Available online at Physica A334(2004)243–254/locate/physaStability and transition inmultiple production linesTakashi Nagatani∗Department of Mechanical Engineering,Shizuoka University,Hamamatsu432-8561,JapanReceived28October2003AbstractWe present the dynamical model of the multiple production lines composed of M parallel and u series machines.We extend the single-series production line model to the multiple production lines.We study the e ect of the multiple lines on the dynamical behavior of the production process.We apply the linear stability analysis to the production process in the multiple lines. The linear stability criterion is derived for the production system with the multiple lines.It is shown that the production process in the multiple lines is more unstable than that in the single line.The phase diagram(region map)is given for the multiple production lines.The nonlinear instability and dynamical transition are investigated by using computer simulation.It is shown that the dynamical transitions occur between the stable and oscillatory productions.c 2003Elsevier B.V.All rights reserved.PACS:05.90.+m;89.90.+m;89.40.+kKeywords:Production process;Instability;Dynamical transition;Transportation;Multiple lines1.IntroductionConcepts from statistical physics and nonlinear dynamics have been very successful in discovering and explaining dynamical phenomena in transportation systems[1–5]. Many of these phenomena are based on mechanisms such as delayed adaptation to changing conditions and competition for limited resources.The delayed adaptation is relevant for production systems as well[6–10].Mathematicians,physicists,tra c sci-entists,and economists have suggested that tra c dynamics has also implications for the dynamical behavior of production process.∗Fax:+81-53-478-1048.E-mail address:tmtnaga@ipc.shizuoka.ac.jp(T.Nagatani).0378-4371/$-see front matter c 2003Elsevier B.V.All rights reserved.doi:10.1016/j.physa.2003.11.002244T.Nagatani/Physica A334(2004)243–254The recently proposed supply-chain model[6]is closely related to the tra c model. The stability of a linear supply chain has been investigated by using the linear stability method and computer simulation,which have been developed in the tra c dynamics [1,11].It has been found that the dynamical transition occurs between the stable and oscillatory productions by varying the adaptation time.The strength of the oscillation increases with the adaptation time.The dynamical transition is very similar to the jamming transition in tra c ow.When the consumption rate is subject to perturbations,the perturbations may cause variations in the production of upstream producers.This is due to delays in their adaptation of the production speed.Under certain conditions,the oscillations in the production and in the resulting inventories(stock levels)of the generated products grow from one producer to the next upstream one.This is called the bullwhip e ect and known,for example,from the“beer distribution game”.Until now,the production process composed of a series line of machines has been proposed and investigated by means of linear stability analysis and computer simula-tions[6,11].A control strategy of the production process has been developed to manage the process of bringing an unstable system into the stable regime.However,there are various production systems in real factories.Products are made through the production processes composed of complex networks.As a result,the products depend highly on the network structure of the production process.In this paper,we consider the dynamical behavior of products produced through the multiple production lines.We present an extended dynamical model of the multiple supply chains to take into account the network structure of M parallel and u series production lines.We study the e ect of multiple lines on the production process.We show that the multiple production lines induce the instability of the production process easier than that of the single production line,and perturbations of consumption grow to higher oscillations of the products.We analyze the stability of multiple production lines using the linear stability analysis method.We show the stability,dynamical transition, and phase diagram(region map).2.ModelWe present the dynamical model of the multiple production lines to take into account the network structure of M parallel and u series machines.The model consists of M parallel chains in which each chain is composed of a series of u production units j, which receive products from the next upstream producer j−1and generate products for the next downstream producer j+1.First,we describe the supply-chain model for late convenience.Recently,Helbing [6]has suggested the following model for the dynamics of supply chains:d N j d t =ub=1(f j b−n j b) b(t)min1;C0b N0(t)c0b;V k b N k(t)c k b−Y j u+1(t);(1)where N j denotes the inventory(stock level)of product j, b is the desired production speed of production unit(machine)b,f j b the number of products generated in eachT.Nagatani/Physica A334(2004)243–254245 production step,n j b the number of products(educts)needed,V k b a limitation of transport capacity in the delivery of the required educts,and Y j u+1the consumption rate of product j.Moreover,there is usually a considerable delay in the adaptation of the (desired)production speeds b,which may be modeled byd b d t =1T[W({N k(t)};{d N k(t)=d t})− b(t)]:(2)Herein,W(···)denotes a control function re ecting the decisions of the production management as a function of the inventories N k(t),their temporal changes d N k=d t,and T is the delay or adaptation time.If we focus on a linear supply chain with f k b= bk and n j b= b;j+1,where b;j represents the Kronecker delta function,the above equations simplify considerably.For j∈{0;1;:::;u}they readd N jd t= j(t)− j+1(t)(3) withd j d t =1T[W({N k(t)};{d N k(t)=d t})− j(t)];(4)where we assume the boundary condition Y j u+1(t)= u+1(t) j;u.In the previous paper [11],we have investigated the stability of the linear supply chain assuming control function of the formW(N)=1−[tanh(N−N c)+tanh(N c)]=2:(5) We have presented the criterion of the strategies to manage to bring into the stable production process.We extend the single linear supply-chain model to the multiple production lines. Fig.1shows the schematic illustration of the multiple production lines.The production system is composed of M parallel and u series machines.Fig.1shows the special case of M=3and u=4for illustration.At each production step,a bu er is set and all the products produced by all machines(i=1;2;:::;M,j)at step j are stocked,transiently,j+1j-1(i-1, j)(i+1, j)Supply DemandFig.1.Schematic illustration of the production process.The system is composed of M(=3)parallel and u(=4)series machines.At each production step,a bu er is set and all the products produced by the machines at step j are stocked,transiently,into bu er j.Then,the products are supplied to the next machines at step j+1.The original materials are supplied on the left boundary.Theÿnal products are consumed on the right boundary.246T.Nagatani/Physica A334(2004)243–254into bu er j.Then,the products are supplied to the next machines(i=1;2;:::;M, j+1).The product N j(t)at bu er j and production rate i;j(t)of machine(i;j)are described by the following equations:d N j d t =Mi=1i;j(t)−Mi=1i;j+1;(6)d i;j d t =1T[W i;j(N j(t))− i;j(t)]:(7)Here,we assume that the control function W i;j(t)of machine(i;j)at time t depends only on the product N j(t)at bu er j and time t.The control function is given by W i;j(N j(t))=1− tanh(N j(t)−N c;i;j)+tanh(N c;i;j) =2;(8) where N c;i;j is the turning point at machine(i;j).We assume that the control function on line i is the same for any machine on the same line:W i;j(N j(t))=W i(N j(t)).In the following sections,we study the dynamical behavior for the production sys-tem described by Eqs.(6)–(8),by using the linear stability analysis and computer simulation.3.Linear stability analysisLet us study the instability of the steady solution for Eqs.(6)–(8).The steady solutions are given by the uniform distribution.The steady solutions are set as N0and i;0,where i;0satisÿes i;0=W i(N0).Let N i(t)and i;j(t)be small deviations from the steady solutions N0and i;0:N j(t)=N0+ N j(t);(9) i;j(t)=W i(N0)+ i;j(t):(10) Then,the linearized equations are obtained from Eqs.(6)and(7)d N j d t =Mi=1i;j(t)−Mi=1i;j+1(t);(11)d i;j d t =1T[W i(N0) N j(t)− i;j(t)];(12)where W i(N0)is the derivative of control function on line i and at N=N0.By expanding N j(t)=X exp(Ikj+zt)and i;j(t)=Y i exp(Ikj+zt),one obtainszX=(1−e Ik)Mi=1Y i;(13) zTY i=W i(N0)X−Y i;(14) where I is the imaginary unit.T.Nagatani/Physica A334(2004)243–254247 By inserting Eq.(13)into Eq.(14),one obtainsY i=W i(N0)(1+zT)X:(15)By solving Eq.(15)with z,oneÿnds that the leading term of z is order of Ik.When Ik→0,z→0.Let us derive the long wave expansion of z,which is determined order by order around Ik≈0.By expanding z=z1(Ik)+z2(Ik)2+···,theÿrst-and second-order terms of Ik are obtained:z1=−Mi=1W i(N0);(16)z2=−Tz21−12Mi=1W i(N0):(17)If z2is a negative value,the steady state becomes unstable for long wavelength modes. When z2is a positive value,the steady state is stable.Therefore,the linear stability condition is given byT¡−12(Mi=1W i(N0)):(18)When the adaptation time(delay)of the production lines satisÿes the above rela-tion(18),the uctuation of consumption decays accordingly as going to the upstream machines.Otherwise,the production lines become unstable and the perturbation of consumption grows with time and accordingly as going to the upstream.The unstable production system results in the oscillating inventories.The neutral stability curve is given byT=−12(Mi=1W i(N0)):(19)The threshold between the stable and oscillating productions is given by the neutral stability curve.We study the e ect of the multiple lines on the neutral stability curve.We compare the neutral stability curve of double production lines with that of the single production line.Fig.2shows the plot of the delay’s threshold T against inventory(product)N where M=2,u=200,N c;1=3,and N c;2=6.In the production system of double lines, the production rate on theÿrst line is one until about N c;1=3and decreases to zero when the inventory is higher than about N c;1=3.While the production rate on the second line is one until about N c;2=6and decreases to zero when the inventory is higher than about N c;2=6.For comparison,we show the two neutral stability curves of the production systems of single and double lines.The solid line on the left-hand side indicates the neutral stability curve of the single production line.The region1 above the solid line represents the unstable state.The solid line on the right-hand side indicates the neutral stability curve of the double production lines.The regions1and 2above the solid line represents the unstable state.The neutral stability curve of the single line has a single valley.While the neutral stability curve of the double lines has two valleys and a mountain.Below each curve,the production system is stable and248T.Nagatani/Physica A334(2004)243–254Fig.2.Neutral stability curve and region map(phase diagram)of the production process with double series (M=2),where N c;1=3and N c;2=6.For comparison,the neutral stability curve of the single series(M=1) is also shown where N c;1=3.Region1above the neutral stability curve shows the unstable state for the single series.Regions1and2above the neutral stability curve show the unstable state for the double series. The stable region exits below each curve.robust for the uctuating consumption.The production system of double lines is more unstable than that of the single line.We study the e ect of the line’s number M on the neutral stability curve.We restrict ourselves to the multiple production lines with the same characteristics.Fig.3shows the plots of the delay’s threshold T against inventory N for M=1–4,where u=200 and N c;1=N c;2=N c;3=N c;4=3.Each curve has a single valley.The minimum value at the valley C1presents a critical threshold.The critical threshold decreases accordingly as the number M of production lines increases.The unstable region above the neutral curve extends with increasing M.Therefore,the production system of multiple lines becomes more unstable according as the line’s number M increases.We study the e ect of the di erence between the two production lines on the in-stability of the production system.We consider the production system of double lines with di erent value of N c;i in the control function.Fig.4shows the neutral stability curves for N c;2=4,6,7where N c;1=3and M=2.Point C1on each curve indicates the ÿrst minimum value of the delay’s threshold.Point C2indicates the maximum value. Point C3indicates the second minimum value.With increasing N c;2,point C3shifts to the right and the second minimum value does not change.While point C2shifts to the right and the maximum value increases with N c;2.Thus,the unstable region extends to higher values of inventory,but the stable region also extends below point C2.Therefore,the stable production process changes highly by varying N c;2in the control function.T.Nagatani/Physica A334(2004)243–254249Fig.3.Neutral stability curves for the line’s number M=1–4,where N c;1=N c;2=N c;3=N c;4=3.Fig.4.Dependence of the neutral stability curve on N c;2for the double series(M=2),where N c;1=3. Three neutral stability curves are shown for N c;2=4,6,7.We study the stability of the triple production lines.Fig.5shows the neutral stability curve in(N;T)parameter space for N c;1=3,N c;2=6,and N c;3=9.The solid line indicates the neutral stability curve of the triple series.The dotted line indicates the neutral stability curve of the double series with N c;1=3and N c;2=6.Points C1,C3, and C5indicate theÿrst,second,and third minimum values,respectively.These points present the same value for the delay time,but the value of the corresponding inventory250T.Nagatani/Physica A334(2004)243–254Fig.5.Neutral stability curve of triple series(M=3)for N c;1=3,N c;2=6,and N c;3=9.Three local minimum values are indicated by points C1,C3,and C5.Two local maximum values are indicated by points C2and C4.increases.Points C2and C4indicate theÿrst and second maximum values.The neutral stability curve of triple series is consistent with that of double series until about N=7. When the inventory is higher than about N=7,the neutral stability curve of triple series deviates from that of double series.The unstable region of triple series extends to the higher values of inventory.Thus,the stability of production systems changes highly with number M of production line.The neutral stability criterion presents the transition line(boundary)between the unstable and stable regions.However,by the linear stability analysis,it is unknown how the production process develops after the production system becomes unstable. In the following section,we carry out computer simulation.We study the dynamical behavior of the production system after the system becomes unstable.4.SimulationWe have carried out computer simulations of Eqs.(6)–(8),using Euler integration with a time discretization of t=0:01,u=200production units,and N c;1=3.The boundary condition has been chosen as follows:N u+1(t)=N0+ (t);(20) where (t)is a white noise with mean value (t) =0and time correlation (t) (t ) = tt =4.We study the dynamical behavior of inventory N j(t)by varying the initial value N0. Fig.6shows the time evolution of the inventory distribution N j(t)of the double series (M=2)for adaptation time(delay)T=1:5at some values(a)N0=3,(b)N0=4,T.Nagatani/Physica A334(2004)243–254251Fig.6.Time evolution of the inventory distribution N j(t)of the double series(M=2)for adaptation time (delay)T=1:5at some values(a)N0=3,(b)N0=4,and(c)N0=6of initial inventory N0for t= 1800–2000,where u=200,N c;1=3and N c;2=6.and(c)N0=6of initial inventory N0for t=1800–2000,where u=200,N c;1=3 and N c;2=6.For(a)initial value N0=3,when the production process starts from the initial uniform distribution,the process becomes unstable,the uctuating consumption propagates upstream,grows accordingly as going to the upstream,and the inventories oscillate highly in time.The oscillation has a constant period.For(b)initial value N0=4,when the production process starts from the initial uniform distribution,the process is stable,the uctuating consumption does not propagates upstream,decays accordingly as going to the upstream,and the inventory distribution remains to be the initial form.For(c)initial value N0=6,starting from the initial uniform distribution, the process becomes unstable,the perturbations of consumption propagate upstream, grow accordingly as going to the upstream,and the inventories oscillate periodically. The space–time pattern of(c)N0=6is consistent with that of(a)N0=3.This is due to the symmetry at N=4:5in the neutral stability curve of Fig.2.At two minimum points,the dynamical behavior of the double series is consistent with one another.The stable uniform distribution occurs below the neutral stability curve in Fig.2.We study the dynamical behavior of oscillating inventories by varying the adaptation time for the production system of double series(M=2).Fig.7shows the plots of the inventory N j(2000)against bu er’s number j at initial value N0=3and t= 2000for adaptation times(a)T=2:5,(b)T=2:25,(c)T=1:75,and(d)T=1:5, where u=200,N c;1=3and N c;2=6.For(a)T=2:5,the inventory oscillates higher accordingly as going to the upstream.With high values of adaptation time T,the width of oscillation is large.Fig.8(a)shows the plot of the maximum and minimum values252T.Nagatani/Physica A334(2004)243–254Fig.7.Plots of the inventory N j(2000)against bu er’s number j at initial value N0=3and t=2000for adaptation times(a)T=2:5,(b)T=2:25,(c)T=1:75,and(d)T=1:5,where u=200,N c;1=3and N c;2=6.(N max and N min)of oscillating inventories against adaptation time T for initial value N0=3where u=200,N c;1=3and N c;2=6.When the inventory distribution exhibits the uniform distribution,the maximum value agrees with the minimum value.The di erence between the maximum and minimum values is consistent with the maximum width of oscillating inventories.At T c;1=1:0,theÿrst dynamical transition occurs. The stable production process changes to the oscillating production process at theÿrst transition point.Furthermore,the second transition occurs at T c;2=2:0.At the second transition point,the oscillation width increases discontinuously and abruptly.Thus,we ÿnd that there are the two transition points in the production process of double lines. Theÿrst transition point is consistent with the neutral stability point,but the second transition point cannot be predicted by the linear stability theory.Fig.8(b)shows the plot of the maximum and minimum values(N max and N min)of oscillating inventories against adaptation time T for initial value N0=2:5,where u=200, N c;1=3and N c;2=6.At T c;1=1:5,theÿrst dynamical transition occurs.The stable production process changes to the oscillating production process at theÿrst transition point.Furthermore,the second transition occurs at T c;2=2:12.At the second transitionT.Nagatani/Physica A334(2004)243–254253Fig.8.Plots of the maximum and minimum values(N max and N min)of oscillating inventories against adaptation time T,where u=200,N c;1=3and N c;2=6.(a)Initial value N0=3.(b)Initial value N0=2:5.(c)Initial value N0=6.point,the oscillation width increases discontinuously and abruptly.The transition points are deÿnitely di erent from those in Fig.8(a).Theÿrst transition point agrees nearly with the neutral stability point.Fig.8(c)shows the plot of the maximum and minimum values(N max and N min)of oscillating inventories against adaptation time T for initial value N0=6,where u=200, N c;1=3and N c;2=6.At T c;1=1:0,theÿrst dynamical transition occurs.The stable production process changes to the oscillating production process at theÿrst transition point.Furthermore,the second transition occurs at T c;2=2:0.At the second transition point,the oscillation width increases discontinuously and abruptly.The transition points agree with those in Fig.8(a).The curve re ected graph(c)at horizontal line N=6 is consistent with that re ected graph(a)at horizontal line N=3.This is due to the symmetry at N=4:5in the neutral stability curve of Fig.2.At two minimum points of the neutral stability curve,the dynamical behavior of the double series is consistent with one another.The stable uniform distribution appears below the neutral stability curve in Fig.2.Thus,weÿnd that theÿrst dynamical transition point is determined by the linear stability theory.However,the second transition point cannot be estimated by the linear stability theory.Also,we are not able to determine the oscillation width analytically,254T.Nagatani/Physica A334(2004)243–254but can calculate the nonlinear dynamical behavior numerically.The production system of multiple series exhibits more complex behavior than that of the single series.It has richer physical context.5.SummaryWe have investigated the dynamical behavior of products in the production process of multiple series analytically and numerically.We have presented the extended dynamical model to take into account the production network composed of M parallel and u series machines.We have clariÿed the e ect of multiple lines on the dynamical characteristics of the production process.We have shown that the dynamical transition occurs between the stable and oscillating production states.We have found that the production process in the multiple series of machines becomes more unstable than that in the single series. We have presented the neutral stability criterion by using the linear stability analysis. We have shown that the neutral stability curve agrees with the dynamical transition line.AcknowledgementsWe would like to thank Professor Dirk Helbing for useful discussions. References[1]T.Nagatani,Rep.Prog.Phys.65(2002)1331.[2]D.Helbing,Rev.Mod.Phys.73(2001)1067.[3]D.Chowdhury,L.Santen,A.Schadschneider,Phys.Rep.329(2000)199.[4]B.S.Kerner,Netw.Spatial Econ.1(2001)35.[5]L.A.Safonov,E.Tomer,V.V.Strygin,Y.Ashhkenazy,S.Havlin,Chaos12(2002)1006.[6]D.Helbing,New J.Phys.5(2003)901.[7]C.Daganzo,A Theory of Supply Chains,Springer,New York,2002.[8]J.D.Sterman,Business Dynamics,McGraw-Hill,Boston,2000.[9]J.A.Buzacott,J.G.Shanthikumar,Stochastic Models of Manufacturing Systems,Prentice-Hall,Englewood Cli s,NJ,1993.[10]D.Helbing,preprint,/abs/cond-mat/0301204,2003.[11]T.Nagatani,D.Helbing,preprint,/abs/cond-mat/0304476,2003.。