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Synchronization Transitions in Supply Chain Networks
Florian Klug*
Department of Business Administration, Munich University of Applied Sciences, Germany
* Corresponding author:
Florian Klug
Department of Business Administration
Munich University of Applied Sciences
Am Stadtpark 20
D - 81243 München
Germany
Tel: +49 89 1265 2778
Fax: +49 89 1265 2714
Email: florian.klug@hm.edu
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Synchronization Transitions in Supply Chain Networks
Abstract
In this paper, we examine the transition process between synchronous and desynchronous states in supply
chain networks, which is strongly correlated to network topology. While most research has focused on
the merits of steady-state synchronization behavior, very little has been published on the
interdependencies between structural and dynamic synchronization properties in supply chain networks.
Synchronization processes are described by a variety of coherent state dynamics that exhibit very
different transition paths. In this research, we developed a new coupled oscillator model to analyze and
evaluate these transient dynamics in more detail. Canonical network models are first described, including
hub-and-spoke, tree-mesh, and tree topologies, which are then applied to real-world problems in the
automotive industry. We found specific synchronization transitions in automotive supply networks such
as explosive and delayed synchronization. In particular, we show that (i) in a hub-and-spoke supply
network, the number of suppliers per hub affects synchronization transitions only at larger frequency
spreads, (ii) scale-free networks with short path lengths generated by the power law connectivity are
unable to synchronize as long as a critical cluster coefficient is exceeded, and (iii) interruptions in tree
networks cause nonlinear synchronization losses for upstream suppliers as a function of the interruption
time.
Keywords: Logistics, Network synchronization, Automotive supply chains, Econophysics
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1. Introduction
A synchronized supply chain network is generated when the individual suppliers of a network move in a
collective rhythm and interact smoothly in a harmonized and coordinated manner without interruptions.
Potential benefits associated with synchronized supply chains include performance improvements
through an increased speed of material flow, inventory reduction, enhanced responsiveness, dampening
of demand amplification, and better long-term planning. Improper synchronization can lead to
instabilities, which cause very complex dynamics. Large-scale supply chain networks consisting of
thousands of geographically dispersed partners with physical or virtual boundaries are difficult to
organize and synchronize (Lim et al., 2017). Therefore, synchronization generation as a transient and
non-equilibrium behavior in a stochastic environment makes the design, planning, and control of complex
supply chains challenging. In this research paper, we focus particularly on the interesting and
sophisticated interaction between synchronization transitions and topology, to achieve a finer
understanding between the non-linear coherence dynamics of interconnected suppliers and the
corresponding network topology. For this purpose, we used an oscillator approach, which has been
already applied very successfully in the description and solution of nonlinear problems in most diverse
fields of application. In his seminal work, Kuramoto (1984) introduced a new oscillator model and
demonstrated the nontrivial relationship between synchronization patterns and system topology.
Nowadays, science and engineering offer a variety of oscillatory models and approaches that can be
applied to different supply chain topologies. Whilst synchronization and topology interactions in physics
(e.g., laser network), engineering (e.g., power network) biology, and neurology (e.g., neural network) are
a common research field, there are only a few studies that refer to the area of operations and supply chain
management. In the literature, the following publications on synchronization network interactions in
supply chains could be located.
A synchronization analysis of material or traffic flows in networks using phase entrainment was presented
by Lämmer et al. (2006). They showed that, under certain assumptions, the control of nodes can be
mapped to a network of phase oscillators, and illustrated the method with an example of traffic signal
control for road networks. Material flows in production systems are mentioned as a possible area of
application for the proposed concept. Helbing et al. (2004) described supply networks as a physical
transport problem with the help of balance equations for the flows of products and for the adaption of
production speeds. Here, a stability analysis was performed by linearizing the model equations, which
generated a coupled set of second-order differential equations. They can be interpreted as a set of
equations for linearly coupled damped oscillators. With the help of this oscillatory model, the bullwhip
effect is described as a connective instability, which corresponds to resonance effects. To the best of our
knowledge, this study by Helbing et al. (2004) is the first description of generic oscillators for supply
networks. Based on this work, Donner et al. (2007) and Donner (2008) studied a decentralized control
approach for the regulation of intersecting material flows within a symmetric grid network. They reported
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that this self-organization process of the system is related to a transition from non-synchronized to phase-
synchronized flows. They found that the network performance and the degree of phase entrainment are
closely related to each other. Although Donner (2008) discussed the Kuramoto model for the degree of
phase entrainment, the main study of phase synchronization was not based on a generic oscillatory model,
where oscillators are explicitly modelled. The solutions were generated on the basis of flow-oriented
balance equations for the temporal evolution of the input and output material flows. Armbruster et al.
(2006) showed that a closed supply chain structure in a re-entrant factory with cycles acts as a damping
device reducing the amplitude of the oscillations in the work-in-progress in the downstream factory.
Scholz-Reiter et al. (2006a) and Scholz-Reiter and Tervo (2006b) investigated the synchronization
phenomena of a linear supply chain with oscillating demand and supply. They found a strong dependence
of the synchronization phenomena on the adaption speed of the production rate. Simchi-Levi and Wei
(2012) proved that adding an arc and closing a manufacturing chain can improve flexibility in a balanced
and therefore synchronized system. Gan and Wang (2013) investigated synchronization problems for a
class of supply chain networks that have nonlinearly coupled identical nodes and an asymmetrical
coupling matrix. The research focused on the construction of an adaptive coupling strength controller,
which generated global synchronization, but without addressing the specific characteristics and
requirements of supply chains.
This brief review indicates that the existing studies are very limited. Furthermore, the use of the analysis
methods shows that oscillator models were used only very sporadically, if at all. Therefore, it is important
that the existing general studies on supply network synchronization are supplemented by concrete
empirical applications for specific network topologies. In addition to the practical need for explanatory
approaches to the existing synchronization dynamics, oscillator models offer new possibilities and can
be understood within a well-established framework based on modern nonlinear dynamics. By combining
the fundamental questions of supply chain topology with the dynamic rhythmic representation of
oscillators, we can develop new methods to analyze synchronization transitions in supply chain networks.
The rest of this paper is organized as follows. In Section 2, we introduce the phase-coupled oscillator
model used, which is then applied to three different network topologies (hub-and-spoke, tree-mesh, and
tree networks). Section 3 examines the synchronization behavior of classical network topologies, which
serves as a benchmark for further empirical investigations. Based on real applications in the automotive
industry, an empirical analysis of synchronization dynamics is discussed in Section 4, from which the
corresponding managerial insights are derived. Section 5 presents a brief summary of the study, its
limitations, and an outlook on further research.
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2. Model formulation
2.1 Node dynamics
We characterized the supply chain network with the help of an oscillator network with continuous
coupling, representing the supply chain dynamics of each partner and its relations. This method posits
that the local supply chain activity can be represented by its phase variable alone. Each node of the supply
chain network vibrates according to its natural frequency and its specific excitations, induced by its
network partners. The oscillatory model was characterized by a population of N non-identical phase
oscillators :
. We used the generalized Kuramoto model with
intrinsic frequencies, where the supply chain network is represented by a set of N coupled differential
equations (Kuramoto 1984):
The phase and its instantaneous frequency (speed) describe the state and its dynamic evolution
of each oscillator in the network, respectively, with its intrinsic frequency , distributed with a given
probability density g(ω). The intrinsic or natural frequency is related to the oscillation period ωi = 2π/Ti.
The oscillator coupling is mapped by a -periodic, anti-symmetric sinusoidal function, which is
specified by an adjacency matrix and generally guided by the phase differences between the
neighboring supply chain partners. The elements represent the links (if ) of the supply
network, and denotes the coupling strength between two supply chain partners. The overall
dynamics of the network model results from a balance between the goal of collaboration in the supply
network (induced by the oscillator coupling) and the tendency of each oscillator to autonomously align
with its personal inherent driving frequency ωi, which represents its self-sustained dynamics. This trade-
off forms the core principle of any supplier network. On the one hand, each supplier acts autonomously
with different goals and is subject to different constraints; on the other hand, cooperation in a supply
chain network is necessary for the fulfillment of customer orders. Despite the apparent simplicity of the
model, equations in (1) are difficult to analyze in general, as the interaction functions can have many
arbitrary Fourier harmonics and the connection topology is unspecified (Strogatz, 2000).
To visualize the network dynamics of the phases, we used the common order parameter defined as
follows:
where corresponds to the centroid of the phases. This phasor representation allows one to
depict the mean phase of the set of N network oscillators (representing the collective
dynamics of the supply network) and with the vector r(t) to measure the normalized phase coherence of
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the oscillators. The order parameter of the centroid vector ranges from r(t) = 1 (coherence), when the
system is fully synchronized and r(t) = 0 (incoherence), when the unsynchronized phases are uniformly
distributed. Thus, the amplitude r(t) of the oscillator network acts as an indicator for synchronization and
as an order parameter of the synchronization transition (see Sections 3 and 4).
2.2 Network topology
The supply chain network and the coupling strength among the oscillators are mapped to a bidirected,
and weighted graph with node set N and edge set , where denotes an
edge between nodes Each edge is weighted with , . This generative
model with local dynamics implies fixed non-identical supply chain partners. Time as a crucial planning
variable is treated intrinsically in this network model, so that an expanded network model with an
enumeration of the model’s relevant time periods is not required (Simchi-Levi et al., 2014). The degree
of each node ki (ni) is defined as the number of edges interlinked with a node ni. Our research focus was
the synchronization dynamics of the existing inbound network structures in the automotive industry. For
further investigations, we selected three major topologies for the following reasons:
• The topologies represent the main inbound supply chain network types from a car manufacturer’s
point of view (OEM: original equipment manufacturer), widely spread in the automotive
industry.
• These partial networks show the maximum intra-interaction within the network and the minimum
inter-interaction between the sub-networks.
• These network types can be found in a variety of other application areas, allowing the analysis
results to be transferred to other industries.
The matching of real inbound automotive supply chains with common network architectures shows a
high affinity with hub-and-spoke, tree-mesh, and tree networks (Sun & Wu, 2005; Parhi, 2008; Keqiang
et al., 2008; Hearnshaw et al., 2013; Mari et al., 2015; Sun et al., 2017). The degree distribution of the
nodes ni is mostly heterogeneous, where the number of neighbors is not uniformly distributed (see Table
2). On the basis of the car manufacturer’s inbound volume as a major driver of the network topology, we
obtained a classic Pareto distribution, which allowed us to split the total inbound supplier network of a
car manufacturer into three different sub-networks.
2.2.1 Hub-and-spoke network
More than half of the direct suppliers (so called C-suppliers) count for around 20% of the total supply
volume, are organized in a hub-and-spoke topology (see Table 2). This break bulk delivery is
characterized by low transport volumes with standardized parts and components. According to the low
inbound volume per supplier, consolidation transport is necessary. Therefore, break bulk cargo is
consolidated in hub-and-spoke transportation networks. Each hub is managed by a logistics service
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provider, operating the consolidation terminal and processing the main-run transport between a hub and
OEM. Initially, short-distance pre-run deliveries from the regional C-suppliers are transported to and
merged in a consolidation hub to create full truck loads (FTLs). These FTLs are then delivered via main-
runs over longer distances directly to the car manufacturer’s assembly plant. The main characteristic of a
hub-and-spoke network is the reduced connectivity between the supply chain partners N. Compared with
an all-to-all coupling with connections or , only supply chain connections are
applied with Load consolidation in nodes with a high degree (hubs) increases the transport volume
of the main runs and reduces the freight rates on the basis of the economies of scale.
2.2.2 Tree-mesh network
Approximately one-third of the supplier base, which counts for around 30% of the total inbound volume
(so called B-suppliers), is organized in a hybrid form between a tree and a mesh network (see Table 2).
Delivery items for this type of inbound supply constitute medium transport volumes for the car
manufacturer. Ordering and delivery cycles are on a daily basis with a mixture of make-to-stock and
assemble-to-order, depending on the specificity and value of the delivery components. The basic supply
chain structure is based on a mesh network for the C-suppliers (2-tier) integrated into a tier system with
the B-suppliers (1-tier). While B-suppliers deliver aggregated assemblies or specific components to the
car manufacturer, they receive standardized parts and components from the sub-suppliers. On the 2-tier
level of standardized components, an n (1-tier supplier) to m (2-tier supplier) supplier relationship
dominates. According to the heterogeneous mesh structure of the network, some nodes have a
disproportionately large number of connections related to other nodes.
2.2.3 Tree network
Around 20% of the suppliers (A-suppliers), counting for half of the total delivery volume, are organized
in a just-in-sequence supply chain network. The further up the supply chain we move, the larger is the
number of suppliers involved in the value creation process, where supply chain partners on the different
delivery tiers interact in a strict hierarchical tree structure (see Table 2). Just-in-sequence (JIS) delivery
is a demand-driven logistics concept, where complete loads are delivered several times a day in sync with
the production (Bennett & Klug, 2012). The prerequisite is a high delivery volume with a constant
delivery frequency. This supply network type is suitable for complex, large-volume, and customer-
specific modules and systems that require late configuration. The modules are supplied by the Tier-1
supplier, in accordance with the sequence call-off, in the same sequence and synchronized with the
OEM’s master production schedules. Freight forwarders are integrated into the information process and
are provided with transport orders, including quantities and collection dates and times. JIS modules and
systems are assembled with sub-modules and components, delivered by 2- and 3-tier suppliers. These
delivery contents are still very specific and therefore mainly usable for the module manufacturing of the
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Tier-1 supplier, so that no other suppliers are delivered with these delivery scopes. On the 3-tier level of
standardized components, an n (2-tier supplier) to m (3-tier supplier) supplier relationship dominates.
Despite the mesh connections on the first level of the value creation process, the hierarchical tree topology
dominates the supply network between the OEM and its Tier-1 and Tier-2 suppliers.
3. Synchronization transition analyses
First, we investigate the synchronization behavior of classical network topologies, which on the one hand,
provide pure results in terms of their prototypical topology, and on the other hand, serve as a benchmark
for empirical investigations in Section 4. Referring to the oscillator model introduced in the previous
chapter, we define supply network synchronization as the adjustment of different supplier phase angles
depending on their phase difference (see Section 2.1 and Table 1).
Table 1. Different synchronization and desynchronization states
Phase synchronized
Phase locked
Desynchronized
DEFINITION 1. Supply network model (1) is said to be frequency synchronized, at a specified coupling
strength, if
for all with For identical
oscillators with equal natural frequency (1) is phase synchronized, if
for all .
DEFINITION 2. Within time T > 0, supply network model (1) is said to be phase locked, at a specified
coupling strength, if their phases are asymptotically identical to with
, for all ,
DEFINITION 3. Supply network model (1) is said to be unsynchronized/incoherent if
π0
π
2
3π
2
�
Y
π0
π
2
3π
2
Y
�
π0
π
2
3π
2
Y
�
9
The solutions of (1) to calculate the critical coupling of the synchronization onset in a network topology
require the solution of simultaneous transcendental equations, which can be solved for the complete graph
with all-to-all coupling.
THEOREM 1. In a supply network topology of non-identical supply chain partners N, with all-to-all
coupling and coupling strength K = /N (with coupling constant ), each supply chain partner interacts
independently with the supply network field .
PROOF OF THEOREM 1.
According to (2):
multiplied with
According to Euler’s formula with
and only considering the imaginary part,
we obtain the following:
. Thus, (1) becomes the following:
(3)
Sinusoidal coupling function (3) can be thought of as the phase response of oscillator i to the supply
network represented by the mean field . The interaction function vanishes when the phases are
identical or differ by , representing the phase and anti-phase synchronization. In the neighborhood of
the phase identity of , the sinusoidal values pull the phase towards the mean phase
, while in the case of the near-antiphase of , the sinusoidal value pushes the phase
towards the network phase . In addition to this transient synchronisation state, a single attracting
synchronous and a single unstable antiphase constellation exist. This dynamic leads to a spontaneous
synchronization with a positive feedback loop, when the greater phase coherence (represented by r)
increases the phase adjustment dynamics in (3), which thus leads to a further increase in the phase
coherence.
We were particularly interested in the phase transition from the desynchronized (incoherent) to the
synchronized (coherent) states, which we first investigated in the limit with all-to-all coupling.
For the empirical investigation in Section 4, we adopted the premises to the finite size case (N < ) with
sparse interconnection topologies. The crucial parameter in (3) are the coupling strength K, coupling
constant , and the distribution of the intrinsic frequencies g(). In general, a weakly coupled and
strongly heterogeneous supply chain network does not display any coherent behavior, whereas a strongly
coupled network with similar intrinsic frequencies is amenable to synchronization.
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THEOREM 2. We assume a symmetric, unimodal, and even probability density g(ω) of the natural
frequencies with a single mean frequency at . In the supply network limit , with all-to-all
coupling, gradually increasing the coupling constant results in a bifurcation or continuous-phase
transition from an asynchronous to a synchronous steady-state solution as the coupling constant reaches
a critical value
PROOF OF THEOREM 2. According to (2), the mean field, representing the global supply network
dynamics, has amplitude r and phase :
With a periodic oscillating mean field with frequency , we obtain , and by variable
transformation of = for (3), we obtain the following:
Two steady-state solutions exist for the supply chain network with constant r as a synchronous solution
for with a stable fixed point where oscillators are phase-locked at frequency with
and as an asynchronous solution for , where oscillators are drifting around the phasor
circle.
For the locked-phase cluster, the phase difference is time-independent and is determined by the intrinsic
frequency according to with the distribution directly derived from the distribution of the
intrinsic frequencies with the following:
Supply network stationarity demands that drifting oscillators form a stationary distribution on the phasor
circle (see Table 1) with constant r. This implies that the proportion of oscillators with intrinsic
frequency that lie between defined by
is inversely proportional to the speed at . The normalization constant is determined by
for each , with
By averaging over , we obtain the following distribution for the drifting oscillators:
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By combining the asynchronous and synchronous populations of the phase oscillators, we obtain for the
network field according to (4), the following:
The drifting part of the distribution has period and therefore does not contribute to the integral
(9), which generates an equation for the frequency of the network field.
The network field is oscillating with , and according to the symmetry of , equation (10) is fulfilled.
For the amplitude of the network field, we obtain the following:
Besides a trivial solution for r = 0 corresponding to incoherence, the transition to synchronization can
be characterized by the critical coupling constant as follows:
where a second branch bifurcates continuously from r = 0.
THEOREM 3. We suppose a symmetric, unimodal, and even probability density g(ω) of the natural
frequencies with a single mean frequency at . In a tree topology, gradually increasing the identical
coupling strength results in a bifurcation or continuous-phase transition from an asynchronous to a
synchronous steady-state solution as the coupling reaches a critical value:
(12)
where
is a subtree of the tree graph
PROOF OF THEOREM 3. See Dekker and Taylor (2013).
THEOREM 4. We suppose a symmetric, unimodal, and even probability density g(ω) of the natural
frequencies with a single mean frequency at . In a hub-and-spoke topology, gradually increasing the
identical coupling strength between the hub and its spokes results in a bifurcation or continuous-phase
transition from an asynchronous to a synchronous steady-state solution as the coupling reaches a critical
value:
(13)
PROOF OF THEOREM 4. See Dekker and Taylor (2013).
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4. Empirical analysis and managerial insights
We extended our analysis to other non-classical network topologies with hybrid structures, as discussed
in Section 2.2. To further investigate the synchronization-topology interactions in a real-world setting,
our model (1) was mapped and validated on the basis of the existing inbound supply networks of a large
German car manufacturer. All of the parameters used were based on the empirical data of supply chain
management, combining a range of different information (order, inventory, and transport). Unless
otherwise specified, all the simulation runs were performed over a period of 72 days. An initial transient
phase of 5 days each was not considered in the calculation. The coupling strength K was assumed to be
indirectly proportional to the number of interacting suppliers N, with K = 1/N, if not otherwise stated.
The inbound supply volume, structure, and frequency, as in other industries, were the main criteria for
the selection of a specific supply chain network topology, as discussed in Section 2. Thus, all the suppliers
involved in the supply chain process were divided into three supply network classes: hub-and-spoke
network, tree-mesh network, and tree network (see Section 2.2). All the parameters used for the empirical
analysis are listed in Table 2. We assumed for the probability density g(ω) of the natural frequencies,
normal distributions with a single mean frequency at and standard deviation .
Table 2. Parameters of automotive supply chain networks
General topology
Hub-and-spoke network
Tree-mesh network
Tree network
Specific degree
distribution
Node
Hub 1
C1
Hub 2
C2
Hub 3
C3
Tier 1
B
Tier 2
C
Tier 1
A
Tier 2
B
Tier 3
C
No. of nodes Ni
Ti [d]
[d]
[rad/d]
[rad/d]
[rad/d]
[rad/d]
32
0.52
0.01
12.08
0.23
4.43
0.46
54
0.44
0.01
14.31
0.32
13.84
0.23
77
0.48
0.01
13.09
0.27
3.75
0.56
5
3.70
0.70
1.70
0.32
30
6.68
3.40
0.94
0.48
1
0.02
0.00
430.84
102.17
6
2.90
0.53
2.17
0.39
30
4.90
4.07
1.28
1.07
0 5 10 15 20 25 30k
0.2
0.4
0.6
0.8
1.0
P(k)C
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To map the supply network clustering, also known as transitivity, of the supply network graph
, we used the global clustering coefficient C. Network transitivity indicates the average
probability of two neighboring nodes that are connected to a given local node, also connected to each
other (forming a triangle). It quantifies closed three-point (triangles) correlations and reflects the
cliquishness among the neighbors. The clustering coefficient is the ratio between the number of close
triadic correlations that actually exists between the nodes ni and the total number of triples (connection
of three nodes):
where each triangle generates three triples, one centered on each of the three nodes (Boccaletti et al.,
2006).
4.1 Hub-and-spoke network
We investigated three typical hubs with different supplier numbers (hub 1 = 32 suppliers, hub 2 = 54
suppliers, and hub 3 = 77 suppliers) and different geographical distances to the OEM (see Section 2.2.1).
The OEM assembly plant was supplied from each hub twice a day ( with small continuous
fluctuations (). C-suppliers within the consolidation area did not only deliver for one vehicle plant but
supplied all European-wide production sites of the vehicle manufacturer. The pre-run times ranged
between = 0.45 d and = 1.68 d.
A comparison of the centroid vector r (2) as main coherence measure in Figure 1 shows that hub 2
synchronizes on a much lower critical coupling strength. The coherence phase transition was nearly linear
up to approximately KC2 = 0.5, where a very high degree of synchronization between a pre-run and the
main-run was achieved. Compared with hub 2 that exhibited a gradual increase in the phase coherence,
hub 3 and hub 1 showed a sudden change when the critical coupling strength (KC3 = 0.41 and KC1 = 0.52)
was reached. A comparison of the intrinsic frequencies of the main- () and pre- () run of hub 2
revealed that a difference of only or guaranteed a phase coupling at a low
coupling strength, which corresponded to the theoretical value of
calculated in (13).
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Figure 1. Development of phase coherence r for hub 1 to hub 3
PROPOSITION 1. Coherence transition to synchronize in a hub-and--spoke supply network changes
from continuous to discontinuous, when the frequency spread between the suppliers and the hub
increases. Rising supplier numbers per hub impact the critical coupling strength only for larger frequency
spreads.
We increased the number of suppliers for each hub from 20 to 80 and mapped the coherence transition in
Figure 2. In the difference between hub 2 (left) and hub 1 (right), synchronization transitions showed the
same dynamic as shown in Figure 1. The coherence transition to a synchronized one in hub 2 with a low
frequency spread ( ) changed to an erratic one in hub 1 with a high frequency spread (
. The influence of the hub size (number of suppliers linked) only affected the synchronization
development, if the frequency spread was large. Such an erratic change in the synchronization of hub-
and-spoke (star) topologies is known as an explosion synchronization transition (Gómez-Gardenes et al.,
2011). This first-order transition occurred when the natural frequency spread exceeded a threshold.
Importantly, the onset of synchronization was delayed when the number of suppliers decreased (see
Figure 2 right). Additionally, the greater the sharp and sudden increase in the supply chain coherence
was, the smaller was the hub ( From a supply network control perspective, we concluded
that the smaller the hub size was, the later but more erratic was the synchronization of the supply chain
partners.
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Figure 2. Development of phase coherence r with different hub sizes for hub 2 (left) and hub 1 (right)
PROPOSITION 2. Hub synchronization can be improved by introducing pre-run milk-runs.
Pre-run transports from the supplier to the hub terminal can be either organized by the supplier or the
operator of the hub terminal, depending on the delivery conditions. By changing the incoterms from Free
Carrier (FCA) to Ex Works (EXW), the area freight forwarder assumes responsibility not only for the
hub operations and the main-runs but also for the pre-runs. The introduction of groupage round-trip
transports, so-called pre-run milk-runs, allows to bundle shipments from several low-volume suppliers.
In the milk-run, the truck starts from and ends at the consolidation center (hub) to successively approach
the individual C-suppliers of the round-trip. The consolidation of break bulk loads reduces the freight
costs and increases the delivery frequency, as compared to individual deliveries. At the same time, goods
receipt capacities and hub processes can be better planned, as truck deliveries are regularly made at the
predefined time periods.
To map the application rate of milk-runs for the pre-run transports, we introduced a milk-run coefficient
ranging from 0.0 (no milk-runs) to 1.0 (all suppliers were linked to one large milk-run). This changed
the hub network topology from a star to a hybrid star-ring structure. Figure 3 depicts the phase coherence
development r(CMR) of hub 1 (left) and hub 2 (right) upon an increase in the milk-run coefficient. In both
the cases (the same was true for hub 3), synchronization improved upon the formation of ring supply
structures in the pre-runs. Obviously, the introduction of milk-runs had a stabilizing effect, and therefore,
the phases could couple more easily than the individual deliveries. Accordingly, isolated deliveries
caused higher fluctuations in the incoming goods, which could be dampened by introducing milk-runs.
Because of the organizational restrictions, the number of suppliers in a milk-run was limited; therefore,
in practice, the full synchronization potential of supplier bundling could not be exploited. Comparing the
synchronization performance in both the cases, we found that the coherence effect of an increasing
coupling strength r(K) was higher than the milk-run effect r(CMR).
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Figure 3. Comparison impact of coupling strength K versus milk-run coefficient CMR on phase coherence
r of hub 1 (left) and hub 2 (right)
PROPOSITION 3. Interlinking of hubs improves inter-hub synchronization, while the effect on spoke
synchronization is inconsistent, depending on the frequency detuning between the hub and the spoke.
In the next step of the phase coherence analysis, we investigated the possibility of interconnecting hubs.
To map the coupling strength between different hubs, we introduced a hub coefficient ranging from
0.0 (no coupling between hubs at all) to 2.0 (extreme strong coupling between all the hubs). This coupling
strength represents the degree of integration of freight flows between hubs. Although each transshipment
point is managed separately by a major freight forwarder within the region, hubs in the automotive
industry are operated in an interconnected system. Here, inter-hub freight relations were established,
which allowed the development of a national, European, and sometimes worldwide break-bulk freight
system, also well-known in the courier, express, and parcel (CEP) services area. Because of the
economies of scale, these multi-level hub systems with central master and local slave hubs are particularly
useful for very large freight volumes. Vehicle manufacturers can thus realize economies of scale in freight
volumes and time advantages in freight relations by increasing transport frequencies, particularly in a
multi-brand environment, as in our area of investigation. In several simulation runs, the relationship
between the pre-run milk-runs and the hub compounds was investigated in more detail. Table 3 shows an
example of the characteristic synchronization transition, as was predominantly demonstrated. In the study
example, a pre milk-run ratio of 70% was assumed (CMR = 0.7). We increased the coupling strength CHub
from 0.0 to 2.0. Despite the additional inter-hub networking, there was initially (CHub up to 0.5) no
significant improvement in hub synchrony (= 0.02). While in the beginning, we observed that the hub
phase velocities influenced each other increasingly (a and b) (without coherence improvement), after
exceeding a critical threshold value of approximately CHub = 0.7; a significant improvement in the
synchronization behavior was observed (c and d).
17
Table 3. Comparison inter-hub coherence development between hub 1 and hub 3 for t = 31 d
(a) CHub = 0.0 r = 0.55
(b) CHub = 0.5 r = 0.57
(c) CHub = 1.5 r = 0.77
(d) CHub = 2.0 r = 0.93
More interesting was the synchronization behavior of hub 1 (same for hub 3), where the peripheral
supplier nodes were strongly detuned against the hub node (= 7.65). Although no changes were
made to the network parameters of hub 1, because of the increased inter-hub connections (= 0.0 to
2.0), the mean coherence level (based on 20 simulation runs) deteriorated from to 0.49
(see black line in Figure 4 right). The improvement in the inter-hub synchronization was thus obtained at
the cost of a deterioration in the intra-hub synchronization. Hub 2 showed no significant degradation of
synchronization due to the very low detuning (= 0.47). Figure 4 left side depicts the evolution
of the phase coherence of all the three hubs. Up to the critical coupling strength of approximately
0.7, there was a cyclic development resulting from the intra-hub interaction of the frequencies. Above a
value of 0.7, a fracturing of the coherence behavior took place, caused by the interference of the intra-
and inter-hub interactions.
18
Figure 4. Comparison of inter-hub coherence development (rh1-3) (left) with box-and-whisker-plot
inter- (rh1-3) and intra-hub (rh1, rh2, rh3) coherence (right) for t = 31 d
4.2 Tree-mesh network
This hybrid network topology is based on a mesh network for the C-suppliers integrated into a tier system
with the B-suppliers (see Section 2.2.2). The mesh topology is best mapped with the help of a scale-free
network pattern, based on a preferential attachment (Barabási & Albert, 1999), where the probability to
connect with a supplier correlates positively with its degree node ki (n). The probability P(ki) that any
node n of the network has ki connections, then follows a power law with
.
This so-called Barabási-Albert model reflects the fact that C-suppliers have a short path length with an
average connectivity that is low relative to the total number of nodes. In our investigation area, a cluster
of 30 national C-suppliers was analyzed, showing a degree exponent of with a normalization
coefficient b = 0.73. This result agreed with that reported by other studies in the automotive industry,
where ranged from 2.02 (Keqianq et al., 2008) to 3.32 (Sun et al., 2017).
Despite the relatively high differences in the delivery times, indicated by a high coefficient of variation
of 0.51 (see Table 2), the 30 C-suppliers showed a full synchronization after a transition phase of
approximately 9 days (see Figure 5 left). This clearly indicated that the establishment of a supplier
hierarchy, where medium-volume B-suppliers assumed a coordination function, had a positive effect on
the synchronization behavior of the entire supply network. A crucial question in supplier network
planning is the critical spread of C-supplier frequencies in a mesh network context, which can be tolerated
without losing significant synchronization behavior.
0
0.2
0.4
0.6
0.8
1.0
rh1-3
19
Figure 5. Instantaneous phase (speed) evolution for all C-suppliers (left) and box-and-whisker plot of the
phase coherence development with increasing frequency spread (right)
PROPOSITION 4. In a supplier tree-mesh network with sparse, scale-free coupling, upon a gradual
increase in the intrinsic supplier frequency spread , there is a phase transition from synchronous to
asynchronous steady-state solutions.
Figure 5 (right) compares an increasing supply process variation, measured by the intrinsic frequency
spread of each supplier, with the supply network coherence (based on i = 20 simulation runs for each
parameter setting ). When the standard deviation of natural frequencies was large as compared to the
coupling (in this case 0.03), the supply chain network behaved incoherently, with each oscillator vibrating
at its intrinsic frequency. With the frequency spread was decreased, the incoherence persisted until a
certain threshold was reached. At this phase transition point, some of the oscillators started synchronizing,
while the others remained incoherent. In particular, we observed a non-linear S-shaped synchronization
transition, indicated by the median values (horizontal white line) of the box-and-whisker plot in Figure 5
(right). The very high min/max value spread indicated a sporadic supply chain dynamic. Table 4 displays
four different frequency regimes with periodic patterns for the same frequency spread. Because of the
stochastic nature of the frequency distribution, there were large differences in the phase coupling. The
number of suppliers decoupling nde from the supply network field ranged from 6 to 0, which varied the
phase coherence from 0.67 to 0.90.
20
Table 4. Comparison coherence development for t = 72 d with = 0.7
r = 0.67
r = 0.76
r = 0.83
r = 0.90
PROPOSITION 5. C-suppliers are not able to generate a significant synchronization transition in a
sparse, scale-free mesh network, without a coordinating B-supplier tier.
A further important issue related to the network topology and synchronization behavior is the ability of
C-suppliers to self-synchronize without the interposition of an additional B-supplier tier. We again used
the same network topology, with the difference that the B-supplier took a simple C-supplier position,
with correspondingly reduced the coupling strength. According to the high number of suppliers and
therefore low coupling strength (0.03) compared with the large spread of natural frequencies (coefficient
of variation = 0.51), the level of coherence with r = 0.15 was non-existent. Because of the small number
of network connections = 1.94 (see Figure 6 – left side), the interaction density was too low to
generate a significant coherence pattern. One way to improve the synchronization behavior would be to
have the C-suppliers collaborate more intensively in the existing supply network structures, which would
put special focus on suppliers with high node degrees. In general, a scale-free network generates a small
average path length as a necessary but not sufficient condition for synchronization behavior. Figure 6
(right side) depicts only a relatively small increase in the network synchronization to a maximum value
of 0.32, despite the substantial increase in the coupling strength from K = 0.03 to 1.00, which is not
sufficient to generate a sustainable supply coherence. This clearly showed the limited possibility of
synchronization within a sparse network, where the number of edges was very small.
21
Figure 6. Degree distribution (left) and order parameter r development (right) of the sparse, scale-free
supplier mesh network for different coupling strengths K
PROPOSITION 6. Onset synchronization of C-suppliers in a scale-free mesh network occurs without the
integration of a coordinating B-supplier tier, when the global clustering coefficient reaches a critical
transitivity.
We successively increased the global cluster coefficient C (14) so that the interaction within the mesh
network enlarged. This was related to the key properties of a successful and modular supply chain
network with a short path length, generated by the power law connectivity, and a high clustering
coefficient. Table 5 depicts two coherence regimes with cluster coefficients of 0.31 to 0.85.
Table 5. Comparison of coherence development for t = 72 d with varying supply network densities
Network para-
meters
Supplier network graph
Instantaneous phase (speed) development
# of edges = 130
7.3
C = 0.31
r = 0.22
0 5 10 15 20 25 30 k
0.2
0.4
0.6
0.8
1.0
P(k)C
0.0 0.2 0.4 0.6 0.8 1.0 K
0.05
0.10
0.15
0.20
0.25
0.30
r
1
23
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18 19
20
21
22
23
24
25 26
27
28
29
30
31
32
33
34
35
22
# of edges = 490
28.0
C = 0.85
r = 0.89
The increasingly dense network structure formed an increasing number of islands of synchronization,
which were reflected in the increasing phase coupling. While at a low network density, only a few phases
coupled and decoupled again (r = 0.22), the regime below showed a strong correlation of 33 of the 35
suppliers with mutual phase entrainment, indicated by the high phase coherence of r = 0.89. The observed
synchronization transition of C-suppliers in a scale-free network was exclusively continuous. No
explosive transitions analogous to the hub-and-spoke topology were observed. Overall, the cluster
coefficient could be considered a necessary condition for a successful synchronization process. A
relatively low coupling strength of K = 0.14 already led to a significant synchronization improvement
from r = 0.4 to 0.9 between a clustering coefficient increasing from 0.1 to 0.4 (see Figure 7 – left).
Comparing this scale-free network to an all-to-all mesh network, as discussed in Section 3, revealed
considerably worse synchronization behavior. In the supply network limit , with all-to-all
coupling, a homogenous mesh network synchronized completely at the expectation value of
= 0.68 (11) and
= 0.02.
Figure 7. Phase coherence development for different cluster coefficients C (left) and disruption times
(right)
1
2
3
4
5
6
7
8
9
10
11 12
13
14
15
16 17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
23
4.3 Tree network
The strict hierarchical tree structure prevailing in just-in-sequence delivery generates a high degree of
synchronization between the automobile manufacturer and the A-suppliers (see Section 2.2.3). In our
example, a classic regional JIS supplier was selected, for which the degree of phase coherence was nearly
perfect (r = 0.99) because of the to-the-minute delivery. This synchronization, which was achieved
through strict organizational integration (e.g., EDI), dropped at the level of the B-suppliers (r = 0.75) and
C-suppliers (r = 0.42). While disruptions at the first delivery level could be easily resolved by a large
number of redundancies (e.g., replacement trucks), the impact of delivery disruptions at the second level
on the synchronization behavior was considerably greater.
PROPOSITION 7. Disruptions in a hierarchical just-in-sequence tree network cause non-linear
synchronization losses for the upstream suppliers related to the disruption time.
Figure 7 (right) depicts the phase evolution of B- and C-suppliers as a function of the disruption duration
measured in hours. With an increase in the interruption duration (up to 7.5 h for B-suppliers and 6 h for
C-suppliers), the phase coherence initially decreased to a minimum value from which the synchronization
behavior improved despite the longer disruption time. This initially counterintuitive effect could be
explained by Table 6 using the example of three C-suppliers. Without disturbance (a–c), the
synchronization behavior was relatively high with r = 0.75 because of the low coefficient of variation of
18%. The supply interruption (d) triggered by a disturbance of 7.5 h caused the green phase to decouple
irreversibly (e); thus, even after 20 d, the disturbed supplier maintained its own supply rhythm, decoupled
from the other two suppliers (blue and orange). Increasing the disturbance time in (g) led to a larger phase
jump between the disturbed (green) and the undisturbed suppliers, but this caused the two remaining
suppliers to couple more strongly (compare e with h). This was a phase-locked situation (see Table 1),
where the phase difference between the supply network partners remained constant and did not exceed a
certain threshold value (see Definition 3.3). The establishment of phase synchronization after a
disruption with time delay (see Figure 7 right) could be considered a lagged synchronization transition.
This showed that not only the question of whether disturbances occurred but also their length and
distribution in the network had an ambivalent effect on the synchronization in the supply network.
24
Table 6. Phase coherence development of B-suppliers with different disruption scenarios
Instantaneous phase (speed)
development
Supplier phase development
Complex phasor
(a) = 0.0 d
(b)
(c) r = 0.75
(d) = 7,5 h
(e)
(f) r = 0.60
(g) = 24 h
(h)
(i) r = 0.75
5. Conclusions
This paper discussed an investigation of the interaction between synchronization transition and network
topologies in supply chains. The aim of this study was to investigate the particular correlation between
the structural and the dynamical synchronization properties to improve the knowledge of how
synchronization is created and maintained in a supply chain network. A network model was proposed
that quantifies supply chain relations on the basis of phase oscillations that are generally difficult to
analyze, because the interaction functions of supply processes in a network can have many arbitrary
Fourier harmonics. Three network topologies widely used in the automotive and other industries were
examined in detail with regard to their synchronization behavior.
First, it could be shown for a low transport volume hub-and-spoke topology that the desynchronization
transition occurred explosively, if the delivery frequencies of the suppliers drifted further and further
25
apart. This phase transition was only influenced by the number of hub suppliers, if the frequency spread
was large. A significant improvement of the individual hub synchronization behavior could be achieved
by introducing pre-run milk-runs (intra-hub). Similar results were reached by further logistical
networking between the hubs (inter-hub), whereby the improvement of inter-hub synchronization was
partly achieved at the expense of the intra-hub synchronization.
Furthermore, for a tree-mesh topology with medium transport volumes, it could be shown that smaller C-
suppliers needed the coordinating function of the larger B-suppliers to generate the coherence in the
delivery frequencies. In addition, it turned out that C-suppliers could generate a significant
synchronization transition in a scale-free mesh network only when the global clustering coefficient
reached a critical transitivity, without integrating a coordinating B-supplier tier.
Finally, it was demonstrated that high transport volume tree structures exhibited very high
synchronization behavior at the top, rapidly falling in the lower supply tiers. Disruptions in a hierarchical
just-in-sequence tree network caused non-linear synchronization losses for the B- and C-suppliers related
to the disruption time. As the interruption time increased, the phase coherence initially dropped to a
minimum value, but from this point on, the synchronization behavior improved despite the longer
interruption time.
An important limitation of our model concerned the reductionist representation of supplier dynamics. By
modeling each supplier in the network as a phase oscillator, we represented the local supply chain activity
by its phase variable alone. However, this simplification ultimately enabled a comprehensive and global
description of collective synchronization dynamics in complex network topologies. The appeal of the
current formulation lies in not only representing steady-state conditions but also representing and
evaluating coherent behavior with a variety of intermediate transient states.
A future goal in the continuation of this work is the extension of the automotive problem-specific
application of the synchronization transition concept in other industry sectors. This raises the question of
how our results can be transferred to other network topologies. Although hub-and-spoke, mesh, and tree
structures, as used in this study, are common network topologies, there are many other structures that are
relevant in the context of supply chain synchronization.
26
References
Armbruster D, Marthaler DE, Ringhofer C, Kempf KG, Tae-Chang J (2006) A continuum model for a
re-entrant Factory, Operations Research 54 (2006) 933-950.
Barabsi AL, Albert R (1999) Emergence of scaling in random networks. Science 286 (5439): 509–512
Bennett D, Klug F (2012) Logistics supplier integration in the automotive industry. International Journal
of Operations and Production Management, 32(11): 1281-1305.
Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and
dynamics, Physics Reports 424 (2006): 175-308.
Dekker AH, Taylor R (2013) Synchronization properties of trees in the Kuramoto model. SIAM Journal
on Applied Dynamical Systems 12(2): 596-617.
Donner R, Hofleitner A, Höfener J, Lämmer S, Helbing D (2007) Dynamic Stabilization and Control of
Material Flows in Networks and its Relationship to Phase Synchronization. The 3rd international IEEE
scientific conference on physics and control, Potsdam, Germany, September 3-7, 2007.
Donner R (2008) Multivariate analysis of spatially heterogeneous phase synchronisation in complex
systems: application to self-organised control of material flows in networks. The European Physical
Journal B 63(2008):349-361.
Gan X, Wang J (2013) The synchronization problem for a class of supply chain complex networks.
Journal of Computers 8(2): 267-271.
Gómez-Gardenes J, Gómez S, Arenas A, Moreno Y (2011) Explosive synchronization transitions in
scale-free networks, Physical review letters, 106(12): 128701 (2011).
Hearnshaw EJ, Wilson MM (2013) A complex network approach to supply chain network theory.
International Journal of Operations & Production Management 33(4): 442-469.
Helbing D, Lmmer S, Seidel T, eba P, Płatkowski T (2004) Physics, stability, and dynamics of supply
networks. Physical Review E 70(6):66–116.
Keqiang W, Zhaofeng Z, Dongchuan S (2008) Structure analysis of supply chain networks based on
complex network theory, In Semantics, Knowledge and Grid, 2008. SKG'08. Fourth International
Conference on IEEE, USA, 493–494.
Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence. Springer Berlin.
Lämmer S, Kori H, Peters K, Helbing D (2006) Decentralized control of material or traffic flows in
networks using phase-synchronization, Physica A 363(1): 39-47.
Lim MK, Mak HY, Shen ZJM (2017) Agility and proximity considerations in supply chain design,
Management Science 63(4): 1026-1041.
27
Mari SI, Lee YH, Memon MS, Park YS, Kim M (2015) Adaptivity of complex network topologies for
designing resilient supply chain networks. International Journal of Industrial Engineering 22(1): 102-
116.
Scholz-Reiter, B., Tervo, J., & Freitag, M. (2006a). Phase-synchronisation in continuous flow models of
production networks. Physica A, 363 (2006), 32-38.
Scholz-Reiter, B., & Tervo, J. T. (2006b). Approach to optimize production networks by means of
synchronization. In Proceedings of the 17th IASTED international conference on Modelling and
simulation (pp. 160-165). ACTA Press.
Simchi-Levi D, Wei Y (2012) Understanding the performance of the long chain and sparse designs in
process flexibility. Operations Research 60(5): 1125-1141.
Simchi-Levi D, Chen X, Bramel J (2014) The logic of logistics – Theory, algorithms, and applications
for logistics management, Springer New York, Third Edition.
Strogatz, SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations
of coupled oscillators. Physica D, 143(2000): 1-20.
Sun H, Wu J (2005) Scale-free characteristics of supply chain distribution networks. Modern Physics
Letters B, 19(17): 841–848.
Sun JY, Tang JM, Fu WP, Wu BY (2017) Hybrid modeling and empirical analysis of automobile supply
chain network. Physica A, 473:377–389.
