**3. Fuzzy model**

44 Fuzzy Inference System – Theory and Applications

This model is very simple, but it has some structural drawbacks. For example the system + which is in Defenseless situation will change to entirely different condition (Endangered), point (**\***), with infinity small deviation in ability to retaliate. Also because of its geometrical structure, this model suffers from the lack of considering additional parameters for risk analysis. Another method which has gained more attraction in the risk analysis literature is the model presented by Eq. (3), based on the linear combination of ability to retaliate and

Fig. 3 represents the continuous increasing surface (risk levels) generated by means of Eq. (3). Two particular levels are shown by the cutting planes K1 and K2. Positions O, + and \* are also presented in this figure. Fig. 4 shows some contours of risky surface. As it is seen, by using this model any small change in the values of impact threat and ability to retaliate will cause a very small deviation on the risky level of system. This model is more realistic than the one presented by Figs. 1 and 2. However, it also has its limitations for real world applications because it simplifies the complicated relation between different factors to a

To have a more applicable model, we can formulate our problem as an input output system

Where X is the set of input variables which affect the level of the risk, R is the level of the

R=F (X) (4)

Risk = (ability to retaliate) × (impact threat) (3)

impact threat.

linear relation.

Fig. 3. Continuous surface for risk levels

risk and F(.) is a nonlinear function (Kreinovich et al., 2000).

by:

Fuzzy inference systems (FIS) are rule-based systems with concepts and operations associated with fuzzy set theory and fuzzy logic (Mendel, 2001; Ross, 2010). These systems map an input space to an output state; therefore, they allow constructing structures that can be used to generate responses (outputs) to certain stimulations (inputs), based on stored knowledge on how the responses and stimulations are related. The knowledge is stored in the form of a rule base, i.e. a set of rules that express the relations between inputs and the expected outputs of the system. Sometimes this knowledge is obtained by eliciting information from specialists. These systems are known as fuzzy expert systems (Takács, 2004). Another common denomination for FIS is fuzzy control systems (see for example (Mendel, 2001)).

FIS are usually divided in two categories (Mendel, 2001; Takagi & Sugeno, 1985): multiple input, multiple output (MIMO) systems, where the system returns several outputs based on the inputs it receives; and multiple input, single output (MISO) systems, where only one output is returned from multiple inputs. Since MIMO systems can be decomposed into a set of MISO systems working in parallel, all that follows will be exposed from a MISO point of view (Mamdani & Assilian, 1999). In our risk analysis model a fuzzy inference system is introduced for calculating the risky situations of systems by considering different factors such as probability, impact threat and ability to retaliate (Cho et al., 2002; Nguene & Finger, 2007). Fig. 5 shows the block diagram of a multi input single output fuzzy risk analysis system for the mentioned factors (Carr & Tah, 2001).

A Fuzzy Approach for Risk Analysis with Application in Project Management 47

(a)

(b) Fig. 7. (a) Risky surface and (b) Counters of the simple example by using fuzzy inference

As it is seen in Fig. 7, organization + which is in appropriately Defenseless situation will change to appropriately Endangered situation, point **(\***), with infinity small deviations in ability to retaliate and in impact threat, which is more realistic comparing to the classic one. To have an idea on utilization of risk management on criticality of activities besides other

criticality criteria, the multi critical PERT is introduced in section 4.

system

Risk Level

Fig. 5. Block diagram of fuzzy inference system for risk analysis

In this work the following Bell shape membership function is used to determine the fuzzy values of inputs for determining the risky levels of activities by FIS.

$$
\mu\_A(\mathbf{x}) = \frac{1}{1 + d(\mathbf{x} - \mathbf{c})^2} \tag{5}
$$

Where µA(x) is the membership of variable x in fuzzy value A, c is the median of the fuzzy value and d is the shape parameter. Fig. 6 shows the bell shape membership functions for different fuzzy verbal values.

Fig. 6. Membership functions of different fuzzy verbal values vs: Very Small, sm: Small, md: Medium, bg: Big, vb: Very Big

The reason for implementing bell shape membership function is that because of its smoothness (comparing Triangular memberships), and simple formula (comparing Gaussian memberships) it is more appropriate for getting qualitative data from experts.

This model is implemented to the simple model of risk analysis, presented in section 2, to have an idea on the main difference between the classic and fuzzy risky levels. Fig. 7 shows the surface and counters of risky levels of organizations +, and O for 50% probability of impact.

FIS

In this work the following Bell shape membership function is used to determine the fuzzy

<sup>1</sup> ( ) 1( ) *<sup>A</sup> <sup>x</sup>*

Where µA(x) is the membership of variable x in fuzzy value A, c is the median of the fuzzy value and d is the shape parameter. Fig. 6 shows the bell shape membership functions for

vs sm md bg vb

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 6. Membership functions of different fuzzy verbal values vs: Very Small, sm: Small, md:

The reason for implementing bell shape membership function is that because of its smoothness (comparing Triangular memberships), and simple formula (comparing Gaussian memberships) it is more appropriate for getting qualitative data from experts.

This model is implemented to the simple model of risk analysis, presented in section 2, to have an idea on the main difference between the classic and fuzzy risky levels. Fig. 7 shows the surface and counters of risky levels of organizations +, and O for 50% probability of impact.

2

(5)

Risk (Vulnerability)

*dx c*

Fig. 5. Block diagram of fuzzy inference system for risk analysis

Probability

Impact

Ability to retaliate

different fuzzy verbal values.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Medium, bg: Big, vb: Very Big

values of inputs for determining the risky levels of activities by FIS.

Fig. 7. (a) Risky surface and (b) Counters of the simple example by using fuzzy inference system

As it is seen in Fig. 7, organization + which is in appropriately Defenseless situation will change to appropriately Endangered situation, point **(\***), with infinity small deviations in ability to retaliate and in impact threat, which is more realistic comparing to the classic one. To have an idea on utilization of risk management on criticality of activities besides other criticality criteria, the multi critical PERT is introduced in section 4.

A Fuzzy Approach for Risk Analysis with Application in Project Management 49

Fig. 9 shows the network representation of a typical project. The data for activities is

To compare the efficiency of multi critical PERT with the classic one, 1000 tests are performed using Mont Carlo simulation by generating uniform distributed random numbers r to be used in Equations (10) and (11). For each activity, two costs of impact are

Expense\_on\_SCA=max {0,r-SCA} (10)

a. SCA is considered as a factor of criticality (Expense\_on\_SCA), by using Eq. (10)

**Step 9.** Classify activities based on MPCs.

Fig. 8. Rule base generated by ANFIS

Fig. 9. Network representation of typical project

represented in table 2.

calculated where:

*MPC w V w PFA w RLA w SFA w SCA w COR* 12 3 4 5 6 (9)
