**4.1 Numerical example**

Suppose three similar dangers produce danger zones DZ1 to DZ3, limited by three serial danger barriers DB1 to DB3 with the following parameters (**Tables 1** and **2**):

Because the danger barriers are supposed on serial, the model accumulates their effects. For this example, using bounded sum fuzzy union

(*μ DB*e <sup>1</sup>⊕*DB*e <sup>2</sup> ð Þ¼ *x* min 1, *μ DB*e <sup>1</sup> ð Þþ *x μ DB*e <sup>2</sup> ð Þ *x* � �), the effective danger barrier is: Therefore, the danger barriers inefficiency zones are (**Table 3**).

Suppose the barriers reduce some specific proportion of the danger, so the residual danger zones after applying the above barriers, by using *EDZ* g ¼ *DZ*f ∩ *IDB* g and the Algebraic product (*μ EDZ* <sup>f</sup> ð Þ¼ *<sup>x</sup> <sup>μ</sup> DZ*<sup>e</sup> ð Þ *<sup>x</sup> <sup>μ</sup> IDB* <sup>f</sup>ð Þ *<sup>x</sup>* ) intersection are (**Table 4**).

Consider 4 targets, using the following protective barriers for targets T1 to T4 (**Table 5**).

The above values indicate the protectives proportion of targets barriers used by different targets in different places. Using the fuzzy complement, the Inefficiency of the above protective measures is (**Table 6**).

The impacted dangers 1 to 3 to target 1, to target 1, after applying target barrier 1 is (**Table 7**).

In the above table, the model uses the bounded sum union to calculate the accumulated danger in each position.

In the same way, the total impacted danger for all the targets is (**Table 8**).


#### **Table 1.**

*The risk entities parameters.*


**Table 2.**

*Membership of different coordinates in effective danger barriers fuzzy set.*


#### **Table 3.**

*Membership of different coordinates in effective danger barrier inefficiency fuzzy set.*


#### **Table 4.**

*Membership of different coordinates in residual fuzzy danger zones.*


#### **Table 5.**

*Membership of different coordinates in fuzzy target barriers.*


#### **Table 6.**

*Membership of different coordinates in residual fuzzy target barriers Inefficiency.*


**Table 7.**

*Membership of different coordinates in residual fuzzy danger zones.*


#### **Table 8.**

*Membership of different coordinates in total fuzzy danger zones.*

Suppose the position of the targets is a random variable with the following 132probabilities (**Table 9**).

Therefore the total risk for every target at each position is (**Table 10**).

The following figure shows the distribution of the danger (**Figure 7**).

The above chart shows that the most dangerous place is point 2, with a total risk index of 1.382, and particularly the risk for target 4 is very high at this point.

In the continue consider a target presence barrier with the following reliabilities (**Table 11**).

It impedes particularly the targets presented in point 2. The Inefficiency of this barrier is (**Table 12**).

Using algebraic product fuzzy intersection, the presence probability of the targets is (**Table 13**).

By applying algebraic product fuzzy intersection on target presence probability and targets membership in danger zones, the total risk for the targets is (**Table 14**).

The following chart illustrates the risk after applying this barrier (**Figure 8**). The results indicate a reduction in the risk significantly.


**Table 9.**

*Presence probability of targets in different coordinates.*


**Table 10.**

*Total risk for targets in different coordinates.*

#### **Figure 7.** *The distribution of the risk for the targets.*


#### **Table 11.**

*The effects of presence barriers for targets in different coordinates.*


#### **Table 12.**

*The effects of presence barriers Inefficiency for targets in different coordinates*


#### **Table 13.**

*The modified presence probability of targets in different coordinates.*


#### **Table 14.**

*The total risk for targets in different coordinates.*

#### **Figure 8.**

*The distribution of the risk for the targets, after applying the presence target.*

## **5. Discussion**

The proposed fuzzy analytical approach attempts to simplify the complexity of the traditional risk analysis by demonstrating the geometric profiles of risk analysis entities. This method models dangers, target presence, material, and immaterial barriers and provides a communication/analysis tool in graphical design platforms.

By using this approach in simulation applications, danger, target, and value attributes may vary during the simulated period. For instance, the magnitude of the HA and target position may vary according to the target and HA position. A software may calculate these parameters for each of the simulation sequences separately. The duration of simulation sequences is essential for calculating a total risk index for a simulated operation.

Shahrokhi and Bernard (2006) presented more discussion about the target vulnerability and worth. An event tree analysis may calculate the probability of occurrence of the simulated conditions. Defining the danger zone and target presence zone provided an index for quantitative risk analysis. The quantitative approaches require carefully scaling factors. For a fixed target, the presence zone will have infinite amplitude. The adaptability of fuzzy operations provides excellent flexibility to tailor the model according to typical situations. The method improves by considering several risks for a group of targets by applying fuzzy operations. Though the impact mode, including the impact duration and direction, is fundamental to estimating the

accident severities, authors believe that in many risk analysis methods, including energy/barrier analysis, the assumptions related to the impact mode are not robust and sufficient.

Most of the barriers have not only protection effects; they relocate or reform DZ or TZ. For example, a protection wall increases the concentration of the DZ and TZ in a limited space.

Other fuzzy union operators can model the modification of the DZ/TZ by the barriers.

This model assumes a linear relationship between the damage and impact time because the presence zone demonstrates the duration of the target presence at each point. However, by defining the presence zone as the population density, the model ignores each target impact time. Like many other risk analysis methods, there is no assumption about other impact mode attributes. Therefore, the model's validity depends on the system's specifications. The model considers the danger zone as a stable and fixed region. In this case, the fundamentals theories are valid only for separated sequence times. A moving and dynamic danger zone is more appropriate if the danger source's harmfulness or position is unstable. Shahrokhi and Bernard (2006) discussed this case.

This model is applicable for calculating the cumulative risk indexes for a group of targets and hazards.
