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Robu et al.

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) = 584

Methods and procedures for environmental risk assessment F xt (x) = P(X 1,t

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