Performance Evaluation of Stochastic Multi-Echelon Inventory Systems : A Survey

Globalization, product proliferation, and fast product innovation have significantly increased the complexities of supply chains in many industries. One of the most important advancements of supply chain management in recent years is the development of models and methodologies for controlling inventory in general supply networks under uncertainty and their widefspread applications to industry. These developments are based on three generic methods: the queueinginventory method, the lead-time demand method and the flow-unit method. In this paper, we compare and contrast these methods by discussing their strengths and weaknesses, their differences and connections, and showing how to apply them systematically to characterize and evaluate various supply networks with different supply processes, inventory policies, and demand processes. Our objective is to forge links among research strands on different methods and various network topologies so as to develop unified methodologies.


Introduction
Many real-world supply chains, such as those found in automotive, electronics, and consumer packaged goods industries, consist of large-scale assembly and distribution operations with geographically dispersed facilities.Clearly, many of these supply chains support the production and distribution of multiple end-products which are assembled from hundreds or thousands of subsystems and components with widely varying lead times and costs.
One challenge in all these supply chains is the efficient management of inventory in a complex network of facilities and products with stochastic demand, random supply and high inventory and transportation costs.This requires one to specify the inventory policy for each mathematical programming models of production-inventory systems.In uncapacitated supply chains, we typically assume a positive exogenous "transit time" for processing a job, where the "transit time" is defined as the total time it takes from job inception to job completion.This transit time may represent manufacturing cycle time, transportation lead time, or warehouse receiving and processing times.The literature on uncapacitated supply chains can be further classified into two categories: i.i.d. or sequential transit time.In the former, the transit times are i.i.d.random variables; while, in the latter, the transit times are sequential in the sense that jobs are completed in the same sequence as they are released.

Optimal Policy Characterization versus Policy Evaluation and Optimization
The focus of the former is on identification and characterization of the structure of the optimal inventory policy.We refer to Federgruen 8 , Zipkin 9 , and Porteus 10 for excellent reviews.Unfortunately, the optimal policy is not known for general supply chains except for some special cases.When the optimal policy is unknown or known but too complex to implement, an alternative approach is to evaluate and optimize simple heuristic policies which are optimal in special cases but not in general.

Guaranteed Service Time Model versus Stochastic Service Time Model
In the former, it is assumed that in case of stockout, each stage has resources other than the on-hand inventory such as slack capacity and expediting to satisfy demand so that the committed service times can always be guaranteed.In the latter, it is assumed that in case of stockout, each stage fully backorders the unsatisfied demand and fills the demand until onhand inventory becomes available.Thus, the delay due to stockout i.e., the stockout delay is random, and the committed service times cannot be 100% guaranteed.A recent comparison between the two models is provided by Graves and Willems 11 .

The Scope and Objective of the Survey
This survey focuses on the stochastic service time model for uncapacitated supply chains.Because we are interested in general supply networks, we focus on policy evaluation and optimization.Given a certain class of simple but effective inventory policies, the specific problem that we address in this survey is how to characterize and evaluate system performance in general structure supply chains.The challenge arises from the fact that the inventory policy controlling one product at one facility may have an impact on all other products/facilities in the network either directly or indirectly.
For guaranteed service time models, Graves and Willems 11 summarize recent development and demonstrate its potential applications in industry-size problems.These developments are based on the lead-time demand method.For the stochastic service time model, Hadley and Whitin 12 provide the first comprehensive review for single-stage systems.Chen 13 reviews the lead-time demand method in serial supply chains, and de Kok and Fransoo 14 discuss some of its applications in more general supply chains.Song and Zipkin 15 provide an in-depth review of the literature on assembly systems, while Axsater 16 presents an excellent survey for serial and distribution systems.Zipkin 9 presents an excellent and comprehensive review for the queueing-inventory method in single-stage systems and the lead-time demand method in single-stage, serial, pure distribution, and pure assembly systems.

Advances in Operations Research
The objective of this paper is to compare the effectiveness of queueing-inventory method, the lead-time demand method, and the flow-unit method in supply chains along the following dimensions: network topology, inventory policy, and demand process.Specifically, we discuss how to apply each method systematically to evaluate various network topologies with either i.i.d. or sequential transit times, either base stock or batch ordering inventory policy, and either unit or batch demand process.The network topology considered includes single-stage see Section 2 , serial Section 3.1 , pure distribution Section 3.2 , pure assembly and 2-level general networks Section 3.3 , and tree and more general networks Section 3.4 .For each network topology, we discuss the three methods side by side and address questions such as, how are different stages connected and dependent?How does each method work?How are the results/methods connected to those of single-stage systems and systems of other topologies?What are the weakness and strength of each method?And what are the differences and connections among the methods?Some open questions are summarized in Section 4.
While some of the materials covered here appeared in previous reviews, we present these materials together with recent results in a coherent way by building connections among different methods and establishing uniform treatment of each method across different network topologies.We also shed some lights on the strengths and limitations of each method.

Single-Stage Systems
In this section, we consider single-stage systems and review the key assumptions and results of the three generic methods.We show how each method can handle different inventory policies, transit times, and demand processes.Following convention, we define a stage a node, equivalently to be a unique combination of a facility and a product, where the facility refers to a processor plus a storage where the latter carries inventory processed by the former.

Inventory Policies
In this paper, we focus on either continuous-review or periodic-review base-stock and batch ordering policies.For any stage in a supply chain, we define inventory position to be the sum of its on-hand inventory and outstanding orders subtracting backorders.Under continuous review, a base-stock policy with base-stock level s works as follows: whenever inventory position drops below s, order up to s.A batch-ordering policy with reorder point r and batch size Q works as follows: whenever the inventory position drops to or below the reorder point r, an order of size nQ is placed to raise the inventory position up to the smallest integer above r.Clearly, a base-stock policy is a special case of the batch-ordering policy with a batch size Q 1. Continuous-review base-stock policies are often used for expensive products facing low-volume but highly uncertain demand e.g., service parts .Batch-ordering policies are often used where economies of scale in production and transportation cannot be ignored commodities .
Under periodic review, the base-stock and batch ordering policies work in similar ways as their continuous-review counterparts except that inventory is reviewed only once in one period.The sequence of events is as follows 12 .At the beginning of a review period, the replenishment is received, the inventory is reviewed, and then an order decision is made.Demand arrives during the period.At the end of the period, costs are calculated.Some work in the literature assumes that all demands arrive at the end of the period; see, for example, Zipkin 9, Chapter 9 .Under this assumption, a single-stage periodic-review inventory system can be viewed as a special case of its continuous-review counterpart with constant demand interarrival times and batch demand sizes.In this survey, we assume demand arrives during the period unless otherwise mentioned.

Transit Times
If the transit times Section 1.1 are sequential and stochastic, namely, "stochastic sequential transit times," then they must be dependent over consecutive orders.Kaplan 17 presents a discrete-time model for the stochastic sequential transit time in a periodic-review single-stage system, where the evolution of the outstanding order vector is modeled by a Markov chain.See Song and Zipkin 18 for a generalization of the model.For continuous-review singlestage systems, Zipkin 19 presents a continuous-time model for stochastic and sequential transit times.
Definition 2.1.The exogenous, stochastic, and sequential transit times are defined as follows: there exits an exogenous continuous-time stochastic process {U t } that is stationary and ergodic with finite limiting moments, such that the sample path of {U t } is left-continuous, the transit time at t, L t U t , and t L t is nondecreasing.
Svoronos and Zipkin 20 apply this model to multistage supply chain with two additional assumptions: 1 the transit times are independent of the system state, for example, demand and order placement and 2 the transit times are independent across stages.
In practice, the transit times can be either parallel or sequential or somewhere in between.Many production and transportation processes in the real world are subject to random exogenous events.Indeed, the orders placed by the systems under consideration may be a negligible portion of their total workload.Thus, the transit times are exogenous and should be estimated from data.While in some practical cases, the sequential transit time model may be more realistic than the i.i.d.transit time model 20 , in cases such as repairing and maintenance, the i.i.d.transit time model may be a better approximation 21 .

Demand Processes
Both unit demand and batch demand processes are studied in the literature.On arrival of a batch demand, one shall address questions such as: should all units of the demand be satisfied together unsplit demand ?Or should each demand unit be satisfied separately split demand ?For a supply system either production or transportation processing a job of multiple units, one needs to address questions like: is the job processed and replenished as an individable entity unsplit supply ?Or is each unit processed and replenished separately split supply ?If the former is true, does the transit time depend on job size?See Zipkin 9 for more discussions on these questions.While the case of split demand is easier to handle and thus widely studied in the literature, the case of unsplit demand is much more difficult; see Section 2.1 for more details.

The Basic Assumption
For the ease of exposition, we make the following assumption throughout the survey unless otherwise mentioned.
Assumption 2.2.The system is under continuous review; unsatisfied demands are fully backordered; outside suppliers have ample stock; the transit times are exogenous either i.i.d. or sequential; demand is satisfied on a first-come first-serve FCFS basis; demand can be split; supply cannot be split; transit times do not depend on job sizes.
Throughout the survey, we use the following notations: a max{a, 0}, a − max{−a, 0}.E • , V • are the mean and variance of a random variable, respectively.If random variables X and Y are independent, we denote X ⊥ Y .We consider base-stock policies with s ≥ 0 and batch-ordering policies with r ≥ 0 unless otherwise mentioned.
We define the basic model for single-stage systems as follows: inventory is controlled by a base-stock policy, demand follows Poisson process with rate λ, and the transit time i.e., lead time L is constant.In the following subsections, we first discuss the methods in the basic model and then extend the results to more general demand process, inventory policies, and supply process.

The Queueing-Inventory Method
Let {IO t , t ≥ 0} be the outstanding order process, {IP t , t ≥ 0} the inventory position process, and {IL t , t ≥ 0} the process of net inventory on-hand minus backorder .Define {I t , t ≥ 0} {B t , t ≥ 0} to be the process of on-hand inventory backorder, resp. .For appropriate initial conditions, the following equations hold under Assumption 2. Note that IO t is the number of jobs in the supply process.The queueing-inventory method characterizes the probability distribution of IO t by identifying the appropriate queueing analogue.One can follow a 3-step procedure to characterize the system performance: 1 the distribution of IP t , 2 the distribution of IO t , and 3 the dependence of IO t and IP t .We focus on steady-state analysis and define IO lim t → ∞ IO t .The same notational rule applies to IL, IP , and I and B.
Clearly, IP s for base-stock policies.For batch-ordering policies, the distribution of IP only depends on the demand process.IP is uniformly distributed in {r 1, r 2, . . ., r Q} for renewal batch demand under mild regularity assumptions 22 .See Zipkin 19 for a discussion of more general demand processes.The distribution of IO depends on the demand process, the inventory policy, and the supply system see discussions below .For batchordering policies, IP depends on IO.Intuitively, the lower the IP , the longer the time since the last order, and therefore the lower the IO.

i.i.d. Transit Time
Consider first the basic model with constant L, the queueing analogue is a M/D/∞ queue.By Palm theorem 23 , IO follows Poisson λL distribution.If L is stochastic, then the queueing analogue is a M/G/∞ queue and IO follows Poisson λE L distribution.Because demand is satisfied on a FCFS basis, the stockout delay differs from L even at s 0; see Muckstadt 24, page 96 for an exact analysis.For renewal unit demand, the queueing analogue is a G/G/∞ queue.For compound Poisson demand, then the queueing analogue is a M Y /G/∞ queue where {Y n } is the demand size process.The distribution of IO is compound Poisson under Assumption 2.2.
Consider the basic model but with a batch ordering policy, the queueing analogue is a Er Q /D/∞ queue where Er stands for Erlang interarrival times.See Galliher et al. 25 for an exact analysis.For batch demand processes, tractable approximations become appealing.One can first assume IP ⊥ IO and then approximate the distribution of IO by results from systems with base-stock policy and batch-demand processes 9, Section 7.2.4 .Consider the batch ordering policy in the basic model with sequential transit times Definition 2.1 .Equation 2.4 does not hold because IO t is clearly not the demand during t − L, t .In addition, IO t depends on IP t .Exact analysis of these systems using the queueing-inventory method is rare.Fortunately, such systems can be easily handled by the lead-time demand method and the flow-unit method.

The Lead-Time Demand Method
Consider the basic model.Observe that at time t, the system receives all orders placed on or before t − L but none of the orders placed after t − L, then The inventory policy includes the batch-ordering policy and the s, S policy, and the demand process includes renewal batch process and the superposition of independent renewal batch processes 19 .We point out that for 2.5 and Proposition 2.4 to hold, the assumptions of sequential transit time, FCFS rule, and split demand are necessary.
In the basic model, the stockout delay, X, for a demand at t, satisfies 26 Pr{X ≤ x} Pr{D t − L x, t < s}, for 0 ≤ x ≤ L.

2.6
To see this, note that, at t x, all orders triggered by demand on or prior to t x − L are replenished.Because the demand at t has priority over demand after t, the demand at t is satisfied on or before t x if and only if the orders triggered by demand during t x − L, t are less than s.By the same logic, for compound Poisson demand, the stockout delay for the kth unit of a demand, X k , is given by Consider now the basic model under periodic review.Let IP n be the inventory position at the beginning of period n after order decision is made and IL n I n and B n the net inventory inventory on-hand and backorder at the end of period n after demand is realized.Let L here be an integer multiple of a review period and D n, m the demand from period n to m inclusive.According to the sequence of events see beginning of Section 2 , 2.

The Flow-Unit Method
For the basic model, suppose a demand arrives at time t, then the order triggered by this demand will satisfy the sth demand after t 28, 29 .Alternatively, the corresponding order that satisfies the demand at time t is placed at t − T s , where T s is determined by starting at time t, counting backwards until the number of demand arrivals reaches s 30 .We call the former the "forward method" because, for each order, it looks forward to identify the corresponding demand.We call the latter the "backward method" because, for each demand, it looks backward to identify the corresponding order.Both methods yield the same result for single-stage systems.For general networks, the two methods may take different angles, and thus one can be more convenient than the other Section 3 .We focus on the backward method unless otherwise mentioned.The stockout delay, X, for the demand at time t and the holding time, W, for the product that satisfies this demand are given by X L − T s , 2.9 Unlike the queueing-inventory method and the lead-time demand method, the flowunit method focuses on the stockout delay the inventory holding time associated with each demand product rather than the on-hand inventory and backorders at a certain time.Equations 2.9 -2.10 hold also for stochastic sequential lead times Definition 2.1 and for any point unit-demand process 31 .We should point out that the assumptions of sequential lead time and FCFS rule are necessary for 2.9 -2.10 .By 2.9 , the distribution of the stockout delay, X, is given by,

2.11
For compound Poisson demand, different units in one demand face statistically different stockout delays 29 .Consider the kth unit of a demand at t, the backorder delay, X k , and the inventory holding time, W k , for the corresponding item that satisfies this unit are where J k is obtained by starting at time t, counting backwards demand arrivals until the cumulative demand becomes greater than s − k in the first time.See Forsberg 32 and Zhao 33 for extended discussions.A comparison between 2.6 -2.7 and 2.11 -2.12 demonstrates the connections between the lead-time demand method and the flow-unit method.Because D t − L, t is the cumulative demand and T s is the sum of interarrival times, the event {T s ≥ L − x} is equivalent to the event {D t − L x, t < s} for unit demand 34, page 406 .Similarly, the event {T J k ≥ L − x} is equivalent to the event {D t − L x, t ≤ s − k} for batch demand.
For the basic model under periodic review, if demand arrives at the end of each period, then the system is a special case of its continuous-review counterpart 33 .If demand arrives during a period, the flow-unit method also applies, see, for example, Axsater 35 .For the basic model with batch ordering policy, by Axsater 36 , where S is a random integer uniformly distributed in {r 1, r 2, . . ., r Q}.See also Zhao and Simchi-Levi 30 .For the basic model with both batch ordering policy and compound Poisson demand, the analysis is more involved but still tractable, see Axsater 37 .

Multistage Supply Chains
Multistage supply chains differ from single-stage systems because the lead time at one stage depends on other stages' stock levels.For a stage, the lead time is the total time needed from order placement to order delivery.Clearly, lead times include but are not limited to the "transit times." Notation 1.Consider a supply chain under Assumption 2.2 with node set N and arc set A. An arc refers to a pair of nodes with direct supply-demand relationship.We define the following.
i {IO j t , t ≥ 0}: the outstanding order process at node j ∈ N.
ii {IP j t , t ≥ 0}: the inventory position process at node j.
iii {IL j t , t ≥ 0}: the net inventory on-hand minus backorder process at node j.
iv {I j t , t ≥ 0} {B j t , t ≥ 0} : the process of on-hand inventory backorder at node j.
v L j L i,j : the processing cycle time at node j transportation lead time over arc i, j ∈ A .
vi IT j IT i,j : the inventory in-transit during L j during L i,j .
vii L j : the total replenishment lead time at node j.
viii X j W j : the stockout delay inventory holding time at node j.
ix τ j α j , β j : the committed service time target type 1, 2 service at node j.
x a i,j : the BOM structure, that is, one unit at node j requires a i,j unit s from node i.
xi h j π j : the inventory holding cost penalty cost per unit item per unit time at node j.
xii s j r j , Q j : base-stock level reorder point, batch size at node j.

Serial Systems
In this section, we extend the methodologies and results of the single-stage systems to a serial supply chain where nodes j ∈ J are numbered by 1, 2, . . ., |J|.Node |J| receives external supply, node j 1 supplies node j, and node 1 supplies external demand.The transit time of node |J| is L |J| , and the transit time between stage j 1 and j is L j .This system can be controlled either by an installation policy or an echelon policy.For an installation policy, the notation is defined as above.For an echelon policy, we need the following notation.
i IP e j : the echelon inventory position at stage j, which is the sum of inventory onhand and on-order at stage j plus inventory on-hand and in-transit at all downstream stages of j subtracting B 1 .
ii IL e j IP e j − IO j : the echelon net inventory at stage j.
iii I e j IL e j B 1 : the echelon on-hand inventory.
iv IT e j IT j IL e j : the echelon inventory in-transit.
v s e j r e j : the echelon base-stock level reorder point .
An echelon batch-ordering policy works as follows: whenever IP e j drops to or below r e j , an order of size nQ j is placed to raise the echelon inventory position up to the smallest integer above r e j .According to convention, we assume that Q j 1 and r j 1 are integer multiples of Q j for all j.
We define the basic model for serial systems as follows: each stage controls its inventory by an installation base-stock policy; external demand follows Poisson process; the transit times are constant, and a j 1,j 1, for all j.We focus on the penalty cost model and refer to Boyaci and Gallego 38 and Shang and Song 39 for discussions on the service constraint model.

Echelon Policies versus Installation Policies
The echelon policies base-stock or batch ordering are equivalent to their installation counterparts under certain conditions.According to Axsater and Rosling 40 , two policies are equivalent if given identical initial conditions, the two policies share the same sample path for their inventory positions at all stages of the supply chain for any external demand sequence.
For serial systems under either continuous review or periodic review with identical periods, one can construct an equivalent echelon batch-ordering policy for each installation batch-ordering policy by setting r e 1 r 1 ; r e j 1 r e j Q j r j 1 , j 1, 2, . . ., |J| − 1.The initial conditions are r j < I j 0 ≤ r j Q j , and I j 0 − r j is an integer multiple of Q j−1 .
For an echelon policy, one may not always find an equivalent installation policy unless the echelon policy is nested: stage j 1 orders only when stage j orders for each j.The initial condition is r e j < I e j 0 ≤ r e j Q j .The result on batch ordering policies remain valid in pure assembly systems but not in distribution systems.Indeed, Axsater and Juntti 41 compare numerically the performance of echelon and installation batch ordering policies in a pure distribution system with Poisson demand and show that either policies can outperform the other and the difference is up to 5%.

Joint Distribution of Inventory Positions
Consider a continuous-review serial system with installation batch-ordering policies and compound Poisson demand, the inventory position vector IP t IP j t , j ∈ J forms a continuous-time Markov chain CTMC with state space S ⊗ j∈J {r j Q j−1 , r j 2Q j−1 , . . ., r j Q j } where Q 0 1.We focus on 3 questions: 1 what is the marginal distribution of IP at each stage? 2 When are the IP s independent across stages? 3 What is the distribution of IP seen by an order placed by a downstream stage?Proposition 3.1.If the CTMC of IP t is irreducible and aperiodic, then as t → ∞ 2 The inventory positions are independent across different stages.
3 Each order of stage j sees IP j 1 in its time averages.
Proof.See the appendix for a proof.
A sufficient condition for IP to be irreducible and aperiodic is that the external demand can equal 1.For a serial supply chain with echelon batch-ordering policies, the inventory position vector has a state space S e ⊗ j∈J {r e j 1, r e j 2, . . ., r e j Q j }.Because inventory positions at different stages are driven by a common demand process, they may not be independent.Proposition 3.1 does not hold here because the CTMC of IP e t may be reducible and depends on initial conditions, see Axsater 42 .Fortunately, if one assumes randomized initial conditions, then IP e is uniformly distributed in S e 43 .So far, the only result on non-Markovian demand process is that Proposition 3.1 holds for renewal unit external demand.See Section 3.2 for more discussions.

The Queueing-Inventory Method
Consider the basic model.Applying 2.1 to each stage, IP j t IO j t IL j t , j ∈ J. Define B |J| 1 t ≡ 0. Because IO j t B j 1 t IT j t , for all j, we must have That is, the inventory position at stage j consists of three elements: backorders at stage j 1, inventory in-transit from stage j 1 to j, and net inventory at stage j.By 3.1 and 2.3 , Note that IP j t is not independent of B j 1 t in general.Equations 3.1 -3.2 hold for any serial system under Assumption 2.2 and extend to periodic-review systems 44 .The queueing-inventory method focuses on characterizing IO j and IT j for each stage.

i.i.d. Transit Time
Consider the basic model with i.i.d.transit times.Other than the special case of s j 0, for all j / 1, where the system forms a Jackson network with mutually independent IT j t , the serial system poses a substantial challenge for exact analysis under the queueing-inventory method because IT j depends on B j 1 .An exact analysis is unknown 9 .Various approximations are proposed, see discussions of the distribution systems Section 3.2.1 .

Sequential Transit Time
For the basic model with stochastic sequential transit times Definition 2.1 , the analysis here is a special case of those of pure distribution systems.We postpone the discussion to Section 3.2.1.For batch ordering systems, the exact analysis by the queueing-inventory method is difficult because B j 1 and thus IO j is not independent of IP j .Fortunately, such systems can be easily handled by the lead-time demand method and the flow-unit method.

The Lead-Time Demand Method
Consider the basic model with sequential transit times Definition 2.1 .We discuss both installation and echelon policies.Extensions to compound Poisson demand is straightforward.

Installation Policies
, for all j.By the lead-time demand method, at time t, all outstanding orders except B j 1 t − L j will be available at stage j.Therefore, Equation 3.3 is similar to 2.5 in single-stage systems.The difference is that here only part of IO j t − L j , that is, IT j t − L j , is available at t.For base-stock policies, IP j t ≡ s j .Equation 3.3 implies the following recursive equations for B j in steady-state: where D ∞ | L j s are mutually independent.We refer the reader to Van Houtum and Zijm 45, 46 , Chen and Zheng 47 , and Gallego and Zipkin 48 for extended discussions.By 3.4 , a serial supply chain can be decomposed into |J| single-stage systems where one can characterize B j from j |J| to j 1 consecutively.Extension to batch-ordering policy is not straightforward because IP j depends on B j 1 .See Badinelli 49 for an exact analysis of systems with Poisson demand and constant lead times.Indeed, echelon policies are easier to handle using the lead-time demand method.

Echelon Policies
First consider echelon base-stock policies.By 2.5 ,

3.5
In

Approximations and Bounds
Policy evaluation based on the exact analysis can be time consuming.One can compute the system performance approximately but fast using two-moment approximations.For instance, one can compute 3.4 by fitting a negative binomial or Gamma distribution to the leadtime demand utilizing the first two moments 20, 51 .Equation 3.4 can also be regarded as incomplete convolutions of the form X 1 − a X 2 .Van Houtum and Zijm 45, 46 fit the incomplete convolutions by mixed Erlang or hyperexponential distributions.
An alternative approach is to develop bounds.The "Restriction-Decomposition" heuristic 48 is based on the observation that by 3.3 -3.4 , The latter is the sum of single-stage cost functions.One can then choose the base-stock levels that optimize the bound.
Shang and Song 52 develop Newsvendor types of close-form bounds and approximations for the optimal base-stock levels.The key idea is to construct a subsystem for each stage that includes itself and its downstream stages then replace the installation holding costs at all stages of the subsystem by either a upper or a lower bound.Such a subsystem effectively collapses into a single-stage system, for which one can use the newsboy model.For batchordering policies, Chen and Zheng 53 develop lower and upper bounds for the total cost by either under-or overcharging a penalty cost for each stage.The resulting bounds are sums of |J| many single-stage cost functions.
Finally, we mention that the performance gap between echelon and installation policies may be minor.Chen 50 compares the best echelon policy with the best installation policy in serial systems.For different number of stages, lead times, batch sizes, demand variabilities, and holding/penalty costs, it is shown, in a numerical study, that the % difference of their performance based on the optimal cost of echelon policies range from 0% to 9% with an average 1.75%.

The Flow-Unit Method
The flow-unit method provides an exact analysis for the basic model with either Poisson or compound Poisson demand.Because the analysis here is a special case of that of pure distribution systems, we postpone the discussion to Section 3.2.3.In the basic model with installation batch ordering policy, applying 2.14 to each j ∈ J yields, X j X j 1 L j − T j S j and W j T j S j − X j 1 − L j , where S j is uniformly distributed in {r j Q j−1 , r j 2Q j−1 , . . ., r j Q j } and S j , j ∈ J are independent Proposition 3.1 .Furthermore, T j • s are not overlapping, and therefore T j S j , j ∈ J are mutually independent.Consequently, a serial system can be decomposed into multiple single-stage systems as in Section 3.1.2.
The flow-unit method can also be applied to serial systems with echelon batch ordering policy 42 or base-stock policy under periodic review 32, 33, 41 .We postpone the discussion to distribution systems Section 3.2.3 .

Pure Distribution
In this section, we focus on 2-level pure distribution systems distribution systems, for brevity , where node 0, the distribution center DC , is the unique supplier for nodes j ∈ J the retailers that face external demand.The transit time of node 0 is L 0 , and the transit time between stage 0 and j is L j .Distribution systems are more complex than serial systems because i the demand process faced by the DC is a superposition of the order processes of all retailers and ii DC needs to allocate inventory among retailers in case of shortages.In this section, we focus on installation policies and FCFS rule unless otherwise mentioned.
Redefine S ⊗ j∈{0} J {r j δ j , r j 2δ j , . . ., r j Q j }, where δ j 1, for all j ∈ J, and δ 0 is the maximum common factor of Q j , j ∈ J, by the proof of Proposition 3.1, see also 54 .

Corollary 3.2. Proposition 3.1 holds for the inventory position vector of the DC and all retailers.
For demand under non-Markovian assumptions, Cheung and Hausman 55 show that if external demand follows independent renewal unit processes, then the first two statements of Proposition 3.1 hold for the inventory position vector of the DC and all retailers.
We define the basic model for distribution systems as follows: each stage utilizes an installation base-stock policy, external demand follow independent Poisson processes with rates λ j , j ∈ J, L j , j ∈ {0} J are constant, and a 0,j 1, for all j.No lateral transshipment is allowed.

The Queueing-Inventory Method
By 2.1 , IO j t IL j t IP j t holds for j ∈ {0} J under Assumption 2.2.Because IO j t B 0,j t IT j t , for all j ∈ J, and B 0,j t is the orders placed by stage j backlogged at stage 0, IP j t B 0,j t IT j t IL j t .

3.8
For the basic model, conditioning on B 0 b, B 0,j follows a binomial distribution with b number of trials and a successful rate of λ j / l∈J λ l per trial the "binomial decomposition," 51, 56 .This is true because the probability that an order received by the DC is placed by retailer j is λ j \ l∈J λ l , and each order is independent of the others.This result holds as long as external demand follows independent Poisson processes, retailers utilize continuous-review base-stock policy, and DC serves retailers' orders on a FCFS basis.For compound Poisson demand or batch ordering policy, it is much more involved to decompose B 0 into B 0,j , see Shanker 57 and Chen and Zheng 43 .

i.i.d. Transit Time
Consider the basic model.Similar to serial systems Section 3.1.1, such a system is difficult for exact analysis unless s 0 0. Various approximations are proposed where the basic idea is to decompose the system into multiple single-stage systems with the input parameters depending on other stages.
A simple approximation METRIC, 21 works as follows: first, apply the single-stage results Section 2.1 to the DC by noting that IO 0 is a Poisson random variable with parameter j∈J λ j •E L 0 .By 2.1 -2.3 , one can characterize IL 0 , I 0 , and B 0 .By Little's law, the expected stockout delay at DC is E X 0 E B 0 / j∈J λ j .Second, for each retailer j, regarding its supply system as an infinite server queue with a mean service time E X 0 E L j , one can again apply the single-stage results to obtain the distribution of IO j , I j , and B j .Clearly, the second step is an approximation because the orders placed by the retailers are satisfied by the DC on a FCFS basis.
Muckstadt 58 generalizes METRIC to include a hierarchical or indentured product structure MOD-METRIC : when an assembly needs repair, then exactly one of its subassemblies modules needs repair.To illustrate the idea, let us consider a single-stage system with a single assembly and its modules k ∈ K. Let s 0 s k be the stock-level of the assembly module k and R 0 R k its repair time.Assume the assembly failure rate is λ with probability p k that module k needs repair, then the expected total repair time for an assembly is , where E X k E B k / p k λ is the expected delay due to stockout of module k.E B k is the expected backorders of module k which can be computed by 2.1 and 2.3 and the fact that IO k follows Poisson E R k p k λ Section 2.1 .Once E R 0 is known, one can use METRIC to compute the performance measure at the assembly.
Sherbrooke 59 considers a similar model as Muckstadt 58 but utilizes a different approximation VARI-METRIC .The key difference is to compute the first 2 moments rather than the first moment of the backorders at the depot and the outstanding orders at each base then fit their distributions by negative binomial distributions.Numerical study shows that VARI-METRIC improves the accuracy of METRIC.For a thorough literature review on inventory control in supply chains with repairable items, see Muckstadt 24 .

Sequential Transit Time
Consider again the basic model.Note that each order placed by the retailers faces statistically the same stockout delay at the DC by the independent Poisson demand and the FCFS rule , the exact analysis works as follows: first, compute the distribution of IO 0 by L 0 and the demand process at DC by Proposition 2.3.Then, determine the distribution of B 0 by 2.3 .The distribution of X 0 can be determined by the fact that demand during X 0 from all retailers has the same probability distribution as B 0 by the proof of Proposition 2.3 .For any retailer j, the total replenishment lead time L j X 0 L j .Given the demand process at retailer j, one can compute the distribution of IO j and then B j and X j in a similar way.Svoronos and Zipkin 20 develop exact expressions of system performance for phrase-type transit times and present a two-moment approximation based on negative binomial distributions.
For compound Poisson demand, although the probability distribution of backorders may differ from that of the demand during stockout delay 29 , the latter serves as a good approximation to the former.Zipkin 29 generalizes the 2-moment approximation of Svoronos and Zipkin 20 to distribution systems and presents an exact analysis based on the flow-unit method for phrase-type transit times and demand sizes see also Section 3.2.3 .

The Lead-Time Demand Method
Consider the basic model with sequential lead times Definition 2.1 .Applying 2.5 to DC yields IL 0 t IP 0 t − L 0 − D 0 t − L 0 , t , where D 0 t − L 0 , t is the lead time demand for DC.By Proposition 2.4 and Corollary 3.2, we can determine the distribution of D 0 ∞ | L 0 , IL 0 , B 0 , and I 0 .For the retailers, we consider two cases.

Base-Stock Policy
By 3.8 , B 0,j t − L j IT j t − L j IL j t − L j IP j t − L j ≡ s j , j ∈ J.By the lead-time demand method, at time t, all outstanding orders except B 0,j t−L j will be delivered to stage j, yielding where D j t−L j , t is the lead-time demand for retailer j.Since the distribution of B 0,j is known "binomial decomposition", Section 3.2.1 , one can exactly characterize the distribution of I j and B j for all j 51, 56 .For fast computation, a two-moment approximation is proposed that fits B 0,j D ∞ | L j by a negative binomial distribution.In a numerical study, Graves 51 shows that the 2-moment approximation is more accurate than "METRIC" which only utilizes the first moment.Exact analysis is feasible for distribution systems where each retailer has multiple supply modes, for example, upon arrival of a demand, a retailer can order a unit either from the DC mode 1 or from mode 2 with constant lead time L 56 .The decision for each order is independent of others, so the total demand at stage j can be split into two independent Poisson processes each is served by a supply mode.Let D j t − L j , t D j t − L j , t be the leadtime demand served by mode 1 2.2 , then IL j t s j −B 0,j t−L j −D j t−L j , t −D j t−L j , t , where all random variables on the right-hand side are independent.
Consider the basic model but assume that each stage utilizes a periodic-review basestock policy.An important issue here is how to allocate DC's on-hand inventory to the retailers when the total demand exceeds the supply.The optimal allocation rule does not have a simple form, see, for example, Clark and Scarf 60 and Federgruen and Zipkin 61 .Therefore, most work so far focuses on heuristic rules, such as the "myopic" allocation rule 61 , the random allocation rule 62, section 3.2.3, and the "virtual allocation" rule 63 .The "virtual allocation" rule works as follows: the DC observes external demand at all retailers and commits its stock in the sequence of external demand arrivals rather than the sequence of retailers' orders.An exact procedure is developed to characterize the inventory levels at all stages.Numerical study shows that virtual allocation has good performance although it is not optimal.

Batch Ordering Policy
As we mentioned at the beginning of Section 3.2, one of the challenges in distribution system is that the DC's demand process is a superposition of the retailers' order processes.This demand process becomes difficult to characterize when the retailers' use batch-ordering policies.Even for a simple system with identical retailers, the DC's demand process is a superposition of |J| many independent Erlang processes by Corollary 3.2 , thus it is nonrenewal 64 .Inspired by the "METRIC" approach, Deuermeyer and Schwarz 64 , Lee and Moinzadeh 65, 66 , and Svoronos and Zipkin 67 decompose the distribution system into single-stage systems and propose various approximations for the retailers' lead-time demand.The key idea here is to characterize the moments of the DC backorders and then approximately determine either the delay due to stock at DC or the retailer j's share of the DC backorder.Finally, utilize either 2.5 or 3.9 to determine the moments of the lead-time demand at each retailer.See Axsater 16 for an extended discussion.
Chen and Zheng 43 consider the basic model with echelon batch ordering policies where the retailers may not be identical.The paper presents an exact analysis for Poisson demand and approximations for compound Poisson demand.To illustrate the idea, let IP e j or IL e j be the echelon inventory position echelon inventory level at stage j ∈ {0} J where IP e 0 IO 0 I 0 j∈J IT j IL j and IL e 0 IP e 0 −IO 0 .First, one has IL e 0 t IP e 0 t−L 0 − D 0 t − L 0 , t and B 0 t j∈J IP e j t − IL e 0 t .The distribution of B 0 can be determined by the fact that IP e j , j ∈ {0} J are independent due to randomized initial conditions .Then, decompose the DC's backorders to each retailer to obtain B 0,j , j ∈ J. Finally, IT e j IP e j − B 0,j and IL e j IT e j − D j ∞ | L j , see 3.6 .

The Flow-Unit Method
The flow-unit method enables exact analysis for a wide range of distribution systems.Consider first the basic model with the sequential lead time Definition 2.1 .Suppose a demand arrives at retailer j ∈ J at time t, the stockout delay for this demand and the inventory holding time for the product that satisfies this demand are given by 2.9 -2.10 , X j L j − T j s j and W j T j s j − L j , where L j is the total replenishment lead time for the order placed by stage j at time t − T j s j .For this order, the stockout delay and the inventory holding time for the corresponding item at the DC are X 0 L 0 − T 0 s 0 and W 0 T 0 s 0 − L 0 .Therefore, L j X 0 L j .Note that T j s j is based on the demand of retailer j while T 0 s 0 is based on the demand at DC.Because of Poisson demand, T 0 s 0 and thus X 0 is statistically the same for all retailer orders.Because T j s j , j ∈ J are not overlapping with T 0 s 0 , T j s j ⊥ T 0 s 0 .This implies that the distribution system can be decomposed into single-stage systems where one can first evaluate the performance of the DC and then the performance of each retailer, see, for example, Axsater 28 , Zipkin 29 , and Simchi-Levi and Zhao 31 .For compound Poisson demand, let us consider the kth unit of a demand at node j.One needs to identify not only the corresponding order placed by stage j but also the corresponding unit in that order that satisfies this demand unit.By Zhao 33 . Here, J j k is the index of the corresponding order defined in Section 2.3, and M j k is the index of the unit in the corresponding order that satisfies the kth demand unit at node j.The analysis extends to a periodic-review systems with base-stock policy and virtual allocation rule see Axsater 35 for Poisson demand and Forsberg 32 for compound Poisson demand .
We point out that for the special case of serial systems, the lead-time demand method handles Poisson demand and compound Poisson demand in the same way 3.4 but the flowunit method becomes considerably more complex.On the other hand, for compound Poisson demand, the flow-unit method handles the serial and distribution systems in the same way but the lead-time demand method becomes much involved the "Binomial decomposition" fails as one moves from serial to distribution systems 57 .
Batch-ordering policy complicates the analysis considerably due to the complex demand process faced by the DC.To see this, let us consider the basic model with identical retailers and installation batch-ordering policy.The number of system demand i.e., the demand of all retailers between two consecutive retailers' orders is now random versus a constant in the case of a single retailer .Forsberg 68 provides an exact analysis for distribution systems with batch ordering policy and Poisson demand.Axsater 36, 54 provides various approximations.
For distribution systems with both batch ordering policy and compound Poisson demand, Axsater 37 presents an exact analysis for installation policies and Axsater 42 considers echelon policies.The exact evaluation is, however, time consuming.Let m be a multiplier of the batch sizes.The computational effort is O |J| 5 and O m 2 68 , O |J| 2 and O m 4 37 , and O |J| 5/2 and O m 2 42 .Cachon 62 provides an exact analysis for a periodic-review system with installation batch ordering policy, identical retailers, and i.i.d.demand, where the DC randomly allocates stock to orders received in the same period but follows the FCFS rule to serve orders in consecutive periods.
Because the flow-unit method requires the FCFS rule and the assumptions that orders are replenished in the sequence as they are placed, it is not clear how to apply this method to problems where these assumptions fail, for example, systems with multiple supply modes Section 3.2.2, systems with reverse material flows 69 , and systems with rationing rules 70 .For these systems, the lead-time demand method still applies.

Assembly Systems
In this section, we consider both pure assembly systems where each stage has at most one customer and two-level general networks where each stage can have multiple customers or suppliers.
In a two-level general network, stages in I are suppliers and stages in J are customers.Supply-demand relationship exists only between sets I and J.It is convenient to call the set I components and the set J products.Let I j {j ∈ J | a i,j > 0} be the component set for product j, and J i {j ∈ J | a i,j > 0} the product set served by component i.Let L i L j the transit time at stage i ∈ I j ∈ J , and L i,j be the transit time e.g., transportation lead time from stage i ∈ I to stage j ∈ J.Note that each stage i ∈ I is performing a distribution operation and each stage j ∈ J is performing an assembly operation.We assume that a product can be assembled only when all necessary components are available.
The two-level general network includes the following important special cases: i pure assembly systems where |J| 1.Here, we index the unique stage in J by 0. ii Assembleto-order ATO systems where L i,j 0 for all i, and j, L j 0 for all j and all stages in J carry zero inventory.This model can be applied to CTO configure-to-order systems, repairable items with multiple failure 71 , and the "pick and ship" systems in B2C e-commerce.
The optimal policies on ordering or allocation in such a network are either not known or state-dependent and thus too complex to implement 72 .In practice, only suboptimal but simple ordering policies e.g., installation policies and simple allocation rules e.g., FCFS are implemented.Here, we focus on installation policies and FCFS rule unless otherwise mentioned.
Assembly systems pose a significant challenge for policy evaluation because of the common demand processes shared by different components.One has to address the question of how to characterize the dependence among components?And what is the impact of the dependence on system performance?
We define the basic model for assembly systems as follows: each stage utilizes an installation base-stock policy, external demand follows independent Poisson processes with 20 Advances in Operations Research rates λ j , j ∈ J, all transit times are constant.Let a i,j be either zero or one unless otherwise mentioned.When a stage j ∈ J places an order and some of its suppliers have on-hand inventory but others do not, we assume that the available stocks are shipped to stage j immediately.Clearly, each stage j ∈ J may hold inventory for components i ∈ I j which is not yet processed due to shortages of other components.We call this inventory the "committed stock" 15 .

The Queueing-Inventory Method
Consider the two-level general network under Assumption 2.2, by 2.1 and 2.3 , IO l t IL l t IP l t and B l t IL l t − , l ∈ I J. Let B i,j be the orders placed by stage j backlogged at stage i.Similar to 3.7 ,

3.10
For each product j ∈ J, let IT i,j be the inventory in-transit from stage i to j during time L i,j , IT j the inventory in-transit during L j , and I i,j the committed stock of component i at stage j.Then,

3.11
In the special case of ATO systems, the backorders at stage j, B j t , and the on-hand plus committed inventory of component i, I i t , are given by B j t max i∈I j B i,j t , 3.12

3.13
If |J| 1 in the ATO systems, then 3.12 -3.13 reduce to

3.15
Because the ATO systems capture the dependence among the components in the twolevel general networks, we focus on ATO systems for the rest of Section 3.3.
Consider first the basic model for ATO systems with i.i.d.transit times and |J| 1.The stages i ∈ I form |I| parallel M/G/∞ queues with common demand arrivals.The objective of the queue-inventory method is to characterize the joint distribution of the outstanding orders i.e., job in queues : IO IO i , i ∈ I .Once IO is known, B 0 is given by 3.14 , I i is given by 3.15 , and the order-based fill rate f 0 Pr{s i − IO i >, ∀i ∈ I}.
The analysis of IO is based on the following observation see, e.g., 73 .For simplicity, let I 2. Define ψ i • or Ψ i • to be the pdf cdf function of L i , for all i.Let Ψ c i u 1−Ψ i u .Consider an arbitrary demand arrival in 0, t .Due to Poisson demand, the arrival time of this demand is uniformly distributed in 0, t .Conditioning on the arrival time 0 ≤ u ≤ t, the probability that both queues i 1, 2 are still processing the job triggered by this demand at Similarly, the probability that only queue 1 or 2 is still processing the job at t is . Finally, the probability that both queues finish the job at t is p 0 u , same logic applies to p 1 , p 2 , and p 0 .
Let N t be the total jobs up to time t.Among these jobs, let N 1,2 t be those in process in both queues j 1 and 2, N 1 t or N 2 t those in process only in queue 1 or 2, resp., and N 0 t those left both queues.Because all arrivals are independent, conditioning on N t n, N Here, N • s are independent Poisson random variables and Lu et al. 74 generalize the result to ATO systems with multiple products and provide the generating function for IO and bounds for the order-based fill rates.Lu et al. 75 present bounds for the order-based backorders.Interestingly, the lower bound on E B j is related to the "binomial decomposition" in distribution systems Section 3.2.1 .Due to independent Poisson demand and FCFS rule, B i,j in 3.10 follows a binomial distribution for any given B i .By 3.12 , E B j ≥ max i∈I j {E B i,j } max i∈I j {E B i λ j / l∈J i λ l }.Lu and Song 76 formulate a nonconstrained cost-minimization problem for the model, where the total cost includes backorder cost and holding cost for both on-hand and committed stock.It is shown that the total cost is submodular in s i , i ∈ I.For other types of ATO systems, Gallien and Wein 77 , Cheung and Hausman 71 , and Dayanik et al. 78 characterize the distribution of IO which leads to either exact analysis or bounds on the key performance measure.See Song and Zipkin 15 for an extended discussion.
To date, it is not clear how to use the queueing-inventory method to characterize ATO systems with either stochastic sequential lead times or batch-ordering policies because the joint distribution of the outstanding orders is difficult to characterize and IO i , i ∈ I depends on IP i , i ∈ I .Fortunately, some of these systems can be handled by the lead-time demand method and the flow-unit method.

The Lead-Time Demand Method
We first consider the basic model for the ATO systems with |J| 1.Let us index the components i ∈ I in a nondecreasing order of their lead times, that is, Since all components face identical demand process, by Zipkin 9, Section 8.4.5 ,

3.17
Advances in Operations Research By 3.14 and 3.17 , For component i, the on-hand inventory is IL i and the committed inventory is B 0 − B i where B i IL − i .Analogous to 3.15 , the total on-hand plus committed inventory of component i is The key idea here is to identify the common lead-time demand shared by different components.
This approach can be generalized to multiproduct ATO systems with constant lead times.Consider the basic model with J |> 1.Because the demand processes for different components may not be completely identical, 3.17 no longer holds.Consider two components, i and ı.There are 4 cases.
2 J i J ı .This case can be handled by 3.17 .
All lead-time demand on the right-hand side of the equations are independent.
4 J i J ı / ∅ but J i / ⊂J ı and J ı / ⊂J i .This case is more complex but still tractable see, for example, 79 .The key idea is to identify the common lead-time demand for both components.
Using convolution, Song 79 presents exact expressions for the order-based fill rates f j Pr{s i − D i t − L i , t > 0, i ∈ I j }.It is also shown that f j ≥ i∈I j Pr{s i − D i t − L i , t > 0}.This inequality implies that ignoring the correlation among components results in underestimating the fill rates.
To determine the expected order-based backorders, Song 80 utilizes the relation between the fill rate and the stockout delay.Let X j be the stockout delay for product j.Clearly, 0 ≤ X j ≤ max{L i , i ∈ I j }, and, by 2.6 , Pr{X j ≤ x} Pr{D i t − L i x, t < s i , i ∈ I j and x < L i }.By Little's law, one can translate the problem of the expected backorders to the problem of the expected stockout delays.See Song 80 for a detailed discussion.
Consider the basic model for the ATO systems except that stage i ∈ I utilizes a batch ordering policy r i , Q i .If external demand follows a compound multivariate Poisson process, Song 81 shows that the inventory position vector of all components, IP i , i ∈ I , is uniformly distributed in ⊗ i∈I {r i 1, r i 2, . . ., r i Q i } if the CTMC of IP i , i ∈ I is irreducible and aperiodic.Therefore, the expected order-based backorders and fill rates, of a batch-ordering ATO system, can be expressed as the average of the counterparts of multiple base-stock systems.
For ATO systems under periodic review, the idea is similar: identify common leadtime demand shared by components.However, the allocation rule for common components becomes an important issue.Hausman et al. 27 consider a multi-item system where D i n , the demand in nth period, follows multivariate normal random distribution.Assuming constant lead times, FCFS rule, and independent demand across periods, the probability of satisfying all demand in period n within τ periods of time is Pr{D i n − L i τ, n ≤ s i , i ∈ I} by 2.8 .Zhang 82 considers a different allocation rule, the "fixed-priority" rule: while demands in consecutive periods are served on a FCFS basis, demands in the same period are served based on their priority.Let j j denote that demand j has higher priority over demand ı.The fill rate for customer type j is given by Pr{D i n−L i , n−1s l∈J,l j a i,l D l n ≤ s i , i ∈ I j }, where D l n is the demand of product l at period n.Since high dimensional multivariate normal distributions are computationally intensive, bounds on the fill rates are developed.Agrawal and Cohen 83 study the "fair-share" rule for demand in the same period: if component i has a shortage in period n, then the fraction of component i's available stock allocated to product j equals to D j n /D i n .The resulting order-based fill rate is identical to that of Hausman et al. 27 .de Kok 84 imposes an "ideal" product structure on the model of Hausmans et al. 27 : if L ı ≤ L i , then either J ı J i ∅ or J ı ⊆ J i .An ATO system is "strongly ideal" if it has an idea product structure and satisfies the condition that for any product j ∈ J ı J i , a i,j a ı,j .Further assume a linear allocation rule and demand occuring at the end of each period, it is shown that the order-based fill rates satisfy Pr{ L i −1 l 0 k∈J a i,k D k n − l ≤ s i , i ∈ I j }, for all j.If the ATO system is strongly ideal, then fill rates have the form of Pr{ m l 1 Z l ≤ c m , m 1, 2, . . ., M} which is a generalized finite horizon nonruin probability studied extensively in the actuarial literature.
Unlike serial and distribution systems Sections 3.1-3.2, extensions from constant lead times to stochastic sequential lead times by the lead-time demand method is not straightforward because it is difficult to determine the common lead time demand.The flow-unit method, which separates demand from the lead time, provides a simpler and cleaner analysis.

The Flow-Unit Method
Consider the basic model for ATO systems with stochastic sequential lead times where the component inventory is managed by either continuous-time base-stock policies or batchordering policies.We refer to the latter as a batch-ordering system and the former as a basestock system.The following discussion is based on Zhao and Simchi-Levi 30 .

Single-Product Base-Stock Systems
Let |J| 1.Consider components i and ı.Without loss of generality, let s i ≤ s ı .Suppose a demand arrives at time t, then the corresponding orders of the components i and ı that satisfy this demand are placed at time t − T s i and t − T s ı , respectively the "backward method", see Section 2.3 .It is easily seen that T s i overlaps with T s ı over the time period t − T s i , t , and therefore T s ı T s i T s ı − s i .The dependence among the arrival times t − T s i L i , i ∈ I, is quite intuitive: if the interarrival times are short for recent demands, and as a result T s i is small for all i ∈ I, then all components are likely to be out of stock.
Indexing the components in the nondecreasing order of their base-stock levels, for any sequence of t 1 ≤ t 2 ≤ • • • ≤ t |I| , the joint probability density function of T s i , i ∈ I, is given by

3.18
For other sequences of t

3.19
The waiting time of component i, i ∈ I, is determined by W i X − L i T s i .The backward method may work better for assembly systems than the forward method because, in the latter, the orders of components triggered by a demand will satisfy different demand in the future; while in the former, we focus on a demand and identify all the orders placed beforehand that satisfy this demand.The flow-unit method separates the demand process from the lead times rather than putting them together as lead time demand.Thus, the demand process determines T s i , i ∈ I, whose joint distribution can be easily characterized, and the supply system determines L i , i ∈ I, which need not be independent.

Multiproduct Base-Stock Systems
Let |I| > 1.Assuming that a demand of product type j ∈ J arrives at time t, then the corresponding order of component i ∈ I j that satisfies this demand is placed at time t − T i,j s i , where T i,j s i is determined by starting at time t, counting backward demand arrivals of all products that require component i until the total number of arrivals reaches s i .Because of the lead time, an order placed at time t − T i,j s i will arrive at time t − T i,j s i L i .For each product j ∈ J, the stockout delay is X j max i∈I j {L i − T i,j s i } , and component i's waiting time, when it is committed to product j, is W i,j X j − L i T i,j s i .Thus, the multiproduct ATO system can be decomposed into |J| single-product subsystems with each subsystem corresponding to a product j ∈ J and its component set I j .It is important to note that these single-product subsystems are not identical to the single-product assembly systems because T i,j s i is associated with the superposition of the demand processes of all products that require component i. Close-form expressions are derived for the covariance matrix of T i,j s i , i ∈ I j .Zhao 33 characterizes their joint probability distribution.
Zhao and Simchi-Levi 30 proposes two numerical methods to evaluate system performance.The first method is based on Monte Carlo simulation while the second method is based on a two-moment approximation.A numerical study of an example inspired by a real world problem, the Dimension 2400 Pentium of Dell, shows that the simulation-based method is scalable and can evaluate large size, real world ATO systems; while the method based on the 2-moment approximation can handle up to medium size ATO systems with multiple products.

Multiproduct Batch-Ordering Systems
Now assume that inventory of each component is controlled by a continuous-time batchordering policy.Let S j ⊗ i∈I j {r i 1, r i 2, . . ., r i Q i }.Based on Song 81 , Zhao and Simchi-Levi 30 prove the following proposition.Proposition 3.3.Assume that the Markov chain of the inventory position vector of the components is irreducible and aperiodic.Suppose that a demand for product j ∈ J arrives at time t, then the corresponding order of component i, i ∈ I j , that satisfies this demand is placed at time t − T i,j S i , where the random vector S i , i ∈ I j is uniformly distributed in S j .
Based on Proposition 3.3, the order-based fill rates and the expected stockout delays can be expressed as the averages of their counterparts in the base-stock systems.However, the number of the corresponding base-stock systems is exponential in the number of components.By exploring the problem structure, Zhao and Simchi-Levi 30 develop efficient numerical methods based on Monte Carlo simulation.Given the sample size, the number of products, and the reorder points, the computational complexity of the methods is no more than that of sorting a set of real numbers, where the set size equals to the sum of the batch sizes of all components.

General Supply Networks
In this section, we discuss extensions of the three generic methods to general supply chains.

Supply Chain Characteristics
A supply chain consists of facilities and products.To specify a network, we need to know the processing cycle time for each product at each facility and the transportation lead time between every two facilities.We also need to know the BOM structure, external demand processes, target service levels e.g., the committed service times and the target fill rates , and the value added at each facility for each product.

Network Classification
A node or a stage refers to a unique combination of facility and product, and an arc refers to a pair of nodes with direct supply-demand relationship.A tree network is the one where breaking any arc results in two separate subnetworks.A tree network includes serial distribution and pure assembly as special cases.Networks with at most one directed path between every two nodes include tree as a special case but are not limited to tree, for example, the two-level general networks Section 3.3 .An acyclic network is more general which allows multiple directed paths between two nodes.Finally, supply chains may have feedback loops or reverse flows which form into close loop networks.
Unless otherwise mentioned, we assume that Assumption 2.2 holds.In addition, each node utilizes an installation base-stock policy, a i,j equals either zero or one, and external demand follows independent Poisson processes in case of continuous review or is i.i.d.random variables in case of periodic-review.

The Lead-Time Demand Method
We follow the development of the literature by first considering the lead-time demand method.The idea here is the same as "METRIC": breaking a network into multiple single-stage systems with the input parameters depending on each other.
Lee and Billington 1 analyze the Hewlett-Packard DeskJet printer supply chain with the objective of providing tools for managers to evaluate various stock positioning strategies.Each stage in the supply chain utilizes a periodic-review base-stock policy, the transit times in manufacturing and transportation processes are stochastic and sequential.Demand process at each stage can be obtained by aggregation of the BOM.For each node j, the total replenishment lead time L j consists of three parts: the processing time at node j, transportation lead times, and stockout delays from immediate suppliers.For assembly systems, it is assumed that at most one supplier can be out of stock in each period.Let f i be the fill rate at stage i.Hence, Similarly, V L j can be determined by the first two moments of X i , L j , and L i,j .
Let RP be the length of one review period.One can compute the first 2 moments of the lead-time demand at node j by E L j RP j μ j and E L j RP j σ 2 j μ 2 j V L j , respectively.Here, μ j or σ j is the mean standard deviation of demand in one period.Approximating the lead time demand by a normal random variable, then the on-hand inventory is determined by Proposition 2.4; E X j and V X j are computed based on 2.8 where L is replaced by E L j .

The Queueing-Inventory Method
Ettl et al. 85 applies the queueing-inventory method to supply chains where each stage utilizes a continuous-time base-stock policy, all transit times are i.i.d.random variables, and the external demand follows compound Poisson process.For each node j, L j is given by

3.21
To compute the moments of L j , it is assumed that at most one supplier can be out of stock at any time 1 .Then, the supply process at node j is approximated by a M Y /G/∞ queue with L j being the service time.By queueing theory, one can derive expressions for the moments of IO j , which in turn yields the statistics of I j , B j see 2.1 -2.3 , and customer service levels.Since it is a challenge to determine E X j , an upper bound based on M/M/∞ queue is utilized.
In addition to performance evaluation, Ettl et al. 85 optimize the total inventory investment, that is, the sum of expected work-in-process and finished goods inventory, in the supply chain subject to meeting certain service requirements of the external customers.Using the safety factors service levels as decision variables, the authors developed analytic expressions for the gradients, and therefore the constrained nonlinear optimization problem can be solved by the conjugate gradient method.Numerical studies show that this problem has many local optimal solutions, and the strategy of setting high fill rates at all stages can perform poorly relative to the optimal solution an average of roughly 20% gap is recorded .

The Flow-Unit Method
Applying the flow-unit backward method, one can provide exact analysis of supply chains with exogenous, stochastic, and sequential transit times Definition 2.1 .
Simchi-Levi and Zhao 31 consider tree networks with independent Poisson demand and continuous-review base-stock policies and develop exact recursive equations for the stockout delays at all stages of the supply chain.At node j, we must have, see 2.9 ,

3.22
Clearly, X i L i − T i s i where T i s i may be dependent see, e.g., 3.18 .The key idea here is that, for each external demand, we look backward in time to identify the corresponding order placed by each stage in the supply chain that eventually satisfies the demand.Thus, the recursive equations hold not only for systems in steady state, but also for systems in transient states with time varying and/or temporally correlated demand.
If supplier i is in turn supplied by other node s in the system, then L i may be correlated across nodes i where i, j ∈ A, and L i may also be correlated with T ı s ı for i / ı.Indeed, the stockout delays of parallel branches in a multistage assembly system can be correlated.See Simchi-Levi and Zhao 31 for an indepth analysis of the correlations.The following proposition characterizes the impact of the correlations on system performance.Proposition 3.4.Consider a tree-structure supply chain.If external demand follows independent Poisson processes, then any assembly node in the system has stochastically shorter backorder delay and longer inventory holding time than their counterparts in an analogous system with independent lead times.
Based on the recursive equations, Simchi-Levi and Zhao 31 prove the following properties.
Theorem 3.5.Given two serially linked nodes, node 2 (supplier) and node 1 (customer), in a tree supply network.Let s 2 > 0.Then, moving one unit of inventory from node 2 downstream to node 1 yields i stochastically shorter backorder delay (equivalently, stockout delay) at node 1 and ii stochastically shorter inventory holding time for any item traveling through both nodes.
This theorem holds for any tree-structure supply chain facing point demand processes under the assumption that demand and supply can be split.One application of this theorem is that moving inventory from all upstream stages to the most downstream stage reduces stochastically the total inventory holding time for any item in the system as well as the backorder delays to the external customers.Proposition 3.6.Under Definition 2.1 and the assumption of independent Poisson demand processes, L j is independent of T j s j at every node j ∈ N.
Guided by the exact analysis, Simchi-Levi and Zhao 31 present two-moment approximations and tractable decompositions that lead to an efficient evaluation and optimization algorithm for general tree-structure supply chains.The algorithm computes the first two moments of the stockout delay at each stage of the network according to Proposition 3.6 and 3.22 .To identify the optimal or near optimal stock levels in the supply chain that minimize system-wide inventory cost subject to service level constraints, the algorithm employs a dynamic programming routine to evaluate all stages sequentially.The algorithm is tested in various supply chains including a 22-stage and 21-arc assembly network inspired by a real world problem, the Bulldozer supply chain, see Graves and Willems 11 .Comparing to simulation results, the approximations are sufficiently accurate for a wide range of system parameters, and the algorithm computes the optimal or near optimal stock levels efficiently.It is shown that the lead time uncertainties have significant impact on the stock levels and stock positions, and ignoring lead time uncertainties can lead to substantial errors.Zhao 33 extends the analysis and approximation to compound Poisson demand and networks with at most one directed path between every two nodes.Shi and Zhao 86 consider acyclic supply chain and discover some simple yet unique properties.

Conclusion
We conclude the paper by pointing out some extensions of the models and methodologies and some of the remaining challenges.

General Supply Network with Batch Ordering Policy
In practice, economies of scale in production or transportation costs may drive batch ordering policies across the supply chain.General supply networks, for example, tree, with batch ordering policies and lead times have not been studied in the literature.Indeed, Ettl et al. 85 and Muckstadt 24 call for models and algorithms to handle these systems.

Supply Chains with Multiple Products: Design of Network and BOM
For exact analysis of general structure supply chains with multiple products, two challenges remain: 1 the mapped network may be acyclic and 2 a i,j may be any nonnegative integer.Resolving these challenges requires an extension of the stochastic, sequential lead time model Definition 2.1 to include joint probability distributions for the transit times 33 .
Despite these challenges, inventory positioning in multiproduct supply chains with common components deserves attentions as it holds the promises of jointly optimizing BOM, network, and inventory.Without doubt, the design of network such as selection of suppliers, transportation modes, manufacturing capabilities, and locations of facilities greatly affect the inventory costs and service levels.Moreover, the implementation of strategies such as component commonality, modular design, and postponement has made significant impact on real-world supply chains, see, for example, Feitzinger and Lee 87 .Given recent developments in the inventory positioning literature in particular, in assembly systems and general networks , we see huge opportunities in this direction.

Supply Chains with Reverse Material Flows
The reverse material flows can be caused by returns, recycling, or feedbacks.Supply chains with returns are different from those handling repairable items because, in the latter, a returned defected item is always accompanied by a demand for a workable item and the defected item cannot be reused immediately.So far, researchers have applied the lead-time demand method to supply chains with returns, see, for example, Fleischmann et al. 69 for single-stage systems, DeCroix et al. 88 for serial systems and DeCroix and Zipkin 89 for assembly systems.It is not clear, though, how other two methods can be applied here.

Advances in Operations Research
Proof of Proposition 3.1.To prove the first statement, note that the CTMC of IP is ergodic due to finite state space and it has a unique steady-state distribution.Because the uniform distribution is stationary for the inventory position vector 101, page 64 , IP is uniformly distributed in S. The uniform distribution of IP immediately implies the independence of the inventory positions.
Clearly, the interarrival times between two orders placed by stage j may not follow exponential distribution.Thus, to prove the third statement, we utilize "ASTA" by Melamed and Whitt 102 .By the second statement, IP j 1 is independent of the order placement at stage j.Therefore, the weak lack of anticipation assumption see 102, Definition 1 holds for IP j 1 and the order process by stage j.Because E IP j 1 does not depend on t, it follows from Theorem 2 of Melamed and Whitt 102 that, in steady state, each order placed by stage j sees IP j 1 in its time averages.

Proposition 2.3. IO has
Consider the basic model with sequential transit times Definition 2.1 .Let D t 1 , t 2 be the demand during time interval t 1 , t 2 , where t 1 ≤ t 2 , and let D ∞ | L lim t → ∞ D t − L, t .By Svoronos and Zipkin 20 : the same distribution as D ∞ | L .
IL and IP are measured at different times t or t − L in the lead-time demand method rather than the same time t in the queueing-inventory method.Let {IP t n } be the embedded discrete time Markov chain DTMC formed by observing IP t right after each ordering decision at t n .Zipkin 19 shows the following.
steady-state, IL e |J| t − L |J| − D t − L |J| , t .Given IL e j 1 t , IT e j t is uniquely determined as follows 44 : 1,2 t , N 1 t , N 2 t , N 0 t follows multinomial distribution with parameters n, p 1,2 , p 1 , p 2 , p 0 .Clearly, IO i t N 1,2 t N i t , i 1, 2, and IO i s are dependent due to the common element, N 1,2 .Applying the logic to ATO systems with any |I| and let t → ∞, 1 , t 2 , . . ., t |I| , Pr{T s 1 t 1 , T s 2 t 2 , . . ., T s |I| t |I| } 0. Advances in Operations Research By 3.18 , we can derive the probability distribution for the product stockout delay, X 0 max i∈I {L i − T s i } .For any service time τ ≥ 0 , conditioning on L l l 1 , l 2 , . . ., l |I| yields Pr{X