research article the supply chain triangle: how synchronisation,...

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2013 Article ID 981710 10 pageshttpdxdoiorg1011552013981710

Research ArticleThe Supply Chain Triangle How Synchronisation Stability andProductivity of Material Flows Interact

Florian Klug

Department of Business Administration University of Applied Sciences Munich Am Stadtpark 20 81243 Munchen Germany

Correspondence should be addressed to Florian Klug florianklughmedu

Received 3 July 2013 Accepted 9 August 2013

Academic Editor Hing Kai Chan

Copyright copy 2013 Florian KlugThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Empirical evidence created a commonly accepted understanding that synchronisation and stability of material flows impact itsproductivityThis crucial link between synchronous and stable material flows by time and quantity to create a supply chain with thehighest throughput rates is at the heart of lean thinking Although this supply chain triangle has generally been acknowledged overmany years it is necessary to reach a finer understanding of these dynamics Therefore we will develop and study supply chainswith the help of fluid dynamics Amultistage continuousmaterial flow ismodelled through a conservation law formaterial densityUnlike similar approaches our model is not based on some quasi steady-state assumptions about the stochastic behaviour of theinvolved supply chain but rather on a simple deterministic rule for material flow densityThesemodels allow us to take into accountthe nonlinear dynamical interactions of different supply chain echelons and to test synchronised and stable flow with respect to itspotential impacts Numerical simulations verify that the model is able to simulate transient supply chain phenomena Moreovera quantification method relating to the fundamental link between synchronisation stability and productivity of supply chains hasbeen found

1 Introduction

Lean thinking under a manufacturing perspective has beenwell described in the literature over many years Graduallythe lean principles spread from the shop floor to the entirecompany and further on to the whole supply chain [1 2] Alean supply chain enables high productivity by synchronisedand stablematerial flows across all partners [3] Lean thinkingcreated a commonly accepted understanding that synchroni-sation (eg just-in-time supply) and stability (eg levelledproduction) of material flows impact the effectiveness ofsupply chains Although this link between synchronisationstability and productivity of supply chains is generallyacknowledged it is necessary to reach a finer understandingof these dynamics The supply chain triangle provides anexplanation for this transient and nonequilibrium behaviourexperienced within supply chains The specific contributionof this paper is to investigate the supply chain trianglewith thehelp of dynamicmodelling to provide a framework or under-standing fromwhich a firm can assess its inherent options forimproving supply chains In this paper concepts from fluiddynamics have been applied in discovering and explaining

dynamical phenomena in supply chains The mathematicaltools we are using stem mainly from statistical physics andnonlinear dynamics This nonlinear and transient modellingof supply chains allows for a better description of real-lifebehaviour [4] By treating material items similar to classicalmany-particle systems a new and better understanding ofsupply chains emerge

The remainder of the paper is organised as followsIn Section 2 we first review synchronisation stability andproductivity of material flows Section 3 then describes thecontinuummaterial flowmodel with conservation law whichis derived from discrete parts movement with Newton equa-tions We proceed in Section 4 with measuring the supplychain triangle In Section 5 the analysis and results of ournumerical simulations are presented Finally Section 6 isdevoted to conclusions

2 The Literature Review

21 Synchronisation of Material Flows In order to preventlocal buildups of inventorymaterial flowmust be harmonised

2 Modelling and Simulation in Engineering

so that parts move in a coordinated fashion [5] The goalis that material flows without interruptions in a highlyorchestrated process between the individual nodes of a supplychain Synchronising upstream operations with downstreamoperations allows responding to changing requirements andhelps to stabilise supply chains tremendously Coordinationof material flows by both volume and time is aimed atprocessing the quantity needed by one process from the onethat precedes it Each partner is fed from the next stage upthe chain in just the quantity needed at precisely the righttime Perhaps one of the most significant synchronisationprinciples becoming widely adopted and practised is thatof just-in-time supply where all elements of the deliveryprocess are synchronised ldquoSynchronous supply is essentiallya system where components supplied are matched exactlyto the production requirements of the buyerrdquo [6] Nowadaysjust-in-time supply is a standard delivery approach In thisconcept the entire manufacturing process is dependent uponthe timely delivery of components This requires suppliers todeliver customer-ordered components and modules in thesame sequence and synchronised with the final assemblyprocess [7] Information flows and systems must thereforebe synchronised so that information replaces the need forinventories Synchronous supply necessitates an integratedinformation system which can accommodate the time-critical transfer of data and activate the synchronous man-ufacturing process to deliver zero defect goods at the righttime at the right place and at the right cost [6] It enablesthe supply chain partners to share logistics information suchas productionplans and capacities deliveryorders and stocklevels in real time Transparency of information upstream anddownstream maintains the flow of materials in time to therhythm of the production process

Synchronisation needs a common beat which coordi-nates the activities of all the partners in a supply chain Thissignal is generated by takt time (ldquotaktrdquo is a German word forrhythm or meter) which ensures that each operation per-forms equally If supply chain partners are going faster theywill overproduce if they are going slower theywill create bot-tleneck operations [8] The takt time is used to synchronisethe pace of production and logistics with the pace of customersales Takt is derived by customer demandmdashthe rate at whichthe customer is buying product In terms of calculation itis the available time to process parts within a specified timeinterval divided by the number of parts demanded in thattime interval [9] Customer demand and the derived takt timecan be seen as a pacemaker for the whole supply chain

22 Stability of Material Flows To make synchronous mate-rial flow work material flow stability is needed There areinnumerable reasons for disturbing material flows like ldquomis-taken estimates clerical errors bad or defective parts equip-ment failures absenteeism and so on and so forthrdquo [10] Theplanning issue to ensure stability is the principle of levelledproduction This is where the production of different items(product mix) is distributed evenly to minimise uncertaintyfor upstream operations and suppliers Volume and varietyof items produced are levelled over the span of production

during the manufacturing process so that suppliers have asmooth stable demand stream [11] A mixed productionsystem is the distinctive feature of schedule levelling to adjustsurplus capacity and rejects stock [12] Production levellingby both volume and product mix is aimed at producing thequantity and variety taken by one process from the one thatprecedes it It does not process products according to theactual flow of customer orders which can swing up and downwildly but takes the total volume of orders in a period andlevels them out so the same amount and mix are being madeeach scheduling period [8] Harrison [13] pointed out thatfixing levelled production schedules prior to build day avoidslast-minute panics and confusion causing turbulences tomaterial flows Schedule stability translates into stable mate-rial calloffs which means a smooth material flow pipelineand improved performance in plant operationsThis is partlyperformed by low buffer stock according to the rigoroussynchronisation between scheduling and material deliveryhandling transport and placement at the point of use Aneven flow of material throughout the shift also requiresfrequent supplier deliveries with tightly scheduled windowsfor delivery and dispatch of inbound materials throughoutthe day Delivery time windows during which all parts mustbe received at the delivery plant lead to stable inboundflows In addition the use of transportation systems to handlemixed-load small-lot deliveries in combination with a cross-docking system enables a high frequency delivery with smallquantities By focusing on a small group of selected carriers(core carriers) which provide reliable service in such areas asconsolidation tightly scheduled deliveries shipment tracingand effective communication a stable supply chain can befulfilled [11]

23 Productivity of Material Flows Whilst synchronisationand stability ofmaterial flows aremore operations conditionswhich must be fulfilled productivity can be seen as theoutcome TheTheory of Swift Even Flow by Schmenner andSwink [14] summarises the relations between material flowspeed and variation ldquoThus productivity for any processmdashbe it labor productivity machine productivity materialsproductivity or total factor productivitymdashrises with thespeed by which materials flow through the process andit falls with increases in the variability associated with thedemand on the process or with steps in the process itselfrdquoTheeffects of a synchronous and stable material flow can yieldsignificant performance improvements through an increasedspeed of material flow enhanced responsiveness and higherproductivity This crucial link between synchronous andstable supply chains by time and quantity to create a materialflow with the highest throughput rates is at the heart of theToyota Production System [8] Shingo [12] states thatmaterialflow productivity can be performed by quick productchangeovers where production transport and storage takeplace in the smallest lot sizes using short set-up routinesThisone-piece flow is characterised by manufacturing movingand handling just one piece at a time Parts are consistentlyinterchanged so that cycle time is stable for every job Thishigh batch frequency enables a smooth material flow withminimum lead times and high throughput rates

Modelling and Simulation in Engineering 3

3 The Supply Chain Model

In this model we first derive the dynamics of material flowsfrom the elementary microscopic interactions for individualparts in the form of Newton equations In order to generatea simple universal model for material flows we than assumea macroscopic approximate material stream where the flowof material is described as a continuous system The macro-scopicmodel of this paper refers to hydrodynamicmodellingculminating in a hyperbolicmass conservation equationThisapproach allows very fast evaluations of the supply chaintriangle with dynamical insights The resulting deterministicmodel consists of a closed evolution equation providingsignificantlymore information than generally used stochasticqueuing models

In the following the term parts refer to a microscopicdiscrete description of logistic entities movement whilst theterms material and material flow link to a continuous macro-scopic view Our model is focused on supply chain dynamicsinside the plant boundaries In particular we focus uponprocesses starting with the goods receiving where incominggoods of the supplier are delivered up to the dispatch areawhere finished goods are sent to customer The describedkinetic model can however be seen as one stretch of a widermaterial stream and could be easily enlarged to intercompanysupply chainsThe used notation of the model is presented inTable 1

We state a uni-directional and linear in-plant materialflow The internal supply chain consists of a large numberof similar and discrete parts The starting point for studyingmaterial flows is the individual movement of a single part119894 The parts motion can be described in detail with Newtonequations in Cartesian coordinates The momentum velocityV119894(119905) of part 119894 at space 119909 and time 119905 can be stated as

V119894 (119905) =

119889119909119894

119889119905 (1)

The one-dimensional momentum velocity V119894(119905) could be

easily enlarged to a three-dimensional mapping using vec-tor 119890 describing all three geographical dimensions of theparts movement For purposes of clarity we will focus ourmodel on a simpler one-dimensional case

Our investigated factory is part of a larger supply chainprocess with upstream suppliers and downstream customersHence we model in- and outbound material flows into andout of the factory which is described by the inflow rate 120582

1(119905)

and outflow rate 120583119898(119905) at a given space 119909 measured in parts

per unit timeThe transactions occurring between the successive stages

of the internal supply chain can be described as serialinteractions [15] We can discern different distinct genericprocedures in material flow which we call an echelon of theinternal supply chain The length of the supply chain echelon119896 has to be chosen big enough under a microscope viewto entail enough parts to generate reasonable macroscopicdimensions [16] We partition the supply chain into 119898 equalsubintervals of the length Δ119909 with the starting points 119909

119895=

119895Δ119909 for 119895 isin 0 1 119898 minus 1 The whole material flow can

Table 1 Notation list

Notation Termi Index number of different partsj Index number of different spacesk Index number of different echelonss Index number of disturbances119897119894

Trajectory of part in Number of partsm Number of echelonsx Space variableΔ119909 Length of supply chain echelone Space vectort Time variableT Observation timeA Supply chain diameter in units of area120582119896(119905) Material inflow rate of echelon k at time t

120583119896(119905) Material outflow rate of echelon k at time t

q(119909 119905) Material flow rate at space x at time tu(119909 119905) Material density at space x at time t119878119896(t) Echelon stock of supply chain echelon k at time t

S(t) Bounded total stock at time tV119894(t) Momentum velocity of part i at time t

V119898119896

Maximum velocity in echelon k

V119890(119906) Equilibrium velocity referred to the momentum

material density119906119898119896

Maximum capacity of echelon k119906 Average maximum capacity of all echelons119906119879

Total maximum capacity

TP119896

Throughput referred to echelon k to a certainobservation time T

TPOutAveraged throughput per supply chain echelon to acertain observation time T

TPIn Throughput at 119909 = 0 to a certain observation time T119904 Standard deviation of internal capacities120578 Material flow productivity in 120591119896

Lead time echelon k120591 Total lead time119889119904

Disturbance s1198870

Long-term demand1198871

Midterm master schedule variation1198861

Amplitude of rectangular function 1198871

1198862

Amplitude of sinusoidal oscillation of supplydisruptions

11987911205961

Periodfrequency of rectangular function 1198871

11987921205962

Periodfrequency of sinusoidal oscillation of supplydisruptions

be now subdivided into echelons 119896 with 119896 isin 1 2 119898 ofequal length Δ119909 and constant diameter area 119860 (Figure 1)

Our described discrete model does have the great advan-tage of corresponding to the dynamics of each individual part


0 Xj Xj + ΔXX

120583m

Echelon k

1205821

mmiddotΔX

A

Figure 1 Multi-echelon supply chain

but is on the other hand very time consuming and thereforenonscalable to larger models Besides we would like to studythe aggregate behaviour of supply chains in the framework ofdynamical systems Although we are losing determinism wereplace the individual parts by a continuum and derive a con-tinuous macroscopic model of a unidirectional material flowfrom themicroscopic discrete description of partsmovementThe global behaviour ofmaterial streamwill be described by afluid flow We adopt here a hydrodynamic point of view andreplace an ensemble of parts by a spatially averaged densityand derive an evolution equation for the continuummaterialdensity 119906 from simple rules governing the interaction ofindividual parts The bivariate function 119906(119909 119905) that gives thepart number at every point in continuous space time containsall the information necessary to keep track of material flowevolution [17] It should be clear that this generalised densitydefinition merely averages the part flow collected at eachinstant within the region of interest [18] Our model isbased on fluid dynamics that have been already successfullyapplied in traffic flow modelling [19] In this model parts insupply chains are considered as particles in fluids The mainmodifications and new perspectives of our model are

(i) the focus on the collective behaviour of materialstreams

(ii) the formulation as internal supply chain (in plant)problem with a separate modelling of individual ech-elons (see Figure 1)

(iii) calculating lead times (see (9)) and stock values (see(3)) based on deterministic density regimes over timeand space rather than on stationary performance of astochastic model (see (11) and (12))

We calculate a spatially averaged material density 119906(119909 119905)

from the number of parts 119899 at a given supply chain volume119860 sdot 119889119909 at time 119905 with

119906 (119909 119905) =119899

119860 sdot 119889119909 (2)

By integrating the local material densities over the wholeechelon 119896 we get the echelon stock at time 119905 with

119878119896 (119905) = int

119909119895+Δ119909

119909119895

119906 (119909 119905) 119889119909 (3)

Adding up all echelon stocks 119878119896over the whole supply

chain we generate the bounded total stock at time 119905 with

119878 (119905) =

119898

sum

119896=1

119878119896 (119905) (4)

In accordancewith fluid dynamicswe describe the impor-tant relation between material flow rate 119902 material density 119906and equilibrium velocity V

119890in the material stream with

119902 (119909 119905) = 119906 (119909 119905) V119890 (119906) (5)

The mean velocity V119890= ⟨V119894⟩ is the arithmetic average of

all momentum velocities of parts 119894 at a given supply chainvolume of the related local material density The materialflow rate 119902 of the material flow also referred to as thematerial volume denotes the number of parts that passat a particular space of the supply chain during a specifictime interval We do not consider quality and yield lossesconversion or rework of the material Hence in accordancewith the relevant conservation laws of hydrodynamic [20]a mass conserving process naturally leads to a hyperbolicconservation law for material density

120597119906

120597119905+

120597 (V119890 (119906) 119906)

120597119909= 0 (6)

This means that although the distribution of material willvary with time the overall amount of material will dependon the flow into and out of the supply chain According tothe nonlinear conservation law any time variation in theamount of material within any stretch of the supply chaincomprised between two spaces 119909

1and 119909

2(1199091lt 1199092) is only

due to the difference between the incoming flow rate 1199021(1199091 119905)

and the outgoing flow rate 1199022(1199092 119905) We couple (6) with a

suitable closure relation which expresses the velocity V119890as

a function of the density 119906 The main characteristic of thelogistics process is then described by a state equation relatingvelocity and density This closure of (6)mdashby substituting theexpression of V

119890mdashleads to a so-called first ordermodel where

the dynamic of the material flow is described by a singlestate equation The closure is obtained by a self-consistentmodel suitable to relate the local velocity to local densitypatterns [21] Although first order models provide a relativelyless accurate description of the logistical reality with respectto higher order models this simpler model appears to bepractical to study complex material flow conditions Increas-ing the order of the model also increases the number ofparameters to be assessed [22]We state that the local velocityof material decays with increasing material density from amaximum value V

119898when 119906 asymp 0 to V

119890= 0 when 119906 reaches

its maximum Because 119906(119909 119905) characterises the part numberat every point in continuous space time velocity V

119890at space 119909

depends only on the local stock In analogy to hydrodynamicmodels thematerial velocity adapts instantaneously to a localequilibrium velocity V

119890 which depends on the local material

density 119906 [23]This equilibrium velocity V119890(119906) is described by

a state equation relating material velocity V119890and momentum

material density 119906(119909 119905) through

V119890= V119898119896

(1 minus119906 (119909 119905)

119906119898119896

) (7)

with 119906119898119896

the maximum capacity measured as materialdensity and V

119898119896as the maximum velocity of the supply chain

echelon 119896 This is called the Lighthill-Whitham-Richards


(LWR)model [24 25] which approximates traffic flows usingkinematic wave theory This model has been successfullyapplied in traffic dynamics as a first step in a hierarchy oftraffic models [16] The LWR model states a negative corre-lation between velocity and density which also agrees withobservations in material flows In our logistics model theparameter V

119898119896(gt0) denotes the maximum material velocity

per echelon 119896 which may be observed in an empty factorywhere just one order is released The maximum capacity 119906

119898119896

ensures that material flows are discharged through the supplychain echelon with a maximum possible material densityThe maximum velocity V

119898119896and maximum capacity 119906

119898119896are

purely empirically specified and determined by structuralconditions of the internal supply chain (eg warehouse typeand capacity or used transport system)

The whole material flow can be now formulated as linearcombination of 119898 echelons where the material outflow ofthe precedent echelon 120583

119896(119905) equals the material inflow of the

successive echelon 120582119896+1

(119905) The material throughput TP119896of

echelon 119896 at the endpoints 119909119895+ Δ119909 for 119896 isin 1 2 119898 and

119895 isin 0 1 119898 minus 1 referred to a certain observation time 119879

is described through

TP119896= int

119879

119905=0

119902 (119909119895+ Δ119909 119905) 119889119905 (8)

To calculate the lead time 120591 for one echelon with thelength Δ119909 we use the space-velocity relation

120591119896=

Δ119909

V119890

(9)

where V119890is the varying equilibrium velocity V

119890(119906) profile over

space 119909 and time 119905 of echelon 119896 according to its individualdensity profile Adding up all echelon lead times 120591

119896over the

whole supply chain we generate the total lead time

120591 =

119898

sum

119896=1

120591119896 (10)

It is important to stress that this calculated lead times arebased on deterministic density regimes over time and spaceThe approximate use of Littlersquos law [26] for a steady-statematerial flow process which links lead time 120591

119896with echelon

stock 119878119896and processing rate 120583

119896of the echelon 119896 according to

120591119896=

119878119896

120583119896

(11)

is not necessary and therefore increases the accuracy of thematerial flowmodel The same can be stated for the boundedechelon and total stock calculated in this model by inte-grating density profiles (3) which is described in stochasticmodels by a continuous variable 119878(119905) whose rate of change isgiven by

119889119878

119889119905=

1205821 (119905) minus 120583

119898 (119905) for 119878 (119905) = 0

0 for 119878 (119905) = 0(12)

4 Measuring the Supply Chain Triangle

41 Measuring Material Flow Synchronisation In our modelwe reproduce a harmonised and synchronised material flow(see Section 21) by capacity variations between the differentsupply chain echelons 119896 To measure disturbances caused bynonsynchronous capacities we use the standard deviation 119904

according to

119904 = radic1

119898

119898

sum

119896=1

(119906119898119896

minus 119906)2 (13)

with the maximum echelon capacities 119906119898119896

and the averagemaximum capacity 119906 of all echelons 119898 Capacity is definedas the potential of the material flow system to allow physicalmaterials to be processed and moved within supply chains[27] Therefore it is necessary for the following numericalanalysis to define a total maximum capacity

119906119879=

119898

sum

119896=1

119906119898119896 (14)

that is fixed so that variations in supply chain response aresolely caused by different synchronisation scenarios

42 Measuring Material Flow Stability We measure a stablematerial flow (see Section 22) with the help of material flowdensity 119906(119909 119905) Each activity independent if value addingmanufacturing process or nonvalue adding logistics processleads to disruptions in the material flow and hence tovariations in material flow density 119906 Without describing thehuge number of disturbances 119889

119904we state that

119889119906

119889119905= 119891 (119889

1 1198892 119889

119904) (15)

describes thematerial flow density variation by time in accor-dance to all relevant direct disturbances 119889

119904 In close analogy

with fluid dynamics we define a totally stablematerial flow asa laminarmaterial streamwith constantmaterial flow density119906 by time (119889119906119889119905 = 0)

The external density disturbances of material flow arereproduced by harmonic oscillations with level variationswhich represent short- mid- and long-term supply chaindisruptions We state an inbound material flow rate 120582

1(119905) =

119906(0 119905)V(119905) into the factory which is used as initial conditionto solve (6) with

119906 (0 119905) = 1198870+ 1198871 (119905) + 119886

2sin (120596

2119905) (16)

This inflow function is composed of three independentcomponents (Figure 2)

The first addend describes a stationary material flowdensity with a constant value 119887

0and refers to the average

inbound flow of delivered material according to the long-term market demand The second component 119887

1(119905) is a peri-

odic rectangular function with amplitude 1198861and period 119879

1=

21205871205961 which is generated by Fourier transform according to

1198871 (119905) =

41198861

120587

infin

sum

119896=1

sin (2119896 minus 1) 1205961119905

2119896 minus 1 (17)


020

015

010

005

u(0t)

5 10 15 20t

T2 = 21205871205962

a2

T1 = 21205871205962

b0

a1

Figure 2 External material flow density disturbances

The addend 1198871relates to the midterm master schedule

variation based on actual customer demand The third com-ponent refers to short-term material flow variations causedby supply disruptions (eg material call-off variation truckdelays supplier behavior etc) and is described by a sinusoidaloscillation with amplitude 119886

2and period 119879

2= 2120587120596

2

43 Measuring Material Flow Productivity To characterisethe material flow productivity of the supply chain (seeSection 23) we first calculate the averaged throughput TPOutper supply chain echelon referred to a certain observationtime 119879 as

TPOut =1

119898

119898

sum

119896=1

TP119896 (18)

TPOut allows a better evaluation of the throughput per-formance than using merely TP

119898 which varies according to

the maximum echelon capacity 119906119898119898

The use of 119906(0 119905) (16)with oscillations by levelling and periodicity (see Section 42)induces different inflow volumes into the supply chain whichwe calculate with

TPIn = int

119879

119905=0

119902 (0 119905) 119889119905 (19)

Material flow productivity 120578 in can be now measuredas the relation between the output- and input-throughput ofthe supply chain with

120578 =TPOutTPIn

sdot 100 (20)

5 Analysis and Results

51 Numerical Simulations In this section we simulate thesystemunder various scenarios and provide numerical resultsthat evaluate the impact of synchronisation and stability onsupply chain productivity Due to the nonlinearity of thegoverning equation (6) in combination with varying ini-tial conditions (16) analytical solutions are precluded For

numerical treatment discretisation of the time-space domainis required To solve the partial differential equation (6) weuse the method of lines This numerical method discretisesthe spatial dimension 119909 and then integrates the semidiscreteproblem by time 119905 as a system of ordinary differential equa-tions The solution in between the discretised space is foundby interpolation To implement this method we first partitionthe space grid into119873 equal subintervals of width ℎ 0 le 119895 le 119873

with spacing ℎ = 1119873 such that the start points are 119909119895= 119895ℎ

The temporal dimension is discretised independently and thetime step 119901 is chosen such that the Courant-Friedrich-Levy(CFL) condition

0 leV (119905119899) 119901

ℎle 1 (21)

is saturated where 119905119899is the current time [28] This condition

prevents the numerical solution from travelling faster thanthe true solution Obtaining the time step we may advancethe solution at each grid point 119909

119895 by using a second-order

finite difference for the space derivative at position 119909119895 The

finite differencemethod proceeds by replacing the derivativesby finite difference approximations [29] In particular we areusing the central difference formula for the second derivative[30] and get the recurrence equation fromTaylorrsquos series witha local error 119874 according to

11989110158401015840(119909119895) =

119891 (119909119895+ ℎ) minus 2119891 (119909

119895) + 119891 (119909

119895minus ℎ)

ℎ2+ 119874 (ℎ

2)

(22)

We partition the supply chain into five equal subintervalsof the length Δ119909 = 05 Boundary conditions of the in-housesupply chain are formulated for 119909 = 0 which correspondsto the goods receiving where incoming goods of the supplierare delivered and for 119909 = 25 which corresponds to thedispatch area where finished goods are sent to customer (seeSection 3) To advance the solution at the left boundary weset our initial conditions according to (16) at 119909 = 0 and119906(119909 0) = 0 at 119905 = 0 This leads to the desired numericalscheme for the internal supply chain model

Setting the values of external control parameters ofour numerical simulation model one can generate differentflow regimes [31] Unless otherwise indicated the parametervalues used in the numerical experiments (with 119899 differentparameter sets) are reported in Table 2 For all simulationruns the maximum material velocity V

119898119896per echelon 119896 was

set at 140 and the totalmaximumcapacity119906119879was set at 1400

Each simulation run 119899 lasts for 20 time units As the supplychain needs to adjust according to the initial conditions (seeFigure 4) we start our response variable calculation at 119905 = 5

so that all results in Table 2 are based on a time interval of 15time units

The first step is to start with a baseline model whichserves as the standard for comparison with alternative supplychain scenarios in the following analysisTherefore we state aperfectly synchronised and stable material flow with anoptimum value both in synchronisation and stability Thiscorresponds to a stationary material flow system where


Table 2 Parameter settings and simulation results

Set Synchronisation

Capacities Stability

119906119879

1199061198981

1199061198982

1199061198983

1199061198984

1199061198985

High Medium Low1198870= 060

1198861= 005

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 010

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 015

1198862= 003

1198791= 300

1198792= 025

1198781

119904 = 000 1400 280 280 280 280 280 120578 = 10000AINV = 027

120578 = 10000AINV = 052

120578 = 10000AINV = 077

1198782

119904 = 027 1400 290 265 290 265 290 120578 = 9897

AINV = 028120578 = 9891

AINV = 054120578 = 9884

AINV = 079

1198783

119904 = 055 1400 300 250 300 250 300 120578 = 9790AINV = 028

120578 = 9778AINV = 055

120578 = 9766AINV = 081

1198784

119904 = 082 1400 310 235 310 235 310 120578 = 9678

AINV = 029120578 = 9661

AINV = 057120578 = 9642

AINV = 083

1198785

119904 = 110 1400 320 220 320 220 320 120578 = 9559AINV = 030

120578 = 9536AINV = 058

120578 = 9512AINV = 085

1198786

119904 = 137 1400 330 205 330 205 330 120578 = 9431

AINV = 030120578 = 9401

AINV = 059120578 = 9371

AINV = 088

the material inflow rate 1205821(119905) is constant over time without

any oscillations by levelling or periodicity (see blue line inFigure 4) In addition internal capacity variation between thedifferent supply chain echelons does not exist thus we set thestandard deviation 119904 = 0 according to (13) At this point weare using a different perspective compared to classic logisticsresearch Traditional material flow theory (eg queuingtheory)maps logistics processes by startingwith a given set ofprocessing entities (egmachines warehouses and transportfacilities) and asking how material flow has to be controlledoptimally to pass through the system Whereas our modelview starts with perfect synchronised and homogenousmate-rial flows and investigates what fluctuations in material flowdensity occur which lead to yield losses of the process

We are especially interested in modelling and analysingthe transient behaviour of material flows We therefore donot focus on steady-statemodelling Highermaterial in-flowsin comparison to the average capacity lead to piling stockswhilst lower inflows generate a stable equilibriumgiven by thestate equations We therefore concentrate on varying time-dependentmaterial flowswith special interest to the nonequi-librium or transient behaviour The transitions are tuned byour control parameters to generate different scenarios

52 Quantifying the Supply Chain Triangle According to theexternal density disturbances ofmaterial flow (reproduced byharmonic oscillations with level variations) in combinationwith the internal nonsynchronous capacities (reproduced bycapacity variations) we define different parameter sets (119878

1

to 1198786) Synchronisation measured by standard deviation s

according to (13) ranges from a maximum synchronisationwith 119904 = 000 to a minimum synchronisation with 119904 = 137

(see Table 2) Each synchronisation step is combined withthree different stability scenarios ranging from high to low

xx

0

1

20

510

1520

t

02040608

q

Figure 3 Space-over time plot of thematerial flow rate 119902 fromgoodsreceiving (119909 = 0) to dispatch area (119909 = 25)

To characterise the principle dynamic response of themodel we first discuss the outcome of simulation experimentwith data set 119878

6in combination with low stability (Figure 3)

Figure 3 displays the computedmaterial flow rates 119902 of thelowest synchronised regime along the supply chain with thelowest stability The figure illustrates the complex spatiotem-poral patterns of a nonstationary and nonperiodic materialflow In this experiment we generate external disruptions by

119906 (0 119905) = 06 +06

120587

infin

sum

119896=1

sin (2119896 minus 1) (21205873) 119905

2119896 minus 1

+ 003 sin (8120587119905)

(23)

Besides external disruptions we add internal capacityvariations (13) According to the varying capacities 119906

119898119896at

each echelon (1199061198981= 330 119906

1198982= 205 119906

1198983= 330 119906

1198984= 205


08

06

04

02

q

5 10 15 20t

Stationary solutionTransient solution

Figure 4 Comparison of the stationary (blue) and transient (red)material throughput at dispatch area

and 1199061198985

= 330) material flows are restricted at differentlevels through the supply chain echelon Capacity variationof the supply chain echelons 119906

119898119896implies varying equilibrium

velocities V119890according to (7) and induces changing material

flow levels As expected this nonsynchronous regime casegenerates the lowest material flow productivity This is partlydue to permanent period and level changes of the materialinflow and partly due to capacity variation

In general material flow productivity 120578 decreases from100 for the maximum synchronisation (119878

1) to a minimum

value of 9371 (1198786) for minimum synchronisation Although

total capacity of the internal supply chain is constant (119906119879=

1400) quantitative performance decreases about 6 due to alack of synchronisation and stability Whilst internal capacityvariation does have a major impact on the quantitative out-put external stability influences the productivity results onlymarginally On the other hand material flow productivity isonly an indicator of the throughput performance referred toa certain observation time 119879 Furthermore it is a majorobjective of supply chain management to minimise thenegative consequences of material flow variations on theoutput performance of the supply chain (eg adherence todelivery dates) To measure a stable output material flowwe use the Actual INVentory Integral of Time multipliedby Absolute Error (AINV ITAE) Originally developed tomeasure hardware systems design [32] this criterion wasalready applied to evaluate material flows [33] The AINVITAE criterion measures the material flow deviation from atarget level that is weighted in the time domain Our targetlevel is the stationary solution of the partial differentialequation in (6)This represents the optimal synchronised andlaminar output of our baselinemodel with a constantmaterialinput at 119909 = 0 (see Section 51) According to our internalsupply chain focus the AINV ITAE can be visualised as thearea between the transient and stationary material outputat dispatch area over simulation time To quantify the totaleffect we evaluate the total difference of the integral over bothoutput curves (Figure 4)

The goal is to minimise the AINV ITAE value indepen-dent if the deviation of material output is positive or negativeA positive error (transient material output is higher than thedemanded stationary material) means that material at dis-patch area is earlier available than demanded by customerswhich causes additional stocking costs A negative error(transient material output is lower than the demanded sta-tionary material) means that material at dispatch area is lateravailable than demanded by customers which causes orderdelay costs This performance measure maps well the overalllogistics goal to make material available at the right time andat the right place So each deviation of the demandedmaterialflow leads inevitably according to the lean approach towaste generation The AINV ITAE criterion can be thereforeinterpreted as a waste indicator

Our simulation results show that stability of the outputmaterial flow at dispatch area measured by AINV ITAEincreases from a minimum of 027 (119878

1) to a maximum of

088 (1198786) Contrary to material flow productivity the com-

parison of all AINV ITAE results shows that the AINV ITAEvalues vary greatly between the different stability levels (lowmedium and high) whereas the impact of synchronisationis more marginal Hence we can state that a high internalsynchronisationwith low capacity variations favoursmaterialflow productivity whilst stable input material flows mainlyinduces output material flow stability This outcome wasalso confirmed in further simulation runs with differentparameter settings compared to the standard experimentsshown in Table 2 Linking the different synchronisation levelswith the material flow productivity 120578 and the AINV ITAEvalues allows for a quantification method of the universalrelation between synchronisation stability and productivityof the supply chain triangle

An additional sensitivity analysis of the inflow parametershows that midterm variations (119879

1 1198861) influence the flow

profiles muchmore than short-term variations (1198792 1198862) As 119879

1

and 1198861reflect master schedule variation (see Section 42) this

outcome does stress the importance of a levelled productionsystem (see Section 22) Further simulations also showedthat a separate variation of the maximum velocity V

119898119896and

the long-term market demand described by 1198870 while the

other parameter configuration remained constant does notchange the main characteristic of the stated flow regimes inTable 2 Simulation results also indicated that a change of thetime horizon 119879 did not influence the fundamental behaviourof the supply chain These results correspond well to otherhigh-order nonlinear systems where one can move manyparameters within a certain regime of operations with littleeffect on essential behaviour [4]

6 Conclusions

Designingmechanisms to analyse evaluate and control dyn-amic phenomena in supply chains allows us to manage themeffectively In this paper we examined the supply chain tri-angle as a nonlinear and multivariate (spatial and temporal)phenomenon which can be quantitatively reproduced bysimulations using fluid dynamics modelling Unlike similar


approaches this model is not based on some quasi steady-state assumptions about the stochastic behaviour of theinvolved supply chain echelons but rather on a simple deter-ministic rule for material flow density Using a deterministicconservation law to describe material flow allows better eval-uation compared to the usually ergodic measures based onstationary performance of the system Supply chainmeasureslike lead times and throughput can be calculated based ondeterministic density profiles rather than on extrapolationsfrom a steady-state situation Numerical simulations verifythat the model is able to simulate transient supply chainphenomena Contrary to existing models the specificity ofour new approach is not only its ability to describe effectivelysupply chain dynamics but also its simplicity to implementand to operate Moreover a quantificationmethod relating tothe fundamental link between synchronisation stability andproductivity of material flows has been found It is importantto understand this link as it gives essential insights into thebigger picture of relating operations management to supplychain performance

A linear material flow with multiple supply chain eche-lons like used in this paper relate to a great number of oper-ations management settings (eg linear assembly processes)Therefore we can state that our used simulation modelgenerates an empirical basis to apply our model in a realworld scenario although there are some limitations A majorlimitation of the model is that it applies to linear sequentialsupply chains Internal and external material flow processescorrespond quite often to a network structure Therefore itis necessary to enlarge fluid models to nonlinear networkstructures Two major changes are required translating non-linear scenarios into a fluid model The first one is to modelseparate incoming and outgoingmaterial flows at each supplychain echelon which can be seen as a node in a supply chainnetwork To map this properly the continuity equation (6) inthe existingmodel needs to be enlargedwith additional termsrelating to the in- and outflow of material at each node Thisapproach already has been successfully applied in modellingfluid transport networks [16] A second modification is tomodel heterogeneous supply chains with multiple materialvariants The reproduction of fine details however willrequire a more refined measurement of the material dynam-ics like transfer functions between multiple supply chainpaths according to multiple variants This can be performedby different material flow densities 119906 (5) depending on theused supply chain echelon so that material can be switchedThe densities are linked via their boundary conditions [34]The second approach which is actually preferable in the caseof a more complex network topology is to introduce virtualsupply chain echelons So depending on the incoming oroutgoing path of material at network nodes different virtualechelons are used Armbruster et al [23] already mappeda fluid dynamics reentrant production process of differentsemiconductor wafers where after one layer is finished awafer returns to the same set ofmachines for processing of thenext layer According to the scale independence of continuummodels a large-scale simulation of a reentrant Intel factorywith 100 machines and 250 simulation steps for about threemonths production was mapped The authors showed that

modelling factory supply chains via hyperbolic conservationlaws can lead to very fast and accurate simulation results

A further limitation of the model is that it does not takein account the turbulences in the material flow These tur-bulences have been already investigated applying the laws offluid dynamics and similitude theory [35] Within a certainrange of values for Reynolds number there exists a region ofgradual transition where the flow is neither fully laminar norfully turbulent and thus fluid behaviour can be difficult topredict These regions consequently have to be avoided whenoptimising the material flow velocity The velocity term inthe Reynolds number can be interpreted as the velocity offlows through the supply chain According to this analogy itis possible to adjust all factors of the supply chain that mayinfluence theReynolds number like the structural complexitydimensions

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

References

[1] D T Jones P Hines and N Rich ldquoLean logisticsrdquo InternationalJournal of Physical Distributionamp Logistics Management vol 27no 3-4 pp 153ndash173 1997

[2] M Holweg ldquoThe genealogy of lean productionrdquo Journal of Ope-rations Management vol 25 no 2 pp 420ndash437 2007

[3] F Klug ldquoWhat we can learn from Toyota on how to tackle thebullwhip effectrdquo in Proceedings of the Logistics Research NetworkConference B Waterson Ed pp 1ndash10 Southampton UK 2011

[4] JW Forrester ldquoNonlinearity in high-ordermodels of social sys-temsrdquo European Journal of Operational Research vol 30 no 2pp 104ndash109 1987

[5] A Harrison and R van Hoek Logistics Management and Strat-egy FT Prentice Hall Harlow UK 4th edition 2011

[6] D Doran ldquoSynchronous supply an automotive case studyrdquoEuropean Business Review vol 13 no 2 pp 114ndash120 2001

[7] A Lyons A Coronado and Z Michaelides ldquoThe relationshipbetweenproximate supply and build-to-order capabilityrdquo Indus-trial Management and Data Systems vol 106 no 8 pp 1095ndash1111 2006

[8] J K LikerTheToyotaWaymdash14Management Principles from theWorldrsquos Greatest Manufacturer McGraw-Hill New York NYUSA 2004

[9] J K Liker and DMeierTheToyotaWay FieldbookmdashA PracticalGuide for Implementing Toyotarsquos 4Ps McGraw-Hill New YorkNY USA 2006

[10] T Ohno ldquoHow the Toyota production system was createdrdquo inTheAnatomy of Japanese Business K Sato and Y Hoshino Edspp 197ndash215 Croom Helm Beckenham UK 1984

[11] J K Liker and Y Ch Wu ldquoJapanese automakers US suppliersand supply-chain superiorityrdquoMIT Sloan Management Reviewvol 21 no 1 pp 81ndash93 2000

[12] S Shingo Study of Toyota Production System from IndustrialEngineering Viewpoint Japan Management Association TokyoJapan 1981


[13] A Harrison ldquoInvestigating the sources and causes of scheduleinstabilityrdquo The International Journal of Logistics Managementvol 8 no 2 pp 75ndash82 1997

[14] R W Schmenner and M L Swink ldquoOn theory in operationsmanagementrdquo Journal of Operations Management vol 17 no 1pp 97ndash113 1998

[15] R Wilding ldquoThe supply chain complexity trianglemdashuncertain-ty generation in the supply chainrdquo International Journal of Phys-ical Distribution and Logistics Management vol 28 no 8 pp599ndash616 1998

[16] M Treiber and A Kesting Traffic Flow DynamicsmdashData Mod-els and Simulation Springer Heidelberg Germany 2013

[17] Y Makigami G F Newell and R Rothery ldquoThree-dimensionalrepresentation of traffic flowrdquo Transportation Science vol 5 no3 pp 302ndash313 1971

[18] M J Cassidy ldquoTraffic flow and capacityrdquo inHandbook of Trans-portation Science R Hall Ed pp 151ndash186 Kluwer AcademicPublishers Norwell Mass USA 1999

[19] E de Angelis ldquoNonlinear hydrodynamic models of traffic flowmodelling and mathematical problemsrdquo Mathematical andComputer Modelling vol 29 no 7 pp 83ndash95 1999

[20] C M Dafermos Hyperbolic Conservation Laws in ContinuumPhysics Springer Berlin Germany 2005

[21] N Bellomo and V Coscia ldquoFirst order models and closure ofthe mass conservation equation in the mathematical theory ofvehicular traffic flowrdquo Comptes Rendus Mecanique vol 333 no11 pp 843ndash851 2005

[22] N Bellomo M Delitala and V Coscia ldquoOn the mathematicaltheory of vehicular traffic flow I Fluid dynamic and kineticmodellingrdquo Mathematical Models and Methods in Applied Sci-ences vol 12 no 12 pp 1801ndash1843 2002

[23] D Armbruster D EMarthaler C Ringhofer K Kempf and T-C Jo ldquoA continuum model for a re-entrant factoryrdquo OperationsResearch vol 54 no 5 pp 933ndash950 2006

[24] M J Lighthill and G B Whitham ldquoOn kinematic waves II Atheory of traffic flow on long crowded roadsrdquo Proceedings of theRoyal Society A vol 229 no 1178 pp 317ndash345 1955

[25] P Richards ldquoShock waves on the highwayrdquoOperations Researchvol 4 no 1 pp 42ndash51 1956

[26] J D C Little ldquoA proof for the queuing formula L=120582Wrdquo Opera-tions Research vol 9 no 3 pp 383ndash387 1961

[27] R A Novack L M Rinehart and S A Fawcett ldquoRethink-ing integrated concept foundations a just-in-time argumentfor linking productionoperations and logistics managementrdquoInternational Journal of Operations and Production Manage-ment vol 13 no 6 pp 31ndash43 1993

[28] R J LeVeque Numerical Methods for Conservation LawsBirkhauser Basel Switzerland 2nd edition 1992

[29] R J LeVeque Finite DifferenceMethods for Ordinary and PartialDifferential Equations Steady State and Time Dependent Prob-lems Society for Industrial and Applied Mathematics (SIAM)Philadelphia Pa USA 2007

[30] U D von Rosenberg Methods for the Numerical Solution ofPartial Differential Equations American Elsevier New YorkNY USA 1969

[31] R Filliger and M-O Hongler ldquoCooperative flow dynamics inproduction lines with buffer level dependent production ratesrdquoEuropean Journal of Operational Research vol 167 no 1 pp 116ndash128 2005

[32] D Graham and R C Lathrop ldquoThe synthesis of optimumtransient responsemdashcriteria and standard formsrdquo Transactions

of the American Institute of Electrical Engineers II vol 72 pp273ndash288 1953

[33] S M Disney M M Naim and D R Towill ldquoDynamic simula-tion modelling for lean logisticsrdquo International Journal of Phys-ical Distribution and Logistics Management vol 27 no 3-4 pp174ndash196 1997

[34] Ch Ringhofer ldquoTraffic flow models and service rules for com-plex production systemsrdquo in Decision Policies for ProductionNetworks D Armbruster and K G Kempf Eds pp 209ndash233Springer London UK 2012

[35] H Schleifenbaum J Y Uam G Schuh and C Hinke ldquoTurbu-lence in production systemsmdashfluid dynamics and ist contribu-tions to production theoryrdquo in Proceedings of theWorld Congresson Engineering and Computer Science vol 2 San FranciscoCalif USA October 2009

[36] J V Morgan Numerical methods for macroscopic traffic models[Doctor thesis] Department of Mathematics University ofReading 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



so that parts move in a coordinated fashion [5] The goalis that material flows without interruptions in a highlyorchestrated process between the individual nodes of a supplychain Synchronising upstream operations with downstreamoperations allows responding to changing requirements andhelps to stabilise supply chains tremendously Coordinationof material flows by both volume and time is aimed atprocessing the quantity needed by one process from the onethat precedes it Each partner is fed from the next stage upthe chain in just the quantity needed at precisely the righttime Perhaps one of the most significant synchronisationprinciples becoming widely adopted and practised is thatof just-in-time supply where all elements of the deliveryprocess are synchronised ldquoSynchronous supply is essentiallya system where components supplied are matched exactlyto the production requirements of the buyerrdquo [6] Nowadaysjust-in-time supply is a standard delivery approach In thisconcept the entire manufacturing process is dependent uponthe timely delivery of components This requires suppliers todeliver customer-ordered components and modules in thesame sequence and synchronised with the final assemblyprocess [7] Information flows and systems must thereforebe synchronised so that information replaces the need forinventories Synchronous supply necessitates an integratedinformation system which can accommodate the time-critical transfer of data and activate the synchronous man-ufacturing process to deliver zero defect goods at the righttime at the right place and at the right cost [6] It enablesthe supply chain partners to share logistics information suchas productionplans and capacities deliveryorders and stocklevels in real time Transparency of information upstream anddownstream maintains the flow of materials in time to therhythm of the production process

Synchronisation needs a common beat which coordi-nates the activities of all the partners in a supply chain Thissignal is generated by takt time (ldquotaktrdquo is a German word forrhythm or meter) which ensures that each operation per-forms equally If supply chain partners are going faster theywill overproduce if they are going slower theywill create bot-tleneck operations [8] The takt time is used to synchronisethe pace of production and logistics with the pace of customersales Takt is derived by customer demandmdashthe rate at whichthe customer is buying product In terms of calculation itis the available time to process parts within a specified timeinterval divided by the number of parts demanded in thattime interval [9] Customer demand and the derived takt timecan be seen as a pacemaker for the whole supply chain

22 Stability of Material Flows To make synchronous mate-rial flow work material flow stability is needed There areinnumerable reasons for disturbing material flows like ldquomis-taken estimates clerical errors bad or defective parts equip-ment failures absenteeism and so on and so forthrdquo [10] Theplanning issue to ensure stability is the principle of levelledproduction This is where the production of different items(product mix) is distributed evenly to minimise uncertaintyfor upstream operations and suppliers Volume and varietyof items produced are levelled over the span of production

during the manufacturing process so that suppliers have asmooth stable demand stream [11] A mixed productionsystem is the distinctive feature of schedule levelling to adjustsurplus capacity and rejects stock [12] Production levellingby both volume and product mix is aimed at producing thequantity and variety taken by one process from the one thatprecedes it It does not process products according to theactual flow of customer orders which can swing up and downwildly but takes the total volume of orders in a period andlevels them out so the same amount and mix are being madeeach scheduling period [8] Harrison [13] pointed out thatfixing levelled production schedules prior to build day avoidslast-minute panics and confusion causing turbulences tomaterial flows Schedule stability translates into stable mate-rial calloffs which means a smooth material flow pipelineand improved performance in plant operationsThis is partlyperformed by low buffer stock according to the rigoroussynchronisation between scheduling and material deliveryhandling transport and placement at the point of use Aneven flow of material throughout the shift also requiresfrequent supplier deliveries with tightly scheduled windowsfor delivery and dispatch of inbound materials throughoutthe day Delivery time windows during which all parts mustbe received at the delivery plant lead to stable inboundflows In addition the use of transportation systems to handlemixed-load small-lot deliveries in combination with a cross-docking system enables a high frequency delivery with smallquantities By focusing on a small group of selected carriers(core carriers) which provide reliable service in such areas asconsolidation tightly scheduled deliveries shipment tracingand effective communication a stable supply chain can befulfilled [11]

23 Productivity of Material Flows Whilst synchronisationand stability ofmaterial flows aremore operations conditionswhich must be fulfilled productivity can be seen as theoutcome TheTheory of Swift Even Flow by Schmenner andSwink [14] summarises the relations between material flowspeed and variation ldquoThus productivity for any processmdashbe it labor productivity machine productivity materialsproductivity or total factor productivitymdashrises with thespeed by which materials flow through the process andit falls with increases in the variability associated with thedemand on the process or with steps in the process itselfrdquoTheeffects of a synchronous and stable material flow can yieldsignificant performance improvements through an increasedspeed of material flow enhanced responsiveness and higherproductivity This crucial link between synchronous andstable supply chains by time and quantity to create a materialflow with the highest throughput rates is at the heart of theToyota Production System [8] Shingo [12] states thatmaterialflow productivity can be performed by quick productchangeovers where production transport and storage takeplace in the smallest lot sizes using short set-up routinesThisone-piece flow is characterised by manufacturing movingand handling just one piece at a time Parts are consistentlyinterchanged so that cycle time is stable for every job Thishigh batch frequency enables a smooth material flow withminimum lead times and high throughput rates






V119894 (119905) =

119889119909119894

119889119905 (1)




1(119905)




119895=








V119898119896






TP119896









1198862


11987911205961


11987921205962





0 Xj Xj + ΔXX

120583m

Echelon k

1205821

mmiddotΔX

A








119906 (119909 119905) =119899

119860 sdot 119889119909 (2)


119878119896 (119905) = int

119909119895+Δ119909

119909119895

119906 (119909 119905) 119889119909 (3)



119878 (119905) =

119898

sum

119896=1

119878119896 (119905) (4)



119902 (119909 119905) = 119906 (119909 119905) V119890 (119906) (5)



120597119906

120597119905+

120597 (V119890 (119906) 119906)

120597119909= 0 (6)


1and 119909

2(1199091lt 1199092) is only








119890= 0 when 119906 reaches


119890at space 119909






V119890= V119898119896

(1 minus119906 (119909 119905)

119906119898119896

) (7)

with 119906119898119896








119898119896



119898119896are









TP119896= int

119879

119905=0

119902 (119909119895+ Δ119909 119905) 119889119905 (8)


120591119896=

Δ119909

V119890

(9)




119896over the


120591 =

119898

sum

119896=1

120591119896 (10)


119896with echelon



120591119896=

119878119896

120583119896

(11)


119889119878

119889119905=

1205821 (119905) minus 120583

119898 (119905) for 119878 (119905) = 0

0 for 119878 (119905) = 0(12)



according to

119904 = radic1

119898

119898

sum

119896=1

(119906119898119896

minus 119906)2 (13)



119906119879=

119898

sum

119896=1

119906119898119896 (14)



119904we state that

119889119906

119889119905= 119891 (119889

1 1198892 119889

119904) (15)





1(119905) =


119906 (0 119905) = 1198870+ 1198871 (119905) + 119886

2sin (120596

2119905) (16)





1(119905) is a peri-


1=


1198871 (119905) =

41198861

120587

infin

sum

119896=1

sin (2119896 minus 1) 1205961119905

2119896 minus 1 (17)


020

015

010

005

u(0t)

5 10 15 20t

T2 = 21205871205962

a2

T1 = 21205871205962

b0

a1




2and period 119879

2= 2120587120596

2


TPOut =1

119898

119898

sum

119896=1

TP119896 (18)





TPIn = int

119879

119905=0

119902 (0 119905) 119889119905 (19)


120578 =TPOutTPIn

sdot 100 (20)






0 leV (119905119899) 119901

ℎle 1 (21)






11989110158401015840(119909119895) =

119891 (119909119895+ ℎ) minus 2119891 (119909

119895) + 119891 (119909

119895minus ℎ)

ℎ2+ 119874 (ℎ

2)

(22)



119898119896per echelon 119896 was







Set Synchronisation


119906119879

1199061198981

1199061198982

1199061198983

1199061198984

1199061198985


1198861= 005

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 010

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 015

1198862= 003

1198791= 300

1198792= 025

1198781

119904 = 000 1400 280 280 280 280 280 120578 = 10000AINV = 027

120578 = 10000AINV = 052

120578 = 10000AINV = 077

1198782

119904 = 027 1400 290 265 290 265 290 120578 = 9897

AINV = 028120578 = 9891

AINV = 054120578 = 9884

AINV = 079

1198783

119904 = 055 1400 300 250 300 250 300 120578 = 9790AINV = 028

120578 = 9778AINV = 055

120578 = 9766AINV = 081

1198784

119904 = 082 1400 310 235 310 235 310 120578 = 9678

AINV = 029120578 = 9661

AINV = 057120578 = 9642

AINV = 083

1198785

119904 = 110 1400 320 220 320 220 320 120578 = 9559AINV = 030

120578 = 9536AINV = 058

120578 = 9512AINV = 085

1198786

119904 = 137 1400 330 205 330 205 330 120578 = 9431

AINV = 030120578 = 9401

AINV = 059120578 = 9371

AINV = 088





1




xx

0

1

20

510

1520

t

02040608

q





119906 (0 119905) = 06 +06

120587

infin

sum

119896=1

sin (2119896 minus 1) (21205873) 119905

2119896 minus 1

+ 003 sin (8120587119905)

(23)


119898119896at

each echelon (1199061198981= 330 119906

1198982= 205 119906

1198983= 330 119906

1198984= 205


08

06

04

02

q

5 10 15 20t



and 1199061198985






1) to a minimum






1) to a maximum of






1



119898119896and



6 Conclusions








References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














V119894 (119905) =

119889119909119894

119889119905 (1)




1(119905)




119895=








V119898119896






TP119896









1198862


11987911205961


11987921205962





0 Xj Xj + ΔXX

120583m

Echelon k

1205821

mmiddotΔX

A








119906 (119909 119905) =119899

119860 sdot 119889119909 (2)


119878119896 (119905) = int

119909119895+Δ119909

119909119895

119906 (119909 119905) 119889119909 (3)



119878 (119905) =

119898

sum

119896=1

119878119896 (119905) (4)



119902 (119909 119905) = 119906 (119909 119905) V119890 (119906) (5)



120597119906

120597119905+

120597 (V119890 (119906) 119906)

120597119909= 0 (6)


1and 119909

2(1199091lt 1199092) is only








119890= 0 when 119906 reaches


119890at space 119909






V119890= V119898119896

(1 minus119906 (119909 119905)

119906119898119896

) (7)

with 119906119898119896








119898119896



119898119896are









TP119896= int

119879

119905=0

119902 (119909119895+ Δ119909 119905) 119889119905 (8)


120591119896=

Δ119909

V119890

(9)




119896over the


120591 =

119898

sum

119896=1

120591119896 (10)


119896with echelon



120591119896=

119878119896

120583119896

(11)


119889119878

119889119905=

1205821 (119905) minus 120583

119898 (119905) for 119878 (119905) = 0

0 for 119878 (119905) = 0(12)



according to

119904 = radic1

119898

119898

sum

119896=1

(119906119898119896

minus 119906)2 (13)



119906119879=

119898

sum

119896=1

119906119898119896 (14)



119904we state that

119889119906

119889119905= 119891 (119889

1 1198892 119889

119904) (15)





1(119905) =


119906 (0 119905) = 1198870+ 1198871 (119905) + 119886

2sin (120596

2119905) (16)





1(119905) is a peri-


1=


1198871 (119905) =

41198861

120587

infin

sum

119896=1

sin (2119896 minus 1) 1205961119905

2119896 minus 1 (17)


020

015

010

005

u(0t)

5 10 15 20t

T2 = 21205871205962

a2

T1 = 21205871205962

b0

a1




2and period 119879

2= 2120587120596

2


TPOut =1

119898

119898

sum

119896=1

TP119896 (18)





TPIn = int

119879

119905=0

119902 (0 119905) 119889119905 (19)


120578 =TPOutTPIn

sdot 100 (20)






0 leV (119905119899) 119901

ℎle 1 (21)






11989110158401015840(119909119895) =

119891 (119909119895+ ℎ) minus 2119891 (119909

119895) + 119891 (119909

119895minus ℎ)

ℎ2+ 119874 (ℎ

2)

(22)



119898119896per echelon 119896 was







Set Synchronisation


119906119879

1199061198981

1199061198982

1199061198983

1199061198984

1199061198985


1198861= 005

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 010

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 015

1198862= 003

1198791= 300

1198792= 025

1198781

119904 = 000 1400 280 280 280 280 280 120578 = 10000AINV = 027

120578 = 10000AINV = 052

120578 = 10000AINV = 077

1198782

119904 = 027 1400 290 265 290 265 290 120578 = 9897

AINV = 028120578 = 9891

AINV = 054120578 = 9884

AINV = 079

1198783

119904 = 055 1400 300 250 300 250 300 120578 = 9790AINV = 028

120578 = 9778AINV = 055

120578 = 9766AINV = 081

1198784

119904 = 082 1400 310 235 310 235 310 120578 = 9678

AINV = 029120578 = 9661

AINV = 057120578 = 9642

AINV = 083

1198785

119904 = 110 1400 320 220 320 220 320 120578 = 9559AINV = 030

120578 = 9536AINV = 058

120578 = 9512AINV = 085

1198786

119904 = 137 1400 330 205 330 205 330 120578 = 9431

AINV = 030120578 = 9401

AINV = 059120578 = 9371

AINV = 088





1




xx

0

1

20

510

1520

t

02040608

q





119906 (0 119905) = 06 +06

120587

infin

sum

119896=1

sin (2119896 minus 1) (21205873) 119905

2119896 minus 1

+ 003 sin (8120587119905)

(23)


119898119896at

each echelon (1199061198981= 330 119906

1198982= 205 119906

1198983= 330 119906

1198984= 205


08

06

04

02

q

5 10 15 20t



and 1199061198985






1) to a minimum






1) to a maximum of






1



119898119896and



6 Conclusions








References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 Xj Xj + ΔXX

120583m

Echelon k

1205821

mmiddotΔX

A








119906 (119909 119905) =119899

119860 sdot 119889119909 (2)


119878119896 (119905) = int

119909119895+Δ119909

119909119895

119906 (119909 119905) 119889119909 (3)



119878 (119905) =

119898

sum

119896=1

119878119896 (119905) (4)



119902 (119909 119905) = 119906 (119909 119905) V119890 (119906) (5)



120597119906

120597119905+

120597 (V119890 (119906) 119906)

120597119909= 0 (6)


1and 119909

2(1199091lt 1199092) is only








119890= 0 when 119906 reaches


119890at space 119909






V119890= V119898119896

(1 minus119906 (119909 119905)

119906119898119896

) (7)

with 119906119898119896








119898119896



119898119896are









TP119896= int

119879

119905=0

119902 (119909119895+ Δ119909 119905) 119889119905 (8)


120591119896=

Δ119909

V119890

(9)




119896over the


120591 =

119898

sum

119896=1

120591119896 (10)


119896with echelon



120591119896=

119878119896

120583119896

(11)


119889119878

119889119905=

1205821 (119905) minus 120583

119898 (119905) for 119878 (119905) = 0

0 for 119878 (119905) = 0(12)



according to

119904 = radic1

119898

119898

sum

119896=1

(119906119898119896

minus 119906)2 (13)



119906119879=

119898

sum

119896=1

119906119898119896 (14)



119904we state that

119889119906

119889119905= 119891 (119889

1 1198892 119889

119904) (15)





1(119905) =


119906 (0 119905) = 1198870+ 1198871 (119905) + 119886

2sin (120596

2119905) (16)





1(119905) is a peri-


1=


1198871 (119905) =

41198861

120587

infin

sum

119896=1

sin (2119896 minus 1) 1205961119905

2119896 minus 1 (17)


020

015

010

005

u(0t)

5 10 15 20t

T2 = 21205871205962

a2

T1 = 21205871205962

b0

a1




2and period 119879

2= 2120587120596

2


TPOut =1

119898

119898

sum

119896=1

TP119896 (18)





TPIn = int

119879

119905=0

119902 (0 119905) 119889119905 (19)


120578 =TPOutTPIn

sdot 100 (20)






0 leV (119905119899) 119901

ℎle 1 (21)






11989110158401015840(119909119895) =

119891 (119909119895+ ℎ) minus 2119891 (119909

119895) + 119891 (119909

119895minus ℎ)

ℎ2+ 119874 (ℎ

2)

(22)



119898119896per echelon 119896 was







Set Synchronisation


119906119879

1199061198981

1199061198982

1199061198983

1199061198984

1199061198985


1198861= 005

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 010

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 015

1198862= 003

1198791= 300

1198792= 025

1198781

119904 = 000 1400 280 280 280 280 280 120578 = 10000AINV = 027

120578 = 10000AINV = 052

120578 = 10000AINV = 077

1198782

119904 = 027 1400 290 265 290 265 290 120578 = 9897

AINV = 028120578 = 9891

AINV = 054120578 = 9884

AINV = 079

1198783

119904 = 055 1400 300 250 300 250 300 120578 = 9790AINV = 028

120578 = 9778AINV = 055

120578 = 9766AINV = 081

1198784

119904 = 082 1400 310 235 310 235 310 120578 = 9678

AINV = 029120578 = 9661

AINV = 057120578 = 9642

AINV = 083

1198785

119904 = 110 1400 320 220 320 220 320 120578 = 9559AINV = 030

120578 = 9536AINV = 058

120578 = 9512AINV = 085

1198786

119904 = 137 1400 330 205 330 205 330 120578 = 9431

AINV = 030120578 = 9401

AINV = 059120578 = 9371

AINV = 088





1




xx

0

1

20

510

1520

t

02040608

q





119906 (0 119905) = 06 +06

120587

infin

sum

119896=1

sin (2119896 minus 1) (21205873) 119905

2119896 minus 1

+ 003 sin (8120587119905)

(23)


119898119896at

each echelon (1199061198981= 330 119906

1198982= 205 119906

1198983= 330 119906

1198984= 205


08

06

04

02

q

5 10 15 20t



and 1199061198985






1) to a minimum






1) to a maximum of






1



119898119896and



6 Conclusions








References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119898119896



119898119896are









TP119896= int

119879

119905=0

119902 (119909119895+ Δ119909 119905) 119889119905 (8)


120591119896=

Δ119909

V119890

(9)




119896over the


120591 =

119898

sum

119896=1

120591119896 (10)


119896with echelon



120591119896=

119878119896

120583119896

(11)


119889119878

119889119905=

1205821 (119905) minus 120583

119898 (119905) for 119878 (119905) = 0

0 for 119878 (119905) = 0(12)



according to

119904 = radic1

119898

119898

sum

119896=1

(119906119898119896

minus 119906)2 (13)



119906119879=

119898

sum

119896=1

119906119898119896 (14)



119904we state that

119889119906

119889119905= 119891 (119889

1 1198892 119889

119904) (15)





1(119905) =


119906 (0 119905) = 1198870+ 1198871 (119905) + 119886

2sin (120596

2119905) (16)





1(119905) is a peri-


1=


1198871 (119905) =

41198861

120587

infin

sum

119896=1

sin (2119896 minus 1) 1205961119905

2119896 minus 1 (17)


020

015

010

005

u(0t)

5 10 15 20t

T2 = 21205871205962

a2

T1 = 21205871205962

b0

a1




2and period 119879

2= 2120587120596

2


TPOut =1

119898

119898

sum

119896=1

TP119896 (18)





TPIn = int

119879

119905=0

119902 (0 119905) 119889119905 (19)


120578 =TPOutTPIn

sdot 100 (20)






0 leV (119905119899) 119901

ℎle 1 (21)






11989110158401015840(119909119895) =

119891 (119909119895+ ℎ) minus 2119891 (119909

119895) + 119891 (119909

119895minus ℎ)

ℎ2+ 119874 (ℎ

2)

(22)



119898119896per echelon 119896 was







Set Synchronisation


119906119879

1199061198981

1199061198982

1199061198983

1199061198984

1199061198985


1198861= 005

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 010

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 015

1198862= 003

1198791= 300

1198792= 025

1198781

119904 = 000 1400 280 280 280 280 280 120578 = 10000AINV = 027

120578 = 10000AINV = 052

120578 = 10000AINV = 077

1198782

119904 = 027 1400 290 265 290 265 290 120578 = 9897

AINV = 028120578 = 9891

AINV = 054120578 = 9884

AINV = 079

1198783

119904 = 055 1400 300 250 300 250 300 120578 = 9790AINV = 028

120578 = 9778AINV = 055

120578 = 9766AINV = 081

1198784

119904 = 082 1400 310 235 310 235 310 120578 = 9678

AINV = 029120578 = 9661

AINV = 057120578 = 9642

AINV = 083

1198785

119904 = 110 1400 320 220 320 220 320 120578 = 9559AINV = 030

120578 = 9536AINV = 058

120578 = 9512AINV = 085

1198786

119904 = 137 1400 330 205 330 205 330 120578 = 9431

AINV = 030120578 = 9401

AINV = 059120578 = 9371

AINV = 088





1




xx

0

1

20

510

1520

t

02040608

q





119906 (0 119905) = 06 +06

120587

infin

sum

119896=1

sin (2119896 minus 1) (21205873) 119905

2119896 minus 1

+ 003 sin (8120587119905)

(23)


119898119896at

each echelon (1199061198981= 330 119906

1198982= 205 119906

1198983= 330 119906

1198984= 205


08

06

04

02

q

5 10 15 20t



and 1199061198985






1) to a minimum






1) to a maximum of






1



119898119896and



6 Conclusions








References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










020

015

010

005

u(0t)

5 10 15 20t

T2 = 21205871205962

a2

T1 = 21205871205962

b0

a1




2and period 119879

2= 2120587120596

2


TPOut =1

119898

119898

sum

119896=1

TP119896 (18)





TPIn = int

119879

119905=0

119902 (0 119905) 119889119905 (19)


120578 =TPOutTPIn

sdot 100 (20)






0 leV (119905119899) 119901

ℎle 1 (21)






11989110158401015840(119909119895) =

119891 (119909119895+ ℎ) minus 2119891 (119909

119895) + 119891 (119909

119895minus ℎ)

ℎ2+ 119874 (ℎ

2)

(22)



119898119896per echelon 119896 was







Set Synchronisation


119906119879

1199061198981

1199061198982

1199061198983

1199061198984

1199061198985


1198861= 005

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 010

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 015

1198862= 003

1198791= 300

1198792= 025

1198781

119904 = 000 1400 280 280 280 280 280 120578 = 10000AINV = 027

120578 = 10000AINV = 052

120578 = 10000AINV = 077

1198782

119904 = 027 1400 290 265 290 265 290 120578 = 9897

AINV = 028120578 = 9891

AINV = 054120578 = 9884

AINV = 079

1198783

119904 = 055 1400 300 250 300 250 300 120578 = 9790AINV = 028

120578 = 9778AINV = 055

120578 = 9766AINV = 081

1198784

119904 = 082 1400 310 235 310 235 310 120578 = 9678

AINV = 029120578 = 9661

AINV = 057120578 = 9642

AINV = 083

1198785

119904 = 110 1400 320 220 320 220 320 120578 = 9559AINV = 030

120578 = 9536AINV = 058

120578 = 9512AINV = 085

1198786

119904 = 137 1400 330 205 330 205 330 120578 = 9431

AINV = 030120578 = 9401

AINV = 059120578 = 9371

AINV = 088





1




xx

0

1

20

510

1520

t

02040608

q





119906 (0 119905) = 06 +06

120587

infin

sum

119896=1

sin (2119896 minus 1) (21205873) 119905

2119896 minus 1

+ 003 sin (8120587119905)

(23)


119898119896at

each echelon (1199061198981= 330 119906

1198982= 205 119906

1198983= 330 119906

1198984= 205


08

06

04

02

q

5 10 15 20t



and 1199061198985






1) to a minimum






1) to a maximum of






1



119898119896and



6 Conclusions








References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Set Synchronisation


119906119879

1199061198981

1199061198982

1199061198983

1199061198984

1199061198985


1198861= 005

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 010

1198862= 001

1198791= 300

1198792= 050

1198870= 060

1198861= 015

1198862= 003

1198791= 300

1198792= 025

1198781

119904 = 000 1400 280 280 280 280 280 120578 = 10000AINV = 027

120578 = 10000AINV = 052

120578 = 10000AINV = 077

1198782

119904 = 027 1400 290 265 290 265 290 120578 = 9897

AINV = 028120578 = 9891

AINV = 054120578 = 9884

AINV = 079

1198783

119904 = 055 1400 300 250 300 250 300 120578 = 9790AINV = 028

120578 = 9778AINV = 055

120578 = 9766AINV = 081

1198784

119904 = 082 1400 310 235 310 235 310 120578 = 9678

AINV = 029120578 = 9661

AINV = 057120578 = 9642

AINV = 083

1198785

119904 = 110 1400 320 220 320 220 320 120578 = 9559AINV = 030

120578 = 9536AINV = 058

120578 = 9512AINV = 085

1198786

119904 = 137 1400 330 205 330 205 330 120578 = 9431

AINV = 030120578 = 9401

AINV = 059120578 = 9371

AINV = 088





1




xx

0

1

20

510

1520

t

02040608

q





119906 (0 119905) = 06 +06

120587

infin

sum

119896=1

sin (2119896 minus 1) (21205873) 119905

2119896 minus 1

+ 003 sin (8120587119905)

(23)


119898119896at

each echelon (1199061198981= 330 119906

1198982= 205 119906

1198983= 330 119906

1198984= 205


08

06

04

02

q

5 10 15 20t



and 1199061198985






1) to a minimum






1) to a maximum of






1



119898119896and



6 Conclusions








References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










08

06

04

02

q

5 10 15 20t



and 1199061198985






1) to a minimum






1) to a maximum of






1



119898119896and



6 Conclusions








References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article the supply chain triangle: how synchronisation,...

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