Fuzzy indices of environmental conditions

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Abstract

Fuzzy logic provides a powerful and convenient formalism for classifying environmental conditions and for describing both natural and anthropogenic changes. Whereas traditional indices are based either on crisp sets with discontinuous boundaries between them (e.g. pristine vs. polluted), or on continuous variables whose values are only meaningful to experts (such as so many ppm of a toxin), fuzzy sets make it possible to combine these approaches. Conceptually the use of fuzzy logic is simple (for example, one can describe a site as 20% pristine and 80% polluted), but the real power of the methodology comes from the ability to integrate different kinds of observations in a way that permits a good balance between favourable and unfavourable observations, and between incommensurable effects such as social, economic, and biological impacts. In addition, fuzzy logic can be used to classify and quantify environmental effects of a subjective nature, such as bad odours, and it even provides a formalism for dealing with missing data. The fuzzy memberships can be used as environmental indices, but it is also possibly to ‘defuzzify’ them and obtain a more traditional type of index. The fuzzy methodology is illustrated by examples based on research to evaluate of the effects of finfish mariculture on coastal zone water quality.

Section snippets

Fuzzy memberships as environmental indices

Environmental indices rarely have much significance by themselves. Knowing that some variable like pollutant concentration or soil acidity has a specific value is usually meaningful only in the context of knowledge of natural background levels, regulatory policy, and the vulnerability of key environmental components. For this reason it is useful, and probably more practical, to relate these indices to some sort of acceptability measure, which can be interpreted as the membership in a fuzzy set

Mathematical formalism

Traditionally the symbol μ has been used to represent fuzzy memberships. If x represents the value of an environmental variable, then μ(x) is the corresponding membership in the set of acceptable conditions, and takes a value between zero and one. For example, if a lake becomes hypoxic and all the fish die, then μ would presumably be zero, indicating that this situation is totally unacceptable.

In most situations more than one environmental variable is important, and we can define μi(xi), which

Examples

Several applications of this approach have been implemented, or are under consideration. Two are described below, one of which is very specific and has been worked out in detail, the other of which is more general and addresses issues which have not yet been fully resolved.

Combining fuzzy indices

One aspect of the problem of applying fuzzy indices to complex situations is the need to combine different indices representing different impacts. Perhaps the strongest positive feature of fuzzy logic in developing environmental indices is the ability to combine such indices much more flexibly than one can combine discrete measures, which are often simply binary indices corresponding to ordinary (‘crisp’) sets, such as ‘acceptable versus unacceptable’. For this reason it is important to discuss

Multi-objective decision making

As pointed out earlier, society is not always able to reach consensus on the value of certain components of the environment, so that effects which are acceptable to some segments may be far less acceptable to others. Examples include the abundances of certain birds and marine mammals, which are highly prized for their beauty and entertainment value by recreational users of the environment, but are seen as predators and competitors by fishers and farmers. Many complex issues deal with the

Consensus building with fuzzy logic

Multi-objective decision making is at the heart of the political process, which involves trying to build a consensus among groups with different values and goals. The formalism described above can be used to identify key areas of disagreement and may possibly contribute to the resolution of conflict in complex situations by providing a language for quantifying these disagreements.

I propose a three-step procedure to deal with these kinds of disagreements:

  • 1.

    Identify environmental variables on which

Conclusion

Fuzzy logic can be applied to the development of environmental indices in a way that resolves many common problems, such as incompatible observations and implicit value judgements. It bridges the gap between scientific measurement and the fulfilment of social objectives and provides a way to translate a wide variety of information — objective data, qualitative information, subjective opinions, and social needs — into a common language for characterising environmental effects.

Because it offers a

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