A new hybrid method using evolutionary algorithms to train Fuzzy Cognitive Maps

www.elsevier.com/locate/asoc

Applied Soft Computing 5 (2005) 409–431

A new hybrid method using evolutionary algorithms

to train Fuzzy Cognitive Maps

Elpiniki I. Papageorgiou*, Peter P. Groumpos

Department of Electrical and Computer Engineering, Laboratory for Automation and Robotics,

Artificial Intelligence Research Center (UPAIRC), University of Patras, Rion 26500, Greece

Received 1 September 2003; received in revised form 2 August 2004; accepted 5 August 2004

Abstract

A novel hybrid method based on evolutionary computation techniques is presented in this paper for training Fuzzy Cognitive

Maps. Fuzzy Cognitive Maps is a soft computing technique for modeling complex systems, which combines the synergistic

theories of neural networks and fuzzy logic. The methodology of developing Fuzzy Cognitive Maps relies on human expert

experience and knowledge, but still exhibits weaknesses in utilization of learning methods and algorithmic background. For this

purpose, we investigate a coupling of differential evolution algorithm and unsupervised Hebbian learning algorithm, using both

the global search capabilities of Evolutionary strategies and the effectiveness of the nonlinear Hebbian learning rule. The use of

differential evolution algorithm is related to the concept of evolution of a number of individuals from generation to generation

and that of nonlinear Hebbian rule to the concept of adaptation to the environment by learning. The hybrid algorithm is

introduced, presented and applied successfully in real-world problems, from chemical industry and medicine. Experimental

results suggest that the hybrid strategy is capable to train FCM effectively leading the system to desired states and determining an

appropriate weight matrix for each specific problem.

# 2004 Elsevier B.V. All rights reserved.

Keywords: Fuzzy Cognitive Maps; Learning algorithms; Nonlinear Hebbian rule; Evolutionary computation; Differential evolution

algorithms; Evolutionary training

1. Introduction

Fuzzy Cognitive Maps (FCMs) were proposed by

Kosko to represent the causal relationship between

* Corresponding author. Tel.: +30 2610997293;

fax: +30 26120997309.

E-mail addresses: [email protected]

(E.I. Papageorgiou), [email protected] (P.P. Groumpos).

1568-4946/$ – see front matter # 2004 Elsevier B.V. All rights reserved

doi:10.1016/j.asoc.2004.08.008

concepts and analyze inference patterns [23,24].

FCMs represent knowledge in a symbolic manner

and relate states, processes, events, values and inputs

in an analogous manner. Compared either expert

system or neural networks, it has several desirable

properties, such as it is relatively easy to use for

representing structured knowledge, and the inference

can be computed by numeric matrix operation. FCMs

are appropriate to explicit the knowledge which has

.

E.I. Papageorgiou, P.P. Groumpos / Applied Soft Computing 5 (2005) 409–431410

been accumulated for years on the operation of a

complex system.

In recent years, there is an increase of published

journals and conference papers on FCMs. FCMs have

already been applied in many scientific areas, such as

medicine, manufacturing, organization behaviour,

political science, industry [3,10,22,27,30,34,35,39,

40,48,49,55].

This research work proposes a hybrid FCM

learning procedure based on the combination of an

unsupervised learning rule, the nonlinear Hebbian

learning (NHL) rule and an evolutionary computation

(EC)-based algorithm, the differential evolution, to

improve the FCM structure, to eliminate the defi-

ciencies in the usage of FCMs and to enhance the

dynamical behavior and flexibility of the FCM model.

Very few research efforts have been made till today to

propose an appropriate learning algorithm suitable for

FCMs [1,24,33,36–38].

Artificial neural networks and evolutionary compu-

tation establish two major research and application

areas in artificial intelligence. In analogy to biological

neural networks, artificial neural networks (ANNs) are

composed of simple processing elements that interact

using weighted interconnections and are of particular

interest because of their robustness, their parallelism,

and their learning abilities [14,15]. Evolutionary

computation is typically considered in the context of

evolutionary algorithms (EAs). Evolutionary algo-

rithms are a very rich class of multi-agent stochastic

search algorithms based on the neo-Darwinian para-

digm of natural evolution, which can perform

exhaustive searches in complex solution spaces. These

techniques start with searching a population of feasible

solutions generated stochastically. Then, stochastic

variations are incorporating into the parameters of the

population in order to evolve the solution to a global

optimum [12,18]. EAs establish a very general and

powerful search, optimization and learning method that

bases, in analogy to biological evolution, on the

application of evolutionary operators like mutation,

recombination and selection. Like no other computa-

tional method, EAs have been applied to a very broad

range of problems [2,4,28,43, 44,45,47].

Recently, the idea of combining ANNs and EAs has

received much attention [7,11,51], and now there is a

large body of literature on this subject. Weiss [52] in

his research work, provides a comprehensive and

compact overview of hybrid work done in artificial

intelligence, and shows the state of the art of

combining artificial neural networks and evolutionary

algorithms. There are two ways of synthesizing the

fields of ANNs and EAs; one way is to use EAs instead

of standard neural learning algorithms for training

ANNs; see e.g. [7], and the other way deals with the

approaches to an ‘‘evolution-based’’ design of appro-

priate structures of ANNs [17,42]. Furthermore, other

hybrid evolutionary approaches have been proposed

recently [25,29,52,53,54].

In this work, we use the differential evolution (DE)

algorithms, which can be easily implemented and they

are computationally inexpensive, since their memory

and CPU speed requirements are low [41]. Moreover,

they do not require gradient information of the

objective function under consideration, but only its

values, and they use only primitive mathematical

operators. DE algorithms can also handle nondiffer-

entiable, nonlinear and multimodal objective func-

tions efficiently, and require few easily chosen control

parameters. Experimental results have shown that DE

algorithms have good convergence properties and

outperform other evolutionary algorithms [46,47].

This paper proposes a new hybrid evolutionary

algorithm (HEA) for FCM learning. The HEA could

conceptually be split-up into two stages. In the first

stage, nonlinear Hebbian learning is adopted using a

recently proposed approach [36]. In the second stage,

a differential evolution (DE) algorithm, [47], is used

for FCM retraining. The usage of DE algorithm is

based on the assumption that the first stage has

produced a ‘‘good’’ solution that can be incorporated

directly into the genes and inherited by offspring. By

this manner, the initial expert knowledge (a priori),

incorporated in FCMs, can be extracted and incorpo-

rated in evolutionary computation, initializing the DE

population.

It is used in two termination conditions for the

nonlinear Hebbian learning algorithm on the first stage

of the proposed learning process and a fitness function

appropriate for the specific problem in the second

stage of the algorithm. If the optimization criterion

(minimization of fitness function) is reached in the

second stage, the algorithmic process has terminated.

This two-stage learning algorithm is used to train three

different FCM models with increasing complexity,

two FCM models for chemical process control

E.I. Papageorgiou, P.P. Groumpos / Applied Soft Computing 5 (2005) 409–431 411

Fig. 1. A simple fuzzy cognitive map model.

problems and one FCM model from medical area

describing the radiotherapy process, proving the

efficiency of the hybrid algorithm.

The rest of the paper is organized as follows: the

FCM modelling technique is described in Section 2.

The use of learning methods and evolutionary

algorithms in ANNs and FCMs learning is discussed

in Section 3. In Section 4, the new hybrid evolutionary

FCM learning algorithm is introduced. Two FCM

models describing industrial process control problems

and one FCM model describing the radiotherapy

process in medicine, on which the proposed algorithm

is tested, are described in Section 5, while the obtained

results are reported in Section 6. In Section 7,

conclusions and a short discussion of future work are

presented.

2. Overview of Fuzzy Cognitive Maps

Fuzzy Cognitive Maps have their roots in graph

theory. Euler formulated the first graph theory in 1736

[8]. Harary et al. in 1965 presented the theory of

directed graphs (digraphs) for studying structures of

empirical world [13]. Axelord in 1976 first used

signed digraphs to represent the assertions of

information [3]. He adopted the term ‘‘cognitive

map’’ for these graphed causal relationships among

variables as defined and described by people. The term

‘‘Fuzzy Cognitive Map’’ was coined by [23] in order

to describe a cognitive map model with two significant

characteristics.

� C
ausal relationships between nodes are fuzzified.
Instead of only using signs to indicate positive or

negative causality, a number is associated with the

relationship to express the degree of relationship

between two concepts.

� T
he system is dynamic involving feedback, where
the effect of change in the concept node affects

other nodes, which in turn can affect the node

initiating the change, the presence of feedback adds

a temporal aspect to the operation of the FCM.

Concepts reflect attributes, characteristics, qualities

and senses of the system. Interconnections among

concepts of FCM signify the cause and effect relati-

onship that a concept has on the others. These wei-

ghted interconnections represent the direction and

degree with which concepts influence the value of the

interconnected concepts. Fig. 1 illustrates a graphical

representation of Fuzzy Cognitive Maps.

The cause and effect interconnection between two

nodes Cj and Ci is described with the weight wji, with

wji taking value in the range �1 to 1.

There are three possible types of causal relation-

ships between concepts:

� w
ji > 0, which indicates positive causality between
concepts Cj and Ci. That is, the increase (decrease)

in the value of Cj leads to the increase (decrease) on

the value of Ci.

� w
ji < 0, which indicates negative causality between
concepts Cj and Ci. That is, the increase (decrease)

in the value of Cj leads to the decrease (increase) on

the value of Ci.

� w
ji = 0, which indicates no relationship between Cj
and Ci.

Human knowledge and experience on the system

determines the type and the number of nodes, as well

as the initial weights of the FCM. The value Ai of a

concept Ci, expresses the quantity of its correspond-

ing physical value and is derived by the transforma-

tion of the fuzzy values assigned by the experts, to

numerical values. Having assigned values to the c-

oncepts and the weights, the FCM converges to an

equilibrium point, through the interaction process

subsequently described.

At each step, the value Ai of a concept is calculated,

computing the influence of other concepts to the

specific concept, according to the following calcula-

tion rule:


Aðkþ1Þi ¼ f A

ðkÞi þ

XN

j 6¼ ij¼1

AðkÞj � wji

0BBB@

1CCCA (1)

where Aðkþ1Þi is the value of concept Ci at simulation

step k + 1, AðkÞj is the value of concept Cj at simulation

step k, wji is the weight of the interconnection between

concept Cj and concept Ci and f is the sigmoid thresh-

old function:

f ¼ 1

1 þ e�lx(2)

where l > 0, is a parameter that determines its steep-

ness. In our approach, the value l = 1 has been used.

This function is selected since the values Ai of the

concepts, must lie within [0, 1]. The interaction of the

FCM results after a few iterations in a steady state, i.e.

the values of the concepts are not modified further.

Desired values of the output concepts of the FCM

guarantee the proper operation of the simulated

system.

The methodology for developing FCMs is based on

a group of experts who are asked to define concepts

and describe relationships among concepts and use

IF–THEN rules to justify the cause and effect

relationship among concepts and infer a linguistic

weight for each interconnection [49,50]. Every expert

describes each one of the interconnection with a fuzzy

rule; the inference of the rule is a linguistic variable,

which describes the relationship between the two

concepts according to everyone expert and determines

the grade of causality between the two concepts. Then,

the inferred linguistic weights suggested by the group

of experts are composed and an overall linguistic

weight is produced through the SUM technique

[21,26]. Finally, the Center of Area (CoA) defuzzi-

fication method [21], is used for the transformation of

the linguistic weight to a numerical value of weight

wji, belonging to the interval [�1, 1] and representing

the overall suggestion of experts. This methodology

has the advantage that experts are not required to

assign directly numerical values to causality relation-

ships, but rather to describe qualitatively the degree of

causality among the concepts. Thus an initial weight

matrix Winitial =[wji], i,j = 1, . . ., N, with wii = 0, i = 1,

.., N, is obtained. Using the initial concept values, Ai,

the matrix Winitial is used for the determination of the

steady state of the FCM, through the application of Eq.

(1).

The most significant weaknesses of the FCMs are

their dependence on the experts’ opinion and the

potential uncontrollable convergence to undesired

states. The desired steady state is characterized by

values of the FCM output concepts accepted by the

experts, ex post. A novel learning procedure that

alleviates the problem of the potential convergence to

an undesired state for the output concepts values and

incorporating the knowledge in evolutionary com-

putation is proposed in this paper. This approach is

based on a hybrid method using differential evolution

algorithms and unsupervised learning algorithms,

and is described extensively in the Section 4. Also,

through the learning procedure, the optimum

available set of weights of each FCM model is

derived.

3. Learning algorithms in artificial neural

networks

Generally, in artificial neural networks (ANNs), the

weight–learning rule requires the definition and

calculation of an objective function (usually an error

function) examining when the objective function

reaches a minimum error that corresponds to a set of

weights of ANN. When the objective function is zero

or conveniently small; then, a steady state for the ANN

is reached; the weights of ANN that correspond to

steady-state define the learning process and the ANN

model [14,56].

The selection of the objective function and the

optimization method is crucial since it may promote

stability, instability, or a solution trapped in a local

minimum. Usually, during learning process, the

objective function decreases or increases, depending

on the specific assumptions of the problem, and the

weights are updated. The decrease may be accom-

plished with different optimization techniques, includ-

ing the Delta Rule, gradient-descent methods,

Boltzman’s algorithm, the back-propagation learning

algorithm, simulated annealing, and evolutionary

computation. Unsupervised Hebbian learning is

designed to maximize the variance of the output of

a given unit, and reinforcement learning is designed to

maximize the average reinforcement signal [14,26].


Many attempts have been made within the artificial

intelligence community to integrate ANNs and

evolutionary algorithms (EAs). A number of attempts

have concentrated on applying evolutionary principles

to improve the generalization of ANNs, discover the

appropriate network topology, and the best available

set of weights [7,11,52]. The majority of approaches in

which evolutionary principles are used in conjunction

with ANN training formulates the problem of finding

the weights of a fixed neural architecture, when the

whole set of examples is available, as an optimization

problem.

Evolutionary algorithms differ substantially from

more traditional search and optimization methods.

The most significant differences are: (a) EAs search a

population of points in parallel, not a single point, (b)

They do not require derivative information or other

auxiliary knowledge; only the objective function and

corresponding fitness levels influence the directions of

search, (c) they use probabilistic transition rules, not

deterministic ones, (d) they are generally more

straightforward to apply and (e) they can provide a

number of potential solutions to a given problem. The

final choice is left to the user [5,6].

Here, we use the differential evolution (DE)

algorithms, which have been designed as stochastic

parallel direct search methods that can handle

nondifferentiable, nonlinear and multi-modal objec-

tive functions efficiently and require few easily chosen

control parameters [46,47]. Storn and Price [47] has

reported impressive results that show DE outper-

formed other evolutionary methods and the stochastic

differential equations approach on solving some

benchmark continuous parameter optimization pro-

blems.

3.1. Description of the differential evolution

algorithm

The DE algorithm, which has been developed by

Storn and Price [47], utilizes N, n-dimensional

parameter weight vectors wi;G i = 1, . . ., N, as a

population for each iteration, called generation, of the

algorithm. The initial population is taken to be

uniformly distributed in the search space. At each

iteration, the mutation and crossover operators are

applied on the individuals, and a new population

arises. Then, the selection phase starts, where the N

best points from both populations are selected to

comprise the next generation.

According to the mutation operator, for each

weight vector, wi;G, i = 1, . . ., N, a mutant vector is

determined through the equation:

vi;Gþ1 ¼ wr1;G þ mðwr2;G � wr3;GÞ (3)

where r1, r2, r3 2 {1, . . ., N}, are mutually different

random indexes and also different from the current

index, i. m 2 (0, 2] is a real constant parameter that

affects the differential variation between two vectors,

and N must be greater than or equal to 4, in order to

apply mutation. Following the mutation phase, the

crossover operator is applied on the population, com-

bining the previously mutated vector,

vi;Gþ1 ¼ ½v1i;Gþ1; v2i;Gþ1; . . . ; vDi;Gþ1� (4)

with a so-called target vector,

wi;Gþ1 ¼ ½w1i;Gþ1;w2i;Gþ1; . . . ;wDi;Gþ1� (5)

Thus a so-called trial vector,

ui;Gþ1 ¼ ½u1i;Gþ1; u2i;Gþ1; . . . ; uDi;Gþ1� (6)

is generated, according to:

uji;Gþ1 ¼ vji;Gþ1; if ðrandbðjÞ � CRÞ or j ¼ rnbrðiÞ(7)

or

uji;Gþ1 ¼ wji;G; if ðrandbðjÞ>CRÞ or j 6¼ rnbrðjÞ (8)

where i = l, . . ., N, randb(j) 2 [0, 1] is the jth evalua-

tion of a uniform random number generator, for j 2 1,

2, . . ., D, and rnbr(i) 2 1, 2, . . ., d is a randomly

chosen index. CR 2 [0, 1] is the crossover constant

(user defined), a parameter that increases the diversity

of the individuals in the population. The three algo-

rithm parameters that steer the search of the algorithm

are the population size (N), the crossover constant

(CR) and the differential variation factor (m). They

remain constant during an optimization.

To decide whether or not the vector ui,G+1 should be

a member of the population comprising the next

generation, it is compared to the initial vector wi;G.

Thus,

wi;Gþ1 ¼ ui;Gþ1; f ðui;Gþ1Þ< f ðwi;GÞwi;G; otherwise

�(9)


The procedure described above is considered as the

standard variant of the DE algorithm. Different

mutation and crossover operators have been applied

with promising results [47]. In addition, DE algo-

rithms have a property that Price has called a universal

global mutation mechanism or globally correlated

mutation mechanism, which seems to be the main

property responsible for the appealing performance of

DE as global optimizers.

To apply DE algorithms to ANNs training someone

starts with a specific number (N) of n-dimensional

weight vectors, as an initial weight population, and

evolve them over time; N is fixed throughout the

training process, and the weight population is

initialized randomly following a uniform probability

distribution. At each iteration, called generation, new

weight vectors are generated by the combination of the

weight vectors randomly chosen from the population.

This operation is called mutation. The derived weight

vectors are then mixed with another predetermined

weight vector, the ‘‘target’’ vector, through the

crossover operation. This operation yields the so-

called trial vector. The trial vector is accepted for the

next generation if and only if it reduces the value of the

error function E. This last operation is called selection.

The above-mentioned operations introduce diversity

in the population and are used to help the algorithm

escape the local minima in the weight space. The

combined action of mutation and crossover is

responsible for much of the effectiveness of DE’s

search, and allows them to act as parallel, noise-

tolerant, hill-climbing algorithms, which efficiently

search the whole weight space.

3.2. Learning methods in Fuzzy Cognitive Maps

FCM learning involves updating the strengths of

causal links. Combining multiple FCMs is the

simplest form of learning. An alternative learning

strategy is to improve FCMs by fine-tuning its initial

causal link or edge strengths through training similar

to that in artificial neural networks. This way of

learning is more effective and more efficient to

accurately adjust the FCM cause–effect relationships.

Up-to-date, there are just a few FCM learning

algorithms [1,23,24,36–38] and they are mostly based

on ideas coming from the field of artificial neural

networks training. Kosko has developed the first

algorithm, named differential Hebbian learning, as a

form of unsupervised learning, but without mathe-

matical formulation and implementation in real

problems [23]. There is no guarantee that DHL will

encode the sequence into the FCM and till today no

concrete procedures exist for applying DHL in FCMs.

Another proposed algorithm is the Adaptive Random

for FCMs learning based on the theoretical aspects of

Random Neural Networks [1]. These algorithms start

from an initial state and an initial weight matrix of the

FCM and adapt the weights in order to compute a

weight matrix that leads the FCM to a desired state.

Also, recently, two unsupervised learning techni-

ques, the active Hebbian learning (AHL) and the

nonlinear Hebbian learning (NHL) algorithms, have

been proposed to fine-tune FCM causal links in order

to succeed desired values for output concepts and

acceptable operation for the system keeping its

constraints [36,37]. These algorithms were imple-

mented successfully in a practical control problem.

The AHL algorithm introduced a sequence of

activation concepts depending on the problem’s

configuration and characteristics [37]. Thus, activation

and activated concepts were bringing on. Experts

determined the mode of activation between concepts

(asynchronous or not) and the number and sequence of

activation concepts. When the AHL algorithm

terminates and the system converges in desired state

(means that the values of decision-output concepts are

accepted), a new updated weight matrix is derived,

determining new cause–effect relationships between

the concepts of FCM model, which are desired for

each case-study scenario.

Also, except the unsupervised learning-based

techniques for FCMs, methods based on evolutionary

computation techniques have been investigated.

Particle Swarm Optimization (PSO) method has been

used for first time and has been proposed for FCM

learning giving very promising results [38]. Using this

learning approach, a number of sub-optimum weight

matrices is derived leading the values of output

concepts within the desired regions and alleviating the

problem of the potential convergence to an undesired

steady state. More investigation is needed for this

method and generally for evolutionary computation

methods [38].

The present work focuses on the development of an

FCM learning procedure based on a hybrid evolu-


tionary algorithm. The purpose is to determine the

appropriate or optimum values of the cause effect

relationships among the concepts, i.e. the optimum set

of the weights of the FCM, that produce the desired

behavior of the system. The determination of the

appropriate set of the weights is of major significance

and it contributes towards the establishment of FCMs

as a robust methodology. The desired behavior of the

system is characterized by output concept values that

lie within desired bounds prespecified by the experts.

These bounds are in general problem-dependent.

4. The hybrid evolutionary algorithm

In this section, we present the hybrid method

combined of differential evolution algorithm and

nonlinear Hebbian learning (NHL) algorithm for

FCMs. The DE algorithm works on the termination

point of NHL algorithm. Thus the method consists of a

NHL-based FCM training stage and a differential

evolutionary strategy-based FCM retraining stage.

4.1. First stage: nonlinear Hebbian learning

algorithm for FCMs

The proposed learning procedure is based on the

nonlinear Hebbian-type learning rule for ANNs

learning [36]. This unsupervised learning rule have

been adapted and modified for the FCM case,

determining the nonlinear Hebbian learning (NHL)

algorithm.

This learning algorithm is based on the premise that

all the concepts in FCM model are triggering at each

iteration step and change their values. Let us assume

an FCM model consisting of three concepts. Fig. 2

illustrates the synchronously triggering of all concepts

Fig. 2. Interaction of all concepts for the same iteration step k.

of the FCM model for the iteration step k. During this

triggering process, the weights wji, wij, wli and wjl of

the causal interconnection of the related concepts are

updated and the modified weights wðkÞji w

ðkÞij w

ðkÞli , and

wðkÞjl are derived for iteration step k.

The value Aðkþ1Þi of concept Ci, at iteration step

k + 1, is calculated, computing the influence of

interconnected concepts with values Aj to the specific

concept Ci due to modified weights wðkþ1Þji at iteration

step k, through the following equation:

Aðkþ1Þi ¼ f A

ðkÞi þ

XN

j 6¼ ij¼1

AðkÞj w

ðkÞji

0BBB@

1CCCA (10)

Furthermore, experts have defined some concepts

of FCM as Desired Output Concepts (DOCs). These

concepts stand for the factors and characteristics of the

system that interest us, and we want to estimate their

values, which represent the final state of the system.

The distinction of FCM concepts as inputs or outputs

is determined by the group of experts for each specific

problem. Any of the concepts of the FCM model may

be inputs or outputs. However, experts select the

output concepts and they consider the rest as initial

stimulators or interior concepts of the system. The

learning algorithm that extracts hidden and valuable

knowledge of experts can increase the effectiveness of

FCMs and their implementation in real problems.

Taking the advantage of the general nonlinear

Hebbian-type learning rule for NNs, [16,31,32], we

introduce the mathematical formalism incorporating

this learning rule for FCMs and the determination of

input and output concepts. This algorithm relates the

values of concepts and values of weights in the FCM

model. The proposed learning rule has the general

mathematical form:

Dwji ¼ hAðk�1Þi ðAðk�1Þ

j � wðk�1Þji A

ðk�1Þi Þ (11)

where the coefficient h is a very small positive scalar

factor called learning rate parameter and is determined

using experimental trial and error method in order to

optimize the final solution. Aj is the value of concept

Cj.

Eq. (11) have been modified and adjusted to train

FCMs. The following form of the nonlinear weight–

learning rule for FCMs have been proposed [36]:


wðkÞji ¼ g w

ðk�1Þji þ hA

ðk�1Þj ðAðk�1Þ

j

� sgn ðwjiÞwðk�1Þji A

ðk�1Þi Þ (12)

where the h is the learning rate parameter and g is the

weight decay parameter. The term sgn (wji) is used to

maintain the sign of the corresponding weight (and

further to keep the physical meaning of relationships

for the examined problem), and the term

�sgn ðwjiÞwðk�1Þji ðAk�1

i Þ2 is used to avoid potential

undesired increase on the weight values from the

initial experts knowledge. Based on the previous

equation, the values of weights are modified at a small

amount comparing with the initial weights because the

second term of Eq. (12) is small in correlation to the

first term gwðk�1Þji .

Using the NHL algorithm only, the initially

nonzero weights suggested by experts are updated

for each iteration step. The value of each concept of

FCM is updated, through the Eq. (10) where the value

of weight wðkÞji is calculated using Eq. (12). Also, we

use two termination functions for the termination of

the proposed algorithm.

The first termination function F1 have been

proposed for the NHL algorithm for examining the

accepted values of outputs concepts, which are the

values of DOCs, we are interested about [36].

The termination function F1 have been defined as:

F1 ¼ jDOCi � Tij2 (13)

where Ti is the mean target value of the interesting

concept DOCi. This type of criterion function (which

has the classic form of Euclidean distance) is appro-

priate for the NHL rule for FCMs. The minimization

of the termination function F1 is the goal, updating the

weights and determining the learning process.

Let us assume that we are interested to calculate F1

of concept Ci with value DOCi. It is supposed that

DOCi take values in the range DOCi ¼ ½Tmini ; Tmax

i �.Then, the target value Ti of the interesting concept Ci is

determined as:

Ti ¼Tmin

i þ Tmaxi

2(14)

If we consider the case of an FCM-model, where

there are m-DOCs, then, for the calculation of F1, we

take the sum of the square differences between the m-

DOCs values with the m-Ts (mean values of DOCs),

and the Eq. (13) takes the following form:

F1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXm

i¼1

ðDOCi � TiÞ2

s(15)

The second termination condition is determined by

the variation of the subsequent values of DOCi, for

iteration step k, yielding a value e, which has to be as

less as possible, taking the form:

F2 ¼ jDOCðkþ1Þi � DOC

ðkÞi j< e ¼ 0:005 (16)

where DOCi is the value of ith concept.

Through this process and when the termination

conditions are met, the final weight matrix WNHL, is

derived.

In NHL algorithm, upper and lower bounds for the

learning parameters g and h have been determined

using trial and error experiments, because we

proposed small modifications on the values of weights

according to their initial values. So, constant values for

each one are calculated for specific case-study

problem. The values of parameters h and g, for each

specific case, ensure that the learning process

converges fast in a desired state.

It was observed after a large number of simulations

for different values for parameters h and g, that using

large values for parameter h, the weights change their

values at a significant amount causing also changes on

their signs. For same simulation using small constant

values for the parameter g, the weights continue to

change their values significantly, and the FCM

concepts cannot take values within a desired region.

The bounds of learning rate parameter h have been

determined as 0 < h < 0.1, and for the weight decay

parameter have been determined as 0.9 < g < 1, after

simulation results for each case problem [36].

A generic description of the proposed hybrid

algorithm is given in Algorithm 1.

Algorithm 1 (Generic Model of the Hybrid FCM

Learning Algorithm).

Stage 1: Nonlinear Hebbian learning

Step 1a:
Read input concept state A0 and
initial weight matrix W0.
Step 2a: Repeat for each iteration k.
Step 3a:
Calculate A(k) according to the Eq. (10).


Step 4a:
Update the weights:
wðkÞji ¼ g w

ðk�1Þji þ hA

ðk�1Þi ðAðk�1Þ

j

� sgn ðwðkþ1Þji A

ðk�1Þi Þ:

Step 5a:
Calculate the two termination functions.
Step 6a:
Until the termination conditions are met.
Step 7a:
Return the final weights Wðkþ1ÞNHL to the
Stage 2.

Stage 2: Differential evolution

Step 1b:
Initialize the DE population in
the neighbourhood of Wðkþ1ÞNHL and

within the suggested weight

constraints (ranges).

Step 2b:
Repeat for each input concept
state (k).

Step 3b:
For i = 1 to NP.
Step 4b:
MUTATION (wðkÞi )! Mutant_Vector.
Step 5b:
CROSSOVER (Mutant_Vector)!Trial_Vector.
Step 6b:
If F (Trial_Vector) � F(wðkÞi ),
accept Trial_Vector for

the next generation.

Step 7b:
End For.
Step 8b:
Until the termination condition is met.
First, the NHL algorithm is outlined in the Stage 1

of Algorithm 1, and provides convergence of

concepts’ values in a desired state. The key features

of the NHL method are the low storage requirements

and the inexpensive computations. In Stage 2 of

Algorithm 1, the differential evolution (DE) algo-

rithm, responsible for the FCM retraining is outlined.

4.2. Second stage: the differential evolution

algorithm

Applying DE algorithms for FCM learning, the

same process described for ANN training is used,

starting with a specific number (N) of n-dimensional

weight vectors, as initial population, and evolve them

over time; N is fixed throughout the training process,

and the weight population is initialised by perturbing

the appropriate solution provided by the NHL

algorithm. Also, the appropriate fitness function is

determined. In this case, the NHL algorithm seeds the

DE, i.e. a preliminary solution is available by the NHL

algorithm; so, the initial population might be

generated by adding normally distributed random

deviations to the nominal solution. However, in the

experiments reported in the next section, we have also

used the constraints on weights defined initially by

experts to perturb the approximated solution provided

by NHL algorithm.

Let us now give some details about the version of

DE algorithm used here. Steps 4b and 5b implement

the mutation and crossover operators, respectively,

while Step 6b is the selection operator.

For each weight vector wðkÞi the new vector called

mutant vector is generated according to the following

relation:

Mutant vector

¼ vðkþ1Þi ¼ w

ðkÞi þ mðwðkÞ

best � wðkÞi þ wr1 � wr2Þ;

i ¼ 1; . . .NP; (17)

where wðkÞbest is the best population member of the

previous iteration, m > 0 is a real parameter (mutation

constant) which regulates the contribution of the

difference between weight vectors, and wr1, wr2 are

weight vectors randomly chosen from the population

with r1, r2 2 {l, 2, . . ., i�l, i + 1, . . ., N}, i.e. r1, r2 are

random integers mutually different from the running

index i. Aiming at decreasing the diversity of the

weight vectors further, the crossover-type operation

yields the so-called trial vector, uðkþ1Þi , i = 1, . . ., N.

This operation works as follows: the mutant weight

vectors (vðkþ1Þi , i = 1, . . ., N) are mixed with the

‘‘target’’ vectors, wðkþ1Þi , i = l, . . ., N. Specifically,

we randomly choose a real number r in the interval

[0, 1] for each component j, j = 1, 2, . . ., n, of the

vðkþ1Þi . This number is compared with CR 2 [0, 1]

(crossover constant), and if r � CR; then, the jth

component of the trial vector uðkþ1Þi gets the value

of the jth component of the mutant vector vðkþ1Þi ;

otherwise, it gets the value of the jth component of

the target vector, wðkþ1Þi . The trial vector is accepted

for the next generation if and only if it reduces the

value of the following proposed fitness function F;

otherwise the old value, wðkÞi is retained. This last

operation is the selection, and due to the moving

‘‘optimum’’ nature of the differential evolution task,


it ensures that the fitness F starts steadily decreasing at

some iteration.

The purpose is to determine the values of the

weights of the FCM that produce a desired behaviour

of the system. The determination of the weights is of

major significance and it contributes towards the

establishment of FCMs as a robust methodology, and

improves the performance of FCMs.

4.2.1. Fitness function F

The desired behaviour of the system is character-

ized by output concept values that lie within desired

bounds prespecified by the experts. These bounds are

in general problem-dependent.

Let Ci, . . ., CN, be the concepts of an FCM, and let A1,

. . ., Am, 1 � m � N, be the values of output concepts,

while the remaining concepts are considered input, or

interior, concepts. The user is interested in restricting

the values of the output concepts in strict bounds:

Amini � Ai � Amax

i ; i ¼ 1; . . . ;m;

predetermined by the experts, which are crucial for the

proper operation of the modelled system. The main

goal is to detect a weight matrix, W = [wji], i, j = 1,

. . ., N, that leads the FCM to a desired state at which,

the output concepts lie in their corresponding bounds,

while the weights retain their physical meaning. The

latter is attained by imposing constraints on the values

of weights. To do this, we consider the following

fitness function:

FðWÞ ¼Xm

i¼1

½jAmini � Aij þ jAi � Amax

i j� (18)

where Ai, i = 1, . . ., m, are the calculated values of the

output concepts at equilibrium point, that are obtained

through the application of the procedure of Eq. (1),

using the weight matrix W. Obviously, the global

minimizer(s) of the fitness function F, is (are) weight

matrix (matrices) that lead the FCM to a desired state,

i.e. all output concepts are bounded within the desired

regions. The fitness function F suits straightforwardly

the problem, however, it is nondifferentiable, and thus,

gradient-based methods are not applicable for its

minimization. On the other hand, in the proposed

approach, DE algorithm is used for the minimization

of the fitness function defined by Eq. (18). The non-

differentiability of function F poses no problems on

our approach since DE algorithm, like all evolutionary

algorithms, requires function values solely, and can be

applied even on discontinuous functions. The weight

matrix W is presented by a vector, which consists of

the rows of W in turn, excluding the elements of the

main diagonal, which are by definition equal to zero.

Thus, an FCM model with N fully interconnected

concepts corresponds to a N(N � 1)-dimensional

minimization problem. If some interconnections do

not exist; then, their corresponding weights are zero

and they are omitted, reducing the dimensionality of

the problem.

The type of fitness function is dependent on

problem’s characteristics and can be different for

discipline problems in order to describe the desired

system operation. Each interconnection of an FCM

has a specific physical meaning, and thus, several

constraints can also be posed initially by the experts on

the values of the weights. Such constraints may

enhance the overall performance of the algorithm.

As soon as a weight configuration that globally

minimizes F is reached, the algorithm is terminated.

Generally, there is a plethora of weight matrices that

lead the values of FCM output concepts within the

desired regions for the proper operation of the system.

This is quite natural due to the fact that DE is a

stochastic algorithm.

Any information available a priori, may be

incorporated to enhance the procedure, either by

modifying the fitness function F, in order to exploit the

available information, or by imposing further con-

straints on the weights. The proposed approach has

proved to be very efficient in practice. In the following

section, its operation on three different problems is

illustrated.

5. Three case studies: problems with increasing

complexity levels

Three real-life problems with different complexity

levels are described in this section and are used for the

simulations.

5.1. First case study: a simple industrial process

control problem

A simple process control problem encountered in

chemical industry is selected to illustrate the workings


Fig. 3. The first process control problem.

of the proposed learning algorithm [48,49]. The

process control problem, illustrated in Fig. 3, consists

of one tank and three valves that influence the amount

of a liquid in the tank. Valves 1 and 2 pour two

different liquids into the tank. During the mixing of the

two liquids, a chemical reaction take place in the tank,

and a new liquid is produced. Valve 3 empties the tank

when the new liquid produced reaches a specific level.

A sensor is placed inside the tank to measure the

specific gravity of the produced liquid. When the

value, G, of the specific gravity is within [Gmin, Gmax],

the desired liquid has been produced. There is also a

limit on the height, T, of the liquid in the tank, i.e. it

cannot exceed a lower limit Tmin and an upper limit

Tmax. The control target is to keep the two variables, G

and T, within their bounds:

Gmin � G � Gmax; Tmin � T � Tmax

A group of experts constructs the FCM for the

simulation of the system, following the procedure

described in [48]. The FCM that models and controls

the specific system is depicted in Fig. 4. It consists of

the five concepts, which are defined as:

Fig. 4. The FCM model of the first process problem.

Concept 1: the amount of the liquid that Tank 1

contains is dependent on the operational state of

Valves 1, 2 and 3.

Concept 2: the state of Valve 1 (it may be closed, open

or partially opened).





Concept 5: the specific gravity of the liquid into the

tank.

There is a consensus among the concepts regarding

the direction of the arcs among the concepts. For each

weight, the overall linguistic variable and its corre-

sponding fuzzy set, are also determined by experts.

The ranges of the weights implied by the fuzzy lin-

guistic variables [50], are:

�0:50 � w12 � �0:3; �0:4 � w13 � �0:2;

0:2 � w15 � 0:5; 0:2 � w21 � 0:5;

0:3 � w31 � 0:5; �1:0 � w41 � �0:7

� 1:0<w52 � �0:5; 0:2 � w54 � 0:5: (19)

The initial weight matrix, derived through the CoA

defuzzification method, is:

Winitial1 ¼

0 �0:4 �0:25 0 0:30:36 0 0 0 0

0:45 0 0 0 0

�0:9 0 0 0 0

0 0:6 0 0:3 0

266664

377775

For this specific problem, experts define the DOCs

and their desired regions. The DOCs for this problem

are the concepts C1 and C5. The desired regions for the

two output concepts, which are crucial for the proper

operation of the modeled system, have been defined:

0:67 � C1 � 0:70; 0:73 � C2 � 0:76 (20)

The most important component in developing the

FCM is the determination of the concepts that best

describe the system and the direction and grade of

causality between concepts.

5.2. Second case study: a more complex industrial

process control problem

A second more complex process problem encoun-

tered in chemical industry is also selected. The process


Fig. 5. The second process control problem.

control problem is a part of a chemical plant that

consists of two tanks, three valves, one heating

element and two thermometers for each tank, as

depicted in Fig. 5.

Each tank has an inlet valve and an outlet valve.

The outlet valve of the first tank is the inlet valve of the

second tank. The objective of the control system is first

to keep the height of liquid, in both tanks, between

some limits, an upper limit Hmax and a low limit Hmin,

and second the temperature of the liquid in both tanks

must be kept between a maximum value Tmax and a

minimum value Tmin. The temperature of the liquid in

Tank 1 is regulated through a heating element. The

temperature of the liquid in Tank 2 is measured

through a sensor thermometer; when the temperature

of the liquid two decreases, Valve 2 needs opening; so,

hot liquid comes into Tank 2 from Tank 1. The control

objective is to keep values of these variables in the

following range of values:

H1min � H1 � H1

max; H2min � H2 � H2

max;

T1min � T1 � T1

max; T2min � T2 � T2

max

(21)

For this system, a FCM model has been constructed

in [49]. Variables and states of the system, such as the

height of the liquid in each tank or the temperature, are

the concepts of an FCM, which describes the system.

Then, concepts have been assigned for the system’s

elements that affect the variables, such as the state of

the valves. For this simple system, eight concepts have

been proposed, and they give a good model of the

system:

Concept 1: the height of the liquid in Tank 1. The

height of liquid is dependent on state of Valves 1 and 2.

Concept 2: the height of the liquid in Tank 2. The

height of liquid is dependent on state of Valves 2

and 3.

Concept 3: the state of the Valve 1. The valve is open,

closed or partially open.





Concept 6: the temperature of the liquid in Tank 1.

Concept 7: the temperature of the liquid in Tank 2.

Concept 8: describes the operation of the heating

element, which has different levels of operation and

which increases the temperature of the liquid in

Tank 1.

These concepts are connected with each other.

The signs and the weight of each connection have

been determined in [49]. For this procedure, the

human experience on the system’s operation is

used. Three experts constructed the FCM. The

experts, used the methodology described in [50],

determine the cause–effect relationship between

concepts. As an example, experts describe the

influence of Valve 1 (concept 3) on the amount of

liquid in Tank 1 (concept 1) using a set of fuzzy

rules from which it is inferred that there is positive

influence, transformed in weight 0.76. Each con-

nection between concepts has a weight, which

ranges between [�1, 1] and was determined by a

group of experts. Each concept of the FCM takes a

value, which ranges in the interval [0, 1], and it is

obtained after threshold the real measurement of

the variable or state, which each concept repre-

sent.


Fig. 6. The FCM model of the second process problem.

Fig. 6 depicts the Fuzzy Cognitive Map that models

this system. The values of the concepts correspond to

the real measurements of the physical process. The

DOCs for this problem, as have been suggested by

experts, are the concepts: C1 of ‘‘Tank 1’’, C2 of ‘‘Tank

2’’, C6 of ‘‘Temperature 1’’ and C7 of ‘‘Temperature

2’’.

Experts have also suggested linguistic variables for

the cause–effect relationships of the FCM model and

through the CoA deffuzification method the following

weight matrix was derived:

W initial2 ¼

0 0 0:21 0:38 0 0 0

0 0 0 0:70 0:60 0 0

0:76 0 0 0 0 0 0

�0:80 0:70 0 0 0 0 0:09

0 �0:42 0 0 0 0 0

0 0 0:40 0 0 0 0

0 0 0 0:30 0 0 0

0 0 0 0 0 0 0:40

266666666664

The ranges of the 13 weights implied by the fuzzy

linguistic variables are:

0:0 � w13 � 0:5; 0:0 � w14 � 0:5;

0:5 � w24 � 0:75; 0:25 � w25 � 0:75;

� 1:0 � w41 � �0:5; 0:0 � w42 � 0:75;

0:0 � w47 � 0:25; �0:75 � w52 � 0:0;

0:5 � w31 � 1:0; 0:0 � w63 � 0:5;

0:25 � w68 � 0:7; 0:0 � w74 � 0:5;

0:25 � w86 � 0:7: (23)

0

0

0

0

0

0:50

0

377777777775

(22)


For the FCM model of the process control problem,

all values of concepts are changed synchronously, at

the same iteration step. The desired regions for the

DOCs, which reflect the proper operation of the

modeled system, have been prespecified:

0:5 � C1 � 0:7; 0:75 � C2 � 0:8;

0:65 � C6 � 0:70; 0:65 � C7 � 0:70: (24)

5.3. Third case study: a generic model of medical

problem

Another FCM model of the medical area, the

Supervisor-FCM proposed in [39], which supervises

the radiotherapy process and estimates the acceptance

or no of the treatment therapy is also used to illustrate

the working of the proposed algorithm. The Super-

visor-FCM have been developed from expert’s

knowledge, and have been described in details in

[39]. The Supervisor-FCM consists of seven concepts

to supervise the decision making process during the

radiation therapy and 14 linkages among the concepts

(Fig. 7). The concepts of Supervisor-FCM that have

been suggested, are the following:

C1: tumor localization. It is dependent on patient

contour, sensitive critical organs and tumor volume. It

embodies the value and influence of these three

Factor-concepts that are concepts of first-level FCM.

C2: dose prescribed from treatment planning (TPD).

This concept describes the delivered doses to target

volume, normal tissues and critical organs that are

calculated at the treatment planning model of the first

level FCM.

C3: machine factors. This concept describes the

equipment characteristics.

C4: human factors. A general concept describing the

experience and knowledge of medical staff.

C5: patient positioning and immobilization. This

concept describes the cooperation of the patient with

the doctors and the potential of follow instructions.

C6: quality assurance (QA). Quality assurance

includes demands on staff, the therapeutic procedures

and the technical systems for complying with the

preset standards.

C7: final dose given to the target volume (FD). A

measurement of the radiation dose received by the

target tumor.

The methodology that was proposed in [50], was

used to develop the FCM. The experts were asked

to describe the relationships among concepts, and

they used IF–THEN rules to justify the effect

relationship among concepts and inferred a linguistic

weight for each interconnection [11]. The degree

of the influence was represented by a member of

the fuzzy set {positive very high, positive high,

positive medium, positive weak, zero, negative

weak, negative medium, negative low, negative very

low}.

The connections among the above-described

concepts of Supervisor-FCM have been determined

by experts and related protocols, and the following 14

linkages have been suggested.

Linkage 1: connects C1 with C6; it relates the tumour

localization with the delivered final dose.

Linkage 2: relates C2 with the C1; when the dose

derived from treatment planning is high, the value of

tumour localization increases at a small amount.

Linkage 3: connects C2 with C7; when the dose from

treatment planning is high, the final dose given to the

patient will be also high.

Linkage 4: relates C3 with C2; when the machine

parameters increase the dose from treatment planning

decreases.

Linkage 5: connects C3 with C7; any change to

machine parameters influences negatively the final

dose given to target volume decreases.

Linkage 6: relates C4 with C7; the human factors

causes decrease in final dose.

Linkage 7: connects C4 with C5; the presence of

human factors causes decrease in patient positioning.

Linkage 8: relates C5 with C4; any change on the

patient positioning influences negatively the factors

related to humans.

Linkage 9: connects C5 with C7; when the patient

positioning increases the final dose also increases.

Linkage 10: connects C6 with C2; any change on the

quality assurance (control) checks influence positively

the treatment planning.

Linkage 11: connects C6 with C7; any change on the

quality assurance (control) checks influence positively

the final dose.

Linkage 12: connects C7 with C5; when the final dose

reaches an upper value the patient positioning

influenced positively.


Linkage 13: connects C7 with C1; any change in final

dose causes change in tumour localization.

Linkage 14: connects C7 with C2; when the final dose

increases to an acceptable value, the dose from

treatment planning increases to a desired one.

After the determination of the linkages among c-

oncepts, three radiotherapists-doctors (experts) sug-

gested the fuzzy linguistic variables for the weights of

the linkages, and through the corresponding member-

ship functions [39], the bounds for these weights have

been defined:

0:3 � w17 � 0:5; 0:2 � w21 � 0:4

0:5 � w27 � 0:8; �0:5 � w32 � �0:2;

� 0:5 � w37 � 0; �0:5 � w45 � �0:2;

� 0:6 � w47 � �0:2; �0:6 � w54 � 0:0;

0:4 � w57 � 0:8; 0:5 � w62 � 0:75;

0:5 � w67 � 0:7; 0:0 � w71 � 0:4;

0:5 � w72 � 0:9; 0:5 � w75 � 0:9 (25)

These linguistic variables were defuzzified and

transformed in numerical weights using the construc-

tion methodology of FCMs. Thus, the following

weight matrix for the Supervisor-FCM was produced:

W initial3 ¼

0 0 0 0 0 0 0:50:4 0 0 0 0 0 0:60 �0:3 0 0 0 0 �0:25

0 0 0 0 �0:3 0 �0:30 0 0 �0:4 0 0 0:60 0:55 0 0 0 0 0:5

0:3 0:7 0 0 0:55 0 0

2666666664

3777777775

Also, the desired limits for the parameters TPD and

FD, which represent the Treatment Planning Dose and

the Final Dose, respectively, have been determined by

the three radiotherapists-doctors and the related ICRU

and IAEA protocols [9,19,20], and they are the control

objectives of the Supervisor-FCM. The desired limits

for these parameters are:

0:90 � FD � 0:98; (26)

0:85 � TPD � 0:95; (27)

The Supervisor-FCM evaluates the success or

failure of the treatment by monitoring the values of FD

and TPD concepts. Successful treatment corresponds

to values of the final dose and dose from treatment

planning that lie within the desired bounds. These

values identify the supervisor model [20,34]. The

Supervisor-FCM can be incorporated to an integrated

two-level hierarchical decision making system for the

description and determination of the specific treatment

outcome and for the scheduling of the treatment

process before its treatment execution [20]. Thus,

optimizing the Supervisor-FCM, i.e., detecting the

weights that correspond to the maximum values of the

concepts FD and TPD, within their prespecified

ranges, the result is an enhanced control system which

models the radiotherapy procedure more accurately

and makes decision-making more reliable.

The detection of the optimum FCM’s weights that

correspond to the maximum value of the parameters

FD and TPD, within prescribed ranges, is the

important task. The parameters FD and TPD are

determined as the desired output concepts. For this

purpose, the DE algorithm has been used for the

optimization of the Supervisor-FCM, through the

minimization of an objective function.

The fitness function for this problem, of each

individual weight matrix Wi, can be straightforwardly

defined as:

FðWiÞ ¼ �FDðW iÞ � TPDðWiÞ (28)

where FD(Wi) and TPD(Wi) are the values of the final

dose and the dose prescribed from the treatment

planning, respectively, that correspond to the weight

matrix Wi. The minus signs are used to transform the

maximization problem to its equivalent minimization

problem. Thus, the main optimization problem under

consideration is the minimization of the objective

function F(Wi), such that the constraints of Eqs.

(26) and (27) hold.

The weight matrix W can, in general, be

represented by a vector, which consists of the rows

of W in turn, excluding the elements of the main

diagonal, w11, . . ., wNN, which are by definition equal

to zero:

W¼½w12; . . . ;w1N ;w21; . . . ;w2N ; . . . ;wN1; . . . ;wN;N�1�:

In the Supervisor-FCM, the experts determined 14

linkages and thus, the corresponding minimization

problem is 14-dimensional. The fuzzy linguistic

variables, describing the cause–effect relationships


Fig. 7. Supervisor-Fuzzy Cognitive Map for decision-making process.

among the concepts, are taken into consideration in

the optimization process. These weight ranges, given

in Eq. (24), are used as constraints for the parameter

vector W, in the following experiments.

6. Simulation results

We have tested the proposed hybrid algorithm in

the previously described case studies. The experi-

ments concern the training of the FCM for each case

study: problem to determine the appropriate values of

cause–effect relationships among concepts (weights)

that drive the values of output concepts to be within the

desired bounds.

6.1. Simulation results for first problem

First, before the implementation and testing of the

proposed hybrid algorithm, we apply the typical Eq.

(1) to find the equilibrium state after the FCM

modeling interactions, in the first described simple

process control problem. The initial values of concepts

given in vector A01 = [0.4 0.7077 0.6120 0.7171 0.3],

represent the real data of the physical process (after

thresholding). These values and the initial weight

matrix W initial1 are used in Eq. (1) to calculate the

equilibrium of the process. After nine iteration steps,

the equilibrium state is reached and Table 1 gives the

subsequent values of calculated concepts. It is

observed that the values of concepts C1 and C5, in

final state, are out of the suggested desired regions in

Eq. (20). Then, we continue implementing the hybrid

learning technique. On the first stage of the learning

approach, the NHL algorithm is applied to train the

FCM and control the process. The training process

starts by applying the initial values of concepts A01 and

weights Winitial1 . For the specific control problem, the

suggested value of learning rate parameter h is 0.04,

for weight decay learning parameter g is 0.98, after

trial and error experiments, in order to fulfill the

requirements of the specific system and to converge

the values of DOCs in the desired region. If the

learning parameter h, takes different value than the

proposed one, for the case-study scenario; then, the

output concepts values will converge in undesired

regions.

The proposed NHL procedure continues, until the

two termination conditions are satisfied. The result of

training the FCM is a set of connection weights,

described in the weight matrix WNHL1 . The conver-

gence steady state is achieved after 17 iteration steps

and the updated weight matrix is the following:

WNHL1 ¼

0 �0:1736 �0:0265 0 0:479

0:5103 0 0 0 0

0:5753 0 0 0 0

�0:90 0 0 0 0

0 0:707 0 0:493 0

266664

377775


Table 1

The values of concepts at each step of FCM interaction

Steps Tank 1 Valve 1 Valve 2 Valve 3 Cauger

1 0.40 0.7077 0.612 0.717 0.30

2 0.5701 0.6743 0.6253 0.6921 0.6035

3 0.6157 0.6918 0.6184 0.7054 0.6845

4 0.6244 0.7019 0.6141 0.7132 0.7046

5 0.6252 0.7058 0.6125 0.7160 0.7093

6 0.6249 0.7071 0.6121 0.7168 0.7103

7 0.6248 0.7075 0.6120 0.7171 0.7105

8 0.6247 0.7076 0.6120 0.7171 0.7105

9 0.6247 0.7077 0.6120 0.7171 0.7105

which leads the FCM to the following desired state:

C1 ¼ 0:6830; C2 ¼ 0:7632; C3 ¼ 0:6538;

C4 ¼ 0:7534; C5 ¼ 0:7440

Some of the weights have changed significantly their

values from the initial ones in order to succeed the

desired output values of concepts. More specifically, the

weights w31, w52 and w47, have updated their values at a

significant amount to satisfy the constraints for DOCs.

These new values for weights describe new cause–

effect relationships between the concepts of FCM.

Now, the second stage of the Algorithm 1 is implied

and the FCM is retrained by the DE algorithm. The

preliminary solution available by the NHL seeds the

DE and the population is initialised in the neighbour-

hood of the WNHL1 and within the suggested weight

constraints.

Default fixed values of the mutation and crossover

constants (DE parameters) were used; values m = 0.5

and CR = 0.5 have been selected after a trial and error

process. The population size has been always equal to

50. For each algorithm variant, 100 independent

experiments have been performed, to enforce the

reliability of the results, and the algorithm was

allowed to perform 1000 iterations (generations) per

experiment. The best weight matrix detected in these

regions, in terms of its fitness function value F (i.e. the

matrix that corresponds to the smallest fitness function

value) is:

Woptim1 ¼

0 �0:35 �0:20 0 0:40

0:40 0 0 0 0

0:50 0 0 0 0

�0:80 0 0 0 0

0 0:75 0 0:20 0

266664

377775

which leads the FCM to the desired state:

C1 ¼ 0:6723; C2 ¼ 0:7417; C3 ¼ 0:6188;

C4 ¼ 0:6997; C5 ¼ 0:7311

that clearly satisfies the constraints for both C1 and C5,

defined in Eq. (20).

It is clear from the obtained results that there is a

significant divergence for some of the weights from

the initial weight values suggested by the experts.

Only, the weights w21, w31, and w13 keep almost the

same values with the initial ones. Thus the weight

adaptation method adjusts the FCM cause–effect

relationships and controls the values of DOCs to be

within the accepted regions.

Also, we have tested the FCM model for 1000 test

cases with random initial values of concepts, using the

weight matrix Woptim1 and we end up at the same result

for DOCs values (Fig. 8). So, the appropriate values of

weights describing the FCM model of the process

control example can be used for any initial set of

concepts values driving the system in desired region

for the output concepts values.

6.2. Simulation results for second problem

The hybrid learning approach is implemented in the

second process control problem consisting of eight

concepts and 13 linkages among concepts to

determine the appropriate values of cause–effect

relationships among concepts (FCM weights).

First, the NHL algorithm is applied to train the

FCM and modify the weights. The training process

starts by applying the initial values of concepts A02 =

[0.628 0.738 0.747 0.827 0.748 0.75 0.676 0.742] and

weights Winitial2 . Notably, only the initially nonzero

weights that connect the triggering concepts are

updated at each iteration step using the Eq. (12). All

other weights remain zero. The suggested value of

learning rate parameter h is 0.02 and the associate

value for weight decay learning parameter g is 0.98,

after trial and error experiments, in order to fulfill the

requirements of the system and to converge the values

of output concepts in the desired region.

The proposed algorithmic procedure continues,

until the two termination criteria are met. The result of

training the FCM is a set of connection weights wji that

minimize the first termination function and satisfy


Fig. 8. The variation of concepts for 10 cycles: convergence region.

synchronously the second termination condition. This

algorithm iteratively updates the connection weights

based on the Eq. (12), and Eq. (10) calculates the

output values of concepts based on the described

updating mode.

A desired state is achieved after 12 iteration steps,

and the updated weight matrix is the following:

WNHL2 ¼

0 0 0:201 0:35 0 0 0 0

0 0 0 0:637 0:49 0 0 0

0:619 0 0 0 0 0 0 0

�0:594 0:66 0 0 0 0 0:14 0

0 �0:364 0 0 0 0 0 0

0 0 0:363 0 0 0 0 0:44

0 0 0 0:27 0 0 0 0

0 0 0 0 0 0 0:525 0

266666666666664

377777777777775

which leads the FCM to the state:

C1 ¼ 0:5481; C2 ¼ 0:7353; C3 ¼ 0:7002;

C4 ¼ 0:8275; C5 ¼ 0:7527; C6 ¼ 0:6968;

C7 ¼ 0:6918; C8 ¼ 0:7463

Now, the second stage of the Algorithm 1 is implied

and the FCM is retrained by the DE algorithm. The

preliminary solution available by the improved NHL

seeds the DE and the population is initialised in the

neighbourhood of the WNHL and within the suggested

weight constraints.

The same values for the DE parameters have been

used. The population size has been always equal to 50.

For each algorithm variant, 100 independent experi-

ments have been performed, to enforce the reliability

of the results, and the algorithm was allowed to

perform 1000 iterations (generations) per experiment.


The appropriate weight matrix detected in these

regions, in terms of its fitness value F in Eq. (18)

and accomplishing the initial expert’s suggestions

is:

Wappropriate2 ¼

0 0 0:404 0:554 0 0 0 0

0 0 0 0:636 0:438 0 0 0

0:303 0 0 0 0 0 0 0

�0:554 0:553 0 0 0 0 0:11 0

0 �0:49 0 0 0 0 0 0

0 0 0:375 0 0 0 0 0:38

0 0 0 0:24 0 0 0 0

0 0 0 0 0 0 0:10 0

266666666666664

377777777777775

which leads the FCM to the desired state:

C1 ¼ 0:586; C2 ¼ 0:7026; C3 ¼ 0:7813;

C4 ¼ 0:8565; C5 ¼ 0:7224; C6 ¼ 0:6797;

C7 ¼ 0:6856; C8 ¼ 0:7284

that clearly satisfies the constraints for all four output

concepts, defined in Eq. (24).

It is clear from the obtained results that there are

div-ergences for some of the weights from the initial

weight values suggested by the experts. Only, the

weights w24, w47, and w63 keep near values with the

initial ones.

WNHL3 ¼

0 0 0 0 0 0 0:54

0:465 0 0 0 0 0 0:61

0 �0:100 0 0 0 0 �0:11

0 0 0 0 �0:105 0 �0:078

0 0 0 �0:23 0 0 0:611

0:409 0:681 0 0 0:54 0 0

2666666664

3777777775

(29)

The derived weight matrix Wappropriate2 represents

the appropriate cause–effect relationships among

concepts of the specific FCM model.

6.3. Simulation results for third problem

Before the implementation and testing of the

proposed hybrid algorithm, the initial values of

concepts given in vector A03 = [0.4 0.67 0.3 0.25

0.32 0.4 0.35] and the initial weight matrix Winitial3 are

used in Eq. (1) to calculate the .equilibrium state of the

process.

After 10 iteration steps, the equilibrium state is

reached and Fig. 9 gives the subsequent values of

calculated concepts. It is observed that the values of

concepts C2 and C7, n final state, are equal to the

values 0.8033 and 0.89, which are out of the suggested

desired regions in Eqs. (25) and (26). Then, we

continue implementing the hybrid learning technique.

The NHL is applied on the Supervisor-FCM using

the initial concepts vector A03 and weight matrix

Winitial3 . When the two termination conditions are met,

the process stops and the updated weight matrix is

derived:

Which leads the Supervisor-FCM to the following

state:

C1 ¼ 0:8278; C2 ¼ 0:8029; C3 ¼ 0:3;

C4 ¼ 0:6987; C5 ¼ 0:7642; C6 ¼ 0:4;

C7 ¼ 0:9033


Fig. 9. Equilibrium state for Supervisor-FCM model.

Then, the process continues implementing the DE

algorithm. The same default fixed values of the

mutation and crossover constants (DE parameters)

with the previous experiments were used. The

population size was the same (50) and 100 independent

experiments have been performed, to enforce the

reliability of the results. Each algorithm was allowed to

perform 1000 iterations (generations) per experiment.

After some number of generations the program

converges, while the best individual is considered as

representing a near-optimum (reasonable) solution.

Fig. 10. Equilibrium state of Supe

All experimental results have obtained the same

solution:

Woptimum3 ¼

0 0 0 0 0 0 0:5

0:4 0 0 0 0 0 0:8

0 �0:2 0 0 0 0 0

0 0 0 0 �0:2 0 �0:2

0 0 0 �0:6 0 0 0:8

0 0:75 0 0 0 0 0:7

0:4 0:9 0 0 0:9 0 0

2666666666664

3777777777775

rvisor-FCM using Woptimum3 .


which leads the FCM to the desired state given in

Fig. 10. We observe that in the equilibrium state the

values of output concepts are C2 = 0.8777 and

C7 = 0.9482.

It is clear that the values of C2 and C7 lie within the

desired regions defined by Eqs. (25) and (26), while the

weights fulfil the ranges posed initially by the group of

experts. This supports the claim that the obtained

solution is the true optimal solution. Some of the

weights have changed significantly their values from

the initial ones in order to succeed the desired output

values of concepts. We remark that there is a very high

positive influence of the concept C7 (final dose) to the

concept C2 (dose prescribed from the treatment

planning) as well as to the concept C5 (patient

positioning). This means that if we succeed to deliver

the maximum dose to the target volume; then, the initial

calculated dose from treatment planning is the desired

and the same happens with the patient positioning. It

must be mentioned that the estimated weights take their

optimum values at the edges of the suggested bounds.

The optimal values of FD and TPD are acceptable

according to the ICRU and IAEA protocols [19,20].

Thus, radiotherapists can follow the suggested

values and the treatment will be executed with

successful results. The proposed approach is efficient

and very useful for the FCM-controlled clinical

radiotherapy process. The recalculation of all weights

that participate in the simulation process constitutes the

most important drawback of the simple FCM model as

initially determined by experts-doctors. Its importance

to the radiotherapists is underlined by the fact that they

will be able to introduce clinical cases based on a range

of accepted values for the output concepts of the model.

The limitations of our work are the following.

� E
xperts should know the proper operation of the
system or the desired behavior of problem in order

to provide certain regions for the values of output

concepts.

� N
o new interconnections are derived using the
proposed learning technique and there is not any

influence on the architecture of the FCM model.

Future experiments and trials can overcome these

limitations. For the specific problems, the experts had

known the proper operations for the problem and the

derived weight matrices are the optimum ones.

7. Conclusions

A new hybrid method to train Fuzzy Cognitive

Maps, based on a two stage learning approach, first

stage the nonlinear Hebbian learning algorithm and

second stage the differential evolution algorithm, has

been developed, tested and applied in three problems

with different complexity levels.

Three real-life problems with increasing complex-

ity levels have been used for the illustration of the

proposed learning algorithm. The results appear to be

very promising, verifying the effectiveness of the

learning procedure. The physical meaning of the

obtained results is retained. Using the proposed

learning procedure, the deficiencies of FCMs, such

as the potential convergence to undesired states for the

output concepts values overcome. Also, the FCM

might change its fuzzy causal web as causal patterns

change and as experts update their causal knowledge.

In this work, it is also investigated how well novel

and easy to understand evolutionary algorithms,

referred to as differential evolution (DE) algorithms,

and claimed to be very efficient when they are applied

to solve optimization problems. The results show that

the DE algorithms are very robust, effective and highly

efficient in solving the studied class of process control

problems. Thus they are able of mitigating the

drawback of long computation times commonly

associated with evolutionary algorithms.

Simulation results suggest that the new method

exhibits fast and stable learning, good generalization,

and therefore, a great possibility of good performance.

The proposed algorithm is able to train Fuzzy

Cognitive Maps fast and efficiently, determining the

appropriate weight matrix that leads the values of

output concepts within desired convergence regions.

Future work is directed towards more investigation

on evolutionary computation techniques and the

development of new learning techniques for adapting

FCM causal links.

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A new hybrid method using evolutionary algorithms to train Fuzzy Cognitive Maps

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