ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
62 
Table 1. Concentration of Contaminants in the monitoring well 
Contaminant Name 
Concentration (mg/L) 
Benzene 
0.31 
Ethyl-Benzene 
0.0004 
Toluene 
0.1 
Xylenes 
0.039 
Fuzzy Risk Assessment 
In the majority of the past and existing risk assessment 
projects, Federal and statewide environmental guidelines are 
widespreadly used as the basis of environmental quality 
evaluation criteria. The Comprehensive Environmental 
Response, Compensation and Liability Act (CERCLA) 
authorized the federal government to respond directly to 
releases of hazardous substances that may endanger public 
health, welfare and the environment. In addition, The National 
Oil and Hazardous Substances Pollution Contingency Plan 
establishes a framework for implementing CERCLA by 
outlining the process for developing and evaluation appropriate 
response action for Superfund Sites. And USEPA has also 
developed risk assessment procedures to address the public 
health concerns and to ensure that Superfund response 
actions limit the concentration of hazardous substances in the 
environment to avoid unacceptable risks to human health 
(USEPA, 1986). In practical site management, however, these 
guidelines are mostly conservative and sometimes impractical. 
Normally, slope factor, is a chemical-specific constant that 
describes the carcinogenicity of a compound. It generally 
derived from animal experiments data and is widely applied to 
estimate human risk accused by the toxic contaminants 
(USEPA, 1989). This interspecies conversion factor is a simple 
number based on the plausible assumption that these toxicities 
are as harmful to the human being as what they applied to the 
animals. Because of the assumptions made and methodology 
used in its derivation, SF values estimated are inherently 
uncertainty involved. Evidently, serious uncertainty problems 
exist between the contaminant exposure and potentiality of 
causing human health cancer risk. A quantitative method, 
therefore, is required to be employed. 
(1) Fuzzy Set Theory 
Fuzzy set were first introduced in 1965 by Lotfi Zadeh (1965) 
to describe imprecisely defined classes or sets that play an 
important role in human thought processes and 
communication. In essence, the theory of fuzzy sets is aimed 
at the development of a body of concepts and techniques for 
dealing with sources of uncertainty or imprecision that are non- 
statistical in nature. In classical set theory, an object either 
belongs to a set or does not, whereas fuzzy set theory allows 
an object to have partial membership of a set. Using fuzzy 
sets, it is possible to represent a set A by a membership 
function value. 
The theory of fuzzy sets deals with sets in a universe of 
discourse U. A fuzzy set x e U is a generalization of the 
concept of an ordinary set; it is being defined by membership 
function 
X = l4“x xe R,M x w 6 l 0 ’ 1 ]} ( 3 ) 
x is a particular value of X; ¡j x (*) represents a membership 
function of X. interval [0,1]. The closer ^ ( JC ) is to 1, the more 
“certain” one is about the value of x. 
Fig. 1 (a) and (b) Types of fuzzy membership functions 
Fig.1 presents two types of fuzzy membership functions 
(triangular and trapezoidal) for illustrating uncertainties 
associated with an parameter. For example, The triangular 
membership function means that (i) X is most likely equal to C, 
and (ii) values lower than a or greater than b are consider 
impossible for X. A membership function is normally defined 
based on characteristics of the uncertain information. 
Fuzzy logic can be considered as a generalization of the 
Boolean logic, by extended Boolean logic to handle the notion 
of partial-truth-truth-values between and including 'completely 
true' and 'completely false'. Fuzzy logic uses a soft linguistic 
type of variables, which are defined by continuous range of 
truth-values or fuzzy membership functions in the interval [0, 1] 
instead of the strict binary (T or F) decisions and assignments. 
It is the best tool to analyses and simplifies data, which 
characterized by vague conception or are subjective by 
incorporation of fuzzy sets (Zadeh,1965). 
(2) Fuzzy Relations 
Fuzzy relationships between fuzzy variables defined on 
different universes of discourse through the fuzzy conditional 
statement or linguistic implication 
X=>Y or “if X (u) then Y (v)” 
Which links the conditional or antecedent set X defined by 
H A ( u ), u e U with the consequence or output set Y defined 
by //g(v),ve V ■ 
[/ 
R = 
{x y) c X x Y l as Cartesian product of 
XxY, is called a fuzzy relation on XxY. 
Let A={ai\i = U2,...m) be a m-dimension fuzzy vector and 
r = {rij\i = 1,2,...m; j -1,2,...«] be a mxn fuzzy relation matrix. 
Then an m-dimension fuzzy vector /? can be obtained as: 
According to the principle of fuzzy set operation, g can be 
determined by a max-min or max-* composition (Zimmermann, 
1985). For the max-min composition, 
(5) 
For the max-* composition, 
ÜU*r« 
(6) 
(3) Fuzzy Risk Analysis 
Two fuzzy sets U for toxic contaminants and V for different age 
group are initially defined as following: