Optimality and Nonoptimality of the Base-stock Policy in Inventory Problems with Multiple Delivery Modes

Transcription

1 Optimality and Nonoptimality of the Base-stoc Policy in Inventory Problems with Multiple Delivery Modes Qi Feng School of Management, the University of Texas at Dallas Richardson, TX , USA Guillermo Gallego Dept. of Industrial Engineering and Operations Research The Columbia University in the City of New Yor, USA Suresh P. Sethi School of Management, the University of Texas at Dallas Richardson, TX , USA Houmin Yan School of Management, the University of Texas at Dallas Richardson, TX , USA and Hanqin Zhang Institute of Applied Mathematics Academy of Mathematics and Systems Sciences Academia Sinica, Beijing, , China February 1, 2004 We present a periodic review inventory model with multiple delivery modes and demand forecast updates. We generalize the notion of the base-stoc policy for inventory system with multiple delivery modes. While base-stoc policies are optimal for one or two consecutive delivery modes, it is not so otherwise. For multiple consecutive delivery modes, we show that only the fastest two modes have optimal base stocs, and provide simple counter examples to show that the remaining ones do not. We investigate why the base-stoc policy is or is not optimal in different situations. Subject Classifications: Inventory, base-stoc policies, multiple delivery modes. 1. Introduction It is common in practice that a buyer chooses among different alternatives to replenish his inventory for certain goods. Typically, the shorter is the lead time, the higher is the procurement cost. Thus, the buyer may begin with ordering early with long lead times to save 1

2 money. As demand information unfolds gradually over time, the buyer is able to place several more expensive orders to supplement the earlier order. The strategic importance of having faster delivery options is underscored by the case of the fashion industry, in which many firms have moved their major manufacturing facilities to Asian countries to tae advantage of lower production costs there. However, it taes wees for container ships to reach North America. When it is close to the selling season and new information indicates a demand surge, it may then not be feasible for the firm to replenish its inventory through overseas orders and still meet the increased demand. For this reason, many fashion industry firms still maintain some of their domestic factories which, while producing goods at higher cost, can respond quicly to demand surges. An important factor contributing to the higher cost of fast delivery options is the transportation cost. An emergency order might be filled by air shipment, which is certainly faster and more costly than the slower rail or maritime shipments. Moreover, sourcing from multiple suppliers may also reduce the buyers procurement riss. It is quite usual that different suppliers are characterized by different prices and lead times for the same product. Suppliers who are capable of delivering faster may reasonably as for higher prices. In most of the studies on inventory models with replenishment lead time options, it is assumed that there are two consecutive procurement modes available. That is, the lead times of the two modes vary by exactly one period. This assumption is very restrictive. Under many circumstances, there are more than two supply modes (Zhang 1996), and/or the modes are not consecutive (Beyer and Ward 2000). To explore the general problem, we consider N consecutive delivery modes. In the case of non-consecutive delivery modes, it is easy to insert fictitious delivery modes as suggested in Sethi et al. (2001) to transform the problem into one with consecutive modes. All one needs to do is to set the cost for any fictitious mode to equal that for the next faster mode that is real. It should be clear that setting costs in this way would mean that we can consider policies which do not issue orders using any of the fictitious modes. Also, the case of N modes with lead times L+1,L+2,...,L+N can be reduced to our case by the standard device used in converting a single mode problem with a lead time to one with no lead time. Thus, our N-consecutive-mode formulation enables us to treat cases where fast and slow modes could be several orders of magnitude apart. A typical case is that of deliveries by air and surface shipments. A real-life instance of such a case is reported in an HP study by Beyer and Ward (2000). It concerns an important component of HP Windows NT networ server MOD0 box. The MOD0 box is pre-assembled in Singapore and then shipped to distribution centers, where the servers are finally assembled 2

3 and configured according to re-sellers specifications. The factory in Singapore can ship to distribution centers in France, Canada and Mexico by either air or ocean. Suppose that the air and sea mode lead times for a given distribution center are T a and T s, respectively. Then we can add (T s T a 1) fictitious delivery modes between air and sea shipments to solve the replenishment problem faced by the distribution center. In this paper, we provide a general formulation of a finite-horizon periodic review inventory system with multiple delivery modes. In each period, the decision maer can place N types of orders. A type j order is characterized by its unit ordering cost and a lead time of j periods. In the case of single delivery mode, it is well nown that the optimal ordering policy is a base-stoc policy. A fundamental characteristic of the base-stoc level in the classical single delivery mode inventory problem is that the level is independent of the preorder inventory position, since any ordering policy can be converted to an order-up-to policy simply by adding the order quantity to the inventory position. In the case of two consecutive modes, it has also been shown that the optimal policy is of base stoc type. When we move to more than two modes, the issues become substantially more complicated because there are N different inventory positions to consider in each period. How do we define base-stoc policies in this case? When is a base-stoc policy optimal? We try to address these questions in this paper. The remainder of the paper is organized as follows. The next section briefly reviews the related literature. In Section, we establish the model and discuss the condition under which certain modes are not used. In Section 4, we summarize the discussion in the literature on inventory models with single mode and two modes, and propose two different definitions of the base-stoc policies. Section 5 presents an example which shows that the optimal ordering policy in inventory models with two non-consecutive modes is not a base-stoc policy in general. In Section 6, we discuss the policy structure of inventory models with more than two modes. Finally, we conclude the paper in Section Literature Review Since the early 50 s, papers by Arrow, Harris and Marscha (1951) and Dvoretzhy, Kiefer and Wolfowitz (1952a,1952b) have stimulated a great deal of research on dynamic inventory models. Since the space does not permit us to summarize the enormous inventory literature that has accumulated over the last fifty years, we limit ourselves to giving an overview of

4 the related wor on base-stoc policies with particular emphasis on single product inventory systems with multiple procurement modes. 2.1 Base-stoc policies in problems with single delivery mode The early discussions on the base-stoc policy for systems involving a single product and a single delivery mode have focused on proving its optimality under the situation that the demands are independent. Examples can be found in Gaver (1959, 1961), Karlin (1958), Karlin and Scarf (1958), etc. Bellman et al. (1955) and Karlin (1960) are the classical papers for the stationary and nonstationary demand cases, respectively. Optimality of a base-stoc policy has been established in these situations. Iglehart and Karlin (1962) first address the problem with dependent demand. They consider an inventory model with the demand process governed by a discrete-time Marov chain. In each period, the current value of the state of the chain decides the demand density for that period. They first propose a state-dependent base-stoc policy for a model with linear ordering cost. Song and Zipin (1997) and Sethi and Cheng (1997) show optimality of a state-dependent base-stoc policy to be optimal when the demand is modeled as a Marovmodulated process, and when the fixed setup cost in their models is zero. Cheng and Sethi (1999) extend the result of Marov-modulated demand models by introducing promotional decisions. Another stream of related research focuses on a product with multi-class demands pioneered by Veinott (1965a). Topis (1968) considers an extended model with one replenishment opportunity at the beginning of the horizon, convex holding cost in each period, and n classes of demands each with a different penalty cost. He shows that a rationing level policy is optimal. Under such a policy, one tries to satisfy demands with higher penalty costs first. Moreover, the optimal post-action inventory position for each demand class is determined by satisfying as much demand for that class as possible with existing stoc, without letting the inventory position drop below a certain critical number associated with that class. The rationing level policy shares certain similarities with base-stoc policies. The only difference is that in Topis s model, the order is placed only at the beginning of the horizon, and the demand is filled in each period with on-hand inventory. Thus, the direction of inventory position change after an action is different from a usual ordering system. We shall point out that the approach used in models with multi-class demands cannot be adopted to study models with multiple delivery modes, since the structures of the cost functions are different. 4

5 Also there are discussions on base-stoc policies for multi-product inventory systems (e.g. Veinott 1965b,1970). Again, the problem structure is different from the one considered here. 2.2 Inventory models with two consecutive modes The earliest discussion on inventory models with two delivery modes can be traced bac to Baranin (1961), who studies a single period problem. Daniel (196) analyzes a multiperiod model with one regular order and one emergency order. Fuuda (1964) uses dynamic programming approach to derive the optimal ordering policy for inventory models with two delivery modes. Parallel to Fuuda, Neuts (1964) also proves the structure of the optimal policy for a similar model. Veinott (1966) gives an alternative proof of Fuuda s result in a survey paper, where he applies an observation by Karush (1957). Recently, several authors have studied different variations of the problem. As examples, we mention three such studies. Sethi et al. (2001) introduce an information updating scheme into the model. Lawson and Porteus (2000) address policy structure of a multi-echelon system with an option of expediting at each location. Muharremoglu and Tsitsilis (200) extend the model of Lawson and Porteus (2000) by allowing for expediting from any stage in the supply chain to any downstream stage. Under a supermodularity assumption on the ordering costs, they show the optimality of an extended echelon base-stoc policy. However, the assumption breas down when converting a single-echelon, multi-mode inventory model with other than two consecutive delivery modes. Thus, these studies show that in the case of two consecutive delivery modes, a base-stoc policy or a modified base-stoc policy is optimal. That is, there are two critical numbers, independent of the initial inventory position, such that the optimal ordering policy is to order up to the critical numbers. As we will see in Section 4.2, the critical number for the slower order is not unique. Depending on how this critical number is chosen, the meaning of the base-stoc level is different. Aside from the discussions on policy structure, computational studies for inventory models with two modes also appear in the literature. For example, Moinzadeh and Nahmias (1988) study a continuous-review inventory system, and Tagaras and Vlachos (2001) show cost reduction by introducing a second replenishment option. 2. Inventory models with more than two modes An inventory model with three consecutive delivery modes is first examined by Fuuda (1964). He considers a special case when orders are placed only every other period. Under 5

6 this assumption, a base-stoc policy is shown to be optimal. Implicitly, he shows that the optimal order-up-to levels are independent of the initial inventory position. Zhang (1996) extends Fuuda s model by allowing three consecutive modes ordered every period. While she claims the optimality of a base-stoc policy under certain conditions, her claim is untrue because of the errors in Lemma 1 and 2 in her paper. Feng et al. (200) study an inventory model with three consecutive delivery modes and demand forecast updates. They show that there exist optimal base-stoc levels for the two faster modes.. Problem Formulation and Effective Modes In this section, we present our basic model and discuss the conditions under which some of the delivery mode are never used in the ordering system..1 Problem Formulation We consider a finite horizon periodic review inventory system with N delivery modes. An order via the i th delivery mode, i = 1, 2,...,N, is an order associated with a lead time of i periods, and is termed the type i order. The notation is summarized as follows: 1,T = {1, 2,...,T }, the time periods; (Ω, F, P) = the probability space; Q i = the order quantity of type i order in period, 1 T, 1 i N; c i = the unit procurement cost of type i order in period, 1 T, 1 i N; D = the random demand in period, 1 T; R i = the i th determinant of demand in period, 1 T, 1 i N; x = the inventory/baclog level at the beginning of period, 1 T; y = the initial inventory position at the beginning of period, 1 T; z i = the post-order inventory position for period + i 1 viewed at the beginning of period,1 T, 1 i N; H ( ) = the inventory holding cost (assuming convex) in period, 1 T; H T+1 ( ) = the inventory and/or penalty costs for the ending inventory/baclog. Fig. 1 shows the inventory positions viewed at the beginning of period (We denote ql i as the realized value of Qi l ). There are N inventory positions to consider, each of which 6

7 corresponds to one type of order. First, we review the initial inventory position y, including the inventory/baclog level x and in-transit orders to be delivered at the end of period. The post-order inventory position in period is simply the sum of y and Q 1. Then we examine the post-order inventory for the subsequent N 1 periods. We observe that in-transit orders {q i +j i }N i=j+1 to be delivered in period +j 1 affect the ordering decision only through their sum p j. The post-order inventory position zj for period +j 1 contains all previous and current orders to be delivered by that time. Hence, we have the relation z j = zj 1 + p j + Qj. Figure 1: Post-order inventory positions viewed at the beginning of period. Period l Post-order inventory position in period l : Order placed in period to be delivered in period l : y N 2 + N 1 z 1 z 2 z z N 1 Q 1 q 2 1 Q 2 z N q 2 q 1 Q History of orders to be delivered in period l : q 4 q q N 1 N+2 q 4 2 q q N 1 N+ q 4 1 q q N 1 N Q N 1 In-transit orders to be delivered in period l q N N+1 q N N+2 q N N+ q N 1 = p 1 = p 2 = p = p N 1 Q N We use the demand forecast updating scheme introduced in Sethi et al. (2001). Accordingly, the demand D is hidden in the core of an N-layered onion containing uncertainties modeled by random variables {R 1,...,RN, R }. The demand signal R j is observed at the beginning of period ( N +j). At the end of period, the last determinant R is observed and demand is realized. Thus, the demand can be written as a function of the random variables {R 1,...,RN, R }, i.e., D = d (R,...,R 1 N, R ). (1) For convenience in exposition, we assume that R 1,...,RN, R are independent random variables. We also define a set of variables called relevant demand history at the beginning of 7

8 period as follows R = R 1, R R N, R 1 +1, R R N 1 +1, R 1 +N 1. At the beginning of period, the values of {{R j l }N j=1, R l } 1 l=1 R are nown. However, the signals for the realized demands in the first ( 1) periods are irrelevant for the ordering decisions (Q 1,...,QN ). Only the information R about future demands is relevant. In this setting, {R 1,...,RN, R } 1 T capture completely the randomness of the underlying process. Let {F } 1 N be the filtration of the underline process. Then we can tae F as the sigma algebra generated by {{R j l }N j=1, R l } 1 l=1 R. We denote the corresponding lower case r j to be the observed value of the signal Rj. Explicitly, if we evaluate D at period l, then d (R 1,...,RN, R ), for l N, D = d (r 1,...,rl +N 1,R l +N,...,R N, R ), for N < l, d (r 1,...,rN,ṙ ), for l + 1. Remar 1 In reality, the demand forecast may not necessarily update exactly N + 1 times. For example, in some period j ( N < j ) there is no update for D, or information for D may be acquired before period N +1. In the first case, we can simply set R N ( j) to any constant. In the second case, we can tae R 1 to contain all the information about D gathered before the beginning period N + 1, since the first time we can order for period is at the beginning of period N + 1 when R 1 (2) is realized. Note that the conditional distribution of R 1 given F j is different for different j N + 1, which affects ordering decisions before period N + 1. We tae N + 1 forecast updates here just for the purpose of exposition. Now we can write the dynamic programming equation as follows: U (x,p 1,...,p N 1 ; r ) () { N = H (x ) + min c i Q i Q 1 0,...,QN 0 + i=1 ( E [U +1 Ψ(Q 1 );p 2 + Q 2,...,p N 1 + Q N 1,Q N ;R+1,...,R N K+N, r )] } 1, 8 = 1, 2,...,T,

9 where Ψ(Q 1 ) = x + p 1 + Q 1 d (r 1,...,r N, R ), and r = r {r 1,...,r N }. In terms of inventory positions (z 1,...,zN ), () is equivalent to W (y,p 2,..,p N 1 ; r ) = U (x,p 1,...,p N 1, r ) EH (x ) (4) { N = min c 1 (z 1 z 1 y y ) + c j (zj zj 1 p j ) + EH +1(z 1 D ), z j zj 1 +p j j=2 +EW +1 ( z 2 D,z z 2,...,z N z N 1 ;R N +1,...,R 1 K+N, r ) }. Throughout this paper, we denote J (z 1,...,zN ; r ) or J (Q 1,...,QN ; r ) to be the objective function for period, i.e., the function inside min of () or (4). Also, ( z 1,..., zn ) is the unconstrained minimizer to J, (z 1,...,zN ) are the optimal post-order inventory positions, (Q 1,...,QN ) are the optimal order quantities, and zi is the optimal base-stoc level for the type i order (if exists)..2 Ordering Costs and Effective Modes In reality, some of the delivery modes in certain period may never be used. In what follows, we identify conditions under which certain modes are never ordered. Proposition 1 If c j with no type j order placed in period. That is, Q j Proof. cj i +i for some j > 1 and 1 i < j, then there is an optimal solution = 0 or zj = zj 1 + p j. Without loss of generality we set i = 1. Suppose the optimal solution in period is (Q 1,...,QN ) with Qj > 0. Then, = = J (Q 1,...,Q N ; r ) N c i Q i + EH +1 (x + Σ N i=2q i+1 i + Q 1 D ) i=1 ( +EU +1 Ψ(Q 1 );p 2 + Q 2,...,p N 1 N i=1 + Q N 1,Q N ; R ) +1 c i Q i + EH +1 (x + Σ N i=2q i i+1 + Q 1 D ) 9

10 i j +E min Q i +1 0, i=1,...,n [ N i=1 c i +1Q i +1 ( +EU +2 Ψ(Q 1 +1);p + Q + Q 2 +1,...,p N 1 + Q N 1 + Q N 2 +1,QN + Q N 1 +1,QN +1; R )] +2 c i Q i + EH +1 (x + Σ N i=2q i i+1 + Q 1 D ) +E min Q i +1 0, i=1,...j 2,j...,N, Q j Qj 0 [ i j c i +1Q i +1 + c j 1 +1 (Qj Q j ) (6) ( +EU +2 Ψ(Q 1 +1);p + Q + Q 2 +1,...,p N 1 + Q N 1 + Q N 2 +1,QN + Q N 1 +1,QN +1; R +2 )]. To see the last inequality, we first note that c j Q j c j i +i Q j. Also, observe that Qj and Q j 1 +1 affect the value of U +2 only through their sum. Now compare the functions inside min of (5) and (6). Any (Q 1 +1,...,QN +1 ) feasible in (5) is also feasible in (6). Hence, we conclude the proposition. Proposition 1 indicates that a slow mode is only utilized when it saves ordering costs. The orders Q j and Qj i +i are delivered at the same time. Because of demand uncertainty, if ordering early in period does not save anything, then we should certainly postpone the ordering decision until period + i when we now more about demand. In this case, we say that the j th mode in period is not an effective mode. On the other hand, when demand is certain, we would lie to place only the cheap early orders. The proof of the next result is similar to that of Proposition 1, and hence it is omitted. Proposition 2 Suppose the demands for periods s,t are deterministic. Then the j th mode in period t is not effective if c j i t i cj t for some i 1,t s. In the single mode case, it is well-nown that if holding/baclog costs are linear and if ordering cost in a given period is larger than the baclog cost plus the ordering cost in the next period, then it is not optimal to place any order in the given period. In the case of multiple delivery modes, we have a similar result. Proposition If the holding/baclog cost for each period is given by { h + x if x 0, H (x) = x if x < 0, h 10 (5)

11 and either or c j c j s 1 h +j+i + cj+s, (7) i=0 s i=1 h + +j i + cj s, (8) then the j th mode in period is not effective. The proof of Proposition is straightforward and is therefore omitted. The proposition simply says that if ordering one unit Q j for period +j 1 is more costly than baclogging this unit for s periods and ordering one more unit of Q j+s for period + j + s 1 (resp. more costly than ordering one more unit of Q j s for period + j s 1 and carrying this unit for s periods), then we should not order any positive Q j. 4. Definition of Base-stoc Policies 4.1 Inventory Models with Single Delivery Mode In the situation with single delivery mode, the problem is essentially a one dimensional convex function minimization problem. The cost function (4) for period becomes W (y ;r 1 ) = min z y {c (z y ) + EH [z d (r 1, R )] + EW +1 (z d (r 1, R ),R 1 +1)}. Denote z as the unconstrained minimizer to the function inside the min. The optimal post-order inventory position is simply z = z y. Since z is independent of the inventory position y, it is called the base-stoc level for period. Note that the result holds even when we allow convex ordering cost c ( ) (e.g., Karlin 1962). 4.2 Inventory Models with Two Consecutive Delivery Modes As we mentioned in Section 2, several authors have shown the policy structure for inventory models with two consecutive modes. These models are similar to the one described here in spirit, and the optimal ordering policies maintain the same structure. The cost function (4) for period becomes W (y ;r 1,r 2,r 1 +1) = { min c 1 (z z 2 z1 y 1 y ) + c 2 (z 2 z) 1 + EH +1 [z 1 d (r,r 1, 2 R )] } +EW +1 (z 2 d (r,r 1, 2 R );r+1,r 1 +1,R 2 +2) 1. 11

12 The function inside the min J (z 1,z2 ;r1,r2,r1 +1 ) is clearly jointly convex and separable in (z 1,z2 ). The following proposition indicates that the optimal ordering policy for period is a base-stoc policy. The proof can be found in Feng et al. (200). Proposition 4 Consider the problem P(y) = min{f 1 (z 1 ) + f 2 (z 2 ) z 2 z 1 y}, with f 1 (z 1 ) and f 2 (z 2 ) to be convex in z 1 and z 2, respectively. There exist critical numbers z 1 and z 2, independent of y, such that the solution to P(y 1 ) taes the form z 1 = z 1 y, z 2 = z 2 z 1. (9) Remar 2 Note that a linear ordering cost is necessary for (9). The objective function J (z 1,z2 ;r1,r2,r1 +1 ) in this case is convex and separable. When objective is convex but not separable, (9) is not always true. Proposition 4 suggests an ordering policy that can be implemented in the following fashion. At the beginning of period, we first review the initial inventory position y and compare it with the base-stoc level z 1. If y < z 1, we place a type 1 order, and the postorder inventory position for period increases to z 1 ; otherwise, we do not place a type 1 order. In the second step, we consider the reference inventory position for period + 1, taing into account the type 1 order decision. Thus, the reference inventory position for period + 1 becomes z 1 = z 1 y. If z 1 is less than z 2, we place a type 2 order and bring the inventory position up to z 2 ; otherwise, the optimal policy calls for no ordering. Such a policy is named a base-stoc policy or a modified-base stoc policy in the literature. However, the meaning of the base stocs is not clarified. To see this, let us define z i = arg min z i f i (z i ) and ẑ 1 = arg min z 1{f 1 (z 1 ) + f 2 (z 1 )}. Let z 1A = z 1 ẑ 1, z 2A = z 2 ; (10) z 1B = z 1 ẑ 1, z 2B = z 2 ẑ 1. (11) From Lemma 4.2 in Feng et al. (200), one can easily deduce that both ( z 1A, z 2A ) and ( z 1B, z 2B ) satisfy (9). z 1A and z 1B are independent of their reference inventory positions y. However, the situation for the second mode z 2 is somewhat subtle. In the first case, z 2A does not depend on how z 1 is decided. While in the second case, z 2B does have some relation 12

13 with z 1, since both depend on the value ẑ 1. If z 1 = ẑ 1, then z 2B = ẑ 1. Thus, the critical number in (11) for the second mode z 2B does depend on its reference inventory position z 1. Whether this number can be called a base-stoc level depends on how we define base-stoc policies. 4. Definitions of Base-stoc Policies We propose two definitions of base-stoc policies. Definition 1 A decision rule is called a base-stoc policy, if there exist critical numbers ( z 1,..., z N ), called base-stoc levels, such that the post-action inventory positions (z 1,...,z N ) are as close to the base-stoc levels as possible. Moreover, the critical numbers should be independent of the initial inventory position. In our model, this definition indicates that the orders (Q 1,...,Q N ) or the post-order inventory positions (z 1,...,z N ) satisfy Q j or z j = = max { z j x } j 1 j p i Q i, 0 for j = 1,...,N; (12) i=1 i=1 z 1 y if j = 1, z j (z j 1 + p j ) for j = 2,...,N 1 z N z N 1 for j = N. Also, ( z 1,..., z N ) should be independent of initial inventory position y. Definition 2 A decision rule is called a base-stoc policy, if there exist critical numbers ( z 1,..., z N ), called base-stoc levels, such that the post-action inventory positions (z 1,...,z N ) are as close to the base-stoc levels as possible. Moreover, the critical numbers are independent of their respective reference inventory positions. In our model, Definition 2 indicates that the orders (Q 1,...,Q N ) and the post-order inventory positions (z 1,...,z N ) should satisfy (12). Also, z 1 is independent of y, z j is independent of z j 1 + p j for j = 2,...,N 1, and z N is independent of z N 1. The two definitions are equivalent when only one procurement mode is available, because there is only one inventory position to consider. In the case of two consecutive delivery modes, previous studies in the literature focus on a solution satisfying the requirement in Definition 1 (e.g., Neuts 1964). Recalling the discussion at the end of Section 4.2 where we 1

14 had two candidate base-stoc policies, the levels ( z 1A, z 2A ) are base stocs according to both definitions, while ( z 1B, z 2B ) are not base stocs according to Definition 2. When the two modes are not consecutive or there are more than two modes, the policy structure is more complex because of in-transit orders to be delivered in the future. Definition 2 seems to be more appealing in these situations. By Definition 2, at the beginning of period, one should not require any information of the inventory position y and the history of orders (p 2,...,pj ) to be delivered before or in period +j 1 to decide the base-stoc level z j for period + j 1. On the other hand, the base-stoc level zj defined in Definition 1 is only independent of the inventory position y. Thus, Definition 2 is more restrictive than Definition 1. Remar The definitions proposed here can be generally applied to many other problems, e.g., the situation described by Topis (1968), multiple-product ordering systems, etc. 5. Two Non-Consecutive Modes: An Example In the previous section, we have seen that the base-stoc policy is optimal for inventory system with two consecutive delivery modes. Is the base-stoc policy optimal for inventory system with two non-consecutive modes? Unfortunately, the answer is no. In this section, we try to explore the reason through an example. For simplicity, we do not consider demand forecast updates in the numerical example. Consider a three period problem with the following settings: There are two types of orders, namely, a fast order Q 1 and a slow order Q, available in each period. The lead times of the fast and the slow orders are 1 period and periods, respectively. Note that this situation is equivalent to one with three consecutive modes in which Q 2 is not effective; see Proposition 1. Ordering costs for the fast and slow modes are c 1 = 10 and c = 1, respectively, and they are stationary over time. The holding/baclog costs are also stationary over time, i.e., H 2 (x) = H (x) = H 4 (x) = H(x) = x 2. 14

15 Figure 2: The cost function for period 2. q1 25 q1 + 2y 2 = W y s C A D q1 + y 2 = 15 B y 2 Note that H 4 (x) = x 2 indicates that unsatisfied demand at the end of the horizon is charged a penalty which is quadratic function in the baclog quantity. Also, the disposal cost for leftover inventory is quadratic. Demand is D 1 = D 2 = D = D = 10 units in each period. We first examine the optimal cost functions and ordering policies for the problem, and then provide some discussion. The details of the calculations are delegated to Appendix. 5.1 The Optimal Cost Functions and Ordering Policies i) Period. W (y ) = { 75 10y if y 5, (y 10) 2 if y > 5. (1) The optimal base-stoc level is z = z 1 = 5 ii) Period 2. W 2 (y 2, q1) = (14) y 2 10q1 if y 2 10 and q1 5, A 275 0y 2 + (y 2 ) 2 10q1 if 10 < y 2 15 q1, B y 2 15q (q 1 )2 if y q1 and q1 > 5, C y 2 + 2(y 2 ) 2 + 2y 2 q1 if y 2 > max{15 q1, q 1 }. D 40q1 + (q 1 )2 15

16 The optimal base-stoc level is { 10 if 0 q z 2 1 = 1 5, q1 if q1 > 5. iii) Period 1. W 1 (y 1, q0) = (15) y 1 10q0 if 0 q0 < 5.5, y 1 10, I(a) y q (q 0 )2 if 5.5 q0 <.5, y q0, I(b) 1175/ 10y 1 80/q0 + 2/(q 0 )2 if q0.5, y 1 (55 2q0 )/, I(c) y 1 + (y 1 ) 2 10q0 if 10 < y q0, II y 1 + 2(y 1 ) 2 + 2q0 y 1 if max{15.5 q0, q 0 } < y 1 41q0 + (q 0 )2, 29.5 q0, III y 1 + (y 1 ) 2 + 4q0 y 1 if y 1 > max{(55 2q0 )/, 29.5 q 0 }. IV 100q0 + 2(q 0 )2 Figure : The cost function for period 1. q 0 2q0 + y 1 = 55 I(c) q0 =.5 q0 + y 1 = W y s0 10 q0 + 2y 1 = 25.5 I(b) q0 = 5.5 I(a) 0 IV q0 III + y 1 = 15.5 II y 1 The optimal base-stoc level for the fast mode in period 1 is 10 if 0 q0 < 5.5, z 1 1 = q0 (55 2q0)/ if 5.5 q0 <.5, if q0.5. The optimal post-order inventory positions are (z 1 1,z 1 ) = 16

17 ( 10, 24 + q0 ) if y 1 10, ( y 1, y 1 + q ) if 10 < y q0, ( y 1, 29.5 ) if 15.5 q0 < y q0, ( y 1, y 1 + q0 ) if y q0 ( q0, 29.5 ) if y q0, ( y 1, 29.5 ) if q0 < y q0, ( y 1, y 1 + q0 ) if y q0 ( (55 2q0)/, 29.5 ) if y 1 (55 2q0)/, ( y 1, y 1 + q0 ) if y 1 > (55 2q0)/, if 0 q0 < 5.5, if 5.5 q 0 <.5, } if q 0.5. We notice that when 0 q0 < 5.5, there does not exist a base-stoc level for the slow mode in period 1. To see why, we tae a closer loo at what is happening in each period. 5.2 Discussion We first examine how the demand is satisfied in each period. Observe that since the fast ordering cost is the same for each period, it is never optimal to order on fast mode in an earlier period and hold inventory to a later period. Liewise, it is never optimal to order on fast mode in a later period and baclog demands from an earlier period. Suppose, for the moment, that the initial inventory position y 1 and in-transit order q0 are both zero. The demand in period can be satisfied via the slow order Q 1 in the first period and the fast order Q 1 in the third period. Since demand is deterministic, we now that Q 1 = 0 from Proposition 2. The quantity Q 1 needed in period is the minimizer to q + (q 10) 2, which turns out to be 9.5. That is, at the end of the horizon, we would lie to have 0.5 units of baclog. Hence, we need a total of 29.5 units of the product during the problem horizon. The demand in period 2 can be satisfied by the fast order Q 1 2 in period 2. Also, we can baclog some demand from period 2 to period by ordering less Q 1 2 but more Q 1. To do this, we save 9 dollars per unit ordering cost and pay the baclog cost. Thus, the optimal quantity to baclog is the maximizer to 9q q 2, which is 4.5. Hence, Q 1 2 should bring the inventory position for period 2 to at least 5.5 = (10 4.5). Now we come to period 1. Since the maximizer to 9q q 2 (4.5 + q) 2 is zero, it is not optimal to baclog the demand in period 1 for two periods and satisfy it via Q 1. We should order 10 units of Q

18 It seems that the ideal order-up-to level for z1 would be 29.5, if the ordering policy for the slow mode were to follow a base-stoc policy. However, this cannot always be achieved when q0 is low. Table 1 shows different inventory positions and corresponding optimal order quantities over time when 0 q0 < 5.5. Table 1: When 0 q 0 < 5.5. Period y 1 (, 10] (10, 15.5 q 0) [15.5 q 0, 24.5 q 0] (24.5 q 0, 29.5 q 0] (29.5 q 0, ] 1 z Q y y 1 + Q q q 0 y 1 + q 0 y 1 + q 0 y 1 + q 0 y 1 + q 0 Q y 1 q y 1 q 0 0 (z1 1, z1 ) (10, 24 + q0) (y 1, y 1 + q0 + 14) (y 1, 29.5) (y 1, 29.5) (y 1, y 1 + q0) Period y 2 q0 y 1 + q0 10 y 1 + q0 10 y 1 + q0 10 y 1 + q z y q Q q y 1 q z y 1 + q0 10 y 1 + q0 10 y 1 + q0 10 Period y y 1 + q0 20 z Q z y 1 + q0 20 End y y 1 + q0 0 When y 1 (, 15.5 q 0) and q 0 [0, 5.5), we have to order a positive Q 1 2 in period 2 and order Q 1 = 14(= ) in period 1. In this case, the inventory position z 1 does not account for the value of Q 1 2, since it only includes what is on-hand and what has been ordered. So the optimal slow order Q 1 in period 1 does not follow a base-stoc policy. On the other hand, when y q 0, there is no fast order placed in period 2. All the demands have to be satisfied by the decision in period 1. In this case, the optimal inventory position z 1 should be as close to 29.5 as possible. One may thin that if we modify the reference inventory position y 1 +q 0 +Q 1 1 to include the anticipated order Q 1 2, then we could set 29.5 to be the base-stoc level for z 1, and have the optimal ordering policy to be a base-stoc type policy. While this could wor in this deterministic demand example, it would not wor in the case of general stochastic demand. In this case, Q 1 2 is a random variable at the beginning of period 1, and no simple deterministic equivalence of this decision can be included in the reference inventory position to restore the base-stoc policy. We examine the stochastic demand case in Section 6. 18

19 Table 2: When 5.5 < q 0.5. Period 1 y 1 (, q0] ( q0,29.5 q0] (29.5 q0, ] z q q q0 Q q0 y y 1 + Q q q0 y 1 + q0 y 1 + q0 Q q y 1 q0 0 (z1 1,z1 ) ( q0,29.5) (y 1,29.5) (y 1,y 1 + q0) Period 2 y q0 y 1 + q0 10 y 1 + q0 10 z y q y q Q z q0 y 1 + q0 10 y 1 + q0 10 Period y y 1 + q0 10 z Q z y 1 + q0 10 End y y 1 + q0 0 As shown in Table 2, when 5.5 q0.5, demand in period 2 can be satisfied through 5.5 units from q0 and 4.5 units from Q 1. Thus, no order in period 2 is needed and Q 1 2 = 0. Also, the remaining q0 5.5 can be allocated to satisfy the demand of either period 1 or of period. In the former case, we baclog demand in period 1 and save the ordering cost for Q 1 1; in the latter case, we hold inventory at the end of period 2 and save the ordering cost Q 1. The marginal cost curve turns out be to the same for the two scenarios. Thus, we split q0 5.5 evenly. As a result, we have 0.5q units of baclog at the end of period 1, and order only q0(= 14 (0.5q0 2.75)) units of Q 1. When q0 5.5, the fast order Q 1 2 in period 2 is never placed, and optimal post-order inventory position z1 should be as close to 29.5 as possible. Later we will see that such a relationship between Q 1 2 and z1 is not accidental. Since the value of q0 decides whether such a situation would happen, 29.5 cannot be called a base-stoc level rigourously according to Definition 2. However, z 1 = 29.5 is a base stoc in the sense of Definition 1 when q We now summarize our findings from this example: In general, the optimal ordering policy for an inventory system with two non-consecutive modes is not a base-stoc policy. Our example indicates that the base-stoc policy fails on the slow mode (the type order). In our example, the base-stoc policy fails to be optimal when 19

20 1. there are two non-consecutive delivery modes, and/or 2. demand is deterministic, and/or. all costs are stationary. The policy structure for period 1 is closely related to the size of the in-transit order q0 to be delivered after period 1. When q0 is large enough, the optimal ordering policy follows a base-stoc policy in the sense of Definition 1. Finally, we remar that the occurrence of the base-stoc policy in period 1 coincides with no fast order in period 2 under the optimal policy for this example. 6. Inventory Models with More than Two Modes The example described in the last section is a special case of an inventory system with multiple modes. The optimal ordering policies for inventory models with more than two modes are no longer base-stoc policies in general. The complexity of the problem with more than two modes increases because the decision has to tae into account the in-transit order to be delivered in the future. A natural question to as is when the optimal policy follows a base-stoc policy? If not, does the optimal ordering policy have some structure? 6.1 Separability and The Base-stoc Policy The following proposition states that in an optimal order policy, the first two modes follows a base-stoc policy. Proposition 5 Define P(y,p 2,..p N 1 ) = min{f(z 1,...,z N ) z 1 y,z j z j 1 + p j for j = 2,...,N} where F(z 1,...,z N ) = G 1 (z 1 ) + G 2 (z 2,...,z N ) with G 1 (z 1 ) convex in z 1 and G 2 (z 2,...,z N ) jointly convex in (z 2,...,z N ). Let (z 1,...,z N ) be any solution to P(y,p 2,..p N 1 ). Then there exist two reals z 1 and z 2 such that z 1 = z 1 y, z 2 = z 2 (z 1 + p 2 ). The proof of Proposition 5 will use the following lemma. 20

21 Lemma 1 Suppose G(x 1,...,x n ) is jointly convex in (x 1,...,x n ). Define G (x 1,...,x ) = min{g(x 1,...,x n ) x j x j 1 + a j for j = + 1,...,n} for = 1,...,N. Then G (x 1,...,x ) is jointly convex in (x 1,...,x ). Proof. We show that G (x 1,...,x ) is jointly convex by induction on. Clearly, when = N, G (x 1,...,x N ) = G 2 (x 1,...,x N ) is jointly convex. Suppose that G (x 1,...,x +1 ) is jointly convex in (x 1,...,x +1 ). Then, G (x 1,...,x ) = min{g(x 1,...,x N ) x j x j 1 + a j for j = + 1,...,N} = min{g (x 1,...,x +1 ) x +1 x + a +1 }. Now define x +1 (x 1,...,x ) = arg min x +1 G (x 1,...,x +1 ), then ) G (x 1,...,x ) = G (x 1,..., x +1 (x 1,...,x ) (x + a +1 ). ) Clearly, G (x 1,..., x +1 (x 1,...,x ) is convex since the lower envelope of a convex function is convex. Thus, G (x 2,...,x ) is convex. Proof of Proposition 5. By Lemma 1, we can rewrite problem P(y,p 2,..p N 1 ) as P(y,p 2,..p N 1 ) = min{g 1 (z 1 ) + G 2(z 2 ) z 1 y,z 2 z 1 + p 1 }. Denote z 1, z 2 and ẑ 1 as the unconstrained minimizer to G 1 (z 1 ), G 2(z 2 ), and G 1 (z 1 ) + G 2(z 1 ), respectively. Let z 1 = z 1 ẑ 1 and z 2 = z 2. Also note that { } min G 1 (z 1 ) + G 2(z 2 ) z 1 y z 2 z 1 +p 1 { ( )} = min z 1 y G 1 (z 1 ) + G 2 z 2 (z 1 + p 2 ). The function inside min on the right-hand side is convex in z 1. We consider two cases. Case 1: z 1 + p 2 z 2. First note that in this case z 1 ẑ 1, z 1 = z 1. We have z 1 + p 1 z 2. Thus, for any z 1 y G 1 ( z 1 y) + G 2 ( ) ( ) z 2 (y z 1 + p 2 ) G 1 (z 1 y) + G 2 z 2 (y z 1 + p 2 ). To see the last inequality, we note that the right-hand side is convex in z 1. If y z 1, the right-hand side is minimized at z 1 = z 1 = z 1. If y > z 1, the right-hand side is minimized at z 1 = y. Hence, the minimizer is z 1 = z 1 y = z 1 y and z 2 = z 2 (z 1 + p 2 ). 21

22 Case 2: z 1 + p 2 > z 2. In this case the minimizer must be such that z 1 + p 2 = z 2. Note that ẑ 1 < z 1 and z 2 < ẑ 1 + p 2. So z 1 = ẑ 1 and z 2 = z 2 < ẑ 1 + p 2. Thus, for any z 1 y, ( ) G 1 (ẑ 1 y) + G 2(ẑ 1 y + p 2 ) = G 1 (ẑ 1 y) + G 2 z 2 (ẑ 1 y + p 2 ) ( ) G 1 (z 1 y) + G 2 z 2 (z 1 + p 2 ). Similar to case 1, if y ẑ 1, then the right-hand side is minimized at ẑ 1. Otherwise, it is minimized at y. Hence, the minimizer is z 1 = ẑ 1 y = z 1 y and z 2 = z 1 + p 2 = z 2 (z 1 + p 2 ). Remar 4 Clearly, z 1 is independent of y, and it depends on p 2,...,p N 1. z 2 is independent of y and p 2, and it depends on p,...,p N 1. Since z 2 is decided without any concern of the first mode z 1, ( z 1, z 2 ) are base-stoc levels in the sense of both Definitions 1 and 2. Note that one could also tae z 2 as z 2 ẑ 1. In this case, z 2 = z 2 ẑ 1 is a base stoc according to Definition 1, but not according to Definition 2. The objective function J is clearly separable in the first two post-order inventory positions z 1 and z2. This is why the base-stoc policy is optimal when there are less than two modes. The next result indicates that the optimality of a base-stoc policy can always be established when the objective function is separable in the post-order inventory positions. Proposition 6 Consider the minimization problem { N } P(y,p 2,...,p N 1 ) = min f i (z i ) z 1 y;z j+1 z j + p j+1 for 1 j < N;z N z N 1, i=1 where f i (z i ) for 1 i N are convex functions. There exists a vector ( z 1,..., z N ), such that the solution (z 1,...,z N ) to problem P(y,p 2,...,p N 1 ) taes the following form z 1 = z 1 y, (16) z j = z j (z j 1 + p j ) for 1 < j < N, z N = z N z N 1. Moreover, z 1 can be taen to be independent of y, and z j can be taen to be independent of (y,p 2,...,p j ). 22

23 Proof. Define f,n (z ) = min{ N j= f j(z j ) z j+1 z j + p j+1 for j < N;z N z N 1 }. By Lemma 1, f,n (z ) is convex in z. Now we can show the result by induction. When = 1, tae z 1 as the unconstrained minimizer to f 1,n (z 1 ). Then z 1 = z 1 y. Note that z 1 is independent of y. Suppose that z j is the unconstrained minimizer to f j,n (z j ) and z j = z j (z j 1 + p j ) for 1 < j i. It is clear from Lemma 1 that (z 1,...,z i ) is the minimizer to min{ i 1 j=1 f j(z j ) + f i,n (x i ) z 1 y;z j+1 z j + p j+1 for 1 j < i}. Now tae z i+1 as the unconstrained minimizer to f i+1,n (z i+1 ). Then, { i } P(y,p 2,...,p N 1 ) = min z 1 y, z j z j 1 +p j, j=1,...,i+1 = min z 1 y, z j z j 1 +p j, j=1,...,i = min z 1 y, z j z j 1 +p j, j=1,...,i j=1 f j (z j ) + f i+1,n (z i+1 ) { i f j (z j ) + j=1 min fi+1,n (z i+1 ) z i+1 z i +p i+1 { i } f j (z j ) + f i+1,n ( z i+1 (z i + p i+1 )). j=1 Since the minimum on the right-hand side is attained at (z 1,...,z i ), we deduce that z i+1 = z i+1 (z i + p j ). Also note that z i+1 is independent of (p i+2,...,p N 1 ). Remar 5 The critical numbers ( z 1,..., z N ) are base-stoc levels in the sense of both definitions. Similar to the situation in two-mode case, one can tae ( z 1,..., z N ) as the solution to P(,p 2,...,p N 1 ), and show that the relationship in (16) still holds. In this case, the critical numbers are base stocs only in the sense of Definition Generalization of Fuuda The paper by Fuuda (1964) is probably the first and the only discussion on the policy structure for inventory ordering system with more than two modes. He considers an inventory system with three consecutive delivery modes. He shows that the optimal ordering policy follows a base-stoc policy when orders are only placed every other period. Fuuda s model is slightly different from the one described below. He does not consider demand forecast updates. Instead, he assumes stationary ordering costs for each period, and costs are discounted over time. In fact, the policy structure remains unchanged even without these } 2

24 assumptions. The optimality equation in this case can be written as W (y ; r ) = min {c 1 (z z z2 z1 y 1 y ) + c 2 (z 2 z) 1 + c (z z) 2 + EH +1 (z 1 D ) +EH +2 (z 2 D D +1 ) + EW +2 (z D D +1 ; R +2 )}. It is clear that the decision variable for the second mode z 2 and the third mode z are separable in the cost function when orders can be placed only every other period. Thus, what Fuuda does amounts to forcing separability, and therefore he gets the base-stoc policy. His result can actually be generalized by the following observation. Proposition 7 If the first mode in period + 1 is not used, then the first three modes in period have optimal base-stoc levels. Proof. can be written as If the first mode is not used in period + 1, then the cost function for period J (z 1,...z N ; r ) = c 1 (z 1 y ) + N j=2 c j (zj zj 1 p j ) + H (z 1 D ) +W +1 (z 2 D,z z,...,z 2 N z N 1 ; R +1 ) N = c 1 (z 1 y ) + c j (zj zj 1 p j ) + EH (z 1 D ) + j=2 +E min z +1 z4 D, z j +1 zj zj+1 z j, { c +1(z +1 z 4 + D ) + N j=4 c j +1 (zj +1 zj 1 +1 zj+1 + z j ) +EH +1 (z 2 D D +1 ) +EW +2 (z D D +1,z+1 z + D + D +1,...,z+1 N z N 1 +1 ; R } +1 ). Clearly, the last expression is separable in (z 1,z2,z ). Hence, the result follows from Proposition 6. Remar 6 Note that this result cannot be generalized for more than three modes. That is, if the first two modes in period + 1 are not used, then the fourth mode in period may not have an optimal base stoc. The reason is that EW +2 depends on z. Corollary 1 If the demand for period is deterministic, and the second mode in period is an effective mode, then the first three modes in period have base-stoc levels. 24

25 We have seen an example in Section 5, in which the demand in each period is deterministic, and the second mode is not effective. In that example, the base-stoc policy fail to be optimal. In the general case, the objective function is not completely separable in the post-order inventory positions. For example, when we have three consecutive modes (z 1,z 2,z ), the objective function is of the form G 1 (z 1 ) + G 2 (z 2,z ). Suppose that the unconstrained minimizers to G 1 (z 1 ) and G 2 (z 2,z ) are z 1 and ( z 2, z ), respectively. If the optimal ordering policy is a base-stoc policy, then ( z 1, z 2, z ) should be the optimal base-stoc levels. A necessary condition for the optimality of a base-stoc policy is that z = arg min z G 2 (z 2,z ) for each z 2 z 2. That is, the optimal base-stoc level for type order should not be affected by increasing of the second inventory position from its base-stoc level. However, when G 2 (z 2,z ) is jointly convex but not separable, this is hardly true. We illustrate the idea through an example. 6. An Example with Three Consecutive Modes Consider a three period example with following settings: There are three consecutive orders available in each period. We name them fast Q 1, medium Q 2, and slow Q. The demand D 1 for the first period follows uniform distribution over [0, 20], and the demands for the last two periods are deterministic D 2 = D = 20. Holding and baclog costs at the end of each period are given by { { 0 if x 0, x if x 0, H 4 (x) = H 10x if x < 0; (x) = H 4x if x < 0; 2 (x) = { 2x if x 0, x if x < 0. Note that at the end of the horizon, a penalty of 10 is charged for each unit of unsatisfied demand, and the leftover inventory is of no value. Ordering costs are stationary over time and are given by c 1 =, c 2 = 2 c = 1. Note that the costs do not satisfy the conditions described in Proposition. We first loo at the cost function and optimal ordering policy in each period (details of this example are available on Sethi s website sethi). 25

26 i) Period. Since the penalty cost is higher than the ordering cost and the leftover inventory is worthless, it is optimal to eep the inventory position as close to the demand as possible. Thus, W (y ) = { 0 if y 20, 60 y if y < 20. The optimal base-stoc level is z 1 = z 1 = 20. ii) Period 2. We first evaluate the objective function. where J 2 (z 1 2,z 2 2) = c 1 (z 1 2 y 2 ) + c 2 (z 2 2 z 1 2 q 1) + H (z 1 2 D) + W (z 2 2 D) = c 1 y 2 c 2 q 1 + G 1 2(z 1 2) + G 2 2(z 2 2), G 1 2(z2) 1 = (c 1 c 2 )z2 1 + H (z2 1 D 2 ) { 80 z 1 = 2 if z2 1 < 20, 60 4z2 1 if z2 1 20, G 2 2(z2) 2 = c 2 z2 2 + W (z2 2 D 2 ) { 400 8z 2 = 2 if z2 2 < 40, 2z2 2 if z The unconstrained minimizers to G 1 2(z 1 2) and G 2 2(z 2 2) are z 1 2 = 20 and z 2 2 = 40, respectively. We can further compute the optimal cost function for period 2 as W 2 (y 2,q 1) = = min z 1 2 y 2, z 2 2 z1 2 +q 1 J 2 (z 1 2,z 2 2) 100 2q1 y 2 if q1 20,y 2 20, (A) 20 2q1 + y 2 if 20 < y 2 40 q1, (B) 60 y 2 if q1 > 20,y 2 20, (C) 60 + y 2 if y 2 > max{20, 40 q1}. (D) The optimal base-stoc levels are ( z 1 2, z 2 2) = ( z 1 2, z 2 2) = (20, 40). Note that the demand for period 2 and are just satisfied in this situation. 26

27 Figure 4: The cost function for period 2. q1 100 W y s C A D q1 + y 2 = 40 B y 2 iii) Period 1. The objective function for period 1 is given by J 1 (z1,z 1 1,Q 2 1) = c 1 (z1 1 y 1 ) + c 2 (z1 2 z1 1 q0) + c Q 1 + EH 2 (z2 1 D) + EW 2 (z2 2 D,Q 1) = c 1 y 1 c 2 q0 + G 1 1(z1) 1 + G 2 2(z1,Q 2 1), where G 1 1(z1) 1 = (c 1 c 2 )z1 1 + EH 2 (z1 1 D 1 ) 40 z (z1) 1 2 if 0 z1 1 20, = 40 z1 1 if z1 1 < 0, 20 + z1 1 if z1 1 > 20, and G 2 1(z1,Q 2 1) = c 2 z1 2 + c Q 1 + EW 2 (z1 2 D 1,Q 1) 10 z1 2 Q 1 if Q 1 < 20,z1 2 < 20, (a) 20 9z (z1) 2 2 Q z1Q 2 1 if Q 1 < 20,20 z1 2 < 40 Q 1, (b) 250 9z (z1) 2 2 5Q z1Q (Q 1) 2 if Q 1 < 20,40 Q 1 z1 2 < 40, (c) = 90 z (z1) 2 2 5Q z1Q (Q 1) 2 if Q 1 < 20,40 z1 2 < 60 Q 1, (d) 90 z1 2 + Q 1 if Q 1 20,z1 2 < 20, (e) z (z1) Q 1 if Q 1 20,20 z1 2 < 40, (f) z1 2 + Q 1 if z1 2 max{40,60 Q 1}, (g) The unconstrained minimizers to G 1 1(z 1 1) and G 2 1(z 2 1,Q 1) are z 1 1 = 10 and ( z 2 1, Q 1) = (70/, 20), respectively. Since the holding/baclog costs are linear, and we are not 27