**3. The structure of Fuzzy Cognitive Maps**

Fuzzy Cognitive Mapping methodology is a symbolic representation for the description and modeling of complex systems. Fuzzy Cognitive Maps (FCMs) describe different aspects of the behavior of a complex system in terms of concepts. Each concept represents a state or a characteristic of the system and interacts with each other showing the dynamics of the system. FCMs have been introduced by Kosko, (1986) as signed directed graphs for representing causal reasoning and computational inference processing, exploiting a symbolic representation for the description and modeling of a system.

In fact, FCM could be regarded as a combination of Fuzzy Logic and Neural Networks (Kosko, 1992). Graphically, FCM seems to be an oriented graph with feedback, consisting of nodes and weighted arcs. Nodes of the graph stand for the concepts that are used to describe the behavior of the system, connected by signed and weighted arcs representing the causal relationships that exist between the concepts (see Figure 1). It must be mentioned that all the values in the graph are fuzzy, so concepts take values in the range between [0,1] and the weights of the arcs are in the interval [-1,1]. Observing this graphical representation it becomes clear which concept influences other concepts by showing the interconnections between them. Moreover, FCM allows updating the construction of the graph, such as the adding or deleting of an interconnection or a concept. FCMs are used to represent both

Using Fuzzy Cognitive Mapping in Environmental

the following equation:

( ) *<sup>B</sup> yq* of the fuzzy output *B*.

index borrowed from social networks analysis.

**4. Data analysis using graph theory indices** 

Decision Making and Management: A Methodological Primer and an Application 431

Graph theory methods help analyzing the structural properties of cognitive maps (Ozesmi & Ozesmi, 2003). During the interviews, participants develop a FCM of the critical variables by drawing and circling the considerations they believe are important in relation to the topic under consideration. Then the main factors are defined and coded as concepts. The fuzzy directional arrows to one or more preceding factors are represented by fuzzy linguistic weights (see section 6), which after defuzzification produce a representative numerical weight. Using the defuzzification method of Centre of Gravity (COG) (Zadeh, 1976) a numerical weight is produced for each connection between concepts. CoG is computed from

1

*q Nq*

*Nq*

( )

*COG B*

 

( )

*Bqq*

*y y*

( )

(1)

*y*

*B q*

1

*q*

where *Nq* is the number of quantization used to discretize the membership function

Thus they make-up a continuity map whereby concept Ci is preceded by concept Cj indicating a cause-and-effect relationship. Each individual map is analyzed in relation to the number of concepts, connections, connection-to-concept ratio, and density (calculated by dividing the number of connections in the map by the square value of concepts). To allow for identification of key criteria within the process of cognitive mapping, an analysis of domain and centrality is also conducted. The complexity level of each individual concept is revealed through a number of structural measures of cognitive maps, e.g. the centrality

According to graph theory an effective way to better understand the structure of complex cognitive maps is condensing them. *Condensation* is achieved by replacing subgraphs (consisting of a group of variables connected with lines) with a single unit (Harary et al. 1965). Once the individual cognitive maps are drawn, they are qualitatively aggregated using clustering concepts to produce a condensed map named the *collective* FCM. Due to the complexity in FCM graphs (as the number of nodes and connections is often very large) the

Analyzing the structure of cognitive maps is to look how connected or sparse the maps are. This is expressed by an index of connectivity, called *density* of a cognitive map (D). The density is equal to the number of connections divided by the maximum number of connections possible between N variables, thus N2. If the density of a map is high then the

The structure of a cognitive map apart from number of variables and connections can best be analyzed by finding *transmitter variables* (forcing functions, givens, tails, independent variables), *receiver variables* (utility variables, ends, heads, dependent variables) and *ordinary variables* (Bougon et al. 1977; Eden et al. 1992; Harary et al. 1965). These variables are defined by their *outdegree* and *indegree.* Outdegree is the row sum of absolute values of a variable in the adjacency matrix and shows the cumulative strengths of connections (Eij). It is a measure of

most central variables with their weighted connections are usually illustrated.

interviewee sees a large number of causal relationships among the variables.

qualitative and quantitative data. The construction of a FCM requires the input of human experience and knowledge on the system under consideration. Thus, FCMs integrate the accumulated experience and knowledge concerning the underlying causal relationships amongst factors, characteristics, and components that constitute the system.

A FCM consists of nodes or concepts, Ci, i = 1…N, where N is the total number of concepts. Each interconnection between two concepts Ci and Cj has a weight, a directed edge Wij, which is similar to the strength of the causal links between Ci and Cj. Wij from concept Ci to concept Cj measures how much Ci causes Cj. In simple FCMs, directional influences take on trivalent values {-1; 0; +1}, where -1 indicates a negative relationship, 0 no causal relation, and +1 a positive relationship. In general, Wij indicates whether the relationship between the concepts is directed or inverse. The direction of causality indicates whether the concept Ci causes the concept Cj or vice versa. Thus, there are three types of weights:


It is important to note that Wij≠Wji in that causal relationship are not necessarily reversible. In Figure 1, an example FCM representation of the public health system is illustrated which has seven generic vertices (C1 to C7) and the weights (weighted edges) showing the relationships between concepts.


Fig. 1. Example of FCM model of the public health system: (a) FCM graph, and (b) connection matrix (adapted from Montazemi & Conrath, 1986).

#### **4. Data analysis using graph theory indices**

430 International Perspectives on Global Environmental Change

qualitative and quantitative data. The construction of a FCM requires the input of human experience and knowledge on the system under consideration. Thus, FCMs integrate the accumulated experience and knowledge concerning the underlying causal relationships

A FCM consists of nodes or concepts, Ci, i = 1…N, where N is the total number of concepts. Each interconnection between two concepts Ci and Cj has a weight, a directed edge Wij, which is similar to the strength of the causal links between Ci and Cj. Wij from concept Ci to concept Cj measures how much Ci causes Cj. In simple FCMs, directional influences take on trivalent values {-1; 0; +1}, where -1 indicates a negative relationship, 0 no causal relation, and +1 a positive relationship. In general, Wij indicates whether the relationship between the concepts is directed or inverse. The direction of causality indicates whether the concept


It is important to note that Wij≠Wji in that causal relationship are not necessarily reversible. In Figure 1, an example FCM representation of the public health system is illustrated which has seven generic vertices (C1 to C7) and the weights (weighted edges) showing the

Fig. 1. Example of FCM model of the public health system: (a) FCM graph, and (b)

connection matrix (adapted from Montazemi & Conrath, 1986).

amongst factors, characteristics, and components that constitute the system.

Ci causes the concept Cj or vice versa. Thus, there are three types of weights:


relationships between concepts.

Graph theory methods help analyzing the structural properties of cognitive maps (Ozesmi & Ozesmi, 2003). During the interviews, participants develop a FCM of the critical variables by drawing and circling the considerations they believe are important in relation to the topic under consideration. Then the main factors are defined and coded as concepts. The fuzzy directional arrows to one or more preceding factors are represented by fuzzy linguistic weights (see section 6), which after defuzzification produce a representative numerical weight. Using the defuzzification method of Centre of Gravity (COG) (Zadeh, 1976) a numerical weight is produced for each connection between concepts. CoG is computed from the following equation:

$$\text{COG}(\mathbf{B}) = \frac{\sum\_{q=1}^{Nq} \mu\_{\mathbf{B}}(y\_q) y\_q}{\sum\_{q=1}^{Nq} \mu\_{\mathbf{B}}(y\_q)} \tag{1}$$

where *Nq* is the number of quantization used to discretize the membership function ( ) *<sup>B</sup> yq* of the fuzzy output *B*.

Thus they make-up a continuity map whereby concept Ci is preceded by concept Cj indicating a cause-and-effect relationship. Each individual map is analyzed in relation to the number of concepts, connections, connection-to-concept ratio, and density (calculated by dividing the number of connections in the map by the square value of concepts). To allow for identification of key criteria within the process of cognitive mapping, an analysis of domain and centrality is also conducted. The complexity level of each individual concept is revealed through a number of structural measures of cognitive maps, e.g. the centrality index borrowed from social networks analysis.

According to graph theory an effective way to better understand the structure of complex cognitive maps is condensing them. *Condensation* is achieved by replacing subgraphs (consisting of a group of variables connected with lines) with a single unit (Harary et al. 1965). Once the individual cognitive maps are drawn, they are qualitatively aggregated using clustering concepts to produce a condensed map named the *collective* FCM. Due to the complexity in FCM graphs (as the number of nodes and connections is often very large) the most central variables with their weighted connections are usually illustrated.

Analyzing the structure of cognitive maps is to look how connected or sparse the maps are. This is expressed by an index of connectivity, called *density* of a cognitive map (D). The density is equal to the number of connections divided by the maximum number of connections possible between N variables, thus N2. If the density of a map is high then the interviewee sees a large number of causal relationships among the variables.

The structure of a cognitive map apart from number of variables and connections can best be analyzed by finding *transmitter variables* (forcing functions, givens, tails, independent variables), *receiver variables* (utility variables, ends, heads, dependent variables) and *ordinary variables* (Bougon et al. 1977; Eden et al. 1992; Harary et al. 1965). These variables are defined by their *outdegree* and *indegree.* Outdegree is the row sum of absolute values of a variable in the adjacency matrix and shows the cumulative strengths of connections (Eij). It is a measure of

Using Fuzzy Cognitive Mapping in Environmental

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

concept *Cj* is *positively very high*.

Decision Making and Management: A Methodological Primer and an Application 433

{negatively very very strong, negatively very strong, negatively strong, negatively medium, negatively weak, negatively very weak, zero, positively very weak, positively weak, positively

The corresponding membership functions for these terms are shown in Fig. 2 and they are

T(influence)


The inference of the rule *T*{*influence*} means that the linguistic weight y (*wij) is μΒ*, where *μ<sup>B</sup>* is a linguistic variable from the set *T*. Also, experts and/or system stakeholders can directly assign the fuzzy linguistic weight y (that describes the strength of connection between concepts *Ci* and *Cj*) with no use of fuzzy rules. For example, someone can assign the strength of the connection between concepts *Ci* and *Cj* as follows: The influence from concept *Ci* to

Finally, the linguistic variables *D* from the set *T(influence)* **-** proposed by the experts for each interconnection - are aggregated using the SUM method and so an overall linguistic weight is produced (Papageorgiou & Stylios, 2008). Finally, the Center of Gravity (CoG) defuzzification method (Zadeh, 1986) is used for the transformation of the linguistic weight to a numerical value within the range [-1, 1]. This methodology has the advantage that experts are not required to assign directly numerical values to causality relationships, but rather to describe qualitatively the degree of causality among the concepts. Thus, an initial matrix Winitial = [Wij], i, j = 1,…,N, with Wii = 0, i = 1,…,N, is obtained. Using the initial concept values, Ai, which are also provided by the experts, the matrix Winitial is used for the determination of the steady state

Using artificial intelligent techniques, the dynamics of a fuzzy cognitive map can be traced analytically through a specific inference and simulation process. Each one of the *Cj* concepts

medium, positively strong, positively very strong, positively very very strong}.

*μnvvs, μnvs, μns, μnm, μnw, μnvw, μz, μpvw, μpw, μpm, μps μpvs* and *μpvvs*.

Fig. 2. The thirteen membership functions describing *T(influence)*

of the FCM, through the application of the rule of Eq. (2) or (5).

**6. The FCM inference and simulation processes** 

how much a given variable influences other variables. Indegree is the column sum of absolute values of a variable and shows the cumulative strength of variables entering the unit.

*Transmitter variables* are units whose outdegree is positive and their indegree is 0. *Receiver variables* are units whose outdegree is 0 and their indegree is positive. Other variables, which have both non-zero outdegree and indegree, are *ordinary variables* (means) (Eden et al., 1992, Ozesmi & Ozesmi 2004). This type of variables reveals how people think about the causal relationships. For instance, if someone views a variable as a transmitter, this means that he perceives of the relative causal relationship as forcing function, which cannot be controlled by any other variables. In contrast, a receiver variable is seen as not affecting any of the other variables in the system. The total number of receiver variables in a cognitive map can be considered an index of its complexity. Larger number of receiver variables indicates that the cognitive map considers many outcomes and implications that are a result of the system (Eden, 1992). Many transmitter units show the "flatness" of a cognitive map where causal arguments are not well elaborated (Eden et al. 1992).

Centrality is the most important measure for map complexity, borrowed from social networks analysis, and is the summation of variable's indegree and outdegree (Bougon et al., 1977; Eden et al., 1992). Actually the centrality shows how connected the variable is to other variables and what the cumulative strength of these connections is. Another structural measure of cognitive maps is the *hierarchy index* (h), which is a function of the out-degrees and number of variables in a given map and represents the type of system as fully hierarchical, or democratic (see Ozesmi & Ozesmi, 2004, pp. 50–51 for formulas).

#### **5. Development of Fuzzy Cognitive Maps**

The design of a fuzzy cognitive map is a process that heavily relies on the input from experts and/or stakeholders (Hobbs et al., 2002). This methodology extracts the knowledge from the stakeholders and exploits their experience of the system's model and behaviour. FCM is fairly simple and easy to understand for the participants, which opens up the possibility for involving lay people as well as planners, managers and experts (Isaac et al. 2009). Even though the cognitive nature of a FCM makes it inevitably a subjective representation of the system, Tan & Özesmi (2006) emphasize that the model is not arbitrary as it is built carefully and reflexively with stakeholders (in groups or individually).

According to the FCM development process, at the first step of the construction process, the number and kind of concepts are determined by a group of experts and/or system stakeholders that comprise the FCM model. Then, a domain expert and/or stakeholder describe each interconnection either with an if-then rule that infers a fuzzy linguistic variable from a determined set or with a direct fuzzy linguistic weight, which associates the relationship between the two concepts and determines the grade of causality between the two concepts.

For example, someone can assign the strength of influence of concept *Cj* on concept *Ci* using the following form: "The strength of influence of concept *Cj* on concept *Ci* is T{*influence}"*  where the variable *T*{*influence*} declares the causal inter-relationships among concepts (i.e. the degree of influence from concept *Cj* to *Ci*). Its term set *T*{*influence*} is suggested to comprise thirteen variables and takes values in the universe U=[-1, 1]. Using thirteen linguistic variables, an expert can describe in detail the influence of one concept on another and can discern between different degrees of influence. The thirteen variables used here are: *T(influence)* =

how much a given variable influences other variables. Indegree is the column sum of absolute

*Transmitter variables* are units whose outdegree is positive and their indegree is 0. *Receiver variables* are units whose outdegree is 0 and their indegree is positive. Other variables, which have both non-zero outdegree and indegree, are *ordinary variables* (means) (Eden et al., 1992, Ozesmi & Ozesmi 2004). This type of variables reveals how people think about the causal relationships. For instance, if someone views a variable as a transmitter, this means that he perceives of the relative causal relationship as forcing function, which cannot be controlled by any other variables. In contrast, a receiver variable is seen as not affecting any of the other variables in the system. The total number of receiver variables in a cognitive map can be considered an index of its complexity. Larger number of receiver variables indicates that the cognitive map considers many outcomes and implications that are a result of the system (Eden, 1992). Many transmitter units show the "flatness" of a cognitive map where causal

Centrality is the most important measure for map complexity, borrowed from social networks analysis, and is the summation of variable's indegree and outdegree (Bougon et al., 1977; Eden et al., 1992). Actually the centrality shows how connected the variable is to other variables and what the cumulative strength of these connections is. Another structural measure of cognitive maps is the *hierarchy index* (h), which is a function of the out-degrees and number of variables in a given map and represents the type of system as fully

The design of a fuzzy cognitive map is a process that heavily relies on the input from experts and/or stakeholders (Hobbs et al., 2002). This methodology extracts the knowledge from the stakeholders and exploits their experience of the system's model and behaviour. FCM is fairly simple and easy to understand for the participants, which opens up the possibility for involving lay people as well as planners, managers and experts (Isaac et al. 2009). Even though the cognitive nature of a FCM makes it inevitably a subjective representation of the system, Tan & Özesmi (2006) emphasize that the model is not arbitrary

According to the FCM development process, at the first step of the construction process, the number and kind of concepts are determined by a group of experts and/or system stakeholders that comprise the FCM model. Then, a domain expert and/or stakeholder describe each interconnection either with an if-then rule that infers a fuzzy linguistic variable from a determined set or with a direct fuzzy linguistic weight, which associates the relationship between the two concepts and determines the grade of causality between the

For example, someone can assign the strength of influence of concept *Cj* on concept *Ci* using the following form: "The strength of influence of concept *Cj* on concept *Ci* is T{*influence}"*  where the variable *T*{*influence*} declares the causal inter-relationships among concepts (i.e. the degree of influence from concept *Cj* to *Ci*). Its term set *T*{*influence*} is suggested to comprise thirteen variables and takes values in the universe U=[-1, 1]. Using thirteen linguistic variables, an expert can describe in detail the influence of one concept on another and can discern between different degrees of influence. The thirteen variables used here are: *T(influence)* =

hierarchical, or democratic (see Ozesmi & Ozesmi, 2004, pp. 50–51 for formulas).

as it is built carefully and reflexively with stakeholders (in groups or individually).

values of a variable and shows the cumulative strength of variables entering the unit.

arguments are not well elaborated (Eden et al. 1992).

**5. Development of Fuzzy Cognitive Maps** 

two concepts.

{negatively very very strong, negatively very strong, negatively strong, negatively medium, negatively weak, negatively very weak, zero, positively very weak, positively weak, positively medium, positively strong, positively very strong, positively very very strong}.

The corresponding membership functions for these terms are shown in Fig. 2 and they are *μnvvs, μnvs, μns, μnm, μnw, μnvw, μz, μpvw, μpw, μpm, μps μpvs* and *μpvvs*.

Fig. 2. The thirteen membership functions describing *T(influence)*

The inference of the rule *T*{*influence*} means that the linguistic weight y (*wij) is μΒ*, where *μ<sup>B</sup>* is a linguistic variable from the set *T*. Also, experts and/or system stakeholders can directly assign the fuzzy linguistic weight y (that describes the strength of connection between concepts *Ci* and *Cj*) with no use of fuzzy rules. For example, someone can assign the strength of the connection between concepts *Ci* and *Cj* as follows: The influence from concept *Ci* to concept *Cj* is *positively very high*.

Finally, the linguistic variables *D* from the set *T(influence)* **-** proposed by the experts for each interconnection - are aggregated using the SUM method and so an overall linguistic weight is produced (Papageorgiou & Stylios, 2008). Finally, the Center of Gravity (CoG) defuzzification method (Zadeh, 1986) is used for the transformation of the linguistic weight to a numerical value within the range [-1, 1]. This methodology has the advantage that experts are not required to assign directly numerical values to causality relationships, but rather to describe qualitatively the degree of causality among the concepts. Thus, an initial matrix Winitial = [Wij], i, j = 1,…,N, with Wii = 0, i = 1,…,N, is obtained. Using the initial concept values, Ai, which are also provided by the experts, the matrix Winitial is used for the determination of the steady state of the FCM, through the application of the rule of Eq. (2) or (5).

### **6. The FCM inference and simulation processes**

Using artificial intelligent techniques, the dynamics of a fuzzy cognitive map can be traced analytically through a specific inference and simulation process. Each one of the *Cj* concepts

Using Fuzzy Cognitive Mapping in Environmental

Papageorgiou, 2011). Thus, the eq. (2) is transformed to the eq. (5).

FCM, which are progressively formed according to given considerations.

suggestions and available knowledge.

applications is equal to 0.001). Thus, a final vector *A\_f* is obtained.

= *A<sup>k</sup>* .

(Stach et al., 2010).

Decision Making and Management: A Methodological Primer and an Application 435

cannot describe efficiently the initial state of a variable (Papageorgiou et al., 2010,

*i i j ji*

The FCM simulation process is initialized through assigning a value between 0 and +1 to the activation level of each of the nodes of the map, based on experts/stakeholder opinion for the current state; then the concepts are free to interact. The value of zero suggests that a given concept is not present in the system at a particular iteration, whereas the value of one indicates that a given concept is present to its maximum degree. Other values correspond to intermediate levels of activation. The activation level of each concept depends on its value at the preceding iteration as well as on the preceding values of all concepts that exert influence on it through non-zero relationships. The simulation, which with regard to its content is mainly qualitative, is not intended to produce exact quantitative values. It aims at identifying the pattern of system's behaviour via the achieved values of the concepts of the

After defining all variables and necessary values, as well as the relationships between them, the simulation is carried out by use of the simulator consisting of the following five steps: **Step 1.** Definition of the initial vector *A* that corresponds to the concepts identified by

**Step 2.** Multiply the initial vector *A* and the matrix *W* defined through equation (2) or (5) **Step 3.** The resultant vector *A* at time step *k* is updating using eqs. (2) or (5) and (4). **Step 4.** This new vector *A<sup>k</sup>* is considered as an initial vector in the next iteration.

**Step 5.** Steps 2–4 are repeated until *A A k k* <sup>1</sup> *e* = 0.001 (where *e* is a residual describing the

In each step of the cycling the values of concepts change according to the equation (2) or (5). This interaction between concepts continues until: i) a fixed equilibrium is reached, ii) a limited cycle is reached or iii) a chaotic behavior is exhibited. Actually, in most cases, the iteration stops when a limit vector is reached, i.e., when *k k -1 A =A* or when *A A k k* <sup>1</sup> *e* ; where *e* is a residual, whose value depends on the application type (and in most

In the previous analysis, all type of information has numerical values. FCM allows us to perform qualitative simulations and experiment with a dynamic model. Simulations allow for analysis of several aspects of FCMs, such as concepts activation levels at the final state (if there are any) and changes/trends in the activation levels throughout the simulation concerning either all concepts or a subset of concepts that is of interest to the user, and discovery of cycles (intervals, concepts activation levels within the cycle). This type of analysis allows investigating "what-if" scenarios by performing simulations of a given model from different initial state vectors. Once an FCM has been subjected to an initial stimulus, it is possible to gain insight into a system's behaviour by studying the resulting stable state or cycle of states. Simulations offer description of dynamic behaviour of the system that can be used to support decision-making or predictions about its future states

minimum error difference among the subsequent concepts) or *A A k k* <sup>1</sup> . Thus *A\_f*

( 1) (2 ( ) 1) (2 ( ) 1)

 

*j i j*

*N*

1

*A k f( A k A k E )* (5)

can take values in the unit interval [0,1], also called the 'activation level'. The activation level can be interpreted as relative abundance (Hobbs et al. 2002). More rigorously, the activation level can represent membership in fuzzy set describing linguistic measures of relative abundance (e.g. low, average, high) (Kosko, 1986).

Values of the concept *Ci* in time *t* are represented by the state vector *Ai(t)* while the state of the whole fuzzy cognitive map can be described by the state vector *1 n AAA (t)= [ (t),..., (t)]* representing a point within a fuzzy hypercube that the system achieves at a certain point. The whole system with an input vector *A*(0) describes a time trace within a multidimensional space *<sup>n</sup> I* that can gradually converge to an equilibrium point, or a chaotic point or a periodic attractor within a fuzzy hypercube. To which attractor the system will converge depends on the value of the input vector *A*(0) .

The value *Ai* of each concept *Ci* in a moment *t+1* is calculated by the sum of the previous value of *Ai* in a precedent moment *t* with the product of the value Aj of the cause node *Cj* in precedent moment *t* and the value of the cause-effect link w*ij*. The mathematical representation of the inference process of a fuzzy cognitive map has the following matrix form (Papageorgiou & Stylios, 2008):

$$\mathcal{A}^{(k)} = f(\mathcal{A}^{(k \cdot l)} + \sum \mathcal{A}^{(k \cdot l)} \cdot W \,) \tag{2}$$

Thus, the value *Aj* for each concept *Cj* is calculated by:

$$A\_i^{\{k+1\}} = f\left(A\_i^{\{k\}} + \sum\_{\substack{j \neq i \\ j=1}}^N A\_j^{\{k\}} \cdot w\_{ji}\right) \tag{3}$$

where ( 1) *<sup>k</sup> Ai* is the value of concept *Ci* at simulation step 1 *<sup>k</sup>* , ( ) *<sup>k</sup> Aj* is the value of concept *Cj* at step *k* , *wji* is the weight of the interconnection between concept *Cj* and concept *Ci* and *f* is a threshold (activation) function (Bueno & Salmeron, 2008). Sigmoid threshold function gives values of concepts in the range [0,1] and its mathematical type is:

$$f'(\mathbf{x}) = \frac{1}{1 + e^{-m\mathbf{x}}} \tag{4}$$

where *m* is a real positive number and *x* is the value ( ) *<sup>k</sup> Ai* on the equilibrium point. A concept is turned on or activated by making its vector element 1 or 0 in (0,1). The sigmoid threshold function is used to reduce unbounded weighted sum to a certain range, which hinders quantitative analysis, but allows for qualitative comparisons between concepts (Bueno & Salmeron, 2008).

A modified FCM inference algorithm, which updates the common FCM simulation process as initially suggested by Kosko (1986) can be used to avoid the conflicts that emerge in cases where the initial values of concepts are 0 or 0.5, thus overcoming the limitation present by the sigmoid threshold function. This rescaled algorithm is implemented especially for the cases where there is no information about a certain concept/state or the expert/stakeholder

can take values in the unit interval [0,1], also called the 'activation level'. The activation level can be interpreted as relative abundance (Hobbs et al. 2002). More rigorously, the activation level can represent membership in fuzzy set describing linguistic measures of relative

Values of the concept *Ci* in time *t* are represented by the state vector *Ai(t)* while the state of the whole fuzzy cognitive map can be described by the state vector *1 n AAA (t)= [ (t),..., (t)]* representing a point within a fuzzy hypercube that the system achieves at a certain point. The whole system with an input vector *A*(0) describes a time trace within a

point or a periodic attractor within a fuzzy hypercube. To which attractor the system will

The value *Ai* of each concept *Ci* in a moment *t+1* is calculated by the sum of the previous value of *Ai* in a precedent moment *t* with the product of the value Aj of the cause node *Cj* in precedent moment *t* and the value of the cause-effect link w*ij*. The mathematical representation of the inference process of a fuzzy cognitive map has the following matrix

> 

*j i j*

*N*

1

<sup>1</sup> ( )

where ( 1) *<sup>k</sup> Ai* is the value of concept *Ci* at simulation step 1 *<sup>k</sup>* , ( ) *<sup>k</sup> Aj* is the value of concept *Cj* at step *k* , *wji* is the weight of the interconnection between concept *Cj* and concept *Ci* and *f* is a threshold (activation) function (Bueno & Salmeron, 2008). Sigmoid threshold function

<sup>1</sup> ( ) <sup>1</sup> *mx f x*

where *m* is a real positive number and *x* is the value ( ) *<sup>k</sup> Ai* on the equilibrium point. A concept is turned on or activated by making its vector element 1 or 0 in (0,1). The sigmoid threshold function is used to reduce unbounded weighted sum to a certain range, which hinders quantitative analysis, but allows for qualitative comparisons between concepts

A modified FCM inference algorithm, which updates the common FCM simulation process as initially suggested by Kosko (1986) can be used to avoid the conflicts that emerge in cases where the initial values of concepts are 0 or 0.5, thus overcoming the limitation present by the sigmoid threshold function. This rescaled algorithm is implemented especially for the cases where there is no information about a certain concept/state or the expert/stakeholder

*e*

*k k k i i j ji*

 

*I* that can gradually converge to an equilibrium point, or a chaotic

*k k -1 (k -1) A A AW = f( + )* (2)

*A fA A w* (3)

(4)

abundance (e.g. low, average, high) (Kosko, 1986).

converge depends on the value of the input vector *A*(0) .

Thus, the value *Aj* for each concept *Cj* is calculated by:

gives values of concepts in the range [0,1] and its mathematical type is:

multidimensional space *<sup>n</sup>*

(Bueno & Salmeron, 2008).

form (Papageorgiou & Stylios, 2008):

cannot describe efficiently the initial state of a variable (Papageorgiou et al., 2010, Papageorgiou, 2011). Thus, the eq. (2) is transformed to the eq. (5).

$$A\_i(k+1) = f(\{2A\_i(k) - 1\} + \sum\_{j=1 \atop j=1}^{N} (2A\_j(k) - 1) \cdot E\_{ji}) \tag{5}$$

The FCM simulation process is initialized through assigning a value between 0 and +1 to the activation level of each of the nodes of the map, based on experts/stakeholder opinion for the current state; then the concepts are free to interact. The value of zero suggests that a given concept is not present in the system at a particular iteration, whereas the value of one indicates that a given concept is present to its maximum degree. Other values correspond to intermediate levels of activation. The activation level of each concept depends on its value at the preceding iteration as well as on the preceding values of all concepts that exert influence on it through non-zero relationships. The simulation, which with regard to its content is mainly qualitative, is not intended to produce exact quantitative values. It aims at identifying the pattern of system's behaviour via the achieved values of the concepts of the FCM, which are progressively formed according to given considerations.

After defining all variables and necessary values, as well as the relationships between them, the simulation is carried out by use of the simulator consisting of the following five steps:


In each step of the cycling the values of concepts change according to the equation (2) or (5). This interaction between concepts continues until: i) a fixed equilibrium is reached, ii) a limited cycle is reached or iii) a chaotic behavior is exhibited. Actually, in most cases, the iteration stops when a limit vector is reached, i.e., when *k k -1 A =A* or when *A A k k* <sup>1</sup> *e* ; where *e* is a residual, whose value depends on the application type (and in most applications is equal to 0.001). Thus, a final vector *A\_f* is obtained.

In the previous analysis, all type of information has numerical values. FCM allows us to perform qualitative simulations and experiment with a dynamic model. Simulations allow for analysis of several aspects of FCMs, such as concepts activation levels at the final state (if there are any) and changes/trends in the activation levels throughout the simulation concerning either all concepts or a subset of concepts that is of interest to the user, and discovery of cycles (intervals, concepts activation levels within the cycle). This type of analysis allows investigating "what-if" scenarios by performing simulations of a given model from different initial state vectors. Once an FCM has been subjected to an initial stimulus, it is possible to gain insight into a system's behaviour by studying the resulting stable state or cycle of states. Simulations offer description of dynamic behaviour of the system that can be used to support decision-making or predictions about its future states (Stach et al., 2010).

Using Fuzzy Cognitive Mapping in Environmental

Sea area.

**8. An application of FCM in the Black Sea** 

between two factors, positive and negative (see Table 1).

obtained by clustering and augmented the 29 individual FCMs.

illustrates the most central variables with their weighted connections.

Ukraine for further assessment.

lj.si/pub/networks/pajek/]).

Decision Making and Management: A Methodological Primer and an Application 437

In order to illustrate in brief the methods described so far, we present a FCM exercise designed and implemented in the Northern Black Sea with Ukrainian stakeholders (for details see Kontogianni et al., forthcoming). In this application we were interested in investigating how the citizens perceive the future prospects and risks of the Black Sea marine environment; creating and analyzing their FCMs this can be achieved. We employed 29 in-depth lay people interviews (see Appendix A). Based on specific guidelines, each interviewee was asked to identify at first the main factors which come to his/her mind when he/she is asked about the Black Sea as a system where humans, marine animals and plants are all living together. During the interviews participants developed thus a FCM of the critical variables (important considerations) by drawing and circling the considerations they believe are important for environmental health of marine state ecosystem in the Black

After the identification of the main factors affecting the environmental health of the marine ecosystem in the Black Sea, each stakeholder was asked to describe the existence and type of the causal relationships among these factors and then, the strength of the causal relationships-influences that may exist between these factors. This phase was implemented 13 grades scale, numbering from -6 to + 6, capable to describe any kind of relationship

Thus, a FCM from each interviewee was established presenting the main factors/variables and the relationships among them illustrating the individual's perceptions about the future prospects and the risks about the ecological health of the marine environment in the Black Sea. Figure 3 illustrates the produced FCM defined by an individual/stakeholder from

The initial number of important factors identified by stakeholders was 52. Since it was decided to produce a collective FCM providing detail for future risks-related issues we limited the number of factors having the same meaning through clustering. Using marine experts' judgement, the importance of the original 52 factors was discussed and then clustered in a total of 26 concepts (Table A in Appendix). The mean number of variables in the individual cognitive maps of the Black Sea ecosystem drawn by the 29 respondents was 7.86 ± 1.7, with 11± 6.513 connections on average between the variables that they defined. There were a total of 26 variables with 145 connections in the collective cognitive map

The process of condensation enabled aggregation of variables into high-level concepts, which then feed into the construction of a collective FCM for Ukrainian stakeholders. The collective FCM (consisting of 26 concepts and 145 relationships among concepts) is thus obtained (see Figure 4 developed in pajek software [http://vlado.fmf.uni-

The collective FCM was then coded as adjacency matrix *E*=[e*ij*] and its structure was analyzed using the indices derived from graph theory (see section 4). Due to the complexity of the collective FCM graph (as the number of nodes and connections is very large) Figure 5

It is observed from graph indices calculations that the density of collective FCM is high and a mentioned complexity is present. A relatively high complexity is considered in the cases where the receiver variables are more than the transmitter variables, and in our case, the complexity is equal to 1.5 (complexity>1 means relatively high complexity). The most
