#### Abstract

Empirical evidence created a commonly accepted understanding that synchronisation and stability of material flows impact its productivity. This crucial link between synchronous and stable material flows by time and quantity to create a supply chain with the highest throughput rates is at the heart of lean thinking. Although this supply chain triangle has generally been acknowledged over many years, it is necessary to reach a finer understanding of these dynamics. Therefore, we will develop and study supply chains with the help of fluid dynamics. A multistage, continuous material flow is modelled through a conservation law for material density. Unlike similar approaches, our model is not based on some quasi steady-state assumptions about the stochastic behaviour of the involved supply chain but rather on a simple deterministic rule for material flow density. These models allow us to take into account the nonlinear, dynamical interactions of different supply chain echelons and to test synchronised and stable flow with respect to its potential impacts. Numerical simulations verify that the model is able to simulate transient supply chain phenomena. Moreover, a quantification method relating to the fundamental link between synchronisation, stability, and productivity of supply chains has been found.

#### 1. Introduction

Lean thinking under a manufacturing perspective has been well described in the literature over many years. Gradually the lean principles spread from the shop floor to the entire company and further on to the whole supply chain [1, 2]. A lean supply chain enables high productivity by synchronised and stable material flows across all partners [3]. Lean thinking created a commonly accepted understanding that synchronisation (e.g., just-in-time supply) and stability (e.g., levelled production) of material flows impact the effectiveness of supply chains. Although this link between synchronisation, stability, and productivity of supply chains is generally acknowledged, it is necessary to reach a finer understanding of these dynamics. The supply chain triangle provides an explanation for this transient and nonequilibrium behaviour experienced within supply chains. The specific contribution of this paper is to investigate the supply chain triangle with the help of dynamic modelling to provide a framework, or understanding, from which a firm can assess its inherent options for improving supply chains. In this paper concepts from fluid dynamics have been applied in discovering and explaining dynamical phenomena in supply chains. The mathematical tools we are using stem mainly from statistical physics and nonlinear dynamics. This nonlinear and transient modelling of supply chains allows for a better description of real-life behaviour [4]. By treating material items similar to classical many-particle systems, a new and better understanding of supply chains emerge.

The remainder of the paper is organised as follows. In Section 2, we first review synchronisation, stability, and productivity of material flows. Section 3 then describes the continuum material flow model with conservation law, which is derived from discrete parts movement with Newton equations. We proceed in Section 4 with measuring the supply chain triangle. In Section 5 the analysis and results of our numerical simulations are presented. Finally, Section 6 is devoted to conclusions.

#### 2. The Literature Review

##### 2.1. Synchronisation of Material Flows

In order to prevent local buildups of inventory, material flow must be harmonised so that parts move in a coordinated fashion [5]. The goal is that material flows without interruptions in a highly orchestrated process between the individual nodes of a supply chain. Synchronising upstream operations with downstream operations allows responding to changing requirements and helps to stabilise supply chains tremendously. Coordination of material flows by both volume and time is aimed at processing the quantity needed by one process from the one that precedes it. Each partner is fed from the next stage up the chain in just the quantity needed at precisely the right time. Perhaps one of the most significant synchronisation principles becoming widely adopted and practised is that of just-in-time supply, where all elements of the delivery process are synchronised. “Synchronous supply is essentially a system where components supplied are matched exactly to the production requirements of the buyer” [6]. Nowadays just-in-time supply is a standard delivery approach. In this concept, the entire manufacturing process is dependent upon the timely delivery of components. This requires suppliers to deliver customer-ordered components and modules in the same sequence and synchronised with the final assembly process [7]. Information flows and systems must therefore be synchronised, so that information replaces the need for inventories. Synchronous supply necessitates an integrated information system which can accommodate the time-critical transfer of data and activate the synchronous manufacturing process to deliver zero defect goods, at the right time, at the right place, and at the right cost [6]. It enables the supply chain partners to share logistics information such as productionplans and capacities, deliveryorders, and stock levels in real time. Transparency of information upstream and downstream maintains the flow of materials in time to the rhythm of the production process.

Synchronisation needs a common beat, which coordinates the activities of all the partners in a supply chain. This signal is generated by takt time (“takt” is a German word for rhythm or meter), which ensures that each operation performs equally. If supply chain partners are going faster, they will overproduce; if they are going slower, they will create bottleneck operations [8]. The takt time is used to synchronise the pace of production and logistics with the pace of customer sales. Takt is derived by customer demand—the rate at which the customer is buying product. In terms of calculation, it is the available time to process parts within a specified time interval, divided by the number of parts demanded in that time interval [9]. Customer demand and the derived takt time can be seen as a pacemaker for the whole supply chain.

##### 2.2. Stability of Material Flows

To make synchronous material flow work, material flow stability is needed. There are innumerable reasons for disturbing material flows, like “mistaken estimates, clerical errors, bad or defective parts, equipment failures, absenteeism and so on and so forth” [10]. The planning issue to ensure stability is the principle of levelled production. This is where the production of different items (product mix) is distributed evenly to minimise uncertainty for upstream operations and suppliers. Volume and variety of items produced are levelled over the span of production during the manufacturing process so that suppliers have a smooth, stable demand stream [11]. A mixed production system is the distinctive feature of schedule levelling to adjust surplus capacity and rejects stock [12]. Production levelling by both volume and product mix is aimed at producing the quantity and variety taken by one process from the one that precedes it. It does not process products according to the actual flow of customer orders, which can swing up and down wildly, but takes the total volume of orders in a period and levels them out, so the same amount and mix are being made each scheduling period [8]. Harrison [13] pointed out that fixing levelled production schedules prior to build day avoids last-minute panics and confusion causing turbulences to material flows. Schedule stability translates into stable material calloffs, which means a smooth material flow pipeline and improved performance in plant operations. This is partly performed by low buffer stock according to the rigorous synchronisation between scheduling and material delivery, handling, transport, and placement at the point of use. An even flow of material throughout the shift also requires frequent supplier deliveries with tightly scheduled windows for delivery and dispatch of inbound materials throughout the day. Delivery time windows, during which all parts must be received at the delivery plant, lead to stable inbound flows. In addition, the use of transportation systems to handle mixed-load, small-lot deliveries in combination with a cross-docking system enables a high frequency delivery with small quantities. By focusing on a small group of selected carriers (core carriers) which provide reliable service in such areas as consolidation, tightly scheduled deliveries, shipment tracing, and effective communication, a stable supply chain can be fulfilled [11].

##### 2.3. Productivity of Material Flows

Whilst synchronisation and stability of material flows are more operations conditions, which must be fulfilled, productivity can be seen as the outcome. The Theory of Swift, Even Flow by Schmenner and Swink [14] summarises the relations between material flow, speed, and variation. “Thus, productivity for any process—be it labor productivity, machine productivity, materials productivity, or total factor productivity—rises with the speed by which materials flow through the process, and it falls with increases in the variability associated with the demand on the process or with steps in the process itself.” The effects of a synchronous and stable material flow can yield significant performance improvements through an increased speed of material flow, enhanced responsiveness, and higher productivity. This crucial link between synchronous and stable supply chains by time and quantity to create a material flow with the highest throughput rates is at the heart of the Toyota Production System [8]. Shingo [12] states that material flow productivity can be performed by quick product changeovers, where production, transport, and storage take place in the smallest lot sizes using short set-up routines. This one-piece flow is characterised by manufacturing, moving, and handling just one piece at a time. Parts are consistently interchanged so that cycle time is stable for every job. This high batch frequency enables a smooth material flow with minimum lead times and high throughput rates.

#### 3. The Supply Chain Model

In this model we first derive the dynamics of material flows from the elementary microscopic interactions for individual parts in the form of Newton equations. In order to generate a simple universal model for material flows we than assume a macroscopic approximate material stream, where the flow of material is described as a continuous system. The macroscopic model of this paper refers to hydrodynamic modelling, culminating in a hyperbolic mass conservation equation. This approach allows very fast evaluations of the supply chain triangle with dynamical insights. The resulting deterministic model consists of a closed evolution equation, providing significantly more information than generally used stochastic queuing models.

In the following the term parts refer to a microscopic discrete description of logistic entities movement, whilst the terms material and material flow link to a continuous macroscopic view. Our model is focused on supply chain dynamics inside the plant boundaries. In particular, we focus upon processes starting with the goods receiving, where incoming goods of the supplier are delivered, up to the dispatch area, where finished goods are sent to customer. The described kinetic model can however be seen as one stretch of a wider material stream and could be easily enlarged to intercompany supply chains. The used notation of the model is presented in Table 1.

We state a uni-directional and linear in-plant material flow. The internal supply chain consists of a large number of similar and discrete parts. The starting point for studying material flows is the individual movement of a single part . The parts motion can be described in detail with Newton equations in Cartesian coordinates. The momentum velocity of part at space and time can be stated as

The one-dimensional momentum velocity could be easily enlarged to a three-dimensional mapping using vector , describing all three geographical dimensions of the parts movement. For purposes of clarity we will focus our model on a simpler one-dimensional case.

Our investigated factory is part of a larger supply chain process with upstream suppliers and downstream customers. Hence, we model in- and outbound material flows into and out of the factory, which is described by the inflow rate and outflow rate at a given space , measured in parts per unit time.

The transactions occurring between the successive stages of the internal supply chain can be described as serial interactions [15]. We can discern different distinct generic procedures in material flow, which we call an echelon of the internal supply chain. The length of the supply chain echelon has to be chosen big enough under a microscope view to entail enough parts to generate reasonable macroscopic dimensions [16]. We partition the supply chain into equal subintervals of the length , with the starting points for . The whole material flow can be now subdivided into echelons , with of equal length and constant diameter area (Figure 1).

Our described discrete model does have the great advantage of corresponding to the dynamics of each individual part but is on the other hand very time consuming and therefore nonscalable to larger models. Besides, we would like to study the aggregate behaviour of supply chains in the framework of dynamical systems. Although we are losing determinism, we replace the individual parts by a continuum and derive a continuous macroscopic model of a unidirectional material flow from the microscopic discrete description of parts movement. The global behaviour of material stream will be described by a fluid flow. We adopt here a hydrodynamic point of view and replace an ensemble of parts by a spatially averaged density and derive an evolution equation for the continuum material density from simple rules governing the interaction of individual parts. The bivariate function , that gives the part number at every point in continuous space time, contains all the information necessary to keep track of material flow evolution [17]. It should be clear that this generalised density definition merely averages the part flow collected at each instant, within the region of interest [18]. Our model is based on fluid dynamics that have been already successfully applied in traffic flow modelling [19]. In this model, parts in supply chains are considered as particles in fluids. The main modifications and new perspectives of our model are(i)the focus on the collective behaviour of material streams,(ii)the formulation as internal supply chain (in plant) problem with a separate modelling of individual echelons (see Figure 1), (iii)calculating lead times (see (9)) and stock values (see (3)) based on deterministic density regimes over time and space rather than on stationary performance of a stochastic model (see (11) and (12)).

We calculate a spatially averaged material density from the number of parts at a given supply chain volume at time with

By integrating the local material densities over the whole echelon we get the echelon stock at time with

Adding up all echelon stocks over the whole supply chain we generate the bounded total stock at time with

In accordance with fluid dynamics we describe the important relation between material flow rate , material density , and equilibrium velocity in the material stream with

The mean velocity is the arithmetic average of all momentum velocities of parts at a given supply chain volume of the related local material density. The material flow rate of the material flow, also referred to as the material volume, denotes the number of parts that pass at a particular space of the supply chain during a specific time interval. We do not consider quality and yield losses, conversion, or rework of the material. Hence, in accordance with the relevant conservation laws of hydrodynamic [20] a mass conserving process naturally leads to a hyperbolic conservation law for material density:

This means that although the distribution of material will vary with time, the overall amount of material will depend on the flow into and out of the supply chain. According to the nonlinear conservation law, any time variation in the amount of material within any stretch of the supply chain, comprised between two spaces and , is only due to the difference between the incoming flow rate and the outgoing flow rate . We couple (6) with a suitable closure relation, which expresses the velocity as a function of the density . The main characteristic of the logistics process is then described by a state equation relating velocity and density. This closure of (6)—by substituting the expression of —leads to a so-called first order model, where the dynamic of the material flow is described by a single state equation. The closure is obtained by a self-consistent model suitable to relate the local velocity to local density patterns [21]. Although first order models provide a relatively less accurate description of the logistical reality, with respect to higher order models, this simpler model appears to be practical to study complex material flow conditions. Increasing the order of the model also increases the number of parameters to be assessed [22]. We state that the local velocity of material decays with increasing material density from a maximum value when to when reaches its maximum. Because characterises the part number at every point in continuous space time, velocity at space depends only on the local stock. In analogy to hydrodynamic models, the material velocity adapts instantaneously to a local equilibrium velocity , which depends on the local material density [23]. This equilibrium velocity is described by a state equation relating material velocity and momentum material density through with the maximum capacity, measured as material density and as the maximum velocity of the supply chain echelon . This is called the Lighthill-Whitham-Richards (LWR) model [24, 25], which approximates traffic flows using kinematic wave theory. This model has been successfully applied in traffic dynamics as a first step in a hierarchy of traffic models [16]. The LWR model states a negative correlation between velocity and density, which also agrees with observations in material flows. In our logistics model, the parameter (>0) denotes the maximum material velocity per echelon , which may be observed in an empty factory where just one order is released. The maximum capacity ensures that material flows are discharged through the supply chain echelon with a maximum possible material density. The maximum velocity and maximum capacity are purely empirically specified and determined by structural conditions of the internal supply chain (e.g., warehouse type and capacity or used transport system).

The whole material flow can be now formulated as linear combination of echelons where the material outflow of the precedent echelon equals the material inflow of the successive echelon . The material throughput of echelon at the endpoints for and referred to a certain observation time is described through

To calculate the lead time for one echelon with the length we use the space-velocity relation where is the varying equilibrium velocity profile over space and time of echelon according to its individual density profile. Adding up all echelon lead times over the whole supply chain we generate the total lead time

It is important to stress that this calculated lead times are based on deterministic density regimes over time and space. The approximate use of Little’s law [26] for a steady-state material flow process, which links lead time with echelon stock and processing rate of the echelon according to is not necessary and therefore increases the accuracy of the material flow model. The same can be stated for the bounded echelon and total stock, calculated in this model by integrating density profiles (3), which is described in stochastic models by a continuous variable whose rate of change is given by

#### 4. Measuring the Supply Chain Triangle

##### 4.1. Measuring Material Flow Synchronisation

In our model we reproduce a harmonised and synchronised material flow (see Section 2.1) by capacity variations between the different supply chain echelons . To measure disturbances, caused by nonsynchronous capacities, we use the standard deviation according to with the maximum echelon capacities and the average maximum capacity of all echelons . Capacity is defined as the potential of the material flow system to allow physical materials to be processed and moved within supply chains [27]. Therefore it is necessary for the following numerical analysis to define a total maximum capacity that is fixed, so that variations in supply chain response are solely caused by different synchronisation scenarios.

##### 4.2. Measuring Material Flow Stability

We measure a stable material flow (see Section 2.2) with the help of material flow density . Each activity, independent if value adding manufacturing process or nonvalue adding logistics process, leads to disruptions in the material flow and hence to variations in material flow density . Without describing the huge number of disturbances we state that describes the material flow density variation by time in accordance to all relevant direct disturbances . In close analogy with fluid dynamics, we define a totally stable material flow as a laminar material stream with constant material flow density by time .

The external density disturbances of material flow are reproduced by harmonic oscillations with level variations, which represent short-, mid-, and long-term supply chain disruptions. We state an inbound material flow rate into the factory, which is used as initial condition to solve (6), with

This inflow function is composed of three independent components (Figure 2).

The first addend describes a stationary material flow density with a constant value and refers to the average inbound flow of delivered material according to the long-term market demand. The second component is a periodic rectangular function with amplitude and period , which is generated by Fourier transform according to

The addend relates to the midterm master schedule variation based on actual customer demand. The third component refers to short-term material flow variations caused by supply disruptions (e.g., material call-off variation, truck delays, supplier behavior, etc.) and is described by a sinusoidal oscillation with amplitude and period .

##### 4.3. Measuring Material Flow Productivity

To characterise the material flow productivity of the supply chain (see Section 2.3) we first calculate the averaged throughput TP_{Out} per supply chain echelon, referred to a certain observation time as

TP_{Out} allows a better evaluation of the throughput performance than using merely , which varies according to the maximum echelon capacity . The use of (16) with oscillations by levelling and periodicity (see Section 4.2) induces different inflow volumes into the supply chain, which we calculate with

Material flow productivity in % can be now measured as the relation between the output- and input-throughput of the supply chain with

#### 5. Analysis and Results

##### 5.1. Numerical Simulations

In this section we simulate the system under various scenarios and provide numerical results that evaluate the impact of synchronisation and stability on supply chain productivity. Due to the nonlinearity of the governing equation (6), in combination with varying initial conditions (16), analytical solutions are precluded. For numerical treatment, discretisation of the time-space domain is required. To solve the partial differential equation (6) we use the method of lines. This numerical method discretises the spatial dimension and then integrates the semidiscrete problem by time as a system of ordinary differential equations. The solution in between the discretised space is found by interpolation. To implement this method we first partition the space grid into equal subintervals of width , with spacing such that the start points are . The temporal dimension is discretised independently, and the time step is chosen such that the Courant-Friedrich-Levy (CFL) condition is saturated, where is the current time [28]. This condition prevents the numerical solution from travelling faster than the true solution. Obtaining the time step, we may advance the solution at each grid point , by using a second-order finite difference for the space derivative at position . The finite difference method proceeds by replacing the derivatives by finite difference approximations [29]. In particular, we are using the central difference formula for the second derivative [30] and get the recurrence equation from Taylor’s series with a local error according to

We partition the supply chain into five equal subintervals of the length . Boundary conditions of the in-house supply chain are formulated for , which corresponds to the goods receiving where incoming goods of the supplier are delivered and for , which corresponds to the dispatch area, where finished goods are sent to customer (see Section 3). To advance the solution at the left boundary we set our initial conditions according to (16) at and at . This leads to the desired numerical scheme for the internal supply chain model.

Setting the values of external control parameters of our numerical simulation model one can generate different flow regimes [31]. Unless otherwise indicated, the parameter values used in the numerical experiments (with different parameter sets) are reported in Table 2. For all simulation runs the maximum material velocity per echelon was set at 1.40, and the total maximum capacity was set at 14.00. Each simulation run lasts for 20 time units. As the supply chain needs to adjust according to the initial conditions (see Figure 4), we start our response variable calculation at so that all results in Table 2 are based on a time interval of 15 time units.

The first step is to start with a baseline model, which serves as the standard for comparison with alternative supply chain scenarios in the following analysis. Therefore we state a perfectly synchronised and stable material flow with an optimum value both in synchronisation and stability. This corresponds to a stationary material flow system, where the material inflow rate is constant over time without any oscillations by levelling or periodicity (see blue line in Figure 4). In addition internal capacity variation between the different supply chain echelons does not exist; thus, we set the standard deviation according to (13). At this point we are using a different perspective compared to classic logistics research. Traditional material flow theory (e.g., queuing theory) maps logistics processes by starting with a given set of processing entities (e.g., machines, warehouses, and transport facilities) and asking how material flow has to be controlled optimally to pass through the system. Whereas our model view starts with perfect synchronised and homogenous material flows and investigates what fluctuations in material flow density occur which lead to yield losses of the process.

We are especially interested in modelling and analysing the transient behaviour of material flows. We therefore do not focus on steady-state modelling. Higher material in-flows in comparison to the average capacity lead to piling stocks whilst lower inflows generate a stable equilibrium given by the state equations. We therefore concentrate on varying time-dependent material flows with special interest to the nonequilibrium or transient behaviour. The transitions are tuned by our control parameters to generate different scenarios.

##### 5.2. Quantifying the Supply Chain Triangle

According to the external density disturbances of material flow (reproduced by harmonic oscillations with level variations) in combination with the internal nonsynchronous capacities (reproduced by capacity variations), we define different parameter sets ( to ). Synchronisation measured by standard deviation s according to (13) ranges from a maximum synchronisation with to a minimum synchronisation with (see Table 2). Each synchronisation step is combined with three different stability scenarios ranging from high to low.

To characterise the principle dynamic response of the model we first discuss the outcome of simulation experiment with data set in combination with low stability (Figure 3).

Figure 3 displays the computed material flow rates of the lowest synchronised regime along the supply chain with the lowest stability. The figure illustrates the complex spatiotemporal patterns of a nonstationary and nonperiodic material flow. In this experiment we generate external disruptions by

Besides external disruptions we add internal capacity variations (13). According to the varying capacities at each echelon (= 3.30, = 2.05, = 3.30, = 2.05, and = 3.30) material flows are restricted at different levels through the supply chain echelon. Capacity variation of the supply chain echelons implies varying equilibrium velocities according to (7) and induces changing material flow levels. As expected this nonsynchronous regime case generates the lowest material flow productivity. This is partly due to permanent period and level changes of the material inflow and partly due to capacity variation.

In general, material flow productivity decreases from 100% for the maximum synchronisation to a minimum value of 93.71% for minimum synchronisation. Although total capacity of the internal supply chain is constant ( = 14.00), quantitative performance decreases about 6% due to a lack of synchronisation and stability. Whilst internal capacity variation does have a major impact on the quantitative output, external stability influences the productivity results only marginally. On the other hand material flow productivity is only an indicator of the throughput performance referred to a certain observation time . Furthermore, it is a major objective of supply chain management to minimise the negative consequences of material flow variations on the output performance of the supply chain (e.g., adherence to delivery dates). To measure a stable output material flow we use the Actual INVentory Integral of Time multiplied by Absolute Error (AINV ITAE). Originally developed to measure hardware systems design [32] this criterion was already applied to evaluate material flows [33]. The AINV ITAE criterion measures the material flow deviation from a target level that is weighted in the time domain. Our target level is the stationary solution of the partial differential equation in (6). This represents the optimal synchronised and laminar output of our baseline model with a constant material input at (see Section 5.1). According to our internal supply chain focus, the AINV ITAE can be visualised as the area between the transient and stationary material output at dispatch area over simulation time. To quantify the total effect we evaluate the total difference of the integral over both output curves (Figure 4).

The goal is to minimise the AINV ITAE value independent if the deviation of material output is positive or negative. A positive error (transient material output is higher than the demanded stationary material) means that material at dispatch area is earlier available than demanded by customers, which causes additional stocking costs. A negative error (transient material output is lower than the demanded stationary material) means that material at dispatch area is later available than demanded by customers, which causes order delay costs. This performance measure maps well the overall logistics goal to make material available at the right time and at the right place. So each deviation of the demanded material flow leads inevitably, according to the lean approach, to waste generation. The AINV ITAE criterion can be therefore interpreted as a waste indicator.

Our simulation results show that stability of the output material flow at dispatch area, measured by AINV ITAE, increases from a minimum of 0.27 to a maximum of 0.88 . Contrary to material flow productivity the comparison of all AINV ITAE results shows that the AINV ITAE values vary greatly between the different stability levels (low, medium, and high), whereas the impact of synchronisation is more marginal. Hence, we can state that a high internal synchronisation with low capacity variations favours material flow productivity, whilst stable input material flows mainly induces output material flow stability. This outcome was also confirmed in further simulation runs with different parameter settings compared to the standard experiments shown in Table 2. Linking the different synchronisation levels with the material flow productivity and the AINV ITAE values allows for a quantification method of the universal relation between synchronisation, stability, and productivity of the supply chain triangle.

An additional sensitivity analysis of the inflow parameter shows that midterm variations influence the flow profiles much more than short-term variations . As and reflect master schedule variation (see Section 4.2), this outcome does stress the importance of a levelled production system (see Section 2.2). Further simulations also showed that a separate variation of the maximum velocity and the long-term market demand described by , while the other parameter configuration remained constant, does not change the main characteristic of the stated flow regimes in Table 2. Simulation results also indicated that a change of the time horizon did not influence the fundamental behaviour of the supply chain. These results correspond well to other high-order nonlinear systems, where one can move many parameters within a certain regime of operations with little effect on essential behaviour [4].

#### 6. Conclusions

Designing mechanisms to analyse, evaluate, and control dynamic phenomena in supply chains allows us to manage them effectively. In this paper, we examined the supply chain triangle as a nonlinear and multivariate (spatial and temporal) phenomenon, which can be quantitatively reproduced by simulations using fluid dynamics modelling. Unlike similar approaches, this model is not based on some quasi steady-state assumptions about the stochastic behaviour of the involved supply chain echelons but rather on a simple deterministic rule for material flow density. Using a deterministic conservation law to describe material flow allows better evaluation compared to the usually ergodic measures based on stationary performance of the system. Supply chain measures, like lead times and throughput, can be calculated based on deterministic density profiles rather than on extrapolations from a steady-state situation. Numerical simulations verify that the model is able to simulate transient supply chain phenomena. Contrary to existing models, the specificity of our new approach is not only its ability to describe effectively supply chain dynamics, but also its simplicity to implement and to operate. Moreover, a quantification method relating to the fundamental link between synchronisation, stability, and productivity of material flows has been found. It is important to understand this link, as it gives essential insights into the bigger picture of relating operations management to supply chain performance.

A linear material flow with multiple supply chain echelons, like used in this paper, relate to a great number of operations management settings (e.g., linear assembly processes). Therefore we can state that our used simulation model generates an empirical basis to apply our model in a real world scenario, although there are some limitations. A major limitation of the model is that it applies to linear sequential supply chains. Internal and external material flow processes correspond quite often to a network structure. Therefore it is necessary to enlarge fluid models to nonlinear network structures. Two major changes are required translating nonlinear scenarios into a fluid model. The first one is to model separate incoming and outgoing material flows at each supply chain echelon, which can be seen as a node in a supply chain network. To map this properly the continuity equation (6) in the existing model needs to be enlarged with additional terms relating to the in- and outflow of material at each node. This approach already has been successfully applied in modelling fluid transport networks [16]. A second modification is to model heterogeneous supply chains with multiple material variants. The reproduction of fine details, however, will require a more refined measurement of the material dynamics, like transfer functions between multiple supply chain paths according to multiple variants. This can be performed by different material flow densities (5) depending on the used supply chain echelon, so that material can be switched. The densities are linked via their boundary conditions [34]. The second approach, which is actually preferable in the case of a more complex network topology, is to introduce virtual supply chain echelons. So depending on the incoming or outgoing path of material at network nodes different virtual echelons are used. Armbruster et al. [23] already mapped a fluid dynamics reentrant production process of different semiconductor wafers, where after one layer is finished a wafer returns to the same set of machines for processing of the next layer. According to the scale independence of continuum models a large-scale simulation of a reentrant Intel factory with 100 machines and 250 simulation steps for about three months production was mapped. The authors showed that modelling factory supply chains via hyperbolic conservation laws can lead to very fast and accurate simulation results.

A further limitation of the model is that it does not take in account the turbulences in the material flow. These turbulences have been already investigated applying the laws of fluid dynamics and similitude theory [35]. Within a certain range of values for Reynolds number there exists a region of gradual transition where the flow is neither fully laminar nor fully turbulent, and thus fluid behaviour can be difficult to predict. These regions consequently have to be avoided when optimising the material flow velocity. The velocity term in the Reynolds number can be interpreted as the velocity of flows through the supply chain. According to this analogy it is possible to adjust all factors of the supply chain that may influence the Reynolds number, like the structural complexity dimensions.

As part of future research it would be also interesting to extend this model to other continuum traffic flow models (high order models) to describe logistics processes. Although the LWR model used is robust with a suitable choice of flow function [36], it does not predict stop-and-go instabilities often observed in material flows [18].