and
(see Appendix). Then risk and
reliability are defined by means of appropriate fuzzy measures, which are
introduced in the following section.
Jacques Ganoulis, Petros
Anagnostopoulos and Iraklis Mbimbas
Hydraulics
Laboratory, Department of Civil Engineering,
Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece
Tel: +30-31-99.56.82, Fax: +30-31-99.56.81, E-mail: iganouli@civil.auth.gr
Abstract: Risk-based water resources management is a very important alternative methodology for analysing water resources problems, mainly because of different uncertainties inherent to real situations. In fact, large natural variabilities, such as those due to possible climate change, may induce important uncertainties, which together with uncertainties related to human activities may result in failure to achieve the project objectives. The analysis of such uncertainties and the study of the possibility of occurrence of incidents or failures are necessary in water quality projects. In water resources engineering risk may be defined as being of two different types: (1) Probabilistic Risk, which is estimated by using the theory of probability, and (2) Fuzzy Risk, which is analysed by use of fuzzy logic-based calculus. The fuzzy set theory, in combination with mathematical modelling is proposed in this paper in order to assess uncertainties in estimating output variables in water pollution problems. Uncertainties in input variables and values of physical parameters are first introduced as fuzzy numbers. Then, uncertainties are propagated by using fuzzy calculus and output variables are estimated in the form of fuzzy numbers. Examples of application in simplified and real cases illustrate the capabilities of the above methodology to evaluate risks in water pollution engineering problems.
An accurate description of water pollution problems is usually impossible because available data is incomplete. The different type of processes involved, such as hydrodynamic, physico-chemical and biological interactions, create difficulties for mathematical modelling. Furthermore, the multitude of parameters, which are necessary to describe the ecosystem’s kinetics, their physical meaning and variability in space and time raise a multitude of challenging and intriguing questions.
Adequate tools are required for studying changes in water quality and environmental impacts from water resources related projects. Engineering risk and reliability analysis provides a general framework to identify uncertainties and quantify risks. As shown in this paper, two main methodologies have been developed so far to assess risks (Ganoulis, 1994): (a) the stochastic approach, and (b) the fuzzy set theory. Stochastic variables and probability concepts are based on frequency analysis and require large amounts of data. Questions of independence between random variables and validation of stochastic relations, such as the well known statistical regression, are often difficult to resolve. Fuzzy set theory and fuzzy calculus may be used as a background to what could be called “imprecision risk analysis”. In this paper it is demonstrated how fuzzy numbers and variables may be used for estimating risks in cases where there is a lack of information or very little data available.
As explained in Ganoulis (1994) we should define as load l a variable reflecting the behaviour of the system under certain external conditions of stress or loading. There is a characteristic variable describing the capacity of the system to overcome this external load. We should call this system variable resistance r. A failure or an incident occurs when the load exceeds this resistance, i.e.,
FAILURE or INCIDENT: l > r
SAFETY or RELIABILITY : l r
In a probabilistic framework, l and r are taken as random or stochastic variables. In probabilistic terms, the chance of failure occurring is generally defined as risk. In this case we have
RISK= probability of failure= P(l > r)
RELIABILITY= probability of success= P(l r)
If fuzzy logic is used, l and r are
considered as fuzzy numbers, noted as
and
(see Appendix). Then risk and
reliability are defined by means of appropriate fuzzy measures, which are
introduced in the following section.
Consider now that the system has a resistance
and a load
, both represented by fuzzy numbers. A reliability measure or a safety
margin of the system may be defined as being the difference between load and
resistance (Shrestha et al., 1990; Ganoulis, 1994). This is also a fuzzy number
given by
Taking the h-level intervals of
and
as
R(h)=[R1(h), R2(h)], L(h)=[L1(h), L2(h)],
then, for every h Î [0, 1], the safety margin M(h) is obtained by subtracting L(h) from R(h), i.e.
.
Two limiting cases may be distinguished, as shown in Fig. 1:
(a) There is absolute safety if:
M(h) 0 " h [0,1]
(b) whereas absolute failure occurs when:
M(h) < 0 " h [0,1] .
A fuzzy measure of risk, or fuzzy risk index Ri may be defined as the area of the fuzzy safety margin, where values of M are negative. Mathematically, this may be shown as:
(1)
The fuzzy measure of reliability, or fuzzy reliability index Re is the complement of (1), i.e.
(2)
Fig.1 Absolute safety (a), absolute failure (b) and fuzzy risk (c).
Fuzzy modelling has not yet been developed extensively, although fuzzy numbers and fuzzy relations have found many applications in control engineering and industrial devices. Fuzzy set theory (Zadeh, 1965; Klir and Folger, 1988; Zimmerman, 1991), and its derivative fuzzy arithmetic (Kaufmann and Gupta, 1985), may be used in order to introduce imprecise data into a mathematical model in a direct way with minimal input data requirements (Ferson et al., 1994). In fuzzy modelling only the range and the most confident values of the input variables are required, so it can be used successfully when the available data is too sparse for a probabilistic method to be applied (Ganoulis, 1994; Silvert, 1997).
In the present study a water pollution model composed of the well-known advection-dispersion equation is used. The system of equations which describes the rate of change of the concentration of n different pollutants under biochemical interactions, can be expressed in two dimensions as:
k=1,2,..n (3)
where:
ck is the concentration for the kth pollutant;
u and v are the water velocities in the x and y directions (m/s);
Dx and Dy are the dispersion coefficients in the x and y directions (m2/s);
Sk are the source and sink terms describing the biochemical reactions.
In this model the various parameters and loads from pollutant sources are considered as triangular fuzzy numbers (T.F.N.). In order to calculate the concentration of all pollutants at each node using finite differences or finite elements, a system of fuzzy equations needs to be solved. This is difficult from a mathematical point of view, and has stimulated a lot of interest because whatever possible technique is used, only enclosures for the range of the output variables can be produced.
Shafike (1994)
introduced the fuzzy set theory coupled with the finite element method into a
groundwater flow model. The algebraic system of equations with fuzzy
coefficients was solved with an iterative algorithm (Moore, 1979). Dou et al.
(1995) applied the fuzzy set theory into a steady-state groundwater flow model
with fuzzy parameters combined with the finite difference method. A non-linear
optimisation algorithm was used for the solution of the groundwater flow
equations with fuzzy numbers as coefficients for the hydraulic heads.
Ganoulis et al. (1995; 1996) used fuzzy arithmetic to simulate imprecise relations in ecological risk assessment and management. Specifically the technique was applied to a simplified domain with coastal circulation, in order to evaluate the risk of coastal pollution. For the solution of the algebraic system of equations with fuzzy coefficients direct interval operations were employed, instead of the iterative methods or non-linear optimisation techniques used in previous studies. Since triplets cannot be used for the multiplication and division operations, as explained by Kaufmann and Gupta (1985), mathematical operations have been performed at various h-level cuts (see Appendix) by the use of the interval of confidence at each h-level (Dou et. al., 1995). It is also important to mention that the solution of an interval equation using interval operations is always an enclosure of the exact solution (Hansen, 1969; Moore, 1979; Neumaier, 1990). The best possible enclosure for an interval function, which is defined as the “hull” of the solution, is a fundamental problem of Interval Analysis and should be treated with care, as the solution accuracy depends on the shape of the interval function (Rall, 1986).
The technique was tested initially with a one-dimensional advection-dispersion model in the non-conservative form, using the finite difference method (see Fig.2). The results derived from the numerical computation considering the dispersion coefficient as fuzzy parameter are very similar to those of the analytical solution, confirming the accuracy of the numerical technique.
A finite element algorithm combined with fuzzy analysis was used for the solution of the advection-dispersion equation, with source terms for the ten different pollutants. The model was applied for the study of pollution in the Thermaikos Gulf, located in Northern Greece, for water velocities obtained from a wind-induced circulation model.
Fig.2
Distribution of fuzzy
for
one dimensional convective-dispersion.
(a)
Fig.3 Fuzzy phytoplankton concentration (b) at nodes 42 and 200 (a).
The finite element grid covering the Bay of Thessaloniki and the location of land-based pollutant sources are shown in Fig.3(a). The model contains ten different pollutants, namely: chlorophyll-a, coliforms, organic nitrogen, ammonia nitrogen, nitrite nitrogen, nitrate nitrogen, organic and inorganic phosphorus, BOD, and dissolved oxygen shortage (or deficit). Results of fuzzy computational modelling are shown in Fig.3(b) for phytoplankton concentrations at nodes numbered 42 and 200. Knowing the allowed by the national specifications phytoplankton concentration values, the risk of eutrophication can be evaluated using the fuzzy risk index, as given by Eq. (1).
When limited information or only few data are available on model parameters and boundary conditions, fuzzy modelling can be used in order to propagate uncertainties from land-based pollutant loads and values of dispersion coefficient. It is shown that for water pollution problems, a fuzzy risk and reliability analysis combined with fuzzy modelling can be applied in real situations in order to estimate the risk of environmental water pollution.
Appendix: Fuzzy Numbers
A fuzzy number
may be formally defined as
a set of ordered pairs
= {( x,
(x)) : x
R;
(x)
[ 0, 1 ]} (A.1)
where x is a particular
value of
and
(x) represents its membership
function. Values of the membership function are located in the closed interval
[0,1]. The closer
(x) is to 1, the more “certain”
one is about the value of x. A fuzzy number
is normal and convex when its membership
function takes one maximum value equal to 1 and is always increasing to the left
of the peak, and decreasing to the right.
The simplest type of fuzzy number is the
triangular, that is, one having linear membership functions on either side of
the peak. A fuzzy triangular number can be characterized by three real numbers:
two values of x where the membership function reaches zero and one where it
reaches a value of 1. Fig. (A.1) gives an example of a triangular fuzzy number (TFN).
This may be described by the values of x at points x1, x2
and x3, i.e.
= (x1, x2, x3)
Fig.A1 A triangular fuzzy number X=(x1, x2, x3).
The h-level set of a fuzzy number
is the ordinary set or interval
(h), defined as
(h)={x: m
(x) ³h)}
(A.2)
Fig. (A.1) illustrates the above definition.
Let us consider two triangular fuzzy numbers A and B given by the triplets A=(a1, a2, a3) and B=(b1, b2, b3). We have
(1) Addition: A+B =
(2) Subtraction: A-B=
Multiplication or division between two fuzzy numbers is not giving always a fuzzy number. However we can approximate them as follows
(3) Multiplication
A*B=
,
(4) Division
A/B=
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