**4. Fuzzy DEMATEL (fuzzy decision-making trial and evaluation laboratory) method**

The DEMATEL method is a multi-criteria decision-making method and is used to solve many complex problems. With this method, the relationships between the variables are evaluated and these relationships are visualized through diagrams showing cause-and-effect relationships. Thanks to this method, all variables are determined as influencing and affected variables, or in other words, cause-effect relationships and the structural relationship between the variables is revealed [19]. The DEMATEL method has a superior feature compared to other multi-criteria decision-making methods as it deals with the interrelationships between variables.

Fuzzy logic was first introduced by Lotfi A. Zadeh in 1965. Fuzzy logic is an approach that is based on thinking like a human and adopts that the key elements of human thought are linguistic variables [20]. The differences in perception arising from the way of thinking of people and the uncertainties in their subjective behaviors and goals are explained by the concept of blurriness, and in this respect, it is defined as the application of fuzzy mathematics to the real world. In fuzzy logic, variables are classified without precise evaluations. Unlike classical logic, it models the data by using linguistic variables such as "very little, little, medium, high, very high" instead of definite propositions such as true-false or yes-no. Afterward, these expressions are converted into fuzzy numbers and more realistic solutions are obtained [21].

The DEMATEL method reveals the relationship between variables in complex systems and it is not always possible to evaluate these variables with definite propositions. At this point, fuzzy logic is used and expert opinions about the variables are

#### *Risk Management, Sustainability and Leadership*


#### **Table 1.**

*Main risk group and sub-risks.*

converted into fuzzy numbers. In summary, the Fuzzy DEMATEL method is obtained by transferring the DEMATEL method to the fuzzy environment [22].

When the studies in the literature with the fuzzy DEMATEL method are examined, the fuzzy DEMATEL method was used to investigate the factors affecting the *Development of a Risk Management Model by the Fuzzy DEMATEL Method in the Evaluation… DOI: http://dx.doi.org/10.5772/intechopen.110018*

adoption of new technology and to determine the relationship between the factors in the study conducted by Zargar et al. [23]. In the study by Chang et al., fuzzy DEMATEL method was used to determine supplier selection criteria [24]. In the study conducted by Chou et al., fuzzy AHP and fuzzy DEMATEL methods were applied integrated in order to evaluate human resources in the field of science and technology [25]. Çelik and Akyüz used the fuzzy DEMATEL method to evaluate the critical hazards in the gas release process in oil tankers [26]. Seker and Zavadskas used the fuzzy DEMATEL method in the analysis of occupational risks in the construction industry [27]. Mahmoudi et al. used the fuzzy DEMATEL method to determine the critical success factors for the self-care process in heart failure [28]. Feng and Ma determined the factors affecting service innovation in the manufacturing sector with fuzzy DEMATEL [29].

#### **4.1 Steps of the method of fuzzy DEMATEL**

Although the steps of the fuzzy DEMATEL method are similar to the steps of the DEMATEL method, fuzzy numbers are used in this method and these numbers need to be defuzzification in order to convert them into definite results. At this point, unlike the DEMATEL method, the defuzzification process is integrated into the steps of the method. Although various methods are used in defuzzification, the CFCS (Converting Fuzzy Data into Crisp Scores) method used in a study by Opricovic and Tzeng in 2003 was used within the scope of our study [30].

Zhou et al. used the fuzzy DEMATEL method to determine critical success factors in emergency management in 2011. The steps followed in the study by Zhou et al. are listed below [31]. In this study, Zhou et al. used the CFCS method, developed by Opricovic and Tzeng [30], which is used to defuzzifying fuzzy numbers. The steps and demonstrations presented within the scope of Zhou et al.'s work were also used in our study [31];

Step 1: Determine the initial direct-relation matrix.

At this stage, a group of experts is formed in order to determine the relationships between variables, criteria, or factors. Linguistic variables and fuzzy numbers in **Table 2** are used when group members make pairwise comparisons.

At this stage, the relations between the criteria or factors are evaluated by experts by making pairwise comparisons. As a result of the evaluation, an initial direct matrix consisting of triangular fuzzy numbers is obtained. Defuzzification processes are applied to obtain the initial direct matrix with the crisp values.

Step 2: Defuzzification.

In this study, CFCS (converting fuzzy data into crisp scores) method was used in order to convert fuzzy numbers into crisp values.


**Table 2.**

*Triangular fuzzy numbers according to the degree of effect.*

$$z\_{\vec{\imath}}^k = \left(l\_{\vec{\imath}\mathfrak{j}}, m\_{\vec{\imath}\mathfrak{j}}, r\_{\vec{\imath}\mathfrak{j}}\right) \tag{1}$$

1 ≤ k ≤ K.

K: Number of experts.

*zk ij*: Evaluation of the effect of the i criterion on the j criterion by the kth expert in a fuzzy environment.

The following formulas are used for normalization, calculation of left and right normalized value, calculation of total normalized value, and calculation and integration of crisp value for defuzzification operations.

*4.1.1 Normalization*

$$\mathcal{L}l\_{\vec{\eta}}^k = \left(l\_{\vec{\eta}}^k - \min\_{1 \le k \le K} l\_{\vec{\eta}}^k\right) / \Delta\_{\min}^{\max} \tag{2}$$

$$
\infty m\_{\vec{ij}}^k = \left( m\_{\vec{ij}}^k - \min\_{1 \le k \le K} l\_{\vec{ij}}^k \right) / \Delta\_{\min}^{\max} \tag{3}
$$

$$
\Delta \mathbf{r}\_{\vec{ij}}^k = \left( r\_{\vec{ij}}^k - \min\_{1 \le k \le K} l\_{\vec{ij}}^k \right) / \Delta\_{\min}^{\max} \tag{4}
$$

$$
\Delta\_{\rm min}^{\rm max} = \max r\_{\vec{ij}}^k - \min l\_{\vec{ij}}^k \tag{5}
$$

*4.1.2 Computing of left (ls) and right (rs) normalized values*

$$
\omega ls\_{\vec{ij}}^k = \infty m\_{\vec{ij}}^k / \left(\mathbf{1} + \boldsymbol{\varkappa} m\_{\vec{ij}}^k - \boldsymbol{\varkappa} l\_{\vec{ij}}^k\right) \tag{6}
$$

$$
\omega r s\_{\vec{ij}}^k = \omega r\_{\vec{ij}}^k / \left(\mathbf{1} + \omega r\_{\vec{ij}}^k - \omega m\_{\vec{ij}}^k\right) \tag{7}
$$

#### *4.1.3 Computing total normalized crisp values*

$$\boldsymbol{\omega}\_{\vec{\boldsymbol{\eta}}}^{k} = \left[ \boldsymbol{\varkappa} \boldsymbol{l} \boldsymbol{s}\_{\vec{\boldsymbol{\eta}}}^{k} \left( \mathbf{1} - \boldsymbol{\varkappa} \boldsymbol{l} \boldsymbol{s}\_{\vec{\boldsymbol{\eta}}}^{k} \right) + \boldsymbol{\varkappa} \boldsymbol{r} \boldsymbol{s}\_{\vec{\boldsymbol{\eta}}}^{k} \boldsymbol{\varkappa} \boldsymbol{r} \boldsymbol{s}\_{\vec{\boldsymbol{\eta}}}^{k} \right] / \left( \mathbf{1} + \boldsymbol{\varkappa} \boldsymbol{r} \boldsymbol{s}\_{\vec{\boldsymbol{\eta}}}^{k} - \boldsymbol{\varkappa} \boldsymbol{l} \boldsymbol{s}\_{\vec{\boldsymbol{\eta}}}^{k} \right) \tag{8}$$

#### *4.1.4 Computing crisp values*

$$BNP^k\_{\vec{\eta}} = \min l^k\_{\vec{\eta}} + \varkappa^k\_{\vec{\eta}} \Delta^{\max}\_{\min} \tag{9}$$

#### *4.1.5 Integrating crisp values*

$$a\_{\vec{\boldsymbol{w}}}^k = \frac{1}{K} \sum\_{k} {\boldsymbol{1} \le k \le K} \text{BNP}\_{\vec{\boldsymbol{w}}}^k \tag{10}$$

As a result of the operations performed, the initial direct-relation matrix is obtained. Step 3: Obtaining the normalized direct-relation matrix

By means of the formula below, the normalized direct-relation matrix is obtained.

$$\mathbf{D} = \mathbf{A}/\mathbf{s} \tag{11}$$

s = max (max <sup>P</sup>*<sup>n</sup> j*¼1 *aij*, max <sup>P</sup>*<sup>n</sup> i*¼1 *aij*) i,j = 1,2, … … n Step 4: Obtaining the total-relation matrix *Development of a Risk Management Model by the Fuzzy DEMATEL Method in the Evaluation… DOI: http://dx.doi.org/10.5772/intechopen.110018*

When the normalized direct-relation matrix D is obtained, the total-relation matrix T is calculated using the formula below. "I" stands for the unit matrix

$$\mathbf{T} = \mathbf{D} + \mathbf{D}^2 + \mathbf{D}^3 + \dots \\ \dots \\ \dots \\ \dots \\ \sum\_{i=1}^{\infty} D^i = \mathbf{D} (\mathbf{I} - \mathbf{D})^{-1} \tag{12}$$

Step 5: Identifying cause and effect groups

The sum of the rows in the T matrix is determined by ri and the sum of the columns by cj. Cause and effect groups are determined by calculating "ri – cj" and "ri + cj" values.

The "r" obtained as a result of row sums shows the effect of the ith factor on other factors. The sum of the columns "cj" shows the effect of other factors on the ith factor. "ri + cj" values show the total effect and the effective value of the relevant factor, in other words, the degree of relations with other criteria.

Among the "ri – cj" values, those with positive values express those that affect other criteria, while those with negative values express those who are affected by other criteria. In other words, the value of "ri – cj" expresses the effect of that criterion on the system [32].

Step 6: Producing diagrams of cause and effect groups

Diagrams are obtained by showing "ri + cj" values on the horizontal axis and "ri – cj" values on the vertical axis on the coordinate plane. If the (ri � cj) axis is positive, the factor is in the cause group. Otherwise, if the (ri � cj) axis is negative, the factor is in the effect group.

A threshold value is determined in order to get rid of the complexity of the criteria with a small effect level. The threshold value is determined by averaging the values in the total correlation matrix or by an expert group. The criteria below the threshold value are determined as the affected (effect) criteria, and the criteria above the threshold value are determined as the affecting (cause) criteria [33].

Step 7: Calculating criterion weights

The following formula was used to calculate the criterion weights [34].

$$w\_i = \sqrt{\left[\left(r\_i + c\_j\right)\right]2 + \left[\left(r\_i - c\_j\right)\right]2} \tag{13}$$

$$\text{Wi} = \frac{wi}{\sum\_{i=1}^n wi}$$

Step 8: Operating the steps for the main criteria

All the steps described above are operated to determine the main criterion weights. Step 9: Operating the steps for sub-criteria.

All the steps described above are operated for the sub-criteria under each main criterion group in order to calculate the sub-criteria weights, and as a result, the sub-criteria weights are calculated.

Step 10: Integrating main criterion and sub-criteria weights

The final weights are calculated by multiplying the weights of the main criteria with the weights of the sub-criteria.
