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Flexibility in manufacturing systems: A relational and a dynamic approach
European Journal of Operational Research 130 (2001) 70±82
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Theory and Methodology
Flexibility in manufacturing systems: A relational and a dynamic approach q Javier Pereira b
a,*
, Bernard Paulre
b
a Departamento de Inform atica de Gesti on, Universidad de Talca, Avenida Lircay s/n, Talca, Chile 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, eort 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 organisational management is evolution. Management must be capable of ¯exible responses to constantly changing oers of new products or product variants. In the same way, the increasing pressure exercised by technological development and the demand variability, compel ®rms to be ¯exible. Ways of looking at problems and their potential solutions must be adapted to ®t the nature and 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).
rate of change in the organisation's environment [25]. It is in this context that the notion of ¯exibility acquires its importance, since it is thought to contribute to the capacity of selection, of modi®cation and innovation of the programs of organizational action. Because an organization needs several types of ¯exibility, there are multiple de®nitions and evaluation schemes. The origins of this diversity may be linked to the variety of uncertainty factors, the possible term perspectives or the dierent possible dimensions to evaluate for ¯exibility. Thus, several authors claim that a speci®c type of ¯exibility maps to a given source of uncertainty [10]. Likewise, ¯exibility is classi®ed in the literature as short-term, mid-term or long-term [3]. Furthermore, many authors use three dimensions to de®ne
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
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
and to evaluate ¯exibility [10]: the variety of potential system's responses, the eort and the time of adaptation. Indeed, in considering these three principal axes, it is sometimes very dicult to use and compare two literature's proposed approaches. There are several operational diculties in achieving an unambiguous understanding of notions of ¯exibility [10±12,36]: 1. an intuitive de®nition, sometimes not operational, of the notion of ¯exibility [13]; 2. lack of uniformity in the classi®cation, de®nition and evaluation of a given type of ¯exibility [1±5,19,27,32]; 3. a diculty in the establishment of the single-dimensional or multidimensional character of ¯exibility [10]; 4. a misconceived relationship by which ¯exibility directly enhances the system's performance [7,28]. In this paper, we propose a new approach in which the system's ¯exibilities are speci®ed by means of a frame of analysis. Our intention is to de®ne clearly and simply what ¯exibility is, what a dimension of ¯exibility is and how it must be evaluated. Our aim is to characterize ¯exibility in a relational, multidimensional and dynamic way. We argue that this perspective conduces researchers and practitioners to an unambiguous de®nition of ¯exibility which enables them to better evaluating schemes. Firstly, we must talk about ¯exibility when one or several particular system dimensions have been speci®ed for change: a ®eld of variations must represent the space of states on which the system's changes are considered. Thus, what are the system properties, conditioning changes on this ®eld?: we talk about the tensions of the system and then we de®ne the ®eld of tensions. Secondly, although dierent articles propose variety as a measure of ¯exibility, we propose a measure called the adjustment degree which captures the system±environment relationship embedded in the ¯exibility notion: it considers the ®eld of variations. Accordingly, because of tensions, the eort and time of adjustment are introduced as two necessary measures of ¯exibility. In this manner, we render consistent the
71
adjustment, eort and time dimensions of ¯exibility in a perspective called here the longitudinal approach. The paper is organized as follows: in Section 2 our approach and the relevant concepts to be used are introduced. In Section 3, a simple stochastic model enables us to present the adjustment degree as one of the right dimensions of ¯exibility. The second example of this section shows the critical dierences between the static and the dynamic approaches to explain ¯exibility. In Section 4, we analyze the problem of determining the product lot sizes in a manufacturing con®guration, submitted to a variable demand, and we identify the dierent factors of tension de®ning the system's eort to obtain an optimal adjustment to the demand process. In the example of Section 5, we characterize what we call the inner logic, i.e., the logic which translates the environment state into the system expected state. The conclusions are presented in Section 6. 2. The proposed approach The capability of a system to adapt to changes that occur in its environment is called here ¯exibility. This intuitive de®nition brings two questions: adaptation to what? and how? In order to answer these questions, we are going to consider the system as any logical and/or a physical device which we might wish to evaluate for ¯exibility. The system could be a machine, a mechanism, a logic, a procedure, a process, a program, a technique, an apparatus, an organization, etc. All these entities may be described as operationally closed [8]: they have boundaries that we can de®ne by their functional and control limits. So, all that it is not modeled as part of this closure, is located in the environment of an entity. Primary, we must clarify what is to be comprehended by adaptation. Indeed, the question of adaptation or not is properly a subject of an observer O1 (the modeller 1) who evaluates the degree of congruence between the system and its environment. In this evaluation O1 actually uses (implicitly or explicitly) a deviation function which tells him or her if the system state is adapted to the
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environment state. But, we must not forget that another observer O2 may use his/her own model of the same system, with its own particular deviation function. Consequently, the adaptation capability is a relative property attributed by an observer to a modeled system. The important question is, when is the system adapted to its environment? The answer is, when there is no more system to study, it ceased to be adapted. In other words, it is the degree of adaptation, which depends on the model that an observer makes of reality [22]. In fact, he represents one thing labeled the system, another thing labeled the environment and any kind of relationships between them. In this paper, we are going to write about models. In consequence, we will use the idea of the degree of adaptation as the local adjustment measure (cf. Section 2.1) between the current and the expected states of the system, such as it is determined by an inner logic. For a given model, we will investigate what is to change in the environment and the system, what is to be de®ned as the congruence between them, what is to be de®ned as the deviation function and how we can measure this deviation.
2.1. The ®eld of variations Suppose that we are able to characterize the behavior of a system and its environment through the trajectories that they take in a set of states. Let S be this set of states; we will call it the ®eld of variations. Let R S be the subset of realizable states of the system. Additionally, let E be the set of the states in which the environment moves. Note that these set de®nitions must all be given by a speci®c observer. Now, let st 2 R, et 2 E and st 2 S be the observed current state of the system, the observed current state of the environment and the expected current state of the system, respectively. We will suppose that there exists, in the system, a L logic such that L et ; st st ; kst ÿ st k;
1
i.e., given st and et the L logic allows us to determine the expected state and the norm between st and st . Then, we can say that the system is in partial equilibrium when L et ; st st ; 0 or not in partial equilibrium if kst ÿ st k 6 0. De®nition 1. If kst ÿ st k 6 0, then ¯exibility is the property that tends to realize the partial equilibrium in the system. Thus, a ¯exible system has the capability to adjust its current state in response to the deviation kst ÿ st k. Therefore, we assume that the relationship between the system and its environment is directly established by the L logic. This is a valid assumption because we have established that the implicated observer models the system as a closed entity (cf. Section 2). Indeed, the adjustment could be understood as the system's expected state tracking process. It means that, in our approach, the system does not adjust to the environment, but to its own expected states. Let D s1 ; . . . ; sn be the ®nite succession of expected states as determined by L between some reference periods t 1 and t n n > 1. We will not restrictively suppose that the relationship D R must hold true. Additionally, let F s1 ; . . . ; sn be the succession of states adopted by the system when it seeks to adjust to the D succession. Thus, we will demonstrate in this paper the necessity of using a measure of adjustment between D and F. 2.2. The ®eld of tensions Time and ease of adjustment can both vary considerably. The fact that adjustment requires an eort, and a certain time interval, points to the existence of one or more factors of resistance to change that we call the factors of tension. For instance, the time and costs of setup operations of a ¯exible machine, depends on the resistance to the change; we argue that this resistance may represent the system's tension. Similarly, the level of acceptability of an urgent order that compels a ®rm to use their machines after the normal day of work gives an idea of the level of system's tension. Alike,
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
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any change in project planning (of construction, engineering, computer system development, etc.) implies an eort (cost); this is an irreversibility effect that represents again the system's tensions. Suppose that n factors of tension represent the system's resistances to change on R S. Let XS i be the set de®ning the ®eld of variations of the i factor of tension when the ®eld of variations S is considered (remember that changes on the ®eld of variations do not correspond to those on the ®eld of tensions: variations on the latter induce or resist to those on the former). We de®ne the ®eld of S . Additionally, we de®ne tensions as XS ni XS i the level of tension by a function T such as:
explain time and eort of adjustment of the system. We claim that this is a congruent dynamical approach to conceive ¯exibility. Thus, we propose a change of perspective that relates F (and not R) to the system's eorts and times of transition. This implies that ¯exibility is a relative property: it depends on a well-speci®ed environment. The following sections present a number of examples to illustrate how this approach may be used. The similarities and the dierences between our approach and other perspectives found in the literature are signaled.
T : S ! XS ;
3. Examples of adjustment capability: The transversal and the longitudinal approaches
2
which establishes a mapping between any system state and the corresponding factors of tension on their own ®elds of variations. Thus, any state transition on S implies a change of the level of tension. Therefore, time and eort are necessary and we argue than they must be considered in relation to a speci®c adjustment situation, what we call the longitudinal approach, and not independently of it, what we call here the transversal approach. In fact, in considering the eects of the factors of tension over the ®eld of variations, we can appreciate the dierence between these two approaches.
2.3. The dynamic and relational ¯exibility In a dynamic approach, we represent the state expectations as a succession D of states determined by a L logic, and the system's responses to these demands as a F succession, tracing the effective system moves. Note that, in F, whatever the transition from a state st to a state stDt be, it demands eort and time. Therefore, the system dynamically adjusts to the demanded changes de®ned by the D succession. The static approaches in literature propose to relate the transition eorts to the R set (cf. Section 2.1). In such perspectives, we cannot correctly associate dynamic eorts or time to R. In our approach, only a speci®c environmental process may
Researchers have traditionally assumed that ¯exibility may be evaluated by three kinds of measures: the variety, the eort and the time of system adjustment considered for adaptation [10]. Under this assumption, variety is evaluated by measures such as the number of responses which the system may potentially generate. In those cases, the ¯exibility is de®ned as a system's property which does not depend on a particular environment process. Nevertheless, we argue that the adjustment capability of a system must be primary related to the system±environment evolution. In the next example, we show that variety does not completely capture the relational sense of ¯exibility. Consider a system and its environment. We want to model their interaction by two completely ergodic Markovian processes [14]. Let P pij nn and Q qij nn respectively be the transition probability matrix of the environment and the system. Suppose that the set of states represented in P and Q are identical; they are both subsets of S (cf. Section 2.1), the ®eld of variations. Let us consider the steady state for the system and its environment. To model the interaction, we assume that the environment undertakes transitions and the system try to follow them. Indeed, P models the transitions of the expected states as they are decoded for one L logic in the system.
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Then, in a perfect adjustment capability model, we have Q P:
3
Moreover, suppose we are going to determine the entropy, H, of these processes [9,19]. Thus, we have HQ HP . Nevertheless, it is easy to conclude that, even if entropy is reduced (or increased), regarding a reference value, the system and the environment processes are identical. Consequently, in that case, we are not able to evaluate the system ¯exibility by entropy (or variety), but by the relative entropy system entropy : environment entropy Here, this ratio is called the adjustment degree. Thus, in the preceeding example, the relative entropy is maximal and the adjustment degree also; note this measure depends on the system±environment relationship. Now, let us consider a two-period sequential decision problem [31,37] in which one needs to make two temporally related decisions. At the beginning of t 1, the ®rst period, there is a set A of alternatives or actions. At the beginning of t 2, the second period, exists a set B of alternatives. There are two decisions related in the following way: in t 1, if an action a 2 A is chosen, then in t 2 the decision maker could choose an action b 2 B a. Actually, we are going to apply a little change to this problem. Let us suppose only one action set, A. Thus, when an action a 2 A is chosen, a remaining set B a A is available; whatever the decision be, it is 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 the environment and the whole necessary information to select this program, including those probabilities, is available upon beginning t 1. In other words, we do not have a real second period decision because it is completely determined in the ®rst period. In contrast, we have an uncertain decision if upon beginning t 2, additional information is received. In an uncertain universe, in t 1, we have the necessary information to select ai , but not
all the information regarding aj 2 B ai . Then, the latter decision is moved to the beginning of the second period. How can we evaluate the adjustment capability of the choice system? First of all, in this problem, the optimizer (observer) considers the rational approach as the right point of view to evaluate the policies of choice. Thus, whatever the optimal solution be, he or she accepts it without hesitation, in the risky or the uncertain situation. Hence, a completely rational decision maker believes that the optimal action is the best response to the system's environment. From this perspective, the adjustment capability must be necessarily perfect. However, if we consider two kinds of decision makers, the ®rst one accepting the risky situation and the second one accepting the uncertain point of view, the optimal solutions may be potentially dierent. Indeed, when the number of second period actions in the whole remaining set is the same for all ®rst period actions, the uncertain solution dominates the risky choice in the sense of gains (or losses [21]). In that case, to select the optimal risky action is evidently not the best policy and the uncertain decision maker would say that ¯exibility is reduced because of the decreased gain. But, we should not forget that the risky and the uncertain decision makers use dierent models to make a choice. We argue that those models can not be compared because the adjustment degrees would be de®ned by two dierent perspectives. Therefore, let S be the set of states of the environment, where the decision maker assigns a subjective probability p s; s 2 S. Likewise, let ut ai ; s be the gain, relative to s, when the action ai is chosen at the t period. Also, let I2 fy1 ; y2 ; . . . ; ym g be the set of probable messages arriving upon the beginning of the second period, 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 X p su1 ai ; s Eai j I2 S
X I2
( qy
max
aj 2B ai
X
) p s j yu2 aj ; s ;
S
4
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where p s j y is the posterior conditional probability of s. This expected gain is maximized by an action a1 , that is, Ea1 maxai 2A Eai j I2 . Let us assume that the decision maker chooses this action and subsequently the message yr occurs upon the beginning of the second period. In that period, he/ she may consider a new decision problem where Eaj j I3 is the expected gain of the second period action aj when a set of messages, I3 , is considered for the third period. Then, we have X p s j yr u2 aj ; s Eaj j I3 S
X I3
( qy
max
ak 2B aj
X
) p s j y; yr u3 ak ; s :
S
5 The term p s j y; yr corresponds to the conditional probability of s, when the yr message results in the second period; B aj is the set of available actions from aj and u3 ak ; s de®nes the gain of ak in the third period, relative to the state s. The ¯exibility eect would be considered when the decision maker contrasts the best second pe0 riod action a2 , given a1 , against the best second period action if all the past routes (the ®rst period alternatives) were considered, a2 . Thus, we have 2 6 Ea j I3 max max Eaj j I3 ; ai 2A
aj 2B ai
20
Ea j I3 max Eaj j I3 ; aj 2B a1
7
and the opportunity cost of choosing the wrong alternative in the ®rst period is given by 0
DE Ea2 j I3 ÿ Ea2 j I3 :
75
native approach called the longitudinal point of view. When ai is chosen at the ®rst period, there is a probability distribution on B ai which denotes the probable transitions from ai . These probabilities are completely determined by the optimization problem which searches for the optimal action aj 2 B ai . In our formulation of the second period optimization problem, the probability distribution may be approximated by the following function: , X Eaj j I3 : 9 p aj j ai Eaj j I3 aj 2B ai
Thus, the entropy of the second period action aj 2 B ai , when the information I3 is considered, could be de®ned: H aj j I3 . In general, two random sets X; Y have the property X Y ) H X 0 6 H Y. Thus, if we have B a2 B a2 , then 0 H a2 j I3 6 H a2 j I3 . 0 Now, let us consider two actions, a2 and a2 , with remaining sets B a2 fa1 ; a2 ; a3 g and 0 B a2 fa1 ; a2 g. Let us suppose Ea1 j I3 10, Ea2 j I3 20, Ea3 j I3 0. In that case, 0 Additionally, DE 0. H a2 j I3 H a2 j I3 . 0 Clearly, we should not use jB a2 j as an interesting measure because the action a3 is totally irrelevant. On the other hand, if Ea3 j I3 30, 0 H a2 j I3 > H a2 j I3 and DE > 0, then we establish the opportunity cost of choosing the wrong way in the ®rst period. Finally, if we set Ea1 j I3 Ea2 j I3 Ea3 j I3 , then H a2 j I3 > 0 H a2 j I3 , but DE 0. Whence, we may conclude that the more appropriate adjustment degree measure is the opportunity cost of optimal actions and not the relative entropy level.
8
Note that DE P 0. Thus, there is a perfect adjustment situation when DE 0. In the literature [21,31], dierent models propose that the ¯exibility of ai is evaluated by jB ai j. Those models are coherent with an approach called here the transversal point of view. Because these approaches do not consider the relative character of ¯exibility they do not consider those actions aj 2 B ai which may have a zero probability to be chosen from ai . We propose an alter-
4. The setup costs in a manufacturing system: The factors of tension Here, we analyze the problem of determining the product lot sizes in a manufacturing system submitted to a variable demand [16,17,34]. A set of products being given, the optimization problem consists of determining the lot sizes minimizing the total production cost function per unit of time. In modeling this problem, DeGroote [7] has proposed
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that the ¯exibility is a binary relation evaluated by the comparison of marginal performance of two systems obeying the same marginal change of environmental variety. Q1 ; Q2 ; . . . ; Qn be the lot sizes vector Let Q of the product set. Then, the optimization cost problem is modeled by min Q
n X mj qj cs S
Qj
j1
icj Qj 2
where, the variables in the model are as follows:
Qj rj p S cs qj cj i a v
number of product types in manufacturing, periodical demand rate for product j 1; . . . ; n, lot size for product j, production rate for product j, availability of the facility, nominal setup time, cost per unit of setup time, fraction of S used in the setup of product j, unit direct labor and material cost for product j, fractional (per period) opportunity cost of capital, fraction of time required to meet the production processing requirements for all products, excluding the setup times, product variety.
This problem may be solved by the Lagrangian relaxation. Thus, in determining the Lagrange multipliers, we may ®nd the measure of product variety proposed by DeGroote: 1 v cm
!2 n p X 2icj mj qj ;
10
j1
Pn Pn where m j1 mj and cm j1 cj mj =m. Accordingly, the optimal cost is de®ned by p 2cs Sicmv cm C Sicmv cs p ÿ a 2 pÿa
Let T be a technology described by the S; cs ; qj ; cj and i parameters. If T1 and T2 are two technologies, with optimal costs C1 and C2 , such that oC1 oC2 6 8v P 0; ov ov
s:t: a 6 p;
n mj
where v is a critical value of v. DeGroote de®nes the ¯exibility of a T technology in the following way:
if v 6 v ; if v > v ;
then, T1 is more ¯exible than T2 . Indeed, three dimensions are considered in this de®nition: 1. The adjustment in relation to the environmental change. 2. The cost of this adjustment, seemingly represented 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 the production lot sizes, has a perfect adjustment property. The only dierence between two technologies, in the ¯exibility sense, is determined by the derivative of the optimal cost. Although the DeGroote's de®nition is well structured, we propose to consider the relative derivative of cost to variety, de®ned as
oC v : ov C
13
This is the elasticity of cost to variety, a measure which has been used in other contexts to evaluate the ¯exibility of a ®rm [23,28]. Actually, it has been argued that human beings perceive gains or losses in relation to the levels from which the move starts [20]. Thus, this measure appears to be better adapted to comparisons than the single derivative of cost to variety. p Let c1 2cs Sicm; c2 Sicm=2 p ÿ a; c3 cs p ÿ a. Then, one can show (
11
12
1=2 p 11= vc1 =cm 1 11= vc2 =c3
if v 6 v ; if v > v :
14
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
77
But, note limv!0 0 and limv!1 1. As a consequence, we have
applied in order to know the ¯exibility comparison between technologies:
0 < < 1:
15
rij P k ^ rji P k ) Ti If Tj ;
In his modeling, DeGroote has shown that the derivative of costs depends on dierent factors such as the nominal setup time, the direct cost per unit of setup time, the production rate and the maximal number of setups that can be performed per unit of time. In our approach, these variables could be called the factors of tension because they make the system's changes more or less dicult. Let us analyze the eect of variables cs and S on the elasticity measure of Eq. (14). Firstly, we know 2 c3 =c2 v [7], that is, v 2cs p ÿ a =Sicm. Then, for a ®xed value of v, rises in S imply rises in . On the other hand, if we consider the variable cs , the eect will depend on the value of v. In fact, for v 6 v , the elasticity increases with cs , whilst it decreases for v > v . Consequently, we may establish these variables as factors of tension. Now, let Mt mt1 ; mt2 ; . . . ; mtn be a vector where the component mtj corresponds to the demand rate for product j during the t period. Consider a sequence of vectors M1 ; M2 ; . . . ; Mk representing the evolution of the manufacturing system environment between the periods t 1 and t k. Let vit be the variety of technology Ti at the t t 1; . . . ; k period and it be the elasticity of the technology Ti when the variety level is vit . In order to compare the technologies Ti and Tj , we de®ne the variable rij as
rij P k ^ rji < k ) Ti Sf Tj ;
rij
k 1X dijt ; k t1
where 1 dijt 0
if it > jt ; otherwise:
16
17
Now, let If ; Sf ; Rf be the binary relations representing the indierence, the outranking and the incomparability [30], respectively, in the sense of ¯exibility. For instance, the expression Ti If Tj means ``Ti as ¯exible as Tj ''. Similarly, Ti Sf Tj means ``Ti at least as ¯exible as Tj ''. If a threshold k 2 0:5; 1 is de®ned, the following rules may be
rij < k ^ rji P k ) Tj Sf Ti ;
18
rij < k ^ rji < k ) Ti Rf Tj : Note the binary relationships are established within the perfect adjustment and the common adjustment time assumptions. Thus, the results contained in r have signi®cances only when the eort dimension is considered. We may consider the perfect adjustment capability as a property to be guaranteed: it is the robustness criterion. Thus, any system using the mathematical model above, is a robust system. Nevertheless, if ¯exibility and robustness are two very similar properties, they are essentially dierent. Meanwhile the former involves three principal dimensions, the latter is related only to the adjustment dimension.
5. Flexibility in a pull-based manufacturing management system: The inner logic In a manufacturing system, we can identify two principal ways to achieve the production and inventory management: the pull and push methods [6]. In a push approach, the production rates over the manufacturing stages are ®xed by production orders determined in an estimated demand base. In contrast, in a pull approach, the production orders are de®ned on a real demand base. Although the performances of both systems have extensively been studied in the literature [18,24,26], we propose to analyze ¯exibility in a particular model of the pull method. To achieve it, we consider a serial disposition of production and inventory stages in the manufacturing system. A model of the manufacturing system dynamics is constructed; we use mathematics to represent the production rates, the inventory levels, the production orders and the demand signal. Furthermore, in order to illustrate certain ¯exibility measures, we simulate the push
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J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
method, when the manufacturing system is submitted to a normal distributed random demand.
5.1. The pull ordering method Let us consider three production stages and their respective inventories. Let Pti be the producthe inventory level of tion rate of stage i; Biÿ1 t product manufactured by the stage i; Oit the production order on the stage i, at time t; and L be the constant representing the production and deliver delay on each stage. Additionally, let us suppose the excedentary production availability, on each stage, and an in®nite level of raw material stock in back of the stages arrangement. Therefore, the following set of equations represents the dynamics of a pull-ordered manufacturing system [28]: Oit
Dt
if i 1;
Dtÿ iÿ1L
if i > 1;
Pti OitÿL ; ( i Biÿ1 tÿ1 Ptÿ1 ÿ Dt iÿ1 Bt i iÿ1 Biÿ1 tÿ1 Ptÿ1 ÿ Pt
19 20 if i 1; if i > 1:
21
Using Eqs. (19) and (20), we can evaluate the production and demand dissimilarity which inform us on the capability of management system to adjust the production rate to the real demand signal. Accordingly, let us de®ne the deviation variable hit as hit Pti ÿ DtÿiL :
22
Therefore, we de®ne the adjustment degree of the production stage i by the following equation: #i
V hit ; V Dt
8i P 1;
23
where V represents the variance of the argument. This measure establishes the capability of the management system to adjust, on each stage, the production rate to the delayed demand signal. Indeed, the pull method searches for a ®xed stock level on each production stage i. If this level decreases, the production order Oit is de®ned just
equal to the decreased amount of stock, that is, the demand rate. The necessary time to adjust the production rate of the stage i is iL and this is a perfect delayed capability of adjustment because Pti ÿ DtÿiL 0 i 1; 2; 3; 8t. Note this equality is justi®ed by Oit Dtÿ iÿ1L and Pti OitÿL , which implies #i 0 i 1; 2; 3. In our framework, the inner logic L is represented by the ordering system, that is the pull approach. It determines the expected states in the manufacturing system. In fact, we may consider the current state of the manufacturing system as a vector st B0t ; B1t ; B2t . Consequently, we may de®ne the vector of expected stock levels st B0 ; B1 ; B2 on stages; in a pull-ordering system, these levels are independent of time (at least, in a short or medium time period [33]). Therefore, the adjustment action consists of calculating the production orders and process them on stages. In the pull method, described above, the orders are completely satis®ed, but with a delay iL on the stage i. Dierent inner logics generate dierent adjustment actions. These are implemented by the production orders. Hence, when a push method is considered, a production order equation considers the feedback corrections of inventory and work-inprocess levels [6]. In such case, we must use another kind of measure to evaluate the adjustment degree because there are several types of expected states. An example, preserving our approach, is provided by Takahashi et al. [35] which have proposed an alternative multidimensional measure to evaluate ¯exibility where production/demand and inventory/demand ratios are calculated on each stage. 5.2. Illustration As we have shown, the adjustment degree in a pull method is perfect and dierences between stages are not appreciated. In order to have a perspective on the ordering methods' ¯exibility, we illustrate the ¯exibility evaluation, by simulation, in three methods. First, let us consider the push method. In this case, the production order equa-
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
tion is concerned with corrections feedback of inventory and work-in-process, and to adjust the production rate to the demand rate. The equation describing the production order on the ®rst stage is [28] ! L X ^ t;tj ÿ EC 1 ; ^ t;tL1 S 0 ÿ B0 D O1 D t
t
j1
t
24 ^ t;tj corresponds to the demand estimate, where D calculated at the end of t, for the period t j; S 0 is the security level of the stock B0 ; ECt1 the level of work-in-process on the stage 1, calculated at the ^ t;tL1 is the feedback signal end of t. The term D adjusting the production rate to the demand rate. The other terms correspond to corrections of the inventory and work-in-process levels. In consequence, we have a multiobjective feedback structure where only the ®rst term keeps tracing on the demand rate. If we would have to contrast the pull and the push methods, using the measure proposed in Section 5.1, we would consider the adjustment capability of production to the demand rate. An elementary transformation permits us to rede®ne the production order equation of the push method. We have [29] Oit Dtÿ iÿ1L
i X j1
^j DD tÿ iÿjL ;
i P 1; 8t 2 Z: 25
Thus, the deviation variable (cf. Section 5.1) is de®ned by hit
i X j1
^j DD tÿ i1ÿjL ;
i P 1; 8t 2 Z:
hiÿ1 tÿL
iÿ1 X j1
^j DD tÿ i1ÿjL ;
i P 1; 8t 2 Z:
As a consequence, one ®nds ^i hit hiÿ1 tÿL DDtÿL ;
^ 1 . In order to Additionally, we have h1t DD tÿL contrast the ordering methods, we have considered a third system in which the ®rst stage works in push and the upstream stages in pull: the hybrid method. In Table 1 these results are shown. With hit , we may establish the adjustment degree expressions. Only the push method imposes certain complexity, for i > 1. In fact, we have iÿ1 ^i ^i V hit V hiÿ1 tÿL V DDtÿL 2 cov htÿL ; MDtÿL :
However, knowing hiÿ1 tÿL , this variance is expressed as a function of the demand estimates. The following factors may be de®ned: ^1 V DD tÿL ; V Dt P ^ i 2 cov iÿ1 DD ^j V DD
G
tÿL
Hi
i P 2:
^i tÿ i1ÿjL ; DDtÿL
j1
V Dt 8i P 2;
which give us the adjustment degree on each stage (Table 2). ^ i 0 8i; t, we ®nd G 0 and Observe, if DD t Hi 0. In other words, the three ordering methods have the pull system's behavior. In order to calculate the value of Hi , the push method has been simulated when an AR(1) stochastic process is considered. In this case, we use ^ ti ki Dt 1 ÿ ki l, the estimation function D where k 2 ÿ1; 1 is the autocorrelation index of the demand process. Here, we assume Dt N l; r, with known distribution parameters. Furthermore, L 1. Table 1 Deviation hit for three ordering methods Stage
It is known that t ÿ i 1 ÿ jL t ÿ L ÿ i 1 ÿ 1 ÿ jL, then it may be established:
79
i1 i>1
Push ^1 DD tÿL iÿ1 ^ itÿL DD htÿL
Hybrid ^1 DD tÿL iÿ1 htÿL
Pull 0 0
Table 2 Adjustement degrees #i Stage
Push
Hybrid
Pull
i1 i>1
G #iÿ1 Hi
G #iÿ1
0 0
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J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
Fig. 1. Hi for i 2; . . . ; 7. Fig. 3. #i for k 2 ÿ1; 0.
In Fig. 1, the Hi 2 6 i values are presented for a serial manufacturing system with seven stages. The curves indexed i correspond to Hi i 2; . . . ; 7. Note that Hi represents the adjustment degree degradation of #i in relation to #iÿ1 . Thus, the Fig. 1 shows how the adjustment degree is degraded in the upstream direction. In Fig. 2, the adjustment degree values are presented. One can see that, for k 2 0; 0:5, # increases over all the stages, but the adjustment degree decreases when k approximates to 0:5. The same kind of graphic has been depicted in Fig. 3, where k 2 ÿ1; 0. The qualitative behavior is very similar to the ®rst sample case, although the curves for a same stage are not symmetric. It is very interesting to note that the hybrid method, represented by the curve 1 is robust (in the ad-
Fig. 2. # in stages i 1; . . . ; 7.
justment degree k 2 ÿ1; 1.
sense)
in
all
the
interval
6. Conclusion We propose an approach that introduces a relational, dynamic and multidimensional conception of ¯exibility in the manufacturing systems. In this approach, we de®ne two ®elds of analysis: the ®eld of variations and the ®eld of tensions. The ¯exibility notion, conceived in a relational perspective, emerges when an observer models the system and the environment process interaction. Thus, this notion additionally incorporates a dynamic character. In our approach, any analysis, evaluation or de®nition of a particular system ¯exibility, must begin by introducing two important inquiries: what is the ®eld of variations on which ¯exibility is going to be observed? and, what is the ®eld of tensions grouping the factors imposing resistances to changes on the ®eld of variations. In order to obtain an operational model of ¯exibility, the analyst (observer) must de®ne the vectors of the current and the expected states of the observed system. Therefore, a model of the factors of tension relationships and their in¯uences on the state of the system must be achieved to determine the three ¯exibility dimensions proposed in our approach: the adjustment degree, the eort (normally a cost function in the manufacturing system
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
modeling) and the time of this adjustment. We argue that the model construction subtends the analyst's belief in a system's inner logic which calculates the expected states of the system. Thus, the analyst must be advised about this logic belief. Through the examples, we have shown that a good measure of the adjustment capability is a proximity function relating the system and the environment processes such as the latter is decoded by the inner logic de®ning the system's expected states. In the example of Section 3, we contrast two decision making rationalities. The ®rst one, called the transversal point of view, considers the ¯exibility as a single measure: the number of available options from a system state, independently from the environment conditions. The second one, which we subscribe, called here the longitudinal point of view, considers the set of available options as relative to the system±environment relationship, such as it is conceived by the modelling believes. The environment process, codi®ed in the action return expectations, must be considered. In the example of Section 4, we have shown that the factors of tension relationships determine the expected state of the analyzed system. Additionally, the eort, represented by an elasticity measure, is characterized as a function of those factors. A simple example shows that two technologies submitted to the same demand process search for dierent adjustment process, each one determined by their own factors of tension. An aggregate elasticity function is proposed to compare technologies. Finally, in Section 5.1, a pull manufacturing management system is analyzed to demonstrate that an inner logic determines the current and the expected state deviation of system. We show that the adjustment action is achieved by a control variable which diers from one inner logic to another. A simulation experience shows that the ordering methods have dierent capabilities to adjust the production rate to the demand rate. The best one seems to be the pull method. In particular, the hybrid method is better than push when the demand process is highly autocorrelated, but they are very similar when autocorrelation is low.
81
Acknowledgements We wish to acknowledge those people in the LAMSADE laboratory at the Universite ParisDauphine, France, who made decisive contributions in helping us to structure these ideas. We also thank Martin Schaernicht, at the Universidad de Talca, for his critical comments on several aspects of this paper. References [1] M. Barad, D. Sipper, Flexibility in manufacturing systems: De®nitions and Petri-net modelling, International Journal of Production Research 26 (2) (1988) 237±248. [2] J. Bernardo, Z. Mohamed, The measurement and use of operational ¯exibility in the loading of ¯exible manufacturing systems, European Journal of Operational Research 60 (1992) 144±155. [3] B. Carlsson, Flexibility and the theory of the ®rm, International Journal of Production Research 25 (7) (1987) 957±966. [4] P. Chandra, M. Tombak, Models for the evaluation of routing and machine ¯exibility, European Journal of Operational Research 60 (1992) 156±165. [5] T. Cox, Toward the measurement of manufacturing ¯exibility, Production and Inventory Management Journal (®rst quarter) (1989) 6±11. [6] A. Crespo, R. Ruiz, New production planning systems: A system dynamics perspective, in: The Proceedings of the 1992 International Conference of the System Dynamics Society, Utrecht, Netherlands, System Dynamics Society, 1992, pp. 415±424. [7] X. DeGroote, Flexibility and product variety, European Journal of Operational Research 75 (1992) 264±274. [8] J.W. Forrester, Industrial Dynamics, The MIT Press, Cambridge, MA, 1969. [9] O. Garro, P. Martin, Towards new architectures of machine tools, International Journal of Production Research 31 (10) (1993) 2403±2414. [10] D. Gerwin, Manufacturing ¯exibility: A strategic perspective, Management Science 39 (4) (1993) 395±410. [11] A. Gunasekaran, P. Martikainen, P. Yli-Olli, Flexible manufacturing systems: An investigation for research and application, European Journal of Operational Research 66 (1993) 1±26. [12] Y. Gupta, S. Goyal, Flexibility of manufacturing systems: Concepts and measurements, European Journal of Operational Research 43 (1989) 119±135. [13] Y. Gupta, T. Sommers, The measurement of manufacturing ¯exibility, European Journal of Operational Research 60 (1992) 166±182. [14] R. Howard, Dynamic Probabilistic Systems, vol. I: Markov Models, Wiley, New York, 1971.
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[15] R. Jones, J. Ostroy, Flexibility and uncertainty, Review of Economic Studies LI (1984) pp. 13±32. [16] U.S. Karmarkar, Lot sizing and lead-time performance in a manufacturing cell, Interfaces 15 (March/April) (1985) 1±9. [17] U.S. Karmarkar, Lot sizes, lead times and in process inventories, Management Science 33 (3) (1987) 409±418. [18] L.J. Krajewski, B.E. Jing, L.P. Ritzman, D. Wong, Kanban, MRP and shaping the manufacturing environment, Management Science 33 (1) (1987) 39±57. [19] V. Kumar, Entropic measures of manufacturing ¯exibility, International Journal of Production Research 25 (7) (1987) 957±966. [20] F.A. Lootsma, A model for the relative importance of the criteria in the Multiplicative AHP and SMART, European Journal of Operational Research 94 (1996) 467±476. [21] M. Mandelbaum, J. Buzacott, Flexibility and decision making, European Journal of Operational Research 94 (1990) 17±27. [22] H. Maturana, F. Varela, El arbol del conocimiento, Editorial Universitaria, Santiago de Chile, 1984. [23] D. Mills, L. Schumann, Industry structure with ¯uctuating demand, American Economic Review 75 (4) (1985) 758± 767. [24] H. Molet, Une Nouvelle Gestion Industrielle, Hermes, 1993. [25] G. Morgan, Images of Organization, Sage Publications, London, 1986. [26] R. Muramatsu, K. Ishii, K. Takahashi, Some ways to increasing ¯exibility in manufacturing systems, International Journal of Production Research 23 (4) (1985) 691±703. [27] R. Parker, A. Wirth, Manufacturing ¯exibility: Measures and relationships, European Journal of Operational Research 118 (1999) 429±449.
[28] J. Pereira, Flexibilite dans les systemes de production: Analyse et evaluation par simulation, These 3e cycle Universite Paris IX Dauphine, France, 1995. [29] J. Pereira, Flexibility in Manufacturing Processes: A relational, dynamic and multidimensional approach, in: R.Y. Cavana, J. Vennix, E. Rouwette, M. StevensonWright, J. Candlish (Eds.), The Proceedings of the 17th International Conference of the System Dynamics Society and 5th Australian & New Zealand Systems Conference, Wellington, New Zealand, System Dynamics Society and ANZSYS, 20±23 July 1999, p. 63. [30] B. Roy, Multicriteria Methodology for Decision Aiding, Kluwer Academic Publishers, Hingham, MA, 1996. [31] A. Schmutzler, Flexibility and adjustment to information in decision sequential problems: A systematic approach, in: Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 1991. [32] J. Shewchuk, A set of generic ¯exibility measures for manufacturing applications, International Journal of Production Research 37 (13) (1999) 3017±3042. [33] S. Shingo, Ma^õtrise de la production et methode Kanban: le cas Toyota, Les Editions de l'Organisation, Paris, 1983. [34] A. Spence, E. Porteus, Setup reduction and increased eective capacity, Management Science 33 (10) (1987) 1291±1301. [35] K. Takahashi, S. Hiraki, M. Soshiroda, Flexibility of production ordering systems, International Journal of Production Research 32 (7) (1994) 1739±1752. [36] D. Upton, Process range in manufacturing: An empirical study of ¯exibility, Management Science 43 (8) (1997) 1079±1092. [37] J. VanMieghem, Investment strategies for ¯exible resources, Management Science 44 (8) (1998) 1071±1078.
www.elsevier.com/locate/dsw
Theory and Methodology
Flexibility in manufacturing systems: A relational and a dynamic approach q Javier Pereira b
a,*
, Bernard Paulre
b
a Departamento de Inform atica de Gesti on, Universidad de Talca, Avenida Lircay s/n, Talca, Chile 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, eort 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 organisational management is evolution. Management must be capable of ¯exible responses to constantly changing oers of new products or product variants. In the same way, the increasing pressure exercised by technological development and the demand variability, compel ®rms to be ¯exible. Ways of looking at problems and their potential solutions must be adapted to ®t the nature and 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).
rate of change in the organisation's environment [25]. It is in this context that the notion of ¯exibility acquires its importance, since it is thought to contribute to the capacity of selection, of modi®cation and innovation of the programs of organizational action. Because an organization needs several types of ¯exibility, there are multiple de®nitions and evaluation schemes. The origins of this diversity may be linked to the variety of uncertainty factors, the possible term perspectives or the dierent possible dimensions to evaluate for ¯exibility. Thus, several authors claim that a speci®c type of ¯exibility maps to a given source of uncertainty [10]. Likewise, ¯exibility is classi®ed in the literature as short-term, mid-term or long-term [3]. Furthermore, many authors use three dimensions to de®ne
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
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
and to evaluate ¯exibility [10]: the variety of potential system's responses, the eort and the time of adaptation. Indeed, in considering these three principal axes, it is sometimes very dicult to use and compare two literature's proposed approaches. There are several operational diculties in achieving an unambiguous understanding of notions of ¯exibility [10±12,36]: 1. an intuitive de®nition, sometimes not operational, of the notion of ¯exibility [13]; 2. lack of uniformity in the classi®cation, de®nition and evaluation of a given type of ¯exibility [1±5,19,27,32]; 3. a diculty in the establishment of the single-dimensional or multidimensional character of ¯exibility [10]; 4. a misconceived relationship by which ¯exibility directly enhances the system's performance [7,28]. In this paper, we propose a new approach in which the system's ¯exibilities are speci®ed by means of a frame of analysis. Our intention is to de®ne clearly and simply what ¯exibility is, what a dimension of ¯exibility is and how it must be evaluated. Our aim is to characterize ¯exibility in a relational, multidimensional and dynamic way. We argue that this perspective conduces researchers and practitioners to an unambiguous de®nition of ¯exibility which enables them to better evaluating schemes. Firstly, we must talk about ¯exibility when one or several particular system dimensions have been speci®ed for change: a ®eld of variations must represent the space of states on which the system's changes are considered. Thus, what are the system properties, conditioning changes on this ®eld?: we talk about the tensions of the system and then we de®ne the ®eld of tensions. Secondly, although dierent articles propose variety as a measure of ¯exibility, we propose a measure called the adjustment degree which captures the system±environment relationship embedded in the ¯exibility notion: it considers the ®eld of variations. Accordingly, because of tensions, the eort and time of adjustment are introduced as two necessary measures of ¯exibility. In this manner, we render consistent the
71
adjustment, eort and time dimensions of ¯exibility in a perspective called here the longitudinal approach. The paper is organized as follows: in Section 2 our approach and the relevant concepts to be used are introduced. In Section 3, a simple stochastic model enables us to present the adjustment degree as one of the right dimensions of ¯exibility. The second example of this section shows the critical dierences between the static and the dynamic approaches to explain ¯exibility. In Section 4, we analyze the problem of determining the product lot sizes in a manufacturing con®guration, submitted to a variable demand, and we identify the dierent factors of tension de®ning the system's eort to obtain an optimal adjustment to the demand process. In the example of Section 5, we characterize what we call the inner logic, i.e., the logic which translates the environment state into the system expected state. The conclusions are presented in Section 6. 2. The proposed approach The capability of a system to adapt to changes that occur in its environment is called here ¯exibility. This intuitive de®nition brings two questions: adaptation to what? and how? In order to answer these questions, we are going to consider the system as any logical and/or a physical device which we might wish to evaluate for ¯exibility. The system could be a machine, a mechanism, a logic, a procedure, a process, a program, a technique, an apparatus, an organization, etc. All these entities may be described as operationally closed [8]: they have boundaries that we can de®ne by their functional and control limits. So, all that it is not modeled as part of this closure, is located in the environment of an entity. Primary, we must clarify what is to be comprehended by adaptation. Indeed, the question of adaptation or not is properly a subject of an observer O1 (the modeller 1) who evaluates the degree of congruence between the system and its environment. In this evaluation O1 actually uses (implicitly or explicitly) a deviation function which tells him or her if the system state is adapted to the
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J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
environment state. But, we must not forget that another observer O2 may use his/her own model of the same system, with its own particular deviation function. Consequently, the adaptation capability is a relative property attributed by an observer to a modeled system. The important question is, when is the system adapted to its environment? The answer is, when there is no more system to study, it ceased to be adapted. In other words, it is the degree of adaptation, which depends on the model that an observer makes of reality [22]. In fact, he represents one thing labeled the system, another thing labeled the environment and any kind of relationships between them. In this paper, we are going to write about models. In consequence, we will use the idea of the degree of adaptation as the local adjustment measure (cf. Section 2.1) between the current and the expected states of the system, such as it is determined by an inner logic. For a given model, we will investigate what is to change in the environment and the system, what is to be de®ned as the congruence between them, what is to be de®ned as the deviation function and how we can measure this deviation.
2.1. The ®eld of variations Suppose that we are able to characterize the behavior of a system and its environment through the trajectories that they take in a set of states. Let S be this set of states; we will call it the ®eld of variations. Let R S be the subset of realizable states of the system. Additionally, let E be the set of the states in which the environment moves. Note that these set de®nitions must all be given by a speci®c observer. Now, let st 2 R, et 2 E and st 2 S be the observed current state of the system, the observed current state of the environment and the expected current state of the system, respectively. We will suppose that there exists, in the system, a L logic such that L et ; st st ; kst ÿ st k;
1
i.e., given st and et the L logic allows us to determine the expected state and the norm between st and st . Then, we can say that the system is in partial equilibrium when L et ; st st ; 0 or not in partial equilibrium if kst ÿ st k 6 0. De®nition 1. If kst ÿ st k 6 0, then ¯exibility is the property that tends to realize the partial equilibrium in the system. Thus, a ¯exible system has the capability to adjust its current state in response to the deviation kst ÿ st k. Therefore, we assume that the relationship between the system and its environment is directly established by the L logic. This is a valid assumption because we have established that the implicated observer models the system as a closed entity (cf. Section 2). Indeed, the adjustment could be understood as the system's expected state tracking process. It means that, in our approach, the system does not adjust to the environment, but to its own expected states. Let D s1 ; . . . ; sn be the ®nite succession of expected states as determined by L between some reference periods t 1 and t n n > 1. We will not restrictively suppose that the relationship D R must hold true. Additionally, let F s1 ; . . . ; sn be the succession of states adopted by the system when it seeks to adjust to the D succession. Thus, we will demonstrate in this paper the necessity of using a measure of adjustment between D and F. 2.2. The ®eld of tensions Time and ease of adjustment can both vary considerably. The fact that adjustment requires an eort, and a certain time interval, points to the existence of one or more factors of resistance to change that we call the factors of tension. For instance, the time and costs of setup operations of a ¯exible machine, depends on the resistance to the change; we argue that this resistance may represent the system's tension. Similarly, the level of acceptability of an urgent order that compels a ®rm to use their machines after the normal day of work gives an idea of the level of system's tension. Alike,
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
73
any change in project planning (of construction, engineering, computer system development, etc.) implies an eort (cost); this is an irreversibility effect that represents again the system's tensions. Suppose that n factors of tension represent the system's resistances to change on R S. Let XS i be the set de®ning the ®eld of variations of the i factor of tension when the ®eld of variations S is considered (remember that changes on the ®eld of variations do not correspond to those on the ®eld of tensions: variations on the latter induce or resist to those on the former). We de®ne the ®eld of S . Additionally, we de®ne tensions as XS ni XS i the level of tension by a function T such as:
explain time and eort of adjustment of the system. We claim that this is a congruent dynamical approach to conceive ¯exibility. Thus, we propose a change of perspective that relates F (and not R) to the system's eorts and times of transition. This implies that ¯exibility is a relative property: it depends on a well-speci®ed environment. The following sections present a number of examples to illustrate how this approach may be used. The similarities and the dierences between our approach and other perspectives found in the literature are signaled.
T : S ! XS ;
3. Examples of adjustment capability: The transversal and the longitudinal approaches
2
which establishes a mapping between any system state and the corresponding factors of tension on their own ®elds of variations. Thus, any state transition on S implies a change of the level of tension. Therefore, time and eort are necessary and we argue than they must be considered in relation to a speci®c adjustment situation, what we call the longitudinal approach, and not independently of it, what we call here the transversal approach. In fact, in considering the eects of the factors of tension over the ®eld of variations, we can appreciate the dierence between these two approaches.
2.3. The dynamic and relational ¯exibility In a dynamic approach, we represent the state expectations as a succession D of states determined by a L logic, and the system's responses to these demands as a F succession, tracing the effective system moves. Note that, in F, whatever the transition from a state st to a state stDt be, it demands eort and time. Therefore, the system dynamically adjusts to the demanded changes de®ned by the D succession. The static approaches in literature propose to relate the transition eorts to the R set (cf. Section 2.1). In such perspectives, we cannot correctly associate dynamic eorts or time to R. In our approach, only a speci®c environmental process may
Researchers have traditionally assumed that ¯exibility may be evaluated by three kinds of measures: the variety, the eort and the time of system adjustment considered for adaptation [10]. Under this assumption, variety is evaluated by measures such as the number of responses which the system may potentially generate. In those cases, the ¯exibility is de®ned as a system's property which does not depend on a particular environment process. Nevertheless, we argue that the adjustment capability of a system must be primary related to the system±environment evolution. In the next example, we show that variety does not completely capture the relational sense of ¯exibility. Consider a system and its environment. We want to model their interaction by two completely ergodic Markovian processes [14]. Let P pij nn and Q qij nn respectively be the transition probability matrix of the environment and the system. Suppose that the set of states represented in P and Q are identical; they are both subsets of S (cf. Section 2.1), the ®eld of variations. Let us consider the steady state for the system and its environment. To model the interaction, we assume that the environment undertakes transitions and the system try to follow them. Indeed, P models the transitions of the expected states as they are decoded for one L logic in the system.
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J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
Then, in a perfect adjustment capability model, we have Q P:
3
Moreover, suppose we are going to determine the entropy, H, of these processes [9,19]. Thus, we have HQ HP . Nevertheless, it is easy to conclude that, even if entropy is reduced (or increased), regarding a reference value, the system and the environment processes are identical. Consequently, in that case, we are not able to evaluate the system ¯exibility by entropy (or variety), but by the relative entropy system entropy : environment entropy Here, this ratio is called the adjustment degree. Thus, in the preceeding example, the relative entropy is maximal and the adjustment degree also; note this measure depends on the system±environment relationship. Now, let us consider a two-period sequential decision problem [31,37] in which one needs to make two temporally related decisions. At the beginning of t 1, the ®rst period, there is a set A of alternatives or actions. At the beginning of t 2, the second period, exists a set B of alternatives. There are two decisions related in the following way: in t 1, if an action a 2 A is chosen, then in t 2 the decision maker could choose an action b 2 B a. Actually, we are going to apply a little change to this problem. Let us suppose only one action set, A. Thus, when an action a 2 A is chosen, a remaining set B a A is available; whatever the decision be, it is 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 the environment and the whole necessary information to select this program, including those probabilities, is available upon beginning t 1. In other words, we do not have a real second period decision because it is completely determined in the ®rst period. In contrast, we have an uncertain decision if upon beginning t 2, additional information is received. In an uncertain universe, in t 1, we have the necessary information to select ai , but not
all the information regarding aj 2 B ai . Then, the latter decision is moved to the beginning of the second period. How can we evaluate the adjustment capability of the choice system? First of all, in this problem, the optimizer (observer) considers the rational approach as the right point of view to evaluate the policies of choice. Thus, whatever the optimal solution be, he or she accepts it without hesitation, in the risky or the uncertain situation. Hence, a completely rational decision maker believes that the optimal action is the best response to the system's environment. From this perspective, the adjustment capability must be necessarily perfect. However, if we consider two kinds of decision makers, the ®rst one accepting the risky situation and the second one accepting the uncertain point of view, the optimal solutions may be potentially dierent. Indeed, when the number of second period actions in the whole remaining set is the same for all ®rst period actions, the uncertain solution dominates the risky choice in the sense of gains (or losses [21]). In that case, to select the optimal risky action is evidently not the best policy and the uncertain decision maker would say that ¯exibility is reduced because of the decreased gain. But, we should not forget that the risky and the uncertain decision makers use dierent models to make a choice. We argue that those models can not be compared because the adjustment degrees would be de®ned by two dierent perspectives. Therefore, let S be the set of states of the environment, where the decision maker assigns a subjective probability p s; s 2 S. Likewise, let ut ai ; s be the gain, relative to s, when the action ai is chosen at the t period. Also, let I2 fy1 ; y2 ; . . . ; ym g be the set of probable messages arriving upon the beginning of the second period, 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 X p su1 ai ; s Eai j I2 S
X I2
( qy
max
aj 2B ai
X
) p s j yu2 aj ; s ;
S
4
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
where p s j y is the posterior conditional probability of s. This expected gain is maximized by an action a1 , that is, Ea1 maxai 2A Eai j I2 . Let us assume that the decision maker chooses this action and subsequently the message yr occurs upon the beginning of the second period. In that period, he/ she may consider a new decision problem where Eaj j I3 is the expected gain of the second period action aj when a set of messages, I3 , is considered for the third period. Then, we have X p s j yr u2 aj ; s Eaj j I3 S
X I3
( qy
max
ak 2B aj
X
) p s j y; yr u3 ak ; s :
S
5 The term p s j y; yr corresponds to the conditional probability of s, when the yr message results in the second period; B aj is the set of available actions from aj and u3 ak ; s de®nes the gain of ak in the third period, relative to the state s. The ¯exibility eect would be considered when the decision maker contrasts the best second pe0 riod action a2 , given a1 , against the best second period action if all the past routes (the ®rst period alternatives) were considered, a2 . Thus, we have 2 6 Ea j I3 max max Eaj j I3 ; ai 2A
aj 2B ai
20
Ea j I3 max Eaj j I3 ; aj 2B a1
7
and the opportunity cost of choosing the wrong alternative in the ®rst period is given by 0
DE Ea2 j I3 ÿ Ea2 j I3 :
75
native approach called the longitudinal point of view. When ai is chosen at the ®rst period, there is a probability distribution on B ai which denotes the probable transitions from ai . These probabilities are completely determined by the optimization problem which searches for the optimal action aj 2 B ai . In our formulation of the second period optimization problem, the probability distribution may be approximated by the following function: , X Eaj j I3 : 9 p aj j ai Eaj j I3 aj 2B ai
Thus, the entropy of the second period action aj 2 B ai , when the information I3 is considered, could be de®ned: H aj j I3 . In general, two random sets X; Y have the property X Y ) H X 0 6 H Y. Thus, if we have B a2 B a2 , then 0 H a2 j I3 6 H a2 j I3 . 0 Now, let us consider two actions, a2 and a2 , with remaining sets B a2 fa1 ; a2 ; a3 g and 0 B a2 fa1 ; a2 g. Let us suppose Ea1 j I3 10, Ea2 j I3 20, Ea3 j I3 0. In that case, 0 Additionally, DE 0. H a2 j I3 H a2 j I3 . 0 Clearly, we should not use jB a2 j as an interesting measure because the action a3 is totally irrelevant. On the other hand, if Ea3 j I3 30, 0 H a2 j I3 > H a2 j I3 and DE > 0, then we establish the opportunity cost of choosing the wrong way in the ®rst period. Finally, if we set Ea1 j I3 Ea2 j I3 Ea3 j I3 , then H a2 j I3 > 0 H a2 j I3 , but DE 0. Whence, we may conclude that the more appropriate adjustment degree measure is the opportunity cost of optimal actions and not the relative entropy level.
8
Note that DE P 0. Thus, there is a perfect adjustment situation when DE 0. In the literature [21,31], dierent models propose that the ¯exibility of ai is evaluated by jB ai j. Those models are coherent with an approach called here the transversal point of view. Because these approaches do not consider the relative character of ¯exibility they do not consider those actions aj 2 B ai which may have a zero probability to be chosen from ai . We propose an alter-
4. The setup costs in a manufacturing system: The factors of tension Here, we analyze the problem of determining the product lot sizes in a manufacturing system submitted to a variable demand [16,17,34]. A set of products being given, the optimization problem consists of determining the lot sizes minimizing the total production cost function per unit of time. In modeling this problem, DeGroote [7] has proposed
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that the ¯exibility is a binary relation evaluated by the comparison of marginal performance of two systems obeying the same marginal change of environmental variety. Q1 ; Q2 ; . . . ; Qn be the lot sizes vector Let Q of the product set. Then, the optimization cost problem is modeled by min Q
n X mj qj cs S
Qj
j1
icj Qj 2
where, the variables in the model are as follows:
Qj rj p S cs qj cj i a v
number of product types in manufacturing, periodical demand rate for product j 1; . . . ; n, lot size for product j, production rate for product j, availability of the facility, nominal setup time, cost per unit of setup time, fraction of S used in the setup of product j, unit direct labor and material cost for product j, fractional (per period) opportunity cost of capital, fraction of time required to meet the production processing requirements for all products, excluding the setup times, product variety.
This problem may be solved by the Lagrangian relaxation. Thus, in determining the Lagrange multipliers, we may ®nd the measure of product variety proposed by DeGroote: 1 v cm
!2 n p X 2icj mj qj ;
10
j1
Pn Pn where m j1 mj and cm j1 cj mj =m. Accordingly, the optimal cost is de®ned by p 2cs Sicmv cm C Sicmv cs p ÿ a 2 pÿa
Let T be a technology described by the S; cs ; qj ; cj and i parameters. If T1 and T2 are two technologies, with optimal costs C1 and C2 , such that oC1 oC2 6 8v P 0; ov ov
s:t: a 6 p;
n mj
where v is a critical value of v. DeGroote de®nes the ¯exibility of a T technology in the following way:
if v 6 v ; if v > v ;
then, T1 is more ¯exible than T2 . Indeed, three dimensions are considered in this de®nition: 1. The adjustment in relation to the environmental change. 2. The cost of this adjustment, seemingly represented 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 the production lot sizes, has a perfect adjustment property. The only dierence between two technologies, in the ¯exibility sense, is determined by the derivative of the optimal cost. Although the DeGroote's de®nition is well structured, we propose to consider the relative derivative of cost to variety, de®ned as
oC v : ov C
13
This is the elasticity of cost to variety, a measure which has been used in other contexts to evaluate the ¯exibility of a ®rm [23,28]. Actually, it has been argued that human beings perceive gains or losses in relation to the levels from which the move starts [20]. Thus, this measure appears to be better adapted to comparisons than the single derivative of cost to variety. p Let c1 2cs Sicm; c2 Sicm=2 p ÿ a; c3 cs p ÿ a. Then, one can show (
11
12
1=2 p 11= vc1 =cm 1 11= vc2 =c3
if v 6 v ; if v > v :
14
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
77
But, note limv!0 0 and limv!1 1. As a consequence, we have
applied in order to know the ¯exibility comparison between technologies:
0 < < 1:
15
rij P k ^ rji P k ) Ti If Tj ;
In his modeling, DeGroote has shown that the derivative of costs depends on dierent factors such as the nominal setup time, the direct cost per unit of setup time, the production rate and the maximal number of setups that can be performed per unit of time. In our approach, these variables could be called the factors of tension because they make the system's changes more or less dicult. Let us analyze the eect of variables cs and S on the elasticity measure of Eq. (14). Firstly, we know 2 c3 =c2 v [7], that is, v 2cs p ÿ a =Sicm. Then, for a ®xed value of v, rises in S imply rises in . On the other hand, if we consider the variable cs , the eect will depend on the value of v. In fact, for v 6 v , the elasticity increases with cs , whilst it decreases for v > v . Consequently, we may establish these variables as factors of tension. Now, let Mt mt1 ; mt2 ; . . . ; mtn be a vector where the component mtj corresponds to the demand rate for product j during the t period. Consider a sequence of vectors M1 ; M2 ; . . . ; Mk representing the evolution of the manufacturing system environment between the periods t 1 and t k. Let vit be the variety of technology Ti at the t t 1; . . . ; k period and it be the elasticity of the technology Ti when the variety level is vit . In order to compare the technologies Ti and Tj , we de®ne the variable rij as
rij P k ^ rji < k ) Ti Sf Tj ;
rij
k 1X dijt ; k t1
where 1 dijt 0
if it > jt ; otherwise:
16
17
Now, let If ; Sf ; Rf be the binary relations representing the indierence, the outranking and the incomparability [30], respectively, in the sense of ¯exibility. For instance, the expression Ti If Tj means ``Ti as ¯exible as Tj ''. Similarly, Ti Sf Tj means ``Ti at least as ¯exible as Tj ''. If a threshold k 2 0:5; 1 is de®ned, the following rules may be
rij < k ^ rji P k ) Tj Sf Ti ;
18
rij < k ^ rji < k ) Ti Rf Tj : Note the binary relationships are established within the perfect adjustment and the common adjustment time assumptions. Thus, the results contained in r have signi®cances only when the eort dimension is considered. We may consider the perfect adjustment capability as a property to be guaranteed: it is the robustness criterion. Thus, any system using the mathematical model above, is a robust system. Nevertheless, if ¯exibility and robustness are two very similar properties, they are essentially dierent. Meanwhile the former involves three principal dimensions, the latter is related only to the adjustment dimension.
5. Flexibility in a pull-based manufacturing management system: The inner logic In a manufacturing system, we can identify two principal ways to achieve the production and inventory management: the pull and push methods [6]. In a push approach, the production rates over the manufacturing stages are ®xed by production orders determined in an estimated demand base. In contrast, in a pull approach, the production orders are de®ned on a real demand base. Although the performances of both systems have extensively been studied in the literature [18,24,26], we propose to analyze ¯exibility in a particular model of the pull method. To achieve it, we consider a serial disposition of production and inventory stages in the manufacturing system. A model of the manufacturing system dynamics is constructed; we use mathematics to represent the production rates, the inventory levels, the production orders and the demand signal. Furthermore, in order to illustrate certain ¯exibility measures, we simulate the push
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J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
method, when the manufacturing system is submitted to a normal distributed random demand.
5.1. The pull ordering method Let us consider three production stages and their respective inventories. Let Pti be the producthe inventory level of tion rate of stage i; Biÿ1 t product manufactured by the stage i; Oit the production order on the stage i, at time t; and L be the constant representing the production and deliver delay on each stage. Additionally, let us suppose the excedentary production availability, on each stage, and an in®nite level of raw material stock in back of the stages arrangement. Therefore, the following set of equations represents the dynamics of a pull-ordered manufacturing system [28]: Oit
Dt
if i 1;
Dtÿ iÿ1L
if i > 1;
Pti OitÿL ; ( i Biÿ1 tÿ1 Ptÿ1 ÿ Dt iÿ1 Bt i iÿ1 Biÿ1 tÿ1 Ptÿ1 ÿ Pt
19 20 if i 1; if i > 1:
21
Using Eqs. (19) and (20), we can evaluate the production and demand dissimilarity which inform us on the capability of management system to adjust the production rate to the real demand signal. Accordingly, let us de®ne the deviation variable hit as hit Pti ÿ DtÿiL :
22
Therefore, we de®ne the adjustment degree of the production stage i by the following equation: #i
V hit ; V Dt
8i P 1;
23
where V represents the variance of the argument. This measure establishes the capability of the management system to adjust, on each stage, the production rate to the delayed demand signal. Indeed, the pull method searches for a ®xed stock level on each production stage i. If this level decreases, the production order Oit is de®ned just
equal to the decreased amount of stock, that is, the demand rate. The necessary time to adjust the production rate of the stage i is iL and this is a perfect delayed capability of adjustment because Pti ÿ DtÿiL 0 i 1; 2; 3; 8t. Note this equality is justi®ed by Oit Dtÿ iÿ1L and Pti OitÿL , which implies #i 0 i 1; 2; 3. In our framework, the inner logic L is represented by the ordering system, that is the pull approach. It determines the expected states in the manufacturing system. In fact, we may consider the current state of the manufacturing system as a vector st B0t ; B1t ; B2t . Consequently, we may de®ne the vector of expected stock levels st B0 ; B1 ; B2 on stages; in a pull-ordering system, these levels are independent of time (at least, in a short or medium time period [33]). Therefore, the adjustment action consists of calculating the production orders and process them on stages. In the pull method, described above, the orders are completely satis®ed, but with a delay iL on the stage i. Dierent inner logics generate dierent adjustment actions. These are implemented by the production orders. Hence, when a push method is considered, a production order equation considers the feedback corrections of inventory and work-inprocess levels [6]. In such case, we must use another kind of measure to evaluate the adjustment degree because there are several types of expected states. An example, preserving our approach, is provided by Takahashi et al. [35] which have proposed an alternative multidimensional measure to evaluate ¯exibility where production/demand and inventory/demand ratios are calculated on each stage. 5.2. Illustration As we have shown, the adjustment degree in a pull method is perfect and dierences between stages are not appreciated. In order to have a perspective on the ordering methods' ¯exibility, we illustrate the ¯exibility evaluation, by simulation, in three methods. First, let us consider the push method. In this case, the production order equa-
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
tion is concerned with corrections feedback of inventory and work-in-process, and to adjust the production rate to the demand rate. The equation describing the production order on the ®rst stage is [28] ! L X ^ t;tj ÿ EC 1 ; ^ t;tL1 S 0 ÿ B0 D O1 D t
t
j1
t
24 ^ t;tj corresponds to the demand estimate, where D calculated at the end of t, for the period t j; S 0 is the security level of the stock B0 ; ECt1 the level of work-in-process on the stage 1, calculated at the ^ t;tL1 is the feedback signal end of t. The term D adjusting the production rate to the demand rate. The other terms correspond to corrections of the inventory and work-in-process levels. In consequence, we have a multiobjective feedback structure where only the ®rst term keeps tracing on the demand rate. If we would have to contrast the pull and the push methods, using the measure proposed in Section 5.1, we would consider the adjustment capability of production to the demand rate. An elementary transformation permits us to rede®ne the production order equation of the push method. We have [29] Oit Dtÿ iÿ1L
i X j1
^j DD tÿ iÿjL ;
i P 1; 8t 2 Z: 25
Thus, the deviation variable (cf. Section 5.1) is de®ned by hit
i X j1
^j DD tÿ i1ÿjL ;
i P 1; 8t 2 Z:
hiÿ1 tÿL
iÿ1 X j1
^j DD tÿ i1ÿjL ;
i P 1; 8t 2 Z:
As a consequence, one ®nds ^i hit hiÿ1 tÿL DDtÿL ;
^ 1 . In order to Additionally, we have h1t DD tÿL contrast the ordering methods, we have considered a third system in which the ®rst stage works in push and the upstream stages in pull: the hybrid method. In Table 1 these results are shown. With hit , we may establish the adjustment degree expressions. Only the push method imposes certain complexity, for i > 1. In fact, we have iÿ1 ^i ^i V hit V hiÿ1 tÿL V DDtÿL 2 cov htÿL ; MDtÿL :
However, knowing hiÿ1 tÿL , this variance is expressed as a function of the demand estimates. The following factors may be de®ned: ^1 V DD tÿL ; V Dt P ^ i 2 cov iÿ1 DD ^j V DD
G
tÿL
Hi
i P 2:
^i tÿ i1ÿjL ; DDtÿL
j1
V Dt 8i P 2;
which give us the adjustment degree on each stage (Table 2). ^ i 0 8i; t, we ®nd G 0 and Observe, if DD t Hi 0. In other words, the three ordering methods have the pull system's behavior. In order to calculate the value of Hi , the push method has been simulated when an AR(1) stochastic process is considered. In this case, we use ^ ti ki Dt 1 ÿ ki l, the estimation function D where k 2 ÿ1; 1 is the autocorrelation index of the demand process. Here, we assume Dt N l; r, with known distribution parameters. Furthermore, L 1. Table 1 Deviation hit for three ordering methods Stage
It is known that t ÿ i 1 ÿ jL t ÿ L ÿ i 1 ÿ 1 ÿ jL, then it may be established:
79
i1 i>1
Push ^1 DD tÿL iÿ1 ^ itÿL DD htÿL
Hybrid ^1 DD tÿL iÿ1 htÿL
Pull 0 0
Table 2 Adjustement degrees #i Stage
Push
Hybrid
Pull
i1 i>1
G #iÿ1 Hi
G #iÿ1
0 0
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J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
Fig. 1. Hi for i 2; . . . ; 7. Fig. 3. #i for k 2 ÿ1; 0.
In Fig. 1, the Hi 2 6 i values are presented for a serial manufacturing system with seven stages. The curves indexed i correspond to Hi i 2; . . . ; 7. Note that Hi represents the adjustment degree degradation of #i in relation to #iÿ1 . Thus, the Fig. 1 shows how the adjustment degree is degraded in the upstream direction. In Fig. 2, the adjustment degree values are presented. One can see that, for k 2 0; 0:5, # increases over all the stages, but the adjustment degree decreases when k approximates to 0:5. The same kind of graphic has been depicted in Fig. 3, where k 2 ÿ1; 0. The qualitative behavior is very similar to the ®rst sample case, although the curves for a same stage are not symmetric. It is very interesting to note that the hybrid method, represented by the curve 1 is robust (in the ad-
Fig. 2. # in stages i 1; . . . ; 7.
justment degree k 2 ÿ1; 1.
sense)
in
all
the
interval
6. Conclusion We propose an approach that introduces a relational, dynamic and multidimensional conception of ¯exibility in the manufacturing systems. In this approach, we de®ne two ®elds of analysis: the ®eld of variations and the ®eld of tensions. The ¯exibility notion, conceived in a relational perspective, emerges when an observer models the system and the environment process interaction. Thus, this notion additionally incorporates a dynamic character. In our approach, any analysis, evaluation or de®nition of a particular system ¯exibility, must begin by introducing two important inquiries: what is the ®eld of variations on which ¯exibility is going to be observed? and, what is the ®eld of tensions grouping the factors imposing resistances to changes on the ®eld of variations. In order to obtain an operational model of ¯exibility, the analyst (observer) must de®ne the vectors of the current and the expected states of the observed system. Therefore, a model of the factors of tension relationships and their in¯uences on the state of the system must be achieved to determine the three ¯exibility dimensions proposed in our approach: the adjustment degree, the eort (normally a cost function in the manufacturing system
J. Pereira, B. Paulre / European Journal of Operational Research 130 (2001) 70±82
modeling) and the time of this adjustment. We argue that the model construction subtends the analyst's belief in a system's inner logic which calculates the expected states of the system. Thus, the analyst must be advised about this logic belief. Through the examples, we have shown that a good measure of the adjustment capability is a proximity function relating the system and the environment processes such as the latter is decoded by the inner logic de®ning the system's expected states. In the example of Section 3, we contrast two decision making rationalities. The ®rst one, called the transversal point of view, considers the ¯exibility as a single measure: the number of available options from a system state, independently from the environment conditions. The second one, which we subscribe, called here the longitudinal point of view, considers the set of available options as relative to the system±environment relationship, such as it is conceived by the modelling believes. The environment process, codi®ed in the action return expectations, must be considered. In the example of Section 4, we have shown that the factors of tension relationships determine the expected state of the analyzed system. Additionally, the eort, represented by an elasticity measure, is characterized as a function of those factors. A simple example shows that two technologies submitted to the same demand process search for dierent adjustment process, each one determined by their own factors of tension. An aggregate elasticity function is proposed to compare technologies. Finally, in Section 5.1, a pull manufacturing management system is analyzed to demonstrate that an inner logic determines the current and the expected state deviation of system. We show that the adjustment action is achieved by a control variable which diers from one inner logic to another. A simulation experience shows that the ordering methods have dierent capabilities to adjust the production rate to the demand rate. The best one seems to be the pull method. In particular, the hybrid method is better than push when the demand process is highly autocorrelated, but they are very similar when autocorrelation is low.
81
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