Understanding the Performance of the Long Chain and Sparse ... · Understanding the Performance of the Long Chain and Sparse Designs in Process Flexibility ... The long chain has

OPERATIONS RESEARCHVol. 60, No. 5, September–October 2012, pp. 1125–1141ISSN 0030-364X (print) � ISSN 1526-5463 (online) http://dx.doi.org/10.1287/opre.1120.1081

© 2012 INFORMS

Understanding the Performance of the LongChain and Sparse Designs in Process Flexibility

David Simchi-LeviEngineering Systems Division, Department of Civil and Environmental Engineering, and the Operations Research Center,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected]

Yehua WeiOperations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected]

The long chain has been an important concept in the design of flexible processes. This design concept, as well as othersparse designs, have been applied by the automotive and other industries as a way to increase flexibility in order to bettermatch available capacities with variable demands. Numerous empirical studies have validated the effectiveness of thesedesigns. However, there is little theory that explains the effectiveness of the long chain, except when the system size islarge, i.e., by applying an asymptotic analysis.

Our attempt in this paper is to develop a theory that explains the effectiveness of long chain designs for finite sizesystems. First, we uncover a fundamental property of long chains, supermodularity, that serves as an important buildingblock in our analysis. This property is used to show that the marginal benefit, i.e., the increase in expected sales, increasesas the long chain is constructed, and the largest benefit is always achieved when the chain is closed by adding the lastarc to the system. Then, supermodularity is used to show that the performance of the long chain is characterized bythe difference between the performances of two open chains. This characterization immediately leads to the optimalityof the long chain among 2-flexibility designs. Finally, under independent and identically distributed (i.i.d.) demand, thischaracterization gives rise to three developments: (i) an effective algorithm to compute the performances of long chainsusing only matrix multiplications; (ii) a result that the gap between the fill rate of full flexibility and that of the long chainincreases with system size, thus implying that the effectiveness of the long chain relative to full flexibility increases asthe number of products decreases; (iii) a risk-pooling result implying that the fill rate of a long chain increases with thenumber of products, but this increase converges to zero exponentially fast.

Subject classifications : flexible manufacturing; capacity pooling; facility planning: design; network flow; supermodularfunction; random walk.

Area of review : Operations and Supply Chains.History : Received May 2011; revisions received September 2011, February 2012; accepted March 2012. Published online

in Articles in Advance September 26, 2012.

1. IntroductionFor many manufacturing firms, the ability to match demandand supply is key to their success. Failure to do so couldlead to loss of revenue, reduced service levels, impact onreputation, and decline in the company’s market share.Unfortunately, recent developments such as intense mar-ket competition, product proliferation, and the increase inthe number of products with short life cycle have cre-ated an environment where customer demand is volatileand unpredictable. In such an environment, traditional oper-ations strategies such as building inventory, investing incapacity buffers, or increasing committed response time toconsumers do not offer manufacturers a competitive advan-tage. Therefore, many manufacturers have started to adoptan operations strategy known as process flexibility to betterrespond to market changes without significantly increasingcost, inventory, or response time (see Simchi-Levi 2010).

Process flexibility is defined as the ability to “build dif-ferent types of products in the same manufacturing plantor on the same production line at the same time” (Jordan

and Graves 1995, p. 577). For example, in “full” (process)flexibility, each plant is capable of producing all products.In this case, when the demand for one product is higherthan expected while the demand for a different product islower than expected, a flexible manufacturing system canquickly make adjustments by shifting production capacitiesappropriately. By contrast, in a “dedicated” strategy (some-times called “no flexibility”), each plant is responsible fora single product and hence does not have the same abilityto match supply with demand.

Because of its effectiveness in responding to uncertain-ties, process flexibility has gained significant attention inseveral industries, in particular, in the automotive indus-try. Indeed, the plants for most of the automobile giantsare much more flexible (in terms of process flexibility)today compared to twenty years ago (Boudette 2006). Evi-dently, it is often too expensive to achieve a high degreeof flexibility, for example full flexibility, and as a result,sparse or partial flexibility, is implemented instead. Oneset of sparse flexibility designs is the 2-flexibility designs.

1125

Simchi-Levi and Wei: Long Chain in Process Flexibility1126 Operations Research 60(5), pp. 1125–1141, © 2012 INFORMS

A flexibility design is a 2-flexibility design if each plantproduces exactly two products and demand for each prod-uct can be satisfied from exactly two plants.

Of course, there are many ways to implement sparsedesigns and the challenge is to identify an effective one.An important concept analyzed in the literature and appliedin practice by various companies is the concept of the longchain. The first to observe the power of the long chain wereJordan and Graves (1995) who, through empirical analy-sis, showed that the long chain design can provide almostas much benefit as full flexibility. In particular, Jordan andGraves (1995) found that for randomly generated demand,the expected amount of demand that can be satisfied by along chain design is very close to that of a full flexibil-ity design. Unfortunately, with a few exceptions (see Chouet al. 2010b), there is very little theory to explain why longchain works so well. One objective of our research is toprovide an answer to this question.

Throughout the paper, we consider balanced manufac-turing systems, i.e., manufacturing systems with an equalnumber of plants and products, and all plants have thesame capacity. Although this is rarely the case in the realworld, as Jordan and Graves (1995) argues, understandingbalanced systems can provide insights into very realisticscenarios.

Given a balanced manufacturing system, a flexibilitydesign A is represented by the arc set of a directed bipartitegraph, where an arc from plant node i to product node jimplies that plant i is capable of producing product j . Forexample, if A is a dedicated design, then A has exactly narcs such that each plant node is incident to one arc andeach product node is incident to one arc. By contrast, if Ais a full flexibility design, then A has arcs connecting everyplant node to all product nodes.

Because A is represented by a bipartite graph, apply-ing standard graph theory notation, we define an undirectedcycle in A to be a set of arcs that forms a cycle when thearc directions are ignored. A flexibility design A is a longchain if its arcs form exactly one undirected cycle contain-ing all plant and product nodes (see Figure 1 for an exam-ple). A closed chain is defined as an induced subgraph in A

Figure 1. Configurations for flexibility designs.

Long chain

Closed chain

Open chain

that forms an undirected cycle, whereas an open chain is aninduced subgraph in A that forms an undirected line (onearc less than an undirected cycle). In Figure 1, an exam-ple of an open and a closed chain is presented. It can beseen that any 2-flexibility design, where each product/plantnode is incident to two arcs, is the union of a number ofclosed chains.

This paper studies the performance of flexibility designson balanced systems with stochastic demand. In this model,given a random demand instance d, the level of sales asso-ciated with a flexibility design A is characterized by themaximum demand satisfied without exceeding plant capac-ities and without violating the constraints associated withthe flexibility design. This quantity can be determined bysolving a simple max-flow problem. We use D to denotethe random distribution of demand, and the performance ofa flexibility design is measured by the expected sales overall random realizations. In what follows, we use the termsperformance and expected sales interchangeably. Over theyears, expected sales has been a popular metric for studyingand evaluating the effectiveness of flexibility designs; seethe survey by Chou et al. (2008). Finally, the fill rate of aflexibility design is defined as the ratio between the perfor-mance of a given flexibility design and the total expecteddemand.

The first part of our paper, §4, is motivated by an obser-vation that has been made in the literature (Graves 2008,Hopp et al. 2004) regarding the performance of the longchain for a balanced system when product demands areindependent and identically distributed (i.i.d.). The obser-vation states that if one starts with a dedicated design andadds arcs to create the long chain, the incremental benefits,or the change in performance, associated with each addedarc is increasing.

To illustrate this observation, consider an example withsix plants and six products, where the demand for eachproduct is equal to either 008, 1, or 102 with equal probabil-ities, and the capacity of each plant is one. Then, we startwith a dedicated flexibility design (the dashed arcs in Fig-ure 2(a)), and add arcs 411251 421351 0 0 0 1 45165, and 46115one at a time, until we complete the long chain. Each timewe add such an arc, we determine the expected sales asso-ciated with the resulting design at that time. Figure 2(b)displays the performance of the flexibility designs at dif-ferent stages, as well as the incremental benefits when anew arc is added. As you can see, the incremental benefitsincrease as we add more arcs. The biggest impact, surpris-ingly, occurs when we add the last arc and close the longchain. To formally prove this observation, we identify in §4an important property of the long chain, supermodularity,and apply it as the long chain is constructed.

Our second main result, §5, characterizes the perfor-mance of a long chain as the difference between the per-formances of two open chains. Interestingly, this new resultalso depends on the supermodularity property identifiedin §4. A direct consequence of this characterization of the

Simchi-Levi and Wei: Long Chain in Process FlexibilityOperations Research 60(5), pp. 1125–1141, © 2012 INFORMS 1127

Figure 2. The increase in incremental benefit.

66

55

33

44

11

22

Plants Products

(a) Flexibility structure (b) Incremental benefits of creating long chain

Structure Performance Incr. benefit

Dedicated 5.6Added arc (1, 2) 50622 00022Added arc (2, 3) 50652 00030Added arc (3, 4) 50686 00035Added arc (4, 5) 50724 000379Added arc (5, 6) 50765 000403Added arc (6, 1) 50842 00077Full flexibility 50842

performance of the long chain is that the long chain designis optimal among all 2-flexibility designs.

Finally, in §6, we further apply this characterization inthree different ways. First, we develop an effective methodto compute the performance of long chains of finite size.Second, we show that the difference between the fill rateof full flexibility and that of the long chain increases withsystem size, thus implying that the effectiveness of the longchain is closer to that of full flexibility as the number ofproducts decreases. Lastly, we prove a risk-pooling resultshowing that the fill rate of the long chain increases withthe number of products, but this increase converges to zeroexponentially fast.

Next, we provide a literature review. This is followed by§3 where we introduce notations and definitions.

2. Literature ReviewThe study of process flexibility, also known as “mix flex-ibility” or “short-term flexibility,” began in the 1980s. Werefer readers to the survey of Sethi and Sethi (1990) for adetailed introduction and literature review on process flexi-bility prior to the 1990s. This research typically focused onthe benefits, challenges, and trade-offs between fully flexi-ble and dedicated systems. Unfortunately, most companiesare not interested in full flexibility because of its enormousimplementation cost.

The seminal paper of Jordan and Graves (1995) is thefirst to consider the design and effectiveness of limiteddegree of (process) flexibility. Applying numerical analy-sis (simulation) to a stochastic demand model, Jordan andGraves demonstrate two important insights in process flexi-bility. First, they show that a sparse flexibility design, whichthey called the long chain, can provide almost as much ben-efit as full flexibility. Second, they introduce the concept ofchaining, which generalizes the idea of the long chain, andshow that this concept leads to sparse designs that performextremely well in numerical studies.

The paper of Jordan and Graves (1995) paved waysfor two important lines of research in the studies of pro-cess flexibility. The first is to apply and extend the ideasof Jordan and Graves to other settings. Indeed, since thepublication of Jordan and Graves (1995), there has beenextensive research that has applied the concept of the longchain and limited flexibility in a variety of applications. Forinstance, Graves and Tomlin (2003) study process flexibil-ity in multistage supply chains; Sheikhzadeh et al. (1998)and Gurumurthi and Benjaafar (2004) in queueing systems;Iravani et al. (2005) in queueing networks and Wallace andWhitt (2005) in call centers. We also note the work of Hoppet al. (2004), Bish et al. (2005), Muriel et al. (2006), Makand Shen (2009), Chou et al. (2010a), and Bassamboo et al.(2010) for research on flexibility in other settings.

The other line of research tries to establish a theoreticalfoundation that explains the effectiveness of limited degreeof flexibility, and in particular, the insights provided by thework of Jordan and Graves (1995). Surprisingly, very lit-tle theory has been developed thus far. Two of the rareexceptions are Chou et al. (2010b, 2011). In Chou et al.(2010b), the authors develop a method to compute the ratiobetween the performance of the long chain to that of fullflexibility in asymptotic regime. They also present a con-straint sampling method to find a sparse flexibility designthat performs almost as well as full flexibility. In a com-panion paper, Chou et al. (2011) prove that when demandsare bounded by a constant, there exists a sparse flexibilitywith graph expansion property that achieves sales close tothat of full flexibility in the worst-case demand scenario.Moreover, Chou et al. (2011) describe a heuristic to findsuch structures. We note that both the constraint samplingmethod and results in Chou et al. (2011) hold in the generalsystem where the number of plants and products need notbe equal, with arbitrary plant capacities and random distri-bution of demands. Finally, Aksin and Karaesmen (2007)show that there is a decrease in marginal benefit associatedwith the increase in either the degree of flexibility or thecapacities of the manufacturing plants.


For more information about the literature on flexibility,we refer readers to surveys by Buzacott and Mandelbaum(2008) and Chou et al. (2008). The first survey intro-duces different applications of flexibility, and the secondfocuses on the recent theoretical advancements in processflexibility.

3. Definitions and NotationConsider a balanced manufacturing system of size n facingrandom demand. In such a system, there are n plants, eachwith unit capacity and n different products. Throughout thepaper, the vector D is used to denote the random demanddistribution, and d is a particular random instance. Becausein practice demand is never negative, this paper assumesthat D is nonnegative.

For a balanced system of size n, we say that itsdemand, D, is exchangeable if 6D11 0 0 0 1Dn7 equals to6D�4151 0 0 0 1D�4n57 in distribution for any � that is a per-mutation of 81121 0 0 0 1 n9. We note that any i.i.d. demandis exchangeable but not all exchangeable demand are i.i.d.For example, consider a random vector D = 6D11 0 0 0 1Dn7that is uniformly distributed on the linear polyhedron{

4x11 0 0 0 1 xn5∣

∣

∣

n∑

i=1

xi = n1 xi ¾ 01 ∀ i = 11 0 0 0 1 n}

0

Clearly D is exchangeable, but the random variables in Dare not independent, because they always sum up to n.

Next, we define several classes of flexibility designs forbalanced manufacturing systems. For any integer n ¾ 2,the dedicated design for a balanced system of size n, Dn,is defined as Dn = 84i1 i5 � i = 1121 0 0 0 1 n9; the long chainflexibility design for a balanced system of size n, Cn, isdefined as Cn = Dn ∪ 84i1 i + 15 � i = 1121 0 0 0 1 n − 19 ∪

84n1159; and full flexibility design for a balanced system ofsize n, Fn, is defined as Fn = 84i1 j5 � i1 j = 1121 0 0 0 1 n9. Inflexibility designs, we refer to an arc 4i1 i5 as a dedicatedarc and arc 4i1 j51 i 6= j as a flexible arc.

Given a random instance of the demand, d, the maximumsales that can be achieved by a flexibility design A, denotedby P4d1A5, is defined as

P4d1A5= max∑

1¶i1 j¶n

fij

s.t.n∑

i=1

fij ¶ dj1 ∀1 ¶ j ¶ n1

n∑

j=1

fij ¶ 11 ∀1 ¶ i¶ n1

fij ¾ 01 ∀ 4i1 j5 ∈A1

fij = 01 ∀ 4i1 j5yA1

f ∈�n20

Throughout the paper, this optimization problem isreferred to as the optimization problem associated with

P4d1A5, or simply, P4d1A5, when there is no ambiguity.It is not difficult to see that this optimization problem is amax-flow problem, and as a result, we refer to fij as theflow on arc 4i1 j5.

Under random demands D, we define the performance,also referred to as expected sales, of A to be Ɛ6P4D1A57,where Ɛ6 · 7 is the expectation of a random variable. Forsuccinctness, we also use 6A7 to denote this quantity, whenthe random vector D is given.

Finally, for any integer k¾ 0, we define open chain Lk

as Lk = Dk ∪ 84i1 i + 15 � i = 11 0 0 0 1 k − 19. One can thinkof Lk as the open chain that connects plant 1 to product k.Note that Lk is simply Ck\84k1159.

4. Supermodularity in Long ChainsIn this section, we establish an important building blockin our analysis by identifying the supermodularity of flex-ible arcs in the long chain design. Throughout the section,we consider a balanced system of size n facing exchange-able random demand D. In this section we first define andprove the supermodularity property (§4.1), and then applythe supermodularity property (§4.2) to show that the incre-mental benefits are nondecreasing when the long chain isconstructed.

4.1. Proof of Supermodularity

We start by formally defining the notion of super-modularity.

Definition 1. A function f 4x1 y5 is said to be supermod-ular in x and y if for any real numbers x′, x′′, y′, y′′,

f(

max8x′1 x′′91max8y′1 y′′9)

+ f(

min8x′1 x′′91min8y′1 y′′9)

¾ f 4x′1 y′5+ f 4x′′1 y′′50

Next, consider a design A, a demand instance d, and twospecific arcs � and � with given nonnegative capacities u�

and u�. Define

P�1�4u�1 u�1d1A5= max∑

i1 j

fij

s.t.∑

i

fij ¶ dj1

∑

j

fij ¶ 11

fij ¾ 01 ∀ 4i1 j5 ∈A1

fij = 01 ∀ 4i1 j5yA1

f� ¶ u�1 f� ¶ u�1

f ∈�n20

We prove that if A ⊂ Cn, then for any two flexiblearcs � and � in A, P�1�4u�1 u�1d1A5 is supermodularin u� and u�. Note that if � and � are not in A, then


Figure 3. G4C55 for the max-weight circulation asso-ciated to P�1�4u�1 u�1d1Cn5.

Products

s

Plants

1

2

3

4

5

1

2

3

4

5

P�1�4u�1 u�1d1A5 is equal to P4d1A5 regardless of u� andu� and the supermodularity property holds. The interestingcase arises when � and � are in A.

For this purpose, we show that P�1�4u�1 u�1d1A5 isequivalent to a max-weight circulation problem, whichallows us to apply a classical result from Gale and Politof(1981). Define G4A5 to be the underlying graph for themax-weight circulation problem, which contains A, anadditional node s, an arc from s to each of the plant nodes,and an arc from each of the product nodes to s. The under-lying graph of the max-weight circulation problem, G4A5,is illustrated in Figure 3 for A = C5, of long chain for abalanced system of size five.

To complete the description of the max-weight circula-tion problem, we set the weight of each plant to product arc(that is, the arcs in A) to one and the weight of every otherarc to zero. The upper bound (capacity) on the flow on anarc from s to plant i is set to be one for all i = 1121 0 0 0 1 n;the upper bound for the flow on an arc connecting productj to s is set to be dj for all j = 1121 0 0 0 1 n; and the upperbound for the flow on every arc in A is also set to be one.Finally, we set the lower bound for the flow on every arcin G4A5 to be zero.

In their seminar paper, Gale and Politof (1981) presentthe following definition.

Definition 2. In a directed graph G, two arcs �, � aresaid to be in series, if for any cycle C containing both �and �, � and � have the same direction when we fix anorientation of C.

Next, we show that any two flexible arcs from the setCn are in series in graph G4Cn5.

Figure 4. Illustration for the proof of Lemma 1.

s X2 contains 2k–1 arcs

�

�

Lemma 1. Let � and � be two flexible arcs in Cn. Then� and � are in series in G4Cn5, where G4Cn5 is theunderlying graph of the max-weight circulation problem forP�1�4u�1 u�1d1Cn5.

Proof of Lemma 1. Let C be an arbitrary undirected cyclein G4Cn5. If C does not contain node s, then C must be theundirected cycle that contains every plant to product arcsin Cn. In that case, it is easy to verify that � and � havethe same direction in C. Otherwise, suppose C contains s.In such a case, C can be decomposed into four pieces, X1,X2, �, and �, where X1, X2 are the two paths between �and �. Without loss of generality, we assume X1 contains s.This assumption implies that all arcs in X2 are plant toproduct arcs (i.e., X2 ⊂ Cn). The structure of Cn impliesthat X2 must contain an odd number of arcs. Moreover, thepath in X2 ∪ 8�9∪ 8�9 has alternating directions for everytwo consecutive arcs, and therefore � and � have the samedirection in C. This is illustrated by Figure 4. Because thisis true for any arbitrary undirected cycle C, � and � are inseries in G4Cn5. �

Lemma 1 allows us to apply the following importantresult of Gale and Politof (1981). They show that if twoarcs, � and �, in the underlying graph are in series, then theoptimal flow of the max-weight circulation is supermodularwith respect to the capacities of both arcs.

Theorem 1. Let A be a flexibility design for a balancedsystem of size n, and A⊂Cn. For any flexible arcs �, � inA, P�1�4u�1 u�1d1A5 is supermodular in u� and u�. Hence,

P4d1A5+P4d1A\8�1�95¾ P4d1A\8�95+P4d1A\8�950

Proof of Theorem 1. By construction, P�1�4u�1 u�1d1A5can be computed by solving the max-weight circulationproblem. Because A⊂Cn, the set arcs in G4A5 is a subsetof the set of arcs in G4Cn5. By Lemma 1, � and � arein series in G4Cn5. Thus, � and � are in series in G4A5.Applying the main theorem in Gale and Politof (1981), wehave that P�1�4u�1 u�1d1A5 is supermodular in u� and u�.Hence

P41111d1A5+P40101d1A5¾P40111d1A5+P41101d1A5

⇒ P4d1A5+P4d1A\8�1�95

¾ P4d1A\8�95+P4d1A\8�950 �


Because the theorem holds for any realization ofdemand, it must be also true in expectation. Thus,

Corollary 1. For any flexible arcs �, � in A ⊂ Cn,Ɛ6P�1�4u�1 u�1D1A57 is supermodular in u� and u� for anyrandom distributions D.

The corollary thus suggests that any two flexible arcs inthe long chain complement each other. That is, the exis-tence of one flexible arc increases the marginal benefit thatcan be gained when the other flexible arc is added.

Interestingly, the supermodular result of Gale and Politof(1981) can be extended for two sets of arcs X and Y ,where any pair of arcs in X ∪ Y are in series with eachother. Although that was not stated in the paper of Galeand Politof (1981), it was proven by Granot and Veinott Jr.(1985) in a more general setting. Here, we state Corol-lary 2, which is a special case from (Granot and Veinott Jr.1985, Theorem 17 ).

Corollary 2. Let A be a flexibility design for a balancedsystem of size n, and A ⊂ Cn. For any X1Y ⊆ S, where Sis the set of all flexible arcs in A, and demand instance d,

P4d1A\4X ∩ Y 55+P4d1A\4X ∪ Y 55

¾ P4d1A\X5+P4d1A\Y 50

4.2. Incremental Benefits in Long Chains

In this section, we apply Corollary 1 to formally provethe observation made in the introduction that the incremen-tal benefits associated with adding arcs to the long chainis increasing. Consider the following sequence of flexibil-ity designs: Ln

1 , Ln2 , Ln

31 0 0 0 1Lnn, Cn, where we define

Ln1 =Dn and Ln

k =Lk∪84i1 i5 � i = k+11 0 0 0 1 n9. In words,Ln

k is simply the open chain from plant 1 to product k plusthe dedicated arcs connecting plants i to products i for allk < i¶ n. Finally, recall that Cn is the long chain of size n.

In the example of Graves (2008) and the table in Figure 2of this paper, one starts at Ln

1 and add arcs sequentially tocreate Ln

21 0 0 0 1Lnn1Cn. Now, we apply the supermodularity

result to show that the incremental benefit, 6Lnk7− 6Ln

k−17,is nondecreasing with k.

Theorem 2. For any balanced system of size n withexchangeable demand, we have

6Ln27− 6Ln

17¶ 6Ln37− 6Ln

27

¶ · · ·¶ 6Lnn7− 6Ln

n−17¶ 6Cn7− 6Lnn70

Proof of Theorem 2. Fix any 1 ¶ k ¶ n − 1. Let � =

41125, �= 4k1 k+ 15. By Corollary 1, we have

Ɛ6P�1�41111D1Lnk+157+ Ɛ6P�1�40101D1Ln

k+157

¾ Ɛ6P�1�41101D1Lnk+157+ Ɛ6P�1�40111D1Ln

k+1570 (1)

Setting u� = 0 is equivalent to deleting arc � in theoptimization problem associated with P�1�4u�1 u�1d1A5,

whereas setting u� = 1 implies that this arc exists in thesame model and its capacity is redundant. As a result, wehave that,

Ɛ6P�1�41111D1Lnk+157= Ɛ6P4D1Ln

k+157= 6Lnk+171 (2)

and

Ɛ6P�1�41101D1Lnk+157= Ɛ6P4D1Ln

k57= 6Lnk70 (3)

Let D� = 6D21D31 0 0 0 1Dn1D17, then

Ɛ6P�1�40101D1Lnk+157= Ɛ6P4D� 1L

nk−157

= Ɛ6P4D1Lnk−157= 6Ln

k−171 (4)

and

Ɛ6P�1�40111D1Lnk+157= Ɛ6P4D� 1L

nk57

= Ɛ6P4D1Lnk57= 6Ln

k71 (5)

where the second to last equality in (4) and (5) holdsbecause the random vector D is exchangeable. Substitut-ing Equations (2)–(5) into Inequality (1), we obtain that6Ln

k+17− 6Lnk7¾ 6Ln

k7− 6Lnk−17, for k = 21 0 0 0 1 n− 1.

Finally, to show 6Lnn7 − 6Ln

n−17 ¶ 6Cn7 − 6Lnn7, let � =

41125, �= 4n115 and let D� = 6D21D31 0 0 0 1Dn1D17. Then

Ɛ6P�1�41111D1Cn57= 6Cn71

Ɛ6P�1�41101D1Cn57= 6Lnn71

Ɛ6P�1�40101D1Cn57= Ɛ6P4D� 1Lnn−157= 6Ln

n−171

and

Ɛ6P�1�40111D1Cn57= Ɛ6P4D� 1Lnn57= 6Ln

n70

By Corollary 1,

Ɛ6P�1�41111D1Cn57+ Ɛ6P�1�40101D1Cn57

¾ Ɛ6P�1�41101D1Cn57+ Ɛ6P�1�40111D1Cn571

we have that 6Cn7− 6Lnn7¾ 6Ln

n7− 6Lnn−17. This completes

the proof.

Observe that the proof of Theorem 2 requires the applica-tion of the supermodularity result (Theorem 1), which holdsdeterministically for any fixed demand instance. By con-trast, Theorem 2 holds only stochastically under exchange-able demand but does not hold for any fixed demandinstance.

5. Characterizing the Performanceof the Long Chain

In this section, we show that in a balanced system of sizen with exchangeable demand, the performance of the longchain can be characterized by the difference between theperformances of two open chains. Like the previous sec-tion, we start the section by developing several propertiesof the long chain when the demand is deterministic.


5.1. Long Chain with Deterministic Demand

In this subsection, we fix an arbitrary demand instance d.Throughout the subsection, when some integer k appears ina statement, we are in fact referring to some i ∈ 811 0 0 0 1 n9congruent to k modulo n. For example, if plant n + 3appears in a statement, then we are referring to plant 3;and if fn+11n+2, the flow from plant n+ 1 to product n+ 2appears in a statement, then we are referring to f112, theflow from plant 1 to product 2.

First, we start with the following lemma.

Lemma 2. Suppose P4d1Cn\8�95 = P4d1Cn5, where � isa flexible arc in Cn. Then, for any set X ⊆ S, where S isthe set of all flexible arcs in Cn, we have that

P4d1Cn\4X ∪ 8�955= P4d1Cn\X50

Proof of Lemma 2. If � ∈ X, the result is trivial as X ∪

8�9=X. Otherwise, By Corollary 2,

P4d1Cn\4X ∪ 8�955+P4d1Cn5

¾ P4d1Cn\X5+P4d1Cn\8�95

⇒ P4d1Cn\4X ∪ 8�955¾ P4d1Cn\X51

since P4d1Cn5= P4d1Cn\8�950

But by definition of P4 · 5, P4d1Cn\4X ∪ 8�955 ¶P4d1Cn\X5, hence

P4d1Cn\4X ∪ 8�955= P4d1Cn\X50 �

Next, we show that the sales associated with Cn can beexpressed as a sum of n quantities, where each quantity isthe difference of the sales associated with two open chainsin Cn.

Theorem 3. For any fixed demand instance d on balancedsystem of size n, we have

P4d1Cn5=

n∑

i=1

(

P4d1Cn\84i1 i+ 1595

−P4d1Cn\84i− 11 i51 4i1 i51 4i1 i+ 1595)

0

Proof of Theorem 3. For each 1 ¶ k1 ¶ k2 ¶ n, defineLk1→k2

= 84i1 i5 � i = k11 k1 + 11 0 0 0 1 k29 ∪ 84i1 i + 15 � i =

k11 k1 + 11 0 0 0 1 k2 − 19, and for each 1 ¶ k2 < k1 ¶ n,define Lk1→k2

= 84i1 i5 � i = k11 k1 +11 0 0 0 1 n11121 0 0 0 1 k29∪84i1 i + 15 � i = k11 0 0 0 1 n1 0 0 0 1 k2 − 19. One can think ofLk1→k2

as the open chain connecting plant k1 to productk2 in the balanced system of size n. Also, because demandinstance d is fixed, for the sake of succinctness, we useP4A5 to denote P4d1A5. Finally, we define �i = 4i1 i+ 15and �i = 4i1 i5 for i = 1121 0 0 0 1 n (note that �n = 4n115 asn+ 1 is congruent with one modular n).

By definition of � and �, we can rewrite Cn\84i1 i+ 159and Cn\84i − 11 i5, 4i1 i5, 4i1 i + 159 as Cn\8�i9 and

Cn\8�i−11�i1�i9. For any 1¶i¶n, because Cn\8�i−11�i9= 8�i9 ]Cn\8�i−11�i1�i9, where ] represent the symbolfor disjoint union,

P4Cn\8�i95−P4Cn\8�i−11�i1�i95

= P4Cn\8�i95−P4Cn\8�i−11�i95+ min811 di90 (6)

Lemma 9 in Appendix B shows that there is some i∗

such that P4Cn5= P4Cn\8�i∗95. Without loss of generality,we assume that i∗ = n, because we can always relabel eachplant (and product) i by i− i∗. Now, we have that for i =21 0 0 0 1 n− 1,

P4Cn\8�i95−P4Cn\8�i−11�i1�i95

= P4Cn\8�i95−P4Cn\8�i−11�i95+ min811 di9

4by Equation (6)5

= P4Cn\8�i1�n95−P4Cn\8�i−11�i1�n95+ min811 di9

4by Lemma 250

Because Cn\8�i1�n9 = L1→i ] L4i+15→n, and Cn\8�i−11�i1�n9 = L1→4i−15 ] L4i+15→n ] 8�i9, we have for i =

21 0 0 0 1 n− 11

P4Cn\8�i95−P4Cn\8�i−11�i1�i95

= P4Cn\8�i1�n95−P4Cn\8�i−11�i1�n95+ min811 di9

=P4L1→i5+P4L4i+15→n5−4P4L1→4i−155+P4L4i+15→n5

+ min811 di95+ min811 di9

= P4L1→i5−P4L1→4i−1550 (7)

Also,

P4Cn\8�195−P4Cn\8�n1�11�195

= P4Cn\8�195−P4Cn\8�n1�195+ min811 d19

4by Equation (6)5

= P4Cn\8�11�n95−P4Cn\8�11�n95+ min811 d19

4by Lemma 25

= min811 d191 (8)

and

P4Cn\8�n95−P4Cn\8�n−11�n1�n95

= P4L1→n5−P4L1→4n−1550 (9)

Now, applying Equations (7)–(9), we obtain thatn∑

i=1

4P4Cn\8�i95−P4Cn\8�i−11�i1�i955

= min811 d19+

n∑

i=2

4P4L1→i5−P4L1→4i−1555

= min811 d19+P4L1→n5−P4L1→15

= P4L1→n5

= P4Cn\8�n95

= P4Cn50 �


We note that Cn\84i1 i+159 is an open chain connectingplant i+1 to product i, and Cn\84i−11 i51 4i1 i51 4i1 i+159is an open chain connecting plant i + 1 to product i − 1.Also, Theorem 3 can be extended to a more general setting.This extension of Theorem 3 is presented in Appendix A.

5.2. Characterization and Optimality of theLong Chain

With Theorem 3, we can now characterize the performanceof the long chain using the performances of open chains.

Theorem 4. For any balanced system of size n withexchangeable demand D, we have

6Cn7= n46Ln7− 6Ln−1750

Proof of Theorem 4. Theorem 3 states that for any d thatis an instance of D,

P4d1Cn5=

n∑

i=1

4P4d1Cn\84i1 i+ 1595

−P4d1Cn\84i− 11 i51 4i1 i51 4i1 i+ 159550 (10)

Because D is exchangeable, for any 1 ¶ i¶ n,

Ɛ6P4D1Cn\84i1 i+ 15957= 6Ln71

Ɛ6P4D1Cn\84i− 11 i51 4i1 i51 4i1 i+ 15957= 6Ln−170

Thus, integrating over all random instances of D onEquation (10), we have

6Cn7= n46Ln7− 6Ln−1750 �

Theorem 4 provides insights on the performance of longchains. Indeed, it relates the expected performance of along chain, 6Cn7 with the difference in the expected per-formances of two open chains, 6Ln7 and 6Ln−17, which aremuch easier to compute and analyze.

An immediate corollary of Theorem 4 is that the longchain is optimal among all 2-flexibility designs.

Corollary 3. Consider a balanced system of size n withexchangeable demand. Let �2 be the set of all 2-flexibilitydesigns of the system. That is, �2 is the set of all flexibilitydesigns where each plant node and each product node areincident to exactly two arcs. Then, we have

6Cn7= arg maxA∈�2

6A70

In words, the long chain maximizes expected sales amongall 2-flexibility designs in the system.

Proof of Corollary 3. Consider a 2-flexibility designA ∈ �2. Recall that a closed chain in A is an inducedsubgraph in A that forms an undirected cycle. Becauseevery node is incident to two arcs in A, A must be acollection of several closed chains, which we denote asSC11 SC21 0 0 0 1 SCk. Let ni be the number of products and

plants in the closed chain SCi. Because the system size isn,∑k

i=1 ni = n. Now, by Theorem 4, we have

6A7=k∑

i=1

ni46Lni7− 6Lni−175

=

k∑

i=1

ni46Lnni7− 6Ln

ni−17+ Ɛ6min811D1975

4by definition of Lnk5

¶k∑

i=1

ni46Lnn7− 6Ln

n−17+ Ɛ6min811D1975

4by Theorem 25

=

k∑

i=1

ni46Ln7− 6Ln−175

= n46Ln7− 6Ln−175= 6Cn70 �

6. Applications to Systems withi.i.d. Demand

In this section, we present three interesting applicationsof Theorem 4 under i.i.d. demand. The organization ofthis section is as follows. The first subsection describesan effective method of computing the performance of thelong chain; the second subsection studies the effectivenessof the long chains relative to that of full flexibility; andthe last subsection presents a risk-pooling result associatedwith long chains.

Throughout this section, we consider balanced systemswith i.i.d. demand. Because we consider systems of arbi-trary sizes, we let D be an infinite random vector with i.i.d.entries, where Di is the random demand for product i gen-erated by a given distribution D for all i¾ 1. Note that forany n¾ 2, Ɛ6D1Cn7 (also denoted by 6Cn7), only dependson the first n entries of the random vector D.

6.1. Computing the Performance ofthe Long Chain

This section presents a method to compute 6Cn7, the perfor-mance of the long chain, using matrix multiplication. First,we introduce Algorithm 1, a greedy algorithm that findsthe optimal solution of the linear program associated withP4d1Ln5, where d is an instance of D. Then, we applyAlgorithm 1 to develop an efficient procedure to computethe quantity 6Ln7 − 6Ln−17, which is equal to 6Cn7/n byTheorem 4.

Algorithm 1 (Finding optimal solution f∗ for P4d1Ln5)1. procedure Solve 4P4d1Ln552. f ∗

111 ← min411 d153. for k = 21 0 0 0 1 n do4. f ∗

k−11k ← min81 − f ∗k−11k−11 dk9

5. f ∗k1k ← min811 dk − f ∗

k−11k96. end for7. return f∗

8. end procedure.


We note that a similar greedy style algorithm for comput-ing the maximum sales in an open chain was also used byChou et al. (2010b). We omit the proof for the correctnessof Algorithm 1, because P4d1Lk5 is simply a max-flowproblem on a path and it is well known that this problemcan be solved using a greedy algorithm.

Given a random demand vector D, let Fij be the randomflow on arc 4i1 j5 returned by Algorithm 1, for 1 ¶ i1 j ¶ n.For each integer 1 ¶ k ¶ n − 1, define Wk = 1 − Fkk andW0 = 0. One can think of Wk as the remaining capacityin plant k after the production of product k at plant k isdetermined.

To develop a method to compute the performance of thelong chain, assume that the support of D lies in 8i/N �

i = 011121 0 0 0 1 9 for some N ¾ 1. Under this assumption,we let pi = �6D = i/N 7, for any i = 0111 0 0 0 12N − 1, andp2N = �6D ¾ 27, where �6 · 7 denotes the probability massfunction.

Because the support of D lies in 8i/N � i = 011121 0 0 09and 0 ¶ Fkk ¶ ∑

i Fik ¶ 1, we have that the support ofFkk lies in 8i/N � i = 011121 0 0 0 1N 9. But Wk = 1 −

Fkk, so Wk also have a support set of 8i/N � i = 011121 0 0 0 1N 9. As a result, the distribution of Wk can bedescribed by a row vector qk with N + 1 elements, whereqki =�6Wk = i/N 7, for i = 0111 0 0 0 1N . Then, we have

Lemma 3. qk+1 = qkA= q0Ak+1 for 0 ¶ k¶ n− 1, where

A=

2N∑

i=N

pi pN−1 pN−2 · · · p1 p0

2N∑

i=N+1

pi pN pN−1 · · · p2 p0 +p1

000000

000000

000000

p2N−1 +p2N p2N−2 p2N−3 · · · pN+1

N−1∑

i=0

pi

p2N p2N−1 p2N−2 · · · pN+1

N∑

i=0

pi

and q0= 61 0 0 · · ·070

Proof for Lemma 3. Because W0 is zero with probabilityone, q0 = 61 0 0 · · · 07. Because the demand is independentand Wk only depends on D11 0 0 0 1Dk, Wk is independent ofDk+1. Hence we have,

qk+1i =�6Wk+1 = i7=

N∑

j=0

�6Wk = j7�6Dk+1 =N − i+ j7

=

N∑

j=0

qkj pN−i+j1 for 1 ¶ i¶N − 11

qk+10 =�6Wk+1 = 07=

N∑

j=0

�6Wk = j7�6Dk+1 ¾N + j7

=

N∑

j=0

qkj

2N∑

l=N+j

pN+l1

and

qk+1N =�6Wk+1 =N7=

N∑

j=0

�6Wk = j7�6Dk+1 ¶ j7

=

N∑

j=0

qkj

j∑

l=0

pl0

This implies that qk+1 = qkA. �A direct consequence of Lemma 3 is that the following

matrix multiplications can be used to determine the per-formance of the long chain, when demands are i.i.d. andthe support of a product demand is a subset of 8i/N � i =

011121 0 0 0 1 9.

Theorem 5. 6Cn7/n= 6Ln7− 6Ln−17= qn−1Ï = q0An−1Ï,where Ï is a vector of size N + 1 and

�i =

N+i∑

j=1

jpj + 4N + i52N∑

j=N+i+1

pj1 ∀1 ¶ i¶N0

Proof for Theorem 5. By Algorithm 1, 6Ln7 − 6Ln−17can be written as the expectation of Fn−1n + Fnn, which isequal to Ɛ6min81 +Wn−11Dn97, thus

Ɛ6min81 +Wn−11Dn97

=

N∑

i=0

�6Wn−1 = i7Ɛ

[

min{

Dn1N + i

N

}]

=

N∑

i=0

qn−1i

( N+i∑

j=1

jpj + 4N + i52N∑

j=N+i+1

pj

)

0

Hence, we have that 6Ln7 − 6Ln−17 = qn−1Ï. ApplyTheorem 4, and we are done. �

The matrix multiplication method developed here tocompute the performance of the long chain is polyno-mial in N and n. Indeed, computing q0An−1Ï requiresO4nN 25 operations if one sequentially evaluates q0Ai fori = 11 0 0 0 1 k, or O4N 20807 logn5 operations if one startsby determining An−1 using the classical algorithm fromStrassen (1969). To the best of our knowledge, this is thefirst polynomial time algorithm that computes the perfor-mance of a finite size long chain exactly when demand isdiscrete and i.i.d. The other known algorithm to computethe performance of the long chain exactly is to solve themax-flow problem for all demand instances and sum themto determine the expected performance. Unfortunately, thismethod is exponential in n.

The matrix multiplication method can be applied forgeneral i.i.d. demands as an approximation algorithm tocompute the performance of long chains. In this case, onecan approximate the performance of the long chain bydiscretizing the demand distribution on the set of 8i/N � i =011121 0 0 0 1 9 for some integer N . Clearly, as N increases,the error of the approximation decreases while the runningtime grows. Specifically, it is straightforward to show thatthe error of the approximation is bounded by n/2N . How-ever, our computational experience suggests that the erroris much smaller than this bound.


Moreover, the matrix multiplication method is fairly fasteven for large N . For example, when N = 11000 and n =

100, q0An−1Ï can be computed within two seconds usingMatlab on a standard 2.1 GHz laptop. Hence, even for gen-eral i.i.d. demands, the matrix multiplication method canquickly approximate the performance of a large size longchain very accurately.

Figure 5 presents computational results obtained usingthe matrix multiplication method for three different i.i.d.demand distributions:

1. Normal. Demand for a product is a discretized nor-mal random variable with mean 1 and standard deviationof 0.33 on the support set of 8i/14 � i = 0111 0 0 0 1289;this distribution was originally applied in Chou et al.(2010b) for their analysis of asymptotic behavior of longchains;

2. Uniform. Demand for a product is uniformly dis-tributed on the set 8i/10 � i = 011121 0 0 0 191111121 0 0 0 1209;

3. Asymmetric. Demand for a product is equal to 45 with

probability 0.4, one with probability 0.5 and two with prob-ability 0.1.

For each distribution, Figure 5 depicts 6Fn7/n (the perproduct performance of full flexibility), 6Cn7/n (the perproduct performance of the long chain), and 6Cn7/6Fn7 (theratio between the performance of the long chain and theperformance of full flexibility design) for n= 11 0 0 0 130.

Figure 5. The performance of long chains vs. the performance of full flexibility.

10.80

0.85

0.90

0.95

1.00

Normal Uniform

3 5 7 9 11 13 15 17 19 21 23 25 27 29 10.70

0.75

0.80

0.85

0.90

0.95

1.00

3 5 7 9 11 13 15 17 19 21 23 25 27 29

10.84

0.86

0.90

0.88

0.92

0.94

0.96

0.98

1.00

3 5 7 9 11 13 15 17 19 21 23 25 27 29

Full flex

Asymmetric

Long chain

Long chain/Full flex

Figure 5 reveals several interesting observations. First,6Fn7/n − 6Cn7/n, i.e., the gap between the fill rates offull flexibility and the long chain, is increasing, whereasthe ratio, 6Cn7/6Fn7, is decreasing. A similar observationon the ratio, using simulation results, is reported in Chouet al. (2008). In addition, Figure 5 suggests that the quan-tity 6Cn7/n, the fill rate of the long chain, is increasing butconverges to a constant very quickly. These observationsare discussed in detail in the next two subsections.

Finally we discuss two important extensions of thematrix multiplication method: (i) computing the per prod-uct performance of the long chain for infinite size system,and (ii) dealing with nonidentical but independent demanddistributions.

Observe that the matrix A is the transition matrix ofa Markov chain with states i/N for each i = 0111 0 0 0 1N .It can be shown that in the matrix A, the communica-tion class that contains state 0 is irreducible and aperiodic.Then, by the Perron-Frobenius theorem (see Grimmett andStirzaker 1992) we have that limn→� q0An−1 = q∗, whereq∗A = q∗ and q∗

0 > 0. Thus, to compute limn→� 6Cn7/n,one can solve for q∗ by finding the eigenvectors of A, andthen compute q∗Ï, which equals to limn→� 6Cn7/n. Thisprovides another method for computing limn→� 6Cn7/n inaddition to the result of Chou et al. (2010b). Interestingly,their procedure also involves discretizing demand and solv-ing a system of linear equations.


Figure 6. Illustration of Wi4d5 and W̃i4d5.

00 1 2 3 4

Wi (d )

5 6 0 1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Wi (d )~

We also note that this matrix multiplication method canbe applied for long chain in balanced systems with inde-pendent but nonidentical product demands (and even non-identical plant capacities). In that case, the performance ofthe long chain, 6Cn7, can be no longer characterized using46Ln7− 6Ln−175. Instead, we have that 6Cn7 is equal to

n∑

i=1

([

Cn\84i1 i+ 159]

−[

Cn\84i− 11 i51 4i1 i51 4i1 i+ 159])

0

Similar to the multiplication procedures we described,for each i, 6Cn\84i1 i + 1597 − 6Cn\84i − 11 i51 4i1 i514i1 i+1597 can be evaluated by computing q0∏n−1

k=1 A4k5 forn− 1 N ×N matrices A415, A4251 0 0 0 1A4n− 15. Becausecomputing q0∏n−1

k=1 A4k5 requires O4nN 25 operations, com-puting the sum of 46Ln7 − 6Ln−175 for 1 ¶ i ¶ n requiresO4n2N 25 operations.

6.2. Long Chain vs. Full Flexibility

In Figure 5, it was observed that the gap between the fillrates of full flexibility and that of the long chain, 6Fn7/n−

6Cn7/n, is increasing, whereas the ratio, 6Cn7/6Fn7 isdecreasing. In this section, we will formally prove the firstpart of the observation, and discuss some partial resultsrelated to the second part. We start by defining two randomwalks in §6.2.1. These random walks are applied to analyzethe difference between the fill rates of the long chain andfull flexibility in §6.2.2, as well as the ratio of the fill rateof long chain to that of full flexibility in §6.2.3.

6.2.1. Random Walks. We define two random walks,Wi and W̃i, as follows:

Definition 3. Let W0 = W̃0 = 0. For i¾ 1, define

Wi =

0 if Wi−1 + 1 −Di−1 < 0

1 if Wi−1 + 1 −Di−1 > 1

Wi−1 + 1 −Di−1 otherwise1

W̃i =

{

0 if W̃i−1 + 1 −Di−1 < 0

W̃i−1 + 1 −Di−1 otherwise0

Wi and W̃i are generalized random walks with randomsteps 1 − D11 0 0 0 11 − Di and different sets of reflectingboundaries. The random walk Wi has reflecting bound-aries of zero and one, whereas W̃i has a reflecting bound-ary only at zero. For any fixed vector d that is aninstance of D, define Wi4d5 (and W̃i4d5) to be the instanceof Wi (and W̃i) corresponding to d. Figure 6 illustratesan example of Wi4d5 and W̃i4d5, with i = 6 and d =

6006100211021109110410057.The next lemma states several simple observations

regarding Wi4d5 and W̃i4d5 for any fixed vector d.

Lemma 4. For any fixed vector d and i¾ j ¾ 1,

Wi4d5¶ W̃i4d51 (11)

Wi−j4dj5¶Wi4d51 and (12)

W̃i−j4dj5¶ W̃i4d51 (13)

where dj = 6dj1 dj+11 0 0 07.

Proof of Lemma 4. Because W̃i has no reflecting bound-ary at one, Wi4d5¶ W̃i4d5. For Equation (12), observe thatWi−j4d

j5 has the same step lengths as the last i − j stepsof Wi4d5. Because Wj4d5, the position of the random walkWi4d5 after j steps, is greater or equal to zero, we havethat Wi−j4d

j5 ¶ Wi4d5. Similarly, we can also show thatW̃i−j4d

j5¶ W̃i4d5. �

The rest of this subsection establishes the connectionsbetween these random walks and sales (and performances)of the long chain and full flexibility. These connectionswould be used for the comparisons between the long chainand full flexibility in §6.2.2.


Similarly to §5.1, in the rest of this subsection, whensome integer k appears in a statement, we will in fact bereferring to some i ∈ 811 0 0 0 1 n9 congruent to k modulo n.First, we show the relationship between P4d1Cn5 and arandom walk on d.

Lemma 5. Let di �n = 6di1 di+11 0 0 0 1 dn1 d11 0 0 0 1 di−17, then

P4d1Cn5=

n∑

i=1

min81 +Wn−14di �n51di−190

Proof for Lemma 5. For each 1 ¶ i¶ n, it is not difficultto check that min81+Wn−14d

i �n51di−19 is equal to the quan-tity f ∗

n1n + f ∗n−11 n returned by Algorithm 1 on P4di �n1Ln5.

By Algorithm 1’s greedy property, f ∗n1n + f ∗

n−11 n =

P4di �n1Ln5 − P4di �n1Ln−15, which implies min81 +

Wn−14di �n51di−19 = P4di �n1Ln5 − P4di �n1Ln−15. Finally,

by definition, P4di �n1Ln5 = P4d1Cn\84i − 11 i595 andP4di �n1Ln−15 = P4d1Cn\84i − 21 i − 151 4i − 11 i − 1514i− 11 i595 and hence

min81 +Wn−14di �n51di−19

= P4d1Cn\84i− 11 i595

−P4d1Cn\84i− 21 i− 151

4i− 11 i− 151 4i− 11 i5950 (14)

Substitute Equation (14) to Theorem 3, and we have

P4d1Cn5=

n∑

i=1

min81 +Wn−14di �n51di−190 �

We note that a similar observation to Lemma 5was stated in Chou et al. (2010b). In particular, Chouet al. observed that min81 + Wn−14d51dn9 = P4d1Ln5 −

P4d1Ln−15. Our Lemma 5 goes a step further by applyingTheorem 3.

Establishing the relationships between P4d1Fn5 and arandom walk on d is more difficult. We do this by proving alemma that shows that the sales associated with Fn is equalto the sales of Cn under a new demand �4d5, which is alinear transformation of d. Specifically, we define �4di5 =

4di + 4n− 155/n, for i = 1121 0 0 0 1 n.

Lemma 6. For any demand instance d,

P4�4d51Cn5= P4�4d51Fn51

where �4di5= 4di + 4n− 155/n, for i = 1121 0 0 0 1 n.

Proof for Lemma 6. By duality of linear programs,

P4�4d51Cn5= min∑

1¶i¶n

pi +∑

1¶j¶n

qj�4dj5 4VC5

s.t. pi + qj ¾ 11 ∀ 4i1 j5 ∈Cn

pi¾01 qj ¾01 ∀1¶ i¶n1 1¶ j¶n

p1 q ∈�n0

The linear program denoted by (VC) is an LP-relaxationof a min-weight bipartite vertex cover problem. Becausethe LP-relaxation of min-weight bipartite vertex cover istight, it has an optimal solution 4p∗1q∗5, where entries inp∗ and q∗ are either zero or one. Let S = 8i � p∗

i = 09 andS ′ = 8j � q∗

j = 19. Note that N4S5 ⊆ S ′, where N4S5 is theset of neighbors of S in Cn. First, suppose S 6= �, and S ′ 6=

81121 0 0 0 1 n9. Then, we must have �S ′� − 1 ¾ �N4S5� − 1 ¾�S�. Let p0

i = 1, q0j = 0 for all 1 ¶ i1 j ¶ n. Clearly 4p01q05

is a feasible solution of (VC). Also as �S ′� − 1 ¾ �S�,

n¶ 4n− �S�5+ 4�S ′� − 15

< 4n− �S�5+∑

j∈S′

4n− 15n

4because �S ′�<n5

¶∑

1¶i¶n

p∗

i +∑

j∈S′

dj + 4n− 15

n4because dj ¾ 05

=∑

1¶i¶n

p∗

i +∑

1¶j¶n

�4dj5q∗

j 0

But∑

1¶i¶n p0i +

∑

1¶j¶n �4dj5q0j = n, and this contradicts

the optimality of 4p∗1q∗5. Thus, one must have that eitherS = � or S ′ = 81121 0 0 0 1 n9. Therefore, P4�4d51Cn5 =

min8n1∑

1¶j¶n �4dj59= P4�4d51Fn5. �Now, we can prove the lemma that establishes the rela-

tionship between P4d1Fn5 and W̃ .

Lemma 7. Let di �n = 6di1 di+11 0 0 0 1 dn1 d11 0 0 0 1 di−17, then

P4d1Fn5=

n∑

i=1

min81 + W̃n−14di �n51di−190

Proof of Lemma 7. By Lemma 5 and 6, P4�4d51Fn5 =

P4�4d51Cn5=∑n

i=1 min81+Wn−14�4di �n551 �4di−159. Note

that for any 1 ¶ j ¶ n− 1, and 1 ¶ i¶ n,

Wj4�4di �n55¶

∑

i¶k¶i+j

max{

011 −dk + 4n− 15

n

}

¶ 4n− 15 ·1n< 10

This implies that Wj4�4di �n55 never touches the reflecting

boundary at one. Hence,

Wn−14�4di �n55= W̃n−14�4d

i �n55 ∀1 ¶ i¶ n0 (15)

Because 1 − �4dk5= 41/n541 −dk5, for any 1 ¶ k¶ n, wehave

W̃j4�4di �n55=

1nW̃j4d

i �n5 ∀1 ¶ j ¶ n− 10 (16)

Thus,

P4�4d51Fn5=

n∑

i=1

min81 +Wn−14�4di �n551 �4di−159

=

n∑

i=1

min81 + W̃n−14�4di �n551 �4di−159

4by Equation (15)5


=

n∑

i=1

min{

1 +W̃n−14d

i �n5

n1di−1 + n− 1

n

}

4by Equation (16)5

=

n∑

i=1

(

n− 1n

+ min{

1 + W̃n−14di �n5

n1di−1

n

})

= n− 1 +1n

n∑

i=1

min81 + W̃n−14di �n51di−190

On the other hand,

P4�4d51Fn5= min{

n1∑

1¶i¶n

di + n− 1n

}

= n− 1 +1n

min{

n1∑

1¶i¶n

di

}

= n− 1 +1nP4d1Fn50

Therefore, we have that

n− 1 +1n

n∑

i=1

min81 + W̃n−14di �n51di−195

= n− 1 +1nP4d1Fn51

which implies

P4d1Fn5=

n∑

i=1

min81 + W̃n−14di �n51di−1950 �

Integrating the equations in Lemma 5 and Lemma 7 overall instances in D, we obtain the Lemma 8, which relatesthe expectation of the two random walks with the perfor-mance of the long chain and that of full flexibility.

Lemma 8. Under i.i.d. demand, we have

6Cn7

n= Ɛ6min81 +Wn−14D51D97

6Fn7

n= Ɛ6min81 + W̃n−14D51D970

Proof of Lemma 8. Because D is i.i.d., we have that forany 1 ¶ i¶ n,

Ɛ6min81 +Wn−14Di �n51Di−197= Ɛ6min81 +Wn−14D51D97

Ɛ6min81 + W̃n−14Di �n51Di−197= Ɛ6min81 + W̃n−14D51D970

After integrating the equations in Lemma 5 and Lemma 7over D, we get

6Cn7= nƐ6min81 +Wn−14D51D97

6Fn7= nƐ6min81 + W̃n−14D51D970 �

6.2.2. Difference in Fill Rates. With Lemma 8 at hand,we now prove that the quantity 6Fn7/n− 6Cn7/n is nonde-creasing with n.

Theorem 6. For any integer n¾ 2 and i.i.d. demand,

6Fn7

n−

6Cn7

n¶ 6Fn+17

n+ 1−

6Cn+17

n+ 1¶ min811Ɛ6D79−�1

where � = limk→� 6Ck7/k.

Proof of Theorem 6. First, we show that for any demandinstance d of D, we have,

min81 + W̃n−14d251dn+19− min81 +Wn−14d

251dn+19

¶ min81 + W̃n4d51dn+19− min81 +Wn4d51dn+191 (17)

where d2 = 6d21 d31 0 0 07. One can think of Wn−14d25 (and

W̃n−14d25) as the walk that started one time unit later than

Wn4d5 (and W̃n4d5). To prove Inequality (17), consider thefollowing two cases.

Case 1. Wi4d25 = 1 for some 1 ¶ i ¶ n − 1. Then

Wi4d25=Wi+14d5= 1, and by the definition of W , we must

have that Wn−14d25 = Wn4d5. By Lemma 4, W̃n−14d

25 ¶W̃n4d5, and therefore we have

min81 +Wn−14d251dn+19= min81 +Wn4d51dn+19

min81 + W̃n−14d251dn+19¶ min81 + W̃n4d51dn+191

which implies that Inequality (17) holds.Case 2. Wi4d

25 < 1 for all 1 ¶ i¶ n−1. By definition ofW and W̃ , we have that Wn−14d

25= W̃n−14d25. By Lemma

4, Wn4d5¶ W̃n4d5, and therefore we have

min81 +Wn−14d251dn+19= min81 + W̃n−14d51dn+19

min81 +Wn4d51dn+19¶ min81 + W̃n4d51dn+191

which again implies that Inequality (17) holds.Because demand is i.i.d., after integrating over D for both

sides of Inequality (17), we get

Ɛ6min81 + W̃n−14D51Dn97− Ɛ6min81 +Wn−14D51Dn97

¶Ɛ6min81+W̃n4D51Dn+197−Ɛ6min81+Wn4D51Dn+1970

Applying Lemma 8, we have that

6Fn7

n−

6Cn7

n¶ 6Fn+17

n+ 1−

6Cn+17

n+ 11 for all n¾ 20 (18)

Finally, Equation (18) implies that

6Fn7

n−

6Cn7

n¶ lim

k→�

(

6Fk7

k−

6Ck7

k

)

= limk→�

(

Ɛ

[

min{∑k

i=1 Di

k11}])

−�

= min8Ɛ6D7119−�1


where the last equality holds because of the weak law oflarge numbers. �

Note that the fill rate of Cn (and Fn) is equal to6Cn7/nƐ6D7 (and 6Fn7/nƐ6D7). Thus, Theorem 6 impliesthat the smaller the system size, the smaller the gapbetween the fill rate of full flexibility and that of the longchain. This suggests that the long chain is more effectiverelative to full flexibility for smaller size systems. More-over, Theorem 6 can be used to bound the gap betweenthe fill rate of full flexibility and that of the long chain forsystems of any size. For this purpose, we point out thatChou et al. (2010b) shows that for many i.i.d. demand withƐ6D7= 1, � is close to one, implying that for any size sys-tem, the performance of the long chain is close to that offull flexibility.

For example, when D is normal with mean 1 and stan-dard deviation of 0033. Chou et al. (2010b) shows that inthis case � = 0096. Therefore, for this demand distribution,we have that the gap between the fill rate of full flexibil-ity and that of the long chain for systems of any size is atmost 4%.

6.2.3. Performance Ratio. The difference between thefill rate of the long chain and that of full flexibility is onlyone metric to evaluate the effectiveness of the long chain.A different metric, discussed in Chou et al. (2008, 2010b),is to consider the ratio of the performance of long chain tothat of full flexibility. Partial results related to the ratio arediscussed next.

Applying the same argument as in the proof of Theo-rem 6, one can show that

min81 +Wn−14d251dn+19

min81 + W̃n−14d251dn+19¾ min81 +Wn4d51dn+19

min81 + W̃n4d51dn+190

Unfortunately, one cannot integrate this inequality over Dto obtain 6Cn7/6Fn7¾ 6Cn+17/6Fn+17, as expectation doesnot preserve over multiplication and division. Indeed, itis not known whether 6Cn7/6Fn7 ¾ 6Cn+17/6Fn+17 holds,though this inequality has been observed empirically inChou et al. (2008) and in this paper (see Figure 5). Notethat 6Cn7/6Fn7¾ 6Cn+17/6Fn+17 for all n¾ 2 implies The-orem 6 and hence is a stronger statement.

Of course, if 6Cn7/6Fn7 ¾ 6Cn+17/6Fn+17 holds, then itfollows that

6Cn7

6Fn7¾ lim

k→�

6Ck7/k

6Fk7/k= lim

k→�

�

min8Ɛ6D71191 ∀n¾ 20 (19)

Where as before � = limk→� 6Ck7/k, and limk→� 6Fk7/k =

min8Ɛ6D7119 by the weak law of large numbers. Thiswould provide a lower bound on the ratio of the perfor-mance of the long chain to that of full flexibility for anysystem size.

Again, using the example from Chou et al. (2010b), � =

0096 when D is normal with mean 1 and standard deviationof 0033. Thus, if Inequality (19) holds, it would indicate

that the performance of the long chain is at least 96% ofthat of full flexibility for any size system.

Although we do not have a proof for 6Cn7/6Fn7 ¾ �when Ɛ6D7 = 1, we provide a lower bound for 6Cn7/6Fn7that is almost equal to �.

Corollary 4. Suppose demand is i.i.d. and Ɛ6D7 = 1,then

6Cn7

6Fn7¾ 1 −

41 −�5n

6Fn71

where � = limk→� 6Ck7/k.

To explain the power of the lower bound in Corollary4, let �n = n/6Fn7 − 1, which implies that 1 − 41 −�5n/6Fn7= � −�n41 −�5. It can be shown that n/6Fn7 is non-increasing with n (by applying for example Lemma 8),and hence, �n is nonincreasing. Thus, if �k ≈ 0 for somesmall integer k, then Corollary 4 provides a lower boundfor 6Cn7/6Fn7 that is close to � for all n ¾ k. Indeed,for many distributions with Ɛ6D7 = 1, �k ≈ 0 for small k.For example, suppose the distribution of D is normal withmean 1 and standard deviation 0.33, then 3/6F37 = 1008,which implies �3 = 0008 and Chou et al. (2010b) showsthat � = 0096. Applying Corollary 4, we have that

6Cn7

6Fn7¾ � − �341 −�5= 0096 − 0004 × 0008 = 0095681

∀n¾ 30

That is, when demand is normal with mean 1 and stan-dard deviation 0.33, the long chain of any size greater thantwo achieves at least 95.68% of the performance of fullflexibility. Next, we provide a proof for Corollary 4.

Proof for Corollary 4. By Theorem 6,

6Fi7

i−

6Ci7

i¶ 6Fi+17

i+ 1−

6Ci+17

i+ 1∀ i¾ 1 (20)

⇒6Fi7

i−

6Fi+17

i+ 1¶ 6Ci7

i−

6Ci+17

i+ 1∀ i¾ 10 (21)

Now, add Inequality (21) for all i¾ n, we have that

6Fn7

n− lim

k→�

6Fk7

k¶ 6Cn7

n− lim

k→�

6Ck7

k0 (22)

But

limk→�

6Fk7

k= 11 lim

k→�

6Ck7

k= �1

and substituting those into Inequality (22), we have

6Fn7

n− 1 ¶ 6Cn7

n−�

⇒6Cn7

6Fn7¾ 1 −

41 −�5n

6Fn70 �


6.3. Risk Pooling of the Long Chain

In this subsection we focus on the per product performance(and fill rate, which is linearly proportional to the per prod-uct performance) of the long chain as a function of systemsize. We start by showing that 6Cn7/n is nondecreasingwith n under i.i.d. demand.

Theorem 7. Under i.i.d. demand D, we have 6Cn7/n ¶6Cn+17/n+ 1, for any integer n¾ 2.

Proof of Theorem 7. Because D is i.i.d., the first n (andn + 1) entries in D is exchangeable for balanced systemof size n (and n + 1). Thus, by Theorem 2, we havethat 6Ln

n7 − 6Lnn−17 ¶ 6Ln+1

n+17 − 6Ln+1n 7, which is equiva-

lent to 6Ln7 − 6Ln−17 − Ɛ6min8D1197 ¶ 6Ln+17 − 6Ln7 −

Ɛ6min8D1197 and hence implies 6Ln7− 6Ln−17¶ 6Ln+17−6Ln7. Apply Theorem 4 completes the proof.

The theorem thus states that 6Cn7/n, as well as the fillrate associated with a long chain, increases with the num-ber of products, n. This phenomenon is analogous to theclassical risk-pooling effect associated with demand aggre-gation, except that here we aggregate across capacities.

Interestingly, Figure 5 suggests that the fill rate of thelong chain quickly converges to a constant. This is shown inthe next theorem, where we prove that the convergence rateis exponential for arbitrary i.i.d., nondegenerate demands.

Theorem 8. When demands are i.i.d. and nondegenerate,there exist constants c < 0 and K > 0 such that

6Cn+17

n+ 1−

6Cn7

n¶Kecn1

for any n¾ 2.

Proof of Theorem 8. From Lemma 8 we have, 6Cn7/n=

Ɛ6min81 + Wn−14D51Dn97. Recall that D2 = 6D21D31 0 0 07.We have,

6Cn+17

n+ 1−

6Cn7

n

=Ɛ6min81+Wn4D51Dn+197−Ɛ6min81+Wn−14D251Dn+197

= Ɛ6min81 +Wn4D51Dn+19− min81 +Wn−14D251Dn+197

¶�6Wn4D5 6=Wn−14D2571

where the last inequality is true because min81 +

Wn4D51Dn+19 − min81 + Wn−14D251Dn+19 never exceeds

one. Note that for any particular random instance d,Wn4d5 = Wn−14d

25 if Wi4d5 = 0 for some 1 ¶ i ¶ n orWi4d

25= 1 for some 1 ¶ i¶ n− 1. Thus,

�6Wn4D5 6=Wn−14D257

¶�6Wi4D5 > 01 Wi4D25 < 11 ∀1 ¶ i¶ n70

Therefore, we have

6Cn+17

n+ 1−

6Cn7

n¶�6Wi4D5 > 01Wi4D

25 < 11 ∀1 ¶ i¶ n70

Now, because D is nondegenerate and i.i.d., there existssome t such that p =�6

∑tj=14Dj − 15¾ 17 > 0. If some

instance d satisfies the condition Wi4d5 > 0, Wi4d25 < 1,

∀1 ¶ i¶ n, then we must have that∑kt+1

j=2+4k−15t4dj −15 < 1for any 1 ¶ k¶ �4n− 15/t�. Hence,

6Cn+17

n+ 1−

6Cn7

n

¶�6Wi4D5 > 01Wi4D25 < 11 ∀1 ¶ i¶ n7

¶�

[ kt+1∑

j=2+4k−15t

4Dj − 15 < 11 ∀1 ¶ k¶⌊

n− 1t

⌋]

= 41 −p5�4n−15/t�

¶Kecn1

for some constants K > 0 and c < 0. �Figure 5 and Theorem 8 show that 6Cn7

n≈ 6Cn+t7/4n+ t5

for any t provided that n is large. Hence, it implies thatin a system with a large number of plants and products, itis not necessary to have a long chain design that connectsall the plants and products. A collection of several chains,each of which with a large number of plants and productscan be as effective.

Finally, we note that Theorem 8 can be applied to showthat 6Cn7/6Fn7¾ 6Cn+17/6Fn+17 when n is large and D hasmean 1 and finite variance. To see this, one needs to applythe central limit theorem to show that 6Fn+17/4n+ 15 −

6Fn7/n¾K141/√n−1/

√n+ 15 for some constant K1 > 0.

Then, we have that

6Cn+17

6Fn+17=

6Cn+17

n+ 1

/

6Fn+17

n+ 1

¶(

6Cn7

n+Kecn

)/(

6Fn7

n+K1

(

1√n

−1

√n+1

))

because Kecn �K1

(

1/√n− 1/

√n+ 1

)

for large n,

¶(

6Cn7

n

)/(

6Fn7

n

)

for large n,

=6Cn7

6Fn7for large n.

7. ConclusionThis paper presents two theoretical results that complementexisting literature on long chain flexibility designs. The firstresult, the supermodularity of flexible arcs, provides the-oretical justification to the idea of “closing the chain,” aconcept that was empirically observed by authors such asHopp et al. (2004) and Graves (2008). The supermodularityproperty reveals that flexible arcs in the long chain comple-ment each other, and this provides some explanation for thepower of the long chain. In addition, the supermodularityproperty motivates the second key result of the paper: thecharacterization of the performance of the long chain as thedifference between the performances of two open chains.


This characterization of the performance of the longchain leads to four important developments: First, we showthat the long chain is optimal among all 2-flexibility sys-tems, a property of the long chain that has been widelyheld but, to the best of our knowledge, never proven before.Second, it allows the development of a simple matrix mul-tiplication algorithm to determine the performance of thelong chain. Third, it leads to a proof that the differencebetween the fill rate of full flexibility design and the fill rateof the long chain is increasing with the number of prod-ucts. This suggests that, relative to full flexibility design,the long chain is more effective for smaller size systems.Finally, it motivates a risk-pooling result where the fill rateof a long chain increases with system size. This increasein fill rate, however, converges to zero exponentially fast.Thus, our analysis suggests that although the long chain isthe optimal 2-flexibility design, a design with several closedchains, where each chain connects a substantial number ofplants and products, performs just as well as the long chain.

Some of the results and techniques developed in thepaper can be used to study a wider range of process flexi-bility systems. For example, in Appendix A, we show thatthe characterization of a long chain (Theorem 4) can beextended to a more general setting where (i) plants havedifferent capacities; (ii) sales is replaced by linear profitthat depends on a plant-product combination; (iii) eachplant-product combination has a given capacity limit; and(iv) demand follows a general random distribution.

It is appropriate to point out an important limitation ofour model: we focus on a balanced system where the num-ber of plants equals the number of products. Many real-world systems do not satisfy this assumption, and typically,the number of products is much larger than the number ofplants. One way to address this limitation is to apply theclustering method from Jordan and Graves (1995) to cre-ate an approximately balanced system and then follow theinsights and guidelines developed here. Thus, the designprinciples emphasized in this paper are also important forunbalanced systems as well. This includes the importanceof closing the chain, and that a design with several closedchains, where each chain connects a substantial number ofplants and products, performs well for large size systems.

Appendix A. Extension for the Characterization

In this section, we present an extension to Theorem 3 in a moregeneral model where we add different capacities across the plant,different profits, and capacity constraint for each plant to productlink. We start off the section by defining P4p1 c1d1u1A5, for anyflexible design A, and nonnegative vectors p, c, d, and u:

P4p1 c1d1u1A5= max∑

1¶i1 j¶n

pijfij

s.t.n∑

i=1

fij ¶ dj 1 ∀1 ¶ j ¶ n

n∑

j=1

fij ¶ ci1 ∀1 ¶ i¶ n

0 ¶ fij ¶ uij 1 ∀ 4i1 j5 ∈A

fij = 01 ∀ 4i1 j5yA

f ∈�n20

One can interpret P4p1 c1d1u1A5 as the maximum profitachieved by flexibility design A, given demand d, plants capacityvector c, linear profit vector p, and a flexibility capacity vector u(that is, the production for each arc 4i1 j5 in A is bounded by uij ).Also, we assume that for each 1 ¶ i ¶ n, pii ¾ pji for all j , anduii = ci. This assumption is intuitive in practice, as plant i can bethought of as primary production plant for product i.

Now, we state Corollary 5, an extension of Theorem 3.

Corollary 5. Let �k = 4k1 k + 15, �k = 4k1 k5 for any integerk = 1121 0 0 0 1 n. For nonnegative vectors p ∈�n2

, c ∈�n, d ∈�n,and u ∈�n2

, if p�k¾ p�k

and ukk = ck ∀1 ¶ k¶ n, we have

P4p1 c1d1u1Cn5=

n∑

i=1

4P4p1 c1d1u1Cn\8�i95

−P4p1 c1d1u1Cn\8�i−11�i1�i9550

Proof for Corollary 5. Similar to §4.1, P4p1 c1d1u1A5 isequivalent to a max-weight circulation problem. Thus we againapply Gale and Politof (1981) and have

P4p1 c1d1A5+P4p1 c1d1A\8�t1�t′95

¾ P4p1 c1d1A\8�t95+P4p1 c1d1A\8�t′951 (A1)

for any A ⊂ Cn and any 1 ¶ t1 t′ ¶ n. With Equation (A1), wecan prove an extension of Lemma 2 in §5.1, which states that

P4p1 c1d1Cn\4X ∪ 8�t955= P4p1 c1d1Cn\X5 (A2)

for any X ∈ 8�11�21 0 0 0 1�n9 and any 1 ¶ t ¶ n.Because p�k

¾ p�kand ukk = ck ∀1 ¶ k ¶ n, we can

apply Lemma 10 in Appendix B and have that P4p1 c1d1u1Cn\8�i∗95= P4p1 c1d1u1Cn5, for some 1 ¶ i∗ ¶ n.

With these, we have established all the results used for provingTheorem 3 in this extended model. The rest of the proof can becompleted following the proof of Theorem 3. �

Given any random demand distribution D, we can integrate theequation presented in Corollary 5 over all instances of D. Thisallows us to obtain a characterization for the expected profit of along chain, which we present next.

Corollary 6. Let �k = 4k1 k + 15, �k = 4k1 k5 for any integerk = 1121 0 0 0 1 n. For nonnegative vectors p ∈�n2

, c ∈�n, d ∈�n,and u ∈�n2

, if p�k¾ p�k

and ukk = ck ∀1 ¶ k¶ n, we have

Ɛ6P4p1 c1D1u1Cn57

=

n∑

i=1

4Ɛ6P4p1 c1D1u1Cn\8�i957

− Ɛ6P4p1 c1D1u1Cn\8�i−11�i1�i95750


Appendix B. Characteristics of theOptimal Solutions to P 4d1Cn5

We prove that with a fixed demand instance d, there is alwaysan optimal solution for P4d1Cn5 that does not use all the flexiblearcs in Cn.

Lemma 9. Let �i = 4i1 i+15 for i = 11 0 0 0 1 n−1, and �n = 4n115.Then, there exists some 1 ¶ i∗ ¶ n such that P4d1Cn\8�i∗95 =

P4d1Cn5, for any demand instance d.

Proof of Lemma 9. Recall that

P4d1Cn5= max∑

1¶i1 j¶n

fij

s.t.n∑

i=1

fij ¶ dj 1 ∀1 ¶ j ¶ n

n∑

j=1

fij ¶ 11 ∀1 ¶ i¶ n

fij ¾ 01 ∀ 4i1 j5 ∈Cn

fij = 01 ∀ 4i1 j5yCn

f ∈�n20

Thus, the optimization problem associated with P4d1Cn5 islinear, bounded, and feasible, which implies that it must have anoptimal solution. Let f∗ be an optimal solution of P4d1Cn5. Iff ∗ij = 0 for some 4i1 j5 ∈ 8�11 0 0 0 1�n9, then there is some i∗ such

that f ∗�i∗

= 0 and this implies that P4d1Cn\8�i∗95= P4d1Cn5.Otherwise, f ∗

ij > 0 for all 4i1 j5 ∈ 8�11 0 0 0 1�n9. Let g be thevector that

gij =

−1 if 4i1 j5 ∈ 8�11 0 0 0 1�n9

1 if 4i1 j5 ∈ 8411151 421251 0 0 0 1 4n1n59

0 otherwise.

In network flow theory, g is known as an augmenting cycle forf∗. Let �∗ = min8fij 1 4i1 j5 ∈ 8�11 0 0 0 1�n99. Note that f∗ + �∗gis a feasible and optimal solution of P4d1Cn5. Moreover, f ∗

ij +

�∗gij = 0 for some 4i1 j5 ∈ 8�11 0 0 0 1�n9. Thus, there is some i∗

such that f ∗�i∗

+�∗g�i∗ = 0 and this implies that P4d1Cn\8�i∗95=

P4d1Cn5. �Lemma 10. Let �i = 4i1 i + 15 for i = 11 0 0 0 1 n − 1, and �n =

4n115. For nonnegative vectors p ∈ �n2, c ∈ �n, d ∈ �n, and

u ∈ �n2, if

∑ni=1 p�i

¶ ∑ni=1 pii, and ukk = ck ∀1 ¶ k ¶ n,

there exists some 1 ¶ i∗ ¶ n such that P4p1 c1d1u1Cn\8�i∗95 =

P4p1 c1d1u1Cn5.

Proof for Lemma 10. From the assumption on p,∑

i1j pijgij ¾ 0,where g is the augmenting cycle vector defined in the proof ofLemma 9. Thus, we can follow the proof from Lemma 9 to proveLemma 10. Let �∗ = min8fij 1 4i1 j5 ∈ 8�11 0 0 0 1�n99. Then again,f∗ + �∗g is a feasible and optimal solution of P4p1 c1d1u1Cn5and f ∗

ij + �∗gij = 0 for some 4i1 j5 ∈ 8�11 0 0 0 1�n9. This impliesthat P4p1 c1d1u1Cn\8�i∗95= P4p1 c1d1u1Cn5. �

AcknowledgmentsThe authors thank the anonymous referees and the associate edi-tor for their suggestions, which have greatly improved the qualityof the paper. This research was supported by the National Sci-ence Foundation [Contract CMMI-0758069], Masdar Institute ofScience and Technology, the Ford-MIT Alliance, and the NaturalSciences and Engineering Research Council of Canada Postgrad-uate Scholarship.

ReferencesAksin OZ, Karaesmen F (2007) Characterizing the performance of process

flexibility structures. Oper. Res. Lett. 35(4):477–484.Bassamboo A, Randhawa RS, Van Mieghem JA (2010) Optimal flexibility

configurations in newsvendor networks: Going beyond chaining andpairing. Management Sci. 56(8):1285–1303.

Bish EK, Muriel A, Biller S (2005) Managing flexible capacity in a make-to-order environment. Management Sci. 51(2):167–180.

Boudette NE (2006) Chrysler gains edge by giving new flexibility to itsfactories. Wall Street J. (April 11).

Buzacott J, Mandelbaum M (2008) Flexibility in manufacturing and ser-vices: Achievements, insights, and challenges. Flexible Services andManufacturing J. 20(1):13–58.

Chou M, Teo C-P, Zheng H (2008) Process flexibility: Design, evaluation,and applications. Flexible Services and Manufacturing J. 20(1):59–94.

Chou M, Teo C-P, Zheng H (2011) Processs flexibility revisited: The graphexpander and its applications. Oper. Res. 59(5):1090–1105.

Chou MC, Chua GA, Teo C-P (2010a) On range and response: Dimen-sions of process flexibility. Eur. J. Oper. Res. 207(2):711–724.

Chou MC, Chua GA, Teo C-P, Zheng H (2010b) Design for process flex-ibility: Efficiency of the long chain and sparse structure. Oper. Res.58(1):43–58.

Gale D, Politof T (1981) Substitutes and complements in network flowproblems. Discrete Appl. Math. 3(3):175–186.

Granot F, Veinott AF Jr (1985) Substitutes, complements, and ripples innetwork flows. Math. Oper. Res. 10(3):471–497.

Graves SC (2008) Flexibility principles. Building Intuition: Insights fromBasic Operations Management Models and Principles, Chap. 3(Springer, New York), 33–51.

Graves SC, Tomlin BT (2003) Process flexibility in supply chains. Man-agement Sci. 49(7):907–919.

Grimmett GR, Stirzaker DR (1992) Probability and Random Process,2nd ed. (Oxford University Press, New York).

Gurumurthi S, Benjaafar S (2004) Modeling and analysis of flexiblequeueing systems. Naval Res. Logist. 51(5):755–782.

Hopp WJ, Tekin E, Van Oyen MP (2004) Benefits of skill chaining inserial production lines with cross-trained workers. Management Sci.50(1):83–98.

Iravani SM, Van Oyen MP, Sims KT (2005) Structural flexibility: A newperspective on the design of manufacturing and service operations.Management Sci. 51(2):151–166.

Jordan WC, Graves SC (1995) Principles on the benefits of manufacturingprocess flexibility. Management Sci. 41(4):577–594.

Mak H-Y, Shen Z-J (2009) Stochastic programming approach to processflexibility design. Flexible Services and Manufacturing J. 21(3):75–91.

Muriel A, Somasundaram A, Zhang Y (2006) Impact of partial manu-facturing flexibility on production variability. Manufacturing ServiceOper. Management 8(2):192–205.

Sethi AK, Sethi SP (1990) Flexibility in manufacturing: A survey. Inter-nat. J. Flexible Manufacturing Systems 2(4):289–328.

Sheikhzadeh M, Benjaafar S, Gupta D (1998) Machine sharing in manu-facturing systems: Total flexibility versus chaining. Internat. J. Flex-ible Manufacturing Systems 10(4):351–378.

Simchi-Levi D (2010) Operations Rules: Delivering Customer Valuethrough Flexible Operations (MIT Press, Cambridge, MA).

Strassen V (1969) Gaussian elimination is not optimal. Numerische Math-ematik 13(4):354–356.

Wallace RB, Whitt W (2005) A staffing algorithm for call centerswith skill-based routing. Manufacturing Service Oper. Management7(4):276–294.

David Simchi-Levi is a professor of engineering systems atMassachusetts Institute of Technology. The work described in thispaper is part of a larger research project that deals with effec-tive supply chain and procurement strategies that improve supplychain performance.

Yehua Wei is a Ph.D. candidate at the Operations ResearchCenter of the Massachusetts Institute of Technology. His primaryresearch interest lies in the field of supply chain management.Currently, he is studying supply chain strategies that enable firmsto cope with uncertainties in demand and supply.

Documents

Understanding the Performance of the Long Chain and Sparse ... · Understanding the Performance of the Long Chain and Sparse Designs in Process Flexibility ... The long chain has