A new method for reaching equilibrium
points in Fuzzy Cognitive Maps
T. L. Kottas , Student Member, IEEE, Y. S. Boutalis, Member, IEEE, Goran Devedzic and Basil G. Mertzios,
Member, IEEE

Abstract A new proposition for computing equilibrium
values in FCMs is presented. The equilibrium values affect the
decisions to be taken and therefore are of great importance.
The proposed method takes into account the fact that in any
complex dynamic system represented by an FCM, there exist
activities that indirectly (through various paths) influence one
another and this influence may not be only positive or only
negative.. By getting the “dominant” influences between nodes
the proposed method suggest getting both positive and
negative “dominant” influences and calculating the
equilibrium values considering both of them. Applying this
approach to a socio-medical problem we observe that we
reach to different equilibrium states and consequently to
different decisions.
Index TermsFuzzy Cognitive Maps, Equilibrium points,
POOL2 algorithm
I. INTRODUCTION
The modern social-financial-political problems have been
studied thoroughly enough the last years from a good many
of scientists. A large number of different methods have
occasionally been used in order to work out this kind of
problems. The scientific community was placed under the
obligation of giving solutions to problems the settlement of
which seemed rather difficult the years before. Up-to-date
methods and ideas were adopted and as a result these
modern problems have been faced with satisfactory
precision.
The contemporary engineer estimated his potential and
knowledge and has grown to believe that paralleling the
social-financial-political problems with man’s way of
thinking and acting, would describe them more effectively.
However, just like any other new methods, this one too,
needs to be thoroughly studied, so as it approximates most
closely the underlying dynamics of any problem studied.
Particularly, the method in question, tallies every single
problem with a cluster of neurons and nodes, the reaction
of which during a shock, seem to be identical with human
neuron’s reactions. Neurons or nodes disturb one another’s
stability in a unique way [9]. Still, that does not mean that
the searcher is absolutely versed in the exact way of them
reacting. But there are so many things that are working
underground of the engineer’s eyes and that’s the problem.
Do we know if the solution we propose approaches the
Theodore L. Kottas and Yiannis S. Boutalis are with the Democritus
University of Thrace, Department of Electr. and Comp. Eng., 67100
Xanthi, Greece (e-mail: [email protected], [email protected])
Goran Devedzic is with University of Kragujevac, School of
Mechanical Engineering 34000 Kragujevac, Yugoslavia, (e-mail:
[email protected])
Basil G. Mertzios is with the Technological Institute of Thessaloniki,
Greece (e-mail: [email protected]).
factuality or it is just a simple advice to our superior?
Fuzzy Cognitive Maps (FCM) can model dynamical
complex systems that change with time following nonlinear
laws [1]. FCMs use a symbolic representation for the
description and modeling of the system. In order to
illustrate different aspects in the behavior of the system, a
fuzzy cognitive map is consisted of nodes with each node
representing a characteristic of the system. These nodes
interact with each other showing the dynamics of the
system in study. An FCM integrates the accumulated
experience and knowledge on the operation of the system,
as a result of the method by which it is constructed, i.e.,
using human experts who know the operation of system
and its behavior.
Fuzzy cognitive maps have already been used to model
behavioral systems in many different scientific areas. For
example, in political science [10], [17] fuzzy cognitive
maps were used to represent social scientific knowledge
and describe decision-making methods. Kosko enhanced
the power of cognitive maps considering fuzzy values for
their nodes and fuzzy degrees of interrelationships between
nodes [1], [2]. After this pioneering work, fuzzy cognitive
maps attracted the attention of scientists in many fields and
they have been used in a variety of different scientific
problems. Fuzzy cognitive maps have been used for
planning and making decisions in the field of international
relations and political developments [17] and to model the
behavior and reactions of virtual worlds. FCMs have been
proposed as a generic system for decision analysis [4], [5]
and as coordinator of distributed cooperative agents. Fuzzy
cognitive maps have been used to model and control a
dynamic plant, to represent failure models and effects
analysis for a system model, and to model the supervisor of
control systems [3]. It is obvious that there is high interest
in the use of FCMs in a wide range of different scientific
fields, but there still remains to see an extensive use of
FCMs on process and manufacturing problems, which are
nonlinear systems requiring such methods.
In this paper, the problem of driving FCMs to “proper”
equilibrium values is addressed. The equilibrium values
affect actually the decisions to be taken and therefore are of
great importance. A new approach is proposed, where the
equilibrium values are calculated using both negative and
positive “dominant” influences between nodes. The paper
is organized as follows. Section II gives a short description
of FCMs and their way of operation. Section III introduces
NPN logic, which is used to capture both positive and
negative influences between nodes. The Pool2 method
modified according to the proposed approach is presented
in section IV. Section V presents a socio-medical system,
where a comparative study of the proposed method versus
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.
the traditional approach in reaching equilibrium points is
made. Finally, in section VI some conclusions are drawn
and a brief interpretation of the results of the method is
given.
II. FUZZY COGNITIVE MAPS REPRESENTATION AND
DEVELOPMENT
Fuzzy cognitive maps approach is a hybrid modeling
methodology, exploiting characteristics of both fuzzy logic
and neural networks theories and it may play an important
role in the development of intelligent manufacturing
systems. The utilization of existing knowledge and
experience on the operation of complex systems is the core
of this modeling approach. Experts develop fuzzy cognitive
maps and they transform their knowledge in a dynamic
cognitive map [11], [12].
The graphical illustration of FCM is a signed directed
graph with feedback, consisting of nodes and weighted
interconnections. Nodes of the graph stand for the nodes
that are used to describe the behavior of the system and
they are connected by signed and weighted arcs
representing the causal relationships that exist among nodes
(Figure 1). Each node represents characteristic of the
system. In general it stands for states, variables, events,
actions, goals, values, trends of the system which is
modeled as an FCM [9]. Each node is characterized by a
number Ai ,which represents its value and it results from
the transformation of the real value of the system's variable,
for which this node stands, in the interval [0, 1]. It must be
mentioned that all the values in the graph are fuzzy, and so
weights of the interconnections belong to the interval
[-1, 1]. With the graphical representation of the behavioral
model of the system, it becomes clear which node of the
system influences other nodes and in which degree.
The most essential part in modeling a system using
FCMs, is the development of the fuzzy cognitive map
itself, the determination of the nodes that best describe the
system, the direction and the grade of causality between
nodes. The selection of the different factors of the system,
which must be presented in the map, will be the result of a
close-up on system's operation behavior as been acquired
by experts. Causality is another important part in the FCM
design, it indicates whether a change in one variable causes
change in another, and it must include the possible hidden
causality that it could exist between several nodes. The
most important element in describing the system is the
determination of which node influences which other and in
what degree. There are three possible types of causal
relationships among nodes that express the type of
influence from one node to the others. The weight of the
interconnection between node Ci and node C j denoted by
Wij , could be positive ( Wij > 0) for positive causality or
negative ( Wij < 0) for negative causality or there is no
relationship between node Ci , and node C j , thus Wij = 0.
The causal knowledge of the dynamic behavior of the
system is stored in the structure of the map and in the
interconnections that summarize the correlation between
cause and effect. The value of each node is influenced by
the values of the connected nodes with the corresponding
Figure 1: A simple fuzzy cognitive map
causal weights and by its previous value. So the value A j
for each node C j is calculated by the following rule [15],
[9]:


N
(1)
Ais  f 
Ais  1Wij  A sj  1 



 i  1, i  j

where Ais , is the value of node C j at step s, Ais  1 is
the value of node Ci , at step s-1, A sj  1 is the value of
node C j at step s-1, and Wij is the weight of the
interconnection between Ci and C j , and f is a threshold
function.
Threshold functions [17]:
1)
f = tanh(x)
converts the node in [-1 , 1]
2)
f 
1
by using c=1 we convert the node in
1  e cx
[0 , 1]. It also called sigmoid function.
The second threshold function is the most common
function which is used in FCM’s.
III. NPN LOGIC AND NPN RELATIONS
NPN logic and relations [4], [5] are defined in this
section. In two-valued logic, if a variable does not assume
the value one (true), it must assume the value zero (false).
In an NPN crisp logic [2], if a variable is not negative, it
may be either neutral or positive; and if a variable is not
positive it may be either negative or neutral. Therefore, we
have three singleton values -1 (negative), 1 (positive), 0
(neutral or unrelated); and three compound values (-1, 0)
(negative or neutral), (0, 1) (neutral or positive), and (-1, 1)
(negative or positive / negative, neutral, or positive). The
compound values are ordered pairs (by  ) of singleton
values. Thus we have the six-valued NPN crisp logic as in
Table I.
Just as fuzzy logic extends two-valued crisp logic by
allowing logic values in the interval [0,1], NPN crisp logic
can be extended to NPN fuzzy logic by using logic values
in the interval [-1,1]. In NPN fuzzy logic, if a variable is
not positive, it may assume any real value in the interval
[- 1.0]; if it is not negative, it may assume any real value m
the interval [0,1]; and if it can be both negative or positive,
an ordered NPN value pair ( x , y ) in [-1,1] indicates a
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.
negative strength and a positive strength (or a lower
boundary and an upper boundary). While the crisp value
pair (-1,1) carries little or no information, an NPN fuzzy
value pair ( x , y ) may carry substantial information. This
structure plays an important role in approximate reasoning.
For instance, if we know that there are both positive and
negative effects of an action, we may know about
the
weights of the effects. If we are told that the
negative and positive effects are weighted as an NPN
fuzzy value pair (with strengths) (-0.1, 0.9), we know that
the positive effect is much larger than the negative one, so a
decision can be reached via thresholding.
Since each singleton value x can also be represented as a
pair (x, y) any NPN logic value can be represented as an
ordered pair. Thus the ordered pairs (by  ) in
[-1, 1]  [- 1,1] form a complete representation space for
all NPN logic values. Using this uniform representation, the
NEG, AND, and OR functions for both NPN crisp and
fuzzy logics can be compactly described by the following
three logic equations [16]:
NEG(x,y) = (NEG(y) , (NEG(x))
( x , y )*( u , v ) = ( min( x*u , x*v , y*u , y*v ) ),
max( x*u , x*v , y*u , y*v ) )
( x , y ) OR ( u , v )= ( min ( x, u ), max ( y, v ) )
IV. THE NEW METHOD FOR REACHING EQUILIBRIUM
STATES IN FCMS
Based on the NPN models developed in Section III, we
will now analyze Pool2 [4], a generic system for cognitive
map development and decision analysis and we will give
our proposition for the way in which the FCM could reach
better equilibrium points. The general architecture and
information flow of Pool2 is depicted in Fig. 2. The system
consists of three subsystems for cognitive mapping,
cognitive map understanding, and decision analysis,
0
TABLE I
NPN CRISP LOGIC TRUTH TABLES
1
-1
(-1,0)
(0,1)
A)
COM
and
COM
(-1,1)
(-1,0)
(0,1)
1
-1
NEG
0
-1
1
(0,1)
(-1,0)
0
0
0
0
0
1
0
1
-1
(-1,0)
(0,1)
(-1,1)
(-1,1)
NEG
0
(-1,1)
B)AND
0
0
-1
0
-1
1
(0,1)
(-1,0)
(-1,1)
(-1,0)
0
(-1,0)
(0,1)
(0,1)
(-1,0)
(-1,1)
(0,1)
0
(0,1)
(-1,0)
(-1,0)
(0,1)
(-1,1)
(-1,1)
0
(-1,1)
(-1,1)
(-1,1)
(-1,1)
(-1,1)
0
0
(0,1)
(-1,0)
(-1,0)
(0,1)
(-1,1)
C) OR
1
(0,1)
1
(-1,1)
(-1,1)
(0,1)
(-1,1)
-1
(-1,0)
(-1,1)
-1
(-1,0)
(-1,1)
(-1,1)
(-1,0)
(-1,0)
(-1,1)
(-1,0)
(-1,0)
(-1,1)
(-1,1)
(0,1)
(0,1)
(0,1)
(-1,1)
(-1,1)
(0,1)
(-1,1)
(-1,1)
(-1,1)
(-1,1)
(-1,1)
(-1,1)
(-1,1)
(-1,1)
respectively, which draw a parallel of knowledge
"pooling," "clearing," and "drawing."
A. Cognitive Mapping
The cognitive mapping subsystem works interactively
with expert(s) to gather knowledge about a world with a
combination of NPN relationships. The knowledge is
pooled together by merging assertions and cognitive maps
from individual experts to form a combined cognitive map,
which is called a primary cognitive map (PCM). Thus
cognitive mapping is used as a basic means for knowledge
pooling.
The importance of knowledge pooling lies in the fact that
no one has perfect and complete knowledge about a large
and complicated world. Expertise from a single expert is
usually limited in both quality and scope. If a body of
partial knowledge could be pooled together from multiple
experts or "semi-experts," knowledge acquisition, the
longstanding bottleneck in knowledge engineering would
be resolved. To make such an approach, CM is a good
representation.
In order to compute conventional PCMs the following
methodology is used. Positive and negative assertions from
all experts on the strength of a relationship are weighted
and summed together. On the contrary, in Pool2, both
negative and positive assertions are weighted and kept
separately to form an NPN compound value. It is obvious
that both positive and negative effects are important in
decision analysis, and that’s why they should not be
summed together if they are not counteractive to each other
(at the same time) or if they are not caused by the same
path. In case there are k experts with the same credibility,
m of them are experts on negative effects who assert that
is negative with strength , respectively, and k-m of them are
experts on positive effects who assert that is positive with
strength respectively, a compound NPN value (0, b) is
computed by the CM mapping function in Pool2, where
a= (Σu)/m and b= (Σv)/ (k-m). It is assumed without loss
that all relational strengths are normalized to the interval
[-1, 1].
B. Cognitive Map Understanding
The cognitive map understanding subsystem is used as the
second phase in Pool2. As a PCM from the cognitive
mapping subsystem is constructed from multiple experts
with partial knowledge, such a CM is considered as raw
knowledge in which implications and inconsistencies must
be clarified. At the end of this phase a new CM is
developed by understanding the PCM. It is called
‘advanced cognitive map’ (ACM). Thus if the word
"muddy" was used to describe a PCM, the word "clear"
might be used for an ACM.
Currently, the CM understanding subsystem consists of
the HTC (Heuristic Transitive Closure) algorithm. The
HTC algorithm computes the heuristic transitive closures of
an NPN relation. The HTCs of NPN relations are actually
the interconnection nodes paths, which could cause the
maximum negative and positive influence of one node to
the other, among all the existing interconnection paths.
The basic steps of the HTC algorithm are tabulated in
pseudo code format as follows:
HTC Algorithm:
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.
1) Given an n x n NPN relation matrix M in X x X
convert M to Mnew by representing each element as a pair
with a lower boundary and upper boundary ì (i , j) = (a,b);
2) for k=1:number of nodes
3)
for i=1: number of nodes
4)
ì (i,k)=(x,y)
5)
if (x,y)~=0
6)
for j=1:number of nodes
7)
ì (k,j)=(u,v)
8)
ì (i,j)=(min(a,x*u,x*v,y*u,y*v),
9)
max(b, x*u,x*v,y*u,y*v))
10)
end
11)
end
12)
end
13) end
C. The new method for calculating equilibrium
With the above way of thinking one converts the FCM to
a new FCM, which has the same number of nodes but
the relations between them are now changed so that they
represent only the maximum positive and negative effect
one node has to the other. So far, in Pool2 approach, the
final decision was taken considering only the positive or
negative interconnection based on its strength. If positive
interconnection was causing the maximum effect this was
the final interconnection for decision making else negative
interconnection was the final one. Maximum effect means
that the absolute value of negative interconnection is bigger
than the value of the positive one [4]. From the above
presentation, one can observe that, using Pool2, the final
decision making after the system reaches its equilibrium
point, reflects the “worst case” (maximum) influences the
nodes could have to the others. But, although this way of
decision making could result in “secure” decisions
regarding the limits of the system, in the authors opinion
does not represent the realistic behaviour of the actual
system. This way, based on the Pool2 approach, one could
take “secure” measures, which however, might not be
optimum regarding other factors. For example, if a
government has to take actions based on such a decision
these actions are probably to cost more that the actual
system would demand. The proposition of this paper is that
there is no reason to examine an FCM with only one
negative or positive simple effect. We have the
computational power to examine the way that FCM
corresponds to a double influence between nodes (both
negative and positive) and find out how the system now
reaches equilibrium points. In the next section an example
of a socio-medical problem is presented and tackled by
using conventional Pool2 approach and the proposed
approach. By comparing the results some useful
conclusions are drawn.
V. SPECIFIC LANGUAGE IMPAIRMENT SYSTEM
Figure 3 is showing the 18 nodes FCM representation of a
language impairement problem, as presented in [8]. The
weight table of the FCM is shown in Table II and the
start-up nodes values are shown in Table III.
Cognitive
mapping
(pooling)
FCM
CM
understanding
(clearing)
Assertions
from experts
ACM
Decision
analysis
(drawing)
Decision
support
Fig 2: General architecture and information flow of Pool2
A) we first calculate the reaching of equilibrium
points
by using the simple way that arises from (1). Writing this
formula in a more compact form one gets:
Anew  f ( A * W )
(2)
Applying (2) repetitively 10 times and using the sigmoid
function f, given in section II, one the results of Table IV
arrive.
B) by using Pool2 modified by the proposed approach of
section IV.C for reaching equilibrium points one gets a new
weight table Wnew which is shown in Table V. For
reaching equilibrium points we use the start-up nodes
prices in Table III and the modified recursive formula:
Where Wnew{1} refers to the negative weight values and
Wnew {2} to the positive ones.
For example in the:
row 1 and column 1 of Wnew : Wnew{1} =
Wnew {2} =
-1
1
row 4 and column 1 of Wnew : Wnew{1} = -0.9
Wnew {2} = 0.9
Applying (3) repetitively 10 times and using the sigmoid
function f, given in section II, one obtains the results of
Table VI.
Taking a close look one observes the following.
Calculating, for example, the influence of node “reading
difficulties” (9) to node “specific language impairment” (1)
we observe that the simple FCM gives us the equilibrium
value 0.58. We see however that, node 9 influences
substantially node1 via node 2 and then via node 3. This
means that the interconnection between 9-1 now become 92-3-1 and this because the last interconnection is bigger
than the direct connection. That means that the change of
node value is bigger and bigger change means better and
more securely results. Naturally the system Pool2 measures
the negative or the positive price interconnection between
two nodes. In the proposed system the nodes changes
derive from the negative as well as from the positive
interconnection, something that gives us more specious
results for the equilibrium points of the system. We see that
our system reaches a completely different balance, with the
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.
use of form (3) contrary to (2), for the nodes we interest
more. The price of the node dyslexia is 0.9789 for the
simple model while for the proposed model is 0.5, autism
node has the price 0.9984 contrary to the 0.5 and finally
specific language impairment node is 0.9972 instead of 0.5
of the proposed model.
VI. CONCLUSIONS
In this paper a new proposition for reaching equilibrium
points in FCMs is presented. So far, the existing
approaches are using the simple method of calculating
values of nodes taking into account only one “dominating”
influence (positive or negative) between nodes. The
diversity of the proposed method lies in the fact that in any
complex dynamic system represented by an FCM, there
exist activities that indirectly (through various paths)
influence one another and this influence may not be only
positive or only negative. By getting the “dominant”
influences between nodes the proposed method suggest
getting both positive and negative “dominant” influences
and calculating the equilibrium values considering both of
them. Applying this approach
Anew  f ( A *Wnew {1}  A *Wnew{2})
[12] Hagiwara M., “Extended Fuzzy Cognitive Maps”, IEEE , 1992
[13] Satur R., Liu Z., “A Contextual Cognitive Map Framework for
Geographic Information Systems”, IEEE transactions on fuzzy
systems, vol 7, no 5, pp-481-494, 1999
[14] Tsadiras A., Margaritis K., “An experimental study of the dynamics
of the certainty neuron fuzzy cognitive maps”, Neurocomputing ,
24 , pp-95-116, 1999
[15] Stylios C., Groumpos P., ”A soft Computing Approach for
Modelling the Supervisor of Manufacturing Systems”, Journal of
Intelligent and Robotic Systems, 26,pp-389-403, 1999
[16] Kundu S., “The min-max composition rule and its superiority over
the usual max-min composition rule”, Fuzzy Sets and Systems, 93,
pp-319-329, 1998
[17] Devedzic G.,”Fuzzy Cognitive Map-A Tutorial “.
[18] G. Eason, B. Noble, and I. N. Sneddon, "On certain integrals of
Lipschitz-Hankel type involving products of Bessel functions," Phil.
Trans. Roy. Soc. London, vol. A247, pp. 529-551, Apr. 1955.
(3)
to a socio-medical problem we observed that we reach to
different equilibrium states and consequently to different
decisions. In the authors’ opinion, the differences in
equilibrium values reflect the fact that the existing
approaches consider the “worst case” influences between
nodes reaching more “secure” but not so realistic
equilibrium states. On the other hand using the proposed
approach the equilibrium states (or decisions to be taken)
reflect more realistically the behaviour of the system but
they are not as “secure” as the traditional approach.
REFERENCES
[1]
Kosko
B.,
“Neural
Networks and
Fuzzy
Systems”,
Prentice-Hall, Englewood Cliffs, NJ, 1992
[2] Kosko B., “Fuzzy Engineering “, Prentice-Hall, NJ, 1997
[3] Stylios C., Groumpos P. ,“The challenge of modeling supervisory
systems using fuzzy cognitive maps”, Journal of Intelligent
Manufacturing, 9, pp-339-345, 1998
[4] Zhang W.,Chen S. Bezdek J. , “ Pool2: A Generic System for
Cognitive Map Development and Decision Analysis “, IEEE
Transactions on Systems,Man,and Cybernetics, vol 19,no 1, pp-3139, 1989
[5] Zhang W., Chen S., Wang W., King R., “A Cognitive Map Based
Approach to the Coordination of Distributed Cooperative Agents”,
IEEE Transactions on Systems, Man, and Cybernetics, vol 22, no 1,
pp-103-114, 1992
[6] Stylios C., Groumpos P., “Fuzzy Cognitive Maps: a model for
intelligent supervisory control systems”, Computers in Industry 39,
pp-229-238, 1999
[7] Tsadiras A., Margaritis K., “Cognitive Mapping and Certainty
Neuron Cognitive Maps”, Information Sciences, 101,
pp-109130, 1997
[8] Georgopoulos, Malandraki G.,Stylios C. , “A fuzzy cognitive map
approach to differential diagnosis language impairment”, Artificial
Intelligence in Medicine, 679, pp-1-18, 2002
[9] Jang J., Sun C., “Neuro-Fuzzy Modelling and Control” , Proceedings
of the IEEE, vol 83, no 3, pp-378-406, 1995
[10] Schneider M., Shnaider E., Kandel A., Chew G., “Automatic
Construction of FCMs”, Fuzzy Sets and Systems, 93, pp-161-172,
1998
[11] Miao Y., Liu Z., Siew C., Miao C., “Dynamical Cogntive Networkan Extension of Fuzzy Cognitive Map” , IEEE transactions on fuzzy
systems , vol 9 , no 5, pp-760-770, 2001
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.
Figure 3: Specific Language Impairment System
TABLE II
WEIGHT TABLE OF SPECIFIC LANGUAGE IMPAIRMENT SYSTEM
W
=
1
-1
-1
0.9
0.9
0.9
0.65
0.58
0.58
0.2
0.8
0.9
0.58
0.58
0.8
0.5
0.58
0.5
-1
1
-1
0.58
0.58
0.58
0.9
0
0.9
0
0.35
0.9
0.7
0.8
0
0.58
0.1
0
-1
-1
1
0.9
0.8
0.8
0.8
0.9
-0.65
0.9
0.9
0.58
0.65
0.9
0.9
0.9
0.9
0.9
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0.2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0.2
0.2
0.2
0.2
0
1
0
0
0
0
0
0.2
0.2
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0.2
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0.2
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0.2
0
0
0
0
0
0
0
0
0
0
0.2
0.2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
TABLE III
START-UP NODES PRICES FOR FCM IN TABLE II
A
=
0
0
0
0.58
0.8
0.8
0.8
0
0.9
0
0
0.65
0
0
0.5
0
0.5
0
TABLE IV
EQUILIBRIUM POINTS BY USING FORMULA (2)
_____________________________________________________________________________________________________________________________
0
0
0
0.5800 0.8000 0.8000 0.8000
0
0.9000
0
0
0.6500
0
0
0.5000
0
0.5000
0
0.9863 0.9686 0.9583 0.6411 0.7231 0.7231 0.6900 0.5000 0.8315 0.5000 0.5290 0.6570 0.5000 0.5290 0.6225 0.5000 0.6717 0.5000
0.9948 0.9710 0.9954 0.6550 0.7043 0.7043 0.6660 0.6225 0.8336 0.6225 0.6586 0.6586 0.6225 0.6586 0.6508 0.6225 0.7358 0.6225
0.9967 0.9770 0.9980 0.6581 0.6995 0.6995 0.6606 0.6508 0.8367 0.6508 0.6878 0.6589 0.6508 0.6878 0.6572 0.6508 0.7543 0.6508
0.9971 0.9784 0.9983 0.6588 0.6984 0.6984 0.6594 0.6572 0.8378 0.6572 0.6941 0.6590 0.6572 0.6941 0.6586 0.6572 0.7591 0.6572
0.9971 0.9788 0.9984 0.6590 0.6980 0.6980 0.6591 0.6586 0.8381 0.6586 0.6955 0.6590 0.6586 0.6955 0.6590 0.6586 0.7602 0.6586
0.9972 0.9788 0.9984 0.6590 0.6980 0.6980 0.6591 0.6590 0.8381 0.6590 0.6958 0.6590 0.6590 0.6958 0.6590 0.6590 0.7605 0.6590
0.9972 0.9789 0.9984 0.6590 0.6980 0.6980 0.6591 0.6590 0.8382 0.6590 0.6958 0.6590 0.6590 0.6958 0.6590 0.6590 0.7606 0.6590
0.9972 0.9789 0.9984 0.6590 0.6979 0.6979 0.6590 0.6590 0.8382 0.6590 0.6959 0.6590 0.6590 0.6959 0.6590 0.6590 0.7606 0.6590
0.9972 0.9789 0.9984 0.6590 0.6979 0.6979 0.6590 0.6590 0.8382 0.6590 0.6959 0.6590 0.6590 0.6959 0.6590 0.6590 0.7606 0.6590
_____________________________________________________________________________________________________________________________
TABLE VI
EQUILIBRIUM POINTS BY USING FORMULA (3)
_____________________________________________________________________________________________________________________________
0
0
0
0.5800 0.8000 0.8000 0.8000
0
0.9000
0
0
0.6500
0
0 0.5000
0
0.5000
0
0.5000 0.5000 0.5000 0.6411 0.7231 0.7231 0.6900 0.5000 0.8315 0.5000 0.5290 0.6570 0.5000 0.5290 0.6225 0.5000 0.6717 0.5000
0.5000 0.5000 0.5000 0.6550 0.7043 0.7043 0.6660 0.6225 0.8336 0.6225 0.6586 0.6586 0.6225 0.6586 0.6508 0.6225 0.7358 0.6225
0.5000 0.5000 0.5000 0.6581 0.6995 0.6995 0.6606 0.6508 0.8367 0.6508 0.6878 0.6589 0.6508 0.6878 0.6572 0.6508 0.7543 0.6508
0.5000 0.5000 0.5000 0.6588 0.6984 0.6984 0.6594 0.6572 0.8378 0.6572 0.6941 0.6590 0.6572 0.6941 0.6586 0.6572 0.7591 0.6572
0.5000 0.5000 0.5000 0.6590 0.6980 0.6980 0.6591 0.6586 0.8381 0.6586 0.6955 0.6590 0.6586 0.6955 0.6590 0.6586 0.7602 0.6586
0.5000 0.5000 0.5000 0.6590 0.6980 0.6980 0.6591 0.6590 0.8381 0.6590 0.6958 0.6590 0.6590 0.6958 0.6590 0.6590 0.7605 0.6590
0.5000 0.5000 0.5000 0.6590 0.6980 0.6980 0.6591 0.6590 0.8382 0.6590 0.6958 0.6590 0.6590 0.6958 0.6590 0.6590 0.7606 0.6590
0.5000 0.5000 0.5000 0.6590 0.6979 0.6979 0.6590 0.6590 0.8382 0.6590 0.6959 0.6590 0.6590 0.6959 0.6590 0.6590 0.7606 0.6590
0.5000 0.5000 0.5000 0.6590 0.6979 0.6979 0.6590 0.6590 0.8382 0.6590 0.6959 0.6590 0.6590 0.6959 0.6590 0.6590 0.7606 0.6590
_____________________________________________________________________________________________________________________________
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.
TABLEV
THENEWWEIGHTTABLE(W
new)byusingPool2
[-1 1] [-1 1] [-1 1] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-1 1] [-1, 1] [-1 1] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-1 1] [-1 1] [-1 1] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [01] [00]
[00]
[00]
[00] [00.2] [00] [00.2] [00]
[00] [00.2] [00]
[00] [00.2] [00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [01]
[00.2]
[00]
[00] [00.2] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00.2] [01]
[00]
[00] [00.2] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[01]
[00] [00.2] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[01]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00]
[01]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00]
[00]
[01]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00]
[00]
[00]
[01]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[01]
[00]
[00]
[00]
[00]
[00]
[00]
[-0.70.7] [-0.70.7] [-0.70.7] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[01]
[00]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[01]
[00]
[00]
[00]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00] [00.2] [00]
[00]
[00]
[00]
[00]
[01]
[00] [00.2] [00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00] [00.2] [00]
[00]
[00]
[00]
[00]
[00]
[01] [00.2] [00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[01]
[00]
[-0.90.9] [-0.90.9] [-0.90.9] [00] [00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[00]
[01]
Created with novaPDF Printer (www.novaPDF.com). Please register to remove this message.