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Suslainabilityof Water Resources under Increasing Uncertainty (Proceedings of Rabat Symposium SI, April 1997). IAHS Publ. no. 240,1997. 3

Risk in sustainable water resources management

SLOBODAN P. SEVIONOVIC Natural Resources Institute and the Department of Civil and Geological Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Abstract Sustainability, as used throughout this work, is defined as the ability to meet the needs of the present without compromising the needs of future generations. Risk is identified as one of the key sustainability issues. A conceptual understanding of the sources of uncertainty is developed. The general concepts surrounding the probabilistic and fuzzy set approaches used to address uncertainty are presented. Risk management is one of the main requirements imposed by the consideration of sustainability. The main characteristics of the available approaches for risk management are presented at the end of the discussion. Three main contributions of the discussion are: (a) the development of a general taxonomy of the sources of uncertainty; (b) use of the fuzzy set theory for risk definition and management; and (c) a detailed review of approaches for risk management.

INTRODUCTION

Sustainability context of water resources management

The new ethic of sustainable development not only reinforces but also extends the main principles of water resources management. Applying principles of sustainability requires major changes in the objectives on which decisions are based and an understanding of the complicated interrelationships between existing ecological, economic and social factors. The broadest objectives for achieving sustainability are: (a) environmental integrity; (b) economic efficiency; and (c) equity (Young, 1992).

The second important aspect of sustainable decision making is the challenge of time (long term consequences). Sustainable development requires forms of progress that meet the needs of the present without compromising the needs of future generations (WCED, 1987). Extension of temporal scale is required if we are to address the needs of future generations. Extension of temporal scale triggers the expansion of spatial boundaries too.

The third aspect of the sustainability context is the change in procedural policies (implementation). Pursuing sustainable project selection will require major changes in both substantive and procedural policies.

In today's complex economy, water resources play a key role. Sufficient supply of fresh water is a necessary condition for economic growth and development. At the same time, preservation of satisfactory water quality in rivers, lakes, reservoirs and aquifers is necessary to protect public health and ecosystems. Economic development of many countries around the world is put under stress due to the increase in water demand by different users and the decrease of the water quality due to pollution. The problem is even more intensified in regions where droughts and floods are further affecting the balance between the demand and water availability.

4 Slobodan P. Simonovic

Sustainability requires innovative approaches for the derivation of appropriate risk measures since the use of the expected value of risk, and/or the conditional expected value, are no longer sufficient. The following section will provide a general taxonomy of the sources of uncertainty necessary for appropriate risk definition, which is discussed afterwards. Knowledge-based risk management is examined in this research. The last section of the paper presents different risk management techniques based on the probabilistic and fuzzy set approaches.

GENERAL TAXONOMY OF THE SOURCES OF UNCERTAINTY

Introduction

Expanded spatial boundaries, a lengthened time scale, comprehensive multi-objective analysis, and other issues related to sustainability place immense demands on science (Simonovic, 1996a). A number of questions raised by the sustainability perspective reveal major deficiencies in the knowledge of the behaviour of a wide range of natural and human systems under consideration. Recognizing that many of these deficiencies cannot be eliminated immediately makes it evident that risk and uncertainty are inherent concepts related to sustainability.

To determine the acceptable level of environmental and social protection, the environmental and social risks associated with economic development must be assessed. Risk has different meaning and is applied differently in various disciplines such as engineering, economics, environmental and social sciences. In this study a combined definition is required which will be based mainly on the quantification of various uncertainties occurring in the evolution of physical processes. The use of modelling to quantify such uncertainties is an essential part of risk management. Furthermore, because sustainability requires long term predictions of how processes might develop under uncertainty in the future, probabilistic approaches are more appropriate than deterministic methods. Probabilities and the fuzzy set theory are suitable tools for quantifying uncertainties which may induce a risk of failure. Understanding uncertainty is essential in risk management in order to narrow the gap between good decisions and good outcomes and the inability to reduce the tension between analysis and action.

Taxonomy of sources

Uncertainty can be divided into two basic forms: uncertainty caused by inherent stochastic variability and uncertainty due to a fundamental lack of knowledge. Awareness of the distinction between these two forms is integral to understanding uncertainty. The first form is labelled as variability and the second one as uncertainty (Ling, 1993). Variability is a result of inherent fluctuations in the quantity of interest (hydrological variables). The three major sources of variability are temporal, spatial and individual heterogeneity. Temporal variability occurs when values fluctuate according to time. Other values which are affected by spatial variability are dependent upon the location of an area. The third category effectively covers all other sources

Risk in sustainable water resources management 5

of variability. In water resources management, variability is mainly associated with the spatial and temporal variation of hydrological variables (precipitation, river flow, water quality, etc.).

The more elusive type of uncertainty is due to a fundamental lack of knowledge. It occurs when the particular values that are of interest cannot be presented with complete confidence because of a lack of understanding or limitation of knowledge. The main sources of uncertainty due to lack of knowledge are depicted in Fig. 1.

uncertainty

variability

temporal spatial individual heterogeneity

model

surrogate variables excluded variables abnormal situations

approximations incorrect form

knowledge

parameter

measurements systematic

sampling error unpredictability ling, imprecision

decision

risk measure social risk cost quantification of social values

Fig. 1 Sources of uncertainty (after Ling, 1993).

Model and structural uncertainties refer to the knowledge of a process. Models are simplified representations of real world processes and model uncertainties can arise from oversimplification or from the failure to capture important characteristics of the process under investigation. Addressing this type of uncertainty is the coarse tuning function of the analysis. This type of uncertainty is best understood by studying its major sources. Modelling in the sustainable water resources management includes surrogate variables (substitute variables for the quantities which are difficult to assess). They are approximations of the real values. The second source of model uncertainty stems from excluded variables (variables deemed insignificant in a model). The removal of certain variables or factors introduces large uncertainties into the model. For example, many environmental risk assessment methods do not consider the propagating effects of hazardous chemicals through vegetation. Attempting to address excluded variables raises a paradox: we do not know when we have forgotten something until it is too late. The


impact of abnormal situations on models is the third source of uncertainty. The very nature of a water resource model requires model calibration and verification using a set of broad circumstances. The problem occurs when a model is used for some other situation outside the set of situations used in the calibration and verification process. Approximation uncertainty is the fourth source of model uncertainty. This source covers the remaining types of uncertainty due to model generalizations. An example of approximation uncertainty in hydrology can be found in the use of discrete probability distributions to represent a continuous process. The final type of model uncertainty, incorrect form (correctness of the model being used to represent the real world), is initially the most obvious. To properly address this source, we must remember that all results are directly dependent on the validity of the assumed model's representation of the true process.

The next general category of uncertainty is parameter uncertainty. It is the fine tuning of a model and cannot cause the large variations found in model uncertainty. The most common uncertainty in this category is caused by a random error in direct measurements. It is also referred to as metric error, measurement error, random error and statistical variation. This error occurs because no measurement in water resources can be exact. Imperfections in the measuring instrument and observation techniques lead to imprecision and inaccuracies of measurements. The second, and largest, source of parameter uncertainty is systematic error (error due to subjective judgement). Measurements involve both random and systematic error. The latter is defined as the difference between the true value and the mean of the value to which the measurements converge. The third type of error is sampling error (error in drawing inferences from a limited number of observations). Sampling causes uncertainty in the degree to which the sample represents the whole. Well developed statistical techniques such as confidence intervals, coefficient of variation, and sample size are used in water resources to quantify this type of uncertainty. The fourth type of parameter uncertainty is caused by the unpredictability of an event. Limitations in knowledge and the presence of inherent unpredictability of the process make it impossible to predict the wind direction and velocity at a future date. The fifth source of uncertainty is caused by linguistic imprecision. Everyday language and communication are rather imprecise. It is possible to reduce linguistic uncertainty through clear specifications of events and values. The final source of uncertainty is derived from disagreement (conflicting expert opinion).

The third category of uncertainty is decision uncertainty which arises when there is controversy or ambiguity concerning how to compare and weigh social objectives. It influences decision making after model and parameter uncertainty have been considered. The first decision uncertainty includes uncertainty in the selection of an index to measure risk. The measure must be as technically correct as possible while still being measurable and meaningful. The second source of decision uncertainty lies in deciding the social cost of risk (transforming risk measures into comparable quantities). The difficulties in this process are clearly illustrated in the concept of developing a monetary equivalent for the value of life in flood control analysis. The quantification of social values is the third source of uncertainty. Once a risk measure and the cost of risk are generated, controversy still remains over what level of risk is acceptable. This level is dependent upon the attitude of society to risk.


RISK ANALYSIS

Introduction

The basic goal of risk assessment is to describe risk. Risk is defined as a measure of the probability and severity of adverse effects (Lowrance, 1976). Risk addresses three questions: (a) What can go wrong? (b) What is the likelihood that it will go wrong? and (c) What are the consequences? (Kaplan & Garrick, 1981). The taxonomy presented in the previous section seeks to improve the information provided in a risk assessment. Identifying the uncertainties in water resources information aids in answering the above questions. The motivation for risk assessment is in determining the current risk state of a system.

Definition of risk

Sustainable management of water resources involves examination of the hydrological basis and management options with special attention given to floods and droughts, but also considering the normal variability of water availability. One difficulty in managing floods and droughts (extreme events) is that there is a lack of experience which makes it difficult to understand fully the impact of these events. Another difficulty in managing these events arises from the complexity of current systems which prevents experienced decision makers from relying only on intuition and experience. In addition to these difficulties, people's perception of the risk of extreme events can be incorrectly influenced by different biases and heuristics (Tversky & Kahneman, 1974). The sus-tainability context is adding one more dimension to risk management: extension of temporal and spatial scales. Traditional quantitative methods of evaluating risk, such as the expected value of an event, do not properly represent the importance of these extremes. Using the product of probability and consequence equates low probability, high damage events with high probability, low damage events, effectively ignoring the impact of extreme events.

A general definition of risk can be based on the concept of load and resistance, terms used in structural engineering. In the field of water resources and environmental sciences these two variables have a more general meaning as shown in Table 1 (modified after Ganoulis, 1994). Load / is a variable reflecting the behaviours of the system under certain external conditions of stress or loading. Resistance r is a characteristic variable which describes the capacity of the system to overcome an external load (Ganoulis, 1994; Plate & Duckstein, 1988). When the load exceeds the resistance (/ > r) there should be a failure or an incident. Safety or reliability state is obtained if the resistance exceeds or is equal to the load (/ < r).

From Table 1 it can be seen that load and resistance may take different meanings, depending on the specific problem domain. The consequences of failure together with the public perception of risks are considered in risk management. In a probabilistic domain, / and r are taken as random variables. In probabilistic terms, the chance of having a failure is defined as risk.

RISK = probability of failure = P(l > r) (1)

Slobodan P. Simonovic

Table 1 Examples of loads and resistance in water resources.

Physical system Scientific discipline

Load

Hydraulic stractureCivil Engineering Force (dam, levee, Wind load gate,...) Flood rate

Stress

Water system (lake, aquifer, river, ...)

Hydrological system (watershed reservoir, ...)

Ecosystem

Human organism

Economic system

Social system

Water Resources Water demand Engineering Pollutant load

Energy demand

Hydrology

Biological Sciences

Health Sciences

Economics

Social Sciences

Flow rate Flood Rainfall Evaporation

Exposure

Exposure

Investment needs Capital Interest rate

Change of system Perception Acceptance

Resistance

Resisting stress Dam height Levee height

Water supply Reservoir capacity Receiving capacity

Threshold flow rate Flood Rainfall

Ecosystem capacity

Human capacity

Money supply Threshold interest rate

Acceptance level Flexibility Resistance capacity

Type of failure

Structural failure

Water shortage Water pollution Energy shortage

Exceedance Floods

Ecosystem damage

Health damage

Fiscal failure Lack of capital

Change of population Culture change War

It is important to note that this definition assumes availability of sufficient data to represent uncertainties in the values of / and r.

Stochastic approach to risk definition

By considering the system variables as random, uncertainties can be quantified using a probabilistic framework. Load I and resistance r are taken as random variables L and R, having the following cumulative distribution and probability density functions:

FL(0, fL(0 FR(r), fR(r)

load resistance

In a probabilistic framework one of the simpler definitions of risk is the probability of the load exceeding the resistance. Risk is given by the following relation:

pF = P(R < L) (2)

where pF is the asymptotic limit. The quantity pF is obtained in terms of the joint probability density function fLR{l, r) of the random variables R and L. As illustrated in Fig. 2, the nskpF may be estimated by integrating the function fLR(l, r) above the line L = R.


L > R L = R

L < R

fLRd,r)

Fig. 2 Definition of probabilistic risk.

By performing the integration in the above domain, the following equation is obtained:

pF = P(L > R) = \kR^rW d/ (3)

This is the general expression for quantifying risk in a probabilistic framework. However, use of equation (3) is rather difficult, because generally the joint density probability function fLR(l, r) is unknown. Simplifications include the assumption of independence between load and resistance. Some examples are provided by Plate & Duckstein (1988).

Fuzzy set approach to risk definition

The uncertainty inherent in water resources data, values of parameters, and boundary conditions of variables used in mathematical modelling may be quantified by use of stochastic variables if sufficient data are available to fit a probability density distribution. However, if the requirements of sustainability are to be addressed (long term consequences, needs of future generations, expanded spatial and temporal scales) the information available is scarce. In this case the fuzzy set approach may be appropriate. Fuzzy set theory (Zadeh, 1965), which is a theory of possibility, is not dissimilar to probability theory. In fact, they can be considered complementary. Fuzzy membership functions, which represent within the interval (0, 1) the degree of confidence one might have for a particular value of the fuzzy number as shown in Fig. 3, have a similar appearance as probability density functions. A fuzzy membership function acknowledges that we may not be completely sure what values we are talking about. Statistical precision can be independent of our classification of events. For example, we may predict 90% probability of the occurrence of a good value. What is a good value? Qualification of good can be very subjective. In general, fuzzy sets


membership values

H(x)

possible values Fig. 3 A fuzzy set.

provide an intuitive and flexible framework for interactively exploring a problem that is either ill-defined or has limited available data.

The basic arithmetic rules of fuzzy sets are different from those of probability theory (Bender & Simonovic, 1996). In order to define fuzzy risk assume that the system has a resistance R* and a loadL*, both represented as fuzzy numbers. A reliability measure (or a safety margin) of the system may be defined by the difference between load and resistance (Schresta et al., 1990). This is also a fuzzy number:

M* = R* - L* (4)

Taking the /z-level intervals as:

R(h) = [Ry(h), R2(h)] L(h) = [Ll(h),L2(h)]

then for every h G [0,1], the safety margin M(h) is obtained by subtraction between R(h) and L(h):

M(h) = R(h) - L(h)

Two possible conditions exist: (a) failure: M(h) < 0; and (b) reliability: M(h) > 0. A fuzzy measure of reliability, or fuzzy reliability index may be obtained as:

R„ UM,(x)dx

(5)

Two limiting cases are of interest as illustrated in Fig. 4: (i) absolute safety when M(h) > 0, Fig. 4(a); and (ii) absolute failure whenM(/z) < 0, Fig. 4(b).

RISK MANAGEMENT

Introduction

Risk management builds on risk assessment by seeking to determine a management policy. To manage risk over time, the following three questions must be addressed (Haimes, 1992): (a) What can be done? (b) Which options are available and what are


x 0 Fig. 4 (a) Absolute safety and (b) absolute failure.

their associated trade-offs in terms of all costs, benefits, and risks? and (c) What are the impacts of current management decisions on future options? When it is possible to assess the risk using either stochastic or fuzzy set approaches, then the process of risk management may follow. Risk management is of primary importance for sustainable water resources analysis. It represents a methodology for giving rational consideration to all factors affecting the safety of the system operation. It identifies, evaluates and executes, in accordance with other social sectors, all aspects of system management, from the identification of loads to the planning of emergency scenarios and relief and rehabilitation, for the case of operational failure and structural failure, respectively. It is a technical and social, and also an economic process based on balancing costs and benefits both in monetary and social terms (UNDRO, 1991).

There are different suggested methodologies of risk management available in the literature. Characteristics of sustainable water resources management can be appropriately addressed through the following steps: (i) identification of feasible alternatives and associated risks; (ii) assessment of all impacts associated with various risk levels; (iii) selection of acceptable options according to the public perception of risk, government policy and other social factors; and (iv) implementation of the optimal choice.

Because of the human and social questions involved, risk management is part of a more general process which is extremely important but also very difficult to develop. In the document by UNDRO (1991) the general process includes: (a) hazard identification (hazard determination and presentation) ; (b) risk assessment (evaluation of consequences, risk calculation, comparison with accepted risks); (c) risk mitigation (investigation of methods of risk control, containment, technical and nontechnical solutions); and (d) risk management (risk control, forecasting, preparedness).

The discussion in this section will focus on the risk management methodology suggested above as a four step process. Theories and algorithms of simulation, optimization, multi-objective analysis and decision making under uncertainty are all applicable including both stochastic and fuzzy set approaches. The following discussion presents available algorithms for risk management.

Tools for risk management

The simple definition of risk (equation ( 1 )) does not reflect completely the characteristics


of the physical water resource system operating under risk. It gives just an indication of the state of, respectively, the safety of the system or the description of system behaviour under uncertainty. In a more general way, we can say that the risk definition is just an index characterizing the performance of the system. In order to describe in more detail the behaviour of water resource systems other performance indices (PI) are used known as resilience, vulnerability, grade of service, availability, quality of service etc. (Fiering, 1982; Hashimoto et al., 1982a, 1982b; Burn etal., 1991; Duckstein& Parent, 1994). In addition to performance indices figures of merit (FM) are also used for the analyses of system behaviour. They are defined as functions of performance indices:

FM|. = FM|.(PI1,PI2,PI3>...,PI„) (6)

Some attempts have been made to measure sustainability using different FMs. One idea views sustainability as a combination of high resilience and low vulnerability. Another idea involves identifying new Pis to directly describe sustainability. Work in progress in the field of water resources is aimed at developing operational definitions for reversibility, equity and risk as sustainability criteria. Preliminary information is made available by Simonovic et al. (1995), Kroeger & Simonovic (1996), Fanai & Burn (1996), and Matheson & Lence (1996).

Stochastic simulation Simulation models "describe" how a system operates, and are used to determine changes resulting from a specific course of action. Such models are sometimes referred to as "cause-and-effect" models. Simulation models describe the state of the system in response to various inputs but give no direct measure of what decisions should be taken to improve the performance of the system. The major components of a simulation model are: (a) input - quantities that "drive" the model (in water resources models a principal input is the set of streamflows, rainfall sequences, pollution loads, water and power demands, etc.); (b) physical relationship - mathematical expressions of the relationship among the physical variables of the system being modelled (continuity, energy conservation, reservoir volume and elevation, outflow relationships, routing equations, etc.); (c) nonphysical relationships - those that define economic variables, political conflicts, public awareness, etc. ; (d) operation rules — the rules that govern operational control; and (e) outputs - the final product of operations on inputs by the physical and nonphysical relations in accordance with operating rules.

The simulation approach can be used in the situations where important components of the system are stochastic processes whose values change with time. A typical example in water resources will be unregulated streamflow. Stochastic simulation is certainly the most flexible and widely used tool for the analysis of complex water resource systems (Loucks et al., 1981). In any stochastic simulation model some provision for the generation of sequences of random variables is included. The process follows with the development of the simulation model. The main part of the process is the output interpretation. It includes the computation of Pis, confidence intervals and risks. With a new operating policy, the process may continue with new iterations.

Stochastic optimization Generally there are two categories of stochastic optimization techniques: (a) implicit (which use deterministic models with the generated sequences of random variables); and (b) explicit (which incorporate uncertainty directly in the objective function and/or constraints).


A general linear optimization problem is:

maximize f(x) (7)

subject to:

gt(x) < bt i = 1, 2, ...,m

where x contains the decision variables. Uncertainty may arise in: (i) the objective function f(x); (ii) one or more constraints g(x); or (iii) right-hand-side values bt. In case (i) the source of uncertainty is imprecise knowledge of future benefits and costs. This type of uncertainty can often be handled by substitution with the expected value. For case (ii), if the uncertainty is small, it may be acceptable to use the expected value of the random variable:

E[gi(x)] < £[£,-] (8)

In case (iii) a chance constraint can be written that defines the probability Pi of the constraint to fail:

Pr[gi(x) < bi\ > 1 - />,. (9)

where bt is a random variable; and 1 - P, indicates that the constraint can be violated no more than 100P,% of time. The major advantage of chance constraints is that they can be converted into deterministic equivalents, given knowledge of the distribution function F,(6;). This concept has been extensively used in water resources (Loucks et al., 1981) and extended to reliability programming (Simonovic & Marino, 1980).

For sequential water resources decision making problems another popular stochastic optimization approach is stochastic dynamic programming (referred to as SDP). SDP models can incorporate uncertainty in the input data directly into the solution procedure through stochastic generalization of the recursive equation:

y, = mm d in 0(0

i - l

Zp(j\i,d)ayj + cid j-o

for all / jt 0 and y0 = 0 ( 1 0 )

where D(i) is the set of possible decisions; yt is the minimum expected present value; p(j\i,d) is the conditional probability that the state of the system is y given that at i decision d is made; a is the one period discount factor; and cid is the expected return in the current period from making decision d at i.

It is important to note the ferocity of the computational burden associated with the recursion equation (10). This problem has given rise to a number of approximate solution techniques, based on the concepts of SDP, that make compromises in either the generality of the problem formulation or in the optimality of the solution, or both. The techniques used to solve these problems are generally either limited to smaller applications or are based on iterative methods which may not converge to a global optimum. The author of this paper is unaware of any work which deals with evaluation of the hydrologie reliability explicitly through SDP algorithms.

Fuzzy optimization Zimmermann (1976) was the first to introduce fuzzy set theory into conventional linear programming optimization. In contrast to a conventional programming problem, Zimmermann (1976) proposed to soften the rigid requirements


to strictly minimize the objective function and to strictly satisfy the constraints. By considering the imprecision or fuzziness of the decision maker's judgment, he softened the usual linear programming problem into the following fuzzy version:

ex < z0

Ax < b < u ) x > 0

where the symbol " < " denotes a relaxed or fuzzy version of the ordinary inequality. To be more explicit, problem (1.1) means that the objective function ex should be essentially smaller than or equal to an aspiration level z0 of the decision maker, and the constraints Ax should be essentially smaller than or equal to b.

This powerful concept may be of great importance in addressing sustainability through risk management of water resources systems. However, there is still a limited number of successful applications of this approach in water resources (Savic & Simonovic, 1991; Kojiri, 1995).

Decision theory This theory is concerned with alternative actions that a decision maker should undertake under different environmental conditions. According to Berger (1985) there are three basic elements in a decision making situation: (a) candidate alternatives or alternative actions (alternative design solutions); (b) states of nature (environmental conditions in which any action should operate); and (c) outcomes (consequences associated with an action and a state of nature). Actions, states of nature and outcomes may be represented in the form of a decision tree or decision table. Decision trees can include uncertainties and preferences. Their combination gives the decision rule which is the tool for taking the final decision.

Water resources applications of decision theory are numerous, from wastewater treatment plant design to spillway capacity analysis. It is important to mention that the approach can be used in decision making under certainty, under risk, and with Bayesian analysis.

Stochastic multi-objective analysis Sustainability is placing much larger weight on replacing single-objective optimization with multi-objective analysis. The importance of considering more then one objective at a time is emphasized by the existence of several conflicting and noncommensurable objectives in every step of sustainable water management process. Multi-objective analysis considers changes in objectives with time due to change in technology, weather, population, etc. The main concept of multi-objective analysis is the replacement of a single optimal solution with a set of tradeoffs (nondominated, noninferior, compromise, Pareto optimal solutions). Sustainable water management is subject to two major sources of complexity regarding the application of multi-objective analysis: (a) quantification of different objectives; and (b) getting preferences from decision makers regarding different objectives.

A multi-objective programming problem is characterized by ap-dimensional vector of objective functions:

Z(x) = [ Z ^ . Z ^ ) , . . . ^ * ) ] (12)

subject to:


x EX and X = {x: x E Rn,gfx) < 0,Xj > 0 v i,j) (13)

where R" is a set of real numbers; gt(x) is a set of constraints; and x is a set of decision variables.

The above formulation of a multi-objective problem may include uncertainties of a different nature. It has been shown that it is possible to deal with random variables in the set of constraints and objective functions of a stochastic multi-objective programming problem (Goicoechea, 1979). However, an extension of the approach to address more realistically a larger variety of problems in water resources management is not available yet.

Another way of using a multi-objective analysis in risk management is through the consideration of risk as one of the objective functions Zp(x). In this way the attitude of the decision maker towards the risk can be incorporated directly into the analysis (Fig. 5). One particular technique of multi-objective analysis, known as the utility function, has been used successfully in different fields but not much in the field of water resources management (Keeney & Raiffa, 1993, first edition 1976).

risk risk indifferent

physical value

Fig. 5 Different forms of utility function.

Fuzzy multi-objective analysis The multi-objective analysis concept has been extended to include imprecise or fuzzy understanding of the nature of the parameters in the problem formulation (12) and (13). A risk management tool with great potential in sustainable water resources management is fuzzy compromise programming (Bender & Simonovic, 1996) which allows various sources of uncertainty and facilitates a flexible form of group decision making. The fuzzy compromise programming is analogous to the discrete "crisp" version of compromise programming. It allows a family of possible scenarios to be reviewed through fuzzy sets designed to reflect collective opinions and conflicting judgments. Evaluating alternative choices to produce rank orderings are accomplished with ranking measures for fuzzy sets. The ranking measures indicate the impact of different levels of decision maker risk aversion. Experimentation is possible with different shapes of input fuzzy sets for values of the objectives, decision makers preferences, and distance metric interpretation.


CONCLUSIONS

Risk is identified as one of the key sustainability issues in water resources. Sustainability requires innovative approaches for the derivation of appropriate risk measures since the use of the expected value of risk, and/or the conditional expected value, is no longer sufficient. A general taxonomy of the sources of uncertainty necessary for appropriate risk definition is developed. Risk has different meaning and is applied differently in various disciplines. In this study a combined definition is presented which is based on the quantification of various uncertainties occurring in the evolution of physical processes. The use of modelling to quantify such uncertainties is an essential part of risk management.

Sustainability requires long term predictions of how processes might develop under uncertainty in the future, and probabilistic approaches are more appropriate for this purpose than deterministic methods. Probabilities and the fuzzy set theory are suitable tools for quantifying uncertainties which may induce a risk of failure.

When it is possible to assess the risk using either stochastic or fuzzy set approaches, then the process of risk management may follow. Risk management is of primary importance for sustainable water resources analysis. It represents a methodology for giving rational consideration to all factors affecting the safety of the system operation. A four step risk management methodology is suggested. Theories and algorithms of simulation, optimization, multi-objective analysis and decision making under uncertainty are all applicable, including both stochastic and fuzzy set approaches.

Acknowledgements This work was made possible by a Natural Sciences and Engineering Research Council grant and support provided by Manitoba Hydro. A special thanks to all my graduate students who eagerly participated in numerous discussions of topics presented in this paper.

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