Modeling a Fuzzy Resource Allocation Mechanism
based on Workflow Nets
Joslaine Cristina Jeske de Freitas
1
, St
´
ephane Julia
1
and Leiliane Pereira de Rezende
1,2
1
Computing Faculty, Federal University of Uberl
ˆ
andia - UFU, Uberl
ˆ
andia - MG, Brazil
2
DMIA Department, Institut Sup
´
erieur de l’A
´
eronautique et de l’Espace - ISAE, Toulouse, France
Keywords:
Petri Net, Workflow Net, Resource Allocation, Possibility Theory, CPN Tools.
Abstract:
Colored Petri Nets (CPN) arose from the need to model very large and complex systems, which are found
in real industrial applications. The idea behind CPN is to unite the ability to represent synchronization and
competition for resources of Petri nets with the expressive power of programming languages, data types and
diverse abstraction levels. This work proposes the use of Hierarchical Colored Petri Nets and CPN Tools
to model a Workflow net with fuzzy sets delimited by possibility distributions associated with the Petri net
models that represent human type resource allocation mechanisms. Additionally, the duration of activities that
appear on the routes (control structure) of the Workflow process, will be represented by fuzzy time intervals.
Besides, firing rules based on a joint possibility distribution will be used in order to express in a more realistic
way the resource allocation mechanisms when human behavior is considered in Workflow activities.
1 INTRODUCTION
The purpose of Workflow Management Systems is
to execute Workflow processes. Workflow processes
represent the sequence of activities that have to be ex-
ecuted within an organization to treat specific cases
and to reach a well-defined goal. Of all notations used
for the modeling of Workflow processes, Petri nets are
very suitable (Aalst and van Hee, 2004), as they rep-
resent basic routings of Business Processes. More-
over, Petri nets can be used for specifying the real
time characteristics of Workflow Management Sys-
tems (in the time Petri net case) as well as complex
resource allocation mechanisms. As a matter of fact,
late deliveries in an organization are generally due to
resources overload.
Many papers have already considered the Petri net
theory as an efficient tool for the modeling and anal-
ysis of Workflow Management Systems. In (Aalst
and van Hee, 2004), Workflow nets, which are acyclic
Petri net models used to represent the Workflow pro-
cess, are defined. Workflow nets have been iden-
tified and widely used as a solid model of Work-
flow processes by several authors (Aalst, 1997), (Ling
and Schmidt, 2000), (Kotb and Badreddin, 2005),
The third author received CAPES Scholarship - Proc. n
◦
.
99999.001925/2015-06.
(van Hee et al., 2006), (Martos-Salgado and Rosa-
Velardo, 2011), (Wang et al., 2009), (Wang and Li,
2013). In (Ling and Schmidt, 2000), an extension
of Workflow nets is presented. This model is called
time Workflow net and associates time intervals with
the transitions of the corresponding Petri net model.
In (Kotb and Badreddin, 2005), an extended Work-
flow Petri net model is defined. Such a model al-
lows for the treatment of critical resources which have
to be used for specific activities in real time. In
(Wang et al., 2009), a resource-oriented Workflow net
(ROWN) based on a two-transition task model was
introduced for resource-constrained Workflow mod-
eling and analysis. Considering the possibility of task
failure during execution, in (Wang and Li, 2013), a
three-transition task model for specifying a task start,
end and failure was proposed.
The majority of existing models put their focus
on the process aspect and do not consider important
characteristics of the Workflow Management System.
In (Aalst, 1997) and (van Hee et al., 2006) for ex-
ample, the resource allocation mechanisms are repre-
sented only in an informal way. In (Ling and Schmidt,
2000), (Kotb and Badreddin, 2005) and (Wang and
Li, 2013) resource allocation mechanisms are repre-
sented by simple tokens in places as is generally the
case in production systems (Lee and DiCesare, 1994).
But a simple token in a place will not represent in a
Freitas, J., Julia, S. and Rezende, L.
Modeling a Fuzzy Resource Allocation Mechanism based on Workflow Nets.
In Proceedings of the 18th International Conference on Enterprise Information Systems (ICEIS 2016) - Volume 2, pages 559-566
ISBN: 978-989-758-187-8
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
559

realistic way human employees who can treat simul-
taneously different cases in a single day, as is usually
the case in most Business processes.
Colored Petri Nets (CPN) can be used graphical
modeling language for managing business processes
(Serral et al., 2014). Business processes often ap-
pear in dynamic environments, for this reason, con-
text adaptation has recently emerged as a new chal-
lenge to explicitly address fitness between Business
process modeling and its execution environment. Ini-
tially, CPNs were supported by Design/CPN. Later,
Design/CPN was replaced by CPN Tools.
This work proposes the use of Hierarchical Col-
ored Petri Nets and CPN Tools to model a Workflow
net with fuzzy sets delimited by possibility distribu-
tions associated with the Petri net models that rep-
resent human type resource allocation mechanisms.
Additionally, the duration of activities that appear on
the routes (control structure) of the Workflow process,
will be represented by fuzzy time intervals. Besides,
firing rules based on a joint possibility distribution
will be used in order to express in a more realistic
way the resource allocation mechanisms when human
behavior is considered in Workflow activities.
The remainder of this paper is set out as follows.
Section 2 shows the concepts the Workflow net. Sec-
tion 3 presents concepts of CPN. Section 4 introduces
the concepts of fuzzy sets and possibility measures.
Section 5 presents resource allocation mechanisms.
Section 6 presents a fuzzy time constraint propaga-
tion mechanism. Section 7 defines firing rules that
consider fuzzy time constraints as well as fuzzy re-
source allocation mechanisms. Section 8 shows the
CPN Model for “Handle Complaint Process”. Finally,
section 9 concludes the paper and provides references
for additional works.
2 WORKFLOW NET
A Petri net that models a Workflow process is called a
Workflow net (Aalst and van Hee, 2004). A Workflow
net satisfies the following properties (Aalst, 1998):
• It has only one source place, named Start and only
one sink place, named End. These are special
places such that the place Start has only outgoing
arcs and the place End has only incoming arcs.
• A token in Start represents a case that needs to be
handled and a token in End represents a case that
has been handled.
• Every task t (transition) and condition p (place)
should be on a path from place Start to place End.
Figure 1: “Handle Complaint Process”.
Figure 1 illustrates the Workflow process which
takes care of the processing of claims related to car
damage (Aalst and van Hee, 2004). An incoming
complaint is first recorded. Then the client who has
complained along with the department affected by the
complaint are contacted. The client is approached for
more information. The department is informed of the
complaint and may be asked for its initial reaction.
These two tasks may be performed in parallel, i.e. si-
multaneously or in any order. After this, data is gath-
ered and a decision is made. Depending upon the de-
cision, either a compensation payment is made or a
letter is sent. Finally, the complaint is filed.
3 CONCEPTS OF CPN
CPN is a graphical modeling language (Jensen and
Kristensen, 2009), which combines the strengths of
Petri nets (Wolfang Reisig, 2013) and of functional
programming languages (Milner et al., 1997). The
formalism of Petri nets is well suited for describing
concurrent and synchronizing actions in distributed
systems. Programming languages can be used to de-
fine data types and to manipulate data. An introduc-
tion to the practical use of CPN can be found in (Kris-
tensen et al., 2004).
The CPN are designed to reduce the size of the
model, allowing individualization of tokens thigh the
use of colors assigned to them. Different processes
or resources can then be represented in the same net-
work. Colors do not mean just colors or patterns.
They can represent complex data types (Jensen and
Kristensen, 2009).
CPN models are formal, in the sense that the CPN
modeling language has a mathematical definition of
its syntax and semantics. This means that they can
ICEIS 2016 - 18th International Conference on Enterprise Information Systems
560

be used to verify system properties, i.e., prove that
certain desired properties are fulfilled or that certain
undesired properties are guaranteed to be absent.
Large and complex models can be built using hi-
erarchical CPN in which modules, which are called
pages in the CPN terminology, are related to each
other in a well-defined way. Without the hierarchi-
cal structuring mechanism, it would be difficult to
create understandable CPN models of real-world sys-
tems (Jensen and Kristensen, 2009).
4 FUZZY SETS AND
POSSIBILITY MEASURES
The notion of fuzzy set was introduced by (Zadeh,
1965) in order to represent the gradual nature of hu-
man knowledge. For example, the size of a man
could be considered by the majority of a population
as small, normal, tall, etc. A certain degree of belief
can be attached to each possible interpretation of sym-
bolic information and can simply be formalized by a
fuzzy set F of a reference set X that can be defined by
a membership function µ
F
(x) ∈ [0, 1]. In particular,
for a given element x ∈ X , µ
F
(x) = 0 denote that x is
not a member of the set F, µ
F
(x) = 1 denotes that x is
definitely a member of the set F, and intermediate val-
ues denote the fact that x is more or less an element of
F. Normally, a fuzzy set is represented by a trapezoid
A = [a1, a2, a3, a4] where the smallest subset corre-
sponding to the membership value equal to 1 is called
the core, and the largest subset corresponding to the
membership value greater than 0 is called the support.
A fuzzy set F can be delimited by a possibility dis-
tribution Π
f
, such as: ∀x ∈ X , Π
f
(x) = µ
F
(x) (Dubois
and Prade, 1988), (Cardoso et al., 1999). Given a pos-
sibility distribution Π
a
(x), the measure of possibility
Π(S) and necessity N(S) that a data a belongs to a
crisp set S of X is defined by Π(S) = sup
x∈S
Π
a
(x) and
N(S) = in f
x6∈S
(1 − Π
a
(x)) = 1 − Π(S). If Π(S) = 0,
it is impossible that a belongs to S. If Π(S) = 1, it is
possible that a belongs to S, but it also depends on the
value of N(S). If N(S) = 1, it is certain that (the larger
the value of N(S), the more the proposition is believed
in). In particular, there exists a duality relationship
between the modalities of the possible and the neces-
sary which postulates that an event is necessary when
its contrary is impossible. Some practical examples
of possibility and necessity measures are presented in
(Dubois and Prade, 1988).
Given two data a and b characterized by two fuzzy
sets A and B the measure of possibility and necessity
of having a ≤ b are defined as:
Π(a ≤ b) = sup
x≤y
(min(Π
a
(x), min(Π
b
(y))) =
max([A, +∞[∩] − ∞, B]) (1)
and
N(a ≤ b) = 1 − sup
x≤y
(min(Π
a
(x), min(Π
b
(y))).
(2)
Given a normalized possibility distribution π
a
,
(Dubois and Prade, 1989) defines the following fuzzy
sets of the time point that are:
• possibly after a: µ
[A,+∞[
(x) = sup
x∈X
π
a
(s)
• necessarily after a: µ
]A,+∞[
(x) = in f
x∈X
(1−π
a
(s))
• possibly before a: µ
]−∞,A]
(x) = sup
x∈X
π
a
(s)
• necessarily before a: µ
]−∞,A[
(x) = in f
x∈X
(1 −
π
a
(s))
A visibility time interval [a, b] is a period of time be-
tween two dates a and b. In the case where a and
b are fuzzy dates A and B (delimited by π
a
and π
b
)
respectively, the interval [a, b] is represented by the
following pair of fuzzy sets:
• [A,B], the conjunctive set of time instants that rep-
resents the set of dates possibly after A and possi-
bly before B;
• ]A,B[, the conjunctive set of time instants that rep-
resents the set of dates necessarily after A and nec-
essarily before B.
The joint possibility admits as upper bound in
(Dubois and Prade, 1988):
∀x ∈ X ∀y ∈ Y π(x, y) = min(π
X
(x), π
Y
(y)) (3)
when the reference sets are non-interactive (the value
of x in X has no influence on the value of y in Y , and
vice versa).
5 RESOURCE ALLOCATION
MECHANISM
Resources in Workflow Management Systems are
non-preemptive (Aalst and van Hee, 2004): once a
resource has been allocated to a specific activity, it
cannot be free before ending the corresponding activ-
ity. As already mentioned, there exists different kinds
of resources in Workflow processes. Some of which
are of the discrete type and can be represented by a
simple token. For example, a printer used to treat
a specific class of documents will be represented as
a non-preemptive resource and could be allocated to
a single document at a same time. On the contrary,
some other resources cannot be represented by sim-
ple tokens. This is generally the case with human
Modeling a Fuzzy Resource Allocation Mechanism based on Workflow Nets
561

type resources. As a matter of fact, it is not unusual
for an employee who works in an administration to
treat several cases simultaneously. For example, in an
insurance company, a single employee can normally
treat several documents during a working day and not
necessarily in a pure sequential order. In this case,
a simple token could not model human behavior in
a proper manner. Fuzzy allocation mechanisms were
presented in (Jeske et al., 2009).
Figure 2: Fuzzy Continuous Resource.
An example of fuzzy continuous resource is given
in Figure 2. For example, this Figure shows that
30% ± 10% of the resource availability R2 is neces-
sary to realize the activity A3 (Contact-Client).
The behavior of a fuzzy continuous resource allo-
cation model can be defined through the concepts of
“enabled transition” and “fundamental equation”.
In an ordinary Petri net, a transition t is enabled
if and only if for all the input places p of the transi-
tion, M(p) ≥ Pre(p, t), which means that the number
of tokens in each input place is greater or equal to the
weight associated to the arcs which connect the input
places to the transition t. With a fuzzy continuous re-
source allocation mechanism, considering a transition
t, the marking of an input place p and the weights as-
sociated to the arc which connects this place to the
transition t are defined through different fuzzy sets.
In this case, a transition t is enabled if and only if (for
all the input places of the transition t):
Π
t
= Π(Pre
FCR
(p, t) ≤ M
FCR
(p)) > 0. (4)
For an ordinary Petri net, once a transition is enabled
by a marking M, it can be fired and a new marking M
0
is obtained according to the fundamental equation:
M
0
(p) = M(p) − Pre(p, t) + Pos(p,t). (5)
With a fuzzy continuous resource allocation model,
the marking evolution is defined through the follow-
ing fundamental equation:
M
0
FCR
(p) = M
FCR
(p)  Pre
FCR
(p, t)  Pos
FCR
(p, t)
(6)
The operation “” corresponds to the fuzzy subtrac-
tion. The operation “”, when considering the sum
of two fuzzy sets, is different from the one given in
fuzzy logic and is defined as:
[a1, a2, a3, a4]  [b1, b2, b3, b4] =
[a1 + b1, a2 + b2, a3 + b3, a4 + b4].
This difference is due to the fact that the fuzzy
operation “⊕” does not maintain the marking of the
fuzzy continuous resource allocation model invariant
(the p-invariant property of the Petri net theory (Mu-
rata, 1989)). As a matter of fact, after realizing differ-
ent activities, the resource’s availability must go back
to 100 %, even in the fuzzy case. To a certain extent,
from the point of view of fuzzy continuous resource
allocation mechanisms, the operation “” can be seen
as a kind of defuzzyfication operation.
6 FUZZY TIME CONSTRAINT
PROPAGATION MECHANISM
As the actual time required by an activity in a Work-
flow Management System is non-deterministic and
not easily predicted, a fuzzy time interval can be as-
signed to every Workflow activity.
The static definition of a fuzzy time Workflow
net is based on fuzzy static intervals [a1, a2, a3, a4]s
which represent the permanency duration (sojourn
time) of a token in places. Before duration a1 the
token is in the non-available state. After a1 and be-
fore a4, the token is in the available state for the fir-
ing of a transition. After a4, the token is again in the
non-available state and cannot enable any transition:
it therefore becomes a dead token. In a real time sys-
tem case, the “death” of a token has to be seen as a
time constraint that is not respected. A transition can-
not be fired with dead tokens as this would correspond
to an illegal action or behavior: a constraint violation.
The dynamic evolution depends on the time situation
of the tokens (date intervals associated with the to-
kens).
In a Workflow Management System, a visibility
interval depends on a global clock associated to the
entire net which calculates the passage of time from
date = 0, which corresponds to the start of the sys-
tems operation. In particular, the existing waiting
times between sequential activities can be represented
by visibility intervals whose minimum and maximum
fuzzy boundaries will depend on the earliest and lat-
est delivery dates of the considered case. Through
correct knowledge of the beginning date of the pro-
cess and the maximum duration of a case, it is pos-
sible to calculate estimated visibility intervals associ-
ated with each token in each waiting place using con-
straint propagation techniques very similar to the ones
used in scheduling problems based on activity-on-arc
graphs without circuits (Gondran et al., 1984).
Figure 3 shows the fuzzy static intervals (intervals
of fuzzy durations) associated to the activity places
ICEIS 2016 - 18th International Conference on Enterprise Information Systems
562

Figure 3: Visibility intervals of the “Handle Complaint Pro-
cess”.
of the process and the fuzzy visibility intervals (inter-
vals of fuzzy dates) associated with the waiting places
(condition places of the Workflow net). It is important
to note that there is no time restriction on resources -
static interval defined for each resource is [0, ∞[s. The
minimal fuzzy bounds of the estimated visibility in-
tervals attached to the waiting places are calculated
applying a forward constraint propagation technique
applied to the different kinds of routings associated
with the “Handle Complaint Process”, and the max-
imum fuzzy bounds of the estimated visibility inter-
vals are calculated by applying a backward constraint
propagation techniques to the different kinds of rout-
ings considering the latest delivery dates of the case.
If the token appears in a place p at date δ and if its
visibility interval is given by [a1, a2, a3, a4], then this
token could be used for the firing of a transition at the
earliest date a1 and at the latest date a4. The global
state of the Workflow net will be then defined by the
current marking of the net and by the time measured
by the clock through the different visibility intervals.
When a transition t is fired at a date which belongs
to its enabling interval, a new marking will be calcu-
lated, the tokens which will not be used for the firing
of the transition will continue with their visibility in-
terval, and new estimated visibility intervals will be
associated to the tokens produced by the firing of t.
Figure 4: Possibilistic Distribution of W6.
For example, if a token is produced in place W 6
at date δ = 50, considering the possibilistic distribu-
tion shown in Figure 4, the firing possibility measure
of transition B-A5 will be equal to µ = 1 and the ac-
tivity associated to place A5 will be initiated normally
and its visibility interval will be [50, 50, 50, 50] (firing
of B-A5) ⊕ [10, 15, 25, 30]s (static interval associated
to A5) = [60, 65, 75, 80]v. If the token in place W 6 is
produced earlier at date δ = 40, for example, the firing
possibility of transition B-A5 will be µ = 0.5 (see Fig-
ure 4) and its visibility interval will be [40, 40, 40,40]
(firing of B-A5) ⊕ [10, 15, 25, 30]s (static interval as-
sociated to A5) = [50, 55, 65, 70]v. However the firing
could eventually be delayed until reaching a date cor-
responding to a possibility equal to µ = 1. Finally, if
the token in place W 6 is produced at date δ = 100,
the firing possibility of transition B-A5 will be equal
to δ = 0.5 (see Figure 4) but with a different meaning.
This situation will correspond to a case where some
of the previous activities on the process were delayed
and its visibility interval will be [100, 100, 100, 100]
(firing of B-A5) ⊕ [10, 15, 25, 30]s (static interval as-
sociated to A5) = [110, 115, 125, 130]v. It will be im-
portant then to immediately fire the transition B-A5
corresponding to the beginning of the next activity
and to inform the responsible resource for executing
this activity of the delay. Eventually, some of the
next activities of this process will be executed with
a high rank priority and the firing possibility of some
of the last transitions in the process will reach a pos-
sibility µ = 1 again, ensuring that the process dead-
line is respected. The complete definition of fuzzy
time constraint propagation mechanism can be found
in (Jeske de Freitas and Julia, 2015).
7 FIRING RULES WITH FUZZY
TIME AND FUZZY RESOURCE
If a transition has n input places and if each one of
these places has several tokens in it, then the enabling
time interval [a1, a2, a3, a4] of this transition is ob-
tained by choosing for each one of these n input places
a token, the visibility interval associated with it. In
this paper, there exists no time restriction on the re-
sources (the static interval) attached to the resource
Modeling a Fuzzy Resource Allocation Mechanism based on Workflow Nets
563

places is always [0, ∞[s and, as a consequence, the
enabling time interval of a transition will simply be
equal to the visibility interval associated with the case
to be treated by the corresponding transition. For ex-
ample, knowing that the visibility interval attached to
the case represented by a token in place W 1 is equal to
[0, 10, 40, 60]v, the enabling time interval of the tran-
sition B-A2 will be [0, 10, 40, 60]v too.
For firing a transition, it is necessary that the ar-
rival date of the token in the input place of the transi-
tion belongs to the fuzzy visibility interval associated
with the input place of the transition (µ > 0) and the
resource availability (equation (1)) necessary to real-
ize the activity initiated by the firing of the transition
must be greater than 0 (Π(a ≤ b) > 0). To evaluate
the availability of resource and time simultaneously,
the joint possibility presented in equation (3) must be
calculated, where π
X
(x) corresponds to the resource
availability and π
Y
(y) to the time location of the cor-
responding activity. A complete definition of firing
rules with fuzzy time and fuzzy resource can be found
in (Jeske de Freitas et al., 2015).
8 CPN MODEL FOR “HANDLE
COMPLAINT PROCESS”
The modeling approach is hierarchical, where each
transition can be replaced by a specific module. The
most abstract level of the modeling of the “Handle
Complaint Process” is depicted in Figure 5.
Figure 5: The most abstract level of “Handle Complaint
Process’.
Observing Figure 5, note that:
• A token in place Start enables the transition that
defines the A1 activity for firing.
• The colset PLACE is a set of colors that defines
the types associated with the places of the model.
In addition, there are two types of resources de-
fined in the model: RESOURCE that defines dis-
crete resource and RESOURCEF that defines the
fuzzy resources.
• Each of the transitions with double borders de-
nominated as “abstract transition” create a link
with one of the sub-networks (it corresponds to
the hierarchy concept) A1, A2, A3, A4, A5, A6, A7
and A8. As already mentioned, A2 and A3 are
parallel activities. At the end of these activities, a
synchronization must be made. In this case, it is
necessary to compare the end time of each activ-
ity and must use the higher value. In the proposed
model, this condition is defined with the subscrip-
tion expression “if xt1 > xt then xt1 else xt” in the
output arc.
Basically, each activity is defined as an input transi-
tion bounded by the guard condition that checks if
there is resource availability and if the available time
is within the visible interval. If the transition is fired
then Startproc function is called and the resource is
used. The SartProc function takes as input the current
value of time and the interval that defines the duration
of the activity, and returns the value of time to be up-
dated after the end of the activity. When the resource
is used, the sub function is called to calculate the new
value of resource availability. When the activity ends,
the value of the resource is returned by soma function.
Figure 6 shows the simulation of the following
scenario. An incoming complaint is first recorded
(Figure 6(a)). Then the client who has complained
along with the department affected by the complaint
are contacted. The client is approached for further in-
formation (Figure 6(b)). The department is informed
of the complaint and may be asked for its initial reac-
tion (Figure 6(c)). These two tasks may be performed
in parallel, i.e. simultaneously or in any order (Figure
6(d)). After this, data is gathered (Figure 6(e)) and a
decision is made. After a decision is reached, a com-
pensation payment is made (Figure 6(f)). Finally, the
complaint is filed (Figure 6(h)).
9 CONCLUSIONS
This article presented how to model fuzzy hybrid re-
sources in Workflow nets with fuzzy time intervals
associated to the activities using a CPN Tools. Be-
sides this, through the definition as well as use of a
joint possibility distribution, it was possible to define
a transition firing definition. This definition takes into
consideration the time constraints associated with the
cases of the process as well as the availability of the
resources used to execute the activities.
ICEIS 2016 - 18th International Conference on Enterprise Information Systems
564

(a) Activity A1 (b) Activity A2
(c) Activity A3 (d) Sync Activity
(e) Activity A4 (f) Activity A5
(g) Activity A7 (h) Activity A8
Figure 6: Simulation of “Handle Complaint Process”.
As a future work proposal, it will be interesting
to build an intelligent model based on event logs to
define the best option for firing a transition. For ex-
ample, the event log will show the possibilities for
each activity firing and may lead to a type of process
quality analysis: if the activities, most of the time, are
working with a possibility equal to 1, then the work
resulting from the process will be of good quality. On
the other hand, if a large number of the activities are
associated with possibilities near to 0, then the qual-
ity of the process will be of poor quality. In addition,
during the execution of process activities, the man-
agement of activities could suffer a certain influence
according to the semantics associated with a low fir-
ing possibility. Finally, in the case of transitions in
conflict, the information concerning the firing possi-
bility can be used to make a decision: for example
if the possibility is low because of delayed activities,
we will give priority to the transition in relation to an-
other that possesses a higher firing possibility.
ACKNOWLEDGMENT
The authors would like to thank FAPEMIG
(Fundac¸
˜
ao de Amparo a Pesquisa de Minas Gerais),
CAPES (Coordenac¸
˜
ao de Aperfeic¸oamento de
Pessoal de N
´
ıvel Superior) and CNPq (Conselho
Nacional de Pesquisa) for their financial support.
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