The extended kanban control system for production coordination of assembly manufacturing systems

7/30/2019 The extended kanban control system for production coordination of assembly manufacturing systems

1/14

The extended kanban control system for production

coordination of assembly manufacturing systems

CLAUDINE CHAOUIYA1, GEORGE LIBEROPOULOS2 and YVES DALLERY3

1Departement Genie Bio Medical (ESIL/GBM), Ecole Superieure d'Ingenieurs de Luminy, Parc Scientique et Technologique deLuminy, case 925, FR-13288 Marseille Cedex 9, FranceE-mail: [email protected] of Mechanical and Industrial Engineering, University of Thessaly, Pedion Areos, GR-38334, Volos, GreeceE-mail: [email protected] Productique et Logistique, Ecole Centrale de Paris, Grande Voie des Vignes, FR-92295 Chatenay-Malabry Cedex,FranceE-mail: [email protected]

Received June 1999 and accepted January 2000

In assembly manufacturing systems there are points in the production process where several component parts are put together inareas called assembly cells so as to form more complex parts called subassemblies. In this paper, we present and compare twovariants of the Extended Kanban Control System (EKCS) a recently developed pull production control mechanism thatcombines base stock and kanban control for the production coordination of assembly manufacturing systems. In both variants,

the production of a new subassembly is authorized only when an assembly kanban is available. Assembly kanbans becomeavailable when nished subassemblies are consumed. If an assembly kanban is available, in the rst variant, each component partof a subassembly is released into the assembly cell as soon as it is available (independent release). In the second variant, however, it

is released only when allother component parts also become available (simultaneous release). In both variants, when a componentpart is released into the assembly cell, it releases its kanban, thus authorizing the production of a new component part.

1. Introduction

In recent years, manufacturing management interestshave turned towards cellular manufacturing. In cellularmanufacturing production is divided into productioncells. Each cell may be thought of as a production/inventory system composed of a manufacturing processand an output buer. The manufacturing process mayconsist of a single machine or a subnetwork of several

machines (e.g., a production line). It contains parts thatare currently being processed (referred to as the Work-In-Process (WIP) of the cell). The output buer is a storagearea that contains parts that have completed processingin the cell (referred to as nished parts of the cell). Themanufacturing system is fed by raw parts and releasesnished parts to customers.

Each cell has a clear production task and is self man-aged. Often, for each customer, one needs to bring to-gether a network of cells, where each cell may have adierent owner. An important managerial concern is howto coordinate production among dierent self-managed

cells so as to create a coherent integrated system. In manycases, coordination is achieved via a pull control policy,

i.e., a policy that decides when to release parts in each cellbased on when customer demands arrive to the system. Asignicant amount of work has been devoted to this issuefor serial systems, i.e., systems consisting of cells in series.Dierent pull control mechanisms have been proposed inthe literature, among which, the Base Stock ControlSystem (BSCS) and the Kanban Control System (KCS)have attracted signicant attention (Buzacott andShanthikumar, 1993; Hopp and Spearman, 1996). The

BSCS is a classical mechanism encountered in inventorytheory, while the KCS was invented in Toyota in the1970s and has since been widely used in industry (Mon-den, 1983). The main advantage of these two controlpolicies is that they are both very simple to understandand implement as each policy depends on only one pa-rameter per cell. Neither policy, however, always achievesa good tradeo between inventory costs and customerservice. As a result, more general pull control policiesleading to better tradeos have been proposed. Amongthese, the Generalized Kanban Control System (GKCS),introduced by Buzacott (1989) and studied in detail by

Buzacott and Shanthikumar (1993), and the ExtendedKanban Control System (EKCS), proposed by Dallery

0740-817X 2000 ``IIE''

IIE Transactions (2000) 32, 9991012


2/14

and Liberopoulos (2000), are of special interest. Each ofthese policies depends on two parameters per cell. Acomparison of these four control policies as well as otherpolicies can be found in Liberopoulos and Dallery (2000).

For industrial applications, it is important to extendthese control policies to systems having more generalstructures than ``cells in series''. Assembly systems are ofparticular interest in manufacturing. In an assemblysystem, a cell may have more than one immediate up-stream cell. Although this situation is highly prevalent inthe industry, not much work has been done in analyzingit. In Sbiti et al. (1999) the BSCS is extended to assemblysystems, and in Di Mascolo and Dallery (1996) the KCSis extended to assembly systems. In the latter case, animportant additional (with respect to the serial systemcase) control element arises having to do with the way inwhich component parts and kanbans are released prior to

assembly. Two cases are considered, namely a simulta-neous release mechanism and an independent releasemechanism leading to the denitions of the so-calledSimultaneous Kanban Control System (SKCS) and theIndependent Kanban Control System (IKCS), respec-tively.

The purpose of this paper is to generalize the EKCS toassembly systems. As in the case of the KCS, we denetwo dierent control policies depending on whethercomponent parts and kanbans are released simulta-neously or independently prior to assembly. This leads tothe denitions of the Simultaneous Extended KanbanControl System (SEKCS) and the Independent ExtendedKanban Control System (IEKCS), respectively.

The rest of the paper is organized as follows. Section 2briey describes the EKCS dened in Dallery and Lib-eropoulos (2000) in the case of serial systems. In Section3, we extend this denition to assembly systems, rstdescribing the SEKCS and then the IEKCS. In Section 4we show special cases in which the EKCS reduces to theclassical KCS and BSCS. Properties of the two controlmechanisms, the SEKCS and the IEKCS, are given inSection 5. These properties concern invariants, bounds,evolution equations, the inuence of the variation of theparameters, and production capacity. Finally, the two

mechanisms are compared in Section 6 and conclusionsare drawn in Section 7.

2. Extended kanban control system for serial systems

The EKCS was recently introduced by Dallery and Lib-eropoulos (2000). This new kanban-based control systemhas the following advantageous features:

Simplicity (compared to the GKCS). WIP limitation in each cell (unlike the BSCS).

Immediate transfer of demands to all cells (unlike theKCS).

Clear separation of roles of the parameters, i.e., thenumber of kanbans and the base stock level (unlikethe GKCS).

In this section, we briey describe the EKCS for a

manufacturing system having cells in series. For a de-tailed description and properties of this system, refer toDallery and Liberopoulos (2000). Figure 1 describes amanufacturing system having N cells in series. Each cellmay be seen as a manufacturing process (e.g., singlemachine, production line, exible manufacturing cell,

job-shop, etc.) with an output buer.The EKCS is a pull control mechanism that can be

viewed as a combination of the BSCS and the KCS (seeBuzacott and Shanthikumar, 1993; Dallery and Libero-poulos, 2000; Liberopoulos and Dallery, 2000). Figure 2shows the queueing network model of an EKCS having Ncells in series.

We use the following notation that was introduced inDallery and Liberopoulos (2000):

piY i 1Y F F F YN = a cell-i nished part;qiY i 1Y F F F YN = a part currently being processed in

cell i;diY i 1Y F F F YN = a demand for the production of a

new pi;dN1 = a demand for a pN;

aiY i 1Y F F F YN = an authorization card (kanban) forthe production of a new pi.

Table 1 describes the contents and initial state of the

queues, or network of queues in the case of MPi, inFig. 2.

The initial number of raw parts in the raw parts buerP0 and the arrival process of new parts into P0 fall outsidethe scope of the control mechanism and are considered asgiven. Each cell i has Ki kanbans ai that authorize theproduction of cell i nished parts. Initially, in cell i, thereare Si kanbans ai attached onto an equal number of parts

pi in PAi, and therefore Ki Si kanbans ai in Ai (initiallyno part is being processed in MPi). When a customerdemand arrives to the system, it is immediately trans-mitted to all cells adding one demand to the contents of

queues Di (i 1YF F F

YN 1).The behavior of the system can be described as follows.

Release of parts into the manufacturing cells

At the rst manufacturing cell, queues P0, A1, and D1 arejoined in a synchronization station. Raw parts in P0 do

Fig. 1. Manufacturing system with N cells in series.

1000 Chaouiya et al.


3/14

not have any kanbans attached to them. Therefore, cell 1can begin processing a part as soon as there is at least onepart p0 in P0, one authorization card a1 in A1, and onedemand d1 in D1. When these conditions are satised,then:

(1) Kanban a1 is attached onto p0, which is relabeledq1, and together they are transferred downstreamto MP1 as a pair (q1, a1).

(2) Demand d1 is satised and is therefore discarded.

The ith manufacturing cell (i 1Y F F F YN) can beginprocessing a part only when there is at least one pair(pi1Y ai1) in PAi1, one authorization card ai in Ai, and

one demand di in Di. When these conditions are satised,then:

(1) Kanban ai1 is detached from pi1 and is trans-ferred upstream to Ai1.

(2) Kanban ai is attached onto pi1, which is relabeledqi, and together they are transferred downstream toMPi as a pair (qiY ai).

(3) Demand di is satised and is therefore discarded.

When part qi (i 1Y F F F YN) nishes its processing inMPi, it is relabeled pi, and, together with kanban ai thatwas attached onto it, they join PAi as a pair (piY ai).

Delivery of nished parts to customers

There is no need for an authorization to release a nishedpart pN to a customer. Therefore, the delivery of a n-ished part can occur as soon as there is at least one pair

(pNY aN) in PAN and one demand dN1 in DN1. Whenthese conditions are satised, then:

(1) Kanban aN is detached from pN and is transferredupstream to AN.

(2) Part pN is released to the customer.(3) Demand dN1 is satised and is therefore dis-

carded.

The EKCS has two parameters for every cell i,Ki and Si.These parameters must be adjusted to achieve a goodcompromise between: (i) keeping a low inventory of partsin the system; and (ii) attaining a high level of immediatecustomer demand satisfaction. In this, their role is clearly

separate. The base stock parameter basically provides abuer against stockouts, which are associated with badcustomer service. The use of kanbans basically ensuresthat the number of parts (WIP plus the nished parts) ineach cell is bounded by the number of kanbans in that cell.

3. Extended kanban control systems for assemblysystems

The EKCS described for the serial system congurationin Fig. 2 can be extended to manufacturing systems

having an assembly conguration. Figure 3 illustrates thetree-structured topology of a system having assemblycells (cells supplied by several raw parts buers) andmanufacturing cells (cells supplied by a single raw partsbuer).

For simplicity, we restrict our study to assembly sys-tems having (R 1) manufacturing cells supplying a sin-gle assembly cell (see Figs. 4 and 5). However, whatfollows can be easily extended to more general assemblytopologies.

The structure that we are considering is the classicalassembly structure where each item of the nal product is

produced by putting together one nished part from eachmanufacturing cell. Extending this to situations where

Fig. 2. Queueing network model of the EKCS.

Table 1. Contents and initial state of the queues in the EKCS

Queue Contents Initial state

MPi i 1Y F F F YN qiY ai 0PAi i 1Y F F F YN piY ai SiAi i 1Y F F F YN ai Ki SiDi i 1Y F F F YN 1 di 0

Extended kanban control system 1001


4/14

more than one item of each manufacturing cell is neededto assemble a single item of the nal product isstraightforward and for the sake of conciseness will not

be considered in this paper.The extension of the EKCS to assembly systems leads

to two kanban release mechanisms as was the case in theextension of the KCS to assembly systems (Di Mascoloand Dallery, 1996). These mechanisms are the Simulta-neous EKCS (SEKCS) and the Independent EKCS(IEKCS) and are described in detail next.

3.1. Simultaneous extended kanban control system

Figure 4 shows the queueing network model of theSEKCS for the case of (R 1) manufacturing cells sup-plying a single assembly cell.

We use the following notation:p0YiY i 1Y F F F YR 1 = a cell-i raw part;

piY i 1Y F F F YR 1 = a cell-i nished part;pR = a cell-R assembled part;

qiY i 1Y F F F YR 1 = a part currently being processedin cell i;

qR = a part currently being assembledin cell R;

diY i 1Y F F F YR 1 = a demand for the production ofa new pi;

dR = a demand for the assembly of anew pR;

dR1 = a demand for the delivery of anassembled part pR;

aiY i 1Y F F F YR 1 = an authorization card (kan-ban) for the production of anew pi;

aR = an authorization card (kanban)for the assembly of a new pR.

Fig. 3. General topology for assembly manufacturing systems.

Fig. 4. Queueing network model for the SEKCS.



5/14

Table 2 shows the contents and initial state of thequeues.

As was the case in the EKCS, queue P0Yi (i 1Y F F F YR 1) represents the raw parts buer supplying manu-facturing cell i. The initial number of raw parts in P0Yi andthe arrival process of new parts into P0Yi fall outside thescope of the control mechanism and are considered asgiven. When a customer demand arrives to the system, itis immediately transmitted to all cells by adding one tothe contents of each of the queues Di (i 1Y F F F YR 1).

The behavior of the SEKCS can be described as fol-lows.


At each manufacturing cell i (i 1Y F F F YR 1), queuesP0Yi, Ai, and Di are joined in a synchronization station.This means that cell i can begin the production of a part

only when there is at least one part p0Yi in P0Yi, one au-thorization card ai in Ai, and one demand di in Di. Whenthese conditions are met, then:

(1) Kanban ai is attached onto p0Yi which is relabeledqi, and together they are transferred downstream toMPi as a pair (qiY ai).

(2) Demand di is satised and is therefore discarded.

When part qi nishes its processing in MPi, it is rela-beled pi, and, together with kanban ai that was attachedonto it, they join PAi as a pair (piY ai).

Release of parts into the assembly cell

At the assembly cell R, queues PAi (i 1Y F F F YR 1), AR,and DR are joined in a synchronization station. Thismeans that the assembly operation can begin only whenthere is at least one pair (piY ai) in PAi for every i 1Y F F F YR 1, one authorization aR in AR, and one demanddR1 in DR1. When these conditions are satised, then:

(1) Kanbans ai are simultaneously detached from the pi(i 1Y F F F YR 1) and are transferred upstream totheir corresponding previous cell.

(2) Kanban aR is attached onto (p1Yp2Y F F F YpR1) whichis relabeled qR, and together they are transferred

downstream to MPR as a pair (qRY aR).

(3) Demand dR is satised and is therefore dis-carded.

When a part qR nishes its assembly process in MPR, itis relabeled pR, and, together with the kanban aR that was

attached onto it, they join PAR as a pair (pRY aR).

Delivery of nished parts to the customer

At the nal cell, queues PAR and DR1 are joined in asynchronization station. There is no need for an autho-rization to release a nished part to the customer.Therefore, the delivery of a nished assembled part canoccur as soon as there is a pair (pRY aR) in PAR and ademand dR1 in DR1. When these conditions are satis-ed, then:

(1) Kanban aR is detached from pR and is transferredupstream to AR.

(2) Part pR is released to the customer.(3) Demand dR1 is satised and is therefore discarded.

3.2. Independent extended kanban control system

Figure 5 shows the queueing network model of theIEKCS for the case of (R 1) manufacturing processessupplying a single assembly process.

We use the same notation as in the SEKCS except thatdR and aR are each split into R 1 components, dRYi andaRYi, respectively, corresponding to each manufacturingcell. Thus,

dRYiY i 1Y F F F YR 1 = a demand for the assembly of anew pR using a cell-i nishedpart pi;

aRYiY i 1Y F F F YR 1 = an authorization card (issuedfrom some aR) for the assemblyof a new pR using a cell-i n-ished part pi.

Table 3 describes the contents and initial states ofqueues ARYi (in place of AR), Bi and DRYi (in place of DR).The contents and initial states of all other queues are thesame as those in the SEKCS (see Table 2).

As was the case in the SEKCS, queues P0Yi (i 1Y F F F YR 1) represent the raw parts buer supplyingmanufacturing cell i. The initial number of raw parts in

P0Yi and the arrival process of new parts into P0Yi falloutside the scope of the control mechanism and areconsidered as given. When a customer demand arrives tothe system, it is immediately transmitted to all cells byadding one to the contents of queues DR1, Di, and DRYi(i 1Y F F F YR 1).

The behavior of the IEKCS can be described as fol-lows.


It is identical to the SEKCS.

Table 2. Contents and initial state of the queues in the SEKCS


MPi i 1Y F F F YR 1 qiY ai 0MPR qRY aR 0PAi i 1Y F F F YR 1 piY ai SiPAR pRY aR SRAi i 1Y F F F YR 1 ai Ki SiAR aR KR SRDi i 1Y F F F YR 1 Di 0DR DR 0

DR1 dR1 0



6/14

Release of parts into the assembly cell

The dierence between the SEKCS and the IEKCS is inthe way kanbans are transferred in cells 1Y F F F YR 1. Inthe SEKCS all kanbans are transferred simultaneously,whereas in the IEKCS they are transferred independentlyof each other. In the IEKCS, each demand for anassembly operation is split into R 1 demands dRYi(i 1Y F F F YR 1). Similarly, each cell-R kanban is splitinto R 1 kanbans aRYi, upon its liberation from a n-ished part pR.

Between the R 1 manufacturing cells 1Y F F F YR 1 andthe assembly cell R there are two layers of synchroniza-

tion stations. The rst layer consists of R 1 synchroni-zation stations in parallel, one for each manufacturingcell, and the second layer consists of a single synchroni-zation station fed by the synchronization stations of therst layer. More precisely:

For each manufacturing cell i (i 1Y F F F YR 1),queues PAi, ARYi, and DRYi are joined in a synchronization

station. The supply of cell i nished parts for the as-sembly operation can occur only when there is at leastone pair (piY ai) in PAi, one authorization aRYi in ARYi, andone demand dRYi in DRYi. When these conditions are met,then:

(1) Kanban ai is detached from pi independently ofwhat is going on in cells 1Y F F F Y i 1Y i 1Y F F F Y

R 1 and is transferred upstream to Ai.(2) Kanban aRYi is attached onto pi, and together they

are transferred downstream to Bi as a pair (piY aRYi).(3) Demand dRYi is satised and is therefore discarded.

Queues Bi (i 1YF F F

YR 1), are joined in a synchro-nization station. When there is at least one pair (piY aRYi) in

Fig. 5. Queueing network model for the IEKCS.

Table 3. Contents and initial state of some queues in theIEKCS


ARYi i 1Y F F F YR 1 aRYi KR SRBi i 1Y F F F YR 1 piY aRYi 0

DRYi i 1Y F F F YR 1 dRYi 0



7/14

each Bi, the assembly process can begin and the followinghappens:

(1) Pairs (piY aRYi), are removed from queues Bi(i 1Y F F F YR 1), (R 1)-tuple (p1Yp2Y F F F YpR1) is

relabeled qR, kanbans (aRY1Y aRY2Y F F F Y aRYR1) aremerged into a kanban aR, and pair (qRY aR) istransferred downstream to MPR.

When a part qR nishes its assembly process in MPR, itis relabeled pR, and, together with kanban aR that wasattached onto it, they join PAR as a pair (pRY aR).

Delivery of nished parts to the customer

It is identical to the SEKCS except that when a kanban aRis transferred back to the input of the assembly cell, it issplit into R 1 kanbans, aRY1Y aRY2Y F F F Y aRYR1, and kanbanaRYi joins queue ARYi (i 1Y F F F YR 1).

4. Special cases of the SEKCS and the IEKCS

Property 1 and Property 2 that follow refer to two specialcases where the SEKCS (respectively the IEKCS) isequivalent to the Simultaneous Kanban Control System(SKCS) and to the Base Stock Control System (BSCS)(respectively to the Independent Kanban Control System(IKCS) and to the BSCS).

Property 1.

(1) The SEKCS with Ki I

and Si !

0

i

1Y F F F YRis equivalent to the BSCS having a base stock of Si

nished parts in cell i i 1Y F F F YR.(2) The SEKCS with Ki Si i 1Y F F F YR is equivalent

to the SKCS having Ki kanbans in cell ii 1Y F F F YR.

The proof, which can be found in Chaouiya et al.(1998), is based on the elimination of the queues thatplay no role in the synchronization stations of thequeueing network model of the SEKCS. Indeed, in case1, queues Ai play no role in the synchronization stationsthey belong to since they have an innite number of

kanbans. Similarly, in case 2, queues Di play no role inthe synchronization stations they belong to since theyalways have a number of demands greater than thenumber of kanbans. Once these queues are eliminated,the remaining queueing network obtained is the same asthat of the BSCS (in case 1) and of the SKCS (in case 2).

Property 2.

(1) The IEKCS with Ki I andSi ! 0 i 1Y F F F YR isequivalent to the BSCS having a base stock of Si

nished parts in cell i i 1Y F F F YR.(2) The IEKCS with Ki Si i 1Y F F F YR is equivalent

to the IKCS having Ki kanbans in cell ii 1Y F F F YR.

The proof is similar to that of Property 1 and can befound in Chaouiya et al. (1998).

5. Properties of the SEKCS and the IEKCS

In this section, we present some basic properties of theSEKCS and IEKCS that are helpful in obtaining a betterunderstanding, as well as some insights, in the behavior ofthe two systems. The purpose is twofold: (i) to under-stand the behavior of each control mechanism and inparticular the inuence of its parameters (the KHi s and theSHi s); and (ii) to compare the behavior of the SEKCS andthe IEKCS in order to emphasize the particularity of eachsystem and to see how the behavior of one system relatesto that of the other. We will establish some relations onthe population of the queues in the system, called invar-iants, which are valid at all times (Section 5.1). Moreover,we will derive bounds that imply a limitation on theWIP and on the number of nished parts in each cell(Section 5.2).

Both the SEKCS and the IEKCS can be modeled asFork-Join Queuing Networks with Blocking (FJQN/B) asdened in Dallery et al. (1997). FJQN/Bs are queueingnetworks composed of a set of servers and a set of buers,such that each buer has exactly one upstream server andone downstream server. Each server may have severalinput buers and/or several output buers, and someservers may have no input (sources) or no output (sinks).

Figures 4 and 5 show the FJQN/B models for the SEKCSand the IEKCS, respectively.

For these FJQN/Bs we will prove results concern-ing:

invariance properties related to the cycles of theFJQN;

conditions for deadlock freeness; recursive evolution equations using the operators

`+' and `max'.

These results are similar to those in Baccelli et al.(1992) and also Dallery et al. (1994, 1997) and are based

on the equivalence of FJQN/Bs to Strongly ConnectedMarked Graphs (SCMGs).

We denote by MQ the current population of anyqueue or network of queues, Q, in the system. MQvaries with time and should therefore be a function oftime. However, for simplicity, and since we are interestedin invariants or instantaneous relations, we will omit thisdependence on time. We rst state some basic relationsfor the SEKCS and the IEKCS.

By denition of the initial state of the SEKCS and theIEKCS, queues Ai have Ki Si free kanbans i 1Y F F F Y

R 1, which means that Ki Si ! 0. At the same time,

queue AR (in the case of the SEKCS) and queues ARYi(i 1Y F F F YR 1, in the case of the IEKCS) have



8/14

(KR SR) free kanbans. The two parameters of cell i(i 1Y F F F YR) are therefore constrained by:

Ki ! SiY i 1Y F F F YRX

By denition of a synchronization station, at least oneof the queues in a synchronization station is alwaysempty. The mathematical expression of this is that theproduct of the populations of the queues in a synchro-nization station is zero.

For the synchronization stations in the SEKCS, thisexpression becomes:

MAiMP0YiMDi 0Y i 1Y F F F YR 1Y 1

MARMDRR1

i1

MPAi 0Y 2

MPARMDR1 0X 3

Again by denition of a synchronization station,Equations (1) and (3) that hold for the SEKCS, also holdfor the IEKCS. In addition, in the IEKCS, the followingexpressions hold:

MARYiMPAiDRYi 0Y i 1Y F F F YR 1Y 4

R1

i1

MBi 0X 5

5.1. Invariants

Next, we present properties in the form of invariantspertaining to the content of various queues of thequeueing network models of the SEKCS and the IEKCS.Some of these invariants express the fact that within thequeueing network model of the SEKCS and the IEKCSthere exist several closed subnetworks with a constantpopulation.

Property 3. In the SEKCS, the following holds:

MAi MMPi MPAi KiY i 1Y F F F YR 1Y 6

MAi MDi MDR Ki SiY i 1Y F F F YR 1Y

7MPAi MDR MMPi MDi SiY

i 1Y F F F YR 1Y 8

MAR MMPR MPAR KRY 9

MAR MDR1 MDR KR SRY 10

MPAR MDR1 MMPR MDR SRX 11

Proof. Equations (6) and (9) are straightforward sincethe total number of kanbans at each cell is constant.When the SEKCS is in its initial state, there are Ki Si

kanbans ai in queue Ai (i 1YF F F

YR 1) and queue Di(i 1Y F F F YR) is empty. Therefore, Equation (7) holds

initially. As the SEKCS evolves starting from its initialstate, Equation (7) remains true since:

(1) When a kanban is transferred to Ai, a demandleaves DR.

(2) When a kanban leaves Ai, a demand also leaves Di.(3) When a demand joins Di, a demand also joins DR.

To prove Equation (8) it suces to substitute MAifrom Equation (6) into Equation (7). Equations (10) and(11) can be proved using similar arguments. j

Property 4. In the IEKCS, the following holds:

MAi MMPi MPAi KiY i 1Y F F F YR 1Y 12

MAi MDi MDRYi Ki SiY i 1Y F F F YR 1Y

13

MPAi MDRYi MMPi MDi SiY i 1Y F F F YR 1Y14

MARYi MBi MARYj MBj 0Y

iYj 1Y F F F YR 1Y 15

MARYi MBi MMPR MPAR KRY

i 1Y F F F YR 1Y 16

MARYi MDR1 DRYi KR SRY

i 1Y F F F YR 1Y 17

MPAR MDR1 MBi MMPR MDRYi SRY

i 1Y F F F YR 1X 18

The proof is similar to that of Property 3 and is thereforeomitted.

5.2. Bounds

The bounds that follow, although quite obvious are im-portant since they establish that the Work-In-Process(WIP) is limited in the SEKCS as well in the IEKCS.

Property 5. In the SEKCS, the following holds for all thecells:

0 MAi KiY i 1Y F F F YRY

0 MPAi KiY i 1Y F F F YRY

0 MMPi KiY i 1Y F F F YRY

0 MMPi MPAi KiY i 1Y F F F YRX

19

Property 6. In the IEKCS, the following holds for all themanufacturing cells:

0 MAi KiY i 1Y F F F YR 1Y

0 MPAi KiY i 1Y F F F YR 1Y

0 MMPi KiY i 1Y F F F YR 1Y

0 MMPi MPAi KiY i 1Y F F F YR 1X

20



9/14

In the IEKCS, the following holds for the assembly cell:

0 MBi KRY i 1Y F F F YR 1Y

0 MARYi KRY i 1Y F F F YR 1Y

0 MMPR KRY

0 MPAR KRY

0 MBi MMPR MDRYi KRY i 1Y F F F YR 1X

21

Property 5 and Property 6 express the fact that, sincethere is no new kanban generated in the productionprocess, the total number of parts can not exceed thenumber of kanbans in each cell. A formal proof can befound in Chaouiya et al. (1998).

Property 7 and Property 8 also present some bounds.

Property 7. In the SEKCS the following holds:

MPAi MDR SiY i 1Y F F F YR 1Y 22

MPAR MDR1 SRX 23

Proof. Equation (22) follows from Equation (8), andEquation (23) follows from Equation (11). j

Property 8. In the IEKCS, the following holds:

MPAi MDRYi SiY i 1Y F F F YR 1Y 24

MPAR MDR1 SRX 25

The proof is similar to that of Property 7 and is therefore

omitted.

5.3. Evolution equations

The purpose of this section is to provide the basic equa-tions that describe the evolution of the SEKCS and theIEKCS. We show that the dynamics of the two systemscan be described by recursive evolution equations thatutilize the operators ``+'' and ``max'' only. These evo-lution equations are of interest because they allow us toestablish some useful properties on the behavior of theSEKCS and the IEKCS (Section 5.4) as well as to com-

pare the two systems (Section 6). Moreover, this ap-proach is very general since it is a sample-path approachthat does not require any assumption on the distributionsof the random variables (processing times and interarrivaltimes of demands).

For the sake of simplicity, we assume that there is aninnite supply of raw parts in piY0 (i 1Y F F F YR 1). Theresults that follow, however, could be extended toincorporate external arrival processes of parts at the ex-pense of more tedious derivations.

5.3.1. Evolution equations for the SEKCS

We introduce the following notation for the times of allpossible events that may take place:

IiYn = the time of the nth arrival in MPi (Input),i 1Y F F F YR;

IR1Yn = the time of the nth delivery to a customer;OiYn = the time of the nth departure from MPi (Out-

put), i 1Y F F F YR;Dn = the time of the nth demand arrival (Demand).

Finally, for the sake of simplicity, we assume that MPiconsists of a single machine, and we let riYn be the pro-cessing time of the nth part at the machine in MPi(i 1Y F F F YR).

Clearly, the following holds:

IiYnm IiYnY i 1Y F F F YR 1 and nY m 1Y 2Y F F F Y

OiYnm OiYnY i 1Y F F F YR and nY m 1Y 2Y F F F Y

Dnm DnY nY m 1Y 2Y F F F Y

where by denition IiYn, OiYn, and Dn are zero for n 0.

We now have the proposition below, which states theevolution equations for the SEKCS.

Proposition 1. In the SEKCS, the times of events arerelated by the following evolution equations:

IiYn maxDnYIRYnKiSiY i 1Y F F F YR 1Y 26

IRYn maxDnY maxi1YFFFYR1

OiYnSiYIR1Y nKRSRY 27

IR1Yn maxDnY ORYnSRY 28

OiYn riYn maxIiYnY OiYn1Y i 1Y F F F YRX 29

Proof. Equation (26) gives an expression forIiYn

which,by denition, represents the time at which the nth pair(qiY ai) is released into MPi (i 1Y F F F YR 1). Indeed, thisrelease occurs only when two conditions are met:

(1) The nth demand di has arrived in Di.(2) The (n Ki Si))th kanban has arrived in Ai,

since initially there are (Ki Si) kanbans in Ai.

We recall that, in the SEKCS, a kanban ai arrives in Aiwhen a pair (qRY aR) is released into MPR.

Because of the assumption that there is an innitesupply of raw parts in P0Yi, no condition involving arrivalsin P0Yi appears in Equation (26).

To prove Equation (27), we will use similar arguments,only now, arrivals in queues PAi must also be taken intoaccount. IRYn is, by denition, the time at which the nthpair (qRY aR) is released into MPR. Indeed, this can occuronly when three conditions are met:

(1) The nth demand dR has arrived in DR.(2) The n Sith cell-i nished part with its attached

kanban has arrived in PAi (i 1Y F F F YR 1), sincethere are initially Si pairs (piY ai) in PAi.

(3) The (n KR SR))th kanban aR has arrived in AR,since initially there are (KR SR) kanbans in AR.

Equation (28) gives an expression for IR1Yn, which is,by denition, the time when the nth part is delivered to



10/14

the customer. This event can occur when the two fol-lowing conditions are met:

(1) The nth demand has arrived in Dn.(2) The (n SRth nished part pR with its attached

kanban has been released in PAR, since initiallythere are SR pairs (pRY aR) in PAR.

Equation (29) gives an expression for OiYn which is, bydenition, the time at which the nth pair (piY ai) hascompleted processing in MPi and is released in PAi. Thistime is equal to the time at which the nth pair (qiY ai)begins its processing in MPi plus its processing time riYn.The nth pair (qiY ai) begins its processing when the twoconditions below are satised:

(1) The nth pair (qiY ai) has been released in MPi.(2) The (n 1th part has completed its processing at

MPi. j

5.3.2. Evolution equations for the IEKCS

In addition to the previous notation, let LiYn be the timewhen the nth pair (piY aRYi) arrives at queue Bi (that is, therelease time of the nth kanban ai).

The following proposition states the evolution equa-tions for the IEKCS.

Proposition 2. In the IEKCS, the times of events are re-lated by the following evolution equations:

IiYn maxDnYLiYnKiSiY i 1Y F F F YR 1Y 30

IRYn maxi1YFFFYR1

LiYnY 31

IR1Yn maxDnY ORYnSRY 32

LiYn maxDnYOiYnSi YIR1YnKRSRY i 1Y F F F YR 1Y

33

OiYn riYn maxIiYnY OiYn1Y i 1Y F F F YRX 34

Proof. Equation (30) diers from Equation (26) only inthe condition for the release of kanbans ai. Otherwise, theargument is similar to that for the SEKCS: the nth pair(qiY ai) enters MPi (i 1Y F F F YR 1) when the two condi-

tions below are met:

(1) The nth demand di has arrived in Di.(2) The (n Ki Si))th kanban has arrived in Ai,

since initially there are (Ki Si) kanbans in Ai. Inthe IEKCS, a kanban ai arrives in Ai when a pair(piY aRYi) is transferred to Bi.

As in the SEKCS, because of the assumption of aninnite supply of raw parts in P0Yi, no condition involvingarrivals in P0Yi appears in Equation (30).

Equation (31) represents the time at which the nth pair(qRY aR) is transferred to MPR. This occurs when the nth

pair (piY aRYi) has arrived in Bi, for all i 1YF F F

YR 1.Equation (32) is the same as Equation (28).

In Equation (33), we consider the time at which thecell-i nth nished part is transferred to Bi. This eventoccurs as soon as the following three conditions are met:

(1) The nth demand has arrived in DRYi.

(2) The (n Si)th pair (piY ai) has arrived in PAi, sinceinitially there are Si pairs in PAi.

(3) The (n KR SR))th kanban aRYi has arrived inARYi, since initially there are (KR SR) kanbans inARYi.

Finally, Equation (34) is the same as Equation (29).j

5.4. Variation of parameters

In this section we study the inuence of parameters Kiand Si (i 1Y F F F YR) on the above-mentioned event times,and we derive some monotonicity properties.

Proofs are similar to the proof of ``StochasticMonotonicity with Respect to the Initial Marking'' inBaccelli et al. (1992) and Baccelli and Liu (1992). We willuse the fact that the time when events occur in the systemcan be computed recursively according to the evolutionequations. Therefore, there exists a total ordering on thetimes described by Proposition 1 and Proposition 2 al-lowing us to use proofs by induction for the followingresults.

We compare two systems: the nominal system denotedby S and the modied system denoted by ~S. These twosystems dier only through their parameters: the number

of kanbans and the base stock level in each cell. Theparameters of the original systems are Ki and Si, theparameters of the modied system are ~Ki and ~Si(i 1Y F F F YR). On the other hand, the two systems havethe same sequence of customer demand times (denoted by

Dn for S, and ~Dn for ~S) and the same sequence of pro-cessing times (denoted by riYn and ~riYn (i 1Y F F F YR)).

Part 1 of Property 9 and Property 10 that follow statesthat increasing the number of kanbans in some cell, de-creases the arrival and departure times in each cell of thesystem. Parts 2 and 3 state that increasing the base stockin any cell q (q 1Y F F F YR), decreases the arrival and de-parture times of all other cells. Moreover, the arrival time

(respectively the departure time) of the nth part from cellq decreases with respect to the arrival time (respectivelythe departure time) of the (n ~Sq Sq)th part from MPq.This means that an increase in the base stock of cell qfrom Sq to ~Sq has the same eect as having ( ~Sq Sq) extraparts enter the cell q and receive processing before therst demand arrives.

Property 9 and Property 10 imply that increasing thenumber of kanbans or the base stock level increases theresponse time of the SEKCS and the IEKCS. We will seein Section 5.5 that the number of kanbans is mainly re-lated to the production capacity of the system, while the

base stock level is related to the customer demands sat-isfaction.



11/14

5.4.1. Variation of parameters in the SEKCS

Property 9. Consider the two systems S and ~S under theSEKCS. Then, we have:

(1) If~

Kq b Kq for some q in f1Y F F F YRgY~

Ki Ki for all iin f1Y F F F YRg fqgY and ~Si Si for all i inf1Y F F F YRgY then for all n:

~IiYn IiYnY i 1Y F F F YR 1Y 35

~OiYn OiYnY i 1Y F F F YRX 36

(2) If ~Ki Ki for all i in f1Y F F F YRgY ~Sq b Sq for some qin f1Y F F F YR 1gY and ~Si Si for all i in f1Y F F F YRgfqgY then for all n:

~IiYn IiYnY i P f1Y F F F YR 1g fqgY 37

~IqYn IqYn~SqSqY 38

~OiYn OiYnY i P f1Y F F F YRg fqgY 39

~OqYn OqYn ~SqSqX 40

(3) If ~Ki Ki i 1Y F F F YRY ~Si Si i 1Y F F F YR 1Yand ~SR b SR, then for all n:

~IiYn IiYn~SRSRY i 1Y F F F YRY 41

~IR1Yn IR1YnY 42

~OiYn OiYn~SRSRY i 1Y F F F YRX 43

The proof can be found in Chaouiya et al. (1998).

5.4.2. Variation of parameters in the IEKCS

The inuence of the variation of the parameters upon theevent times for the IEKCS is basically the same as for theSEKCS. We will only state the corresponding property.Its proof is very similar to the proof of Property 9.

Property 10. Consider the two systemsS and ~S under theIEKCS. Then, we have:

(1) If ~Kq b Kq for some q in 1Y F F F YRf g, ~Ki Ki fori P 1Y F F F YRf g qf g, and ~Si Si i 1Y F F F YR,

then for all n:~IiYn IiYnY i 1Y F F F YR 1Y 44

~OiYnY OiYnY i 1Y F F F YRX 45

(2) If ~Ki Ki for all i in 1Y F F F YRf g, ~Sq b Sq for some qin 1Y F F F YR 1f g, and ~Si Si for all i in

1Y F F F YRf g qf g, then for all n:

~IiYn IiYnY i P 1Y F F F YR 1f g fqgY 46

~IqYn IqYn~SqSqY 47

~OiYn OiYnY i P 1Y F F F YRf g fqgY 48

~OqYn OqYn~SqSqX 49

(3) If ~Ki Ki and ~Si Si i 1Y F F F YR 1 and ~SR bSR,then for all n:

~IiYn IiYn~SRSRY i 1Y F F F YRY 50

~

IR1Yn IR1YnY 51~OiYn OiYn~SRSRY i 1Y F F F YRX 52

5.5. Production capacity

The production capacity of a pull control system is themaximum demand rate that the system can meet. Todetermine the production capacity of a pull control sys-tem, we study the saturatedversion of the system, that isthe original system under the assumption that there is aninnite number of raw parts and customer demands. Theproduction capacity of the original system is then the

throughput of the saturated system.Property 11 and Property 13 that follow are important

since they state that the production capacity of the SE-KCS and of the IEKCS depends only on one parameter,namely Ki. The roles of parameters Si (base stock level)and Ki (number of kanbans) are thus clearly distinct.Parameters Si are related to the satisfaction of demands,whereas parameters Ki are related to the production ofnew parts. Thus, parameters Ki could be designed rst toobtain a desirable production capacity and parameters Sicould be designed subsequently to obtain a desirablecustomer satisfaction level (Dallery and Liberopoulos,

2000)Property 12 and property 14 that follow state that inthe saturated case, the SEKCS (respectively the IEKCS)and the SKCS (respectively the IKCS) are equivalent.

5.5.1. Production capacity of the SEKCS

Figure 6 shows the queueing network model of the sat-urated SEKCS having R cells (R 1 manufacturing cellsand a single assembly cell). Figure 6 is obtained fromFig. 4 as follows:

By denition of the saturated SEKCS, queuesP0YiY i 1Y F F F YR 1 have an innite number of raw

Fig. 6. Queueing network model for the saturated SEKCS.



12/14

parts, and queues DiY i 1Y F F F YR 1 have an in-nite number of demands.

Therefore, these queues play no role in the syn-chronization station they belong to since they neverblock the transfer of parts through that synchroni-zation station, hence, they can be eliminated.

Once P0Yi and Di have been eliminated, Ai remains asthe only queue in the synchronization station at theentry of cell i i 1Y F F F YR 1. Similarly, once DR1has been removed, PAR remains as the only queue inthe synchronization station at the output of cell R.Clearly, if a synchronization station is fed by onlyone queue, this queue can be removed since anycustomer arriving at this queue immediately goesthrough the synchronization station. In the saturatedSEKCS, queues Ai i 1Y F F F YR 1 and PAR cantherefore be eliminated.

The queueing network model that results after theseeliminations is shown in Fig. 6. We now have the fol-lowing two properties.

Property 11. The production capacity of the SEKCS de-pends only on parameters Ki i 1Y F F F YR, and is inde-pendent of Si i 1Y F F F YR.

The proof is based on the following result. Thethroughput of a basic FJQN/B containing N elementaryclosed subnetworks depends only on the xed num-ber of customers in each closed subnetwork, and not on

the initial allocation of these customers along the queuesof this closed subnetwork (Dallery et al., 1994, 1997).In the SEKCS, Ki is the xed number of customers in theclosed subnetwork that includes Ai, MPi, and PAi, whereas

Si determines the initial allocation of customers in each ofthe queues of the closed subnetwork.

Property 12. The production capacity of the SEKCS, withparameters Ki and Si i 1Y F F F YR, is equal to the pro-duction capacity of the SKCS with the same parameters

Ki i 1Y F F F YR, as those in the SEKCS.

Indeed, by comparing Fig. 6 and the queueing networkmodel for the saturated SKCS, it is clear that the satu-rated SEKCS is equivalent to the saturated SKCS, andtherefore their throughputs are equal to each other.

5.5.2. Production capacity of the IEKCS

Figure 7 shows the queueing network model of thesaturated IEKCS having R cells (R 1 manufacturingcells and a single assembly cell). Figure 7 can be ob-tained from the queueing network model in Fig. 5 aftereliminating:

queues P0Yi and queues Di and DRYi i 1Y F F F YR 1,using the fact that they contain an innite number ofentities;

queues Ai i 1Y F F F YR 1 and queue PAR, using thefact that they are the only queues feeding a syn-chronization station.

We now have the following properties. The argumentsfor their proofs are similar to those used for the SEKCSand are therefore omitted.

Property 13. The production capacity of the IEKCSdepends only on parameters Ki i 1Y F F F YR, and is inde-

pendent of Si i 1Y F F F YR.

Fig. 7. Queueing network model for the saturated IEKCS.



13/14

Property 14. The production capacity of the IEKCS, withparameters Ki and Si i 1Y F F F YR, is equal to theproduction capacity of the IKCS with the same parametersKi i 1Y F F F YR, as those in the IEKCS.

6. A comparison between the SEKCS and the IEKCS

An intuitive comparison between the queueing networkmodel of the SEKCS shown in Fig. 4 and the queueingnetwork model of the IEKCS shown in Fig. 5 leads tothe conjecture that the IEKCS responds faster to cus-tomer demands than does the SEKCS with the sameparameters. This is due to the mutual independence ofoperation of the manufacturing cells upstream of theassembly cell and it is stated more precisely in Property15 that follows.

To distinguish equivalent times in the two systems, letus introduce the following notation:

ISiYn and IIiYn i 1Y F F F YR = the time of the nth arrival

in MPi, in the SEKCS andthe IEKCS, respectively;

ISR1Yn and IIR1Yn = the time of the nth delivery

to a customer, in theSEKCS and the IEKCS,respectively;

OSiYn and OIiYn i 1Y F F F YR = the time of the nth depar-

ture from MPi, in theSEKCS and the IEKCS,

respectively;DSiYn and D

IiYn = the time of the nth demand

arrival, in the SEKCS andthe IEKCS, respectively.

Property 15. Consider two systems, the SEKCS and theIEKCS, having the same parameters Ki and Si, the samesequence of service times riYn and the same customer de-mand times Dn for all i 1Y F F FR. Then:

IIiYn ISiYnY i 1Y F F F YR 1Y 53

OIiYn OSiYnY i 1Y F F F YRX 54

The proof is similar to that of Property 9 and can befound in Chaouiya et al. (1998)

Property 15 states that the time at which the nth partbegins its processing (respectively nishes its processing)in MPi in the IEKCS is smaller than the time when the nthpart begins its processing (respectively nishes its pro-cessing) in MPi in the SEKCS. Therefore, customer de-mands are satised earlier in the IEKCS than they are inthe SEKCS. This does not necessarily mean that the IE-KCS has an overall better performance than the SEKCS,since the inventory storage costs are not taken into

account. In fact, the IEKCS is likely to incur higherinventory storage than does the SEKCS.

7. Conclusions

The EKCS is a combination of the Base Stock ControlSystem (BSCS) and the Kanban Control System (KCS)and it includes both systems as special cases. As such, itperforms no worse and is in any case more robust thaneither of the two systems. The EKCS depends on twoparameters per cell, each parameter having a distinct role:the number of kanbans is used basically to limit the WIPin the cell, and the base stock is used basically to providea buer against stockouts, which are associated with badcustomer service.

We extended the EKCS to assembly systems, deningtwo policies and proving some of their properties. Thedierence between the two policies rests on whethercomponent parts and kanbans are released simulta-neously or independently prior to assembly. For sim-

plicity, we restricted our study to assembly systemshaving (R 1) manufacturing cells supplying a singleassembly cell. The results in this paper, however, can beeasily extended to more general assembly topologies. Forinstance, queues P0Yi in Figs. 4 and 5 may be transformedinto queues PA0Yi containing nished parts and kanbansfrom previous cells. In this case, when a part in PA0Yi istransferred downstream, a kanban is released to theprevious cell, in the same way as it happens in cellsi 1Y F F F YR 1, or in cell R.

We assumed that each assembled part is made byputting together one of each of the component parts. Ageneralization of this is straightforward, and can beperformed in two ways. One way is to consider the casewhere the assembly cell requires n items from manufac-turing cell i, so that the latter cell produces batches of nitems, each batch being associated with one kanban.Another way is to consider that n parts of manufacturingcell i are released into the assembly cell, one at a time, andthat n kanbans are transferred upstream to cell i.

An important question when implementing a pull con-trol policy is how to dene its parameters to achieve agood trade o between the customer service levels and theinventory costs. For this purpose, performance evaluationof the EKCS for assembly systems should be performed

using simulation or analytical approximate techniques.

References

Baccelli, P., Cohen, G., Olsder, G.J. and Quadrat, J.-P. (1992) Syn-

chronization and Linearity: an Algebra for Discrete Event Systems,

John Wiley & Sons, New York

Baccelli, F. and Liu, Z. (1992) Comparison properties of stochastic

decision free Petri nets. IEEE Transactions on Automatic Control,

37(12), 19051920.

Buzacott, J.A. (1989) Queueing models of kanban and MRP controlled

production systems. Engineering Cost and Production Economics,

17, 320.

Buzacott, J.A. and Shanthikumar, J.G. (1993) Stochastic Models ofManufacturing Systems, Prentice-Hall, Englewood Clis, NJ.



14/14

Chaouiya, C., Liberopoulos, G. and Dallery, Y. (1998) The extended

kanban system for production control of assembly systems.

Rapport de Recherche LIP6 1998/024, Laboratoire d'Informa-

tique de Paris 6.

Dallery, Y. and Liberopoulos, G. (2000) Extended kanban control

system: combining kanban and base stock. IIE Transactions on

Design and Manufacturing, 32, 369386.

Dallery, Y., Liu, Z. and Towsley, D. (1994) Equivalence, reversibility,

symmetry and concavity in fork/join queueing networks with

blocking. Journal of the ACM, 41(5), 903942.

Dallery, Y., Liu, Z. and Towsley, D. (1997) Properties of fork/join

queueing networks with blocking under various operating mech-

anisms. IEEE Transactions on Robotics and Automation, 13(4),

503518.

Di Mascolo, M. and Dallery, Y. (1996) Performance evaluation of

kanban controlled assembly systems, in Proceedings of the Sym-

posium on Discrete Events and Manufacturing Systems of the

Multiconference on Computational Engineering in Systems

Applications (CESA'96, IMACS), Lille, France.

Hopp, W.J. and Spearman, M.L. (1996) Factory Physics, Irwin/

McGraw Hill, New York, NY.

Liberopoulos, G. and Dallery, Y. (2000) A unied framework for pull

control mechanisms in multi-stage manufacturing systems. Annals

of Operations Research, Vol 93: Performance evaluation and op-

timization of production lines, MacGregor Smith, J., Gershwin,

S.B. and Papadopoulos, C.T. (eds), Baltzer Science, Bussum, The

Netherlands, pp. 325355.

Monden, Y. (1983) Toyota Production System: Practical Approach to

Production Management, Industrial Engineering and Management

Press, Narcross, GA.

Sbiti, N., Di Mascolo, M. and Amghar, M. (1999) Analysis of base-

stock controlled assembly systems, in Proceedings of the 2nd Ae-

gean Conference on Analysis and Modeling of Manufacturing

Systems, Editions Ziti, Thessaloniki Greece, pp. 141151.

Biographies

Claudine Chaouiya received her Ph.D. from the Universite de Nice

Sophia-Antipolis, in 1992. She is currently an Assistant Professor in

Computer Science at the Ecole Superieure d'Ingenieurs de Luminy

(ESIL). In 19931994, she was a Visiting Professor at the UniversidadeFederal do Ceara (Brasil), in 19941999 she was an Assistant Professor

at the Universite de Nouvelle Caledonie. Her research interests are in

modelling and analysis of discrete event systems, with applications in

manufacturing and communication networks.

George Liberopoulos received his B.S. and M.Eng. from Cornell

University in 1985 and 1986, respectively, and his Ph.D. from Boston

University in 1993. In 199496 he was a visiting scientist at Universite

Paris IV. He is currently an Assistant Professor of Production Man-

agement at the University of Thessaly. His research interests are in

operations management with an emphasis on the analysis and control

of manufacturing systems. He serves at the Editorial Board of IIE

Transactions on Design and Manufacturing.

Yves Dallery received his Ph.D. and the degree of Habilitation a Di-riger des Recherches from the Institut National Polytechnique de

Grenoble (INPG) in 1984 and 1989, respectively. He is currently Pro-

fessor of Manufacturing and Logistics at Ecole Centrale de Paris.

Before that, he was Directeur de Recherche at the Centre National de

la Recherche Scientique (CNRS). In 19841985, he was a post-doc-

toral fellow at Harvard University. In 19911992, he was a visiting

scientist at M.I.T. and in 19921993 he was an Associate Professor of

Manufacturing Engineering at Boston University. His research inter-

ests are in operations management, supply chain management, and

stochastic models.


The extended kanban control system for production coordination of assembly manufacturing systems

Documents

Transcript of The extended kanban control system for production coordination of assembly manufacturing systems