and proposed yopic heuristics. In addition, he used a siulator to find the best value of S aong the order-up-to-s or base-stock policies using Fibonacci search for probles with no set-up costs. He observed that the response surface as a function of S was convex for all of the odels considered. Donselaar et al. (1996) derived heuristics for finding the optial S and copared that order-up-to-s policy with a different policy that they proposed. Downs et al. (2001) considered order-up-to-s policies for a lost-sales proble with deterinistic lead ties and proved the convexity of the average cost function with respect to S. They derived nonparaetric estiates of the costs that they used in a linear-prograing-based policy to deterine the optial order-up-to levels for ultiple products in the presence of resource constraints. This policy is coputationally siple and can be used even in the absence of the specification of the deand distribution. Karush (1957) analyzed a continuous-tie inventory syste with Poisson deands and independent and identically distributed lead ties and deonstrated the convexity of the steady-state rate of lost sales as a function of the orderup-to level. Healy (1992) considered s S policies for an inventory proble with backorders and showed that for a fixed value of S s, the n-period cost function is convex in S for every saple path of deands. Fu and Healy (1997) used this convexity result to deterine the optial s S pair for a given saple path of deands. Glasseran and Tayur (1995) used perturbation analysis to deterine base-stock levels in capacitated ultiechelon inventory systes with backorders. Agrawal and Sith (1996) developed a paraeterestiation ethodology to estiate paraeters for negative binoial deand distributions in the presence of unobservable lost sales. Ketzenberg et al. (2000) and Metters (1997; 1998a, b) are soe papers that developed heuristic inventory policies for systes with lost sales. Hill (1999) studied a continuous-review inventory proble with constant lead ties, Poisson deand, and lost sales. He proved that S 1 S policies can never be optial if S is greater than one unit. Johansen (2001) considered the sae proble in a periodic-review environent and proposed a policy called the odified base-stock policy, which has two paraeters S and t. S is the base-stock level and t is the iniu nuber of periods between two successive replenishent orders. Kaplan (1970), Erhardt (1984), and Zipkin (1986) are soe of the papers in the literature that studied inventory probles with stochastic, noncrossing lead ties and backorders. To our knowledge, there is no existing analytical work in the discrete-tie inventory control literature with lost sales and stochastic lead ties. Though order-up-to-s policies are suboptial for lostsales probles, there are two reasons for working with the. One is the siplicity and widespread use of these odels. The second one is that it sees to be ipossible to analytically infer the structure of optial policies in periodic-review, stochastic lead tie, lost-sales odels. 3. Proble Definition Throughout this paper we use the ters increasing and decreasing in the weak sense. We begin with soe notation. n, t, and k are indices for tie periods, where 0 n t k N. N denotes the length of the horizon. D n is the deand that occurs in period n. x n is the aount of inventory on hand after receiving the shipents that arrive in period n. q n is the size of the order placed in period n. L n is the lead tie of that order. l n is the aount of sales lost in period n. i n is the aount of inventory on hand at the end of the period. n is the inventory position (inventory on hand plus in transit) at the end of the period. n is the period index of the latest shipent that arrived in or before period n. Thus, n = ax t t +L t n. Assue that the stochastic process L = L 0 L 1 L N is such that: Assuption 1. Orders do not cross, i.e., n + L n is increasing in n. Next, we describe the sequence of events in period n (0 n N ) when a base-stock policy with paraeter S (also called an order-up-to-s policy) is used.

3 Operations Research 52(5), pp , 2004 INFORMS Place an order of size q n to raise the inventory position, that is, on-hand inventory plus inventory in transit, to S. 2. Receive all shipents that arrive in period n, bringing the on-hand stock to its new value x n. Because an orderup-to-s policy is being followed, it is easy to see that x n = S n <t n q t 3. Observe deand D n. 4. Satisfy as uch deand as possible. Let y n denote the aount of deand satisfied in period n. Note that y n = in x n D n. That is, the inventory on hand drops fro x n by y n to x n D n + and the inventory position drops to S y n. The aount of sales lost is given by l n = D n x n + = D n y n. That is, i n = x n y n and n = S y n Because the inventory position falls by y n in period n, the aount ordered in period n + 1isgivenbyq n+1 = y n. We assue that the syste starts with inventory on hand of size S, and none on order. Consequently, x 0 = S and q 0 = 0. We also define 0 to be zero. We consider a cost odel where a holding cost of h 1 is charged for the inventory position (inventory on hand and in transit) at the end of each period, a storage cost of h 2 is charged for every unit of inventory on hand at the end of each period, and a lost-sales cost of p is charged on every unit of sale lost in each period. There is a discount factor of 0 < 1. Note that there is no discounting if = 1. It can be shown that if purchase costs are linear, the unit purchase cost can be assued to be zero without loss of generality by assuing coplete salvage at the end of the horizon (see Janakiraan and Muckstadt 2004 for the proof of a siilar result for ore general systes). For a given realization of lead ties and deands, and a given order-up-to level S, the discounted cost is N n h 1 n + h 2 i n + p l n (3.1) Using relations derived earlier, we can rewrite this cost as N n h 1 S y n + h 2 x n y n + p D n y n (3.2) The cost of capital is captured by h 1 if payent due dates are based on order placeent dates, or by h 2 if due dates are based on order arrival dates. Costs related to the physical storage of inventory are captured by h 2. In highly industrialized countries, in ake-to-stock inventory systes, payent is usually due when the inventory arrives, not when orders are placed. Consequently, h 1 is often zero. However, third-world iporters often face a different situation. Consider two Costa Rican copanies. One of the iports fine liquors fro the United States and sells the in Costa Rica. The other iports galvanized wire fro the United States and uses it to ake barbed wire for local needs. These copanies buy relatively sall quantities and serve a liited arket. Consequently, they are obliged to purchase inventory in the country of origin and ship it theselves. In these settings, h 1 is often the doinant holding-cost ter. In the following section, we will state and prove properties of the behavior of this syste as S varies. For this reason, we will henceforth use the ore descriptive notation x n S, l n S, q n S, i n S, and n S. 4. Analytical Results This section contains our analytical results. In 4.1, we derive saple-path results by liiting attention to a given realization of the deands D n and lead ties L n, 0 n N. We establish eleentary properties of (i) the inventory on hand at the end of a period, (ii) the inventory position at the end of a period, and (iii) the lost sales incurred in a period, as functions of S. Furtherore, we show that both the discounted su of lost sales and the discounted su of inventory position are convex functions of S. The convexity of these two functions iplies that the discounted su defined in (3.1) is a convex function of S when h 2 is zero. Because expectations of convex functions are convex, this establishes the convexity of the expected cost when h 2 = 0. We provide an exaple showing that the discounted su of lost sales can fail to be convex when lead ties are allowed to cross. In addition, we present an exaple to show that the discounted cost in (3.1) can fail to be convex when h 2 is not zero, eaning that the assuptions in 4.1 are too weak to establish convexity when h 2 > 0. In 4.2, we address the case h 2 > 0. We restrict ourselves to a class of lead-tie processes first proposed by Kaplan (1970). With this restriction, we show that the expected discounted su of inventory on hand at the end of each period is convex in S, where the expectation is taken over the lead-tie process, for any realization of deands. Our ain theore, that the expected value of the discounted su defined in (3.1) is a convex function of S, follows iediately Saple-Path Results The results in this section are derived for a given saple path of lead ties and deands. Many of the saple-path results directly iply corresponding results relating to the expectations. We start by stating a lea that describes properties of the functions x n S, q n S, i n S, and n S. Lea 1. Consider a given saple path of deands and lead ties. For all n 0 1 N, x n S, q n S, i n S, and n S are continuous and increasing. Furtherore, they are all differentiable except at a finite nuber of points. Their derivatives (wherever they exist)belong to the set 0 1.

4 798 Operations Research 52(5), pp , 2004 INFORMS Proof. The proof is by induction and is straightforward, based on the following relations: q n+1 S = y n S = in x n S D n x n+1 S = x n S D n + + = S n+1 <t n+1 t t+l t =n+1 q t S q t S (4.3) i n S = x n S D n + = x n S y n S and (4.4) n S = S y n S (4.5) In the induction proof, the first parts of (4.3) and (4.4) establish that x n+1 S and i n S are nonnegative integers, respectively. The second parts of the sae equations show that x n+1 S 1 and i n S 1. (4.5) shows that n S 0 1. The following stateent about l n S, the nth period lostsales function, is a direct corollary to this lea and is based on the relation l n S = D n in x n S D n = D n y n S (4.6) Corollary 2. Consider a given saple path of deands and lead ties. For all n 0 1 N, l n S is continuous and decreasing. Further, it is differentiable everywhere except at a finite nuber of points. Their derivatives (wherever they exist)belong to the set 1 0. We see fro Lea 1 that Figure 1 shows a typical plot of q n S versus S. Typical plots of x n S, n S, and i n S are siilar. Clearly, these functions are neither convex nor concave. However, we will now establish the concavity of the discounted su of the y n S and q n S functions. The convexity of the discounted su of lost sales and of the inventory position follows iediately. The iportance of these results can be seen by exaining the discounted-cost function defined in the previous section. Lea 3. Consider a given saple path, i.e., a specific realization of all lead ties and deands. For all Figure 1. q n (S) Plot of q n S. 0 S 0 N + 1 and for all nonnegative and decreasing sequences n, W def n S = n q n S = n y n 1 S is concave in S (When n = n 1 for all n, W n S is the discounted su of the sales in periods ) Proof. The proof is by induction. The stateent is trivially true for = 0 because W n 0 S = 0 q 0 S = 0 S. Assue that the stateent is true for soe 0. We will now prove the stateent for + 1. W n +1 S = = = n q n S + +1 q +1 S n q n S + +1 in x S D ( n q n S + +1 in S <t ( = in +1 S + n q n S ) +1 D + n q n S q t S D ) where n = n if n and n = n +1 if < n. Therefore, we can write W n +1 S as +1 S =in ( +1 S +W n S +1 D +W n S ) W n It is easy to verify that n is a nonnegative, decreasing sequence. Therefore, W n S is concave by induction, so W n +1 S is a iniu of two concave functions and is, hence, concave. An alternate proof of this lea can be found in Janakiraan and Roundy (2002). (This alternate proof technique was suggested by a referee.) We now show that the discounted su of lost sales and the discounted su of end-of-period inventory position are convex functions of S. The proof is a direct corollary to Lea 3 and is based on (4.5) and (4.6). Corollary 4. Consider a given saple path, i.e., a specific realization of all lead ties and deands. For all 0 N and for all nonnegative and decreasing sequences n, n l n S and n n S are convex in S. (When n = n for all n, n l n S and n n S are the discounted sus of l n S and n S, respectively.) The first of our ain results is the convexity of the discounted-cost function in (3.1) when h 2 = 0, a direct consequence of Corollary 4.

5 Operations Research 52(5), pp , 2004 INFORMS 799 Table 1. Calculations for Exaple 1. n x n S S 1 + S 2 + S in S 1 S in S 1 q n 0 in S 1 in S in S i n S 1 + S 2 + S 3 + S in S 1 S in S 1 l n 1 S + in 2 S + 1 in 3 S Theore 5. For a given saple path of deands and lead ties, N n h 1 n S + p l n S is convex in S The natural next step is to investigate whether the discounted su of end-of-period inventory on hand is convex or not. Convexity of this function would iply the convexity of the discounted cost in (3.1) for all values of h 1, h 2, and p. Unfortunately, this is not the case, as is shown by the following exaple. Exaple 1. Consider a proble where N = 4. Assue that n = 1 n The aounts of inventory on hand and in the pipeline at the start of period 0 are S and 0, respectively. Assue that L 1 = 2, L 2 = L 3 = L 4 = 3 (so orders do not cross), and that D n = 1, n 0 1 2, and D n = 0, n 3 4. An order-up-to-s policy is followed. Under these assuptions, the variables that describe the syste in the different periods are calculated and given in Table 1. Note that the following functions have discontinuities in their derivatives at S = 1 i 0 S i 3 S i 4 S l 0 S l 1 S. Let h 1 = 0, h 2 = 1, and p = 1 2. The total su of all costs in all periods, the su of holding costs in all periods, and the su of lost-sales costs in all periods are plotted in Figure 2. The holding-cost function and the total-cost function are locally concave at S = 1. Next, we provide an exaple to show that Lea 3, Corollary 4, and consequently Theore 5, can fail when the lead-tie process is such that orders can cross. Figure Total Plots of costs. 0 1 Su of costs in periods 0,1,,4 Holding S Lost Sales 2 3 Exaple 2. Consider a proble where N = 3. Assue that n = 1 n The syste starts with inventory on hand of size S and none on order. Assue that L 1 2, L 2 = 0, and that for soe >0, S D 0 and D k >, k An order-up-to-s policy is followed. Under these assuptions, the variables that describe the syste in the different periods are calculated and given in Table 2. (The values of the? s in the table depend on L 1 and L 3.) We see that 3 n q n S = 2 n+1 D n l n S is equal to S + S D 0 +. This is clearly not concave for S in a sall neighborhood of D The Kaplan Lead-Tie Model In this section, we assue that our lead-tie process belongs to the faily introduced by Kaplan (1970). As we will see, this assuption will let us extend Theore 5 to include holding costs charged on the inventory on hand. We state the Kaplan assuption in a for different fro the original. The for of the assuption we use is very siilar to the one used by Nahias (1979) and Erhardt (1984). Assuption 2. n = ax n 1 n n n where n is a sequence of i.i.d., nonnegative, integer-valued rando variables. Equivalently, the orders received in period n are the orders that are outstanding at the beginning of period n and were placed in period n n or earlier. Observe that this assuption iplies that the lead ties L n are identically distributed. In fact, P L n = 0 = P n = 0 and k 1 P L n = k = P n > 0 P n+1 > 1 P n+k 1 >k 1 P n+k k (This distribution is derived on page 123 of Erhardt 1984.) It should be noted that this odel for stochastic lead ties does not allow arbitrary distributions of the lead tie. For exaple, Zipkin (1986, p. 766) shows that this odel iplies that the failure rate of the lead-tie distribution, P L n = k /P L n k, is nondecreasing and converges to one as k approaches. We are now ready to prove the convexity of a discounted su of the expected aounts of inventory on hand at the start of each period. We use the notation = n n= 0 1 N. We use x n S, q n S, l n S, i n S, n S, n, and L n to ake the dependence of these quantities on explicit.

6 800 Operations Research 52(5), pp , 2004 INFORMS Table 2. Calculations for Exaple 2. Period n q n 0 S S D 0 + S D 0 + S D 0 + x n S S D 0 + S D 0 +? Lost sales in period n D 0 S + D 1 S D 0 + D 2 S D 0 +? Lea 6. Under Assuption 2, for every realization of deands, the function n E x n S is convex for all 0 1 N for every nonnegative and decreasing sequence n, where E denotes the expectation operator with respect to the rando vector. Proof. Recall that x n S = S q t S n <t n Consequently, we have ( n x n S = n S n <t n ) q t S By interchanging the order of suation and taking the expectation, it can be verified that [ E n x n S ( [( in n+l n 1 = n )S E t )q n S t=n (The interchange of suation operators can be seen intuitively using the following arguent. n <t n q t S is the aount of inventory on order in period n and n <t n q t S is the su of the aounts of inventory on order in all the periods. This su can also be coputed by tracking the tie periods in which any given order is outstanding. Note that n n + 1 in n + L n 1 are the periods when q n S is an outstanding order.) Also note that for fixed deands, the rando variable q n S is a deterinistic function of 1 2 n 1 because q n S equals in x n 1 S D n 1 and x n 1 S is copletely deterined by the set of orders that arrive in or before period n 1. Furtherore, note that L n is a function of n n+1, which is probabilistically independent of 1 2 n 1. Consequently, we can write [ E n x n S ( ( in n+l n 1 = n )S E t )E q n S t=n Using the facts that n is a decreasing sequence and the distribution of L n is identical for all n, it can be verified that E in n+l n 1 t=n t is a decreasing sequence in n. Therefore, E n x n S is convex in S by Lea 3. Using the fact that i n S = x n S y n S along with Leas 3 and 6, we can now state an identical convexity result for the discounted su of end-of-period inventories on hand. Corollary 7. Under Assuption 2, for every realization of deands, the function n E i n S is convex for all 0 1 N for every nonnegative and decreasing sequence n, where E denotes the expectation operator with respect to the rando vector. We now state the second of our ain results. Theore 8. Under Assuption 2, for every realization of deands and every set of nonnegative cost paraeters h 1 h 2 p, the function [ N E n h 1 n S + h 2 i n S + p l n S is convex. Proof. A direct consequence of Theore 5 and Corollary 7. Rearks. By exaining the proof of Lea 6 carefully, one can observe that the proof holds even when the n s are not identically distributed. The proof requires only the independence of n s and the property that the lead ties are stochastically decreasing; that is, P L n k is increasing in n for all k. Siilarly, the proofs of Theores 5 and 8 hold even when the cost paraeters are nonstationary under certain conditions. More iportantly, every result proved in this paper so far is true for any saple path of deands. Consequently, these results are valid for nonstationary and/or correlated deand processes as long as the deand process is independent of the lead-tie process and the inventory policy. Although each one of the cases entioned above leads to an easy generalization of our results, it is unnatural to assue that one would use stationary order-up-to levels in these situations. Our convexity results easily extend to the infinite-horizon case as we show in the next section. 5. Infinite-Horizon Results In this section, we use n S, i n S, and l n S to denote unconditional rando variables and n S D, i n S D, and l n S D to denote the respective values for a given realization D = n D n n 0. First, we show that Theore 5 holds for an infinitehorizon, discounted-cost odel.

7 Operations Research 52(5), pp , 2004 INFORMS 801 Theore 9. Let <1. For every set of nonnegative cost paraeters h 1 h 2 p and every saple path of deands and lead ties, the following possibly infinite-valued function is convex: n h 1 n S D + p l n S D is convex in S If the deand D n in each period n is bounded above by M, then this function is finite valued. Siilarly, for every realization of lead ties, the following possibly infinite-valued function is convex: [ E D n h 1 n S D + p l n S D is convex in S where E D denotes the expectation with respect to the deand process D n n 0. If the expected deand in each period n, E D n, is bounded above by M, then this function is finite valued. Proof. We sketch the proof for the second part of this theore. The proof for the first part is siilar, but sipler. For any saple path of lead ties and any S, the expectation E D N n h 1 n S D + p l n S D exists and is increasing in N. By the onotone convergence theore, the (possibly infinite) liit as N exists, and we can interchange the expectation and the liit. Using Theore 5 and the fact that the liits of convex functions are convex, we can see that E D n h 1 n S D + p l n S D exists and is convex. If E D n <M n, then this function is bounded above by ax h 1 S p M / 1 <. Next, we extend Theore 8 to the infinite-horizon discounted case in the sae way. We oit the proof. Theore 10. Let <1. Under Assuption 2, for every set of nonnegative cost paraeters h 1 h 2 p and every saple path of deands, the following possibly infinitevalued function is convex: [ E n h 1 n S D + h 2 i n S D + p l n S D If D n is bounded above by M n, this function is finite valued. The following possibly infinite-valued function is also convex: [ E D n h 1 n S D + h 2 i n S D + p l n S D where E D is the expectation operator over the rando sequence n D n n 0. IfE D n is bounded above by M n, this function is finite valued. Next, we extend these results to the long-run averagecost odel. For these results, we ake three additional assuptions. Assuption 3. First, we assue that n D n n 0 is an i.i.d. sequence of rando vectors. Second, we assue that P D n = 0 >0 n. Third, we assue that the deand in any period is an integer. These assuptions are stronger than necessary, but they facilitate easy proofs. Our arguents for the average-cost equivalents of Theores 9 and 10 are the following. Assue that the inventory on hand at the start of period 0 is S, and there is no pipeline inventory in period 0. Let v n S represent the rando vector of inventory aounts in different stages of the pipeline, including the inventory on hand, at the beginning of period n. For exaple, if the axiu possible lead tie is three periods, v n S is a three-diensional vector. Let us define the state space of the Markov chain v n S n 0 as all possible vectors v n S accessible fro the state at which we start period 0 (S on hand and none in the pipeline). Because we have assued that the probability of zero deand occurring in a period is strictly positive, we can show that the state where v n S is such that inventory on hand is S, and such that the pipeline has no inventory is a positive recurrent state of the Markov chain. This is because a sufficiently long sequence of zero deands will get the syste back to this state. Now, we can observe that v n S n 0 is an irreducible and positive recurrent Markov chain. Consequently, it has a stationary distribution, represented by the rando variable v S. This also ensures the existence of stationary distributions for i n S, n S, and l n S. Furtherore, using Proposition fro Resnick (1992), we can see that the long-run average cost converges to the stationary expected one-period cost alost surely, that is, the expected cost incurred in a period, say n, where v n S has the sae distribution as v S. Theore 11. Under Assuption 3, for every set of nonnegative cost paraeters h 1 h 2 p and alost every saple path of deands and lead ties, li N [ N h 1 n S D + p l n S D /N = h 1 E S + p E l S where S and l S are rando variables whose distributions are the stationary distributions of the rando variables n S and l n S, respectively. Furtherore, this long-run average cost is convex in S. Proof. The arguents preceding the theore establish the equation and the existence of the quantities in the equation. The convexity of the long-run average cost is a direct consequence of the existence of these distributions and the finite-horizon result, that is, Theore 5. Siilarly, Theore 10 can be extended to the averagecost case. We state the result for the sake of copleteness.

8 802 Operations Research 52(5), pp , 2004 INFORMS Theore 12. Under Assuptions 2 and 3, for every set of nonnegative cost paraeters h 1 h 2 p, [ N E D li h 1 n S D N + h 2 i n S D + p l n S D /N = h 1 E S + h 2 E i S + p E l S where S, i S, and l S are rando variables whose distributions are the stationary distributions of the rando variables n S, i n S, and l n S, respectively. Furtherore, this long-run average cost is convex in S. This concludes our discussion of infinite-horizon odels. Note that all the results we have developed in this paper assue that we start period 0 in a very specific way; that is, the inventory on hand is S and there is no inventory in the pipeline. A natural question is whether these results are valid if this assuption is violated. We address this issue in the appendix. 6. Conclusion Soe saple-path properties of a discrete-tie inventory syste with lost sales, with stochastic lead ties such that orders do not cross, and operating under a base-stock policy or order-up-to-s policy have been presented. Using these results, we derive the convexity of the expected discounted su of holding and lost-sales costs in the planning horizon with respect to the order-up-to paraeter S. The proof of this result requires an additional assuption about the leadtie process. This result is an extension of the result shown by Downs et al. (2001) for systes where lead ties are deterinistic and holding costs are charged only on inventory on hand. The convexity result can be exploited (as is done in Downs et al. 2001) to copute optial order-up-to policies (optial within that class of policies) for ultiple products with a budget constraint. In addition, it justifies the use of coon search techniques or infinitesial perturbation analysis for deterining optial order-up-to levels. The convexity result has also been extended to the infinite-horizon, discounted-cost, and average-cost odels. To our knowledge, this paper is the first analytical work on discrete-tie inventory odels with lost sales and stochastic lead ties. It is hoped that this work creates ore interest in analytical results for lost-sales probles. Appendix. Alternate Assuption about the Starting State We assue that the starting state is a fixed vector independent of S. We state this assuption next. Assuption 4. Let v 0 represent the vector of inventory levels at different stages of the pipeline, including inventory on hand, at the start of period 0 before placing an order in period 0. v 0 is independent of S. The diension of v 0 is the axiu possible lead tie. Let x 0 S and 0 S represent the inventory on hand and the total pipeline inventory at the start of period 0 after placing an order and receiving shipents due in period 0, when an order-up-to-s policy is used. Consequently, x 0 S + 0 S S. Lea 1 and Corollary 2 can easily be verified with the alternate assuption about the starting state. Unfortunately, the discounted su of lost-sales costs over a finite or infinite horizon need not be convex in S with this assuption. To see this, consider an exaple where v 0 is such that the pipeline is epty and x 0 = S 0 +. Consider three order-up-to policies with the following order-up-to levels: S 0, S 0 +, and S The first two policies order nothing in period 0, but the third syste orders. Assue d 0 = S 0 +, L 0 = 1, L 1 1, and d 1 =, so l 0 S 0 = l 0 S 0 + = l 0 S = 0, l 1 S 0 = l 1 S 0 + =, and l 1 S = 0. Therefore, l 0 S + l 1 S is not a convex function. However, we argue next that the infinite-horizon results for the undiscounted odel continue to hold. Let us now assue that there is a positive probability of nonzero deand occurring in a period, in addition to Assuption 3. We now clai that Theores 11 and 12 continue to hold with the alternate assuption about the starting state. This is easy to see because it is clear that the tie to reach a state in which the inventory on hand is S and none is on order is finite, with probability one. Consequently, the contribution of the periods before the period in which this state is reached to the long-run average cost is zero, with probability one. Acknowledgents The authors thank the associate editor and the referees for providing any valuable suggestions for significantly iproving the content and exposition of this paper. They thank Paul Zipkin for pointing the to Karush (1957). The first author was a doctoral student in the School of O.R.I.E. at Cornell University when this work was conducted. References Agrawal, N., S. Sith Estiating negative binoial deand for retail inventory anageent with unobservable lost sales. Naval Res. Logist. 43(6) Donselaar, K., T. Kok, W. Rutten Two replenishent strategies for the lost sales inventory odel: A coparison. Internat. J. Production Econo Downs, B., R. Metters, J. 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