3 This measure indicates the level of optimism in a decision
making process. An optimistic decision maker select the alternative which has the maximum similarity to the favourable
solution, while a pessimistic decision maker tends to select the alternative with minimum similarity to the unfavourable
solution Chen, 2011. The optimism degree ranges from 0 to 1. For a pessimistic OWA operator O W =0, while O W =1
indicates an optimistic OWA operator. Yager, 1988 proved that optimism degree is not enough to describe the behavior of
OWA operator. He introduced measure of dispersion presented by Equation 3 Yager, 1988:
∑
3 where DW = dispersion of the weight vector
Larger values of dispersion ensure that different criteria have approximately the same effect on the vulnerability assessment
problem. However, order weights are not equal. Therefore, the weight vector associated to an OWA operator with a predefined
optimism degree is derived by solving a constraint optimization problemYager, 1988. Fullér and Majlender 2003 indicates
that using the minimal variability constraint leads to more efficient results. In order to obtain the weight vector, they
proposed the following optimization problem:
Minimize ∑
∑ ∑
∑ Subject to
∑ ∑
4 where D = standard deviation of weights
W = weight vector w
i
= the weight associated to the ith largest attribute value
n = the number of criteria
3. CASE STUDY
3.1 Study Area
The study area of this research is Tehran Metropolitan Area. Having more than eight million inhabitants, TMA is the most
crowded city of Iran Hashemi and Alesheikh, 2012. Seismologists argue that existence of four great faults including
North Tehran Fault, North and South Rey Faults and Mosha Fault inside and around the city makes it extremely vulnerable
in future earthquake Samadi Alinia and Delavar 2011. 3.2
Data
Although censuses were taken in 2006 and 2010, 1996 census data is used in this paper due to availability of 1996 data. Six
contributing criteria are introduced by experts see Table 2. Based on Iranian Civil Society ICS buildings built before
1966 are considered as non-standard buildings. Moreover, the percentage of weak buildings built between 1966 and 1988 is
considered as another affective criterion. Height of buildings are included into the model using two categories including
percentage of buildings having less than four floors and percentage of buildings having 4 floors and more. The distance
of each statistical unit and their average slope are determined as important factors by experts.
Layer Definition
Slp Average slope of each statistical unit
Ls_4 Percentage of buildings having less than 4 floors
in each statistical unit Mr_4
Percentage of buildings having 4 floors and more in each statistical unit
Bf_66 Percentage of buildings built before 1966 in
each statistical unit Bt66-88
Percentage of weak buildings built between 1966-1988 in each statistical unit
Dst_F The geometric distance of statistical units to
NTF in km Table 2. Contributing criteria and their definitions
3.3 Data Preparation
The attribute values of five randomly selected statistical units are illustrated in Table 3.
id Slp
Ls_4 Mr_4
Bf_66 Bt66-88 Dst_Fkm
1 14
17 8
52 13
6.751 2
1 4
84 6
3.890 3
26 2
91 19
1.435 4
4 88
97 5.476
5 6
9 1
63 5
2.054 Table 3. Attribute values of five randomly selected statistical
units Attribute values of different criteria are not in the same scale.
Therefore, they cannot be aggregated directly. Attribute values are first normalized. Malczewski, 2006 introduced a method to
normalize attribute values. He classified all contributing criteria into two classes including cost and benefit criteria. Benefit
criteria are a class of criteria that their larger values are favourable for the decision maker. However, lower values of
cost criteria are desirable. Through the normalization process all attribute values are divided by their maximum value. Then, the
attribute values of the cost criteria are subtracted bye one. In the seismic vulnerability assessment distance to the North Tehran
Fault is specified as a cost criterion, while five other criteria are benefit criteria. Table 4 demonstrates the normalized attribute
values of five selected statistical units.
id Slp
Ls_4 Mr_4
Bf_66 Bt66-88 Dst_Fkm
1 0.53
0.17 0.22
0.53 0.54
0.71 2
0.03 0.04
0.86 0.25
0.84 3
1 0.02
0.93 0.79
0.94 4
0.15 0.91
1 0.77
5 0.25
0.09 0.02
0.64 0.20
0.92 Table 4. Normalized attribute values of five randomly selected
statistical units
3.4 Sensitivity Analysis for Minimal Variability OWA
