Optimal Control of Assembly Systems with Multiple Stages and Multiple Demand Classes 1

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1 Optima Contro of Assemby Systems with Mutipe Stages and Mutipe Demand Casses Saif Benjaafar Mohsen EHafsi 2 Chung-Yee Lee 3 Weihua Zhou 3 Industria & Systems Engineering, Department of Mechanica Engineering, University of Minnesota, Minneapois, MN 55455, saif@umn.edu 2 The A. Gary Anderson Graduate Schoo of Management, University of Caifornia, Riverside, CA , mohsen.ehafsi@ucr.edu 3 Department of Industria Engineering and Logistics Management, Hong Kong University of Science & Technoogy, Cear Water Bay, Kowoon, Hong Kong, arry@ust.h, cyee@ust.h October 20, 2006 Abstract We consider an assemby system with mutipe stages, mutipe items, and mutipe customer casses. The system consists of m production faciities, each producing a different item. Items are produced one unit at a time. To produce one unit of an item, one unit from each of its predecessor items is needed. Upon production competion, items are paced in inventory. At each decision epoch, we must determine whether or not to produce an item and shoud demand from a particuar cass arise whether or not to satisfy it from existing inventory, if any is avaiabe. Hence, at each epoch, we must mae decisions about both production and inventory aocation. In doing so, we must baance inventory hoding costs against shortage costs (ost saes or bacorders). We formuate the probem as a Marov decision process and use it to characterize the structure of the optima poicy. For production, we show that the optima poicy for each item is a state-dependent base-stoc poicy with the base-stoc eve non-increasing in the inventory eve of items that are downstream and non-decreasing in the inventory eve of a other items. For inventory aocation, we show that the optima poicy is a muti-eve state-dependent rationing poicy with the rationing eve for each demand cass non-increasing in the inventory eve of a non-end items. We describe severa additiona properties for both the production and inventory aocation poicies. Using numerica resuts, we compare the performance of the optima poicy against a heuristic poicy that contros production and inventory aocation using fixed base-stoc and rationing eves. We find that such a poicy is effective in systems with ost saes but can perform poory in systems with bacorders. Key words: assemby systems, production & inventory contro, muti-echeon inventory systems, Marov decision processes, mae-to-stoc queues Authors are isted in aphabetica order.

2 . Introduction Muti-stage assemby, one might argue, is the most common process with which physica goods are produced nowadays. Assemby systems are pervasive in industries as wide ranging as eectronics, automotive, heavy equipment, househod appiances, and many others. Despite their pervasiveness, assemby systems remain notoriousy difficut to anayze and manage. Few resuts exist on how to optimay contro production and inventory in assemby systems, especiay when there is variabiity in either demand or production. In practice, assemby systems are managed for the most part using heuristics, or other ad-hoc procedures, whose effectiveness reative to optima poicies remains difficut to verify. The difficuty of identifying optima poicies appears due to the muti-dimensionaity of the probem (mutipe items, mutipe production faciities and, in some cases, mutipe demand casses). Moreover, there are inter-dependencies between these various dimensions: () demands for different items are correated and (2) the production of one item depends on the avaiabiity of other items. Hence, there is a need to coordinate production and inventory decisions among the different items. For exampe, decisions about whether or not to produce an item probaby shoud tae into account the inventory status of other items (e.g., there may be no point in continuing to produce one component if there is a current shortage of other components). Coordination is however chaenging when items have different production times which may be stochastic, different inventory hoding costs, or reside in different ocations within the assemby process (e.g., coordination between upstream and downstream items). This can be further compounded when there are mutipe demand casses (customers with different priorities). In that case, deciding whether or not to satisfy a demand from a particuar cass must tae into account the abiity to fufi demand from other casses in the future, which may depend not ony on the current avaiabiity of the end-product but on the avaiabiity of a other items in the system. In a recent review of research on assembe-to-order systems, which aso covers assemby systems, Song and Zipin (2003) state: Litte is nown about the forms of optima poicies for muti-period modes. The research to date mosty assumes particuar poicy types. It woud be vauabe to earn more about truy optima poicies. Even partia characterizations woud be interesting. Aso, better heuristic poicy forms woud be usefu. In this paper, we tae a step toward addressing some of these chaenges. We do so in the context of the system described beow by (a) characterizing the structure of the

3 corresponding optima poicy, (b) providing comparisons between optima poicies and heuristics, and (c) proposing a cass of heuristics that cosey mimic the optima poicy. We consider an assemby system with mutipe production and inventory stages and mutipe demand casses. The system consists of mutipe faciities each producing a singe item one at a time. Production times at each faciity are stochastic, with production rates that can vary from item to item. Items can be produced in a mae-to-stoc fashion in anticipation of future consumption. Each item, other than the end product, has one successor item but possiby mutipe predecessors. One unit of each of the predecessor items is needed for the production of a successor item. Demand for the end product arises from different demand casses. Orders from each demand cass tae pace continuousy over time with stochastic inter-arriva times between orders. Demand casses differ in their demand rates and in the shortage penaties incurred if orders are not immediatey satisfied from inventory. At any point in time, the system manager must decide () which items to produce (production decision) and (2) shoud a demand arise, whether or not to satisfy it from avaiabe inventory (inventory aocation decision). We formuate the probem as a Marov decision process (MDP) and characterize the structure of the optima poicy under both the discounted and average cost criteria. For production, we show that the optima poicy for each item is a dynamic (state-dependent) base-stoc poicy, where the state of the system is specified by the vector of inventory eves for items. We show that the base-stoc eve for each item is non-increasing in the inventory eve of downstream items and non-decreasing in the inventory eve of a other items. For inventory aocation, we show that the optima poicy consists of a muti-eve rationing poicy with state-dependent rationing eves. An order from a demand cass is satisfied ony if the current inventory eve of the end item is above the rationing eve for that cass. The rationing eve for each cass is non-increasing in the inventory eves of a items (other than the end product). Furthermore, we identify severa additiona properties of the optima poicy. Using numerica resuts, we compare the performance of the optima poicy against a heuristic poicy that contros production and inventory aocation using fixed base-stoc and rationing eves. We show that such a poicy is effective in systems with ost saes but can perform poory in systems with bacorders. The rest of this paper is organized as foows. In section 2, we offer a brief review of reated iterature. In section 3, we present our mode for systems with ost saes. In section 4, we characterize the structure of the optima poicy. In section 5, we extend our anaysis to a system with bacorders. In section 6, we present numerica resuts and comparisons with the heuristic. In section 7, we offer a summary and a discussion of possibe future extensions. 2

4 2. Reated Literature Literature deaing with optima contro poicies for assemby systems is reativey imited. Schmidt and Nahmias (985) study a system with two components, one end product, a singe demand cass, and constant assemby and procurement eadtimes. They consider a discrete time probem with a finite panning horizon and random independent identicay distributed (i.i.d.) demand. They show that, whie the optima assemby poicy for the end product is essentiay a base-stoc poicy, the optima ordering poicy for the components does not have a simpe form. Rosing (989) provides a characterization of the optima poicy for a muti-stage version of the assemby system considered by Schmidt and Nahmias but with an infinite panning horizon. He shows, under some conditions on initia inventory, that the assemby system reduces to an equivaent series system. The optima poicy for this series system is nown from Car and Scarf (960) to be an echeon base-stoc poicy (the echeon inventory of an item consists of its own oca inventory and the inventory of a downstream items). Chen and Zheng (994) extend the resut of Rosing to systems with continuous review and compound Poisson demand and Chen (2000) extends it to systems with batch ordering. Veatch and Wein (992) provide monotonicity resuts for optima poicies for queueing networs where the time between state transitions is exponentiay distributed. They note that for an assemby system, such as the one we consider here but with a singe customer cass, the optima produce/do not produce regions in the state space are separated by monotone switching curves. Benjaafar and Ehafsi (2006) study a singe stage assembe-to-order (ATO) system with exponentiay distributed component production times, Poisson demand, and mutipe demand casses. They assume assemby of the end product is instantaneous and, therefore, ony components are hed in stoc. They show that the optima component production poicy is a state-dependent base-stoc poicy and the optima inventory is a muti-eve rationing poicy with state-dependent rationing eves. The probem we treat in this paper is of course different; we consider a muti-stage system, aow for positive assemby time for the end product, and for the end product to be hed in inventory ahead demand. Chen et a. (993) consider a specia ATO system with two components, each produced on a singe production faciity, and unimited demand for the end product. They show that the optima poicy for each faciity is to produce as ong as its inventory is beow a certain threshod or if there is positive inventory at the other faciity. This poicy ceases to be optima when the demand rate is finite. 3

5 There is significant iterature that does not dea with optima poicies but instead focuses on the anaysis and performance evauation of heuristics. We refer the reader to the exceent review in Song and Zipin (2003) and aso to Zipin (2000; chapter 8), Axsater (2006; chapter 5) and Chaouiya et a. (2000) for discussion of important resuts. Our paper is reated to the iterature on inventory systems with mutipe demand casses. Here too, there are few resuts for optima poicies. Topis (968) considers a system with a singe item (no assemby) with mutipe demand casses. In a discrete time setting, he shows that the optima ordering poicy is a base-stoc poicy and, when inventory is aocated over severa sub-intervas of a stocing cyce, the optima inventory aocation consists of a set of rationing eves in each sub-interva. Ha (997a) considers a production-inventory system in a continuous time setting with a singe item, Poisson demand and exponentiay distributed production times. He shows for systems with ost saes that the optima production poicy is a base-stoc poicy and the optima inventory aocation poicy is a muti-eve rationing poicy with fixed rationing eves. Ha (997b) and de Véricourt et a. (2002) extend these resuts to systems with bacordering. Our paper can be viewed as merging the streams of iterature on assemby systems and on systems with mutipe demand casses. We do so in the context of production-inventory systems where items are produced on independent faciities with finite production rates and stochastic production times. We provide a unified treatment for both production contro and inventory aocation from which resuts for simper systems can be retrieved as specia cases e.g., singe stage systems, series systems, and simpe two-stage assemby systems, with or without mutipe demand casses in each case. To our nowedge, our paper is the first to consider a muti-stage assemby system with mutipe demand casses. In addition to characterizing the optima poicy and describing severa of its properties, we compare the performance of the optima poicy against a commony used heuristic where the production of items are independenty managed. 3. Probem Formuation We consider an assemby system with mutipe stages, mutipe items, and one end product. The system consists of m production faciities, each producing a different item. Items may correspond to starting components, intermediates, incuding subassembies, or the end-product. Starting components are produced from materia suppied by an externa source whie intermediates are produced from other items 4

6 which are themseves produced internay. We refer to the set of items needed to produce item, =,, m, as P(), the set of predecessors of item, where P ( ) = if is a starting component. We consider pure assemby systems where each item is needed for the production of exacty one other item. We refer to this other item as the successor item. The exception is the end item, item, which does not have a successor. We use the notation SS() to refer to the successor of item for =2,.., m. We aso use the notation SS r () to denote the item obtained by r successive appications of the successor operator SS to item. Hence, for every item, = 2,, m, there exists an r() such that SS r() () =. In other words, successive appications of the operator SS eventuay ead to the end item. We refer to r() as the stage (in the production process) to which item beongs. We refer to any item that can be obtained via r appications of the successor operator SS, r = 0,, r(), as being on the successor path from item to and denote the set of such items as S(), with SS 0 () =. Figure shows an exampe of an assemby system with 4 starting components, 4 intermediates, and one end item; in this exampe, P(3) = {5, 6}, SS(3) = 2, S(3) = {3, 2, } and r(3) = 2. Specia cases of the types of systems we consider incude singe stage systems, seria systems, and two-stage assemby systems where components are produced in the first stage and assembed into the end product in the second production faciity inventory ocation Figure An exampe assemby system Items are produced one unit at a time. To produce one unit of an item, one unit from each of its predecessor items is needed. Upon production competion, items are paced in inventory. Items in inventory incur a hoding cost per unit per unit time (more about this ater). Production times are independent and exponentiay distributed with mean µ for item. Each faciity can hence be viewed as a singe server with finite service rate µ. Demand for the end-product arises from n different demand 5

7 casses. Demand from cass, =,, n, taes pace continuousy over time according to an independent Poisson process with rate λ. Demand for the end product from any cass can be satisfied ony if there is positive inventory avaiabe for the end product. Otherwise the demand is considered ost (or must be expedited through other means such as overtime or outsourcing to a third party). A demand from cass that cannot be immediatey fufied from stoc incurs a ost sae cost c per unit, which can vary from cass to cass. Without oss of generaity, we assume c c 2 c n (we treat the case of bacorders in section 5). Because the ost sae costs can be different for different casses, it may not aways be optima to satisfy demand from a cass even if there is on-hand inventory for the end product. In fact, it might be more desirabe to reject a demand from a cass in order to reserve the avaiabe inventory for future demand from a more important cass (i.e., one with a higher ost sae cost). Consequenty, each time an order is paced, the system manager must decide whether or not to satisfy it from on-hand inventory, if any is avaiabe. In addition to determining whether or not to reject an incoming order, the system manager must decide at any point in time whether or not to produce any of the items. If an item is not currenty being produced, this means deciding whether or not to initiate its production. If the item is currenty being produced, this means deciding whether or not to interrupt its production. Note that an item can be produced ony if there is at east one unit avaiabe in inventory for each of its predecessor items. If the production of an item is interrupted, it can be resumed the next time the production of that item is initiated (because of the memoryess property of the exponentia distribution, resuming production from where it was interrupted is equivaent to initiating it from scratch). We assume that there are no costs associated with interrupting production. This is consistent with earier treatments of production-inventory systems in the iterature; see for exampe Ha (997a, 997b) and Veatch and Wein (992). This assumption is not restrictive since, as we show in Theorem, it turns out that it is never optima to interrupt production of an item once it has been initiated. In our mode, we assume that demand is Poisson and both production times and times between consecutive updates are exponentiay distributed. These assumptions are made in part for mathematica tractabiity as they aow us to formuate the contro probem as an MDP and enabe us to describe the structure of an optima poicy. They are aso usefu in approximating the behavior of systems where variabiity is high. The assumptions of Poisson demand and exponentia production times are consistent with previous treatments of production-inventory systems; see for exampe, Buzacott and Shanthiumar 6

8 (993), Ha (997a, 997b), Zipin (2000), and de Véricourt et a. (2002) among others. In Section 7, we discuss how these assumptions may be partiay reaxed. The state of the system at time t can be described by the vector X ( t) = ( X( t),..., Xm ( t)), where X (t), =,, m, is a non-negative integer denoting the on-hand inventory for item at time t. We et m h( X ( t)) = h ( ( )), = X t where h is an increasing convex function, denote the inventory hoding cost rate when the state of the system is X(t). Note that because of the possibiity of interrupting production, it is not necessary to incude in the state description whether an item is currenty being produced or not. Furthermore, because both order inter-arriva times and production times are exponentiay distributed, the system is memoryess and decision epochs can be restricted to ony times when the state changes (i.e., the competion of an item or the fufiment of an order). The memoryess property aows us to formuate the probem as an MDP and to restrict our attention to the cass of Marovian poicies for which actions taen at a particuar decision epoch depend ony on the current state of the system. In each state, the system manager must decide which item to produce and whether or not to accept an order from a particuar customer cass shoud one arise. A poicy π specifies for each state x = (x,, x m ), π the action a ( x ) = ( u,..., um, w,..., wn), where u = means produce item ( =,, m), u = 0 means do not produce item, w = means satisfy demand from cass ( =,, n), and w = 0 means reject demand from cass. For exampe for the assemby system shown in Figure and assuming there are two demand casses, the action a π ( x ) = (,0,0,0,,0,0,0,0,,0) means that whenever the system is in state x, produce items and 5, do not produce items 2-4 and 6-9, accept demand from cass, and reject demand from cass 2. As we can see, there are two types of decisions, production decisions indicated by the parameters u and inventory aocation decisions indicated by the parameters w. Production decisions determine for each state whether or not an item shoud be produced whie inventory aocation decisions determine in which states it is best not to fufi demand from a particuar cass in order to reserve inventory for future demand from other casses. We refer to this type of aocation as inventory rationing. Let N(t) be an n-dimensiona vector with items N (t), where N (t) denotes the number of orders from cass that have not been fufied from on-hand inventory up to time t and et = ( c c ) c,, n denote the vector of ost saes costs. Then the expected discounted cost (the sum of inventory hoding and bacorder costs) over an infinite panning horizon v π ( x ) obtained under a poicy π and a starting state x=(x,, x m ), can be written as: 7

9 π π αt αt v ( x) = E e h( () t ) dt e d () t x X 0 c N 0 () where α > 0 is the discount rate, and c denotes the transpose of the vector c. Foowing Lippman (975), we wor with a uniformized version of the probem in which the transition rate in each state under any action is n m = = so that the transition times 0 = t 0 t t 2 are such that the times β = λ µ between transitions {t i t i : i 0} is a sequence of i.i.d. exponentia random variabes, each with mean /β. This eads to a Marov chain defined by {X i : i 0} where X i X ( ti) = ( X( ti),..., Xm( ti)) is the state resuting from the i-th transition. The introduction of the uniform transition rate aows us to transform the continuous time decision process into a discrete time decision process, simpifying the anaysis consideraby. If action a is seected in state x, the next state is y with probabiity: p xy, λ I{ x 0, and w } if -, = y = x e β µ I if y = x e { x 0, and } i P( ) i u = -e P( ), ( ) β a = n m β λ I { x 0, and w = } µ I = = { x 0, and } i P( ) i u = if y = x, and β 0 otherwise, (2) where I {z} is an indicator function with I {z} = if z is true and I {z} = 0 otherwise and e is the -th unit vector of dimension m and e P( ) = e. Let ( ), j P( ) j N t i =,, n, denote the cumuative number of unfied orders from cass after the i-th transition. Then, v π ( x ) in () can be rewritten in the equivaent form: i i π π β h( x) β n v ( x ) = E c ( ( ) ( ) ). i 0 i N ti N t = = = i α β α β α β (3) x * Our objective is to choose a poicy π that minimizes the expected discounted cost. The optima cost function * π v v * satisfies the foowing optimaity equation: * h( x) β n λ * v ( x) = min a c ( I{ x ) 0} I{ x 0, w 0} p, ( ) v ( ). = = = xy a y (4) y α β α β β Without oss of generaity, we rescae time so that α β = and rewrite (4) in the equivaent form: where the operators T and T are defined as foows m n * * * = µ λ = = v ( x ) h ( x ) T v ( x ) T v ( x ), (5) 8

10 and v( x) if xt = 0 Tv( ) t P( ) x = min{ v( x e ep( ) ), v( x)} otherwise, v( x) c if x = 0 Tv( x) = (7) min{ v( x e), v( x) c } otherwise. The operator T is associated with decisions about whether or not to produce item and he operator is associated with decisions about whether or not to satisfy an order of type. Note that it is optima to produce item if v * ( x e e ) v * ( x ), provided there is on-hand inventory for each of its P( ) predecessors. Simiary, it is optima to satisfy demand from cass when the system is in state x if v ( x e ) v ( x ) c, provided there is on-hand inventory for the end product. * * (6) T 4. The Structure of the Optima Poicy In this section, we characterize the structure of an optima poicy. In order to do so, we wi show that the optima vaue function v*(x) for a states x satisfies certain properties. First, we introduce the foowing difference operators for functions v defined on integers: and combinations of such operators, incuding v( x) = v( x e ) v( x ), for j =,, m, j j m Z where Z is the set of non-negative i, jv( x) = i jv( x) = jv( x ei) jv( x ), where x = (x,, x m ) and the variabes x j are state variabes associated with an assemby system per the description in the previous section. Note that the order in which the differences are taen does not matter, i.e., i, jv( x) = j, iv( x ). For notationa convenience, we aso define v( x) = v( x e e ) v( x ). P( ) P( ) Definition : Let V be the set of functions defined on A: ( ) 0 (), v x ; i P i A2: (), ( ) ( x ) 0, j i; and v i P i j P j A3: v( x ) c. m Z such that if v V, then for i=,, m: Proposition : If v V, then v aso satisfies the foowing properties for i=,, m: B: (), ( x ) 0, j S() i ; v i P i j B2: (), ( x ) 0, j S() i, j, S( j), j; v i P i j 9

11 B3: v i P(), i j( x ) 0, j S() i ; B4:, ( x ) 0, j S() i ; i j v B5: (), () ( x ) 0; and v i P i i P i B6:, ( x ) 0, i S( j) and j S (). i i j v The proof of Proposition and of a subsequent resuts can be found in the Appendix. Let v V and define the foowing quantities. s( x ) = min{ x 0 P( ) v( x ) 0 }, and r( x ) = min{ x 0 v( x e ) c}, x is the on-hand inventory eve of item and = ( x,, x, x,, x ) where x m is an (m ) dimensiona vector with eements corresponding to the on-hand inventory eves of items j. Then, condition B3 appied to the case i = and j =, ( ), ( x) 0, combined with the above definitions impies v P v( x e ep( ) ) v( x) if and ony if x s( x-), v( x e ep( ) ) < v( x) otherwise, and condition B4 appied to the case i =, j =, v( x ) 0, impies, v( x) c v( x e) if and ony if x r( x-), and v( x) c < v( x e) otherwise. and since c c 2 c n, we have r ( x ) r( x ). Moreover, condition B impies that s( x ) is non-decreasing in x j for j S(); i condition B3 impies that s( x ) is non-increasing in x j for j S( ) and j ; condition B4 appied to the case j = (i.e., ( ) 0) i, v x impies that r ( x ) is non-increasing in each x j for j ; condition A3 impies that r ( x ) = ; condition A2 impies that s ( x e e ) s ( x ) for j ; and condition B2 impies s ( x e e ) s ( x ) or j P( j) equivaenty s ( x e ) s ( x e ). j Proposition 2: If v V, then Tv V, where the optima cost function v * is an eement of V. That is, v* V. We are now ready to state our main resut for this section. m = = j Tv ( x ) = h ( x ) µ T v ( x ) λ T v ( x ). Furthermore, Theorem : There exists an optima stationary poicy that can be specified as foows. The optima production poicy for item, =,, m, is a base-stoc poicy with a dynamic base-stoc eve n s * ( x ) that depends on the inventory eve of a other items, such that it is optima to produce item if x < s * ( x ) and not to produce it otherwise. The optima inventory aocation poicy is a muti-eve 0

12 rationing poicy with a vector of dynamic rationing eves r * ( x ) = ( * ( x ),, * ( x )) r r n such that it is optima to satisfy an order from cass, =,, n, if x r* ( x ) and to reject it otherwise. The optima poicy has the foowing additiona properties. P. The optima base-stoc eve s * ( x ) is non-increasing in the on-hand inventory eves of items that are on the path from item to item (the end item) and non-decreasing in the on-hand inventory eve of a other items. That is, s * ( x ) is non-increasing in x j if j S( ) and j and is non-decreasing in x j if j S( ). P.2 The rationing eve r* ( x ) for cass demand is non-increasing in the on-hand inventory eves of other items. That is, r* ( x ) is non-increasing in each x j for j. P.3 The rationing eves are ordered such that r* ( x ) r* ( x ) =. n P.4 It is aways optima to satisfy cass demand as ong as there is on-hand inventory. P.5 The optima base-stoc eve s * ( x ) for item does not decrease with the production competion of any other item j. P.6 For any item, the base-stoc eve is more iey to decrease with an increase in the inventory eve of items on its successor path that are more cosey ocated. That is, s * ( x e ) s * ( x e ) for j j S( ) and S( j), j. P.7 It is never optima to interrupt the production of an item once it has been initiated. The resuts of Theorem show that the base-stoc eve for an item does not increase with an increase in the inventory eve of items that are on its successor path; inventory in downstream stages can be viewed as substitute for inventory in upstream stages. On the other hand, the base-stoc eve for an item does not decrease with an increase in the inventory eve of items that are not on its successor path; in this case the inventories can be viewed as compements. For exampe, for items that are assembed together at a particuar stage having ess of one item reduces the need for producing the other items, since eventuay one unit from each item wi be needed. For inventory aocation, the resuts of the Theorem show that different demand casses shoud be treated differenty, with each cass assigned an inventory rationing eve beow which the demand from this cass woud be rejected in favor of reserving inventory for casses with higher ost saes costs. Simiar to the base-stoc eves, the rationing eves are dynamic and must be adjusted based on the inventory eve of a items. We iustrate the structure of the optima poicy using the exampe system with four items and three demand casses shown in Figure 2. In Figure 3, we show how the optima poicy is affected by changes in

13 the inventory eves of items 2 and 3 whie the inventory eves of items and 4 are ept fixed. As we can see from Figure 3(a), the optima production poicy divides the state space into four regions: region where both items 2 and 3 are produced, region 2 where item 2 is produced but not 3, region 3 where item 3 is produced but not 2, and region 4 where neither items is produced. It is important to note that other than region, the other three regions are transient under the optima poicy. Note aso, as described in Theorem, that the optima poicy dynamicay adjusts the base-stoc eve of each item based on the inventory eve of the other items. The optima rationing poicy (see Figure 3(b)) divides the state space into three regions: region where demand from any cass is satisfied, region 2 where demand from ony cass and 2 is satisfied, and region 3 where ony demand from cass is satisfied. The boundaries of these regions highight the fact that the rationing eves are sensitive to changes in the inventory eves of any item in the system as described in Theorem. The superposition of Figures 3(a) and 3(b) woud partition the state space into severa sub-regions, each corresponding to a combination of (a) satisfying one or more demand casses and (b) producing one or more items or not producing at a. In Figure 4, we show how the optima production and inventory aocation poicies are simiary affected by changes in the inventory eves of items and 4 whie the inventory eves of items 2 and 3 are ept fixed. In genera, the optima poicy defines regions in the state space where ony certain items are produced and ony certain demand casses are satisfied. These regions are defined by the hyperpanes associated with the muti-dimensiona base-stoc and rationing eve functions. For production, there is ony one recurrent region defined by the subset of the state space: { : x < s ( x )} { : x < s ( x )}...{ : x < s ( x )} x x x m m m Within this subset, a items are aways produced. The intersection of this subset with the subsets defined by the rationing functions (i.e., { : < ( )} more of the demand casses are satisfied. x x r x for =,, n) defines the sub-regions where one or 4 2 cass demand cass 2 demand 3 cass 3 demand Figure 2 An exampe assemby system with 4 items and 3 demand casses (µ = 2, µ 2 = µ 3 = µ 4 = ; λ = λ 2 = λ 3 = 0.5; c = 80, c 2 = 50, c 3 = 30; h =.47, h 2 = 0.7, h 3 = 0.7, h 4 = 0.56) 2

14 Produce item 2 but not 3 Do not produce either items 2 or 3 x 3 x Produce items 2 & 3 Produce item 3 but x 2 (a) The optima production poicy Satisfy casses, 2 & 3 x 3 x Satisfy cass ony Satisfy casses & x 2 (b) The optima inventory aocation poicy Figure 3 - The Structure of the optima poicy as a function of inventory eves for items 2 and 3 (x = 5 and x 4 = 5 for a the cases shown) 3

15 Produce item but not 2 Do not produce either item or 2 x 2 x Produce items & 2 Produce item 2 but not x (a) The optima production poicy 5 x 2 x Satisfy casses ony Satisfy casses & 2 ony Satisfy casses, 2 & x (b) The optima inventory aocation poicy Figure 4 - The Structure of the optima poicy as a function of inventory eves for items 2 and 3 (x 3 = 5 and x 4 = 5 for a the cases shown) 4

16 In Figure 5, we iustrate the fact that the base-stoc eve of each item becomes ess sensitive to the inventory eve of other items the further downstream these items are (Property P.6 in Theorem ). In the exampe, we consider a series system with three items and a singe customer cass, with item 3 being the starting item. As we can see, an increase in x has significanty ess of an impact on the base-stoc eve of item 3 than an increase in x 2. This is an important resut because it shows that an echeon base-stoc poicy, where production is determined by the sum of inventory eves of downstream items, is not optima. In other words, it is not sufficient to monitor ony the sum of downstream inventory eves in maing decisions about whether or not to produce. Instead, the specific contribution to tota inventory from each item matters. This is different from resuts for other types of inventory systems, such as systems with fixed suppy eadtimes, where an echeon base-stoc poicy has been shown to be indeed optima; see for exampe Rosing (989). 9 6 s 3(3, x 2) s 3 (x, 3) x 3 s 3 3 s 3 s(3, 3(x x2, 2 3) ) x or x 2 Figure 5 The impact of downstream inventory eves on the base-stoc eve (µ =µ 2 = µ 3 = ; λ = 0.95; c = 200, h = 2, h 2 =.5, h 3 = ) Theorem characterizes the optima poicy for a genera assemby networ structure with mutipe demand casses. Severa simper probems can be retrieved as specia cases. For instance, for a singe stage system with a singe item, the state of the system can be described by the inventory eve x ony and 5

17 the base-stoc and rationing eves reduce to fixed parameters as stated in the foowing coroary, which corresponds to the main resut in Ha (997a). Coroary : In a system with a singe item produced on a singe production faciity with n demand casses, an optima poicy can be specified in terms of a base-stoc eve s * and rationing eves *,..., * * r r n, such that it is optima to produce if x < s and not to produce otherwise and it is optima to * satisfy an order from customer cass if x r and to reject it otherwise, where r *... r * =. Furthermore, it is aways optima to satisfy cass demand as ong as there is on-hand inventory. In a seria system where each item has a singe predecessor, the optima poicy for the case of a singe customer cass can be specified in terms of a dynamic base-stoc poicy where the base-stoc eve is non-increasing in the inventory eve of items that are downstream and non-decreasing in the inventory eve of items that are upstream. Coroary 2: In a seria system where items are abeed in the sequence in which they are produced, with item being the end product and item m the starting component, the optima production poicy for an item, < < m, can be specified in terms of a base-stoc eve s*( x ) which is non-decreasing in x j for j > and non-increasing in x j for j <. Furthermore, s * ( x ) s * ( x e e ) for j, and s ( x ) s ( x e )... s ( x e ). * * * j j In a two stage assemby system, where in the first stage m- components are produced, each on a separate faciity, and in the second stage the components are assembed into the end product on a different faciity, the optima poicy for the case of a singe customer cass is described in the foowing coroary. Coroary 3: In a two stage assemby system with a singe customer cass where m- components are first produced on independent faciities and then assembed into the end product on a different faciity, the optima poicy for component, = 2,, m, can be specified in terms of a base-stoc eve s * ( x ) which is non-increasing in x and non-decreasing in x j for j and j. For the end item, the optima poicy can aso be specified in terms of a base-stoc eve s * ( x ) but in this case s * ( x ) is non-decreasing in x j for j. It is possibe to extend the anaysis to the case where the optimization criterion is the average cost per unit time instead of the expected discounted cost. Given a poicy π, the average-cost is given by: T T π J ( x ) = imt sup { h( ( t)) dt d ( t) 0 0 }. T X c N (8) n 6

18 A poicy π* that yieds J * ( x) = inf J π ( x ) for a states x is said to be optima for the average cost π criterion. In the foowing theorem, we show that the optima poicy retains a of the properties observed in Theorem under the expected discounted cost criterion. Theorem 2: The optima poicy under the average cost criterion retains a the properties of the optima poicy under the discounted cost criterion, namey that () the production poicy for each item consists of a base-stoc poicy with a state-dependent base-stoc eve, (2) the inventory aocation poicy consists of a muti-eve rationing poicy with state-dependent rationing eves, and (3) the base-stoc and rationing eves satisfy properties P-P6 in Theorem. The optima poicy aso satisfies property P.7 of Theorem. Furthermore, the optima average cost is finite and independent of the initia state; that is, there exists a finite constant J* such that J*(x) = J* for a states x. 5. The Case of Bacorders In this section, we treat the case where, instead of being ost, orders that cannot be fufied immediatey from inventory are bacordered. Orders that are bacordered incur a bacorder cost per unit per unit time. We restrict our treatment to the case of a singe customer cass. Extending the anaysis to systems with mutipe customer casses is difficut and we do not attempt it here (the anaysis is more compex since there is a need to incude in the state space not ony information about net inventory for each item but aso bacorder eve for each cass). For systems with bacorders, we use the notation X() t = ( X (), t, X ()) t to denote the state of the system at time t, where X () t ( ) is a m non-negative integer that corresponds to the on-hand inventory of item at time t; X () t is an integer that corresponds to the net inventory eve of the end product (item ) at time t, such that = ( X t ) corresponds to the bacorder eve at time t and X () t max ( 0, X () t ) X () t max 0, () = to the on-hand inventory eve of the end item (item ). The tota cost rate z( X ( t)) incurred at time t can be written as z ( X () t ) = h( X() t ) b( X() t ) m h ( ()) = 2 X t, where b is an increasing convex function denoting the bacorder cost rate when the net inventory eve of the end product is X (). t The objective of the system manager is to choose a production poicy that determines for each state whether an item shoud be produced or not. The expected discounted cost over an infinite horizon under a given poicy π and a starting state x = (x,, x m ) is given by π π αt v ( x) = E e z( ( t)) dt x. X 0 (9) 7

19 Our objective is to choose a poicy π * which minimizes the expected discounted cost. After rate uniformization and time rescaing (as we did for the ost saes cost), the optima cost function can be shown to satisfy the foowing optimaity equation for any starting state x: * * v v π where v ( x) z( x) v ( x e ) T v ( x), (0) m * * * = λ µ = v( x) if x = 0 = min{ v( x e ep( ) ), v( x)} otherwise. i ( x) i P( ) () Tv The operator T is associated with the decision of whether or not item shoud be produced. This decision is made each time a new order arrives or any item competes production. Theorem 3: There exists an optima stationary poicy that can be specified in terms of a state-dependent base-stoc eve s * ( x ) for item, =,, m, such that it is optima to produce item if x < s * ( x ) and not to produce it otherwise. The optima base-stoc eves satisfy properties P., P.5 and P.6 of Theorem. The optima poicy aso satisfies property P.7 of Theorem. 6. Comparisons to Poicies with Fixed Base-Stoc and Rationing Leves In this section, we compare the performance of the optima poicy against a heuristic poicy that contros production and inventory aocation using fixed base-stoc and rationing eves. We refer to this poicy as the independent base-stoc with rationing (IBR) poicy. Such a heuristic is commony used to manage assemby systems in practice and is aso widey studied in the iterature (see Song and Zipin (2003) and the references therein). This heuristic is perhaps easier to impement and communicate than the optima poicy; the optima poicy is aso difficut to compute for systems with more than few items due to the exponentia growth in the size of the state space (computing the optima poicy is NP-hard because of the we nown curse of dimensionay of dynamic programming). The IBR poicy contros the production of items independenty of each other via a vector s=(s,, s m ) of fixed (i.e., state-independent) base-stoc eves and the acceptance/rejection of orders via a vector r=(r,, r n ) of fixed rationing eves. For systems with ost saes, the expected discounted cost of the IBR poicy can be obtained (for given vectors s and r) via the foowing dynamic programming equation: 8

20 where T m n IBR IBR IBR, IBR IBR ( ) = ( ) µ ( ) ( ), λ = = v x h x T v x T v x (2) IBR v( x e ep( ) ) if x < s and xi 0 v( x) = i P( ) v( x) otherwise, (3) and T v(, ) if x r, IBR x e v( x) = v( x) c otherwise. (4) A simiar equation can be written for the case with bacorders. Aso, simiar equations can be written for the average cost criterion. In contrast to the optima poicy, the base-stoc and rationing eves for items are fixed and not affected by the inventory eves of other items. This is of course sub-optima since there is no attempt to use current information about the state of the system to coordinate the production of different items. To test the performance of the IBR poicy against the performance of the optima poicy, we carried out a series of numerica experiments for a system with three items (items 2 and 3 are assembed into item ) and for a wide range of parameter vaues. For each probem instance, numerica resuts are obtained by soving the corresponding dynamic programming equation using the vaue iteration method. The vaue iteration agorithm we use is a direct adaptation of the agorithm described in Puterman (Chapter 8, 994). We present resuts for the average cost criterion because they are independent of the starting state and the discount factor. The state space is truncated at { n, n } { n, n } { n, n } where n min min max min max min max max and n, =, 2, 3 are positive integers that are graduay increased unti the cost is no onger sensitive min to the truncation eve (for the case of systems with ost saes, n = 0). The vaue iteration agorithm is terminated once five-digit accuracy is obtained. Because the computationa effort grows exponentiay in the dimensions of the state space, soving a arge number of probems with more than 3 or 4 items is generay difficut; see Puterman (Chapter 8, 994) for further discussion of computationa compexity and convergence of the vaue iteration agorithm. To identify good vaues for the contro parameters for the IBR poicy in the case of ost saes, we use the foowing heuristic search procedure (a sighty modified version of this procedure is used in the case of bacorders). For the base-stoc eves, we exhaustivey search over the region {0, s 5} {0, s 5} {0, s 5} where s max is the argest base-stoc vaue for item, =, 2, 3 max max max 2 3 that is observed under the optima poicy in the recurrent region. For each combination of base-stoc 9

21 vaues, we search exhaustivey over a feasibe vaues for the rationing eves. This means examining a vaues that do not exceed the base-stoc eve for item. Representative resuts for systems with ost saes and one demand cass are shown in Tabe. The resuts are consistent with a arger set of exampes where we varied the parameters over a wider range and with resuts where we generated the parameter vaues randomy (for brevity, resuts from this more extensive data set are not incuded but are avaiabe from the authors upon request). As we can see from Tabe, the IBR heuristic is surprisingy effective. The percentage cost difference reative to the optima poicy is sma, with an average for the exampes shown of.43%, a minimum of 0% and a maximum of 2.54%. In fact, if we excude the four argest vaues, the range is from 0% to.95%. The argest vaues correspond to rather extreme scenarios where the utiization of the production faciities is ow so that the base-stoc eves are very sma. In this case, sma differences in the base-stoc eves (due to integraity) can ead to reativey arge differences in cost. These resuts are consistent with those observed for systems with mutipe casses. In Tabe 2, we show representative resuts from a set of randomy generated exampes with two demand casses. The average percentage cost difference is 0.99% with a minimum of 0% and a maximum of 5.6%. Representative resuts for systems with bacorders are shown in Tabe 3. In contrast to the case of ost saes, the IBR poicy can perform poory. For the cases shown, the average percentage cost difference between the IBR heuristic and the optima poicy is 3.06% with a minimum of.8% and a maximum of 34.73%. The percentage difference is argest when the bacorder cost is ow or the utiization of the production faciities is high, due to either a high demand rate or ow production rates. A possibe expanation of why the IBR heuristic can perform poory is as foows. In systems with bacorders the region of the state space over which coordination among the different items under the optima poicy can tae pace is arger (the recurrent region is finite for ost saes but not for bacorders); see Figure 6. This is significant since the difference in inventory eves between the end item and other items aways stays bounded for ost saes whie this difference can grow infinitey arge for systems with bacorders. For systems with bacorders, it is possibe to accumuate a arge bacog at the eve of the end item whie items maintain positive inventory. This is more iey to occur when the demand rate is high, the production rate for the end item is ow, or the bacorder penaty is sma, which are indeed the cases for which we observe the argest percentage cost differences. Systems with bacorders and ost saes differ in another important way. In the case ost saes, it is aways possibe to set a the base-stoc eves at zero 20

22 (do not produce anything) and reject a demand. This eads to an average cost of n λc, which provides a ower bound on the cost of the IBR heuristic. Such a poicy and such bound do not exist in the case of bacorders where the bacog, depending on system parameter vaues, can be arbitrariy arge. = 90 Produce 80 item but not item 3 70 x 3 x3 Produce both items & Produce item 3 but not item Do not produce either item or item x Figure 6 The structure of the optima poicy under bacorders for a system with 3 items where item is assembed from items 2 and 3 (x 2 = 32, µ = 5, µ 2 = µ 3 = 4.3; λ = 4; b = 50; h = 5, h 2 = 2, h 3 = 2.9) 7. Summary and Discussion In this paper, we considered the probem of optima production contro and inventory aocation in a muti-stage assemby system with mutipe customer casses. We formuated the probem as a Marov decision process and used the formuation to study the structure of the optima poicy. For production, we showed that the optima poicy for each item is a dynamic base-stoc poicy with the base-stoc eve of each item dependent on the inventory eve of a other items. Specificay, we showed that the base-stoc eve of each item is non-increasing in the inventory eve of items that are downstream and non-decreasing in the inventory of other items otherwise. For inventory aocation, we showed that the optima poicy is a dynamic muti-eve rationing poicy with the rationing eve for each demand cass dependent on the inventory eve of a items (they are non-increasing in the inventory eve of each item, other than the end product). We aso provided severa additiona properties for both the production and 2

23 Tabe The effect of varying system parameters on the performance of the IBR poicy for systems with max ost saes (for the optima poicy, s is the maximum base-stoc eve observed for item in the recurrent region; for the heuristics, s is the seected base-stoc eve for item ) c λ µ µ 2 µ 3 h h 2 h 3 IBR cost Percentage difference from optima cost Optima poicy max max max [ s, s, s ] 2 3 IBR poicy [ s, s, s ] [2,2,2] [0,0,0] [2,2,2] [7,,] [4,3,3] [4,3,3] [0,7,7] [9,5,5] [3,9,9] [2,7,7] [6,,] [4,9,9] [8,3,3] [6,0,0] [20,4,4] [7,2,2] [23,6,6] [9,3,3] [24,7,7] [20,4,4] [2,2,2] [3,,] [4,4,4] [4,2,2] [7,6,6] [6,4,4] [,9,9] [0,7,7] [20,4,4] [7,2,2] [38,8,8] [43,6,6] [58,9,9] [63,7,7] [80,9,9] [85,7,7] [0,9,9] [06,7,7] [23,9,9] [28,7,7] [73,3,3] [63,3,3] [65,4,4] [7,4,4] [48,7,7] [58,7,7] [32,0,0] [42,0,0] [20,4,4] [7,2,2] [4,6,6] [2,2,2] [,7,7] [0,2,2] [9,8,8] [9,,] [8,8,8] [8,2,2] [7,8,8] [7,2,2] 22