2.2 Bases Teóricas
2.2.36 Clasificación de las Estrategias de Aprendizaje
Ranking of fuzzy numbers can be viewed as an important aspect of decision making in fuzzy environment. In fuzzy decision making situations, fuzzy quantities have been used to define the performance of alternatives in modeling a real life problem. Jain (1976) has first proposed a
158 ranking procedure for ordering the fuzzy quantities in decision making environment. In the literature, many authors have proposed various ranking methods for ranking fuzzy numbers by preference ratio (Modarres and Nezhad, 2001), left and right dominance(Chen and Lu, 2001), area between the centroid point and original point (Chu and Tsao, 2002), sign distance (Abbasbandy and Asady, 2006)and distance minimization(Asady and Zendehnam, 2007). Rao and Shankar (2011) have demonstrated an improved ranking method for ordering fuzzy numbers using the concept of ‘Circumference of centroids’. The method provides a mathematical formulation for ranking the fuzzy numbers based on their crisp score. This concept has been explored in this research towards proposing an efficient risk assessment module. The basic concept of ‘Circumference of centroids’ has been reproduced below.
Fig. 6.1: Circumcenter of centroids
The centroid of a trapezoid is considered as the balancing point of the trapezoid. Firstly, the trapezoid is split into three plane figures like a triangle (APB), a rectangle (BPQC), and again a triangle (CQD), respectively (Fig. 6.1). Then the centroids of these plane figures are calculated followed by the calculation of the Circumcenter of these centroids. The Circumcenter of centroids is considered as the point of reference to define the ranking of generalized fuzzy numbers. The reason for selecting this point as a point of reference is that each centroid point (
1
G of triangle APB, G2of rectangle BPQC, and G3 of triangle CQD) are balancing points of each individual plane figure, and the Circumcenter of these centroid points is equidistant from
159 each vertex (i.e. centroids). Therefore, this point would be a better reference point than Centroid point of the trapezoid.
Consider a generalized trapezoidal fuzzy number A%=
(
a b c d w, , , ;)
. The centroids of the three plane figures are G1=((
a+2b)
/ 3,w/ 3)
,G2=((
b+c)
/ 2,w/ 2)
, and G3=((
2c+d)
/ 3,w/ 3)
, respectively. Equation of the line G G1 3suuuur
is y=w/ 3 and G2 does not lie on the lineG G1 3
suuuur
. Therefore, G G1, 2 and G3 are non-collinear and they form a triangle.
Let us define the Circumcenter SA%
(
x y0, 0)
of the triangle with vertices G G1, 2 and G3 of the generalized trapezoidal fuzzy number A%=(
a b c d w, , , ;)
as(
0 0)
(
)(
)
2 2 3 2 3 5 2 2 , , 6 12 A a b c d c b w a b c d S x y w + + + + − + − + = % (6.2)As a special case, for triangular fuzzy numberA%=
(
a b c d w, , , ;)
, that is, c=b the Circumcenter of centroids is given by(
0 0)
(
)(
)
2 4 5 4 , , 6 12 A a b d b w a b d S x y w + + − − + = % (6.3)The ranking function of the trapezoidal fuzzy number A%=
(
a b c d w, , , ;)
which maps the set of all fuzzy numbers to a set of real numbers is defined as:( )
2 20 0
R A% = x +y (6.4) where R A%
( )
is the Euclidean distance from the Circumcenter of the centroids and the original point.When decision makers’ attitude is considered, then the ranking function has been modified as follows:
( )
0(
1)
0Iα A% =αy + −α x (6.5) where α ∈
[ ]
0,1 is the index of optimism which represents the degree of optimism of a decision maker. The value of α may vary with the change of decision makers view point. If decision makers view point is pessimistic(
α =0)
, moderate(
α =0.5)
, and optimistic(
α =1)
. The lager the value of α is, the higher the degree of optimism.160
6.5 Proposed Methodology
In this work, a hierarchical risk breakdown structure with two distinct levels (Table 6.1) has been used for assessing risks of metropolitan construction project. First level includes various potential risk factors which are identified from the source of different risk dimensions, and the second level highlights different risk dimensions which can be considered as major risk sources affecting to the overall project performance. A more general multi-criteria decision making scenario has been introduced for quantifying the construction project risks effectively. The scenario comprises a committee of k experts
(
E E1, 2,...,Ek)
who are responsible for assessing the risks of n risk influencing factors(
F F1, 2,...,Fn)
, under m risk dimensions(
D D1, 2,...,Dm)
. In fuzzy decision making environment, risk of each influencing factor is quantified by multiplying two evaluating parameters such as likelihood of occurrence and its impact. The following procedural steps have been proposed for calculating fuzzy risk ratings, and also for categorizing as well as managing the identified risk factors.Step 1: Identification of potential risk factors from the hierarchical risk breakdown structure of construction project.
Step 2: Selection of appropriate fuzzy linguistic scale for expressing both likelihood of occurrence and impact of risk.
Step 3: Linguistic data (in relation to likelihood of occurrence and impact of risk) have been collected from the experts through the focus group survey. Thereafter, linguistic data have been translated into appropriate trapezoidal fuzzy numbers.
Step 4: Aggregated fuzzy preferences have been computed by using fuzzy aggregation rules. Fuzzy risk rating of each project risk factor has been calculated by multiplying fuzzy likelihood of occurrence and fuzzy risk impact.
Step 5: Crisp risk rating corresponding to each project risk factor has been calculated using ‘Circumference of centroids’ (Rao and Shankar, 2011) method. Moreover, risk factors have been ranked based on their crisp ratings. Then, a comparative study on the ranking order of risk factors has been presented to analyze the variation of risk rating values with respect to the risk- bearing attitudes of different experts.
Step 6: Risk factors have been categorized based on the concept of risk matrix.
Step 7: An action requirement plan has been suggested for different risk factor categories. The above procedure seems to be a generic one. However, the aforementioned risk ratings may vary due to different decision making environment as well as different risk management policies.
161
6.6 Case Application
In order to validate the proposed risk assessment approach, a case study has been conducted using the data from a metro system construction project in the city of Kolkata, India. The scope of the project includes building an underground station for the Kolkata Metro system, which necessitates deep excavation, excavation support system, and dewatering work. The undertaking project follows cut-and-cover construction plan in a heavy traffic area. A focus group survey has been conducted from construction executives and managers (profile displayed in Table 6.2) who have been actively associated in aforementioned construction project. The group including five experts with more than ten years’ experience in construction project management and being familiar with construction project risks has been selected to participate in the survey. Due to anonymity reasons, experts’ identity have not been wide-opened here and therefore, they have been abbreviated asE E E E1, 2, 3, 4, andE5. Experts have been requested to provide their personal opinion in a detailed questionnaire (Appendix D) referring to a linguistic scale. A structured questionnaire has been provided containing a total of twenty potential project risk factors, and each risk factor has been described by likelihood of occurrence as well as its impact. The selected experts being effectively involved in metro system construction projects; their experience and expertise have aided a lot in pursuit of this case study.