[This paper was presented at Systems Science 2000 in Osnabrück, Germany, March 2000, and will be published in the proceedings of the conference, Integrative Systems Approaches in Natural and Social Sciences, H. Malchow (ed.), Springer-Verlag, Berlin. It is also available as a PowerPoint show that was used in the oral presentation. Not to be reproduced without permission of the author.]

Systems scientists generally use precise numerical models, but in many cases it might be helpful to use models based on fuzzy logic, especially when dealing with qualitative aspects of systems. For example, when dealing with parameter uncertainty it is usual to provide confidence ranges for numerical outputs, but suppose that one carries out a Monte Carlo simulation for parameter uncertainty, and finds that in 40% of the simulations the system is unstable? It is impossible to say that such a system is stable or unstable in crisp terms, but by interpreting the zone of stability as a fuzzy set it is reasonable to say that the system is in a state that is 60% stable, i.e., the membership in the set of stable states is 0.6. The same approach can be applied to experimental studies in which a number of replicates fail to give identical results. Fuzzy terminology can also be used to communicate effectively the results of complex analyses. In the case of oil spill modelling for example, the usual outputs can include the risk of an oil slick reaching a seabird colony, the concentration of oil in the slick and in the water column, and some measure of the toxicity of the oil, but what the public and the mitigation teams need to know is whether the threat to the birds is serious, moderate, or slight. This paper presents several ways in which fuzzy methods could be used in systems science and describes implementation techniques, such as ways of combining multiple variables into fuzzy classes.

Keywords: Fuzzy set theory, Systems analysis, Classification

Fuzzy set theory offers a natural, convenient and powerful means of dealing with many of the problems of system science, so it is puzzling that it has not received wider attention and application. This is the first issue that needs to be addressed, looking both at the definition of fuzzy set theory and possible reasons why it has not been widely accepted and implemented. This will be followed with a discussion of ways in which it could be used productively, with several examples.

Part of the riddle of fuzzy set theory can be understood by the fact that it is so obvious and intuitive that it is hard to take seriously (the term "fuzzy" doesn’t help either). It is a natural extension of the way in which we classify things, and classification is one of the fundamental activities in science. Classification means putting things into sets, but paradoxically the way in which we classify objects when we are doing science does not reflect the way in which we really think. This probably reflects an attitude that science must be precise and exact, but since science seeks to represent the real world, and since the real world is neither precise nor exact, our desire to be scientific can actually weaken our scientific efforts.

Consider how we might characterise a scientific colleague: as a tall man. This associates him with two sets, the set of "men" and the set "tall". The set of "men" is a typical example of a "crisp set", since everyone either belongs to this set or to its complement, "not-men" (normally referred to as "women"). The set "tall" is not as clearly defined, and someone whom the Portuguese might consider tall (i.e. a member of this set) might not be so classified by a Norwegian. This is a standard example of a fuzzy set, so that instead of saying that the man in question is or is not a member of the set "tall", we might say that he is 70% tall, where the fractional membership could represent the result of a poll where 70% of the people polled consider him tall.

A crisp set is clearly a special type of fuzzy set, in which the membership is either 0 or 100%. Although cumbersome, we could describe our colleague as 100% a member of the set "men" and 70% a member of the set "tall". However, there are very few completely crisp sets in the natural world. Even though the division of human beings into the two complementary sets of "men" and "women" is universal and even legally definitive (just look at your drivers license, for example), it is not always meaningful. If our colleague has a sex change operation, will there be some instant of time at which he goes from being 100% male to 100% female? And what about the many plants and animals that undergo natural changes in sex over periods of several months, or which possess both male and female sexual organs at the same time? It is extremely difficult to find realistic examples of sets that are completely crisp, so we must conclude that most of the sets that we use in systems classification are actually fuzzy.

Why then the reluctance to use fuzzy set theory? It must be admitted that fuzzy set theory has sometimes been over-sold, with some proponents claiming that it is the greatest development in mathematics since Galois invented group theory, or perhaps even since the Arabs came up with the concept of zero. This is hardly credible, and it is difficult to find many mathematical constructs in fuzzy set theory that are not simply generalisations of well-known aspects of traditional set theory. The advantages of fuzzy set theory are not that it brings a set of fantastic new mathematical tools into play, but rather that it lets us apply common sense principles that have often been ignored in an effort to appear rigorous and "scientific". This paper will explore some of the ways in which fuzzy set theory can be used, but will not delve into the more mathematical issues, which are adequately covered in some of the literature cited.

There are many ways in which fuzzy set theory can be used in systems science, and a brief survey like this can only cover a few of them. This paper is focussed primarily on applications from the author’s own work, but it should not be thought that the only areas where fuzzy set theory is useful is in marine ecology.

One area where fuzzy sets are widely used is in the interpretation of remote sensing data. This is a natural sort of application, since the data are in the form of a discrete array of pixels, each of which must be classified, but the classifications are not clear-cut and can often best be represented by fuzzy sets.

This is most easily seen if one looks at a problem like land use classification, where a pixel may cover an area which is half residential and half farm land. It is more logical to classify the pixel as 50% of each (i.e., with 50% membership in each of the two sets) than to assign it arbitrarily to one of the two sets, and since the information can often be inferred from spectral analysis of the pixel, it is a much more sensible approach than trying to give each pixel an unambiguous classification. Fuzzy set theory seems to have been widely accepted in remote sensing for this reason, and we can generalise to conclude that in any case where we need to aggregate over scales larger than the natural scale of system heterogeneity, treating the aggregates as fuzzy sets will usually give us better resolution than crisp classification.

Ecology offers many situations where fuzzy sets are or can be used, although it has not been widely accepted under that name. The concept of trophic levels (Lindeman, 1942) illustrates this: plants are at the lowest trophic level, "producers", and are consumed by herbivores which are classified as "primary consumers", which in turn are eaten by carnivores, "secondary consumers", and so on. Although this is a very useful scheme, not all organisms fit it exactly; some plants are carnivorous, and there are many omnivorous animals which feed at more than one trophic level, such as planktivorous fish that consume both algae and zooplankton. It has become common to use fractional trophic levels for this reason, although this is not as good a solution as using fuzzy sets, since it provides less information – there are many ways in which we can obtain a numerical value, such as 2.5, for a trophic level, which can correspond to an omnivore that feeds equally at trophic levels 1 and 2, or to one that feeds three times as much at level 1 as at level 3 (the trophic level of a consumer is defined to be one greater then that of its prey). It makes more sense to assign partial memberships to trophic levels, especially since in the second case the fractional trophic level of the predator would be less than that of one of its prey!

Another potential application of fuzzy set theory to ecology is in the theory of the niche, which Hutchinson (1957) defined as the set of points in "hyperspace" (i.e., the space defined by environmental conditions) in which a population can persist. This is one of the few times that an ecologist has even used the word "set" in a mathematical sense, but the use of a crisp definition for such a fuzzy concept as niche has led generations of ecologists to hair-splitting and confusion. By defining the niche as a crisp set there is no distinction between ideal conditions and those in which a population can only barely persist, but by defining the niche as a fuzzy set it is easy to differentiate points of environmental hyperspace by degree of suitability. This also facilitates applications of niche theory, such as niche packing (the overlap between the niches of competing species), but this topic goes beyond the context of this paper (cf. Christiansen and Fenchel, 1977).

A good example of how fuzzy set theory can be used to combine scientific data with common sense is in the evaluation of environmental impacts of multiple pollutants. We need to define threshold levels of maximum permissible impact, which amounts to classifying these impacts as acceptable or not, i.e., classifying them as members of the set of acceptable impacts or as members of the complementary set of unacceptable impacts. We need to carry out this classification for each pollutant, and then to combine all of the classifications to obtain an overall classification.

With crisp sets one cannot identify borderline cases, which leads to inconsistencies. One must of course define threshold values that are somewhat arbitrary, and any factory or plant which generates pollutant levels just under these thresholds meets the standards. On the other hand, any installation which exceeds just one of these thresholds by a minuscule amount must be considered in violation of the standards, even if in all other respects it is absolutely clean. If we think of all of the thresholds normalised to one, then this means that a vector of pollution outputs (0.99, 0.99, 0.99, …) is acceptable, while (1.01, 0.00, 0.00, …) is not. This does not seem reasonable, but the use of crisp set theory forces us into a framework where it is difficult to avoid regulating pollution in this way. However, with fuzzy set theory we can convert acceptability from a strictly dichotomous yes/no classification to a continuous one, and although ultimately one has to decide whether the operation is within acceptable limits or not, there is more flexibility in the evaluation process. Instead of saying that levels of 99% and 101% of the threshold values are respectively acceptable and unacceptable, one can classify them as 51% and 49% acceptable, and ultimately combine all of these partial memberships to obtain a measure of the overall acceptability of all the pollutants from a particular plant. The final evaluation can then be made on the overall acceptability when all pollutants are taken into account, so that a degree of balance can be achieved between drastically different levels of different pollutants. The specific details of how one combines different levels of acceptability in a reasonable way have been discussed elsewhere (Silvert, 1997), but the basic principle is clear – environmental impacts lie along a continuum, and forcing them into crisp categories is poor science and can lead to inefficient and inconsistent management decisions.

It often happens that systems are characterised by ambiguous and seemingly inconsistent data, which makes crisp classification virtually impossible. The above example of multiple pollution outputs can be seen as an example of this, especially when the outputs of some pollutants are low and some are high, and we try to describe the operation as clean or dirty. The situation is even harder to address when we have to deal with different measures of the impact of single causes affecting a natural system.

For example, studies of the benthic impacts of fish farms involve multiple indicators, some of which may reflect little or no impact (or even a beneficial effect), while others show serious degradation. Combining these with crisp classifications is almost impossible, but fuzzy classification has proved to be both practical and informative, and gives reproducible results (Angel et al., 1998). Part of the utility of the fuzzy approach is the ability to allow for uncertainty in the significance of certain indicators. Some observations are unambiguous – when hydrogen sulphide gas is seen bubbling out from under a fish farm, it is clear evidence of severe degradation and calls for immediate remedial measures (for one thing, it is toxic to the fish). But bacterial mats, although usually indicative of nutrient fluxes exceeding the assimilative capacity of the seabed, are sometimes found under natural conditions and may not always be interpreted as evidence of adverse impacts. We can therefore classify H₂S ebullition as a severe impact, while bacterial mats can be classified as, say, 20% severe, 70% moderate, and 10% no impact (for clarity these are simpler categories than those used in Angel et al., 1998).

Ambiguity can arise from theoretical as well as experimental results. Simulation techniques are increasingly used to study the behaviour of complex systems, but they do not always lead to clear conclusions. For example, Monte Carlo simulations of large-scale model ecosystems show that complex systems tend to be more unstable than simpler systems, although for a given level of complexity some of the systems are stable and some are unstable (for a discussion of whether this has any relevance to the real world, see Silvert, 1983). One can, at least in principle, determine whether a precisely specified dynamical system with exactly specified parameters is stable, but once we start to consider real systems with uncertain structure and approximate parameter values, it is often impossible to be positive about the stability properties and it is best to use the less strict classifications of fuzzy set theory.

It is reasonable to expect that the more complex the systems we deal with and the greater the number of variables that have to be analysed and assimilated, the greater will be the difficulty of obtaining clear-cut answers. Fuzzy set theory offers a way to deal with this sort of complexity and ambiguity.

An important aspect of systems science that we tend to overlook is the necessity to interpret scientific work in terms that are meaningful to non-specialists. Meteorology is typical of the conflicts that this generates, since it is as technical and abstract as any scientific field, and yet it serves an enormous client base that includes not only other scientific professionals, but also airline pilots and children going on picnics. These groups are interested in the answers to specific questions such as "is there adequate visibility at the destination airport?" and "will it rain tomorrow?". It is not easy to provide crisp answers to these questions. This is reflected in the fact that meteorologists have turned to probabilistic predictions, such as "40% chance of rain". This is a form of fuzzy classification, since it is equivalent to saying that the predicted weather is 40% a member of the set of rainy days (in fact, some antagonists of fuzzy set theory claim that it is just repackaged probability theory, although there are many counter-examples to show that this is not the case).

More ambiguity occurs when the questions are broader. "Will it rain tomorrow?" is a more specific question than "Will the weather be nice?", and there is no clear answer if the predicted weather is, say, clear and sunny but cold. And while making weather predictions for picnic planning may seem too trivial to worry about, the same issues arise when we ponder the possibility of environmental disasters. An oil spill or forest fire has many different effects, some of which may be serious while others fall within the range of normal system perturbations. Often the perceived magnitude of the effects depend very much on the perspective of the observer – a red tide event may not kill many fish, which is satisfactory from a general environmental point of view, but if the resultant toxicity makes the surviving fish commercially unacceptable, it would be seen as disastrous from the viewpoint of the fishing industry. Clearly there are major problems in evaluating such events and it is probably impossible to achieve a consensus about them in many cases, but fuzzy set theory can provide a context for developing indices of the level of different effects that can provide a quantitative framework for evaluating the consequences of complex events.

One of the most widely accepted applications of fuzzy set theory is in the area of control, and fuzzy controllers are in use in an incredible range of devices and industries, ranging from video cameras to cement factories. Typical uses are image stabilisation for hand-held cameras, automatic control of washing machines (based on optical properties of the wash water), elevator scheduling, and temperature regulation in kilns. While fuzzy controllers do not do anything that is mathematically impossible by other means, they are simpler, and thus faster and cheaper, and are therefore widely used in consumer devices where cost is a major concern. Of course in the English-speaking world the term "fuzzy" is not a good marketing term, so products containing fuzzy controllers are usually referred to as "smart" or "intelligent" devices, but in other languages it seems that calling something "fuzzy" is a positive sales factor.

The principle underlying fuzzy control can be seen by looking at a hypothetical approach to the management of fish stocks. Regulatory agencies have basically two tools at hand for controlling fishing pressure, the short-term method of imposing catch or effort quotas, and the long-term mechanism of reducing the number of fishers by buying back licenses and suspending the sale of new licenses (this is of course a gross simplification of a field that has produced an enormous number of obscure acronyms for different management strategies). The problem is to apply these two tools in an optimal way to regulate the fishery on the basis of current stock levels and trends. This can be done in principle with the complex mathematical techniques of optimal control theory, but the implementation of a truly optimal management strategy is very demanding (Silvert and Smith, 1977), and given the degree of compliance which most management strategies achieve, probably not worth the effort.

A fuzzy approach could be based on a simple discrete representation of the data, with the stock size characterised as high, medium, and low, and the stock trend as increasing, decreasing, or stable. For each of the nine possibilities ( 3 x 3 ) we need to decide on a management strategy – for example, if the stock is high and stable, we can set high quotas, while if the stock is low and decreasing we set the quota at zero and reduce the number of fishing licenses as much as possible. In other words, based on the data we have on stock size and trend, we can characterise the situation according to membership in each of nine sets, for each of which we have selected a management strategy.

The key to this approach is that the nine sets are fuzzy sets, so we can associate each set of data with a combination of sets. A moderately high stock size can be treated as 60% high and 40% medium, while a slowly decreasing trend can be characterised as 70% stable and 30% decreasing. By weighting the nine discrete strategies according to these memberships we obtain a continuous range of management strategies.

Aside from simplicity, there are two advantages to this approach. One is that it is much more transparent to fishers, factory operators, and most other concerned parties, than a complex system of differential equations. The other is that it is easy to incorporate uncertainties in the data, a common problem in complex systems, where it might, for example, happen that fish stocks in some areas are healthy while catches in other areas are very poor and thus it is difficult to assess the stock size reliably.

One potential application of fuzzy set theory has to do with forecast evaluation, since forecasts can be thought of as fuzzy sets. The basic idea is that to make a forecast one has to consider the set of all possible future events and select a subset which consists of events which are predicted. For example, when a weather forecaster states that tomorrow will be clear and dry, clearly this is correct if there is not a cloud in the sky and zero precipitation. It is equally clear that the forecast is incorrect if there is damp fog all day. But suppose that there is 10% cloud cover and 2 mm of rain – is this consistent with the forecast or not? It is difficult and not very reasonable to set sharp dividing lines between correct and incorrect forecasts, but by treating the set of predicted events (the "prediction set") as fuzzy, this can be avoided. Precipitation less than 1 mm can be considered "dry", more than 5 mm "wet" ("not-dry"), and the success of the prediction can be interpolated between these values so that 2 mm of rain makes the precipitation prediction 75% successful.

Interpretation of forecasts in terms of fuzzy sets involves a number of complications, such as defining the strength of the prediction (the larger the set of predicted outcomes, the weaker the prediction), and the need for a baseline prediction so that the prediction can be interpreted in Bayesian terms, but without such an approach it is difficult to evaluate forecasting success in quantitative terms (Silvert, 1981).

It is difficult to describe and classify complex systems, so we usually simplify the description by aggregating components into broader categories, or bins, or "lumps". Binning is a common way to deal with continuous variables, which are often described by histograms. This can be seen as an alternative way of describing fuzzy variables – we can consider a student’s grade on a examination as membership in a set (e.g., a grade of 60 means that the student is 60% excellent), or we may prefer to describe performance by quartiles, corresponding to "good", "mediocre", "poor", and "bad". Both are fuzzy descriptions.

Medicine offers many examples of this lumping of complex situations into a small number of discrete categories. Burns are classified according to several variables, including percentage of body coverage, depth of damage, and degree of tissue destruction, but the descriptive categories are first, second, and third degree burns. These can clearly be treated as fuzzy variables, although whether it would help the medical profession to do so is a question that only specialists in the field can answer.

Still, fuzzy concepts seem to arise naturally in many fields, even when they are not explicitly referred to as such. Psychology is full of categories such as "schizophrenia", "paranoia", and "neurosis", as well as fuzzy qualifications like "borderline schizophrenia", "extreme paranoia", and "mild neurosis". Nor is this type of qualification associated only with the "soft" sciences – in the astronomical theory of stellar evolution terms like "red giant" are used, but there are no strict rules that define when a star becomes a red giant and when it passes on to the next stage.

It is difficult to choose between the complication of exact specification of a complex system and a much simpler "lumped" description in which much potentially valuable information has been lost in the process of aggregation. Fuzzy classification offers an alternative type of description based on a limited number of categories, but retains much of the information by allowing for the possibility that a system can fit into more than one category at a time.

Fuzzy set theory has many potential applications to systems science, only a few of which have been summarised here. Classification is a fundamental aspect of all scientific activities, and classification means putting things into sets, most of which are more meaningful in fuzzy terms than as traditional crisp sets. The traditional adherence to using crisp sets for purposes such as taxonomic classification of animals has decreased with the realisation that nature does not draw sharp lines, and categories which seem rigorous and scientific may not offer the best and most realistic descriptions of how systems actually behave.

Fuzzy set theory has acquired a misleading reputation as a new and forbidding branch of mathematics, but there is really very little that is new about it, and, far from being forbidding, many fuzzy applications are conceptually much simpler than the traditional alternatives. The purpose of this paper is not to break new ground, but simply to inspire its readers to consider the possibility that scientific issues are not all black and white, and to consider different shades of grey as realistic representations of the natural world.

Angel, D., P. Krost and W. Silvert. 1998. Describing benthic impacts of fish farming with fuzzy sets: theoretical background and analytical methods. J. Appl. Ichthyology 14: 1-8.

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Hutchinson, G. E. 1957. Concluding remarks. Cold Spring Harbor Symp., Quant. Biol. 22: 415-427.

Lindeman, R. L. 1942. The trophic-dynamic aspect of ecology. Ecology 23: 399-418.

Silvert, W. 1981. The formulation and evaluation of predictions. Int. J. General Systems 7: 189-205.

Silvert, W. 1983. Is dynamical systems theory the best way to understand ecosystem stability? In: H. I. Freedman and C. Strobeck (Editors), Population Biology, Springer-Verlag Lecture Notes in Mathematics 52, pp. 366-371.

Silvert, W. 1997. Ecological impact classification with fuzzy sets. Ecological Modelling 96: 1-10.

Silvert, W., and W. R. Smith. 1977. Optimal exploitation of a multi-species community. Math. Biosci. 33: 121-134.

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