Expert-Based Method of Integrated Waste Management Systems for Developing Fuzzy Cognitive Map - Complex System Modelling and Control Through Intelligent Soft Computations - page 119

Fig. 1 The symbolic graph of
a fuzzy cognitive map

W12

C2

C1

W51

W12 W32

C5

W23

W45

C4 C3

W34

exive, i.e., a concept cannot
cause itself, which means that the weight matrix always has

Usually it is accepted that causality is not self re fl

in its diagonal
(Carvalho 2010 ). Otherwise the component would grow without limits.
The description of the inference mechanism, which represents the behaviour of
the physical system, lies in the interpretation of FCM

' 0-s '

s mathematical formulation.
After the initialization of the FCM and the determination of concept activation
values by experts, concepts are ready to interact. As it is obvious, the activation of a
concept in

'

uences the values of concepts that are connected to it. At each step of
interaction (simulation step), every concept acquires a new value that is calculated
according to equations (Eqs. 1 and 2 ) and the interaction between concepts con-
tinues until a

fl

fix equilibrium is reached; a limit cycle is reached; or a chaotic
behaviour is observed (Ketipi et al. 2010 ).
The mathematical description of our FCM system is a simple loop:

V k þ 1 ¼ f ð N V k Þ ð 1 Þ

where V k is the state k of the system; N is the matrix of the system which contains
the weight w i ; j , and

1

f ð x Þ ¼ ð 2 Þ

1 þ e k x

where > 0 determines the steepness of the of the continuous function f.

We have conducted an online survey where each stakeholder was asked to
describe the existence and type of the causal relationships among the determined
factors and then to assess the strength of these using a predetermined simple scale,
capable to describe any kind of relationship between a pair of factors, both positive
and negative ones. It was helpful to draw a guideline in order to describe the terms of
concepts and the basics of the development of a FCM before starting with the survey.
The questionnaire guideline functioned as a support material in answering the
questions. In order to conduct the survey and to draw a suitable FCM, we needed to
apply the steps of designing the process. At

λ

first, we explained to the participant what
a FCM is, what its elements are and what our aim is with the results. As the
interviewees understood the underlying basic information, they were able to assess
the value of the connections. Thus, from each interviewee theoretically a different