机械类英文文献

Available online at

Physica A334(2004)243–254

/locate/physa

Stability and transition in

multiple production lines

Takashi Nagatani∗

Department of Mechanical Engineering,Shizuoka University,Hamamatsu432-8561,Japan

Received28October2003

Abstract

We 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.+k

Keywords:Production process;Instability;Dynamical transition;Transportation;Multiple lines

1.Introduction

Concepts 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.2003.11.002

244T.Nagatani/Physica A334(2004)243–254

The 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.Model

We 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 =

u

b=1

(f j b−n j b) b(t)min

1;

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 each