continuum modeling of supply chain networks using ...design is to model and evaluate the performance...

Available online at www.sciencedirect.com

www.elsevier.com/locate/cma

Comput. Methods Appl. Mech. Engrg. 197 (2008) 1204–1218

Continuum modeling of supply chain networks usingdiscontinuous Galerkin methods

Shuyu Sun a,*, Ming Dong b

a Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USAb Department of Industrial Engineering and Management, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, PR China

Received 28 August 2007; accepted 23 October 2007Available online 4 November 2007

Abstract

Using a connectivity matrix, we establish a continuum modeling approach with partial differential equations of conservation laws forsimulating materials flow in supply chain networks. A number of existing and new constitutive relationships for modeling velocity aresummarized or proposed. To effectively treat strong advection components within the modeling system, we apply discontinuous Galerkin(DG) methods for solving production flow in a supply chain network. In addition, a number of DG properties are analyzed for treatingnetwork flow. In particular, a nearly optimal error estimate is obtained using a new estimating technique that utilizes two physical mean-ingful assumptions on the connectivity matrix. Numerical examples are provided to simulate a single node, a serial supply chain and anentire network as well as to investigate the influence of influx variation and node shut-down to the profiles of work in progress (WIP) andoutflux. It is shown that the proposed modeling approach is applicable to a large number of scenarios including re-entrant lines and theproposed DG algorithm is robust and accurate for predicting WIP and outflux behaviors.� 2007 Elsevier B.V. All rights reserved.

Keywords: Supply chain network; Re-entrant line; Connectivity matrix; Discontinuous Galerkin method; Conservation law; Continuum modeling

1. Introduction

A supply chain can be viewed as a network of suppliers,manufacturing sites, distribution centers, and customerlocations, through which components and products flow.A node in a network can be a physical location, a sub-network, or just an operation process, while links representmaterials (components or products) flow. These networksfind significant applications in manufacturing and logisticsin many fields, such as electronic and automobile industries[15]. A central problem in integrated supply chain networkdesign is to model and evaluate the performance of supplychains. The problem becomes more challenging because ofthe dynamic nature of the supply chains: prolific productvariety, short lifetime products, frequent new product intro-duction, non-stationary customer demand, and frequently

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.10.012

* Corresponding author.E-mail address: [email protected] (S. Sun).

changing service-level requirements. This dynamic natureof complex supply chains causes the models change overtime. In turn, the performance of supply chains must becontinually reevaluated.

Much progress has been made to characterize thedynamics of supply chains. Discrete event and agent basedmodels are routinely developed to study the dynamics offlows through such networks [13,14]. However, throughoutthese networks, there are different sources of uncertain-ties, including supply (availability and quality), process(machine break-down, operator variation), and demand(arrival time and volume). Moreover, these variations willpropagate from upstream to downstream stages. Theseuncertainties degrade the performances of a network caus-ing, for example, longer cycle times and lower fill-rates.Their effective allocation and control impose a great chal-lenge to the managers of supply chains. Performance mod-eling and analysis become increasingly more important butdifficult in the management of such complex supply chains.

mailto:[email protected]

S. Sun, M. Dong / Comput. Methods Appl. Mech. Engrg. 197 (2008) 1204–1218 1205

The traditional modeling and analysis methods such as dis-crete event simulations are prohibitively expensive to main-tain and are not equipped well to answer questions on thebehavior of the networks as a whole. In particular, discreteevent simulators have increasing numerical complexity asthe number of simulated parts increases, leading to compu-tationally difficult or even intractable tasks for simulatinghigh-volume, multistage production flow. Therefore, a con-tinuum modeling approach to be used for modeling andperformance analysis is in need. Furthermore, to answer‘‘what-if’’ questions quickly, such a continuum model hasto be computed efficiently.

Classical continuous models for supply chains (see e.g.,[2]) use rate equations to describe the queueing and flowprocess in the system macroscopically. These continuousapproaches can be solved efficiently by numerical computa-tion, even though they are less exact than discrete eventsimulations. They are computational scalable with respectto the number of parts; in other words, their computationalcomplexity does not depend on the number of parts to beprocessed. Borrowing techniques from gas dynamics, acontinuum modeling approach was established in [3,5] withcompromise made between rate equations and discreteevent simulations. This allows scalable and efficient compu-tation while being capable of providing more informationthan simple rate equations. Similarly, continuous produc-tion flow through a re-entrant factory was modeled usinga continuum model in [4], where a conservation law wasdeveloped for a continuous density variable and a stateequation was assumed for the speed of the flow along theproduction line, allowing fast and accurate simulations.Existence of solutions for continuous models on a networkhas also been analyzed for simple scenarios [17]. Contin-uum models can be also combined with discrete models[20] to take the advantage of both approaches, namelythe accuracy offered by discrete models and the scalablecomplexity of computation offered by continuum models.In addition to their efficient computation, continuum mod-els can be treated with a rich collection of mathematicaltools available for differential equations. For example, adirect application of multiscale algorithm to the solutionof continuum models can be used to build up a multiscaleanalysis of supply chains [33]. Here, unlike discrete models,the modeling of supply chain networks with continuumapproaches does not explicitly incorporate the number ofparts into the equation system. Consequently, continuummodels or combination of continuum and discrete modelsare more suitable for multiscale modeling and cross-scalecomputation than the most microscopic discrete simula-tions. As supply chain networks naturally exhibit multi-scale behaviors, easy extendability to multiscale modelingand simulation is obviously an attractive feature.

To solve continuum models, a differential equation sol-ver is in need. As materials flow in supply chains consists ofmainly advection processes and discontinuous Galerkin(DG) methods have superior numerical performance foradvection-dominated problems, we utilize DG in this

paper. DG methods [6–10,16,18,19,21–24,26,27,29–32] arespecialized finite element methods that utilize discontinu-ous piecewise polynomial spaces to approximate the solu-tions of differential equations, with inter-elementcontinuities (if diffusion presents) and boundary conditionsweakly imposed through bilinear forms. Derived from var-iational principles by integration over local cells, the meth-ods are locally mass conservative by construction. Weakenforcement of boundary conditions and inter-elementcontinuities leads to small numerical diffusion and littleoscillation for DG. In addition, the DG methods handlerough coefficient problems and capture the discontinuityin the solution very well by the nature of discontinuousfunction spaces. For time-dependent problems in particu-lar, their mass matrices are block diagonal, providing asubstantial computational advantage if explicit time inte-grations are used. Examples of DG methods include localdiscontinuous Galerkin [10,11], Symmetric Interior PenaltyGalerkin (SIPG) [25,28,32], Oden–Babuska–Baumann DGformulation (OBB-DG) [21], Non-symmetric InteriorPenalty Galerkin (NIPG) [23] and Incomplete Interior Pen-alty Galerkin (IIPG) methods [12,25,28]. For advection–reaction problems without diffusion, the above five DGschemes coincide.

Random variation exists in all production systems, dueto various sources of uncertainties from supply, processand demand, and it can significantly affect the performanceof supply chains. Incorporation of random processes intocontinuum models will result in stochastic partial differen-tial equation (PDE) systems. In most numerical treatmentsfor stochastic PDEs, including the implementation-friendlyMonte Carlo method, and the more efficient Karhunen–Loeve expansion method, the stochastic PDE problem isapproached by solving a number of associated determinis-tic PDEs. Consequently, deterministic PDE modeling andalgorithms are of interest even for intrinsically stochasticsystems. In this paper, we restrict our attention to deter-ministic PDE modeling of supply chains and its efficientsolutions. Extension of this study to stochastic PDE mod-eling is currently under progress, and it will be presentedelsewhere.

The paper is organized as follows: In the following sec-tion, we formulate the continuum modeling equations forsupply chain networks. Here we first establish the massconservation equation to model a single node in a supplychain network and formulate various constitutive equa-tions for the velocity. We then extend this framework toa serial supply chain and to an entire supply network witha new tool of connectivity matrices. In Section 3, we pro-pose discontinuous Galerkin methods for the numericaltreatment of the modeling system and analyze various algo-rithmic properties of the proposed algorithm for solving anentire supply chain network. These algorithmic propertiesinclude consistency, existence of a solution, element-wiseconservation, and convergence. To our knowledge, theerror analysis of DG here for a PDE system equipped witha connectivity matrix is new. Section 4 is devoted to

1206 S. Sun, M. Dong / Comput. Methods Appl. Mech. Engrg. 197 (2008) 1204–1218

computational simulations and numerical studies, wherewe illustrate how the proposed continuum model can beused to describe various scenarios in supply chain applica-tions, and demonstrate the performance of the proposednumerical algorithm. Finally, in Section 5, our results aresummarized, and future work is described.

2. Modeling equations

2.1. Conservation law within a single node and constitutive

relations between velocity and parts density

We first consider continuum models for a single nodewithin supply chain networks. We remark that the singlenode in the model could represent a factory, warehouse,or retailer. The factory here could consist of a number ofphysical machines and inventory points. Let x be a contin-uous variable representing the completion of the productwithin the node. Parts at x ¼ 0 represent raw materials thathave just entered the node, while parts at x ¼ 1 are the fin-ished products that are ready to exit the node. We denote byqðx; tÞ the density of parts at stage x and at time t. LetrYLðx; tÞ denote the yield loss at stage x and time t, whichcould be a function of parts density: rYL ¼ rYLðqÞ. Thespeed (velocity) of the product movement through the nodeis denoted by u. We can obtain the following modelingequation from the conservation of parts within the node.

oqotþ oðuqÞ

oxþ rYLðqÞ ¼ 0; t 2 ð0;1Þ; x 2 ð0; 1�: ð1Þ

We assume a 100% yield in the remainder of this paper,which yields the following equation:

oqotþ oðuqÞ

ox¼ 0; t 2 ð0;1Þ; x 2 ð0; 1�: ð2Þ

The speed u is assumed to be a function of parts densityq. A general relationship between u and q can be written as:

uðx; tÞ ¼ f ðfqðn; sÞ; 0 6 n 6 1; s0ðx; tÞ 6 s 6 tgÞ; ð3Þ

where s0ðx; tÞ is the entry time for the current part at ðx; tÞ.Similarly, the notation sxðn; tÞ used below refers to thetrace-back time to x (x 6 n) for the part at ðn; tÞ. We im-pose the initial work in progress (WIP) distribution andthe arrival rate through the following initial and boundaryconditions.

qðx; 0Þ ¼ q0ðxÞ;uð0; tÞqð0; tÞ ¼ kðtÞ:

We remark two properties of the velocity. First, by thedefinition of completion degree, the velocity is alwaysnon-negative:

uðx; tÞP 0:

It becomes zero only if the factory breaks down com-pletely. Secondly, as the completion degree increases bythe value of one from entrance to exit, the velocity satisfies

Z te

ta

uðxðtÞ; tÞdt ¼ 1;

where ta is the entry time and te the exit time. As mentionedbefore, this general continuum model, being independentof the number of parts, is computationally efficient to sim-ulate, particularly for high-volume multi-stage factories.The unknown q represents the local WIP density and theflux qu has the physical meaning of the local throughput(TH) at stage x and time t; namely,

ðlocal THÞ ¼ ðlocal WIPÞu:

Comparing this with the famous Little’s Law stating

WIP ¼ TH � CT;

we see that the velocity u here corresponds to 1/CT, whereCT stands for the cycle time.

The form (3) is quite general and we need assume a moreconcrete constitute relationship in order to compute veloc-ities numerically. For example, we could assume that thevelocity is determined for a part entering the factory bythe WIP of that time and it remains constant for this partuntil the part exits:

uðx; tÞ ¼ f ðWIPðs0ðx; tÞÞÞ; ð4Þ

which is a special case of the following form:

uðx; tÞ ¼ f ðfqðn; s0ðx; tÞÞ; 0 6 n 6 1gÞ: ð5Þ

An example of (4) is

uðx; tÞ ¼ 1

P þ P �WIPðs0ðx; tÞÞ

¼ 1

P þ PR 1

0qðn; s0ðx; tÞÞdn

; ð6Þ

where P represents the process time (excluding the timespent in queues) for a part. A nice property of this specificfunction form (6) is that the cycle time agrees with simplequeueing models:

CT ¼Z 1

0

dxuðx; sxð1; teÞÞ

¼ ð1þWIPðs0ð1; teÞÞÞP :

A disadvantage of (6) is the need of computing the trace-back time of a part.

The trace-back time computation is eliminated if we canassume that the velocity is determined by the density infor-mation at the current time. A general form for this scenariocan be expressed by

uðx; tÞ ¼ f ðfqðn; tÞ; 0 6 n 6 1gÞ: ð7Þ

It is natural to assume that the velocity of a part within aregular (non-re-entrant) factory is determined by the partsahead of this part, i.e.

uðx; tÞ ¼ f ðfqðn; tÞ; x 6 n 6 1gÞ: ð8Þ


An example of this is

uðx; tÞ ¼ 1� x

P ð1� xÞ þ PR 1

x qðn; tÞdn: ð9Þ

The cycle time for a part in this velocity model does notfully agree with simple queue models, but still have a clearphysical meaning:

CT ¼Z 1

0

dsuðs; tðsÞÞ

¼Z 1

0

P dsþZ 1

0

P1� s

Z 1

sqðn; tðsÞÞdnds

¼ P þ ðweighted average of WIPÞP :

Velocity model (9) involves computation of numericalintegration. A simplified model without this need of inte-gration is

uðx; tÞ ¼ f ðqð1; tÞÞ; ð10Þwhere we assume that the velocity depends upon only theexit density at current time. Eq. (10) is a special case ofthe following model assuming that the velocity is only afunction of parts density at exit:

uðx; tÞ ¼ f ðfqð1; sÞ; s0ðx; tÞ 6 s 6 tgÞ: ð11ÞFor simplicity, we will utilize (10) in our computationalexamples in this paper. A specific form of (10) is

uðx; tÞ ¼ 1

P þ Pqð1; tÞ : ð12Þ

We comment that the velocity given by (12) is independentof the completion degree within a node, and it agrees withsimple queueing models:

CT ¼Z 1

0


¼Z 1

0

P dxþZ 1

0

Pqð1; sxð1; teÞÞdx

¼ P þ PZ te

ta

qð1; sÞuðxðsÞ; sÞds

¼ P þ PZ te

ta

qð1; sÞuð1; sÞds ¼ P þ P �WIPðs0ð1; teÞÞ;

where we have used the space-independence of the velocityuðx; sÞ ¼ uð1; sÞ, 8x. Another example of (11) is

uðx; tÞ ¼ 1

P maxð1; qð1; tÞÞ ; ð13Þ

which might be more appropriate than (12) for certainmaterials flow.

As a special but important scenario of supply chains, there-entrant line is widely used in the semiconductor indus-try. The modeling and simulation of a re-entrant factoryusually causes substantially additional complexity forqueueing models, but it can be elegantly and efficientlytreated using a continuum model like the one describedin (2) and (3). However, the assumption used in (8) is nolonger valid for a re-entrant factory. In other words, the

velocity of a product is no longer only a function of infor-mation ahead of this product. It can be a function of localWIP distribution ahead and behind this product. Conse-quently, general form (7) instead of (8) must be assumed.An example of (7) is

uðx; tÞ ¼ 1

P þ PR 1

0qðn; tÞdn

¼ 1

P þ P �WIPðtÞ : ð14Þ

The physical meaning of (14) is clear with the following cy-cle time:

CT ¼Z 1

0


¼Z 1

0

P dxþ PZ 1

0

WIPðsxð1; teÞÞdx

¼ P þ ðWIP averageÞP :

The formula below is another example of (7) that might bemore physical than (14) for certain cases:

uðx; tÞ ¼ 1

P maxð1;R 1

0qðn; tÞdnÞ

¼ 1

P maxð1;WIPðtÞÞ : ð15Þ

2.2. A serial supply chain

We consider a supply chain consisting of n nodes con-nected serially, and we assume that materials flow fromnode 1 to 2, and then from node 2 to 3, and so on, andfinally the products exit from the supply chain at node n.In general, node i is located in the upstream of node j fori < j. Let qiðx; tÞ, i ¼ 1; 2; . . . ; n denote the density of partsat node i, at stage x and at time t. Conservation of materi-als yields

oqi

otþ oðuiqiÞ

ox¼ 0; t 2 ð0;1Þ; x 2 ð0; 1�; i ¼ 1; 2; . . . ; n:

The velocity ui of the product movement through the nodei is a function of qi, and various concrete constitutive equa-tions for the velocity discussed above can apply to eachnode individually. We impose the following initial condi-tions for the local WIP densities:

qiðx; 0Þ ¼ qi;0ðxÞ; i ¼ 1; 2; . . . ; n:

For the boundary condition, we only need to impose an in-flux for the first node:

u1ð0; tÞq1ð0; tÞ ¼ k1ðtÞ:

The above n partial differential equations are connected bythe inter-node flux continuity conditions:

uið0; tÞqið0; tÞ ¼ ui�1ð1; tÞqi�1ð1; tÞ þ kiðtÞ; i ¼ 2; 3; . . . ; n;

where kiðtÞ is the extra materials fed to node i from outsideof the chain. For an isolated supply chain, k2 ¼ k3 ¼� � � ¼ kn ¼ 0, and the entire system of n partial differentialequations can be simplified to the following single partialdifferential equation:

Fig. 1. An example network diagram.


oqotþ oðuqÞ

ox¼ 0; t 2 ð0;1Þ; x 2 ð0; n�;

qðxþ i� 1; 0Þ ¼ qi;0ðxÞ; 0 6 x < 1; 1 6 i 6 n;

uð0; tÞqð0; tÞ ¼ k1ðtÞ

with the original solution qiðx; tÞ replaced by qðxþ i� 1; tÞfor 0 < x 6 1; 1 6 i 6 n. We remark that this replacementcan facilitate the analysis of a supply chain and improvethe efficiency of its numerical computation.

2.3. A supply chain network

Together with a connectivity matrix representing link-ages among production nodes, the transport equationsfor multiple nodes can be used to model an entire supplychain network. We again assume that the system has n

nodes, but node i is not necessarily the upstream of nodej for i < j. As before, we let qiðx; tÞ, i ¼ 1; 2; . . . ; n denotethe density of parts at node i, at stage x and at time t,and for notational convenience, we also use piðx; tÞ inter-changeably with qiðx; tÞ for this quantity. The transportequation within each node reads

opi

otþ oðuipiÞ

ox¼ 0; t 2 ð0; T �; x 2 ð0; 1�; i ¼ 1; 2; . . . ; n;

ð16Þ

where T is the final simulation time. The connection of then nodes is described by a connectivity matrix C, which con-tains n� n entries Cij, i ¼ 1; 2; . . . ; n and j ¼ 1; 2; . . . ; n. Weimpose the following boundary conditions:

uið0; tÞpið0; tÞ ¼ kiðtÞ þXn

j¼1

Cijujð1; tÞpjð1; tÞ;

i ¼ 1; 2; 3; . . . ; n: ð17Þ

Cij represents the fraction of the materials flux flowingfrom node j to node i relative to all outflux exiting nodej. The connectivity matrix could be a function of velocityand/or parts density, but for simplicity, we assume that itis a constant matrix in our numerical examples below.We remark a few properties of the matrix C here. If nodei is an entrance node in the network, then Cik ¼ 0, 8k; onthe other hand, if node i is an exit node, we have Cki ¼ 0,8k. We always have Ckk ¼ 0, 8k for obvious reasons. Wenote that, Ckk ¼ 0 even for a re-entrant node, as the re-en-trant process has been incorporated into the constitutiveequation already. We do not consider back flow of materi-als, and thus we have Cij P 0 for all i and j. For a non-exitnode i that does not have outflux leaving the system, thematerials flux fraction should add up to one. In general,we have

Xn

k¼1

Cki 6 1; 8i;

since outflux cannot be negative (while influx is modeledseparately). The initial conditions have the same form asbefore:

piðx; 0Þ ¼ pi;0ðxÞ; i ¼ 1; 2; . . . ; n: ð18Þ

The above system of modeling equations can be conve-niently expressed using a vector notation. Denotingp ¼ ðp1; p2; . . . ; pnÞ

T, pinit ¼ ðp1;0; p2;0; . . . ; pn;0ÞT, f ¼ ðp1u1;

p2u2; . . . ; pnunÞT, f in ¼ ðk1; k2; . . . ; knÞT; we have a conciserepresentation of this system:

op

otþ of

ox¼ 0; t 2 ð0; T �; x 2 ð0; 1�;

fð0; tÞ ¼ f inðtÞ þ Cfð1; tÞ; t 2 ð0; T �;

pðx; 0Þ ¼ pinitðxÞ; x 2 ð0; 1�:

To illustrate the above notation, we provide a simpleexample of the supply chain network system consisting ofseven members as shown in Fig. 1. Nodes 1, 2, and 3 areentry nodes, representing suppliers; nodes 4 and 5 are inte-rior nodes, representing manufacturers; and nodes 6 and 7are exit nodes, representing retailers. The connectivitymatrix for this example is

C ¼

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1:0 0:5 0 0 0 0 0

0 0:5 1:0 0 0 0 0

0 0 0 0:8 0:2 0 0

0 0 0 0:2 0:8 0 0

266666666666664

377777777777775: ð19Þ

The connectivity matrix contains linkage information onmaterials flow across nodes. In this example, all productsfrom node 1 flow into node 4 and all products from node3 flow into node 5. The products from node 2 are split intotwo equal sizes, which flow into nodes 4 and 5, respectively.Eighty percent of products from node 4 go to node 6, withthe rest flowing to node 7. The reason that node 4 has astronger connection with node 6 than with node 7 couldbe that the retailer at node 6 has more demand to node4. The products from node 5 also go to nodes 6 and 7,but with the splitting weights switched.

We note that the single node and the serial supply chainmodeling systems are special cases of the supply chain net-work. For brevity, we only present the results with supplychain networks in the following analysis.


3. Discontinuous Galerkin schemes

3.1. Notation

Let fx0 ¼ 0; x1; x2; . . . ; xm ¼ 1g be a family of partitionsof the domain X ¼ ð0; 1Þ composed of line segmentsðxk�1; xkÞ, k ¼ 1; 2; . . . ;m. We could employ different parti-tions for different nodes (note that in this paper nodes referto factories, not geometric vertices in the spatial discretiza-tion). For convenience of presentation, we assume the samepartition for all nodes. We let h be the maximum length ofall intervals:

h :¼ maxk¼1;2;...;m

ðxk � xk�1Þ:

The set of all intervals is denoted by Eh.We denote by k � ks;R the usual Sobolev norm over a

domain R [1]. The Sobolev norm k � ks;X over the entiredomain X ¼ ð0; 1Þ is also denoted simply by k � ks. Wedefine the broken Sobolev space

H sðEhÞ :¼ / 2 L2ðXÞ : /jE 2 H sðEÞ;E 2 Eh

� �for s P 0. One can show that HsðEhÞ is a normed linearspace with its norm defined by

k/kHsðEhÞ :¼XE2Eh

k/k2s;E

!1=2

:

Following convention, we also use the notation jjj � jjjs todenote the broken norm k � kHsðEhÞ. For a given normedspace X and a number p P 1, we define

Lpð0; T ; X Þ :¼ f/ : /ðtÞ 2 X ; k/kX 2 Lpð0; T Þg:

The space Lpð0; T ; X Þ is also a normed linear space with itsnorm given by

k/kLpð0;T ;X Þ :¼ kðk/kX ÞkLpð0;T Þ:

The broken norm k � kLpð0;T ;HsðEhÞÞ is often also written asjjj � jjjLpð0;T ;HsÞ in the triple bar notation. We denote byð�; �ÞR the inner product in ðL2ðRÞÞd or L2ðRÞ over a domainR. The inner product ð�; �ÞX over the entire domain X is alsodenoted simply by ð�; �Þ. The above notation for scalarfunction spaces extends to vector function spaces in a nat-ural way.

We now define the average and jump for / 2 H sðEhÞ,s > 1=2. For k ¼ 1; 2; . . . ;m, we denote

f/gjxk:¼ 1

2ðð/jðxk�1;xkÞÞjxk

þ ð/jðxk ;xkþ1ÞÞjxkÞ;

½/�jxk:¼ ð/jðxk�1;xkÞÞjxk

� ð/jðxk ;xkþ1ÞÞjxk:

The upwind value of a concentration /�jxkis always

ð/jðxk�1;xkÞÞjxkdue to the non-negativity of the velocity.

3.2. Continuous-in-time schemes

The discontinuous Galerkin finite element space is takento be

DrðEhÞ :¼ / 2 L2ðXÞ : /jE 2 PrðEÞ; E 2 Eh

� �; ð20Þ

where PrðEÞ denotes the space of polynomials of degreeless than or equal to r on E. We then introduce the bilinearform Bðp; q; uÞ defined as

Bðp; q; uÞ :¼Xn

i¼1

�Xm

k¼1

Z xk

xk�1

piuioqi

oxdxþ

Xm�1

k¼1

p�i ui½qi�jx¼xk

þpiuiqijx¼1

!: ð21Þ

The linear functional Lðq; u; pÞ is defined as

Lðq; u; pÞ :¼ LextðqÞ þ Lintðq; u; pÞ; ð22Þwhere

LextðqÞ :¼Xn

i¼1

kiðqijx¼0Þ;

Lintðq; u; pÞ :¼Xn

i¼1

Xn

j¼1

CijðujpjÞjx¼1qijx¼0:

The weak formulation of the supply chain network sys-tem (16) and (17) is given below.

Lemma 3.1 (Weak formulation). Let p ¼ ðp1; p2; . . . ; pnÞbe a solution of (16) and (17). If p is continuous with respect

to x, then p satisfies

op

ot; q

� �þ Bðp; q; uÞ ¼ Lðq; u; pÞ;

8q 2 HsðEhÞð Þn; s >3

2; 8t 2 ð0; T �: ð23Þ

Proof. Let qi 2 H sðEhÞ, s > 3=2 for each i ¼ 1; 2; . . . ; n.Multiplying (16) by qi and integrating by parts over eachinterval, we have

Z 1

0

qi

opi

otdx

þXm

k¼1

piuiqijx¼xk�0�piuiqijx¼xk�1þ0

�Z xk

xk�1

uipi

oqi

oxdx

� �¼0:

Summing this over all nodes, recalling the assumption onthe continuity of pi, and applying the inter-node boundarycondition (17), we obtain (23). h

We remark that, if p is discontinuous with respect to x,(16) is not well defined in the classical sense, but its solutioncan be still interpreted in the weak sense of (23). The con-tinuous-in-time (or semi-discrete) DG algorithm for solv-ing (16)–(18) is to seek for an approximation pDGð�; tÞ 2ðDrðEhÞÞn such that

opDG

ot; q

� �þ BðpDG; q; uDGÞ ¼ Lðq; uDG; pDGÞ;

8q 2 DrðEhÞð Þn; t 2 ð0; T �; ð24ÞpDG; q� �

¼ ðpinit; qÞ; 8q 2 DrðEhÞð Þn; t ¼ 0; ð25ÞuDG ¼ uðpDGÞ: ð26Þ


The last equation above specifies the constitutive relation-ship of the velocity with the parts density. We remark thatthe consistency of the DG scheme follows directly fromLemma 3.1. A unique solvability of the above semi-discreteDG algorithm is stated below for the case of constantvelocities.

Lemma 3.2 (Existence of a solution). If the velocity u is a

constant vector (independent of p, t and x), then the

discontinuous Galerkin scheme (24),(25) has a unique solu-

tion for all t > 0.

Proof. Let fvigMi¼1 be a basis of ðDrðEhÞÞn. We then can

express the DG solution by

pDG ¼XM

i¼1

ciðtÞviðxÞ:

With this, (24) and (25) become

Adc

dt¼ Bcþ b;

Ac ¼ binit;

where cðtÞ ¼ ðc1ðtÞ; c2ðtÞ; . . . ; cMðtÞÞT, and the componentsof other matrices and vectors are ðAÞij ¼ ðvj; viÞ, ðBÞij ¼Lintðvi; u; vjÞ � Bðvj; vi; uÞ, ðbÞi ¼ LextðviÞ, and ðbinitÞi ¼ðpinit; viÞ. It follows from the theory of ordinary differentialequations that cðtÞ exists and is unique for t > 0. h

We remark that the unique solvability of the semi-dis-crete DG algorithm (24) and (25) can be shown under amuch milder assumption of the velocity. Rigorous argu-ments for the cases with general velocity constitutiveassumptions will be investigated in a separate study.

The element-wise mass conservation of the DG schemeis stated in the following Lemma. This is a desired featureof DG since we would like to conserve the amount of mate-rials globally and locally in supply chains.

Lemma 3.3 (Local mass conservation). The DG approxi-

mation of the concentration satisfies the following local mass

balance property.Z xk

xk�1

opDGi

otþ ðpDG;�

i uiÞjx¼xk� ðpDG;�

i uiÞjx¼xk�1¼ 0 ð27Þ

for i ¼ 1; 2; . . . ; n and for k ¼ 2; 3; . . . ;m, andZ x1

x0

opDGi

otþ pDG;�

i uijx¼x1� ki �

Xn

j¼1

CijðujpDGj Þjx¼1 ¼ 0

ð28Þ

for i ¼ 1; 2; . . . ; n.

Proof. We fix k and i, and then construct q 2 ðDrðEhÞÞnsuch that qj � 0 for all j 6¼ i, and qiðxÞ ¼ 1 ifxk�1 < x < xk and qiðxÞ ¼ 0 elsewhere. Substituting the con-structed test function q into the DG scheme (24), we obtain(27) and (28). h

3.3. Convergence of DG

We now analyze the convergence of the above DG algo-rithm for simulating a supply chain network. Our resultsare summarized by the following error estimates for theDG solution.

Theorem 3.4 (Error estimates for the DG solution). Let

p ¼ ðp1; p2; . . . ; pnÞ be the solution of (16)–(18). Assume that

p is continuous with respect to x, and that the velocity u is a

constant vector (independent of p, t and x) satisfyingui P

Pnj¼1Cijuj for all i. In addition, we assume that

Cij P 0 for all i; j andPn

i¼1Cij 6 1 for all j. Then there

exists a constant K independent of h such that

kpDG � pkL1ð0;T ;L2ð0;1ÞÞ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

ui

Xm�1

k¼1

Z T

0

½pDGi �

2jx¼xkdt

vuut

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

Xn

j¼1

Cijuj

Z T

0

ðpDGj � pjÞjx¼1 � ðpDG

i � piÞjx¼0

� �2dt

vuut6 Khrþ1

2:

Proof. We let pI :¼ ðpI1; p

I2; . . . ; pI

nÞT, with pI

i being the L2

projection of the exact solution pi onto the DG spaceDrðEhÞ, and define

E :¼ pDG � p; ð29Þ

EI :¼ p� pI ; ð30Þ

EA :¼ pDG � pI ¼ Eþ EI : ð31Þ

Subtracting the weak formulation (23) from the DGscheme (24), we have for any q 2 ðDrðEhÞÞn,

oE

ot; q

� �þ BðE; q; uÞ ¼ Lintðq; u;EÞ:

Splitting E as E ¼ EA � EI and choosing q ¼ EA, we have

oEA

ot;EA

� �þ BðEA;EA; uÞ ¼ oEI

ot;EA

� �þ BðEI ;EA; uÞþ LintðEA; u;EÞ: ð32Þ

The first term in the error Eq. (32) can be written as

oEA

ot;EA

� �¼ 1

2

d

dtkEAk2

L2ð0;1Þ:

Recalling the bilinear form definition (21), we expand thesecond term of (32) as


BðEA;EA; uÞ ¼Xn

i¼1

�Xm

k¼1

Z xk

xk�1

EAi ui

oEAi

oxdx

þXm�1

k¼1

EA;�i ui½EA

i �jx¼xkþ uiðEA

i Þ2jx¼1

!

¼Xn

i¼1

ui �1

2

Xm

k¼1

ðEAi Þ

2jx¼xk�0x¼xk�1þ0

þXm�1

k¼1

EA;�i ½EA

i �jx¼xkþ ðEA

i Þ2jx¼1

!;

where ðEAi Þ

2jx¼xk�0x¼xk�1þ0 :¼ lim�!0þðEA

i Þ2jx¼xk��

x¼xk�1þ�: Noting that

� 1

2

Xm

k¼1

ðEAi Þ

2jx¼xk�0x¼xk�1þ0 þ

Xm�1

k¼1

EA;�i ½EA

i �jx¼xk

¼ 1

2ðEA

i Þ2jx0� 1

2ðEA

i Þ2jxm

� 1

2

Xm�1

k¼1

ððEAi Þ

2jxk�0 � ðEAi Þ

2jxkþ0Þ

þXm�1

k¼1

EAi jxk�0ðEA

i jxk�0 � EAi jxkþ0Þ

¼ 1

2ðEA

i Þ2jx0� 1

2ðEA

i Þ2jxm

� 1

2

Xm�1

k¼1

ðEAi jxk�0 þ EA

i jxkþ0ÞðEAi jxk�0 � EA

i jxkþ0Þ

þXm�1

k¼1

EAi jxk�0ðEA

i jxk�0 � EAi jxkþ0Þ

¼ 1

2ðEA

i Þ2jx0� 1

2ðEA

i Þ2jxmþ 1

2

Xm�1

k¼1

ðEAi jxk�0 � EA

i jxkþ0Þ2;

we see that

BðEA;EA; uÞ ¼ 1

2

Xn

i¼1

ui

ðEA

i Þ2jx¼0 þ ðEA

i Þ2jx¼1

þXm�1

k¼1

½EAi �

2jx¼xk

!:

We now consider the right-hand side of (32). The firstterm vanishes due to the orthogonality in L2ð0; 1Þ. The sec-ond term on the right-hand side of (32) can be bounded as

BðEI ;EA; uÞ ¼Xn

i¼1

ui �Xm

k¼1

Z xk

xk�1

EIi

oEAi

oxdx

þXm�1

k¼1

EI ;�i ½EA

i �jx¼xkþ EI

i EAi jx¼1

!

¼Xn

i¼1

ui

Xm�1

k¼1

EI ;�i ½EA

i �jx¼xkþ EI

i EAi jx¼1

!

6

Xn

i¼1

ui1

4

Xm�1

k¼1

½EAi �

2jx¼xkþ Kh2rþ1 þ EI

i EAi jx¼1

!;

where the approximation results of the L2 projection havebeen used. Combining the term EI

i EAi jx¼1 into the last term

of (32), we have

LintðEA; u;EÞ þXn

i¼1

uiðEIi E

Ai Þjx¼1

¼Xn

i¼1

Xn

j¼1

CijðujEjÞjx¼1EAi jx¼0 þ

Xn

i¼1

uiðEIi E

Ai Þjx¼1

¼Xn

i¼1

Xn

j¼1

CijujEAj jx¼1EA

i jx¼0 �Xn

i¼1

Xn

j¼1

CijujEIjjx¼1EA

i jx¼0

þXn

i¼1

uiðEIi E

Ai Þjx¼1:

We note that

EAj jx¼1EA

i jx¼0 ¼1

2ðEA

j Þ2jx¼1þ

1

2ðEA

i Þ2jx¼0�

1

2ðEA

j jx¼1�EAi jx¼0Þ

2:

In addition, we recall the two properties of Cij: Cij P 0 andPni¼1Cij 6 1 and let 1�

Pni¼1Cij ¼: dj P 0. We then obtain

�Xn

i¼1

Xn

j¼1

CijujEIjjx¼1EA

i jx¼0 þXn

i¼1

uiðEIi E

Ai Þjx¼1

¼ �Xn

i¼1

Xn

j¼1

CijujEIjjx¼1EA

i jx¼0

þXn

j¼1

Xn

i¼1

Cij þ dj

!ujðEI

jEAj Þjx¼1

¼Xn

i¼1

Xn

j¼1

CijujEIjjx¼1ðEA

j jx¼1 � EAi jx¼0Þ

þXn

j¼1

djujEIjjx¼1EA

j jx¼1

61

4

Xn

i¼1

Xn

j¼1

Cijuj EAj jx¼1 � EA

i jx¼0

2

þ Kh2rþ1

þ 1

2

Xn

j¼1

djujððEAj Þ

2jx¼1Þ:

Consequently, we see

LintðEA; u;EÞ þXn

i¼1

uiðEIi E

Ai Þjx¼1

61

2

Xn

i¼1

Xn

j¼1

CijujðEAj Þ

2jx¼1 þ1

2

Xn

i¼1

Xn

j¼1

CijujðEAi Þ

2jx¼0

þ Kh2rþ1 þ 1

2

Xn

j¼1

djujððEAj Þ

2jx¼1Þ

� 1

4

Xn

i¼1

Xn

j¼1

CijujðEAj jx¼1 � EA

i jx¼0Þ2

61

2

Xn

i¼1

uiðEAi Þ

2jx¼1 þ1

2

Xn

i¼1

uiðEAi Þ

2jx¼0 þ Kh2rþ1

� 1

4

Xn

i¼1

Xn

j¼1


i jx¼0

2

;

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

time(days)

flux(

part

s/da

y)

Fig. 2. Influx imposed on the left of the domain.


where we have used the velocity assumption of ui PPnj¼1Cijuj.Combining all terms, we conclude

1

2

d

dtkEAk2

L2ð0;1Þ þ1

4

Xn

i¼1

ui

Xm�1

k¼1

½EAi �

2jx¼xk

þ 1

4

Xn

i¼1

Xn

j¼1


i jx¼0

2

6 Kh2rþ1: ð33Þ

The initial condition for our DG scheme implies

kEAkL2ð0;1Þ ¼ 0:

Integrating (33) over ð0; tÞ, and taking supremum over allt 2 ð0; T �, we conclude that

kEAk2L1ð0;T ;L2ð0;1ÞÞ þ

Xn

i¼1

ui

Xm�1

k¼1

Z T

0

½EAi �

2jx¼xkdt

þXn

i¼1

Xn

j¼1

Cijuj

Z T

0

ðEAj jx¼1 � EA

i jx¼0Þ2 dt 6 Kh2rþ1:

The theorem follows from this and the triangleinequality. h

Remark 3.5. The above error estimate onffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1ui

Pm�1k¼1

R T0½pDG

i �2jx¼xk

dtq

is optimal in the sense that

it is as good as the approximation with the L2 projectionexcept a possibly different constant K. Our error estimateon kpDG � pkL1ð0;T ;L2ð0;1ÞÞ is nearly optimal with a loss ofonly 1

2.

4. Numerical examples

4.1. Single-node scenarios

We first consider a few examples with a single node. Afactory starts with zero parts density initially at t ¼ 0.Imposed on the boundary x ¼ 0 is a given influx (parts/day) that is displayed in Fig. 2. We simulate up to 300 days.We consider four different constitutive equations (12)–(15)for the velocity calculation. The parameter P (process time)is set to be 10 days for all velocity models.

To investigate the influence of the approximation orderto the simulation of materials flow in a supply chain node,we run simulations using the lowest order DG (r ¼ 0) and ahigh-order DG (r ¼ 2). Fig. 3 provides the two DG simu-lations for the materials flow with the constitutive equation(13). In the lowest order DG simulation, we use the for-ward Euler method and employ piecewise constant approx-imation functions with a uniform spatial interval lengthh ¼ 0:01 and a uniform time step size Dt ¼ 0:05 days. Inthe high-order DG run, we utilize piecewise quadratic func-tions with a uniform spatial interval length h ¼ 0:05 and auniform time step size Dt ¼ 0:1 days, but with the back-ward Euler method. Other combinations of h and Dt arealso investigated (not shown). It is found that high-order

DG tends to give more accurate simulation results, whichagrees with our theoretical error estimates. However, ahigh-order DG might have strong non-physical oscillationunless the mesh is sufficiently refined. In addition, the for-ward Euler method tends to work with the lowest orderDG better than with a high-order DG.

Fig. 3 displays three plots for each DG scheme: (1) localWIP density (or parts density) q as a function of completiondegree x for time t ¼ 1, 10, 100, 200 days; (2) exit flux (out-flux) qujx¼1 as a function of time; (3) integrated WIPR 1

0 qðxÞdx as a function of time. Starting from time zero,influx of 0.05 parts/day flows into the node. During earlytimes, the node is below its saturated load, thus achievingits optimal process time of 10 days. Consequently, a plugflow with 0.5 parts/(unit completion) of local WIP densityis formed. After this plug flow reaches the exit point x ¼ 1at 10 days, the exit flux jumps from 0 to 0.05 parts/day.Obviously, there is no accumulation of WIP between 10and 50 days. However, after 50 days, the influx increases lin-early until 100 days. During this period, cycle time starts todeteriorate due to task overloading, and the velocity u startsto decrease. The exit flux increases to its maximum valueand then remains there flat, and WIP starts to accumulatequickly. After day 100, the influx decreases linearly untilreturning back to its normal condition at day 150, whichremains afterward. WIP continues accumulating for a while(until about 130 days) but with a slower speed, and thenstarts to decrease. Due to the delay from treating past accu-mulated parts, outflux drops back to its normal condition ata much later time (roughly 220 days). It was found that DGschemes are able to predict outflux and WIP profiles accu-rately. Outflux is important as it will propagate to down-stream nodes in supply chains. WIP is also criticalinformation for supply chain management, in particularfor inventory control and utilization management at eachnode and cycle time reduction for the whole supply chain.

The simulation results with the constitutive equation(12) are plotted in Fig. 4, and the simulation results with

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

completion degree

dens

ity(p

arts

/uni

t com

plet

ion)

time= 1.0time=10.0time=100.0time=200.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

completion degree

dens

ity(p

arts

/uni

t com

plet

ion)


0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

time(days)

flux(

part

s/da

y)

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

time(days)

flux(

part

s/da

y)

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

time(days)

WIP

(par

ts)

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

time(days)

WIP

(par

ts)

Fig. 3. DG simulations of a single node using the non-re-entrant constitutive equation (13) (left column: DG with P0; right column: DG with P2).


the re-entrant constitutive equations (15) and (14) are givenin Fig. 5. In these three cases, the forward Euler method isemployed together with piecewise constant approximationfunctions, a uniform spatial interval length h ¼ 0:01 anda uniform time step size Dt ¼ 0:05 days.

The behavior of materials flow with the constitutiveequation (12) is quite similar to that with (13), except theplateau of the overloading outflux in Fig. 3 is replaced bythe curved saturated outflux in Fig. 4. The nodes withthe re-entrant constitutive equations (14) and (15) have

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

completion degree

dens

ity(p

arts

/uni

t com

plet

ion)


0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

time(days)

flux(

part

s/da

y)

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

time(days)

WIP

(par

ts)

Fig. 4. DG simulation of a single node using the non-re-entrantconstitutive equation (12).


behaviors different dramatically from the nodes with non-re-entrant relationships (12) and (13). In particular, theoutflux exhibits multiple modes in the two re-entrant caseswhile it has only a single mode in the non-re-entrant cases.

4.2. A serial supply chain example

In this example, we simulate a system of three nodes.Parts first enter node 1, then flow to node 2, and finallyget to node 3 before exiting the system. The connectivitymatrix C obviously has the following form:

C ¼0 0 0

1 0 0

0 1 0

264

375:

Nodes 1 and 3 are modeled with the non-re-entrant consti-tutive equation (13), both having a process time of 8 days,while node 2 is described with the re-entrant constitutiveequation (15) with a process time of 10 days. The influx(parts/day) imposed on the boundary x ¼ 0 is the sameas that used in the previous example, displayed in Fig. 2.Again, we simulate up to 300 days, and all factories startwith zero parts density initially at t ¼ 0. We employ theforward Euler method, piecewise constant approximationfunctions, a uniform spatial interval length h ¼ 0:05 anda uniform time step size Dt ¼ 0:05 days. The simulation re-sults of outflux and WIP as functions of time are plotted inFig. 6.

Accumulation of WIP and its effect to the flow are evi-dent as displayed in the plots of WIP and outflux, respec-tively. Node 1 starts to accumulate at 50 days and doesnot return back to its normal loading until about 170 days,which is a little later than the return-to-normal time of theinflux due to the delay effect of the dynamics. The down-stream nodes 2 and 3 have later overloading periods thanthat for node 1 as expected. Information propagationdownward from upstream is clearly demonstrated in theplots of WIP and outflux here.

4.3. A supply chain network

We now consider the network example depicted inFig. 1. The inter-factory flow is described by the connectiv-ity matrix specified in (19). Nodes 4 and 5 are modeledusing the re-entrant constitutive equation (15), while therest of nodes are described by the non-re-entrant relation(13). Nodes 1–5 have a process time of 10 days, and nodes6 and 7 have a process time of 8 days. The influxes (parts/day) imposed on the boundary x ¼ 0 are all constants: 0.04parts/day for node 1, 0.05 parts/day for node 2, and 0.06parts/day for node 3. As before, we simulate up to 300days, and all factories start with zero parts density initiallyat t ¼ 0. The forward Euler method is employed with piece-wise constant approximation functions, a uniform spatialinterval length h ¼ 0:05 and a uniform time step sizeDt ¼ 0:05 days. Unlike previous examples, we now try tosimulate the controllable shut-down of a node at timet ¼ 100 days, which is implemented by blocking all influxto this node. We remark that an uncontrollable break-down of a node due to disasters can be simulated byforcing the process time of that node to infinity, and an

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

completion degree

dens

ity(p

arts

/uni

t com

plet

ion)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

completion degree

dens

ity(p

arts

/uni

t com

plet

ion)


0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time(days)

flux(

part

s/da

y)

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time(days)

flux(

part

s/da

y)

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

time(days)

WIP

(par

ts)

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

9

10

time(days)

WIP

(par

ts)

Fig. 5. DG simulations of a single node using the re-entrant constitutive equations (15) (left column) and (14) (right column).


uncontrollable break-down of a link between two nodescan be simulated by modifying the connectivity matrix.The difference between planned shut-down of a node simu-lated here and uncontrollable break-down of a node is thatthe leftover WIP within the node will continue to be pro-

cessed after shut-down time in the former case, but willstop being treated in the latter.

Simulation results from this example are plotted inFig. 7. The WIP of nodes 1, 2 and 3 starts to accumulatefrom time 0, but the outfluxes from these three nodes jump

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time(days)

flux(

part

s/da

y)

node 1node 2node 3

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time(days)

WIP

(par

ts)

node 1node 2node 3

Fig. 6. DG simulation of a serial supply chain.

0 50 100 150 200 250 3000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time(days)

flux(

parts

/day

)

node 1node 2node 3node 4node 5node 6node 7

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time(days)

WIP

(par

ts)

node 1node 2node 3node 4node 5node 6node 7

Fig. 7. DG simulation of a supply chain network.


from zero to feed values at 10 days due to the process timedelay. Again, as a result of the time consumption in pro-cessing parts, the WIP buildup for nodes 4 and 5 occursafter that for the first three nodes, but before that for nodes6 and 7. A similar order sequence is observed for the out-flux jumps. At 100 days, we suppose that node 1 is plannedto shut-down gradually due to equipment maintenance (thenode will be totally shut-down after all leftover WIP is pro-cessed), thus the WIP of node 1 starts to decrease at 100days, but it is not reduced to zero completely until 110 daysdue to the WIP leftover. The outflux of node 1 reduces tozero in a similar way. Nodes 2, 3 and 5 are not linked to,thus not affected by, node 1; their WIP and outflux remainunchanged after the shut-down of node 1. However, down-stream nodes 4, 6, and 7 are connected to node 1 eitherdirectly or indirectly, and their WIP and outflux decreaseby certain values at delayed times. The outflux of node 4does not reduce all the way to zero because it also has sup-ply coming from an alternative node (node 2). The reduc-tion of outflux in node 6 is much greater than that innode 7 because node 6 has a stronger connection (requiring

more products from node 1 via node 4) than node 7 to node4, which is partially fed by the terminated node (node 1).For illustration purpose, we have used a relatively simpleconstitutive model for velocities, but more complex veloc-ity models can be treated by DG without any additionaltechnical difficulty.

5. Conclusions

Continuum modeling has been considered and studiedfor simulating supply chain networks. Partial differentialequations (PDEs) of mass conservation were first estab-lished for a single node, then extended to a supply chainand finally applied to an entire network. Comparison ofthe continuum model to Little’s law has been made to illus-trate physical meanings of employed variables. Someimportant features such as re-entrant nodes and risk analy-sis of supply chain networks with unreliable nodes and linkscould be described easily by the proposed continuum mod-eling method. Another novelty of our modeling approach is


the introduction of a connectivity matrix, allowing us toincorporate the linkage among nodes systematically intothe boundary conditions of PDEs. To close the modelingsystem, we also summarized or proposed a number of exist-ing and new constitutive equations for modeling velocity asa function of the local WIP density. Since materials flow insupply chains often contains strong advection components,we proposed to utilize the discontinuous Galerkin (DG)methods for numerical treatment of the continuum model-ing equations. Various algorithmic properties, includingconsistency, existence of a solution, element-wise conserva-tion, and optimal/nearly optimal convergence, of the pro-posed DG algorithm have been analyzed for simulatingan entire supply chain network. In order to obtain a tighterror bound, two physical meaningful conditions on theconnectivity matrix were assumed. A new error estimatingtechnique was employed to take the advantage of the prop-erties of the connectivity matrix. Numerical examples havebeen provided to simulate a single node, a serial supplychain and an entire network. The influence of influx varia-tion to the behaviors of WIP and outflux has been numeri-cally investigated. Shut-down of a node within a supplychain network has also been simulated. Our results illus-trated the applicability of our models to various scenariosand demonstrated the robustness and accuracy of DG tothe modeling system. An important future extension of cur-rent work is to incorporate random variation componentsinto the modeling system, and to investigate efficient numer-ical algorithms for treating the associated stochastic PDEs.

Acknowledgements

The work presented in this paper has been partially sup-ported by a grant from National Natural Science Founda-tion of China (70571050), a grant from the NationalHigh-Tech Research and Development Program of China(863 Program), and an internal research grant from Clem-son University.

References

[1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.[2] E.J. Anderson, A new continuous model for job shop scheduling, Int.

J. Syst. Sci. 12 (1981) 1469–1475.[3] D. Armbruster, D. Marthaler, C. Ringhofer, Kinetic and fluid model

hierarchies for supply chains, Multiscale Model. Simul. 2 (1) (2003)43–61.

[4] D. Armbruster, D.E. Marthaler, C. Ringhofer, K. Kempf, T.-C. Jo, Acontinuum model for a re-entrant factory, Oper. Res. 54 (5) (2006)933–950.

[5] D. Armbruster, C. Ringhofer, Thermalized kinetic and fluid modelsfor reentrant supply chains, Multiscale Model. Simul. 3 (4) (2005)782–800.

[6] D.N. Arnold, An interior penalty finite element method withdiscontinuous elements, SIAM J. Numer. Anal. 19 (1982) 742–760.

[7] I. Babuska, M. Zlamal, Nonconforming elements in the finite elementmethod with penalty, SIAM J. Numer. Anal. 10 (1973) 863–875.

[8] C.E. Baumann, J.T. Oden, A discontinuous hp finite element methodfor convection–diffusion problems, Comput. Methods Appl. Mech.Engrg. 175 (3–4) (1999) 311–341.

[9] Z. Chen, H. Chen, Pointwise error estimates of discontinuousGalerkin methods with penalty for second-order elliptic problems,SIAM J. Numer. Anal. 42 (3) (2004) 1146–1166.

[10] B. Cockburn, G.E. Karniadakis, C.-W. Shu, The development of thediscontinuous Galerkin methods, in: First International Symposiumon Discontinuous Galerkin Methods, Lecture Notes in Computa-tional Science and Engineering, vol. 11, Springer-Verlag, 2000, pp. 3–50.

[11] B. Cockburn, C.-W. Shu, The local discontinuous Galerkin methodfor time-dependent convection–diffusion systems, SIAM J. Numer.Anal. 35 (6) (1998) 2440–2463 (Electronic).

[12] C. Dawson, S. Sun, M.F. Wheeler, Compatible algorithms forcoupled flow and transport, Comput. Methods Appl. Mech. Engrg.193 (2004) 2565–2580.

[13] M. Dong, Process modeling, performance analysis and configurationsimulation in integrated supply chain network design, PhD thesis,Virginia Polytechnic Institute and State University, 2001.

[14] M. Dong, F.F. Chen, The impacts of component commonality onintegrated supply chain network configurations: a state and resourcebased simulation study, Int. J. Adv. Manuf. Technol. 27 (3–4) (2005)397–406.

[15] M. Dong, F.F. Chen, Performance modeling and analysis ofintegrated logistic chains: an analytic framework, Eur. J. Oper. Res.162 (1) (2005) 83–98.

[16] J. Douglas, T. Dupont, Interior penalty procedures for ellipticand parabolic Galerkin methods, Lect. Notes Phys. 58 (1976)207–216.

[17] S. Gottlich, M. Herty, A. Klar, Network models for supply chains,Commun. Math. Sci. 3 (4) (2005) 545–559.

[18] O.A. Karakashian, F. Pascal, A posteriori error estimates for adiscontinuous Galerkin approximation of second-order elliptic prob-lems, SIAM J. Numer. Anal. 41 (6) (2003) 2374–2399.

[19] M.G. Larson, A.J. Niklasson, Analysis of a nonsymmetric discon-tinuous Galerkin method for elliptic problems: Stability and energyerror estimates, SIAM J. Numer. Anal. 42 (1) (2004) 252–264.

[20] Y.H. Lee, M.K. Cho, S.J. Kim, Y.B. Kim, Supply chain simulationwith discrete-continuous combined modeling, Comput. Ind. Engrg.43 (1–2) (2002) 375–392.

[21] J.T. Oden, I. Babuska, C.E. Baumann, A discontinuous hp finiteelement method for diffusion problems, J. Comput. Phys. 146 (1998)491–516.

[22] J.T. Oden, L.C. Wellford Jr., Discontinuous finite element approx-imations for the analysis of shock waves in nonlinearly elasticmaterials, J. Comput. Phys. 19 (2) (1975) 179–210.

[23] B. Riviere, M.F. Wheeler, V. Girault, A priori error estimates forfinite element methods based on discontinuous approximationspaces for elliptic problems, SIAM J. Numer. Anal. 39 (3) (2001)902–931.

[24] D. Schotzau, C. Schwab, Time discretization of parabolic problemsby the hp-version of the discontinuous Galerkin finite elementmethod, SIAM J. Numer. Anal. 38 (2001) 837–875.

[25] S. Sun, Discontinuous Galerkin methods for reactive transport inporous media, PhD thesis, The University of Texas at Austin, 2003.

[26] S. Sun, M.F. Wheeler, Discontinuous Galerkin methods for coupledflow and reactive transport problems, Appl. Numer. Math. 52 (2–3)(2005) 273–298.

[27] S. Sun, M.F. Wheeler, L2ðH1Þ norm a posteriori error estimation fordiscontinuous Galerkin approximations of reactive transport prob-lems, J. Sci. Comput. 22 (2005) 501–530.

[28] S. Sun, M.F. Wheeler, Symmetric and nonsymmetric discontinuousGalerkin methods for reactive transport in porous media, SIAM J.Numer. Anal. 43 (1) (2005) 195–219.

[29] S. Sun, M.F. Wheeler, Anisotropic and dynamic mesh adaptation fordiscontinuous Galerkin methods applied to reactive transport,Comput. Methods Appl. Mech. Engrg. 195 (25–28) (2006) 3382–3405.

[30] S. Sun, M.F. Wheeler, A posteriori error estimation and dynamicadaptivity for symmetric discontinuous Galerkin approximations of


reactive transport problems, Comput. Methods Appl. Mech. Engrg.195 (2006) 632–652.

[31] S. Sun, M.F. Wheeler, Discontinuous Galerkin methods for simulat-ing bioreactive transport of viruses in porous media, Adv. WaterResourc. 30 (6–7) (2007) 1696–1710.

[32] M.F. Wheeler, An elliptic collocation-finite element method withinterior penalties, SIAM J. Numer. Anal. 15 (1978) 152–161.

[33] Y. Zou, I.G. Kevrekidis, D. Armbruster, Multiscale analysis of re-entrant production lines: An equation-free approach, Phys. A: Statist.Mech. Appl. 363 (1) (2006) 1–13.

continuum modeling of supply chain networks using ...design is to model and evaluate the performance...

Documents