**3. Methodology**

The presented method defines danger and target zones by fuzzy membership and probability density functions, illustrating them by geometric profiles in the 3D model.

#### **3.1 Defining danger zones**

A danger zone (DZ) is a portion of space where a harmful event may occur [10]. Shahrokhi and Bernard define fuzzy space to calculate every point x's membership in the danger zone [6]. According to the fuzzy sets notification, the following formula uses the integral and division symbols to explain that *DZ*f is a continuous fuzzy set and assigns membership *μ* DZ<sup>e</sup> ð Þ <sup>x</sup> to each x in the workplace.

$$\widehat{DZ} = \int\_{\mathfrak{x}} \frac{\mu\_{\widetilde{DZ}}(\mathfrak{x})}{\mathfrak{x}} \quad \mathbf{0} \le \mu\_{\widetilde{DZ}}(\mathfrak{x}) \le \mathbf{1} \quad \forall \mathfrak{x} \in X$$

As **Figure 1** presents, in contrast to the classic definition, a member (e.g., a point) is not just inside or outside of a fuzzy danger zone (*DZ*f ); it may also have a degree of membership. Function *μ DZ*<sup>e</sup> ð Þ *<sup>x</sup>* indicates the membership of point x in danger zone *DZ*<sup>f</sup> .

**Figure 2** illustrates a mono-dimensional danger zone and shows how a HA decreases by increasing the distance from the danger source.

A dangerous source can be an explosion, radiation, or a toxic gas leak. In all cases, a *DZ*f assigns a membership value *μ DZ*<sup>e</sup> ð Þ *<sup>x</sup>* to each point, x. The *DZ*<sup>f</sup> form is a function of the physical characteristics of the danger and the environmental conditions. For

**Figure 1.**

*Demonstration of the dangerous zone around a moving car using (a) the traditional and (b) the fuzzy space definitions [6].*

**Figure 2.** *Schematic demonstrations of a one-dimensional DZ.* f

example, for a punctual radioactive material, the radiation in a uniform environment decreases proportionally to the distance squared from the danger source.

### **3.2 Aggregating several danger zones**

Fuzzy union apples on several dangers to form a total danger zone, as follows:

$$\widetilde{D}\widetilde{Z} = \bigcup\_{i=1}^{n} \widetilde{D}\widetilde{Z}\_{i}$$

The selection of an appropriate union method among standard union (max(a,b)), bounded sum (min(1,a + b)), and other fuzzy union operations are essential to simulate the actual effect of accumulated dangers on the target.

#### **3.3 Defining the target zone**

A fuzzy target zone (*TZ*f) is also a fuzzy space, indicating the geographical distribution of the presence of the target:

*Risk Analysis, a Fuzzy Analytic Approach DOI: http://dx.doi.org/10.5772/intechopen.108535*

$$\widetilde{TZ} = \int\_{\widetilde{\mathfrak{x}}}^{\mu\_{\widetilde{\mathfrak{X}}}(\mathfrak{x})} \frac{\mu\_{\widetilde{\mathfrak{X}}}(\mathfrak{x})}{\mathfrak{x}}$$

The *μ TZ*<sup>e</sup> ð Þ *<sup>x</sup>* is the normalized target population density (P(x)) or the target presence probability f(X) in point x. The following normalization formula converts the population density to a membership value by ensuring that the maximum membership value is 1:

$$\mu\_{\widetilde{TZ}}(\mathfrak{x}) = \left(\frac{P(\mathfrak{x})}{\sup\_{\mathfrak{x}} (P(\mathfrak{x}))}\right)\_{\mathfrak{x}}$$

It divides the population density of every point to their maximum (supremum) value for every x. For a target with random movement or stochastic existence, the model normalizes the probability function of the target presence, f(X), by using the following formula:

$$\mu\_{\widetilde{TZ}}(\mathfrak{x}) = \left(\frac{f(\mathfrak{x})}{\sup\_{\mathfrak{x}}(|f(\mathfrak{x})|)}\right)\_{\mathfrak{x}}$$

The membership function of TZ (i.e., *μ TZ*<sup>e</sup> ð Þ *<sup>x</sup>* ) indicates the distribution of the targets in the workplace. A fixed target creates a singleton fuzzy set target zone.

## **3.4 Modeling the barriers**

The barriers limit danger and target zone(s) in several ways:


The proposed approach models the above barriers by using geometric shapes. Fuzzy barriers (i.e., *TBs* f and *DBs* f ) illustrate the geographical distribution of barriers' capability to impede the danger and target presence.

#### **3.5 Modeling danger barriers**

**Figure 3a–c** exemplify danger barriers for the following cases, respectively:


**Figure 3.** *Some schematic demonstration of danger barriers.*

This approach uses fuzzy complement operation to transfer the danger barriers protection to danger barrier inefficiency (*DBI* g) as follows

$$
\bar{D}\bar{B}I = \bar{\cdot}\bar{D}\bar{B} \quad \mu\_{\widetilde{DBI}}(\mathfrak{x}) = \mathbf{1} - \mu\_{\widetilde{D}\tilde{B}}(\mathfrak{x}) \quad \forall \mathfrak{x} \in X
$$

The risk exists when there is a danger and no barriers to neutralize the threat. Therefore, the practical (residual) danger zone equals the intersection of the original danger zone and the danger barrier inefficiency.

$$
\widetilde{EDZ} = \widetilde{DZ} \cap \left( \neg \widetilde{DB} \right),
$$

The following equation is applied when barrier effectiveness is proportional (e.g., using percentages). It means that the barrier reduces a specified portion of the danger.

$$\mu\_{\widetilde{\mathit{EDZ}}}(\mathbf{x}) = \mu\_{\widetilde{\mathit{DZ}}}(\mathbf{x}) \left(\mathbf{1} - \mu\_{\widetilde{\mathit{DB}}}(\mathbf{x})\right) \qquad \forall \mathbf{x} \in X$$

The following equation is valid when the barrier effectiveness is in absolute values (e.g., the barrier absorbs or filters a specified amount of the hazardous effects).

$$\mu\_{\widetilde{\rm{EDZ}}}(\mathbf{x}) = \mathbf{Max}\left(\mathbf{0}, \mu\_{\widetilde{\rm{DZ}}}(\mathbf{x}) - \left(\mathbf{1} - \mu\_{\widetilde{\rm{DZ}}}(\mathbf{x})\right)\right) \quad \forall \mathbf{x} \in X.$$

#### **3.6 Modeling target barriers**

The exemplified TBs affect the *TZ*f, For example, they may describe the following cases:

a detector, organizational measure or warning signs (**Figure 4a**), or a wall that prevents the presence of the target in a limited zone with specified reliability (**Figure 4b**), and a thermal protective cloth which controls 30% of the outside temperature from colliding with the wearer body (**Figure 4c**).

Some protective measures may affect both the *DZ*f and *TZ*f simultaneously; for example, **Figures 3b** and **4b** model effects of the same protection wall on the *DZ*f and *TZ*f, respectively.

*Risk Analysis, a Fuzzy Analytic Approach DOI: http://dx.doi.org/10.5772/intechopen.108535*

**Figure 4.** *Some schematic demonstration of target barriers.*

#### **3.7 Cumulative effects of several barriers**

There are cumulative effects where two or several barriers are practical at the same place and time. The fuzzy union operator aggregates a set of J danger barriers and a set of K target barriers as effective danger barriers and practical target barriers, using the following formulas:

$$
\widehat{EDB}(\mathbf{x}) = \bigcup\_{j=1}^{J} \widehat{DB}\_j(\mathbf{x}).
$$

$$
\widehat{ETB} = \bigcup\_{k=1}^{K} \widehat{TB}\_k
$$

Fuzzy spaces *EDB* g and *ETB* g are the effective danger zone barrier and effective target barrier, respectively.

The definition of the fuzzy union in the above equations varies according to the cumulative barrier effects. The proposed approach defines serial and parallel barriers. In a serial barrier configuration, the danger must pass from all obstacles to impact the target (e.g., the consecutive antifire doors); in this case, the bounded sum (min(1, a + b)) is one of the appropriate s-norms if the danger reduced after passing from each barrier.

The standard union (i.e., max(a,b)) is a helpful s-norm when the most effective barrier is essential in limiting the danger zone or target presence zones.

In a parallel structure, it is sufficient for the threat to pass through one of the guards to impact the target. The analyst may consider the most unreliable barrier as the weakest link in the protection chain. A fuzzy intersection operator such as the standard intersection (e.g., min(a,b)) aggregates the parallel safety measures as effective danger follows:

$$\overline{EDB} = \prescript{}{\frown}{DB\_j}$$

$$\overline{ETB} = \prescript{}{\frown}{T\overline{B}\_k}$$

#### **3.8 Barriers inefficiency**

Appling fuzzy compliment operator on "effective danger barriers" and "effective target barriers" results in "danger barriers inefficiency" and "target barriers inefficiency," respectively, as follows:

$$
\widetilde{IDB} = \neg \widetilde{DB}
$$

$$
\widetilde{ITB} = \neg \widetilde{TB}
$$

Operator ⌐ means fuzzy complement (fuzzy NOT) operation. The standard complement (⌐a = 1-a) is one of the most suitable fuzzy complementation methods. However, also there are other alternatives for this operator.

$$\mu\_{\widetilde{IDB}}^{\sim}(\mathbf{x}) = \mathbf{1} - \mu\_{\widetilde{ID}}^{\sim}(\mathbf{x}) \qquad \forall \mathbf{x} \in \mathbf{X}$$

$$\mu\_{\widetilde{IDB}}^{\sim}(\mathbf{x}) = \mathbf{1} - \mu\_{\widetilde{ID}}^{\sim}(\mathbf{x}) \qquad \forall \mathbf{x} \in \mathbf{X}$$

#### **3.9 Apply barrier effects to the danger zone**

Danger remains in dangerous places, but there is not enough protection; this means that the residual (effective) hazard at each point equals the intersection of the threat and the barriers inefficiency at that point. Therefore effective danger for every danger zone is:

$$
\widehat{EDZ} = \widehat{DZ} \cap \widehat{IDB}
$$

In the same way, a practical target presence zone is:

$$
\widetilde{ETZ} = \widetilde{TZ} \cap \widetilde{ITB}
$$

Using the *EDZ* g and *ETZ* g, the fuzzy risk zone is:

$$
\widetilde{RZ} = \widetilde{EDZ} \cap \widetilde{ETZ}
$$

In this case, multiplication is one of the alternatives for fuzzy intersection operation.

#### **3.10 Determining the fuzzy risk zone**

The proposed risk analysis approach uses the fundamentals of energy analysis and considers an accident resulting from the impact of a harmful agent (energy) on a target. Therefore, the fuzzy risk zone (*RZ*f)is an intersection area of a *DZ*f and a *TZ*f. The risk analyst should select the most appropriate fuzzy intersection operator (i.e., triangular norms (t-norm)) to reflect the accident consequence best. Triangular norms are indispensable for interpreting the conjunction in the intersection of fuzzy sets. One of the simplest choices is the product intersection, defined as:

$$
\widetilde{RZ} = \widetilde{DZ} \cap \widetilde{TZ}
$$

$$
\mu\_{\widetilde{Z}}(\mathbf{x}) = \mu\_{\widetilde{Z}}(\mathbf{x})\mu\_{\widetilde{TZ}}(\mathbf{x}) \qquad \forall \mathbf{x} \in X
$$

This formula uses the concept that the accident importance equals the multiplication of the hazard amplitude and the target presence probability or density. Other tnorms may be more appropriate for specific cases.

**Figure 5** illustrates the distribution of targets in the neighborhood of a supposed hazard source.

**Figure 6a** shows both *DZ*f and *TZ*f, and **Figure 6b** illustrates the resulting risk zone using the algebraic product t-norm as the fuzzy intersection operator. The horizontal axis corresponds to the distance from the hazard source. The vertical axis illustrates the risk index function; thus, risk zone (*RZ*f) presents the geographic distribution of the risk amplitude.

If the hazardous effects of several dangers are not similar, *RZ*f should be calculated separately for different hazards.

**Figure 5.** *Schematic demonstrations of a one-dimensional TZ.* f

**Figure 6.** *Schematic demonstrations of one-dimensional DZ,* f *TZ, and* f *RZ.* f
