**2. Classic and fuzzy risk analysis**

Fig. 1 shows a classic and simple model of risk analysis. It consists of two factors: Impact threat and ability to retaliate. In this model the risk value is classified in four groups. Each group represents a risky condition for the system (organization, project, activity …). Fig. 2 shows the points (situations) with identical risky levels. The distributions of points with the identical risks (contours of different levels) are also presented in Fig. 2. Points O and + represent the risky situations for two systems with ability to retaliate and impact threats of (1,8) and (4.9,5.1) respectively.

without considering other real world factors such as probability of impacts on project (such as inflation or stagnation) , impact threat and ability to retaliate. Hence a new approach based on fuzzy inference system and fuzzy decision making is introduced to have more realistic procedure for project management in real world applications. Fuzzy set is introduced by Zadeh in 1965 (Zadeh, 1965). Different applications of fuzzy sets are studied by researches in different fields (Jamshidi et al., 1993). T. J. Ross has published an interesting book on fuzzy sets theory and its applications in engineering (Ross, 2010). Several papers are also published on applications of fuzzy sets in project management (Chanas et al., 2002; Shipley et al, 1997). M. F. Shipley et al. have used the fuzzy logic approach for determining probabilistic fuzzy expected values in a project management application (Shipley et al, 1996). An extension of their method is introduced and used for determination of expected values for estimated delays of activities in (Khanmohammadi et al., 2001). The procedure introduced here deals with defining multi-purpose criticalities for activities where some other features such as probability of impact, impact threat and ability to retaliate are considered as criticality factors of activities in project management process. In this way the risky situations (vulnerabilities) of activities are calculated using a fuzzy inference system which will be used for calculating the risky situation for each activity as a main criticality

Considerable quantitative models have been introduced in literature to calculate the level of risk; which is simply defined as the rate of threat or future deficit of any system imposed by controllable or uncontrollable variables (Chavas, 2004; Doherty, 2000). Several factors such as probability of occurrence, impact threat and ability to retaliate are introduced as affecting factors on the risk. Then it is tried to find the mathematical relation between affecting factors and the value (level) of the risk (Li & Liao, 2007; McNeil et al., 2005). The concept of risk is considerably wide. It can contain strategic, financial,

The concept of fuzzy risky conditions for activities is introduced in sections 2 and 3. In section 4 the concept of Multi-Critical PERT by considering risky levels for activities is introduced and a typical project network is considered as a case study for analyzing the effect of imposing the risky level of activities to criticality. The results are compared to classic PERT by means of Mont Carlo simulation using random variables. Another typical example, project management of rescue robot that provides preliminary processes for helping injured people before the arrival of rescue teams, is studied in section 5. Analysis of

Fig. 1 shows a classic and simple model of risk analysis. It consists of two factors: Impact threat and ability to retaliate. In this model the risk value is classified in four groups. Each group represents a risky condition for the system (organization, project, activity …). Fig. 2 shows the points (situations) with identical risky levels. The distributions of points with the identical risks (contours of different levels) are also presented in Fig. 2. Points O and + represent the risky situations for two systems with ability to retaliate and impact threats of

factor.

operational or any other type of risk.

**2. Classic and fuzzy risk analysis** 

(1,8) and (4.9,5.1) respectively.

obtained results and conclusions are presented in section 6.

Fig. 1. Risky situations classified in 4 levels

Fig. 2. Different levels of situations (contours of Fig. 1.)

A Fuzzy Approach for Risk Analysis with Application in Project Management 45

The problem here is to find an appropriate model by which the level of risk of the system can be determined in complex situations where there is no access to all data, or the historical

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

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

data is useless. This problem may be solved by using Fuzzy inference system.

Fig. 4. Some contours of Fig. 3.

**3. Fuzzy model** 

(Mendel, 2001)).

system for the mentioned factors (Carr & Tah, 2001).

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 impact threat.

$$\text{Risk} = \text{(ability to retained)} \times \text{(impact that)} \tag{3}$$

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 linear relation.

Fig. 3. Continuous surface for risk levels

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

$$\mathbb{R} = \mathbb{F} \left( \mathbb{X} \right) \tag{4}$$

Where X is the set of input variables which affect the level of the risk, R is the level of the risk and F(.) is a nonlinear function (Kreinovich et al., 2000).

The problem here is to find an appropriate model by which the level of risk of the system can be determined in complex situations where there is no access to all data, or the historical data is useless. This problem may be solved by using Fuzzy inference system.

Fig. 4. Some contours of Fig. 3.
