Flexibility in manufacturing systems: A relational and a dynamic approach

Theory and Methodology

Flexibility in manufacturing systems: A relational and a dynamicapproach q

Javier Pereira a,*, Bernard Paulr�e b

a Departamento de Inform�atica de Gesti�on, Universidad de Talca, Avenida Lircay s/n, Talca, Chileb ISYS-METIS, Universit�e Paris I Panth�eon-Sorbonne, Centre Pierre Mend�es France, 90 rue de Tolbiac, Paris, France

Received 3 June 1997; accepted 2 December 1999

Abstract

In this paper, we propose that ¯exibility in a manufacturing system is not independent of the environment with which

it interacts. Furthermore, we propose that ¯exibility is a multidimensional concept relating the degree, e�ort and time of

adaptation. In order to arrive at this approach, a dynamic perspective is adopted in which the ¯exibility dimensions are

related. A number of examples are presented to point out some problems where this approach may be used. Ó 2001

Elsevier Science B.V. All rights reserved.

Keywords: Flexible manufacturing systems; Multidimensional approach; Field of variations; Field of tensions

1. Introduction

One of the characteristics of modern organisa-tional management is evolution. Managementmust be capable of ¯exible responses to constantlychanging o�ers of new products or product vari-ants. In the same way, the increasing pressure ex-ercised by technological development and thedemand variability, compel ®rms to be ¯exible.Ways of looking at problems and their potentialsolutions must be adapted to ®t the nature and

rate of change in the organisation's environment[25]. It is in this context that the notion of ¯exi-bility acquires its importance, since it is thought tocontribute to the capacity of selection, of modi®-cation and innovation of the programs of organi-zational action.

Because an organization needs several types of¯exibility, there are multiple de®nitions and eval-uation schemes. The origins of this diversity maybe linked to the variety of uncertainty factors, thepossible term perspectives or the di�erent possibledimensions to evaluate for ¯exibility. Thus, severalauthors claim that a speci®c type of ¯exibilitymaps to a given source of uncertainty [10]. Like-wise, ¯exibility is classi®ed in the literature asshort-term, mid-term or long-term [3]. Further-more, many authors use three dimensions to de®ne

European Journal of Operational Research 130 (2001) 70±82www.elsevier.com/locate/dsw

q This research is supported by the DIUT project No. 463-12,

at the Universidad de Talca.* Corresponding author.

E-mail address: [email protected] (J.

Pereira).

0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 0 2 0 - 5

and to evaluate ¯exibility [10]: the variety of po-tential system's responses, the e�ort and the timeof adaptation. Indeed, in considering these threeprincipal axes, it is sometimes very di�cult to useand compare two literature's proposed approach-es.

There are several operational di�culties inachieving an unambiguous understanding of no-tions of ¯exibility [10±12,36]:1. an intuitive de®nition, sometimes not opera-

tional, of the notion of ¯exibility [13];2. lack of uniformity in the classi®cation, de®ni-

tion and evaluation of a given type of ¯exibility[1±5,19,27,32];

3. a di�culty in the establishment of the single-di-mensional or multidimensional character of¯exibility [10];

4. a misconceived relationship by which ¯exibilitydirectly enhances the system's performance[7,28].In this paper, we propose a new approach in

which the system's ¯exibilities are speci®ed bymeans of a frame of analysis. Our intention is tode®ne clearly and simply what ¯exibility is, what adimension of ¯exibility is and how it must beevaluated. Our aim is to characterize ¯exibility in arelational, multidimensional and dynamic way.We argue that this perspective conduces research-ers and practitioners to an unambiguous de®nitionof ¯exibility which enables them to better evalu-ating schemes.

Firstly, we must talk about ¯exibility when oneor several particular system dimensions have beenspeci®ed for change: a ®eld of variations mustrepresent the space of states on which the system'schanges are considered. Thus, what are the systemproperties, conditioning changes on this ®eld?: wetalk about the tensions of the system and then wede®ne the ®eld of tensions.

Secondly, although di�erent articles proposevariety as a measure of ¯exibility, we propose ameasure called the adjustment degree whichcaptures the system±environment relationshipembedded in the ¯exibility notion: it considersthe ®eld of variations. Accordingly, because oftensions, the e�ort and time of adjustment areintroduced as two necessary measures of ¯exi-bility. In this manner, we render consistent the

adjustment, e�ort and time dimensions of ¯exi-bility in a perspective called here the longitudinalapproach.

The paper is organized as follows: in Section 2our approach and the relevant concepts to be usedare introduced. In Section 3, a simple stochasticmodel enables us to present the adjustment degreeas one of the right dimensions of ¯exibility. Thesecond example of this section shows the criticaldi�erences between the static and the dynamicapproaches to explain ¯exibility. In Section 4, weanalyze the problem of determining the productlot sizes in a manufacturing con®guration, sub-mitted to a variable demand, and we identify thedi�erent factors of tension de®ning the system'se�ort to obtain an optimal adjustment to the de-mand process. In the example of Section 5, wecharacterize what we call the inner logic, i.e., thelogic which translates the environment state intothe system expected state. The conclusions arepresented in Section 6.

2. The proposed approach

The capability of a system to adapt to changesthat occur in its environment is called here ¯exi-bility. This intuitive de®nition brings two ques-tions: adaptation to what? and how? In order toanswer these questions, we are going to considerthe system as any logical and/or a physical devicewhich we might wish to evaluate for ¯exibility. Thesystem could be a machine, a mechanism, a logic, aprocedure, a process, a program, a technique, anapparatus, an organization, etc. All these entitiesmay be described as operationally closed [8]: theyhave boundaries that we can de®ne by their func-tional and control limits. So, all that it is notmodeled as part of this closure, is located in theenvironment of an entity.

Primary, we must clarify what is to be com-prehended by adaptation. Indeed, the question ofadaptation or not is properly a subject of an ob-server O1 (the modeller 1) who evaluates the degreeof congruence between the system and its envi-ronment. In this evaluation O1 actually uses (im-plicitly or explicitly) a deviation function whichtells him or her if the system state is adapted to the

J. Pereira, B. Paulr�e / European Journal of Operational Research 130 (2001) 70±82 71

environment state. But, we must not forget thatanother observer O2 may use his/her own model ofthe same system, with its own particular deviationfunction. Consequently, the adaptation capabilityis a relative property attributed by an observer to amodeled system.

The important question is, when is the systemadapted to its environment? The answer is, whenthere is no more system to study, it ceased to beadapted. In other words, it is the degree of adap-tation, which depends on the model that an ob-server makes of reality [22]. In fact, he representsone thing labeled the system, another thing labeledthe environment and any kind of relationships be-tween them.

In this paper, we are going to write aboutmodels. In consequence, we will use the idea of thedegree of adaptation as the local adjustment mea-sure (cf. Section 2.1) between the current and theexpected states of the system, such as it is deter-mined by an inner logic. For a given model, we willinvestigate what is to change in the environmentand the system, what is to be de®ned as the con-gruence between them, what is to be de®ned as thedeviation function and how we can measure thisdeviation.

2.1. The ®eld of variations

Suppose that we are able to characterize thebehavior of a system and its environment throughthe trajectories that they take in a set of states. LetS be this set of states; we will call it the ®eld ofvariations. Let R � S be the subset of realizablestates of the system. Additionally, let E be the setof the states in which the environment moves. Notethat these set de®nitions must all be given by aspeci®c observer.

Now, let st 2 R, et 2 E and s�t 2S be the ob-served current state of the system, the observedcurrent state of the environment and the expectedcurrent state of the system, respectively. We willsuppose that there exists, in the system, a L logicsuch that

L�et; st� � �s�t ; ks�t ÿ stk�; �1�

i.e., given st and et the L logic allows us to de-termine the expected state and the norm betweens�t and st. Then, we can say that the system is inpartial equilibrium when L�et; st� � �s�t ; 0� or notin partial equilibrium if ks�t ÿ stk 6� 0.

De®nition 1. If ks�t ÿ stk 6� 0, then ¯exibility is theproperty that tends to realize the partial equilib-rium in the system.

Thus, a ¯exible system has the capability toadjust its current state in response to the deviationks�t ÿ stk. Therefore, we assume that the relation-ship between the system and its environment isdirectly established by the L logic. This is a validassumption because we have established that theimplicated observer models the system as a closedentity (cf. Section 2). Indeed, the adjustment couldbe understood as the system's expected statetracking process. It means that, in our approach,the system does not adjust to the environment, butto its own expected states.

Let D � s�1; . . . ; s�n be the ®nite succession ofexpected states as determined by L between somereference periods t � 1 and t � n �n > 1�. We willnot restrictively suppose that the relationshipD � R must hold true. Additionally, letF � s1; . . . ; sn be the succession of states adoptedby the system when it seeks to adjust to the Dsuccession. Thus, we will demonstrate in this paperthe necessity of using a measure of adjustmentbetween D and F.

2.2. The ®eld of tensions

Time and ease of adjustment can both varyconsiderably. The fact that adjustment requires ane�ort, and a certain time interval, points to theexistence of one or more factors of resistance tochange that we call the factors of tension. For in-stance, the time and costs of setup operations of a¯exible machine, depends on the resistance to thechange; we argue that this resistance may representthe system's tension. Similarly, the level of ac-ceptability of an urgent order that compels a ®rmto use their machines after the normal day of workgives an idea of the level of system's tension. Alike,

72 J. Pereira, B. Paulr�e / European Journal of Operational Research 130 (2001) 70±82

any change in project planning (of construction,engineering, computer system development, etc.)implies an e�ort (cost); this is an irreversibility ef-fect that represents again the system's tensions.

Suppose that n factors of tension represent thesystem's resistances to change on R �S. Let XS

ibe the set de®ning the ®eld of variations of the ifactor of tension when the ®eld of variations S isconsidered (remember that changes on the ®eld ofvariations do not correspond to those on the ®eldof tensions: variations on the latter induce or resistto those on the former). We de®ne the ®eld oftensions as XS � Sn

i XSi . Additionally, we de®ne

the level of tension by a function T such as:

T : S! XS; �2�

which establishes a mapping between any systemstate and the corresponding factors of tension ontheir own ®elds of variations. Thus, any statetransition on S implies a change of the level oftension. Therefore, time and e�ort are necessaryand we argue than they must be considered in re-lation to a speci®c adjustment situation, what wecall the longitudinal approach, and not indepen-dently of it, what we call here the transversal ap-proach. In fact, in considering the e�ects of thefactors of tension over the ®eld of variations, wecan appreciate the di�erence between these twoapproaches.

2.3. The dynamic and relational ¯exibility

In a dynamic approach, we represent the stateexpectations as a succession D of states deter-mined by a L logic, and the system's responses tothese demands as a F succession, tracing the ef-fective system moves. Note that, in F, whateverthe transition from a state st to a state st�Dt be, itdemands e�ort and time. Therefore, the systemdynamically adjusts to the demanded changes de-®ned by the D succession.

The static approaches in literature propose torelate the transition e�orts to the R set (cf. Section2.1). In such perspectives, we cannot correctly as-sociate dynamic e�orts or time to R. In our ap-proach, only a speci®c environmental process may

explain time and e�ort of adjustment of the sys-tem. We claim that this is a congruent dynamicalapproach to conceive ¯exibility. Thus, we proposea change of perspective that relates F (and not R)to the system's e�orts and times of transition. Thisimplies that ¯exibility is a relative property: it de-pends on a well-speci®ed environment.

The following sections present a number ofexamples to illustrate how this approach may beused. The similarities and the di�erences betweenour approach and other perspectives found in theliterature are signaled.

3. Examples of adjustment capability: The trans-versal and the longitudinal approaches

Researchers have traditionally assumed that¯exibility may be evaluated by three kinds ofmeasures: the variety, the e�ort and the time ofsystem adjustment considered for adaptation [10].Under this assumption, variety is evaluated bymeasures such as the number of responses whichthe system may potentially generate. In thosecases, the ¯exibility is de®ned as a system's prop-erty which does not depend on a particular envi-ronment process. Nevertheless, we argue that theadjustment capability of a system must be primaryrelated to the system±environment evolution. Inthe next example, we show that variety does notcompletely capture the relational sense of ¯exibil-ity.

Consider a system and its environment. Wewant to model their interaction by two completelyergodic Markovian processes [14]. Let P ��pij��n�n� and Q � �qij��n�n� respectively be thetransition probability matrix of the environmentand the system. Suppose that the set of statesrepresented in P and Q are identical; they are bothsubsets of S (cf. Section 2.1), the ®eld of varia-tions.

Let us consider the steady state for the systemand its environment. To model the interaction, weassume that the environment undertakes transi-tions and the system try to follow them. Indeed, Pmodels the transitions of the expected states asthey are decoded for one L logic in the system.


Then, in a perfect adjustment capability model, wehave

Q � P : �3�Moreover, suppose we are going to determine theentropy, H, of these processes [9,19]. Thus, wehave HQ � HP . Nevertheless, it is easy to concludethat, even if entropy is reduced (or increased), re-garding a reference value, the system and the en-vironment processes are identical. Consequently,in that case, we are not able to evaluate the system¯exibility by entropy (or variety), but by the rela-tive entropy

system entropy

environment entropy:

Here, this ratio is called the adjustment degree.Thus, in the preceeding example, the relative en-tropy is maximal and the adjustment degree also;note this measure depends on the system±envi-ronment relationship.

Now, let us consider a two-period sequentialdecision problem [31,37] in which one needs to maketwo temporally related decisions. At the beginningof t � 1, the ®rst period, there is a set A of alterna-tives or actions. At the beginning of t � 2, the sec-ond period, exists a set B of alternatives. There aretwo decisions related in the following way: in t � 1,if an action a 2 A is chosen, then in t � 2 the deci-sion maker could choose an action b 2 B�a�. Actu-ally, we are going to apply a little change to thisproblem. Let us suppose only one action set, A.Thus, when an action a 2 A is chosen, a remainingset B�a� � A is available; whatever the decision be, itis relative to the same set of actions, A.

Let us consider a program of action �ai; aj�,where ai 2 A; aj 2 B�ai�. A decision is called risky[15,21] if it depends on the state probabilities of theenvironment and the whole necessary informationto select this program, including those probabili-ties, is available upon beginning t � 1. In otherwords, we do not have a real second period deci-sion because it is completely determined in the ®rstperiod. In contrast, we have an uncertain decision ifupon beginning t � 2, additional information isreceived. In an uncertain universe, in t � 1, wehave the necessary information to select ai, but not

all the information regarding aj 2 B�ai�. Then, thelatter decision is moved to the beginning of thesecond period.

How can we evaluate the adjustment capabilityof the choice system? First of all, in this problem,the optimizer (observer) considers the rationalapproach as the right point of view to evaluate thepolicies of choice. Thus, whatever the optimal so-lution be, he or she accepts it without hesitation, inthe risky or the uncertain situation. Hence, acompletely rational decision maker believes thatthe optimal action is the best response to the sys-tem's environment. From this perspective, theadjustment capability must be necessarily perfect.

However, if we consider two kinds of decisionmakers, the ®rst one accepting the risky situationand the second one accepting the uncertain pointof view, the optimal solutions may be potentiallydi�erent. Indeed, when the number of second pe-riod actions in the whole remaining set is the samefor all ®rst period actions, the uncertain solutiondominates the risky choice in the sense of gains (orlosses [21]). In that case, to select the optimal riskyaction is evidently not the best policy and theuncertain decision maker would say that ¯exibilityis reduced because of the decreased gain. But, weshould not forget that the risky and the uncertaindecision makers use di�erent models to make achoice. We argue that those models can not becompared because the adjustment degrees wouldbe de®ned by two di�erent perspectives.

Therefore, let S be the set of states of the en-vironment, where the decision maker assigns asubjective probability p�s�; s 2 S. Likewise, letut�ai; s� be the gain, relative to s, when the action ai

is chosen at the t period. Also, letI2 � fy1; y2; . . . ; ymg be the set of probable mes-sages arriving upon the beginning of the secondperiod, with a probability qy�y 2 I2�. Accordingly,in the uncertain situation, the expected gain of the®rst period action ai is given [31] by the expression

E�ai j I2� �X

S

p�s�u1�ai; s�

�X

I2

qy maxaj2B�ai�

XS

p�s jy�u2�aj; s�( )

;

�4�


where p�s jy� is the posterior conditional proba-bility of s. This expected gain is maximized by anaction a1�, that is, E�a1�� maxai2A E�ai j I2�. Let usassume that the decision maker chooses this actionand subsequently the message yr occurs upon thebeginning of the second period. In that period, he/she may consider a new decision problem whereE�aj j I3� is the expected gain of the second periodaction aj when a set of messages, I3, is consideredfor the third period. Then, we have

E�aj j I3� �X

S

p�s jyr�u2�aj; s�

�X

I3

qy maxak2B�aj�

XS

p�s jy; yr�u3�ak; s�( )

:

�5�

The term p�s jy; yr� corresponds to the conditionalprobability of s, when the yr message results in thesecond period; B�aj� is the set of available actionsfrom aj and u3�ak; s� de®nes the gain of ak in thethird period, relative to the state s.

The ¯exibility e�ect would be considered whenthe decision maker contrasts the best second pe-riod action a20 , given a1�, against the best secondperiod action if all the past routes (the ®rst periodalternatives) were considered, a2�. Thus, we have

E�a2� j I3� � maxai2A

maxaj2B�ai�

E�aj j I3��

; �6�

E�a20 j I3� � maxaj2B�a1��

E�aj j I3�; �7�

and the opportunity cost of choosing the wrongalternative in the ®rst period is given by

DE � E�a2� j I3� ÿ E�a20 j I3�: �8�

Note that DE P 0. Thus, there is a perfect adjust-ment situation when DE � 0.

In the literature [21,31], di�erent models pro-pose that the ¯exibility of ai is evaluated by jB�ai�j.Those models are coherent with an approachcalled here the transversal point of view. Becausethese approaches do not consider the relativecharacter of ¯exibility they do not consider thoseactions aj 2 B�ai� which may have a zero proba-bility to be chosen from ai. We propose an alter-

native approach called the longitudinal point ofview. When ai is chosen at the ®rst period, there isa probability distribution on B�ai� which denotesthe probable transitions from ai. These probabili-ties are completely determined by the optimizationproblem which searches for the optimal actionaj 2 B�ai�. In our formulation of the second periodoptimization problem, the probability distributionmay be approximated by the following function:

p�aj jai� � E�aj j I3�X

aj2B�ai�E�aj j I3�:

,�9�

Thus, the entropy of the second period actionaj 2 B�ai�, when the information I3 is considered,could be de®ned: H�aj j I3�. In general, two randomsets X;Y have the property X � Y) H�X�6H�Y�. Thus, if we have B�a20 � � B�a2��, thenH�a20 j I3�6H�a2� j I3�.

Now, let us consider two actions, a2� and a20 ,with remaining sets B�a2�� fa1; a2; a3g andB�a20 � � fa1; a2g. Let us suppose E�a1 j I3� � 10,E�a2 j I3� � 20, E�a3 j I3� � 0. In that case,H�a2� j I3� � H�a20 j I3�. Additionally, DE � 0.Clearly, we should not use jB�a20 �j as an interestingmeasure because the action a3 is totally irrelevant.

On the other hand, if E�a3 j I3� � 30,H�a2� j I3� > H�a20 j I3� and DE > 0, then we estab-lish the opportunity cost of choosing the wrongway in the ®rst period. Finally, if we setE�a1 j I3� � E�a2 j I3� � E�a3 j I3�, then H�a2� j I3� >H�a20 j I3�, but DE � 0. Whence, we may concludethat the more appropriate adjustment degreemeasure is the opportunity cost of optimal actionsand not the relative entropy level.

4. The setup costs in a manufacturing system: The

factors of tension

Here, we analyze the problem of determiningthe product lot sizes in a manufacturing systemsubmitted to a variable demand [16,17,34]. A set ofproducts being given, the optimization problemconsists of determining the lot sizes minimizing thetotal production cost function per unit of time. Inmodeling this problem, DeGroote [7] has proposed


that the ¯exibility is a binary relation evaluated bythe comparison of marginal performance of twosystems obeying the same marginal change of en-vironmental variety.

Let �Q � �Q1;Q2; . . . ;Qn� be the lot sizes vectorof the product set. Then, the optimization costproblem is modeled by

min�Q

Xn

j�1

mjqjcsSQj

�� icjQj

2

�s:t: a6 p;

where, the variables in the model are as follows:

This problem may be solved by the Lagrangianrelaxation. Thus, in determining the Lagrangemultipliers, we may ®nd the measure of productvariety proposed by DeGroote:

v � 1

cm

Xn

j�1

��2icjmjqj

p !2

; �10�

where m �Pnj�1 mj and cm �Pn

j�1 cjmj=m. Ac-cordingly, the optimal cost is de®ned by

C ��2csSicmvp � cm if v6 v�;Sicmv

2�pÿa� � cs�p ÿ a� if v > v�;

��11�

where v� is a critical value of v. DeGroote de®nesthe ¯exibility of a T technology in the followingway:

Let T be a technology described by theS; cs; qj; cj and i parameters. If T1 and T2 aretwo technologies, with optimal costs C1 andC2, such that

oC1

ov6 oC2

ov8v P 0; �12�

then, T1 is more ¯exible than T2.

Indeed, three dimensions are considered in thisde®nition:1. The adjustment in relation to the environmen-

tal change.2. The cost of this adjustment, seemingly repre-

sented by the derivative of the optimal cost.3. The time of the adjustment, equaling one unit

of time.In other words, any system or technology which

uses the precedent optimization model to set theproduction lot sizes, has a perfect adjustmentproperty. The only di�erence between two tech-nologies, in the ¯exibility sense, is determined bythe derivative of the optimal cost.

Although the DeGroote's de®nition is wellstructured, we propose to consider the relativederivative of cost to variety, de®ned as

� � oCov

vC: �13�

This is the elasticity of cost to variety, a mea-sure which has been used in other contexts toevaluate the ¯exibility of a ®rm [23,28]. Actually, ithas been argued that human beings perceive gainsor losses in relation to the levels from which themove starts [20]. Thus, this measure appears to bebetter adapted to comparisons than the single de-rivative of cost to variety.

Let c1 ��2csSicmp

; c2 � Sicm=2�p ÿ a�; c3 �cs�p ÿ a�. Then, one can show

� �1=2

1�1=� ��vp c1=cm� if v6 v�;1

1�1=�vc2=c3� if v > v�:

(�14�

n number of product types in manufacturing,mj periodical demand rate for product

j � 1; . . . ; n,Qj lot size for product j,rj production rate for product j,p availability of the facility,S nominal setup time,cs cost per unit of setup time,qj fraction of S used in the setup of product j,cj unit direct labor and material cost for

product j,i fractional (per period) opportunity cost of

capital,a fraction of time required to meet the

production processing requirements for allproducts, excluding the setup times,

v product variety.


But, note limv!0� � � 0 and limv!�1 � � 1. As aconsequence, we have

0 < � < 1: �15�In his modeling, DeGroote has shown that thederivative of costs depends on di�erent factorssuch as the nominal setup time, the direct cost perunit of setup time, the production rate and themaximal number of setups that can be performedper unit of time. In our approach, these variablescould be called the factors of tension because theymake the system's changes more or less di�cult.

Let us analyze the e�ect of variables cs and S onthe elasticity measure of Eq. (14). Firstly, we knowc3=c2 � v� [7], that is, v� � 2cs�p ÿ a�2=Sicm. Then,for a ®xed value of v, rises in S imply rises in �. Onthe other hand, if we consider the variable cs, thee�ect will depend on the value of v. In fact, forv6 v�, the elasticity � increases with cs, whilst itdecreases for v > v�. Consequently, we may es-tablish these variables as factors of tension.

Now, let Mt � �mt1;mt2; . . . ;mtn� be a vectorwhere the component mtj corresponds to the de-mand rate for product j during the t period.Consider a sequence of vectors M1;M2; . . . ;Mk

representing the evolution of the manufacturingsystem environment between the periods t � 1 andt � k. Let vit be the variety of technology Ti at thet �t � 1; . . . ; k� period and �it be the elasticity of thetechnology Ti when the variety level is vit. In orderto compare the technologies Ti and Tj, we de®nethe variable rij as

rij � 1

k

Xk

t�1

dijt; �16�

where

dijt � 1 if �it > �jt;0 otherwise:

��17�

Now, let If ; Sf ;Rf be the binary relations repre-senting the indi�erence, the outranking and theincomparability [30], respectively, in the sense of¯exibility. For instance, the expression TiIf Tj

means ``Ti as ¯exible as Tj''. Similarly, TiSf Tj means``Ti at least as ¯exible as Tj''. If a thresholdk 2 �0:5; 1� is de®ned, the following rules may be

applied in order to know the ¯exibility comparisonbetween technologies:

rij P k ^ rji P k) TiIf Tj;

rij P k ^ rji < k) TiSf Tj;

rij < k ^ rji P k) TjSf Ti; �18�rij < k ^ rji < k) TiRf Tj:

Note the binary relationships are establishedwithin the perfect adjustment and the commonadjustment time assumptions. Thus, the resultscontained in r have signi®cances only when thee�ort dimension is considered.

We may consider the perfect adjustment capa-bility as a property to be guaranteed: it is the ro-bustness criterion. Thus, any system using themathematical model above, is a robust system.Nevertheless, if ¯exibility and robustness are twovery similar properties, they are essentially di�er-ent. Meanwhile the former involves three principaldimensions, the latter is related only to the ad-justment dimension.

5. Flexibility in a pull-based manufacturing man-

agement system: The inner logic

In a manufacturing system, we can identify twoprincipal ways to achieve the production and in-ventory management: the pull and push methods[6]. In a push approach, the production rates overthe manufacturing stages are ®xed by productionorders determined in an estimated demand base. Incontrast, in a pull approach, the production ordersare de®ned on a real demand base. Although theperformances of both systems have extensivelybeen studied in the literature [18,24,26], we pro-pose to analyze ¯exibility in a particular model ofthe pull method. To achieve it, we consider a serialdisposition of production and inventory stages inthe manufacturing system. A model of the manu-facturing system dynamics is constructed; we usemathematics to represent the production rates, theinventory levels, the production orders and thedemand signal. Furthermore, in order to illustratecertain ¯exibility measures, we simulate the push


method, when the manufacturing system is sub-mitted to a normal distributed random demand.

5.1. The pull ordering method

Let us consider three production stages andtheir respective inventories. Let P i

t be the produc-tion rate of stage i; Biÿ1

t the inventory level ofproduct manufactured by the stage i; Oi

t the pro-duction order on the stage i, at time t; and L be theconstant representing the production and deliverdelay on each stage. Additionally, let us supposethe excedentary production availability, on eachstage, and an in®nite level of raw material stock inback of the stages arrangement. Therefore, thefollowing set of equations represents the dynamicsof a pull-ordered manufacturing system [28]:

Oit �

Dt if i � 1;

Dtÿ�iÿ1�L if i > 1;

��19�

P it � Oi

tÿL; �20�

Biÿ1t � Biÿ1

tÿ1 � P itÿ1 ÿ Dt if i � 1;

Biÿ1tÿ1 � P i

tÿ1 ÿ P iÿ1t if i > 1:

(�21�

Using Eqs. (19) and (20), we can evaluate theproduction and demand dissimilarity which in-form us on the capability of management systemto adjust the production rate to the real demandsignal. Accordingly, let us de®ne the deviationvariable hi

t as

hit � P i

t ÿ DtÿiL: �22�

Therefore, we de®ne the adjustment degree of theproduction stage i by the following equation:

#i � V �hit�

V �Dt� ; 8i P 1; �23�

where V �� represents the variance of the argu-ment. This measure establishes the capability ofthe management system to adjust, on each stage,the production rate to the delayed demand signal.Indeed, the pull method searches for a ®xed stocklevel on each production stage i. If this level de-creases, the production order Oi

t is de®ned just

equal to the decreased amount of stock, that is, thedemand rate.

The necessary time to adjust the productionrate of the stage i is iL and this is a perfect delayedcapability of adjustment because P i

t ÿ DtÿiL �0 �i � 1; 2; 3; 8t�. Note this equality is justi®ed byOi

t � Dtÿ�iÿ1�L and P it � Oi

tÿL, which implies#i � 0 �i � 1; 2; 3�.

In our framework, the inner logic L is repre-sented by the ordering system, that is the pullapproach. It determines the expected states in themanufacturing system. In fact, we may considerthe current state of the manufacturing system as avector st � �B0

t ;B1t ;B

2t �. Consequently, we may

de®ne the vector of expected stock levelss�t � �B�0;B�1;B�2� on stages; in a pull-orderingsystem, these levels are independent of time (atleast, in a short or medium time period [33]).Therefore, the adjustment action consists of cal-culating the production orders and process themon stages. In the pull method, described above, theorders are completely satis®ed, but with a delay iLon the stage i.

Di�erent inner logics generate di�erent adjust-ment actions. These are implemented by the pro-duction orders. Hence, when a push method isconsidered, a production order equation considersthe feedback corrections of inventory and work-in-process levels [6]. In such case, we must use an-other kind of measure to evaluate the adjustmentdegree because there are several types of expectedstates. An example, preserving our approach, isprovided by Takahashi et al. [35] which haveproposed an alternative multidimensional measureto evaluate ¯exibility where production/demandand inventory/demand ratios are calculated oneach stage.

5.2. Illustration

As we have shown, the adjustment degree in apull method is perfect and di�erences betweenstages are not appreciated. In order to have aperspective on the ordering methods' ¯exibility, weillustrate the ¯exibility evaluation, by simulation,in three methods. First, let us consider the pushmethod. In this case, the production order equa-


tion is concerned with corrections feedback of in-ventory and work-in-process, and to adjust theproduction rate to the demand rate. The equationdescribing the production order on the ®rst stage is[28]

O1t � Dt;t�L�1 � �S0 ÿ B0

t � �XL

j�1

Dt;t�j

ÿ EC1

t

!;

�24�where Dt;t�j corresponds to the demand estimate,calculated at the end of t, for the period t � j; S0 isthe security level of the stock B0; EC1

t the level ofwork-in-process on the stage 1, calculated at theend of t. The term Dt;t�L�1 is the feedback signaladjusting the production rate to the demand rate.The other terms correspond to corrections of theinventory and work-in-process levels. In conse-quence, we have a multiobjective feedback struc-ture where only the ®rst term keeps tracing on thedemand rate. If we would have to contrast the pulland the push methods, using the measure pro-posed in Section 5.1, we would consider the ad-justment capability of production to the demandrate.

An elementary transformation permits us torede®ne the production order equation of the pushmethod. We have [29]

Oit � Dtÿ�iÿ1�L �

Xi

j�1

DDjtÿ�iÿj�L; i P 1; 8t 2 Z:

�25�Thus, the deviation variable (cf. Section 5.1) isde®ned by

hit �

Xi

j�1

DDjtÿ�i�1ÿj�L; i P 1; 8t 2 Z:

It is known that t ÿ �i� 1ÿ j�L � t ÿ Lÿ�i� 1ÿ 1ÿ j�L, then it may be established:

hiÿ1tÿL �

Xiÿ1

j�1

DDjtÿ�i�1ÿj�L; i P 1; 8t 2 Z:

As a consequence, one ®nds

hit � hiÿ1

tÿL � DDitÿL; i P 2:

Additionally, we have h1t � DD1

tÿL. In order tocontrast the ordering methods, we have considereda third system in which the ®rst stage works inpush and the upstream stages in pull: the hybridmethod. In Table 1 these results are shown.

With hit, we may establish the adjustment degree

expressions. Only the push method imposes certaincomplexity, for i > 1. In fact, we have

V �hit� � V �hiÿ1

tÿL� � V �DDitÿL� � 2cov�hiÿ1

tÿL;MDitÿL�:

However, knowing hiÿ1tÿL, this variance is expressed

as a function of the demand estimates. The fol-lowing factors may be de®ned:

G � V �DD1tÿL�

V �Dt� ;

Hi �V �DDi

tÿL� � 2cov�Piÿ1j�1 DDj

tÿ�i�1ÿj�L;DDitÿL�

V �Dt�8i P 2;

which give us the adjustment degree on each stage(Table 2).

Observe, if DDit � 0 �8i; t�, we ®nd G � 0 and

Hi � 0. In other words, the three ordering methodshave the pull system's behavior.

In order to calculate the value of Hi, the pushmethod has been simulated when an AR(1) sto-chastic process is considered. In this case, we usethe estimation function Dt�i � kiDt � �1ÿ ki�l,where k 2 �ÿ1; 1� is the autocorrelation indexof the demand process. Here, we assumeDt � N�l; r�, with known distribution parame-ters. Furthermore, L � 1.

Table 1

Deviation hit for three ordering methods

Stage Push Hybrid Pull

i � 1 DD1tÿL DD1

tÿL 0

i > 1 hiÿ1tÿL � DDi

tÿL hiÿ1tÿL 0

Table 2

Adjustement degrees #i

Stage Push Hybrid Pull

i � 1 G G 0

i > 1 #iÿ1 � Hi #iÿ1 0


In Fig. 1, the Hi�26 i� values are presented for aserial manufacturing system with seven stages. Thecurves indexed i correspond to Hi �i � 2; . . . ; 7�.Note that Hi represents the adjustment degreedegradation of #i in relation to #iÿ1. Thus, theFig. 1 shows how the adjustment degree is de-graded in the upstream direction.

In Fig. 2, the adjustment degree values arepresented. One can see that, for k 2 �0; 0:5�, # in-creases over all the stages, but the adjustment de-gree decreases when k approximates to 0:5.

The same kind of graphic has been depicted inFig. 3, where k 2 �ÿ1; 0�. The qualitative behavioris very similar to the ®rst sample case, although thecurves for a same stage are not symmetric. It isvery interesting to note that the hybrid method,represented by the curve 1 is robust (in the ad-

justment degree sense) in all the intervalk 2 �ÿ1; 1�.

6. Conclusion

We propose an approach that introduces a re-lational, dynamic and multidimensional concep-tion of ¯exibility in the manufacturing systems. Inthis approach, we de®ne two ®elds of analysis: the®eld of variations and the ®eld of tensions. The¯exibility notion, conceived in a relational per-spective, emerges when an observer models thesystem and the environment process interaction.Thus, this notion additionally incorporates a dy-namic character.

In our approach, any analysis, evaluation orde®nition of a particular system ¯exibility, mustbegin by introducing two important inquiries:what is the ®eld of variations on which ¯exibility isgoing to be observed? and, what is the ®eld oftensions grouping the factors imposing resistancesto changes on the ®eld of variations. In order toobtain an operational model of ¯exibility, the an-alyst (observer) must de®ne the vectors of thecurrent and the expected states of the observedsystem. Therefore, a model of the factors of ten-sion relationships and their in¯uences on the stateof the system must be achieved to determine thethree ¯exibility dimensions proposed in our ap-proach: the adjustment degree, the e�ort (normallya cost function in the manufacturing system

Fig. 1. Hi for i � 2; . . . ; 7.

Fig. 2. # in stages i � 1; . . . ; 7.

Fig. 3. #i for k 2 �ÿ1; 0�.


modeling) and the time of this adjustment. Weargue that the model construction subtends theanalyst's belief in a system's inner logic whichcalculates the expected states of the system. Thus,the analyst must be advised about this logic belief.

Through the examples, we have shown that agood measure of the adjustment capability is aproximity function relating the system and theenvironment processes such as the latter is decodedby the inner logic de®ning the system's expectedstates.

In the example of Section 3, we contrast twodecision making rationalities. The ®rst one, calledthe transversal point of view, considers the ¯exi-bility as a single measure: the number of availableoptions from a system state, independently fromthe environment conditions. The second one,which we subscribe, called here the longitudinalpoint of view, considers the set of available optionsas relative to the system±environment relationship,such as it is conceived by the modelling believes.The environment process, codi®ed in the actionreturn expectations, must be considered.

In the example of Section 4, we have shownthat the factors of tension relationships determinethe expected state of the analyzed system. Addi-tionally, the e�ort, represented by an elasticitymeasure, is characterized as a function of thosefactors. A simple example shows that two tech-nologies submitted to the same demand processsearch for di�erent adjustment process, each onedetermined by their own factors of tension. Anaggregate elasticity function is proposed to com-pare technologies.

Finally, in Section 5.1, a pull manufacturingmanagement system is analyzed to demonstratethat an inner logic determines the current and theexpected state deviation of system. We show thatthe adjustment action is achieved by a controlvariable which di�ers from one inner logic toanother. A simulation experience shows that theordering methods have di�erent capabilities toadjust the production rate to the demand rate.The best one seems to be the pull method. Inparticular, the hybrid method is better than pushwhen the demand process is highly autocorrelat-ed, but they are very similar when autocorrelationis low.

Acknowledgements

We wish to acknowledge those people in theLAMSADE laboratory at the Universit�e Paris-Dauphine, France, who made decisive contribu-tions in helping us to structure these ideas. We alsothank Martin Scha�ernicht, at the Universidad deTalca, for his critical comments on several aspectsof this paper.

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Flexibility in manufacturing systems: A relational and a dynamic approach

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