A Common Modeling Framework for Dynamic Traffic Assignment and Supply Chain Management Systems with Congestion Phenomena

Georgios Kalafatas, Purdue University, USA; Srinivas Peeta, Purdue University, USA

Abstract This paper seeks to illustrate the ability of the graph theoretic cell transmission model (GT-CTM), previously developed by the authors, to address some dynamic supply chain management (SCM) problems with congestion phe-nomena using a simple graphical representation. It further shows the conceptual equivalence between SCM and dynamic traffic assignment (DTA) problems using the GT-CTM framework. Thereby, the GT-CTM provides a generalized modeling framework to address dynamic network problems with congestion phenomena.

1. Introduction

Recently, Kalafatas and Peeta (2007) developed the graph theoretic cell transmis-sion model (GT-CTM) and used it to address the single destination dynamic traffic assignment (SD-DTA) problem consistent with the hydrodynamic theory. The GT-CTM is the graph theoretic extension of the cell transmission model (CTM) (Daganzo, 1994; 1995). In this paper, we show that the GT-CTM can be used to model single product supply chain management (SP-SCM) systems characterized by congestion phenomena.

Zawack and Thompson (1987) had proposed a dynamic space-time network flow model with a corresponding graph representation for modeling SD-DTA and the job scheduling problems. However, their approach does not address congestion phenomena realistically. It ignores the fundamental role of traffic density. There-by, congestion phenomena are absent at medium to high densities, and backward propagating traffic waves and queues are not captured realistically. The GT-CTM inherently addresses these issues in the DTA context (Kalafatas and Peeta, 2007), thereby enhancing realism. Here we show that improved realism can also be at-tained for the SCM problem using the GT-CTM modeling framework, as well as its ability to better illustrate the interconnections between DTA and SCM. This is significant because notions like optimal routing (DTA)/scheduling (SCM) and congestion, and solution methodologies in terms of heuristics and decomposition

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techniques, become transferable. This enriches both fields by potentially enabling robust techniques from one to be applied to the other.

As a starting point, we will illustrate the conceptual equivalence between DTA and SCM using their fundamental characteristics. DTA is the problem of routing a set of vehicles from their origins to their destinations through the road links of a traffic network while taking into account the spatio-temporal interactions of their routes. SCM is the problem of planning and implementing the operations related to the movement, processing and storage of raw materials, work-in-process inven-tory (WIP), and finished goods, all measured in equivalent stock keeping units (SKUs), from their initial locations to the processing facilities and finally to con-sumption. In general, both problems study the optimal transshipment of some “units” (vehicles or SKUs) from their origins to their destinations through a net-work of interconnected processes (road links, industrial processes or storage facili-ties) and are modeled across multiple time periods. In DTA, each road link is as-sociated with a free-flow travel time. Link travel time increases as the link traffic density increases, implying congestion. Link performance functions express link travel time as a function of the link flow and model congestion up to the point that maximum flow occurs. Congestion can also be described using the fundamental traffic flow-density diagram which relates link flow to link density. It models the whole spectrum of congestion levels, from the uncongested region to jam densi-ties. In SCM, each operation is associated with a minimum lead (or processing) time, which is achieved when the WIP has a relatively low value. Congestion in SCM is the phenomenon where lead times increase as the WIP increases. Clearing functions express throughput as a function of WIP (Karmarkar, 1989), and repre-sent conceptual analogs to the link performance function and the fundamental traf-fic flow-density diagram in a DTA context. Clearing functions represent conges-tion phenomena in the same region as the link performance functions – up to the congestion level at which maximum flow occurs. However, the functional form of clearing functions (throughput-WIP relation) is equivalent to the functional form of the fundamental traffic flow-density diagram. The difference between clearing functions and the fundamental flow-density diagram is that the notion of clearing function (and its mathematical form) does not cover the highly congested region where flow decreases as a function of density. Table 1 summarizes the analogies between concepts in DTA and SCM. The qualitative equivalence between DTA and SCM elements illustrated heretofore will be used later to develop an analytical framework for SP-SCM through the GT-CTM.

The GT-CTM is selected because it is based on a simple, exact and time-expandable graph theoretic cell representation (Figure 1), capable of modeling congestion phenomena in DTA applications without the need for additional con-gestion-specific constraints beyond the set of constraints that constitute the GT-CTM’s exact graph structure. Further, the GT-CTM generalizes time-expanded graphs with the modeling capabilities of node inflow/outflow capacity restrictions and congestion phenomena (Kalafatas and Peeta, 2008a), and has the computa-tional complexity of the minimum cost flow problem. In the past, time-expanded formulations have also been widely used to model dynamic network flow prob-

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lems, such as SCM problems. In the context of this study, we select the single-product (or single-commodity) formulation (SPF) used by Karmarkar (1989) as a well-accepted and representative formulation for SCM modeling. It is selected be-cause it has a clear graph-based sub-structure, and addresses congestion phenome-na through additional constraints that are not necessarily linear (Asmundsson et al., 2002). We show that SPF’s graph sub-structure is a construct of the GT-CTM, thereby illustrating that the GT-CTM can be used to address the broad class of dy-namic network flow problems with congestion phenomena, such as DTA and SCM.

Table 1. Conceptual equivalencies between DTA and SCM

DTA SCM Description

vehicle SKU the fundamental unit of flow road link process (road link, pro-

cessing unit or storage unit)

the fundamental modeling element (server) forming the network

traffic density work in process invento-ry (WIP)

number of units in a server in a modeling peri-od

jam or maximum density

maximum WIP maximum number of units in a server in a modeling time period

traffic flow throughput number of units (vehicles, SKUs) being served from a server in a modeling time period

link flow capacity maximum throughput maximum number of units being served in a modeling time period (serving capacity)

link travel time process lead time the time that a unit of flow spends in a server free-flow travel time minimum lead time the minimum time that a unit of flow has to

spent in a server (usually experienced when few units are present in the server)

link performance function (flow-travel

time relation)

clearing function (throughput-WIP rela-

tion)

relations (of different functional form) repre-senting congestion phenomena for congestion

levels up to the point which maximum flow oc-curs

fundamental flow-density diagram

(flow-density rela-tion)

clearing function (throughput-WIP rela-

tion)

relations (of the same functional form) between the flow units present in a server and the max-

imum serving capacity

origin source location where flow units are entering the net-work from an external location

destination destination location where flow units are exiting the net-work to an external location

In Section 2 we retain the original structure of the GT-CTM and describe its

analytical formulation in SCM terms. In Section 3 we construct the SPF of Kar-markar (1989) on a GT-CTM base. In Section 4 we discuss how notions like con-

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gestion and sub-assemblies as identified in SCM can be addressed in the GT-CTM modeling framework. Finally, in Section 5 we summarize our research and identi-fy some issues to be addressed in the future.

2. GT-CTM from an SCM perspective

In this section we will describe the exact formulation of the GT-CTM in SCM terms. The GT-CTM was developed and used for applications in traffic engineer-ing. Therefore, the analytical descriptions of its corresponding formulation are provided by Kalafatas and Peeta (2007) in DTA terms. Recently, the authors (2008a) showed that the GT-CTM is a generalized time-expanded graph (G-TEG), which is suitable for enhancing the modeling capabilities of any TEG. Aronson (1989) summarizes a wide variety of TEG applications, including and not limited to SCM models. This indicates that an analytical description of the GT-CTM in SCM terms can be achieved by directly reading the GT-CTM’s graph structure with SCM terms instead of DTA terms. As the GT-CTM descriptions in DTA terms are provided by Kalafatas and Peeta (2007), we will retain the initial GT-CTM graph structure of Figure 1 in the following paragraphs and then re-describe the GT-CTM in SCM terms.

2.1 Cell and cell connector in SCM

The two fundamental modeling elements of the GT-CTM are the cell and the cell connector (Daganzo 1994; 1995).

A cell, from an SCM perspective, is the part of a homogeneous process with duration equal to the modeling time interval τ . For instance, if a process has a lead time of 'τ time intervals, then it is modeled with a number of consecutive cells equal to the appropriate integer near the 'τ τ ratio (by definition 'τ τ≥ ). A cell in its generalized form is illustrated in Figure 1.a. Depending on the type of process, we define two types of cells. If the process has to do with the actual pro-cessing of materials in a machine, then we define a processing-cell (P-Cell). If the process has to do with the storage of some SKUs in a warehouse or on a finished goods stage, then we define an inventory-cell (I-Cell). Both cells are mathemati-cally and graphically equivalent, and the only reason we differentiate them now is to allow a smoother transition to Karkarkar’s (1989) model later on.

A cell connector links sequential cells and propagates SKUs from an upstream cell to the next downstream cell in the next time interval.

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2.2 Parameters

The supply chain (SC) is represented by the set of cells i C∈ , and the set of cell connectors j E∈ . A cell (P-Cell or I-Cell) belongs to one of three cell types de-pending on its position in the SC: i) it is an origin cell RC C⊂ (or source cell), if there are only downstream cells attached to it and its only incoming flows are from external sources (Figure 1.b), ii) it is a destination cell SC C⊂ (or sink cell), if there are only upstream cells attached to it and its only outgoing flows are to ex-ternal destinations (Figure 1.c), or iii) it is an intermediate cell GC C⊂ , if there are no incoming or outgoing flows from or to external sources or destinations ac-cordingly (Figure 1.d). All cell types described above are graphical and mathemat-ical instances of the generalized cell representation of Figure 1.a. The set of the successor cells of cell i C∈ is ( )iΓ and the set of its predecessor cells is 1( )i−Γ . The maximum work in process inventory (WIP) that can exist in a cell i C∈ in time interval t T∈ is t

iN , and the maximum input or throughput of SKUs is tiQ

for a homogeneous cell i C∈ in time interval t T∈ . In the case of non-homogeneous cells, the maximum input is t

IN iQ and the maximum throughput is t

OUT iQ ( )tIN iQ≠ . The cell connector’s capacity is tjq . The incoming materials

from external sources at a source cell Ri C∈ in time interval t T∈ are tid (sup-

ply), and the outgoing materials to external destinations at a destination cell

Si C∈ in time interval t T∈ are tib (demand).

Fig. 1.a. Generalized cell representation in the GT-CTM (Kalafatas and Peeta, 2007) for single destination DTA (SD-DTA) and single product SCM (SP-SCM).

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Fig. 1.b. Origin cell in the GT-CTM (Kalafatas and Peeta, 2007) for SD-DTA and SP-SCM.

Fig. 1.c. Destination cell in the GT-CTM (Kalafatas and Peeta, 2007) for SD-DTA and SP-SCM.

Fig. 1.d. Intermediate cell in the GT-CTM (Kalafatas and Peeta, 2007) for SD-DTA and SP-SCM.

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2.3 Variables

The fundamental variables of the model are the number of SKUs tix (indicative of

WIP) in cell i C∈ in time interval t T∈ , and the number of SKUs tjy propagated

by cell connector j E∈ in time interval t T∈ (indicative of throughput). We highlight here that a cell connector j E∈ , consistent with the CTM literature, al-ways starts from a time interval t T∈ and ends at the next time interval ( )1t T+ ∈ . We will see later in this study that for cell connectors connecting a P-Cell to an I-Cell we make use of a special structure where these cell connectors start and end in the same time interval t T∈ . The total number of SKUs that ad-vance into cell i C∈ at interval t T∈ is t

IN iy , which is by definition equivalent to the sum of the flows 1t

jy− of the incoming cell connectors ( )1j i−∈Γ . According-

ly, the total number of SKUs in cell i C∈ at interval t T∈ that advance to the next cells is t

OUT iy (total throughput), which is by definition equivalent to the sum of the flows t

jy of the outgoing cell connectors ( )j i∈Γ . The number of SKUs in

cell i C∈ in interval t T∈ that do not advance to the next cells is tiz (remaining

WIP), and it is equal to the difference between the total number of SKUs tix in

cell i C∈ in time interval t T∈ minus the number of SKUs tOUT iy in cell i C∈

in time interval t T∈ that advance to the next cells. The variables described here-tofore are for the conservation of flow constraints presented in the following para-graphs.

All variables are non-negative real numbers.

2.4 Constraints

The constraints presented hereafter originate from the GT-CTM formulation (Kalafatas and Peeta, 2007). The equivalent graph representation for the resulting formulation is illustrated in Figure 1.a in the generalized form. The constraints of the formulation are:

( )1

1t tj IN i

j i

y y−

−

∈Γ

=∑ , ,i C t T∀ ∈ ∀ ∈ (1) 1t t t t

i IN i i iz y d x− + + = , ,i C t T∀ ∈ ∀ ∈ (2) t t t ti i OUT i ix z y b= + + , ,i C t T∀ ∈ ∀ ∈ (3)

( )

t tOUT i j

j iy y

∈Γ

= ∑ , ,i C t T∀ ∈ ∀ ∈ (4) t tj jy q≤ , ,j E t T∀ ∈ ∀ ∈ (5)

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t tIN i iy Q≤ , ,i C t T∀ ∈ ∀ ∈ (6) t ti ix N≤ , ,i C t T∀ ∈ ∀ ∈ (7) t ti iz N≤ , ,i C t T∀ ∈ ∀ ∈ (8)

t tOUT i iy Q≤ , ,i C t T∀ ∈ ∀ ∈ (9)

, , , , 0t t t t tj INi i i OUTiy y x z y ≥ , , ,i C j E t T∀ ∈ ∀ ∈ ∀ ∈

(10)

Constraints (1) to (4) are the conservation of flow constraints. They correspond to the conservation of flow equation of each of the four nodes (read from left to right) in the generalized cell representation of Figure 1.a. The conservation of flow at the cell level for incoming traffic from outside the cell is constraint (1), for in-coming traffic at nodes within the cell is constraint (2), for outgoing traffic at nodes within the cell is constraint (3), and for outgoing traffic from the cell is con-straint (4). From an application standpoint, these four sets of constraints are easy to model because each of them refers to any type of cell. The type of cell is differ-entiated from the values of the supply t

id and demand tib parameters and the ad-

jacency matrix of cell connectors to cells. Constraints (5) to (9) are the arc capacity constraints, and constraints (10) are

the non-negativity constraints. Constraint (5) bounds the SKUs propagated by a single cell connector. Constraints (6), (7), (8) and (9) provide a “hard” upper bound on the total inflow, the number of SKUs in a cell (WIP), the number of SKUs not propagated to the next cell (remaining WIP), and the total throughput, respectively.

2.5 Objective function

The generalized objective function of the GT-CTM is:

( ) ( )1 2 3 4 5 ,t t t t t t t t t ti IN i i i i i i OUT i j j

t T i C j Ec y c x c z c y c y

∈ ∈ ∈

⎛ ⎞⋅ + ⋅ + ⋅ + ⋅ + ⋅⎜ ⎟

⎝ ⎠∑ ∑ ∑ (11)

It is a linear combination of all arc flows in the graph corresponding to the GT-CTM multiplied by the corresponding arc weight. The numerical values of the arc weights are selected according to the model objective.

A widely used objective is the System Optimal (SO) objective. SO was formu-lated in a CTM context by Ziliaskopoulos (2000). In DTA it seeks to minimize the total time spent in the network. In SCM, each flow in a cell (total input, WIP, WIP at the end of the time interval, throughput) or a cell connector can be assigned a different cost other than time (usually in monetary units).

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2.6 Shared properties and models between SCM and DTA

In the previous sub-sections we presented the analytical formulation of the GT-CTM in SCM terms by retaining its graph structure. The key advantage of doing this is that a common graph representation can be used to represent both the SD-DTA and SP-SCM problems using GT-CTM’s exact graph structure.

For instance, in DTA the GT-CTM was used as a tool to study the properties of existence and uniqueness of a traffic assignment solution (Kalafatas and Peeta, 2007). The same properties of existence and uniqueness of an SC configuration can be studied in SCM using the same GT-CTM tools. Therefore, the existence of an SC configuration depends on whether the equivalent minimum cost flow prob-lem is feasible, and a feasible and optimal SC configuration is unique only if there is no zero cost cycle in the residual network.

Kalafatas and Peeta (2008b) recently developed a graph-based formulation for the multiple destinations case of the DTA problem incorporating the FIFO proper-ty. However, the FIFO property substantially increases the multiple destinations DTA model complexity. The analytical study of the analogies between multiple destinations DTA and multiple commodities SCM is one of our future research objectives.

2.7 Discussion

In the current section, the GT-CTM retained its original graph structure and was re-described from an SCM perspective. The key advantage is that it provides a preliminary connection between DTA and SCM which enables the sharing of properties like existence and uniqueness, and advancements like the graph-based formulation for the multi-product case incorporating the FIFO property. However, existing and well-accepted formulations like Karmarkar’s (1989) differ when compared to the GT-CTM’s graph structure. In the next section, we will bridge this gap by showing the relationship between the two formulations.

3. A Single Product Formulation (SPF) as a GT-CTM construct

In this section, Karmarkar’s (1989) deterministic SP-SCM formulation is first pre-sented analytically and then constructed on a GT-CTM base. In two simple steps, we first connect a P-Cell and an I-Cell (which both have an exact GT-CTM repre-sentation), and then we assume infinite WIP capacity for the connected P and I cells.

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3.1 Karmarkar’s SCM Formulation

Karmarkar (1989) discussed congestion phenomena in SCM using a simple and representative single product formulation (SPF) for a single facility. As it is a well-accepted formulation in SCM, we focus on illustrating how it can be built as a construct of GT-CTM’s formulation which originates from DTA.

The variables of the formulation are the WIP tW at the end of period t carried over to period 1t + , the new work tR released for period t , the output or actual production possible tX in period t , the inventory tI of finished product at start of period t , the demand tD in period t and the maximum production tP possible in period t . The constraints of the formulation are as follows:

1t t t tW R W X− + = + ,t T∀ ∈ (12)

1t t t tI X I D− + = + ,t T∀ ∈ (13) ( )1, ,t t t tX f W R P−= ,t T∀ ∈ (14)

, , , , 0t t t t tW R X D P ≥ ,t T∀ ∈ (15) Constraints (12) and (13) are the conservation of flow constraints and define a

graph sub-structure. A graphical representation of this sub-structure appears in Figure 2 which resembles Figure 4 of Karmakar (1989) (the difference is that in Figure 2 only the graph sub-structure of Karmarkar’s formulation is represented while Karmarkar’s Figure 4 additionally depicts the facility as a “black box”). Constraint (14) represents a set of constraints which provide better approximations for congestion modeling. Constraint (15) is the non-negativity constraint for all variables.

Fig. 2. Karmarkar’s (1989) graph sub-structure for SCM modeling.

The objective function of this formulation is the summation of the above prod-

uct flows (in the form of new released work, WIP or finished goods inventory) multiplied by an associated production/inventory cost.

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Karmarkar (1989) identified all SKU flows to be variables in this formulation as the objective of his study was to focus on congestion phenomena. In practice, demand tD and the maximum production tP are usually parameters in SCM, just as network outflows at destination nodes and link capacities are typically parame-ters in DTA problems. Furthermore, as Karmarkar (1989) discussed congestion for a single facility the concept of a cell connector was not required, and thus did not appear in that study. In our case, we included cell connectors (as discussed in Sec-tion 2) in order to represent multiple facility problems as well.

3.2 An equivalent GT-CTM based representation for SP-SCM

In this section we will develop an equivalent formulation to Karmarkar’s (1989) SP-SCM formulation starting from the GT-CTM, in two simple steps. We will first connect a P-Cell to an I-Cell and then perform some mathematical (and graphical) reductions.

In the first step we assume an upstream P-Cell and a downstream I-Cell. Then, we connect them with a cell connector, such that the cell connector starts and ends in the same time interval. We note that this modeling detail is not a necessary re-quirement for SCM models and we mainly illustrate it for the consistency of our DTA formulation with Karmarkar’s (1989) SCM formulation. Cell connectors have not been used in this way before in the CTM literature, hence we introduce this modeling detail here as a generalization of the GT-CTM graph modeling ca-pabilities and as an indicative example of GT-CTM’s modeling adaptability. We call the resulting graph representation a joint processing/inventory-cell (P/I-Cell) and illustrate it in Figure 3.a, where subscript i corresponds to the functions of the P-Cell, subscript r corresponds to the functions of the I-Cell and subscript p corre-sponds to the cell connector connecting the P-Cell and the I-Cell. Since the P-Cell and the I-Cell are linked serially with three arcs (total throughput of the P-Cell, cell connector, total input of the I-Cell), we replace all three with a single cell connector with capacity ' tpq equal to the minimum capacity of the three replaced arcs, as illustrated in Figure 3.b. This second graph representation of the P/I-Cell (Figure 3.b) is mathematically equivalent to the first graph representation or the P/I-Cell (Figure 3.a).

In the second step, we assume infinite capacity for the WIP capacity constraint for both the P-Cell and the I-Cell. Therefore, the corresponding arcs ( itx , rtx of Figure 3) are graphically deleted from the P/I-Cell, in the sense that their starting and ending node are reduced into a single node. The mathematical equivalent is the replacement of their flow variables with the use of the corresponding conser-vation of flow equality equations at the starting and ending node. The resulting graphical representation is illustrated in Figure 4, and it is the graph representation of Karmarkar’s (1989) formulation graph sub-structure (of Figure 2) extended for

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multiple facilities (with the support of cell connectors) and the additional model-ing capability of total facility input and output capacity.

Fi

g. 3

.a. J

oint

pro

cess

ing/

inve

ntor

y ce

ll (P

/I-Ce

ll) fo

r SP-

SCM

.

Fig.

3.b

. Mat

hem

atic

ally

equ

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ent P

/I-Ce

ll af

ter g

raph

redu

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ns.

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This second step is indicative of the following key issue in modeling conges-tion phenomena. In the GT-CTM the finite capacity of the itx and rtx arc flow variables is the equivalent of DTA’s finite jam density. Finite jam density in the GT-CTM allows the modeling of congestion phenomena such as highly dense traffic conditions (or even gridlock), queue spillbacks and backward propagating traffic waves. In the context of SCM such congestion phenomena resemble situa-tions where the SC is disrupted and WIP or finished product are queued through a series of facilities. However such congestion phenomena are not captured in the graph sub-structure of Karmarkar (1989) as described by the corresponding con-straints (12) and (13). The broader constraint (14) can potentially capture such congestion phenomena, but it does not integrate them elegantly in a graph sub-structure similar to that provided by constraints (12) and (13). In fact, this issue is the major difference between Karmarkar’s (1989) formulation and ours as depict-ed in Figures 3 and 4. Karmarkar depicts facility level congestion phenomena (constraint 14) as a “black box”, while our Figures 3 and 4 provide more analyti-cal graph representations for such congestion phenomena at the facility level. This further implies that the GT-CTM’s graph-based congestion modeling capabilities can facilitate simpler and more computationally efficient formulations for SCM with such congestion modeling capabilities because more congestion related con-straints become part of the computationally efficient minimum cost flow structure.

Fig. 4. Representation of the graph sub-structure of Karmarkar’s (1989) single-product for-mulation for multiple facilities and total facility input and output capacity.

Finally, the conceptual, graphical and mathematical equivalence we showed for

the GT-CTM based DTA formulation and the SCM formulation may indirectly imply that these formulations are expected to generate the same optimal solutions. The equivalency in the flow conservation constraints (hard constraints) do imply the above notion but in the end it is the (mostly empirical) calibration of the func-tional form (and the corresponding numerical values) of the congestion related constraints (fundamental flow-density diagram, clearing functions) that will de-termine how close the optimal solutions are.

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3.3 Discussion

So far in this section, we showed that the GT-CTM can be used to construct exist-ing graph-based SC representations. The fundamental difference from the original GT-CTM structure is that we allow a cell connector to start and end in the same time interval. Although this is a minor extension of the original GT-CTM struc-ture, it is significant because it reveals GT-CTM’s adaptability to content-specific modeling of dynamic phenomena with the property of retaining (a slightly modi-fied) graph structure, and therefore its minimum cost flow computational com-plexity. However, this modeling approach does not apply in DTA because the concept of “finished goods inventory” suggests but is not equivalent to queued ve-hicles which will remain in the same cell in the next time interval. Furthermore, we identified how congestion phenomena such as highly dense traffic conditions, queue spillback and backward propagating traffic waves can be modeled with the use of the GT-CTM’s graph structure in the context of SCM.

So far, we moved conceptually from DTA to SCM. We showed how DTA properties and models can be applied in SCM and how congestion in this context can be captured in a single graph representation. In the next section, we will seek to identify SCM phenomena and interpret them from a DTA GT-CTM perspec-tive.

4. SCM phenomena from a DTA perspective

In this section, we will use the GT-CTM modeling framework to study SCM phe-nomena from a DTA perspective. These phenomena include congestion (As-mundsson et al., 2002) and interconnection of sub-assemblies.

4.1 Congestion

In DTA, each road link is associated with a minimum link travel time, labeled free-flow travel time. The link travel time will increase as the link’s traffic density increases. This phenomenon is described in DTA as congestion. It is usually cap-tured by link performance functions, which express link travel time as a function of the link flow, and can also be described using the fundamental traffic flow- density diagram. The GT-CTM does not require an additional link performance function to capture congestion, as it is indirectly captured through the graph rela-tions of its internal arc flows; GT-CTM’s graph relations constitute a linear ap-proximation of the fundamental traffic flow-density diagram.

Accordingly, in SCM a process can be associated with a minimum lead (or processing) time, which is achieved when the WIP has a relatively low value.

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Congestion in SCM is the phenomenon where lead time increases as the WIP in-creases. Clearing functions express throughput as a function of WIP (Karmarkar, 1989; Asmundsson et al., 2002), and represent conceptual analogs to the funda-mental traffic flow-density diagram in a DTA context.

However, a careful examination of clearing functions indicates that congestion is also considered the phenomenon where the maximum throughput of a process is not achieved at conditions of minimum lead time. This is also true for traffic oper-ations according to the Highway Capacity Manual (HCM, 2000). In the simplest graph form of the GT-CTM, maximum throughput (maximum traffic flow) is achieved at conditions of minimum lead time (minimum travel time), and only af-ter this point lead times (travel times) increase as a function of WIP (traffic densi-ty). Interestingly, this further indicates the potential for the GT-CTM to be extend-ed (in future research) for traffic operations with the proper adaptation and calibration of some equivalent clearing functions.

4.2 Sub-assemblies

A characteristic phenomenon in SCM is the organization of production in sub-assemblies of different sub-components which are then assembled in a single new component. A similar phenomenon in DTA is not typical and may not exist at all. The reason is that vehicles are independent units in typical traffic systems, and do not require the presence of another vehicle to propagate to the next traffic link.

As sub-assemblies do not exist in DTA, we will address them by an ad-hoc modification on the GT-CTM modeling framework, specifically for SCM. Let’s assume that in order to assemble a unit of component k , it is required to consume lγ units of sub-component l and mγ units of sub-component m . Only when both

a number of tly sub-components l and a number of my sub-components m arrive

at the P-Cell, a number of tIN ky components k will be produced. Thus, the follow-

ing equations have to hold:

t tl l IN ky yγ = (16) t tm m IN ky yγ = (17)

When linear equations (16) and (17) are added to the original formulation (1)-

(11) for the corresponding cells which will link sub-assemblies, the pure minimum cost flow problem becomes a minimum cost flow problem with side constraints (Ahuja et al., 1993). The resulting formulation is more complex, but it is still a well-known mathematical problem.

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4.3 Discussion

In this section, we examined how congestion and sub-assemblies are modeled in SCM. Congestion modeling, with the use of clearing functions, shares similar concepts with DTA, thus advances from both DTA and SCM can be used to en-rich the modeling capabilities of the GT-CTM. The existence of sub-assemblies in SCM does not have an equivalent in typical DTA applications. Therefore, a spe-cial set of constraints (12)-(13) was developed as an ad-hoc modeling approach, which altered the classification of GT-CTM’s formulation to minimum cost flow with side constraints. The two phenomena studied in this section indicate that fur-ther research can provide more insight on the similarities between SCM and DTA.

5. Conclusions and future research

Concluding, this study illustrated the capabilities of the GT-CTM as a direct bridge between DTA and SCM. The GT-CTM can be used for SCM applications in its original form (Section 2, Figure 1), or with a special graph construct (Figures 3, 4) in order to better resemble Karmarkar’s (1989) modeling approach (Section 3.1). In both cases the DTA properties of existence and uniqueness of a traffic as-signment solution can be studied in SCM applications. In the second form of the GT-CTM (Figure 4), we utilize for the first time the concept that cell connectors may start and end in the same time interval. This enriches the modeling capabili-ties of the GT-CTM as a general graph theoretic model and further illustrates its modeling adaptability. However, this approach does not directly apply in DTA. Further, we discussed the definitional analogies between link performance func-tions, the fundamental flow-density diagram and clearing functions (Section 1) and later on we identified how congestion phenomena such as highly dense traffic conditions, queue spillbacks and backward propagating traffic waves can be mod-eled using the GT-CTM’s graph structure in the context of SCM (Section 3.2). Fi-nally, in Section 4 we discussed how congestion and sub-assemblies are viewed from an SCM perspective in the modeling environment of the GT-CTM. In future research, we will seek to adapt and calibrate equivalent clearing functions for the GT-CTM and examine whether the notion of clearing functions for SCM can be meaningfully extended for the highly congested region.

Acknowledgments We would like to acknowledge Professor Reha Uzsoy of North Carolina State University and an anonymous referee for their useful comments.

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