2007_6_Nr6_EEMJ

Robu et al. /Environmental Engineering and Management Journal 6 (2007), 6, 573-592
It is very important to identify the risk and to
estimate it in order to be analyzed. The risk analyzing
process tries to identify all the results of an action.
The risk estimation is done using the analytic
methods or simulation. There are estimated thus the
occurrence probability of every disaster, as well as
the associated magnitude (dimension). The risk
analysis process uses technical information related to
estimations and other additional available
information, for assessing diverse variants of possible
actions. An original model based on multiple criteria
optimization to appropriate financial funds for
atmospheric pollution reduction is presented. For this
model, the solving manner is specified by reduction
to an optimization problem with a single objective
function (Radulescu, 2002).
5.1. Measures for calculus of the risk value
The probability theory offers many adequate
tools for modeling the risk phenomenon. Any activity
exhibits an incertitude element. From mathematical
point of view, the incertitude will be modeled using
random variables or, more generally, using the
stochastic processes. The risk that appears within an
activity may be described with adequate measures.
One of the most used measures is the dispersion of
the random variable, which describes the incertitude
of the respective activity. Another measure of the risk
is given by the repartition function of the random
variable. More precisely, if X is a random variable
that describes the risk associated to a decision, F x is
the repartition function associated to variable X, and f x
is the probability density of the random variable X,
then using Eq. (2):
+∞
∫
−∞
µ = E(X ) = xdF ( x)
, (2)
the risk measures result from Eqs. (3, 4):
+∞
∫ (
−∞
2
2
σ = var( X ) = x − µ ) dF ( x)
(3)
+∞
1/ 2
⎤
2
( x − µ ) dFx
( x)
⎥
−∞
⎦
⎡
σ = ⎢ ∫
(4)
⎣
A measure of the risk may be considered also:
+∞
∫
−∞
| x − µ | dF ( x)
(first order central moment) (5)
x
Stone has shown that all the measures of the
risks above presented are special cases of some
families of risks measures. The first measure of the
risk within the Stone risk measures family that has
three parameters is defined as (Eqs.2-6):
x
q
( F x )
| (k>0 (6)
∫
−∞
k
R S1 (X) = x − p(
F ) | dF ( x)
x
where p\F x ) defines a level of the profit (success) that
is used for measuring the abatement.
The positive number k is a measure of the
relative impact of the small and big abatements. q(F x )
is a level a parameter that specifies the abatements
will be included in the risk measurement. The second
measure of the risk within Stone family with three
parameters is defined as k order root from R S1 (X)
(Eq. 7):
⎡
R S2 (X) = ⎢
⎢⎣
q(
Fx
)
∫
−∞
| x − p(
F
x
) |
k
x
1
k
⎤
⋅dFx
( x)
⎥
⎥⎦
(7)
One may observe that through proper
selection of the parameters p(F X ), q{F x ) and k, the
above presented risk measurement are special cases
of those from the Stone family of risk measurements.
For example, if in (l) k = 2 and p(F x ) = (F x )
= µ are inserted, the semi-dispersion is obtained as a
measure of the risk. A more interesting case related to
risk measures Stone family is the generalized risk
measure Eq. (8):
t
R F1 (X) =
∫
−∞
(t - x) α dF x (x) (α > 0), (8)
proposed by Fishburn, where t is a superior scopelevel
that is fixed. This measure results from (l) if one
choose p(F x )=q(F X ) = t. The parameter “a” of
Fishburn risk measurement R F1 may be interpreted as
“k” parameter from the measures Stone family (l) in
this way: it is a risk parameter, which characterizes
the attitude toward risk. The values α > l describe a
sensitive risk, while the values α ∈(0. l) describe an
insensitive risk.
Another known measure of the risk is
Shannon entropy (Eq. 9):
+∞
∫
−∞
fx( x)ln(
f
x
( x))
dx
(9)
5.2. Probabilistic modeling of pollution
Mathematic modeling of air pollution is
done using the theory of stochastic processes. Thus,
over long periods of time, the pollution degree may
be described by a multidimensional stochastic
process: X = (X t ) t ≥ 0 . For t ≥ 0 attached to the
components of the random vector: X t = (X l,t ;
X 2,t ...X m,t ) represents the concentrations of the
atmospheric pollutant factors. Repartition function of
the stochastic multidimensional process X, F(t,x) =
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