Flexibility in Manufacturing Processes: a relational,
dynamic and multidimensional approach!
Javier Pereira
Departamento de Informatica de Gestion, Universidad de Talca,
Avenida Lircay, Talca, Chile,
Tel: 56 71 200356
e-mail: jpereira@ pehuenche.secom.utalca.cl
Abstract
We propose an approach that introduces a relational, dynamic and multidimensional
conception of flexibility in manufacturing systems. In this approach, two inquiries
must be introduced by the analyst: what is the field of variations on which flexibility is
going to be observed? and, what is the field of tensions grouping the resistences to
changes on the field of variations? In order to obtain an operational model of
flexibility, the analyst must define the current and the expected states of the observed
system and construct a model of the factors of tension's influences on its state. Thus,
three flexibility dimensions are proposed: the degree of adjustment, the effort and the
time necessary to achieve this adjustment. It is our contention that the model
construction substends the analyst's conjecture about a system's inner logic
calculating the expected states.
Key words: flexibility, multidimensional approach, field of variations, field of
tensions, manufacturing.
1. Introduction
Because an organization needs several types of flexibility, there are multiple
definitions and evaluation schemes. The origins of this diversity may be linked to the
variety of uncertainty factors, the possible time perspectives or the different possible
dimensions to evaluate for flexibility (Gerwin,1993; Carlsson,1987). Consequently,
there are several operational difficulties in achieving an unambiguous understanding
of notions of flexibility (Pereira,95): intuitive definitions; misclassification,
misdefinition and misevaluation; single-dimensional and transversal approaches, etc.
We propose an approach in which the system's flexibilities are specified by means of
a frame of analysis. Our intention is to define clearly and simply what flexibility is,
what a dimension of flexibility is and how it must be evaluated. Our aim is to
characterize flexibility in a relational, multidimensional and dynamic way. We argue
that this perspective conduces researchers and practitioners to an unambiguous
definition of flexibility which enables them to better evaluation schemes.
The paper is organized as follows: in Section 2 our approach and the relevant
concepts to be used are introduced. In Section 3, we introduce a single-line push-
based manufacturing ordering system. Subsequently, in Section 4 we characterize
flexibility in that ordering system to examplify our approach. Finally, the conclusions
are presented in Section 5.
2. The proposed approach
The flexibility is the capability of a system to adapt to changes that occur in its
environment. This intuitive definition 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 flexibility. In
according to Maturana and Varela (1984), the adaptation degree is properly a subject
of an observer who evaluates the congruence between the system and its environment.
In this evaluation he/she actually uses an implicit or explicit deviation function which
tells him or her how much the system is adapted to its environment. Consequently, the
adaptation capability will be attributed by an observer to a modelled system and then,
the system flexibility will be a model-related property also.
For a given system's model, an observer will be compelled to investigate what is to
change in the environment and the system, what is to be defined as the congruence
between them, what is to be defined as the deviation function and how he can measure
this deviation.
2.1 The field of variations
Let S be the field of variations, representing the set of states in which an observer
accords to characterize the behaviour of a system and its environment through the
trajectories that they take in. Let's suppose that he accepts to model RCS as the
subset of realizable states of the system. Also, he defines E as the set of the states in
which the environment moves.
Now, let s,e R, ¢,€ E, Ss, eS 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 the observer assumes the existence, in the system,
of a logic L such that:
Lee,.5,)=(5, ). a)
ie., given s, and e,, the L logic allows to determine the expected state and the norm
5, —5,
between 5; and 5. Then, we can say that the system is in partial equilibrium when
L(e,,8,) = (s, ,0), or not in partial equilibrium if |s, —s,| 40.
Definition
If s, —s,|#0, then flexibility is the property that tends to realize the partial
equilibrium in the system.
Thus, a flexible system has the capability to adjust its current state in response to the
deviation |s; = | and the observer models the relationship between the system and its
environment by the construction of a logic L. Note that, in this perspective, the
system does not adjust to the environment, but to the L-defined expected states.
Let DZS 45.055;
two arbitrary periods d and f . In general, D ¢ R. Additionally, let F = SyrrerSy be
be the succession of expected states as determined by L between
the succession of states adopted by the system when it seeks to adjust to the D
succession. Whatever an adjustment degree would be defined, it should consider a
measure of similarity between D and F.
2.2 The field of tensions
Let the factors of tension be the set of one or more factors of resistance to change
which imply an effort and time interval for the adjustment. Let Q) be the set
defining the variations of the factor of tension i=1,...,n when the field of variations
Sis considered. We define the field of tensions as Q* =|)’ Q? . Additionally, we
define the level of tension by a function T such that T: S$ — Q*. Therefore, any
system state transition on S implies a change of the level of tension. Because of
resistences, this change occurs with an effort and a time interval. Thus, those time and
effort must be considered in relation to a specific adjustment situation, what we call
the longitudinal approach, and not independently of it, what we call the transversal
approach.
2.3 The dynamical and relational flexibility
In a dynamic approach, the observer models the logic L to produce a succession D
of expected states. Also, the system's responses to these demands are defined as a F
succession, tracing the actual system moves. Whatever the transition of a s, state to a
5,,,4, State be in F’,, it demands effort and time. Therefore, we say that the system
dynamically adjusts to the demanded changes defined by the D succession.
The static approaches in the literature propose to relate the transition efforts to the R
set (cf. Section 2.1). In such perspectives, we can't correctly associate dynamic efforts
or time to R. In our approach, only a specific environmental process may explain
time and effort of adjustment of the system. We claim that this is a congruent
dynamical approach to conceive flexibility. Thus, we propose a change of perspective
that relates F (and not R ) to the system's efforts and times of transition. This implies
moreover that flexibility is a relative property: it depends on a well specified
environment. The following section presents an example to illustrate how this
approach may be used.
3. Example: a push-based manufacturing ordering system
3.1 Introduction
In this section, in order to illustrate the proposed approach, a model of one push-based
manufacturing ordering system is introduced. The push ordering method is well
known in the literature and different articles have been dedicated to model, evaluate
and simulate it (Krajewski ef al.,1987; Shingo,1983; Molet,1993; Takahashi et
al.,1994; Pereira,1995). First of all, we are going to consider the ordering method as a
management system working over a single-line manufacturing system's model.
Thereby, we introduce the later and the involved variables. Then, the push model is
defined.
3.2 The basic manufacturing system model
Let us suppose a single-line manufacturing system composed by a set of
manufacturing stages and stockage sites in between. This system is represented in
Figure 1:
Figure 1. A single-line manufacturing system
In Figure 1, the rough arrows indicate the product flow direction which is regulated,
in quantity and delay, by production orders O, (i = 1,2) on each manufacturing stage
P,; additionnally, the production rates regulate the stockage sites B,. A specifical
manufacturing management method establishes a regulation model to define these
orders. In this way, two methods will be distinct if the production orders are
calculated in a different manner (Crespo,1992). In order to introduce the model, we
present below the variable notations:
index, i =1,2,...,
the time interval,
the manufacturing delay, the same on each stage,
the demand rate on stockage B,, during t,
the t+i demand estimate, calculated at the end of rt,
the sum of the demand estimate (SDE), calculated at the end of r, for the stage
Ef,
the marginal change of SDE, calculated at the end of r, for the stage P,
the production order on the stage F,
production rate on stage F during rt placed on stock B,_, at the beginning of
#1,
stock level of B, at the end of r,
security level of B,,
work-in-process level on stage P, calculated at the end of t,
accumulated manufacturing delay between stages P and RP.
Therefore, a set of basic equations must be defined:
a) The stock level on site B,_,:
B= Ba +P -. ifi=l Q)
Bol+pi—Po} if i>.
b) Production rate” on stage i:
F=0,,+ (3)
The variable t indicates the necessary delay before the production quantity arrives
to the stockage site.
c) The sum of the demand estimate (SDE):
Noted Di
face?
this variable is defined by (LT° = 0):
tH
Dace = p> ttt LT hej (4)
j=
d) The work-in-process level:
Noted EC,, this variable represents the production quantities ordered to the stage
P., between f—T and t—1, arriving to stockage site B,_, after r—1:
EC, = YO, es (5)
j=l
3.3 The push method
Actually, the push method is a demand-estimate-based system: the production order
for each manufacturing stage is calculated considering to a demand estimate function;
whereas, in other methods like the pull systems (in its ideal-type conception;
Pereira,1995) the production order on each stage is calculated only by the real demand
rate. In this section, we will present the necessary equations to model the former. In
Section 4, we will use these equations to determine the flexibility dimensions.
Proposition:
Let AD! = B!,. -B!
face 1=L,ace
be the marginal change of SDE for the first stage,
calculated at the end of t. If this manufacturing stage is managed under the
push method, the production order is given by
O! = D,+AD}, Vt. (6)
Dem:
1) The production order equation at the end of t¢ is given by (Pereira,1995)
O! = Di. + S°-(B° + EC!), Vt. (7)
face
Then, subtracting O| — 01, we have
fa
O} - 0}, = AD} — AB) — AEC}. (8)
Additionnally, it is easy to show
B) = Br = Py —D,, (2)
EC} - EC], =O1,-O!,,. (10)
2) Therefore, the equations (9) and (10) finally imply 0! = D,+AD'+0!,-P',
which demonstrates the proposition. lll
In a push method, the equations for production orders in the upstream stages have a
similar structure to the first stage, but they include a delay on the real and estimated
demand; the next proposition establishes it.
Proposition:
Let D,(;, be the demand rate on the first stage at the end of t—(i-\)t and
AD! eis be the marginal change of SDE, calculated for the stage
jJ(s ji). If the stage iis managed under the push method, the production
order is given by
OF =D) genes > ADE epee (1)
j=l
Dem:
Let us consider a recurrent procedure:
1) i =1: The equation 6 establish the truth value for the first stage.
2) i=m(m>1): The recurrence hypothesis says:
t—(m—j)t *
OP = Deine + 2Ab}
=
3) i=m+l1: In general, the production order for the stage iis defined by
(Pereira, 1995)
0! =P’ +d), (12)
Then, one has O/"' = P” + AB" . Furthermore, OQ)", = P” . Thus, we obtain
on =0", +b"
ml we A ml
Of" = Deine + LAD) nar ye + AD!
=|
m+t
o"™ =D ¥ ab!
4 = Deine + t—(m+l—j)e*
j=l
which demonstrates the proposition. lll
4. Flexibility in push methods
The results obtained in the precedent section will serve us to specify the constructs representing the
flexibility dimensions. Thus, in the following sections, we develop our framework. Firstly, the field of
variations and one adjustment measure are defined. Secondly, the field of tensions, the effort and the
time dimensions are defined and analyzed.
4.1 The field of variations: an adjustment measure
Actually, in a manufacturing system, we may define several fields of variations, each one related to one
kind of expectations: the coupling of, production and demand rates, real and desired stock or real and
desired work-in-process (Forrester,1969). Additionnally, several objectives may be defined for the
ordering system and it may be evaluated in relation to the success or failing to reach them
(Lenard,1995). Thus, the first choice to be made by the analyst is what is to count as the field of
variations? In this particular instance, we will select the common space of changes of demand and
production r:
as the D succession of expected states and the production rate process as the F succession of the
system responses (cf. Section 2.1). Then, to define an adjustement measure we must find a similarity
function for the demand and production signals.
s. Thereby, let S = be this space. We are going to consider the demand rate process
In Figure 2, three manufacturing stages are managed in a push ordering system. We may observe that
there is no a great similarity between the production curves and the demand process, represented by a
ae
rough line”.
Production and
demand rates
rt nN OM © BAN DO Ow
rrr ere nna
Time interval
Figure 2. Production rates in a push ordering system (three stages)
In contrast, in Figure 3, the same three stages are managed in a pull ordering system. In this case, we
may appreciate an astonishing similarity between curves, excepting the sliding effect caused by the
production delay. According to these examples, an appropriate measure for the adjustment may be a
similarity or dissimilarity demand-production indicator. However, it is important to take into account
the production delay. In fact, we know that P’ = O!_..
120
Zo
§ 100
g 2 80
st
ae 60
BE 40
of 20
a? 0
ry BN OM OD HAN
- rr Nw
25
28
Time interval
Figure 3. Production rates in a pull ordering system (three stages)
Then, the difference (cf. Equation (11)) P’ — D,_;, = SAD) a iene Vt may give us an idea of the
j=l
gap between the expected state D,_;
defined by P' = D,
tit
, and the system's response P’. In general, the production rate is
+0), where 9! depends on the ordering system. In other words, the distance
variable between the production and demand rates corresponds to @' = P' — D,
iz Then, we define
the adjustment degree of the stage 7, in relation to the demand rate, by the following expression (the
condition E[AD/] = 0 = E[0/] = 0, Vi, , must be satisfied):
pf -YO@
= 21
VD)
(13)
Now, we have t—(i+1—j)t =t—t—(i+1—1—)t, then, in order to obtain a developed
expression for the adjustment degree, we can establish:
int
ia a .
6... = DAD! wai-nes i2i,
j=l
(4)
which will lead to
i i-1 Hi .
6! =O" +AD! 122.
Note that a pull method is characterized by the absence of the demand estimate term, that is,
0; =0, Vi,t (Pereira,1995). Also, we may consider an hybrid ordering system in which the first
stage is managed in a push method and the upstream stages in pull. In Table 1, the @/'s for these three
ordering systems are presented. It should be noted that the push method induces an upstream
propagation of the demand estimate signal.
Stage Push Hybride Pull
i=l AD! AD! 0
tt fe
i>l | ot+abdi, 6; 0
Table 1. Distance to demand rate in different ordering systems.
Indeed, Table 1 suggests that, it suffices to determine the variances for the push method and the other
methods are resolved. Thus, the variance on the stage i > 1, for the push ordering system, corresponds
to V(0!) =V@i!) +V(AD!_,) + 2cov(O'!, AD!) . Using the equation (14), let us define the
following terms:
V(Ab}_,)
G =
V(D,)
V(AD)_,) + 2cov()) AD} in) ADL.)
H, = cies
‘ V(D,)
In according to these expressions, the adjustments degrees are specified in the Table 2.
Stage Push Hybride Pull
i=l G G 0
>t | et+H, | oF 0
Table 2. Adjustment degree in different ordering systems.
When the manufacturing system is managed in a push method, the upstream stages potentially raise the
dissimilarity between production and demand rates. As a result, the stages are not synchronized,
whereas, in the pull or the hybrid methods, they are. These are the evident behaviours showed in
Figures 2 and 3. Another conclusion may be obtained. It should be pointed out that the production-
demand distance measure depends on delay T . In consequence, the higher the T delay, the slower the
ordering system will adjust production rates.
4.2 The field of tensions: the effort and time consequences
Muramatsu ef al. (1985) have proposed an amplification measure, V(P')/V(D,) which should
characterize flexibility in a manufacturing stage. The desirable management systems satisfy
1> Amp! >...> Amp", where Amp’ (i= 1.....n). Indeed, the amplification evaluates the relative
average raise or decrease of production rate. In fact, in a single time interval, several lot-related setup
operations may occur. In that case, a lot-sizing problem may be resolved in which one of constraints
imposes the demand satisfaction (Spence y Porteus,1987). Thereby, the amplification ratio indicates the
opportunity cost incurred by the ordering system when it fails to determine the optimal production rate.
Then, the adjustment effort increases with the non unitary amplification.
It is clear that any variable contributing to the production costs may be considered as a factor of
tension’: the facility availability for processing, the nominal and relative time setups, the direct setup
cost, the fixed cost, the unit cost of production, the production lot-size, the opportunity cost of capital
(DeGroote, 1994).
Now, we know that P’ = O;.
), and (cf. Equation (12)):
AI spe
Di~t AD, ifi=l
i Ai pe
PL+AD_, if i>t.
(5)
Thus, an expression for the variance of production rate may be found.
Proposition
Let us consider a push ordering system and a stationary stochastic demand process satisfying
the equation (15), then the variance of the production rate P is given by:
V(D,)+V(AD}) + 2cov(D,,AD!) if i=1
V(P'")+V(AD') + 2cov(P'",AD') if i> 1.
Dem:
1) When i = 1, one has V(P') = V(D,_,) +V(AD}_,) + 2cow(D,_,, AD} ,) . Nevertheless, the
0?
stationarity hypothesis implies
V(AD}_,) =V(AD!),
cov(D, AD} ,) = cov(D,,AD') °
Nat?
2) In the same manner, one concludes on the variance for P’ when j > 1. i
It can be seen that the amplification ratios for the manufacturing stages may be deduced. Moreover,
with this result, it is easy to find the variances expressions for the pull and hybrid cases. Next, to do
this, let us define the following variables:
de V(AD!) +2cov(D,,AD!)
- V(D,) ,
R V(AD') + 2cov(P', AD!)
‘ V(D,)
The ordering systems behaviour (see Table 3) is very similar to the patterns found for the adjustment
degree. It should be noted, however, that the amplification ratio is not null for the pull method. Again,
the push method introduces an additional term in the upstream direction, whereas the hybrid ordering
system introduces an initial term A which propagates upstream in the manufacturing line.
Stage Push Hybride Pull
i=l 1+A 1+A 1
i>l Amp"! + B, Amp’ 1
Table 3. Amplification ratio in different ordering systems.
Because of the potentially higher amplification of the push ordering system, we establish that, under
the specified conditions, this method induces a larger adjustment effort than the other two (Takahashi
et al.,1994).
5. Discussion
We have proposed an approach that introduces a relational, dynamic and multidimensional conception
of flexibility in manufacturing systems. In this approach, we define two fields of analysis: the field of
variations and the field of tensions. In our approach, any analysis, evaluation or definition of a specific
system flexibility, must begin by introducing two important inquiries: what's the field of variations on
which flexibility is going to be observed? and, what is the field of tensions grouping the factors
imposing resistences to changes on the field of variations? In order to obtain an operational model of
flexibility, the analyst (observer) must define the vectors of the current and the expected states of the
observed system. A model of the factors of tension relationships and their influences on the state of the
system must be achieved to determine the adjustment degree, the effort and the time of this adjustment.
It is our contention that the model construction substends the analyst's conjecture in a system's logic
which calculates the expected states of the system.
In the example of manufacturing ordering system, we have shown the necessity of a correct definition
of the field of variations. Subsequently, we have proposed that, in the manufacturing ordering example,
our adjustment degree measure differs from those presented in other articles (Kimura and Terada,
1981; Muramatsu et al.,1985; Takahashi et al.,1987; Takahashi et al.,1994) because the amplification
ratio is better aprehended as an effort indicator and not as a production-demand adjustment measure.
Additionally, we have shown that the time factor is directly considered in the adjustment and
amplification ratios. A further analysis of time aspect may be found in Pereira (1995). The inner logic,
in our case the defined push, pull or hybrid methods, strongly determines the flexibility evaluations.
Thus, in Sections 4.1 and 4.2 we show that adjustment and amplification ratios depend on the demand
estimate functions: the push and hybrid methods are very sensible to these functions, meanwhile the
pull method is sensible only to the real demand rate.
We conclude that, in the manufacturing ordering system example, the proposed approach establishes a
well structured framework to define and evaluate flexibility.
' This research is supported by a DIUT project, at the Universidad de Talca.
> Here, we suppose that there is no production shortages.
> The demand curve corresponds to an AR(1) random processus, autocorrelated ( A=0.8), and the
production delay is a constant number (T = 1).
* An analysis model from these factors goes beyond the scope of this article. A flexibility approach to
this problem has been undertaken in Pereira (1997).
Acknowledgments
I wish to acknowledge Martin Schaffernicht, at the Universidad de Talca, for his critical comments on
several aspects of this article.
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