**4. Multi critical PERT by considering risky levels**

The multi critical PERT uses the data presented by table 1 to define the multi-purpose criticality of activities.


Table 1. Data used by Multi-Critical PERT

The procedure for using these data to calculate the multi-purpose criticalities of activities is as follows:

Algorithm


$$PFA = p(D \le DR) \tag{6}$$

**Step 4.** Use the fuzzy inference system to calculate the Risky Level of each Activity RLA by using the fuzzy values of probability of impact pr, impact treat im, and ability to retaliate ar.

The following experimental data gathered from experts are fed to ANFIS (Artificial Neural Fuzzy Inference System) in MATLAB and 14 appropriate FIS rules (Fig. 8) are generated by means of "genfis3" for the case study.

Probability = [1 .5 1.2 .8 .4 1.7 .8 .2 .2 .7 .5 .5 1 1 .6 .1 .3 .4]

Impact = [10 0 5 5 5 2 7 0 8 8 9 3 10 10 10 8 2 10 2 6 .8]

Ability to retaliate = [4 10 5 5 2 3 3 5 2 7 5 5 3 2 10 0 8 4 6 8 1]

Risky level = [7.5 0 1 3 0 5 1 0 4 2 .5 0 4.5 2.5 0 10 .5 3 0 0 0]

**Step 5.** Normalize the free slack times of activities by dividing them to their maximum value. Calculate the Severity of Free slack times of Activities SFA based on durations of activities by:

$$\text{SFA} = \text{1-normalized FS} \tag{7}$$

**Step 6.** Normalize the total slack times of activities by dividing them to their maximum value. Calculate the Severity of Criticalities of Activities SCA based on durations of activities by:

$$\text{SCA} = \text{1-normalized TS} \tag{8}$$


$$\text{MPCC} = w\_1 \times V + w\_2 \times \text{PFA} + w\_3 \times \text{RLA} + w\_4 \times \text{SFA} + w\_5 \times \text{SCA} + w\_6 \times \text{COR} \tag{9}$$

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

The multi critical PERT uses the data presented by table 1 to define the multi-purpose

Activity a m b V PFA RLA SFA SCA COR MPC

The procedure for using these data to calculate the multi-purpose criticalities of activities is

**Step 1.** Perform classic PERT to calculate Durations of activities D, variances V, Earliest

**Step 3.** Calculate the Probability of Finishing each Activity PFA in duration range DR, by

**Step 4.** Use the fuzzy inference system to calculate the Risky Level of each Activity RLA by

The following experimental data gathered from experts are fed to ANFIS (Artificial Neural Fuzzy Inference System) in MATLAB and 14 appropriate FIS rules (Fig. 8) are generated by

**Step 5.** Normalize the free slack times of activities by dividing them to their maximum

**Step 6.** Normalize the total slack times of activities by dividing them to their maximum

**Step 7.** Perform CPM to calculate total slacks of activities where RLAs are used instead of

**Step 8.** Use V, SFA, SCA, PFA, RLA and COR as criteria with corresponding weighs Wi

Normalize CORs by dividing them to their maximum value.

value. Calculate the Severity of Free slack times of Activities SFA based on

value. Calculate the Severity of Criticalities of Activities SCA based on durations of

durations for activities to calculate the criticalities based on risky levels (COR).

(defined by experts), to calculate Multi-Purpose Criticalities (MPC) of activities,

using the fuzzy values of probability of impact pr, impact treat im, and ability to

where scheduled times ST may be imposed to different events.

**Step 2.** Calculate the Duration Range of activities DR=LF-ES.

considering duration D and standard deviation

Probability = [1 .5 1.2 .8 .4 1.7 .8 .2 .2 .7 .5 .5 1 1 .6 .1 .3 .4] Impact = [10 0 5 5 5 2 7 0 8 8 9 3 10 10 10 8 2 10 2 6 .8] Ability to retaliate = [4 10 5 5 2 3 3 5 2 7 5 5 3 2 10 0 8 4 6 8 1] Risky level = [7.5 0 1 3 0 5 1 0 4 2 .5 0 4.5 2.5 0 10 .5 3 0 0 0]

Start Times ES, Latest Finish times LF, Free slack times FS and Total slack times TS,

*V* .

*PFA p D DR* ( ) (6)

SFA = 1- normalized FS (7)

SCA = 1- normalized TS (8)

**4. Multi critical PERT by considering risky levels** 

Table 1. Data used by Multi-Critical PERT

criticality of activities.

retaliate ar.

means of "genfis3" for the case study.

durations of activities by:

where for each activity:

activities by:

as follows: Algorithm

Fig. 8. Rule base generated by ANFIS

Fig. 9 shows the network representation of a typical project. The data for activities is represented in table 2.

Fig. 9. Network representation of typical project

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 calculated where:

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

$$\text{Expense\\_on\\_SCA} = \max\left\{0, \text{r-SCA}\right\} \tag{10}$$

A Fuzzy Approach for Risk Analysis with Application in Project Management 51

As another interesting application, a heuristic method for simultaneous rescue robot pathplanning and mission scheduling is introduced based on Graphic Evaluation and Review Technique (GERT) (Alan & Pritsker, 1966), along with multi criteria decision making and

Consider some groups of injured people who are trapped in a disastrous situation. These people are categorized into several groups based on the severity of their situation. A rescue robot, whose ultimate objective is to reach injured groups and provide preliminary aid for them through a path with minimum risk, has to perform certain tasks on its way towards targets before the arrival of rescue team. A decision value is assigned to each target based on the whole degree of satisfaction of the criteria and duties of the robot in the way toward the target, and the importance of rescuing each group based on their category and the number of injured people. The resulted decision value defines the strength of the attractive potential field of each target. Dangerous environmental parameters are defined as obstacles whose risk determines the strength of the repulsive potential field of each obstacle. Moreover, negative and positive energies are assigned to the targets and obstacles respectively. These

The potential field method has been studied extensively for mobile robot path planning (Latombe, 1990). The basic idea behind the potential field method is to define an artificial potential field (energy) in the robot's workspace in which the robot is attracted to its goal position and is repulsed away from the obstacles (Alsultan & Aliyu, 1996; Khanmohammadi & Soltani, 2011). Despite the problems in architecture of potential field such as local minima and oscillation in narrow passages, this method is particularly attractive because of its mathematical elegance and simplicity (Casper & Yanco, 2002; Chadwick, 2005; Tadokoro et al, 2000). For simplicity, we assume that the robot is of point mass and moves in a twodimensional (2-D) workspace. Its position in the workspace is denoted by q = [x y]T. The most commonly used attractive potential Uatt and the corresponding attractive force Fatt

> <sup>1</sup> () ( ,) <sup>2</sup> *<sup>m</sup> Uq q q att goal*

( ) *F U att att goal q q*

Where ξ is a positive scaling factor, ρ (qgoal,q) = ║qgoal - q║ is the distance between the robot q and the goal qgoal, and m = 1 or 2. For m = 1, the attractive potential is conic in shape and the resulting attractive force has constant amplitude except at the goal, where Uatt is singular. For m = 2, the attractive potential is parabolic in shape. Also, the attractive force

One commonly used repulsive potential function and the corresponding repulsive force is

(12)

artificial potential fields path-planning.

energies vary with respect to different environmental factors.

converges linearly toward zero as the robot approaches the goal.

**5. Rescue robot path planning** 

**5.1 Potential feld path planning** 

takes the form:

given by:


b. MPC is considered as a factor of criticality (Expense\_on\_MPC), by using Eq. (11)

Table 2. Activities with appropriate data generated in different steps

Considering that the expense of each unit of impact is 1000\$, the mean values of the obtained expenses for 1000 iterations are

Mean value of Expense\_on\_SCA = 2720.3 \$

Mean value of Expense\_on\_MPC = 1356.3 \$

It means that in real world applications, with probabilistic and non precise situations for finishing activities, if we consider MPC as the criticality of activities our project managements will be more realistic causing less expenses.

Fig. 10 represents the two Expenses, for 1000 tests.

Fig. 10. Two Expenses, for 1000 tests

As another interesting application, a heuristic method for simultaneous rescue robot pathplanning and mission scheduling is introduced based on Graphic Evaluation and Review Technique (GERT) (Alan & Pritsker, 1966), along with multi criteria decision making and artificial potential fields path-planning.

## **5. Rescue robot path planning**

50 Fuzzy Inference System – Theory and Applications

Expense\_on\_MPC=max {0,r-MPC} (11)

1-2 2 3 4 0.3906 3.0000 0.5003 0.0071 0.00 0.00 0.0000 0.1503 1-3 1 3 4 0.8789 4.8333 0.9842 0.5840 0.00 0.80 0.4691 0.8806 2-4 1 3 5 1.5625 3.0000 0.5003 1.0000 0.00 0.00 0.0000 0.5356 3-5 1 2 3 0.3906 4.0000 1.000 0.0006 0.00 0.80 1.0000 0.8054 3-6 2 5 7 2.4414 7.3333 0.9458 0.0024 1.00 1.00 0.4691 1.0000 4-6 3 4 6 0.8789 4.1667 0.5003 0.0008 0.00 0.00 0.0000 0.1704 4-7 3 4 5 0.3906 4.5000 0.7887 0.0787 0.00 0.20 0.4421 0.4310 5-7 1 4 5 1.5625 5.6667 0.9458 0.0000 0.60 0.80 1.0000 0.9560 6-8 2 5 6 1.5625 4.6667 0.5003 0.4829 0.00 0.00 0.0000 0.3628 7-8 3 4 7 1.5625 4.8333 0.6559 0.0077 0.40 0.20 0.4421 0.4633

Considering that the expense of each unit of impact is 1000\$, the mean values of the

It means that in real world applications, with probabilistic and non precise situations for finishing activities, if we consider MPC as the criticality of activities our project

0 100 200 300 400 500 600 700 800 900 1000

Test

RLA Step 4 W3=0.9

a m b Step 8

SFA Step 5 W4=0.5 SCA Step 6 W5=0.9

Expense on SCA Expense on MPC COR Step 7 W6=0.7 MPC

b. MPC is considered as a factor of criticality (Expense\_on\_MPC), by using Eq. (11)

PFA Step 3 W2=0.7

Activity Durations V

Step 1 W1=0.3

obtained expenses for 1000 iterations are Mean value of Expense\_on\_SCA = 2720.3 \$ Mean value of Expense\_on\_MPC = 1356.3 \$

Fig. 10. Two Expenses, for 1000 tests

Expense

DR Step 2

Table 2. Activities with appropriate data generated in different steps

managements will be more realistic causing less expenses.

Fig. 10 represents the two Expenses, for 1000 tests.

Consider some groups of injured people who are trapped in a disastrous situation. These people are categorized into several groups based on the severity of their situation. A rescue robot, whose ultimate objective is to reach injured groups and provide preliminary aid for them through a path with minimum risk, has to perform certain tasks on its way towards targets before the arrival of rescue team. A decision value is assigned to each target based on the whole degree of satisfaction of the criteria and duties of the robot in the way toward the target, and the importance of rescuing each group based on their category and the number of injured people. The resulted decision value defines the strength of the attractive potential field of each target. Dangerous environmental parameters are defined as obstacles whose risk determines the strength of the repulsive potential field of each obstacle. Moreover, negative and positive energies are assigned to the targets and obstacles respectively. These energies vary with respect to different environmental factors.

#### **5.1 Potential feld path planning**

The potential field method has been studied extensively for mobile robot path planning (Latombe, 1990). The basic idea behind the potential field method is to define an artificial potential field (energy) in the robot's workspace in which the robot is attracted to its goal position and is repulsed away from the obstacles (Alsultan & Aliyu, 1996; Khanmohammadi & Soltani, 2011). Despite the problems in architecture of potential field such as local minima and oscillation in narrow passages, this method is particularly attractive because of its mathematical elegance and simplicity (Casper & Yanco, 2002; Chadwick, 2005; Tadokoro et al, 2000). For simplicity, we assume that the robot is of point mass and moves in a twodimensional (2-D) workspace. Its position in the workspace is denoted by q = [x y]T. The most commonly used attractive potential Uatt and the corresponding attractive force Fatt takes the form:

$$\mathcal{U}\_{att}(q) = \frac{1}{2} \xi \rho^m(q\_{\text{goal}}, q) \tag{12}$$

$$F\_{att} = -\nabla \mathcal{U}\_{att} = \xi (q\_{\text{goal}} - q)$$

Where ξ is a positive scaling factor, ρ (qgoal,q) = ║qgoal - q║ is the distance between the robot q and the goal qgoal, and m = 1 or 2. For m = 1, the attractive potential is conic in shape and the resulting attractive force has constant amplitude except at the goal, where Uatt is singular. For m = 2, the attractive potential is parabolic in shape. Also, the attractive force converges linearly toward zero as the robot approaches the goal.

One commonly used repulsive potential function and the corresponding repulsive force is given by:

A Fuzzy Approach for Risk Analysis with Application in Project Management 53

values are treated as the virtual durations of activities and are given to CPM. It is obvious that output Es (Earliest starts representing the decision indexes of missions) of CPM can be interpreted as the degree of fulfillment of the activities leading to a certain event. By comparing the Es of the last events of several missions, we can deduce which mission fulfills

Human factors Environmental parameters Parameters Concerning the

destruction of path for the rescue team

danger in the peripheries

R4 Damage negligible

The ultimate objective of rescue mission is to help the injured people. The injured situations are divided into four groups: endangered, vulnerable, defenseless and prepared. To compare different groups of injured people four criteria are considered (refer to Table 6). The weights of criteria along with the degree of satisfaction of different criteria are given to MCDM algorithm and a decision value is calculated for each group of injured people as targets. In fact ξ (the positive scaling factor for attractive force) for each target is calculated

Where *norm* is normalization operator and *ADV*i is the Attraction Decision Value of the ith

Considering environmental situation and defining certain criteria for degree of danger of each obstacle, a similar approach is possible for determining the scaling factor *η* of the repulsive force. The degree of satisfaction of each criterion is fed into MCDM and the

Having obtained the corresponding strength of the attractive and repulsive potential field, the path planning algorithm is established and the optimal path with respect to least time,

resulting decision value equals the positive scaling factor of repulsive force:

Where *RDV*i is the Repulsive Decision Value of the ith obstacle.

least risk and most help to injured people is achieved.

E1 Prevention of air positioning in the surroundings

E2 Prevention

robot

R2 Annihilation of the robot

R3 Repairable damage to the robot

task

ξi = norm (Esi) + norm (ADVi) (15)

ηi = norm (RDVi) (16)

for the robot to be able to continue its

R1 Destruction of accessories

our criteria better than the other ones.

reducing the life risk of the rescue

personal damage to the injured person

Table 3. Main criteria for choosing the path

E3 Prevention of fire

H1 Capacity for

team

H2 Rescuing and preventing

as follows:

target.

$$\mathcal{U}\_{rep} = \left| \frac{1}{2} \eta \left( \frac{1}{\rho(q\_{\prime} q\_{obs})} - \frac{1}{\rho\_0} \right)^2 \right|, \qquad \text{if } \rho(q\_{\prime} q\_{obs}) \le \rho\_0 \tag{13}$$
 
$$\text{if } \rho(q\_{\prime} q\_{obs}) > \rho\_0$$

$$F\_{rep} = -\nabla L l\_{rep} = \begin{cases} \eta \left( \frac{1}{\rho(q\_\prime q\_{obs})} - \frac{1}{\rho\_0} \right) \frac{1}{\rho^2 (q\_\prime q\_{obs})} \nabla \rho(q\_\prime q\_{obs})\_\prime & \text{if } \rho(q\_\prime q\_{obs}) \le \rho\_0 \\ 0 & \text{if } \rho(q\_\prime q\_{obs}) > \rho\_0 \end{cases}$$

Where η is a positive scaling factor, ρ (q,qobs) denotes the minimal distance from the robot q to the obstacle, qobs denotes the point on the obstacle such that the distance between this point and the robot is minimal between the obstacle and the robot, and ρ0 is a positive constant denoting the distance of influence of the obstacle. The total force applied to the robot is the sum of the attractive force and the repulsive force which determines the motion of the robot (Jacoff et al., 2000).

$$F\_{\text{total}} = F\_{\text{attt}} + F\_{\text{rep}} \tag{14}$$

#### **5.2 Graphic evaluation and review technique**

In fact GERT is a generalized form of PERT, where the probability of occurrence of activities of the project is taken into consideration. In other words in PERT, either an activity occurs (probability=1) or it does not occur (probability=0); however, in GERT the probability of occurrence of each activity can be a real number between zero and one. GERT approach addresses the majority of the limitations associated with PERT/CPM technique and allows loops between tasks. The fundamental drawback associated with the GERT technique is that a complex program (such as Monte Carlo simulation) is required to model the GERT system.

#### **5.3 Proposed methodology**

Given the graph representing the sequence of activities in a disastrous situation, the first step is to obtain necessary information for making decision. The mentioned information consists of: a) parameters affecting the decision making, which are mostly predefined and weighted, and b) estimating approximate durations of activities which may occur during the mission. The mentioned parameters are categorized in two main classes; one of them deals with the degree of satisfaction of the criteria defined in tasks of the robot, and the other one is concerned with importance of targets. These parameters are listed in table 3.

Having gained the necessary data via a questionnaire of experts, PERT algorithm is used for the process of durations of activities. The resulted output is a part of the data needed for Multiple Criteria Decision Making (MCDM) analysis which consists of: standard deviation, free slack and total slack for activities, and the probability of occurrence of activities before a certain time.

The outputs of PERT and the degree of satisfaction of criteria defined for intermediate actions of robot, along with the importance of each criterion are given to MCDM algorithm as inputs. MCDM makes a decision and assigns a decision value for each activity. These

0

 

*if q q <sup>U</sup> q q*

0

 

*q q if q q F U q q q q*

*rep obs*

*rep rep obs obs*

**5.2 Graphic evaluation and review technique** 

of the robot (Jacoff et al., 2000).

**5.3 Proposed methodology** 

certain time.

0 , (, )

Where η is a positive scaling factor, ρ (q,qobs) denotes the minimal distance from the robot q to the obstacle, qobs denotes the point on the obstacle such that the distance between this point and the robot is minimal between the obstacle and the robot, and ρ0 is a positive constant denoting the distance of influence of the obstacle. The total force applied to the robot is the sum of the attractive force and the repulsive force which determines the motion

*F FF total att re <sup>p</sup>* (14)

In fact GERT is a generalized form of PERT, where the probability of occurrence of activities of the project is taken into consideration. In other words in PERT, either an activity occurs (probability=1) or it does not occur (probability=0); however, in GERT the probability of occurrence of each activity can be a real number between zero and one. GERT approach addresses the majority of the limitations associated with PERT/CPM technique and allows loops between tasks. The fundamental drawback associated with the GERT technique is that a complex program (such as Monte Carlo simulation) is required to model the GERT system.

Given the graph representing the sequence of activities in a disastrous situation, the first step is to obtain necessary information for making decision. The mentioned information consists of: a) parameters affecting the decision making, which are mostly predefined and weighted, and b) estimating approximate durations of activities which may occur during the mission. The mentioned parameters are categorized in two main classes; one of them deals with the degree of satisfaction of the criteria defined in tasks of the robot, and the other one

Having gained the necessary data via a questionnaire of experts, PERT algorithm is used for the process of durations of activities. The resulted output is a part of the data needed for Multiple Criteria Decision Making (MCDM) analysis which consists of: standard deviation, free slack and total slack for activities, and the probability of occurrence of activities before a

The outputs of PERT and the degree of satisfaction of criteria defined for intermediate actions of robot, along with the importance of each criterion are given to MCDM algorithm as inputs. MCDM makes a decision and assigns a decision value for each activity. These

is concerned with importance of targets. These parameters are listed in table 3.

<sup>111</sup> , (, ) 2 (, )

2

0

(13)

0

*obs*

 

*obs*

*obs*

2 0

*if q q*

<sup>111</sup> ( , ), ( , ) (, ) (, )

0 , (, )

0

*obs obs*

*if q q*

values are treated as the virtual durations of activities and are given to CPM. It is obvious that output Es (Earliest starts representing the decision indexes of missions) of CPM can be interpreted as the degree of fulfillment of the activities leading to a certain event. By comparing the Es of the last events of several missions, we can deduce which mission fulfills our criteria better than the other ones.


Table 3. Main criteria for choosing the path

The ultimate objective of rescue mission is to help the injured people. The injured situations are divided into four groups: endangered, vulnerable, defenseless and prepared. To compare different groups of injured people four criteria are considered (refer to Table 6). The weights of criteria along with the degree of satisfaction of different criteria are given to MCDM algorithm and a decision value is calculated for each group of injured people as targets. In fact ξ (the positive scaling factor for attractive force) for each target is calculated as follows:

$$
\xi\_{\text{i}} = \text{norm } (\text{E}\_{\text{oi}}) + \text{norm } (\text{ADV}\_{\text{i}}) \tag{15}
$$

Where *norm* is normalization operator and *ADV*i is the Attraction Decision Value of the ith target.

Considering environmental situation and defining certain criteria for degree of danger of each obstacle, a similar approach is possible for determining the scaling factor *η* of the repulsive force. The degree of satisfaction of each criterion is fed into MCDM and the resulting decision value equals the positive scaling factor of repulsive force:

$$\mathbf{n}\_{\rm li} = \text{norm}\left(\mathbf{RDV\_{i}}\right) \tag{16}$$

Where *RDV*i is the Repulsive Decision Value of the ith obstacle.

Having obtained the corresponding strength of the attractive and repulsive potential field, the path planning algorithm is established and the optimal path with respect to least time, least risk and most help to injured people is achieved.

A Fuzzy Approach for Risk Analysis with Application in Project Management 55

poisonous gas

explosion by

team for

for CO2 and respiration

oxygen

22-24 Dummy activity

32-34 Dummy activity

42-44 Dummy activity

temperature

team to evaluate the place for possible

conflagration

the rescue team

34-36 ----------

44-46 ----------

the rescue team

4-26 No dangerous gas detected

means of thermal sensors

possibility of explosion

0-2 Building 2-4 Applying the sensor to detect

4-6 Gas detected 6-8 Evaluating the probability of

8-10 Possibility of explosion present 10-26 Signaling warning to the rescue

8-12 No Possibility for explosion 12-14 Considering the data of the sensor

14-18 Human life detected 18-20 Providing the living person with

20-24 Dummy activity 18-24 Signaling assistance message to

26-80 ---------- 2-28 Applying the sensor to detect CO2

46-80 ---------- 2-48 Applying the sensor to measure

48-60 Low temperature 60-62 Signaling message to the rescue

48-50 High temperature 50-54 Signaling assistance message to

24-26 Aggregated tasks 14-16 No Human life detected

28-36 No CO2 detected 28-30 CO2 detected

36-80 ---------- 2-38 Noise detection 38-46 No Noise detected 38-40 Noise detected

Activity Description Activity Description

18-22 Signaling warning to the rescue team to wear gas masks

16-26 Signaling warning to the rescue team to wear gas masks

30-32 Signaling assistance message to the rescue team

30-34 Providing the living person with oxygen

40-42 Providing the living person with oxygen

40-44 Signaling assistance message to the rescue team

#### **5.4 Case study and simulation**

Assume that two groups of injured people with different number of people and different categories of injuring are identified. One of the groups is located near a gas station, where people are endangered by the threat of explosion and the other group is next to a building and is threatened by the collision risk of the building. The rescue robot must choose one of the groups as the priority of its mission. Also it is expected that the rescue robot accomplishes several intermediate tasks such as searching for any injured person isolated from other members of identified group, taking picture of the surroundings and sending it to the rescue team, sensing the environmental factors that can signify explosion, etc. Fig. 11 demonstrates the GERT network for rescue mission.

The list of activities for the network represented in Fig. 11, are listed in table 4. The criteria for intermediate actions of robot in choosing the path are listed in Table 5.

The three optimistic, most likely and pessimistic values for the duration of each activity and the fulfillment of the main criteria (by performing each activity) which are listed in Table 5 are estimated based on the experts' opinions. In this table H, E and R indicate parameters concerning human, environment and the robot, respectively (Khanmohammadi & Soltani, 2011).

Durations of activities (first column of Table 5) are given to the PERT algorithm and standard deviation, free slack and total slack for activities, and the probability of performing activities in the range DR are obtained as the outputs of PERT. The output of the PERT and the degree of the satisfaction of the criteria by intermediate actions (H1, H2, E1, E2, E3, R1, R2, R3 and R4 columns) are fed to MCDM algorithm which yields a decision value for each activity. These decision values are treated as the virtual durations of activities and comprise the inputs of the CPM algorithm. Since there is the possibility of obtaining negative decision values, to avoid assigning negative inputs to CPM, the values are normalized in the range [1,10]. Es in the output of the CPM represents the degree of satisfaction of each activity in each network (mission index). The following values are obtained for the networks of the gas station (target 1) and building (target 2), respectively.

$$E\_{s1} = 52.9434, \; E\_{s2} = 27.0122.1$$

As defined in the previous section, a set of criteria is defined for the injured people to be able to distinguish which group of injured people are more at risk. These criteria are described in Table 6.

The degree of satisfaction of these criteria along with the weight (importance) related to each criterions are the inputs of MCDM and the decision value for each target is the value assigned to *ADV*i.

Similar to the procedure above, a set of criteria is defined for the degree of danger of the obstacles based on the environmental situation. Consider three kinds of obstacles consisting: Risk of fire, Risk of electric shock and Risk of building collision. Table 7 summarizes the factors involved.

Similar to obtaining *ADV*s, *RDV*s (Repulsive Decision Values) are simply obtained by using MCDM algorithm on the importance of each criterion and the degree of satisfaction of them for each obstacle. For comparison purpose, consider two scenarios with different environmental situations and different groups of troubled people.

Assume that two groups of injured people with different number of people and different categories of injuring are identified. One of the groups is located near a gas station, where people are endangered by the threat of explosion and the other group is next to a building and is threatened by the collision risk of the building. The rescue robot must choose one of the groups as the priority of its mission. Also it is expected that the rescue robot accomplishes several intermediate tasks such as searching for any injured person isolated from other members of identified group, taking picture of the surroundings and sending it to the rescue team, sensing the environmental factors that can signify explosion, etc. Fig. 11

The list of activities for the network represented in Fig. 11, are listed in table 4. The criteria

The three optimistic, most likely and pessimistic values for the duration of each activity and the fulfillment of the main criteria (by performing each activity) which are listed in Table 5 are estimated based on the experts' opinions. In this table H, E and R indicate parameters concerning human, environment and the robot, respectively (Khanmohammadi &

Durations of activities (first column of Table 5) are given to the PERT algorithm and standard deviation, free slack and total slack for activities, and the probability of performing activities in the range DR are obtained as the outputs of PERT. The output of the PERT and the degree of the satisfaction of the criteria by intermediate actions (H1, H2, E1, E2, E3, R1, R2, R3 and R4 columns) are fed to MCDM algorithm which yields a decision value for each activity. These decision values are treated as the virtual durations of activities and comprise the inputs of the CPM algorithm. Since there is the possibility of obtaining negative decision values, to avoid assigning negative inputs to CPM, the values are normalized in the range [1,10]. Es in the output of the CPM represents the degree of satisfaction of each activity in each network (mission index). The following values are obtained for the networks of the gas

*Es*1= 52.9434, *Es*2= 27.0122. As defined in the previous section, a set of criteria is defined for the injured people to be able to distinguish which group of injured people are more at risk. These criteria are

The degree of satisfaction of these criteria along with the weight (importance) related to each criterions are the inputs of MCDM and the decision value for each target is the value

Similar to the procedure above, a set of criteria is defined for the degree of danger of the obstacles based on the environmental situation. Consider three kinds of obstacles consisting: Risk of fire, Risk of electric shock and Risk of building collision. Table 7 summarizes the

Similar to obtaining *ADV*s, *RDV*s (Repulsive Decision Values) are simply obtained by using MCDM algorithm on the importance of each criterion and the degree of satisfaction of them for each obstacle. For comparison purpose, consider two scenarios with different

environmental situations and different groups of troubled people.

for intermediate actions of robot in choosing the path are listed in Table 5.

**5.4 Case study and simulation** 

Soltani, 2011).

described in Table 6.

assigned to *ADV*i.

factors involved.

demonstrates the GERT network for rescue mission.

station (target 1) and building (target 2), respectively.


A Fuzzy Approach for Risk Analysis with Application in Project Management 57

\* Values other than durations of activities are normalized in the range [0,1]

Table 5. Durations of activities and satisfaction levels of criteria by performing each activity


\* Activities with the dashed lines in the description do not signify any specific activity. They represent the priority considered in making decision

Table 4. List of activities for Network of Fig. 11.

64-66 Roughness detected 66-68 Considering the data of the sensor

72-74 Dummy activity 70-74 Signaling message to the rescue

13-33 Dummy activity 7-9 Applying the sensor to detect gas

9-11 gas leakage detected 11-15 Signaling warning to the rescue

15-29 Dummy activity 11-17 Using nitrogen to cool down the

19-21 CO2 detected 21-23 Providing the living person with

23-25 Dummy activity 21-25 Signaling assistance message to

\* Activities with the dashed lines in the description do not signify any specific activity. They represent

25-27 ---------- 27-29 ---------- 29-31 ---------- 31-33 ----------

9-31 ---------- 3-33 Sending photos

17-29 Dummy activity 11-29 Applying the extinguisher

of CO2 and Respiration

surroundings

surroundings

temperature

leakage

surroundings

the rescue team

team

19-27 No CO2 detected

oxygen

7-13 Applying the extinguisher

5-7 Moving to the point with highest

responsible for leveling the path

team

54-62 ---------- 48-56 Extremely high temperature

2-64 Detecting bumpy plains 64-76 No Roughness detected

68-70 No alive Human detected 70-72 Leveling the path

74-76 ---------- 68-76 Human life detected 76-80 ---------- 2-78 Taking photos of the

0-1 Gas Station 1-3 Taking photos of the

Activity Description Activity Description 50-52 Applying the extinguisher 52-54 Dummy activity

56-62 Applying the extinguisher 62-80 ----------

78-80 Sending the photos

1-5 Detecting the temperature of the surroundings with sensor

7-33 Using nitrogen to cool down the surroundings

11-19 Applying the sensor to detect

the priority considered in making decision

Table 4. List of activities for Network of Fig. 11.

CO2


\* Values other than durations of activities are normalized in the range [0,1]

Table 5. Durations of activities and satisfaction levels of criteria by performing each activity

A Fuzzy Approach for Risk Analysis with Application in Project Management 59

The introduced procedure has been run twice, once for hot and dry and once for cold and rainy weather. Results are illustrated in Fig. 12. Priority is given to the first target (group1 near gas station) by robot. As it is seen in Fig. 12(a), the rescue robot tries to get as far as possible from the power electric station when it is rainy and it gets a shorter path (near

group1: 15 people near gas station comprised of 15 endangered (injured), 5 vulnerable, 5

group2: 25 people near a damaged building with possibility of collision comprised of 4

We have considered the mentioned environmental conditions and the results are illustrated

The priority is given to the second target (group2 near damaged building) by rescue robot. In case one, when it is cold and rainy, the possibility of explosion is low, so the robot gets closer to the gas station, Fig. 13(a). But when it is hot, robot tries to be far from the gas

Fig. 12. Generated path for the first scenario: (a) cold and rainy condition, (b) hot and dry

Fig. 13. Generated path for second scenario: (a) cold and rainy condition, (b) hot and dry

electric power station) in dry conditions, Fig. 12(b).

Scenario 2

defenseless

in Fig. 12.

condition

condition

injured and 11 defenseless.

Fig. 11. Network of project activities

Category and Number of the troubled people Exposure to dangerous situation


Table 6. Criteria for calculating the priority values of injured groups


Table 7. Criteria for measuring the danger level of obstacles

Scenario 1

group1: 25 people near gas station comprised of 15 endangered (injured) 5 vulnerable, 5 defenseless

group2: 15 people near a building with possibility of collision comprised of 4 injured and 11 defenseless.

The introduced procedure has been run twice, once for hot and dry and once for cold and rainy weather. Results are illustrated in Fig. 12. Priority is given to the first target (group1 near gas station) by robot. As it is seen in Fig. 12(a), the rescue robot tries to get as far as possible from the power electric station when it is rainy and it gets a shorter path (near electric power station) in dry conditions, Fig. 12(b).

#### Scenario 2

58 Fuzzy Inference System – Theory and Applications

Category and Number of the troubled people Exposure to dangerous situation

vicinity – rainy/dry weather

group1: 25 people near gas station comprised of 15 endangered (injured) 5 vulnerable, 5

group2: 15 people near a building with possibility of collision comprised of 4 injured and

possibility of building collision - Humidity – rainy/dry weather

Adjacency of the danger



Fig. 11. Network of project activities

Category of the troubled people:

Health status of the injured people

prepared

 Building collision Electric shock

Fire

Scenario 1

defenseless

11 defenseless.

endangered, defenseless, vulnerable,

Table 6. Criteria for calculating the priority values of injured groups

Type of the obstacle Criteria and factors involved

Table 7. Criteria for measuring the danger level of obstacles

Number of the people in each category

group1: 15 people near gas station comprised of 15 endangered (injured), 5 vulnerable, 5 defenseless

group2: 25 people near a damaged building with possibility of collision comprised of 4 injured and 11 defenseless.

We have considered the mentioned environmental conditions and the results are illustrated in Fig. 12.

The priority is given to the second target (group2 near damaged building) by rescue robot. In case one, when it is cold and rainy, the possibility of explosion is low, so the robot gets closer to the gas station, Fig. 13(a). But when it is hot, robot tries to be far from the gas

Fig. 12. Generated path for the first scenario: (a) cold and rainy condition, (b) hot and dry condition

Fig. 13. Generated path for second scenario: (a) cold and rainy condition, (b) hot and dry condition

A Fuzzy Approach for Risk Analysis with Application in Project Management 61

Alan, A., & Pritsker, B. (1966). GERT: *Graphical Evaluation and Review Technique*, Rand Corp Alsultan, K. S., & Aliyu, M. D. S. (1996). A new potential field-based algorithm for path

Carr, V., & Tah, J. H. M. (2001). A Fuzzy approach to construction project risk assessment

Chadwick, R. A. (2005). The Impacts of Multiple Robots and Display Views: An Urban

Chanas, S., Dubois, D., & Zielinski, P. (2002). On the Sure Criticality of Tasks in Activity

Doherty, N. A. (2000). *Integrated Risk Management: Techniques and Strategies for Managing* 

Jacoff, A., Messina, E., & Evans, J. (2000). A Standard Test Course for Urban Search and

Kerzner, H. (2009). *Project Management: A System Approach to Planning, Scheduling and* 

Khanmohammadi, S., Charmi, M., & Nasiri, F. (2001). Delay Time Estimating in Project

Khanmohammadi, S., & Soltani, R. (2011). Intelligent path planning for rescue robot. *World Academy of Science, Engineering and Technology,* No. 79, (April 2011), pp. (764-769) Kreinovich, V., Nguyen H. T., & Yam Y. (2000). Fuzzy Systems are Universal Approximators

Li, Y., & Liao, X. (2007). Decision support for risk analysis on dynamic alliance. *Decision* 

Mamdani, E. H., & Assilian, S. (1999). An Experiment in Linguistic Synthesis with a Fuzzy

McNeil, A. J., Frey, R., & Embrechts, P. (2005). *Quantitative Risk Management: Concepts,* 

*Support Systems,* Vol. 42, No. 4, (January 2007), pp. (2043-2059)

*cybernetics part B,* Vol. 32, No. 4, (August 2002), pp. (393-407)

Jamshidi M., Vadiee N., & Ross T. J. (1993). *Fuzzy Logic and Control,* Prentice Hall

*Safety,* Vol. 78, No. 2, (November 2002), pp. (173-183)

*Corporate Risk*, McGraw-Hill Professional

Vol. 15, No. 6, (June 2000), pp. (565-574) Latombe, J. C. (1990). *Robot Motion Planning,* Springer

*Techniques and Tools,* Princeton University Press

Chavas, J. P. (2004). *Risk Analysis in Theory and Practice,* Elsevier, ISBN – 10: 0121706214 Cho, H. N., Choi, H. H., & Kim, Y. B. (2002). A risk assessment methodology for

*Software,* Vol. 32, No. 10-11, (October-November 2001), pp. (847-857) Casper, J., & Yanco, H. (2002). AAAI/RoboCup-2001 Robot Rescue. *AI Magazine,* Vol. 23,

planning. *Journal of intelligent and robotic systems*, Vol. 17, No. 3, (November 1996),

and analysis: construction project risk management system. *Advances in Engineering* 

Search and Rescue Simulation. *Human Factors and Ergonomics Society, Annual Meeting Proceeding Cognitive Engineering and Decision Making,* (January 2005), pp.

Networks with Imprecise Durations. *IEEE Transactions on systems, man, and* 

incorporating uncertainties using fuzzy concepts. *Reliability Engineering & System* 

Rescue Robots, *Proceeding of performance metrics for intelligent systems workshop,*

Management Using Fuzzy Delays and Fuzzy Probabilities, *Proceeding of International ICSC Congress on Computational Intelligence: Methods and Applications,*

for a Smooth Function and its Derivatives. *International Journal of Intelligent Systems,*

Logic Controller. *International Journal of Human-Computer Studies,* Vol. 51, No. 2,

**7. References** 

pp. (265–282)

(387-391)

August - 2000

*Controlling*, Wiley

Bangore - UK, June - 2001

(August 1999), pp. (135-147)

No. 1, (Spring 2002)

station where there is the risk of explosion, Fig. 13(b). The simulation results show the fact that the introduced algorithm is flexible in terms of the environmental conditions and the factors involved in targets.

To further illustrate the conceptual basis of the utilized potential field, a 3D representation of the risk potential function and the corresponding optimal path are represented in Fig. 14.

Fig. 14. Artificial potential field and the obtained path with minimum risk

#### **6. Conclusion**

A new fuzzy approach is introduced to perform a more applicable risk analysis in real world applications. This procedure is used to determine the multi-purpose criticalities of activities where six main factors V, SFA, SCA, PFA, RLA and COR are considered as criticality indexes. A fuzzy inference system with three inputs: probability of impact, impact treat, and ability to retaliate is used to calculate the values of RLA for activities. The output of FIS represents the risky level of each activity. The decision values obtained by classic multi criteria decision making problem are then considered as criticality indexes of activities. The obtained results are compared to classic PERT, from the view point of impact expenses, by using the Mont Carlo method. It has been shown that by considering the multipurpose criticalities (instead of total slacks) a considerable amount of expenses caused by different impacts may be saved. The introduced method is applied to simultaneous task scheduling and path planning of rescue robots. Simulation results show that project management technique along with risk analysis by means of artificial potential field path planning is an efficient tool which may be used for rescue mission scheduling by intelligent robots. The algorithm is flexible in terms of environmental situation and the effective factors in risk analysis. In fact the proposed method merges the path planning methods with rescue mission scheduling.
