1	The Inescapable Fuzziness of Being - can Something so Simple be Useful? William Silvert University of the Algarve Faro, Portugal
2	Outline Fuzzy Set Theory is not a great or especially novel mathematical discovery. Exaggerated claims have generated a lot of skepticism and even hostility. But, Fuzzy Sets are still very useful. This talk deals with what Fuzzy Set Theory is and what it can do.
3	What are Fuzzy Sets? Sets are among the most fundamental concepts in mathematics. All sentient creatures classify the things in their environment, i.e. assign them to sets. New-born infants learn about the sets of things that taste good, that cause pain, that are theirs, that are family. Sets are a universal part of perception.
4	Membership in Sets There are two ways to define a set: Enumeration – list members Polling – to find out what is a member Enumeration is most common – we simply list the members of the set. Consider the set of people attending this colloquium. We just go around the room and write down everyone’s name. This list defines the set.
5	Polling for Membership Another way to describe a set is to list all possible members and determine which ones are members of the set. We could make a (long) list of everyone in the world and indicate which ones are at this colloquium. We could use check marks or True/False, but since this is a Computer Science talk we of course use 1 and 0
6	Fuzzy Memberships The problem with using 1 and 0 to indicate membership is that most of the people in the world are not Computer Scientists, so they add dots and confuse the integers 1 and 0 with the real numbers 1. and 0. Then we conclude that memberships can take any value between 0. and 1. And that is all there is to Fuzzy Sets!
7	Criticism This leads to the most common criticism of Fuzzy Set Theory: It is too simple to be taken seriously. Can something so simple really be an important contribution to mathematics, science or anything else? So let’s take a moment to consider whether something has to be complicated to be useful.
8	Trivialities in Math There are some extremely simple ideas in mathematics which are very important. Set Theory itself The number zero Addition Non-Euclidean geometry Simple ideas can be very useful and even powerful.
9	Is Fuzzy Useful? Fuzzy sets obviously have some value. How do we define the set of people attending this colloquium if some of you leave the room? It simplifies matters if we can have partial membership in this set. This could be useful if colloquium attendance is a class requirement for your students.
10	Uniqueness Are fuzzy sets really necessary? Couldn’t we do the same thing with more traditional scoring techniques, or probability theory, etc.? In some cases yes – but many problems can be attacked in different ways: Geometry vs. Algebra (general relativity) Matrices vs. PDEs (quantum mechanics) Programming languages
11	Applications Control Theory Remote Sensing &Terrain Classification Forecast Evaluation Ecological Ranges & Niche Theory Pollution Regulation Environmental Impact Expert Systems Design Decision Support
12	Control Theory Humans use fuzzy control. We eat when we are hungry, drink when we are thirsty, sleep when we are tired. We do not measure glycogen levels, liquid content or lactic acid. IF you are thirsty THEN drink water. The more Є thirsty, the more you drink! This is how Fuzzy Control works.
13	Using Simple Controls Fuzzy control is simple, which makes it computationally easy, fast and cheap. Since membership in a fuzzy set is a continuous variable, fuzzy control allows continuously variable control actions. “IF thirsty THEN drink” is a single rule which implicitly says that the thirstier you are, the more you should drink.
14	The Business View The advantages of fuzzy control are evident in its industrial applications: Cement kilns (1980) Washing machines (IF water is dirty THEN...) Video cameras (image stabilization) Automobiles Medical instruments And many others … It has certainly impressed engineers.
15	Remote Sensing Remote sensing applications use a grid representation where each pixel is classified according to the image. Resolution is constantly improving, but it is not always feasible to make the pixels smaller. An alternative is to classify each pixel by fuzzy sets, e.g. 40% forest and 60% grassland.
16	Forecast Evaluation A lot of science involves prediction. How do we evaluate a weather forecast or fish stock estimate or anything else? If the weatherman forecasts rain, how do we decide whether he was correct? One way is to treat a prediction as a fuzzy set – for each possible outcome a membership value tells us to what extent the prediction is correct.
17	A Fuzzy Prediction Set
18	Ecological Ranges We typically represent the ranges of wildlife by figures like this: It’s OK for bird watchers. It is not very useful though. It does not tell you where to go if you want to be sure to see a barn owl. It also does not guarantee that you will not encounter barn owls outside the range (important if you are a mouse!).
19	The Ecological Niche
20	Pollution Regulation Countries usually regulate pollution by putting limits on how much of each pollutant can legally be emitted. A factory which emits 99% of the allowed amount for every pollutant is allowed to function. One which emits 101% of one pollutant and nothing else gets closed down. Is this right? Can we do better?
21	Two Factories This is what the situation looks like in graphical terms. Emission levels are shown relative to the allowed threshold values.
22	Fuzzy Regulation We can instead define the (fuzzy) set of acceptable operations. Slightly greater emission means slightly less acceptability but no threshold crossing.
23	Tradeoffs In other words, using fuzzy acceptabilities instead of threshold values gives us a greater degree of flexibility. We can easily incorporate tradeoffs between different impacts so that a single problem does not compromise a project that is in all other respects desirable and environmentally friendly. This is how people make decisions!
24	Environmental Impact Many environmental impacts are hard to determine quantitatively: Noise Smell Ugliness or Vulgarity Offensive to Religion Others can be measured but it is expensive and cannot be carried out in a non-destructive and repeated fashion.
25	Seabed Impacts In the work I did with Dror Angel and Peter Krost in Eilat, we wanted to describe the impact of a fish farm on the seabed. How do you measure the amount of sea grass under a fish pen? Do you rip up a patch and take it to the lab to weigh and measure it? Or do you accept the judgement of an experienced diver?
26	“Objectivity” Bias Often in an attempt to appear objective, scientists and engineers rely only on what they can measure, even though it may not be most important. They measure sound level in dB, with no difference between music and noise. They measure the nutrient effluents from feed lots but not the foul smell.
27	Reproducibility An important factor that is often overlooked in scientific measurement is the importance of reproducibility. Most fuzzy measurements are highly reproducible. If you are in a crowd and hear an unpleasant noise, so does the rest of the crowd. If you smell something awful, so do the people with you. An ugly factory looks ugly to most people.
28	Expert Systems Some Expert Systems are precise. For example, to tune a string to middle A we can use the rule (f = frequency in Hz), IF f ≠ 440 THEN decrease by (f–440). We can also use the rules IF f is high THEN loosen string IF f is low THEN tighten string This is a simple illustration of how a fuzzy controller works.
29	Fuzzy Expert Control By defining partial membership in the set of frequencies that are too high/low we can improve the expert system and obtain fuzzy control. If f is a just bit too high, so that membership in the set “high” is low, we loosen the string just a small amount. If f is much too high we loosen the string much more.
30	Fuzzy Decision Support In going from an expert system to a Decision Support System (DSS) we generally need to incorporate fuzzy factors. These are often subjective. An expert system can predict how much SO₂ a factory will emit. How much SO₂ the environment can tolerate is another matter, depending on wind, rain, population, and environmental sensitivity.
31	Driving a Car IF there are children playing near the road THEN slow down How many children? What ages? Playing what? How near the road? All of these are important factors, which makes a precise rule impractical. Driving is just too complex an activity.
32	Complex Systems Efforts to quantify and classify complex systems often fail. Fuzzy descriptions may work much better. Consider medical diagnostics – is the skin pale or yellow? Is the patient fat? These descriptors depend on the patient – by the time we have all the relevant variables in place the definitions are impossibly complicated.
33	Incompatibility This can be summarised by Zadeh’s Principle of Incompatibility: Precision is incompatible with significance. Does it really help to know how many parts per million of arsenic there is in your drinking water? Would you rather know if it is safe, dangerous, or borderline?
34	Advantages Fuzzy Sets can represent all kinds of information to present a comprehensive picture that cannot always be conveyed by precisely measured quantities alone. In decision-making we often tend to balance good points against bad ones, and partial memberships in the set of what is desirable can be averaged to give us sensible outcomes.
35	Technical Points When combining fuzzy memberships there are many ways of combining them, far more than for crisp sets. Fuzzy sets can be weighted. Quantitative data can be used but they are not essential. Qualitative data need not be quantified. Fuzzy sets avoid the need for discontinuous criteria.
36	Summary Fuzzy Set Theory is very simple and it does not require sophisticated math. Even so it has proven very useful and has many applications. Some applications can be handled by other means, but Fuzzy Set Theory is usually the simplest approach and provides a common methodology for addressing many different problems.