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On the structure of capacitated assembly systems
Operations Research Letters 41 (2013) 19–26
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On the structure of capacitated assembly systems Alexandar Angelus a,∗ , Wanshan Zhu b a
Lee Kong Chian School of Business, Singapore Management University, 50 Stamford Road, Singapore 178899, Singapore
b
Department of Industrial Engineering, Tsinghua University, Beijing 100084, China
article
info
Article history: Received 29 May 2012 Received in revised form 4 November 2012 Accepted 4 November 2012 Available online 16 November 2012 Keywords: Multiechelon inventory Capacitated assembly Optimal policy
abstract We consider a finite-horizon inventory model with an assembly structure, stochastic demand, and capacity constraints throughout the system. We explore optimal behavior of this system, with a special focus on stockpiling and stock-withholding of inventory, two novel inventory phenomena that arise due to asymmetric constraints. We identify conditions, such as having the bottleneck at the most downstream stage, that restore structural symmetry to the problem. This allows the state space to be collapsed to that of an equivalent series system. © 2012 Elsevier B.V. All rights reserved.
1. Introduction We consider a finite-horizon, multi-period, assemble-to-stock inventory system where products of a single type are assembled from multiple components, each with a possibly different leadtime. We assume stochastic, possibly non-stationary, demand for the assembled end-product. In addition, we allow production (capacity) constraints to apply to each component at each stage in this assembly system. Those constraints act as a (finite) upper limit on the amount of any component that can be processed at any stage in a single period. We explore the structure of this capacitated assembly system, with the objective of facilitating the development of means to effectively and efficiently manage it. In general, very little is known about capacitated assembly systems, and even less about their structural properties, so that the management of such systems in practice becomes exceedingly difficult. Thus, for example, due to their complexity, capacity constraints on (sub)assemblies are not taken into account by traditional MRP systems, which, as a result, may generate infeasible orders for assembly parts. More generally, companies are struggling to effectively manage capacity-constrained supply chains for assembled products. In analyzing a capacitated assembly system, we assume non-stationary demands and model parameters, as well as full backlogging of unsatisfied demand and carryover of inventory. The
∗
Corresponding author. Tel.: +65 6828 0413. E-mail addresses: [email protected] (A. Angelus), [email protected] (W. Zhu). 0167-6377/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2012.11.001
level of assembled products at the beginning of a period therefore equals the level at the beginning of the previous period plus the amount of newly completed units minus the random demand in that period. We first address a system with a single assembly at the most downstream stage, and provide a formulation of such an assembly system that allows for multiple components and multiperiod leadtimes between successive stages. We numerically explore those systems, and describe two novel inventory phenomena that occur in them due to the asymmetric constraints on the flow of inventory across different components. The first phenomenon acts to maintain a shortage of one component relative to all others, while the second one acts to keep an excess inventory of one component relative to all others. Those phenomena are also shown to prevent the landmark results of [14] from applying to capacitated assembly systems considered in this paper. Next, we identify system configurations, such as having the bottleneck in the system be located at the most downstream stage, that restore structural symmetry to the problem, thus allowing an assembly system with capacity constraints to be reduced to an equivalent series system. One consequence of this state space collapse, of relevance to inventory managers at companies, is that such an assembly system becomes much easier to manage by means of existing heuristics and approximations for capacitated series systems. We further investigate a more general problem, in which (sub)assemblies are allowed throughout the (capacitated) system. The same conditions that restore the structural symmetry to the original assembly model are shown to do the same in this (more) general assembly setting, so that the original state space collapse carries over in a direct manner.
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A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
Fig. 1. Component assembly system.
Other related literature The study of assemble-to-stock systems under stochastic demand can be traced to [15]. This work addresses a two component assembly problem with joint ordering and assembly decisions, and shows that the problem can be decomposed into component ordering decisions and finished goods assembly decisions. The classic paper in the field, [14], considers a stationary, multiechelon assembly under random demand and infinite horizon, and arrives at conditions under which this assembly problem can be reduced to an equivalent series system of Clark and Scarf, first introduced in [5]. Chen and Zheng [4] offer another derivation of this result, while [7] applies this approach to an assembly system with multiple end-products and derives well-performing heuristic policies. DeCroix and Zipkin [6] extend the results of [14] to uncertain product (and component) returns from customers. They describe the item-recovery pattern and restrictions on the inventory policy under which an equivalent series system can be shown to exist. When returns are allowed only for the end product, no policy restrictions are required to reduce the assembly system to an equivalent series one. Angelus and Porteus [1] generalize the results of [24] to include non-stationary, Markov modulated demands and model parameters, as well as production assets not depleted by stochastic demand. In the context of capacitated series systems, [8,9] analyze the stability and sensitivity of such systems under the control of a basestock policy. Parker and Kapuscinski [13] proves that, when the lowest capacity in a two-echelon series system is at the downstream stage, and the leadtime at the upstream stage is one period, the objective function, in the spirit of [8], becomes additively convex. The optimal policy is the modified echelon base stock policy: the downstream station orders the optimal echelon basestock level subject to both stock availability at the upstream station, and its processing capacity. Janakiraman and Muckstadt [11] rederives this result using the customer-unit decomposition approach of [12], and shows that, when the leadtime at the upstream echelon is extended to two periods, the optimal policy becomes a ‘‘two-tier, basestock policy’’: the optimal decision at the bottom echelon is determined by one basestock level when that level is below the inventory in transit, and by another basestock level otherwise; the optimal decision at the upper echelon has a similar two-tiered structure.
Capacitated assembly systems have so far only been studied under the assumption of basestock control. In particular, [10] explores the convexity properties of different cost functions under basestock policies, and shows, among other things, that the shortage cost is jointly convex with respect to the basestock levels and capacity limits. 2. Model description We assume, without loss of generality, that assembling a product requires exactly one of each of n components. Thus, the number of assembled products is simply the minimum number of completed components available, over all component types. Components are managed independently over time and assembled into the final product at the most downstream stage in the system. Each component type i(= 1, 2, . . . , n) has a standard leadtime of Li periods, which is the smallest possible number of periods between placing an order for the component and its becoming a part of an assembled product. We assume L1 ≤ L2 ≤ · · · ≤ Ln . In our basic model (an example is depicted in Fig. 1), assembly can be interpreted to take place only once—in the period immediately preceding completion of the product. Consequently, we refer to this system as the component (assembly) system because it is the components that flow downstream until they are assembled at the finished goods stage. Following [5], we identify stages (also echelons) for the scheduled number of periods that a component is away from being assembled at the bottom stage. When a component is first ordered, it is in stage Li + 1. At the beginning of the next period, this component is in stage Li . Each unit can be moved downstream one stage at a time, until it reaches stage 1, referred to as the finished goods stage, where it is assembled into the final product used to satisfy (stochastic) customer demand. For each component, we allow for the existence of stages where a unit can be delayed (i.e. deferred), if necessary, in which case it stays at the same level of completion until the next period (inventory holding costs accrue accordingly). For each component, we also allow for stages where inventory cannot be deferred (e.g., ocean-going shipping). We refer to any such stage as a flow-through stage. Fig. 1 illustrates a component assembly system: stages 3 and 4 for component n, and stage 2 for component 2, are the flow-
A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
through stages in the system. (Stages that allow deferrals are shaded). The following parameters describe our system: xijt = (initial) on-hand inventory of component i in stage j, at the beginning of period t , prior to making any decisions; Xijt = the decision variable representing the amount of component i ordered into stage j during period t ; Kij = processing capacity for component i at stage j; Dt = the stochastic demand for assembled final products in period t . The order quantities Xijt are limited by both xi,j+1,t , the on-hand inventory available at the upstream stage, and by Kij , the processing capacity for component i at stage j. Thus, Xijt ≤ min (xi,j+1,t , Kij ) for all i and j in period t. In addition, for each flow-through stage j, 1 < j ≤ Li , of component i, Xi,j−1,t = xijt in each period t, by definition. We assume full backlogging of unsatisfied demand and carryover of inventory. The collection of all on-hand inventories xijt , denoted by xt , will be referred to as the on-hand (inventory) state, while the collection of all order quantities Xijt , denoted by Xt , will be referred to as the order schedule. The sequence of events in each period t is as follows: (1) on-hand inventory state xt is observed; (2) order schedule Xt is selected; (3) ordered amounts are received; (4) customer demand is observed and satisfied to the extent made possible by the available stock; and, (5) costs are incurred. The transition equations therefore become xi1t + Xi1t − Dt , xijt + Xijt − Xi,j−1,t
xij,t +1 =
if j = 1, if j = 2, . . . , Li .
(1)
Because assembly requires exactly one unit of each component, the quantity of assembled products at the end of period t is min(x11t + X11t , x21t + X21t , . . . , xn1t + Xn1t ). The most downstream stage can be assumed, without loss of generality, to be able to process exactly the same amount of each component in any period, so that K11 = K21 = · · · = Kn1 = K1 . A unit backlogging cost pt is charged for each unit of demand not satisfied in period t; a unit inventory holding cost, Hijt is incurred on the amount of component i located at stage j at the end of period t. Holding a unit assembled product at the finished goods stage at the end of period t incurs a holding cost HtA . Thus, one-period costs in period t are
pt E
Dt − min(xi1t i
+ HtA E +
Li n
+ + Xi1t )
min(xi1t + Xi1t ) − Dt
+
i
Hijt (xijt − Xi,j−1,t + Xijt ),
i=1 j=1
where Xi0t := 0. The first term is the expected backlogging cost; the second term is the expected holding cost for the assembled product; the third term, the holding costs for unassembled components. We reformulate the problem using the following echelon variables: yijt := xi1t + xi2t + · · · + xijt referred to as the echelon j inventory of component i; Yijt := yijt + Xijt referred to as the echelon j level for component i.
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The updated echelon quantities are given by yij,t +1 = Yijt − Dt . The collection of all echelon inventories, denoted by yt , will be referred to as the echelon (inventory) state; the collection of all echelon levels, denoted by Yt , will be referred to as the echelon (inventory) schedule. Let SD (i) be the set of all stages for each component i where inventory can be deferred, and let SF (i) be the set of all flowthrough stages. Thus, SD (i) ∩ SF (i) = ∅, and SD (i) ∪ SF (i) = {1, . . . , Li }. (We allow SF (i) to be empty; SD (i) cannot be empty for any i, since, by definition, stage 1 is in SD (i) for every i.) We refer to any set of contiguous flow-through stages that are bounded below by an element SD (i), and are either bounded from above by another element of SD (i), or extend all the way to Li , as a (flow-through) string. We partition each SF (i) into flow-through strings S1F (i), . . . , M
SF i (i), for any Mi , as follows: Let ji1 be the smallest element of SF (i), and let ji1 ∈ S1F (i); define S1F (i) as S1F (i) := {ji1 , j ∈ SF (i) | j − 1 ∈ S1F (i)}. Thus, S1F (i) represents the stages of the first flow-through string for component i. Let qi1 be the largest element of S1F (i). We proceed recursively: having defined Sm F (i) for some m < Mi , as well as its smallest element jim , and its largest element qim , let jim+1 , the smallest element of SFm+1 (i), be defined as the smallest element of SF (i) larger than qim ; that is, jim+1 := min{j | j > qim , j ∈ +1 SF (i)}. Then, SFm+1 (i) := {j ∈ SF (i) | j − 1 ∈ Sm (i)}. The F M
strings S1F (i), . . . , SF i (i) are mutually exclusive and collectively exhaustive. We now define the feasible set Y(yt ), given echelon inventory state yt in period t.
y ≤ Y ≤ y ijt i,j+1,t , Yijt − yijt ≤ Kij ijt if { j , j + 1 } ∈ SD (i); yijt ≤ Yijt ≤ yi,j+1,t , Yijt − yijt ≤ min K min Ki µ Y(yt ) := Yijt , 1 ≤ i ≤ n i,jim −1 , µ∈{jim ,...,qim } if ∃m ∈ {1, . . . , M }, j = qim ; Y = ijt yi,j+1,t if j + 1 ∈ SF (i).
The first line in the expression for Y(yt ) applies to any stage j that allows inventory deferrals and is located immediately downstream of another such stage. The second line applies when stage j + 1 allows deferrals, while stage j does not; in that case, each quantity Xijt ordered into stage j has to be sufficiently small to meet not just the processing capacity constraint of stage j, but also the constraints of all other stages in the flow-through string of which stage j is the uppermost stage. The third line describes the flow-through constraint. Let Ft (Yt ) denote the minimum expected present value of the costs over periods t through T , given that decisions Yt are made. Let Ft (yt ) denote the smallest of these: Ft (yt ) =
min Ft (Yt ).
(2)
Yt ∈ Y(yt )
Excluding costs unaffected by any decisions made over the time horizon, we have
Li n Ft (Yt ) = γt min Yi1t + hijt Yijt i
i =1 j =1
+ α E[Ft +1 (Yt − Dt )],
(3)
where γt (y) := pt E[(Dt − y)+ ] + HtA E[(y − Dt )+ ], while Yt − Dt refers to the collection {Yijt − Dt }, hijt := Hijt − Hi,j+1,t , and α is the discount factor. The expectation is taken over Dt . The time horizon is T periods long, with zero salvage value (FT +1 := 0).
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Fig. 2. A system with a flow-through stage.
Assumption 1. For each component i, stage j, and period t, pt ≥ 0 and hijt ≥ 0. The optimality equations given in (2) and (3) bear the curse of n dimensionality: the state space has i=1 Li dimensions. Further, each optimal order quantity Yijt depends on all the variables in the state space, rendering the problem practically impossible to solve numerically, even for systems with a few components and relatively short leadtimes. To make any progress in solving such capacitated systems, further structural results are needed. 3. Optimal behavior In this section, we make use of numerical examples to study the optimal behavior of capacitated assembly systems. We first consider a simple capacitated assembly system, shown in Fig. 2, with two components, and equal capacities throughout the system (K1 = K12 = K22 = K23 = 5). Stage 3 for component 2 is a flow-through stage (i.e., the leadtime to get units of component 2 into stage 2 is two periods). Inventory holding costs are: H11 = H21 = 0.5, H22 = H23 = 0.1, H12 = 0.3. The demand in each period follows an Erlang distribution with the mean of 3, and the coefficient of variation of 0.5. The backlogging cost is pt = 10, and the holding cost for the assembled product is H A = 0.1. Table 1 shows the optimal behavior in period 1 for a set of balanced initial echelon states. Since every initial echelon inventory state shown in Table 1 is balanced (Columns 6–10), there is exactly the same amount of each component at each stage (Columns 1–5). When the initial on-hand inventory at stage 1 starts to exceed 3 (the on-hand inventory at stage 2 is constant at 4), the optimal decisions for components 1 and 2 start to differ: stage 2 begins to withhold inventory of component 1, relative to component 2. Given the flow-through nature of stage 3 for component 2, stage 2 for component 1 can respond more quickly to the changing demands, and can thus carry less inventory than stage 2 for component 2. Having flow-through stages in a capacitated assembly system can thus be expected to give rise to this stock-withholding of inventory: a stage that allows deferrals of some component will, under low demand realizations, carry less inventory of that component relative to all other components. Therefore, if the initial echelon state this period is balanced, the initial echelon state next period will, generally, not be so (bold font entries in the table). Because echelon inventories’ next period at stage 2 cannot be represented by a single number for both components, it becomes impossible to reduce this assembly system to an equivalent series system, and the landmark result of [14] fails to hold. We refer to any such assembly system as irreducible. In order to allow a capacitated assembly system to be potentially managed as a series one, we are thus compelled to assume, going forward, no flow-through stages in the system.
Fig. 3. A system with bottleneck capacity upstream.
Assumption 2. For every component i, SF (i) = ∅. Consequently, the feasible set for the problem becomes
Y(yt ) := Yijt , 1 ≤ i ≤ n, 1 ≤ j ≤ Li |yijt ≤ Yijt ≤ yi,j+1,t ,
Yijt − yijt ≤ K .
(4)
Under Assumption 2, our model still allows for a component to stay at a single physical location for more than one period, as workin-progress (which thus incorporates multi-period leadtimes between adjacent stages). However, the work-in-progress at a particular location can now be deferred at each stage of that progress, potentially staying at the same stage of completion for more than one period. This is equivalent to the leadtime assumptions made in [6,14], where flow-through stages are (implicitly) ruled out. In Fig. 3, we consider another system with two components, each with a two-period leadtime (L1 = L2 = 2). This time, per Assumption 2, there are no flow-through stages. Component 1 has unit holding costs H11 = 0.5 and H12 = 0.2, and stage-2 capacity of K12 = 5; component 2 has unit holding costs H21 = 0.5 and H22 = 0.2, and stage-2 capacity of K22 = 7. The downstream stage is the (final) assembly stage, with capacity K1 = 6, backlogging cost pt = 5, and assembly holding cost H A = 0.1. The demand in each period follows an Erlang distribution with the mean of 4.5, and the coefficient of variation of 0.5. Table 2 shows the optimal behavior for a set of balanced initial inventory states. This time, when the on-hand inventory at stage 1 in period 1 starts to exceed 3 (the on-hand inventory at stage 2 is held at 5), the optimal order decisions for components 1 and 2 start to differ, as stage 2 accrues excess inventory of component 1 (over component 2). This phenomenon, referred to as stockpiling, occurs because the capacity constraint at stage 2 is different for the two components, and because the smallest capacity for the system is at the upstream stage. As a result, and in order to hedge against high future realizations of demand, stage 2 aims to stock up on component 1, relative to component 2. The asymmetry in capacity constraints (across components) thus creates an asymmetry in optimal inventory levels. Due to such stockpiling in systems with the above capacity configurations, balanced initial states can lead to unbalanced decisions, and thus to unbalanced states the next period. Such a system cannot be expected to be reducible to an equivalent series one. To reduce a such an assembly system to an equivalent series one, it becomes necessary to correct the (remaining) asymmetry in the system. The following assumption about the location of the bottleneck capacity in the system is shown to suffice for that purpose.
A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
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Table 1 An assembly system with a flow-through stage. On-hand inventory
Echelon inventory
Order quantity
Echelon levels
x11
x21
x12
x22
x23
y11
y21
y12
y22
y23
X11
X21
X12
X22
X23
Y11
Y21
Y12
Y22
Y23
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
5 6 7 8 9 10 11 12 13 14
5 6 7 8 9 10 11 12 13 14
10 11 12 13 14 15 16 17 18 19
4 4 4 3 2 1 0 0 0 0
4 4 4 3 2 1 0 0 0 0
5 5 5 4 3 2 1 0 0 0
5 5 5 5 5 5 5 5 5 5
5 5 5 4 3 2 1 0 0 0
5 6 7 7 7 7 7 8 9 10
5 6 7 7 7 7 7 8 9 10
10 11 12 12 12 12 12 12 13 14
10 11 12 13 14 15 16 17 18 19
15 16 17 17 17 17 17 17 18 19
Table 2 An assembly system with bottleneck capacity at an upstream stage. On-hand inventory
Echelon inventory
Order quantities
Echelon levels
x11
x21
x12
x22
y11
y21
y12
y22
X11
X21
X12
X22
Y11
Y21
Y12
Y22
2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 11 12
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 11 12
7 8 9 10 11 12 13 14 15 16 17
7 8 9 10 11 12 13 14 15 16 17
5 4 3 2 1 0 0 0 0 0 0
5 4 3 2 1 0 0 0 0 0 0
5 5 5 5 5 4 3 2 2 1 0
5 5 4 3 2 1 1 0 0 0 0
7 7 7 7 7 7 8 9 10 11 12
7 7 7 7 7 7 8 9 10 11 12
12 13 14 15 16 16 16 16 17 17 17
12 13 13 13 13 13 14 14 15 16 17
Assumption 3. Let K := mini,j {Kij }. Then, K1 = K . Lemma 1. For any xt in period t, there exists an optimal order schedule Xt such that xij,t +1 ≤ max (K , xijt − Xi,j−1,t ) for all i and j > 1. This results generalizes Lemma 1 of [13] to the case of capacitated assembly systems. The proof follows the basic approach of [13], and is therefore omitted. One implication of this lemma is that, if stage j has more than K units of inventory after it has sent a shipment downstream, it is optimal for this stage not to order any more stock. We say that initial inventory state xt (order schedule Xt ) is capacitated in period t if, for all i and all j > 1, xijt ≤ K (for all i and j, Xijt ≤ K ). Assumption 4. For all i and j > 1, xij1 ≤ K . Assumption 4 assures that the system begins in a capacitated state: each component i at each stage j > 1 starts with the level of on-hand inventory below bottleneck capacity K . The following result is a corollary of Lemma 1. Corollary 1. In each period t, there exists an optimal policy Yt such that Yijt − Yi,j−1,t ≤ K for all components i and stages j > 1. If the system starts out in a capacitated state, then it is optimal to keep it in that state. Further, all capacities in the system can be replaced with the bottleneck capacity K , without affecting costs. By Corollary 1, Assumptions 2–4 therefore imply that
Y(yt ) := Yijt , 1 ≤ i ≤ n, 1 ≤ j ≤ Li |yijt ≤ Yijt ≤ yi,j+1,t ,
Yi,j+1,t − Yijt ≤ K .
Assumptions 2 and 3 act to restore symmetry to the flow of stock across different components. Thus, the equivalence result of [14] can be said to fail in capacitated assembly problems where structural asymmetries are present in the system.
4. Balancing the system Let C(j) to be the set of components with a leadtime strictly greater than j, for each j = 1, 2, . . . , L − 1, where L := Ln is the longest component leadtime. These are the components for which stage j decisions (and states) apply, and are therefore called the relevant components at stage j. The following definitions are stated in terms of yt , but also apply to Yt . Formally, we say that yt is balanced if, for any echelon j, yijt = ykjt for all i, k ∈ C(j). A balanced inventory state has, for each stage, exactly the same number of units of each relevant component at that stage. Put another way, there are no unmatched components. We say yt and y∗t are echelon similar if, for all j, min yijt = min y∗ijt .
i ∈ C(j)
i ∈ C(j)
That is, two inventory states are echelon similar if any differences between them consist of excesses of components (compared to some other relevant component for the same stage). We say y∗t is the balanced reduction of yt if y∗t and yt are echelon similar, and y∗t is balanced. In this case, we get y∗t by removing any excesses from yt , and, thus, y∗t ≤ yt . Lemma 2. Let yt be capacitated, and y∗t be the balanced reduction of yt . Let Yt ∈ Y(yt ) be a capacitated policy in period t. Let Y∗t be the balanced reduction of Yt . Then Y∗t is capacitated and feasible for y∗t . (Y∗t ∈ Y(y∗t )). Proof. Fix j. Let i1 = i1 (j, Yt ), j ≥ 1, represent a component in C(j) Yi1 jt = min Yijt . i∈C(j)
Fix arbitrary i ∈ C(j). Then y∗ijt = y∗i1 ,jt ≤ yi1 ,jt
≤ Yi1 ,jt = Yi∗1 ,jt
[y∗t is balanced] [y∗t is the balanced reduction of yt ] [Yt ∈ Y(yt )] [definition of i1 ]
= Yijt∗ [Y∗t is balanced].
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A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
Let i2 = i2 (j − 1, Yt ), j > 1, represent a component in C(j − 1) such that Yi2 ,j−1,t = min Yi,j−1,t .
(c) Let yt be an arbitrary capacitated state, and y∗t be the balanced reduction of yt . By Lemma 1 and Corollary 1, there exists an optimal policy Yt , feasible for yt in period t, which is capacitated. Let Y∗t denote the balanced reduction of Yt . Then,
Fix arbitrary i ∈ C(j), j > 1. Then,
Ft (yt ) = Ft (Yt )
i∈C(j−1)
∗ Yijt = Yi∗2 ,jt ≤ Yi2 ,jt
[Y∗t is balanced]
[Y∗t is the balanced reduction of Yt ] ≤ Yi2 ,j−1,t + K [Yt is capacitated (Yt ∈ Y(yt ))] = Yi∗2 ,j−1,t + K [definition of i2 ] =
Yi∗,j−1,t
∗
+ K [Yt is balanced].
Similarly, let i3 = i3 (j + 1, yt ) denote a component in C(j + 1) such that yi3 ,j+1,t = min yi,j+1,t . i∈C(j+1)
Fix arbitrary i ∈ C(j + 1). Note that i ∈ C(j + 1) implies that i ∈ C(j). Then ∗
∗
Yijt = Yi3 jt
∗
[Yt is balanced]
≤ Yi3 jt [Y∗t is the balanced reduction of Yt ] ≤ yi3 ,j+1,t [Yt ∈ Y(yt )] = y∗i3 ,j+1,t [definition of i3 ] = y∗i,j+1,t [y∗t is balanced]. Consequently, Y∗t satisfies the required lower and upper bounds, and thus it is capacitated and feasible for y∗t . By Lemma 2, the balanced reduction of a capacitated policy Yt , feasible for yt , is capacitated and feasible for the balanced reduction of yt . This guarantees that we can construct a balanced policy out of any given capacitated policy Yt , by means of the balanced reduction, and have this policy be feasible for the same construction of a balanced state. We say that Ft is balance inducing if, for arbitrary capacitated yt such that y∗t is the balanced reduction of yt , it follows that Ft (y∗t ) ≤ Ft (yt ). Lemma 3. Fix t and suppose that Ft +1 is balance inducing. (a) If Yt is capacitated, and Y∗t is the balanced reduction of Yt , then Ft (Y∗t ) ≤ Ft (Yt ). (b) If y∗t is balanced and capacitated, then there exists an optimal inventory schedule Y∗t that is balanced and capacitated. (c) Ft is balance inducing. ∗ = Proof. (a) Because Y∗t is the balanced reduction of Yt , mini Yi1t mini Yi1t . Thus,
Ft (Y∗t ) − Ft (Yt ) =
Li n
∗ hijt (Yijt − Yijt )
[Yt is optimal] ∗ ≥ Ft (Yt ) [part (a)] ≥ Ft (y∗t ) [by Lemma 2, Y∗t is feasible for y∗t ].
Consequently, Ft is balance inducing.
Assumption 5. For each j = 0, 1, . . . , L − 1, yij1 = ykj1 for all i, k ∈ C(j). Thus, the system starts out balanced: The initial echelon state has, for each stage j, the same number of relevant components scheduled to complete assembly in j periods. Theorem 1. In each period, there exists an optimal policy that is capacitated and balanced. Proof. Since FT +1 = 0 by assumption, then FT +1 is balance inducing. Thus, by backward induction and Lemma 3(c), Ft is balance inducing for each t. By Assumptions 4 and 5, the initial state y1 is balanced and capacitated. Thus, by Lemmas 1 and 3(b), there exists an optimal decision Y1 in period 1 that is balanced and capacitated. Thus y2 = Y1 − D1 is balanced and capacitated, and, hence, by forward induction, there exists an optimal decision that is balanced and capacitated for each period. If a system starts out balanced then, when model parameters conform to Assumptions 1–5, it is possible (and optimal) to keep it balanced. It is not necessary to manage each component separately; instead, those components that are at the same stage can be managed together as a kit, where the kit for stage j has one each of every component in C(j). We can represent the echelon inventory for each relevant component at stage j by a single variable, zjt for j = 1, . . . , L − 1. We may assume that yijt = zjt for every i ∈ C(j). Let zt denote the L vector referred to as the echelon state. Because the optimal inventory schedule is balanced, Yijt is the same for every i ∈ C(j); it can be represented by a single variable Zjt . Let Zt , referred to as the echelon schedule denote the L-vector of zjt ’s. We now define the new cost parameters hjt for each j and t: hjt := i∈C(j) hijt . We thus obtain an equivalent model with the following optimality equations for each zt .
ft (zt ) = min
Zt ∈Z(zt )
γt (Z1t ) +
L
hjt Zjt + α E[ft +1 (Zt − Dt )] ,
(5)
j =1
where, by Corollary , the feasible set Z for echelon state zt can be written as
Z(zt ) = Zjt | zjt ≤ Zjt ≤ zj+1,t , 1 ≤ j ≤ L;
Zjt − Zj−1,t ≤ K , 1 < j ≤ L .
(6)
i=1 j=1
+ α E[Ft +1 (Y∗t − Dt )] − E[Ft +1 (Yt − Dt )] . ∗
Since Yt is the balanced reduction of a capacitated policy Yt , then: ∗ (i) Yijt ≤ Yijt for all i and j; (ii) Y∗t − Dt is the balanced reduction of Yt − Dt ; and (iii) Y∗t − Dt and Yt − Dt are capacitated. Thus, the first term on the RHS are negative. Since Ft +1 is balance inducing, then by (ii) and (iii), Ft +1 (Y∗t − Dt ) ≤ Ft +1 (Yt − Dt ) for any Dt . Therefore, Ft (Y∗t ) ≤ Ft (Yt ). (b) By Lemma 1 and Corollary 1, there exists an optimal policy Yt for y∗t which is capacitated. Let Y∗t be the balanced reduction of Yt . By part (a), Ft (Y∗t ) ≤ Ft (Yt ); by Lemma 2, Y∗t is capacitated and ∗ feasible for yt .
Corollary 2. Fix zt . If yijt = zjt for all j and i ∈ C(j), then ft (zt ) = Ft (yt ). Under the specified conditions, a capacitated component system can be reduced to a series system. The state space thus collapses from i Li dimensions to only L. 5. System with subassemblies Consider an assembly system where subassemblies are allowed at any stage prior to the final assembly into the finished product. One example of such an assembly system is shown in Fig. 4. We make use of the notation introduced in [14].
A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
25
Table 3 Allocation of assembly costs to components and stages. Li
k=1
2
5
5
β51 c1A
β52 c2A
6
5
β61 c1A
β62 c2A
8
7
β81 c1A
β83 c3A
9
8
β91 c1A
β93 c3A
β97 c7A
10
9
β10,1 c1A
β10,3 c3A
β10,7 c7A
i
3
4
5
6
7
8
9
10
c5A c6A c4A
c8A c9A A c10
Fig. 4. An assembly system with subassemblies.
For each (sub)assembly k = 1, 2, . . . , N, let lk denote the incremental leadtime, required to complete assembly k, where Assembly 1 is the finished product. Let s(k) represent the unique immediate successor assembly to assembly k, for all k > 1. Let L1 := l1 , and, for each k > 1, let Lk := lk + Ls(k) , denote the leadtime for assembly k. Let P (k) denote the set of immediate predecessor assemblies of assembly k. The set of components of such a system is the set of assemblies that have no predecessor assemblies. Let O(k) denote the set of components that are required in the composition of assembly k. For the system shown in Fig. 4, for example, s(6) = 2 and L9 = 8, while P (1) = {2, 3}, P (2) = {5, 6}, P (3) = {4, 7}, P (4) = {8}, P (7) = {9, 10}, P (5) = P (6) = P (8) = P (9) = P (10) = ∅, and O(1) = {5, 6, 8, 9, 10}, O(2) = {5, 6}, O(3) = {8, 9, 10}, O(4) = {8}, O(7) = {9, 10}, and O(k) = {k}, for k = 5, 6, 8, 9, 10. We will refer to such a system as the general assembly system. Capacity constraints carry over from the component system directly: the processing capacity for assembly k represents the smallest of the capacities for each of the components in O(k), since assembly k requires one of each of those components. Assume, for convenience, that the only costs incurred in this A general system are the assembly costs: Let ckt be the discounted present value of the costs related to assembling/purchasing/ transforming assembly k, evaluated at the beginning of period t, if that assembly is begun then. This cost may be a processing or a purchasing cost for components (e.g. 6, 8, 11, 13, 14, 15, 16, and 17), because they have no predecessor assemblies. We now introduce the constraint-relaxation approach from [2] and the cost allocation ideas of [3,4]. Briefly, [2,3] develop the approach of relaxing some of the constraints in the problem to provide lower bounds on the cost for certain multi-period, multistage, production/inventory models with deterministic demand. The underlying idea is that such a relaxed version of the problem will naturally have lower costs due to fewer constraints in the system, and thus provide a reasonable lower bound on the cost for the original problem. With the right choice of constraints to relax, the problems considered in [2,3] can be broken into simpler subproblems that allow a range of possible cost allocation across them. By choosing an optimal cost allocation for a given relaxation, it is possible to derive tight lower bounds on the cost function for the original problem. Chen and Zheng [4] extend this constraint-relaxation method to production-distribution networks with stochastic demands. In particular, the paper makes use of this method to prove the original multiechelon result of [5], and the assembly equivalence result of [14], in a more direct manner. This is accomplished by relaxing the constraints, and finding the optimal policy for the simpler, relaxed problem thus obtained. This optimal policy for the relaxed system is then showed to be feasible in the original system. Since this policy is optimal for the less constrained system, and found to be feasible in the original, more constrained system, then the policy must also be optimal for this original system.
In order to apply the constraint-relaxation and cost-allocation method to our problem, we first relax the constraint(s) that there must be exactly the right number of each component ready when any assembly takes place. This provides a system in which the performance is at least as good as in the original one. Further, this relaxed system has exactly the same constraints as the component assembly system analyzed in Section 4. Next, following [4], we design a cost allocation scheme so that, if the solution is balanced and capacitated, then the total expected cost will be the same in the relaxed system and the original (constrained) system with subassemblies. If there exists an allocation of assembly costs to its components that satisfies the required conditions, then, by Theorem 1, there exists an optimal policy for the relaxed system that is balanced and capacitated, and accounts for the costs incurred. Since it is balanced and capacitated, this policy is feasible in the original constrained system (by being balanced this policy meets the constraint that there must be exactly the right number of each component ready when any (sub)assembly takes place). Because this policy is optimal for the relaxed system, and feasible for the original constrained system, then, following the reasoning in [4], it must be optimal for it. Let βikt denote the portion of the assembly cost for assembly k that is allocated to component i, for each i ∈ O(k) in period t (e.g. allocate the assembly costs equally to each required component, so that β51t = β61t = β81t = β91t = β 10,1t = 1/5). In general, we require full allocations, namely, that i∈O(k) βikt = 1 for each k and t. For example, we require that β84t = 1 because assembly 4 requires only component 8. Thus, each component i is A allocated βikt ckt , for each k for which i ∈ O(k). Thus, if assembly k is initiated at the beginning of period t, it will require one unit of A component i for each i ∈ O(k), and the amount βikt ckt is allocated to each such component i for stage Lk in period t. The allocated assembly costs for the example are summarized in Table 3, for the case of stationary assembly costs. The component 6, for example, is shown as bearing the full cost of assembly 6— this is because it is the only component needed for that particular assembly. The cost of assembly 7 at stage 6, for instance, is shared among components 9 and 10. Theorem 2. If, for each assembly k and component i in each period A t, there exists an allocation of assembly costs such that βikt ckt ≥ A αβik,t +1 ck,t +1 , then there exists an optimal policy that is balanced and capacitated in every period. Proof. Let L(j) := {k |Lk = j} denote the set of assemblies with leadtimeLk equal to j. For each component i and echelon j, let A θijt := k∈Lj βikt ckt . Then, θijt represents all the assembly costs allocated to each unit of component i at stage j in period t. The total cost allocated to component i at stage j in period t is θijt Xijt . For any set of demand realizations {D1 , D2 , . . . , DT }, the discounted
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A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
net present value of all such costs is n
Li
T
n
α t −1 θijt Xijt =
i =1 j =1 t =1
=
Li n T
Li
T
α t −1 θijt (Yijt − yijt )
i =1 j =1 t =1
α t −1 θijt (Yijt − Yij,t −1 + Dt −1 )
i=1 j=1 t =1
=
Li n T
α t −1 (θijt − αθij,t +1 )Yijt + θijt Dt −1 ,
i=1 j=1 t =1
where D0 := 0 and θij,T +1 := 0 all i and j. Let h′ijt := hijt + (θijt − αθij,t +1 ) for j ≥ 2, and h′i1t := hi1t . Since θijt ≥ αθij,t +1 , then h′ijt ≥ 0 and Assumption 1 holds. (There is an additional positive term for j = 1: (θi1t − αθi1,t +1 )Yi1t .) Assumptions 3 and 4 hold directly. By Theorem 1, there exists an optimal policy that is balanced and capacitated. If a policy is balanced, then, under an allocation of costs such as the one above, the actual costs incurred in the general system with subassemblies are the same as those captured by the component assembly model. Since we allow each of the components to be independently managed, then the optimal solution of the general systems is at least as good as the best one in the original (i.e. component assembly) system. Because, under Assumptions 1– 5, there is an optimal policy that is balanced and capacitated, that policy is feasible in the original system, and thus it must be optimal for it. The condition required by Theorem 2 clearly holds for the case of stationary assembly costs. In fact, it is only mildly restrictive: as long as assembly costs are not increasing too fast in time, and across all assemblies at the same time, finding an appropriate allocation should be straightforward. 6. Discussion The conditions needed for the state space collapse shown in this paper may not always hold. Assumption 1 would not apply in those (rare) settings when the inventory holding cost for some component(s) is higher at an upstream stage than at a downstream stage. Every unit of that component would then be moved from the upstream stage to the downstream stage, thus creating an excess of that component at the downstream stage. The set of all relevant components at that downstream stage could then no longer be managed as a kit. Assumption 2 would not hold for certain transportation processes, such as ocean-going shipping. In its absence, a capacitated assembly system with flow-through stages can be expected to exhibit stock-withholding of inventory, which has the effect of unbalancing the system by maintaining a shortage of one component relative to others. Assumption 3 is the assembly-system equivalent of the assumption made by Parker and Kapuscinski [13] in proving the Clark–Scarf decomposition of a capacitated multiechelon system.
Very little is presently known about capacitated systems (either assembly or series) for which this assumption does not hold, and further research is clearly needed to better understand those systems. It is worth noting, however, that Assumption 3 is not the only restriction on capacity constraints that achieves the results of this paper. In a capacitated assembly system where, for every j > 1, and i, i′ ∈ C(j), Kij = Ki′ j , so that, at each stage in the assembly system, all relevant components have the same processing capacity, it is straightforward to verify that all the results of our paper go through (except for Lemma 1). Clearly, there is more than one way to enforce the structural symmetry in a capacitated assembly system. Assumptions 4 and 5 contribute to the ease of analysis. A capacitated assembly system can be shown to achieve a capacitated state on its own (the proof is similar to that in [13]) even if it starts out uncapacitated. Of course, once the system reaches a capacitated state, it is optimal to keep it there, and all the results of this research apply. Similarly, if the system starts in an unbalanced state, it will eventually reach a balanced state (see [14] for an example of such a result), so that Assumption 5 is also not critical to the analysis. Acknowledgments The authors would like to thank Sridhar Seshadri, an anonymous associate editor, and the reviewers for their helpful comments and suggestions. References [1] A. Angelus, E. Porteus, An asset assembly problem, Oper. Res. 56 (2008) 665–680. [2] D. Atkins, A survey of lower bounding methodologies for production/inventory models, Ann. Oper. Res. 26 (1990) 9–28. [3] D. Atkins, A simple lower bound to the dynamic assembly problem, Eur. J. Oper. Res. 75 (1994) 462–466. [4] F. Chen, Y.-S. Zheng, Lower bounds for multi-echelon stochastic inventory systems, Manage. Sci. 40 (1994) 1426–1443. [5] A.H. Clark, H. Scarf, Optimal policies for the multi-echelon inventory problem, Manage. Sci. 6 (1960) 475–490. [6] G.A. DeCroix, P.H. Zipkin, Inventory management for an assembly system with product or component returns, Manage. Sci. 51 (2005) 1250–1265. [7] A.G. De Kok, J.W.C.H. Visschers, Analysis of assembly systems with service level constraints, Int. J. Prod. Econ. 59 (1999) 313–326. [8] P. Glasserman, S. Tayur, The stability of a capacitated, multiechelon production–inventory system under a base-stock policy, Oper. Res. 42 (1994) 913–925. [9] P. Glasserman, S. Tayur, Sensitivity analysis for base-stock levels in multiechelon production–inventory systems, Manage. Sci. 41 (1995) 263–281. [10] W.T. Huh, G. Janakiraman, Base-stock policies in capacitated assembly systems: convexity properties, Naval Res. Log. 57 (2010) 109–118. [11] G. Janakiraman, J.A. Muckstadt, A decomposition approach for a class of capacitated serial systems, Oper. Res. 57 (2009) 1384–1393. [12] A. Muharremoglu, J.N. Tsitsiklis, A single-unit decomposition approach to multiechelon inventory systems, Oper. Res. 56 (2007) 1089–1103. [13] R.P. Parker, R. Kapuscinski, Optimal policies for a capacitated two-echelon inventory system, Oper. Res. 52 (2004) 739–755. [14] K. Rosling, Optimal inventory policies for assembly systems under random demands, Oper. Res. 37 (1989) 565–579. [15] C.P. Schmidt, S. Nahmias, Optimal policy for a two-stage assembly system under random demand, Oper. Res. 33 (1985) 1130–1145.
Contents lists available at SciVerse ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
On the structure of capacitated assembly systems Alexandar Angelus a,∗ , Wanshan Zhu b a
Lee Kong Chian School of Business, Singapore Management University, 50 Stamford Road, Singapore 178899, Singapore
b
Department of Industrial Engineering, Tsinghua University, Beijing 100084, China
article
info
Article history: Received 29 May 2012 Received in revised form 4 November 2012 Accepted 4 November 2012 Available online 16 November 2012 Keywords: Multiechelon inventory Capacitated assembly Optimal policy
abstract We consider a finite-horizon inventory model with an assembly structure, stochastic demand, and capacity constraints throughout the system. We explore optimal behavior of this system, with a special focus on stockpiling and stock-withholding of inventory, two novel inventory phenomena that arise due to asymmetric constraints. We identify conditions, such as having the bottleneck at the most downstream stage, that restore structural symmetry to the problem. This allows the state space to be collapsed to that of an equivalent series system. © 2012 Elsevier B.V. All rights reserved.
1. Introduction We consider a finite-horizon, multi-period, assemble-to-stock inventory system where products of a single type are assembled from multiple components, each with a possibly different leadtime. We assume stochastic, possibly non-stationary, demand for the assembled end-product. In addition, we allow production (capacity) constraints to apply to each component at each stage in this assembly system. Those constraints act as a (finite) upper limit on the amount of any component that can be processed at any stage in a single period. We explore the structure of this capacitated assembly system, with the objective of facilitating the development of means to effectively and efficiently manage it. In general, very little is known about capacitated assembly systems, and even less about their structural properties, so that the management of such systems in practice becomes exceedingly difficult. Thus, for example, due to their complexity, capacity constraints on (sub)assemblies are not taken into account by traditional MRP systems, which, as a result, may generate infeasible orders for assembly parts. More generally, companies are struggling to effectively manage capacity-constrained supply chains for assembled products. In analyzing a capacitated assembly system, we assume non-stationary demands and model parameters, as well as full backlogging of unsatisfied demand and carryover of inventory. The
∗
Corresponding author. Tel.: +65 6828 0413. E-mail addresses: [email protected] (A. Angelus), [email protected] (W. Zhu). 0167-6377/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2012.11.001
level of assembled products at the beginning of a period therefore equals the level at the beginning of the previous period plus the amount of newly completed units minus the random demand in that period. We first address a system with a single assembly at the most downstream stage, and provide a formulation of such an assembly system that allows for multiple components and multiperiod leadtimes between successive stages. We numerically explore those systems, and describe two novel inventory phenomena that occur in them due to the asymmetric constraints on the flow of inventory across different components. The first phenomenon acts to maintain a shortage of one component relative to all others, while the second one acts to keep an excess inventory of one component relative to all others. Those phenomena are also shown to prevent the landmark results of [14] from applying to capacitated assembly systems considered in this paper. Next, we identify system configurations, such as having the bottleneck in the system be located at the most downstream stage, that restore structural symmetry to the problem, thus allowing an assembly system with capacity constraints to be reduced to an equivalent series system. One consequence of this state space collapse, of relevance to inventory managers at companies, is that such an assembly system becomes much easier to manage by means of existing heuristics and approximations for capacitated series systems. We further investigate a more general problem, in which (sub)assemblies are allowed throughout the (capacitated) system. The same conditions that restore the structural symmetry to the original assembly model are shown to do the same in this (more) general assembly setting, so that the original state space collapse carries over in a direct manner.
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A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
Fig. 1. Component assembly system.
Other related literature The study of assemble-to-stock systems under stochastic demand can be traced to [15]. This work addresses a two component assembly problem with joint ordering and assembly decisions, and shows that the problem can be decomposed into component ordering decisions and finished goods assembly decisions. The classic paper in the field, [14], considers a stationary, multiechelon assembly under random demand and infinite horizon, and arrives at conditions under which this assembly problem can be reduced to an equivalent series system of Clark and Scarf, first introduced in [5]. Chen and Zheng [4] offer another derivation of this result, while [7] applies this approach to an assembly system with multiple end-products and derives well-performing heuristic policies. DeCroix and Zipkin [6] extend the results of [14] to uncertain product (and component) returns from customers. They describe the item-recovery pattern and restrictions on the inventory policy under which an equivalent series system can be shown to exist. When returns are allowed only for the end product, no policy restrictions are required to reduce the assembly system to an equivalent series one. Angelus and Porteus [1] generalize the results of [24] to include non-stationary, Markov modulated demands and model parameters, as well as production assets not depleted by stochastic demand. In the context of capacitated series systems, [8,9] analyze the stability and sensitivity of such systems under the control of a basestock policy. Parker and Kapuscinski [13] proves that, when the lowest capacity in a two-echelon series system is at the downstream stage, and the leadtime at the upstream stage is one period, the objective function, in the spirit of [8], becomes additively convex. The optimal policy is the modified echelon base stock policy: the downstream station orders the optimal echelon basestock level subject to both stock availability at the upstream station, and its processing capacity. Janakiraman and Muckstadt [11] rederives this result using the customer-unit decomposition approach of [12], and shows that, when the leadtime at the upstream echelon is extended to two periods, the optimal policy becomes a ‘‘two-tier, basestock policy’’: the optimal decision at the bottom echelon is determined by one basestock level when that level is below the inventory in transit, and by another basestock level otherwise; the optimal decision at the upper echelon has a similar two-tiered structure.
Capacitated assembly systems have so far only been studied under the assumption of basestock control. In particular, [10] explores the convexity properties of different cost functions under basestock policies, and shows, among other things, that the shortage cost is jointly convex with respect to the basestock levels and capacity limits. 2. Model description We assume, without loss of generality, that assembling a product requires exactly one of each of n components. Thus, the number of assembled products is simply the minimum number of completed components available, over all component types. Components are managed independently over time and assembled into the final product at the most downstream stage in the system. Each component type i(= 1, 2, . . . , n) has a standard leadtime of Li periods, which is the smallest possible number of periods between placing an order for the component and its becoming a part of an assembled product. We assume L1 ≤ L2 ≤ · · · ≤ Ln . In our basic model (an example is depicted in Fig. 1), assembly can be interpreted to take place only once—in the period immediately preceding completion of the product. Consequently, we refer to this system as the component (assembly) system because it is the components that flow downstream until they are assembled at the finished goods stage. Following [5], we identify stages (also echelons) for the scheduled number of periods that a component is away from being assembled at the bottom stage. When a component is first ordered, it is in stage Li + 1. At the beginning of the next period, this component is in stage Li . Each unit can be moved downstream one stage at a time, until it reaches stage 1, referred to as the finished goods stage, where it is assembled into the final product used to satisfy (stochastic) customer demand. For each component, we allow for the existence of stages where a unit can be delayed (i.e. deferred), if necessary, in which case it stays at the same level of completion until the next period (inventory holding costs accrue accordingly). For each component, we also allow for stages where inventory cannot be deferred (e.g., ocean-going shipping). We refer to any such stage as a flow-through stage. Fig. 1 illustrates a component assembly system: stages 3 and 4 for component n, and stage 2 for component 2, are the flow-
A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
through stages in the system. (Stages that allow deferrals are shaded). The following parameters describe our system: xijt = (initial) on-hand inventory of component i in stage j, at the beginning of period t , prior to making any decisions; Xijt = the decision variable representing the amount of component i ordered into stage j during period t ; Kij = processing capacity for component i at stage j; Dt = the stochastic demand for assembled final products in period t . The order quantities Xijt are limited by both xi,j+1,t , the on-hand inventory available at the upstream stage, and by Kij , the processing capacity for component i at stage j. Thus, Xijt ≤ min (xi,j+1,t , Kij ) for all i and j in period t. In addition, for each flow-through stage j, 1 < j ≤ Li , of component i, Xi,j−1,t = xijt in each period t, by definition. We assume full backlogging of unsatisfied demand and carryover of inventory. The collection of all on-hand inventories xijt , denoted by xt , will be referred to as the on-hand (inventory) state, while the collection of all order quantities Xijt , denoted by Xt , will be referred to as the order schedule. The sequence of events in each period t is as follows: (1) on-hand inventory state xt is observed; (2) order schedule Xt is selected; (3) ordered amounts are received; (4) customer demand is observed and satisfied to the extent made possible by the available stock; and, (5) costs are incurred. The transition equations therefore become xi1t + Xi1t − Dt , xijt + Xijt − Xi,j−1,t
xij,t +1 =
if j = 1, if j = 2, . . . , Li .
(1)
Because assembly requires exactly one unit of each component, the quantity of assembled products at the end of period t is min(x11t + X11t , x21t + X21t , . . . , xn1t + Xn1t ). The most downstream stage can be assumed, without loss of generality, to be able to process exactly the same amount of each component in any period, so that K11 = K21 = · · · = Kn1 = K1 . A unit backlogging cost pt is charged for each unit of demand not satisfied in period t; a unit inventory holding cost, Hijt is incurred on the amount of component i located at stage j at the end of period t. Holding a unit assembled product at the finished goods stage at the end of period t incurs a holding cost HtA . Thus, one-period costs in period t are
pt E
Dt − min(xi1t i
+ HtA E +
Li n
+ + Xi1t )
min(xi1t + Xi1t ) − Dt
+
i
Hijt (xijt − Xi,j−1,t + Xijt ),
i=1 j=1
where Xi0t := 0. The first term is the expected backlogging cost; the second term is the expected holding cost for the assembled product; the third term, the holding costs for unassembled components. We reformulate the problem using the following echelon variables: yijt := xi1t + xi2t + · · · + xijt referred to as the echelon j inventory of component i; Yijt := yijt + Xijt referred to as the echelon j level for component i.
21
The updated echelon quantities are given by yij,t +1 = Yijt − Dt . The collection of all echelon inventories, denoted by yt , will be referred to as the echelon (inventory) state; the collection of all echelon levels, denoted by Yt , will be referred to as the echelon (inventory) schedule. Let SD (i) be the set of all stages for each component i where inventory can be deferred, and let SF (i) be the set of all flowthrough stages. Thus, SD (i) ∩ SF (i) = ∅, and SD (i) ∪ SF (i) = {1, . . . , Li }. (We allow SF (i) to be empty; SD (i) cannot be empty for any i, since, by definition, stage 1 is in SD (i) for every i.) We refer to any set of contiguous flow-through stages that are bounded below by an element SD (i), and are either bounded from above by another element of SD (i), or extend all the way to Li , as a (flow-through) string. We partition each SF (i) into flow-through strings S1F (i), . . . , M
SF i (i), for any Mi , as follows: Let ji1 be the smallest element of SF (i), and let ji1 ∈ S1F (i); define S1F (i) as S1F (i) := {ji1 , j ∈ SF (i) | j − 1 ∈ S1F (i)}. Thus, S1F (i) represents the stages of the first flow-through string for component i. Let qi1 be the largest element of S1F (i). We proceed recursively: having defined Sm F (i) for some m < Mi , as well as its smallest element jim , and its largest element qim , let jim+1 , the smallest element of SFm+1 (i), be defined as the smallest element of SF (i) larger than qim ; that is, jim+1 := min{j | j > qim , j ∈ +1 SF (i)}. Then, SFm+1 (i) := {j ∈ SF (i) | j − 1 ∈ Sm (i)}. The F M
strings S1F (i), . . . , SF i (i) are mutually exclusive and collectively exhaustive. We now define the feasible set Y(yt ), given echelon inventory state yt in period t.
y ≤ Y ≤ y ijt i,j+1,t , Yijt − yijt ≤ Kij ijt if { j , j + 1 } ∈ SD (i); yijt ≤ Yijt ≤ yi,j+1,t , Yijt − yijt ≤ min K min Ki µ Y(yt ) := Yijt , 1 ≤ i ≤ n i,jim −1 , µ∈{jim ,...,qim } if ∃m ∈ {1, . . . , M }, j = qim ; Y = ijt yi,j+1,t if j + 1 ∈ SF (i).
The first line in the expression for Y(yt ) applies to any stage j that allows inventory deferrals and is located immediately downstream of another such stage. The second line applies when stage j + 1 allows deferrals, while stage j does not; in that case, each quantity Xijt ordered into stage j has to be sufficiently small to meet not just the processing capacity constraint of stage j, but also the constraints of all other stages in the flow-through string of which stage j is the uppermost stage. The third line describes the flow-through constraint. Let Ft (Yt ) denote the minimum expected present value of the costs over periods t through T , given that decisions Yt are made. Let Ft (yt ) denote the smallest of these: Ft (yt ) =
min Ft (Yt ).
(2)
Yt ∈ Y(yt )
Excluding costs unaffected by any decisions made over the time horizon, we have
Li n Ft (Yt ) = γt min Yi1t + hijt Yijt i
i =1 j =1
+ α E[Ft +1 (Yt − Dt )],
(3)
where γt (y) := pt E[(Dt − y)+ ] + HtA E[(y − Dt )+ ], while Yt − Dt refers to the collection {Yijt − Dt }, hijt := Hijt − Hi,j+1,t , and α is the discount factor. The expectation is taken over Dt . The time horizon is T periods long, with zero salvage value (FT +1 := 0).
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A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
Fig. 2. A system with a flow-through stage.
Assumption 1. For each component i, stage j, and period t, pt ≥ 0 and hijt ≥ 0. The optimality equations given in (2) and (3) bear the curse of n dimensionality: the state space has i=1 Li dimensions. Further, each optimal order quantity Yijt depends on all the variables in the state space, rendering the problem practically impossible to solve numerically, even for systems with a few components and relatively short leadtimes. To make any progress in solving such capacitated systems, further structural results are needed. 3. Optimal behavior In this section, we make use of numerical examples to study the optimal behavior of capacitated assembly systems. We first consider a simple capacitated assembly system, shown in Fig. 2, with two components, and equal capacities throughout the system (K1 = K12 = K22 = K23 = 5). Stage 3 for component 2 is a flow-through stage (i.e., the leadtime to get units of component 2 into stage 2 is two periods). Inventory holding costs are: H11 = H21 = 0.5, H22 = H23 = 0.1, H12 = 0.3. The demand in each period follows an Erlang distribution with the mean of 3, and the coefficient of variation of 0.5. The backlogging cost is pt = 10, and the holding cost for the assembled product is H A = 0.1. Table 1 shows the optimal behavior in period 1 for a set of balanced initial echelon states. Since every initial echelon inventory state shown in Table 1 is balanced (Columns 6–10), there is exactly the same amount of each component at each stage (Columns 1–5). When the initial on-hand inventory at stage 1 starts to exceed 3 (the on-hand inventory at stage 2 is constant at 4), the optimal decisions for components 1 and 2 start to differ: stage 2 begins to withhold inventory of component 1, relative to component 2. Given the flow-through nature of stage 3 for component 2, stage 2 for component 1 can respond more quickly to the changing demands, and can thus carry less inventory than stage 2 for component 2. Having flow-through stages in a capacitated assembly system can thus be expected to give rise to this stock-withholding of inventory: a stage that allows deferrals of some component will, under low demand realizations, carry less inventory of that component relative to all other components. Therefore, if the initial echelon state this period is balanced, the initial echelon state next period will, generally, not be so (bold font entries in the table). Because echelon inventories’ next period at stage 2 cannot be represented by a single number for both components, it becomes impossible to reduce this assembly system to an equivalent series system, and the landmark result of [14] fails to hold. We refer to any such assembly system as irreducible. In order to allow a capacitated assembly system to be potentially managed as a series one, we are thus compelled to assume, going forward, no flow-through stages in the system.
Fig. 3. A system with bottleneck capacity upstream.
Assumption 2. For every component i, SF (i) = ∅. Consequently, the feasible set for the problem becomes
Y(yt ) := Yijt , 1 ≤ i ≤ n, 1 ≤ j ≤ Li |yijt ≤ Yijt ≤ yi,j+1,t ,
Yijt − yijt ≤ K .
(4)
Under Assumption 2, our model still allows for a component to stay at a single physical location for more than one period, as workin-progress (which thus incorporates multi-period leadtimes between adjacent stages). However, the work-in-progress at a particular location can now be deferred at each stage of that progress, potentially staying at the same stage of completion for more than one period. This is equivalent to the leadtime assumptions made in [6,14], where flow-through stages are (implicitly) ruled out. In Fig. 3, we consider another system with two components, each with a two-period leadtime (L1 = L2 = 2). This time, per Assumption 2, there are no flow-through stages. Component 1 has unit holding costs H11 = 0.5 and H12 = 0.2, and stage-2 capacity of K12 = 5; component 2 has unit holding costs H21 = 0.5 and H22 = 0.2, and stage-2 capacity of K22 = 7. The downstream stage is the (final) assembly stage, with capacity K1 = 6, backlogging cost pt = 5, and assembly holding cost H A = 0.1. The demand in each period follows an Erlang distribution with the mean of 4.5, and the coefficient of variation of 0.5. Table 2 shows the optimal behavior for a set of balanced initial inventory states. This time, when the on-hand inventory at stage 1 in period 1 starts to exceed 3 (the on-hand inventory at stage 2 is held at 5), the optimal order decisions for components 1 and 2 start to differ, as stage 2 accrues excess inventory of component 1 (over component 2). This phenomenon, referred to as stockpiling, occurs because the capacity constraint at stage 2 is different for the two components, and because the smallest capacity for the system is at the upstream stage. As a result, and in order to hedge against high future realizations of demand, stage 2 aims to stock up on component 1, relative to component 2. The asymmetry in capacity constraints (across components) thus creates an asymmetry in optimal inventory levels. Due to such stockpiling in systems with the above capacity configurations, balanced initial states can lead to unbalanced decisions, and thus to unbalanced states the next period. Such a system cannot be expected to be reducible to an equivalent series one. To reduce a such an assembly system to an equivalent series one, it becomes necessary to correct the (remaining) asymmetry in the system. The following assumption about the location of the bottleneck capacity in the system is shown to suffice for that purpose.
A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
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Table 1 An assembly system with a flow-through stage. On-hand inventory
Echelon inventory
Order quantity
Echelon levels
x11
x21
x12
x22
x23
y11
y21
y12
y22
y23
X11
X21
X12
X22
X23
Y11
Y21
Y12
Y22
Y23
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
5 6 7 8 9 10 11 12 13 14
5 6 7 8 9 10 11 12 13 14
10 11 12 13 14 15 16 17 18 19
4 4 4 3 2 1 0 0 0 0
4 4 4 3 2 1 0 0 0 0
5 5 5 4 3 2 1 0 0 0
5 5 5 5 5 5 5 5 5 5
5 5 5 4 3 2 1 0 0 0
5 6 7 7 7 7 7 8 9 10
5 6 7 7 7 7 7 8 9 10
10 11 12 12 12 12 12 12 13 14
10 11 12 13 14 15 16 17 18 19
15 16 17 17 17 17 17 17 18 19
Table 2 An assembly system with bottleneck capacity at an upstream stage. On-hand inventory
Echelon inventory
Order quantities
Echelon levels
x11
x21
x12
x22
y11
y21
y12
y22
X11
X21
X12
X22
Y11
Y21
Y12
Y22
2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 11 12
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 11 12
7 8 9 10 11 12 13 14 15 16 17
7 8 9 10 11 12 13 14 15 16 17
5 4 3 2 1 0 0 0 0 0 0
5 4 3 2 1 0 0 0 0 0 0
5 5 5 5 5 4 3 2 2 1 0
5 5 4 3 2 1 1 0 0 0 0
7 7 7 7 7 7 8 9 10 11 12
7 7 7 7 7 7 8 9 10 11 12
12 13 14 15 16 16 16 16 17 17 17
12 13 13 13 13 13 14 14 15 16 17
Assumption 3. Let K := mini,j {Kij }. Then, K1 = K . Lemma 1. For any xt in period t, there exists an optimal order schedule Xt such that xij,t +1 ≤ max (K , xijt − Xi,j−1,t ) for all i and j > 1. This results generalizes Lemma 1 of [13] to the case of capacitated assembly systems. The proof follows the basic approach of [13], and is therefore omitted. One implication of this lemma is that, if stage j has more than K units of inventory after it has sent a shipment downstream, it is optimal for this stage not to order any more stock. We say that initial inventory state xt (order schedule Xt ) is capacitated in period t if, for all i and all j > 1, xijt ≤ K (for all i and j, Xijt ≤ K ). Assumption 4. For all i and j > 1, xij1 ≤ K . Assumption 4 assures that the system begins in a capacitated state: each component i at each stage j > 1 starts with the level of on-hand inventory below bottleneck capacity K . The following result is a corollary of Lemma 1. Corollary 1. In each period t, there exists an optimal policy Yt such that Yijt − Yi,j−1,t ≤ K for all components i and stages j > 1. If the system starts out in a capacitated state, then it is optimal to keep it in that state. Further, all capacities in the system can be replaced with the bottleneck capacity K , without affecting costs. By Corollary 1, Assumptions 2–4 therefore imply that
Y(yt ) := Yijt , 1 ≤ i ≤ n, 1 ≤ j ≤ Li |yijt ≤ Yijt ≤ yi,j+1,t ,
Yi,j+1,t − Yijt ≤ K .
Assumptions 2 and 3 act to restore symmetry to the flow of stock across different components. Thus, the equivalence result of [14] can be said to fail in capacitated assembly problems where structural asymmetries are present in the system.
4. Balancing the system Let C(j) to be the set of components with a leadtime strictly greater than j, for each j = 1, 2, . . . , L − 1, where L := Ln is the longest component leadtime. These are the components for which stage j decisions (and states) apply, and are therefore called the relevant components at stage j. The following definitions are stated in terms of yt , but also apply to Yt . Formally, we say that yt is balanced if, for any echelon j, yijt = ykjt for all i, k ∈ C(j). A balanced inventory state has, for each stage, exactly the same number of units of each relevant component at that stage. Put another way, there are no unmatched components. We say yt and y∗t are echelon similar if, for all j, min yijt = min y∗ijt .
i ∈ C(j)
i ∈ C(j)
That is, two inventory states are echelon similar if any differences between them consist of excesses of components (compared to some other relevant component for the same stage). We say y∗t is the balanced reduction of yt if y∗t and yt are echelon similar, and y∗t is balanced. In this case, we get y∗t by removing any excesses from yt , and, thus, y∗t ≤ yt . Lemma 2. Let yt be capacitated, and y∗t be the balanced reduction of yt . Let Yt ∈ Y(yt ) be a capacitated policy in period t. Let Y∗t be the balanced reduction of Yt . Then Y∗t is capacitated and feasible for y∗t . (Y∗t ∈ Y(y∗t )). Proof. Fix j. Let i1 = i1 (j, Yt ), j ≥ 1, represent a component in C(j) Yi1 jt = min Yijt . i∈C(j)
Fix arbitrary i ∈ C(j). Then y∗ijt = y∗i1 ,jt ≤ yi1 ,jt
≤ Yi1 ,jt = Yi∗1 ,jt
[y∗t is balanced] [y∗t is the balanced reduction of yt ] [Yt ∈ Y(yt )] [definition of i1 ]
= Yijt∗ [Y∗t is balanced].
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A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
Let i2 = i2 (j − 1, Yt ), j > 1, represent a component in C(j − 1) such that Yi2 ,j−1,t = min Yi,j−1,t .
(c) Let yt be an arbitrary capacitated state, and y∗t be the balanced reduction of yt . By Lemma 1 and Corollary 1, there exists an optimal policy Yt , feasible for yt in period t, which is capacitated. Let Y∗t denote the balanced reduction of Yt . Then,
Fix arbitrary i ∈ C(j), j > 1. Then,
Ft (yt ) = Ft (Yt )
i∈C(j−1)
∗ Yijt = Yi∗2 ,jt ≤ Yi2 ,jt
[Y∗t is balanced]
[Y∗t is the balanced reduction of Yt ] ≤ Yi2 ,j−1,t + K [Yt is capacitated (Yt ∈ Y(yt ))] = Yi∗2 ,j−1,t + K [definition of i2 ] =
Yi∗,j−1,t
∗
+ K [Yt is balanced].
Similarly, let i3 = i3 (j + 1, yt ) denote a component in C(j + 1) such that yi3 ,j+1,t = min yi,j+1,t . i∈C(j+1)
Fix arbitrary i ∈ C(j + 1). Note that i ∈ C(j + 1) implies that i ∈ C(j). Then ∗
∗
Yijt = Yi3 jt
∗
[Yt is balanced]
≤ Yi3 jt [Y∗t is the balanced reduction of Yt ] ≤ yi3 ,j+1,t [Yt ∈ Y(yt )] = y∗i3 ,j+1,t [definition of i3 ] = y∗i,j+1,t [y∗t is balanced]. Consequently, Y∗t satisfies the required lower and upper bounds, and thus it is capacitated and feasible for y∗t . By Lemma 2, the balanced reduction of a capacitated policy Yt , feasible for yt , is capacitated and feasible for the balanced reduction of yt . This guarantees that we can construct a balanced policy out of any given capacitated policy Yt , by means of the balanced reduction, and have this policy be feasible for the same construction of a balanced state. We say that Ft is balance inducing if, for arbitrary capacitated yt such that y∗t is the balanced reduction of yt , it follows that Ft (y∗t ) ≤ Ft (yt ). Lemma 3. Fix t and suppose that Ft +1 is balance inducing. (a) If Yt is capacitated, and Y∗t is the balanced reduction of Yt , then Ft (Y∗t ) ≤ Ft (Yt ). (b) If y∗t is balanced and capacitated, then there exists an optimal inventory schedule Y∗t that is balanced and capacitated. (c) Ft is balance inducing. ∗ = Proof. (a) Because Y∗t is the balanced reduction of Yt , mini Yi1t mini Yi1t . Thus,
Ft (Y∗t ) − Ft (Yt ) =
Li n
∗ hijt (Yijt − Yijt )
[Yt is optimal] ∗ ≥ Ft (Yt ) [part (a)] ≥ Ft (y∗t ) [by Lemma 2, Y∗t is feasible for y∗t ].
Consequently, Ft is balance inducing.
Assumption 5. For each j = 0, 1, . . . , L − 1, yij1 = ykj1 for all i, k ∈ C(j). Thus, the system starts out balanced: The initial echelon state has, for each stage j, the same number of relevant components scheduled to complete assembly in j periods. Theorem 1. In each period, there exists an optimal policy that is capacitated and balanced. Proof. Since FT +1 = 0 by assumption, then FT +1 is balance inducing. Thus, by backward induction and Lemma 3(c), Ft is balance inducing for each t. By Assumptions 4 and 5, the initial state y1 is balanced and capacitated. Thus, by Lemmas 1 and 3(b), there exists an optimal decision Y1 in period 1 that is balanced and capacitated. Thus y2 = Y1 − D1 is balanced and capacitated, and, hence, by forward induction, there exists an optimal decision that is balanced and capacitated for each period. If a system starts out balanced then, when model parameters conform to Assumptions 1–5, it is possible (and optimal) to keep it balanced. It is not necessary to manage each component separately; instead, those components that are at the same stage can be managed together as a kit, where the kit for stage j has one each of every component in C(j). We can represent the echelon inventory for each relevant component at stage j by a single variable, zjt for j = 1, . . . , L − 1. We may assume that yijt = zjt for every i ∈ C(j). Let zt denote the L vector referred to as the echelon state. Because the optimal inventory schedule is balanced, Yijt is the same for every i ∈ C(j); it can be represented by a single variable Zjt . Let Zt , referred to as the echelon schedule denote the L-vector of zjt ’s. We now define the new cost parameters hjt for each j and t: hjt := i∈C(j) hijt . We thus obtain an equivalent model with the following optimality equations for each zt .
ft (zt ) = min
Zt ∈Z(zt )
γt (Z1t ) +
L
hjt Zjt + α E[ft +1 (Zt − Dt )] ,
(5)
j =1
where, by Corollary , the feasible set Z for echelon state zt can be written as
Z(zt ) = Zjt | zjt ≤ Zjt ≤ zj+1,t , 1 ≤ j ≤ L;
Zjt − Zj−1,t ≤ K , 1 < j ≤ L .
(6)
i=1 j=1
+ α E[Ft +1 (Y∗t − Dt )] − E[Ft +1 (Yt − Dt )] . ∗
Since Yt is the balanced reduction of a capacitated policy Yt , then: ∗ (i) Yijt ≤ Yijt for all i and j; (ii) Y∗t − Dt is the balanced reduction of Yt − Dt ; and (iii) Y∗t − Dt and Yt − Dt are capacitated. Thus, the first term on the RHS are negative. Since Ft +1 is balance inducing, then by (ii) and (iii), Ft +1 (Y∗t − Dt ) ≤ Ft +1 (Yt − Dt ) for any Dt . Therefore, Ft (Y∗t ) ≤ Ft (Yt ). (b) By Lemma 1 and Corollary 1, there exists an optimal policy Yt for y∗t which is capacitated. Let Y∗t be the balanced reduction of Yt . By part (a), Ft (Y∗t ) ≤ Ft (Yt ); by Lemma 2, Y∗t is capacitated and ∗ feasible for yt .
Corollary 2. Fix zt . If yijt = zjt for all j and i ∈ C(j), then ft (zt ) = Ft (yt ). Under the specified conditions, a capacitated component system can be reduced to a series system. The state space thus collapses from i Li dimensions to only L. 5. System with subassemblies Consider an assembly system where subassemblies are allowed at any stage prior to the final assembly into the finished product. One example of such an assembly system is shown in Fig. 4. We make use of the notation introduced in [14].
A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
25
Table 3 Allocation of assembly costs to components and stages. Li
k=1
2
5
5
β51 c1A
β52 c2A
6
5
β61 c1A
β62 c2A
8
7
β81 c1A
β83 c3A
9
8
β91 c1A
β93 c3A
β97 c7A
10
9
β10,1 c1A
β10,3 c3A
β10,7 c7A
i
3
4
5
6
7
8
9
10
c5A c6A c4A
c8A c9A A c10
Fig. 4. An assembly system with subassemblies.
For each (sub)assembly k = 1, 2, . . . , N, let lk denote the incremental leadtime, required to complete assembly k, where Assembly 1 is the finished product. Let s(k) represent the unique immediate successor assembly to assembly k, for all k > 1. Let L1 := l1 , and, for each k > 1, let Lk := lk + Ls(k) , denote the leadtime for assembly k. Let P (k) denote the set of immediate predecessor assemblies of assembly k. The set of components of such a system is the set of assemblies that have no predecessor assemblies. Let O(k) denote the set of components that are required in the composition of assembly k. For the system shown in Fig. 4, for example, s(6) = 2 and L9 = 8, while P (1) = {2, 3}, P (2) = {5, 6}, P (3) = {4, 7}, P (4) = {8}, P (7) = {9, 10}, P (5) = P (6) = P (8) = P (9) = P (10) = ∅, and O(1) = {5, 6, 8, 9, 10}, O(2) = {5, 6}, O(3) = {8, 9, 10}, O(4) = {8}, O(7) = {9, 10}, and O(k) = {k}, for k = 5, 6, 8, 9, 10. We will refer to such a system as the general assembly system. Capacity constraints carry over from the component system directly: the processing capacity for assembly k represents the smallest of the capacities for each of the components in O(k), since assembly k requires one of each of those components. Assume, for convenience, that the only costs incurred in this A general system are the assembly costs: Let ckt be the discounted present value of the costs related to assembling/purchasing/ transforming assembly k, evaluated at the beginning of period t, if that assembly is begun then. This cost may be a processing or a purchasing cost for components (e.g. 6, 8, 11, 13, 14, 15, 16, and 17), because they have no predecessor assemblies. We now introduce the constraint-relaxation approach from [2] and the cost allocation ideas of [3,4]. Briefly, [2,3] develop the approach of relaxing some of the constraints in the problem to provide lower bounds on the cost for certain multi-period, multistage, production/inventory models with deterministic demand. The underlying idea is that such a relaxed version of the problem will naturally have lower costs due to fewer constraints in the system, and thus provide a reasonable lower bound on the cost for the original problem. With the right choice of constraints to relax, the problems considered in [2,3] can be broken into simpler subproblems that allow a range of possible cost allocation across them. By choosing an optimal cost allocation for a given relaxation, it is possible to derive tight lower bounds on the cost function for the original problem. Chen and Zheng [4] extend this constraint-relaxation method to production-distribution networks with stochastic demands. In particular, the paper makes use of this method to prove the original multiechelon result of [5], and the assembly equivalence result of [14], in a more direct manner. This is accomplished by relaxing the constraints, and finding the optimal policy for the simpler, relaxed problem thus obtained. This optimal policy for the relaxed system is then showed to be feasible in the original system. Since this policy is optimal for the less constrained system, and found to be feasible in the original, more constrained system, then the policy must also be optimal for this original system.
In order to apply the constraint-relaxation and cost-allocation method to our problem, we first relax the constraint(s) that there must be exactly the right number of each component ready when any assembly takes place. This provides a system in which the performance is at least as good as in the original one. Further, this relaxed system has exactly the same constraints as the component assembly system analyzed in Section 4. Next, following [4], we design a cost allocation scheme so that, if the solution is balanced and capacitated, then the total expected cost will be the same in the relaxed system and the original (constrained) system with subassemblies. If there exists an allocation of assembly costs to its components that satisfies the required conditions, then, by Theorem 1, there exists an optimal policy for the relaxed system that is balanced and capacitated, and accounts for the costs incurred. Since it is balanced and capacitated, this policy is feasible in the original constrained system (by being balanced this policy meets the constraint that there must be exactly the right number of each component ready when any (sub)assembly takes place). Because this policy is optimal for the relaxed system, and feasible for the original constrained system, then, following the reasoning in [4], it must be optimal for it. Let βikt denote the portion of the assembly cost for assembly k that is allocated to component i, for each i ∈ O(k) in period t (e.g. allocate the assembly costs equally to each required component, so that β51t = β61t = β81t = β91t = β 10,1t = 1/5). In general, we require full allocations, namely, that i∈O(k) βikt = 1 for each k and t. For example, we require that β84t = 1 because assembly 4 requires only component 8. Thus, each component i is A allocated βikt ckt , for each k for which i ∈ O(k). Thus, if assembly k is initiated at the beginning of period t, it will require one unit of A component i for each i ∈ O(k), and the amount βikt ckt is allocated to each such component i for stage Lk in period t. The allocated assembly costs for the example are summarized in Table 3, for the case of stationary assembly costs. The component 6, for example, is shown as bearing the full cost of assembly 6— this is because it is the only component needed for that particular assembly. The cost of assembly 7 at stage 6, for instance, is shared among components 9 and 10. Theorem 2. If, for each assembly k and component i in each period A t, there exists an allocation of assembly costs such that βikt ckt ≥ A αβik,t +1 ck,t +1 , then there exists an optimal policy that is balanced and capacitated in every period. Proof. Let L(j) := {k |Lk = j} denote the set of assemblies with leadtimeLk equal to j. For each component i and echelon j, let A θijt := k∈Lj βikt ckt . Then, θijt represents all the assembly costs allocated to each unit of component i at stage j in period t. The total cost allocated to component i at stage j in period t is θijt Xijt . For any set of demand realizations {D1 , D2 , . . . , DT }, the discounted
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A. Angelus, W. Zhu / Operations Research Letters 41 (2013) 19–26
net present value of all such costs is n
Li
T
n
α t −1 θijt Xijt =
i =1 j =1 t =1
=
Li n T
Li
T
α t −1 θijt (Yijt − yijt )
i =1 j =1 t =1
α t −1 θijt (Yijt − Yij,t −1 + Dt −1 )
i=1 j=1 t =1
=
Li n T
α t −1 (θijt − αθij,t +1 )Yijt + θijt Dt −1 ,
i=1 j=1 t =1
where D0 := 0 and θij,T +1 := 0 all i and j. Let h′ijt := hijt + (θijt − αθij,t +1 ) for j ≥ 2, and h′i1t := hi1t . Since θijt ≥ αθij,t +1 , then h′ijt ≥ 0 and Assumption 1 holds. (There is an additional positive term for j = 1: (θi1t − αθi1,t +1 )Yi1t .) Assumptions 3 and 4 hold directly. By Theorem 1, there exists an optimal policy that is balanced and capacitated. If a policy is balanced, then, under an allocation of costs such as the one above, the actual costs incurred in the general system with subassemblies are the same as those captured by the component assembly model. Since we allow each of the components to be independently managed, then the optimal solution of the general systems is at least as good as the best one in the original (i.e. component assembly) system. Because, under Assumptions 1– 5, there is an optimal policy that is balanced and capacitated, that policy is feasible in the original system, and thus it must be optimal for it. The condition required by Theorem 2 clearly holds for the case of stationary assembly costs. In fact, it is only mildly restrictive: as long as assembly costs are not increasing too fast in time, and across all assemblies at the same time, finding an appropriate allocation should be straightforward. 6. Discussion The conditions needed for the state space collapse shown in this paper may not always hold. Assumption 1 would not apply in those (rare) settings when the inventory holding cost for some component(s) is higher at an upstream stage than at a downstream stage. Every unit of that component would then be moved from the upstream stage to the downstream stage, thus creating an excess of that component at the downstream stage. The set of all relevant components at that downstream stage could then no longer be managed as a kit. Assumption 2 would not hold for certain transportation processes, such as ocean-going shipping. In its absence, a capacitated assembly system with flow-through stages can be expected to exhibit stock-withholding of inventory, which has the effect of unbalancing the system by maintaining a shortage of one component relative to others. Assumption 3 is the assembly-system equivalent of the assumption made by Parker and Kapuscinski [13] in proving the Clark–Scarf decomposition of a capacitated multiechelon system.
Very little is presently known about capacitated systems (either assembly or series) for which this assumption does not hold, and further research is clearly needed to better understand those systems. It is worth noting, however, that Assumption 3 is not the only restriction on capacity constraints that achieves the results of this paper. In a capacitated assembly system where, for every j > 1, and i, i′ ∈ C(j), Kij = Ki′ j , so that, at each stage in the assembly system, all relevant components have the same processing capacity, it is straightforward to verify that all the results of our paper go through (except for Lemma 1). Clearly, there is more than one way to enforce the structural symmetry in a capacitated assembly system. Assumptions 4 and 5 contribute to the ease of analysis. A capacitated assembly system can be shown to achieve a capacitated state on its own (the proof is similar to that in [13]) even if it starts out uncapacitated. Of course, once the system reaches a capacitated state, it is optimal to keep it there, and all the results of this research apply. Similarly, if the system starts in an unbalanced state, it will eventually reach a balanced state (see [14] for an example of such a result), so that Assumption 5 is also not critical to the analysis. Acknowledgments The authors would like to thank Sridhar Seshadri, an anonymous associate editor, and the reviewers for their helpful comments and suggestions. References [1] A. Angelus, E. Porteus, An asset assembly problem, Oper. Res. 56 (2008) 665–680. [2] D. Atkins, A survey of lower bounding methodologies for production/inventory models, Ann. Oper. Res. 26 (1990) 9–28. [3] D. Atkins, A simple lower bound to the dynamic assembly problem, Eur. J. Oper. Res. 75 (1994) 462–466. [4] F. Chen, Y.-S. Zheng, Lower bounds for multi-echelon stochastic inventory systems, Manage. Sci. 40 (1994) 1426–1443. [5] A.H. Clark, H. Scarf, Optimal policies for the multi-echelon inventory problem, Manage. Sci. 6 (1960) 475–490. [6] G.A. DeCroix, P.H. Zipkin, Inventory management for an assembly system with product or component returns, Manage. Sci. 51 (2005) 1250–1265. [7] A.G. De Kok, J.W.C.H. Visschers, Analysis of assembly systems with service level constraints, Int. J. Prod. Econ. 59 (1999) 313–326. [8] P. Glasserman, S. Tayur, The stability of a capacitated, multiechelon production–inventory system under a base-stock policy, Oper. Res. 42 (1994) 913–925. [9] P. Glasserman, S. Tayur, Sensitivity analysis for base-stock levels in multiechelon production–inventory systems, Manage. Sci. 41 (1995) 263–281. [10] W.T. Huh, G. Janakiraman, Base-stock policies in capacitated assembly systems: convexity properties, Naval Res. Log. 57 (2010) 109–118. [11] G. Janakiraman, J.A. Muckstadt, A decomposition approach for a class of capacitated serial systems, Oper. Res. 57 (2009) 1384–1393. [12] A. Muharremoglu, J.N. Tsitsiklis, A single-unit decomposition approach to multiechelon inventory systems, Oper. Res. 56 (2007) 1089–1103. [13] R.P. Parker, R. Kapuscinski, Optimal policies for a capacitated two-echelon inventory system, Oper. Res. 52 (2004) 739–755. [14] K. Rosling, Optimal inventory policies for assembly systems under random demands, Oper. Res. 37 (1989) 565–579. [15] C.P. Schmidt, S. Nahmias, Optimal policy for a two-stage assembly system under random demand, Oper. Res. 33 (1985) 1130–1145.