What is the difference between randomness and fuzziness?

A common question is often posed by new learners of fuzzy logic and membership functions that take values between zero and one. This question is: “I know about probabilities. They also take values between zero and one. Aren’t they the same?”.

This confusion is always expected, especially with the concept of probability, which we learnt at school and got used to dealing with it.

(If you didn't read the previous articles in this series, please take a look at them to get the picture)

To clarify the difference between fuzziness and randomness, let’s take the following example:

Suppose there is a sport club with different facilities in some city. We may choose a random person from that city and ask the question: “What is the probability that this person is a member of that club?”. Here, we use what we know from probability, where we divide the number of the members in the club over the total number of people in the city to get the answer. Note that this answer is always the same since we choose randomly, and we don’t know -before doing the trial- whether we will choose one of the club members or not.

On the other hand, let’s assume that this club allows its members to choose how many facilities they can use if they want to pay less. Here, we may take one of the members and ask him: “What is your degree of membership in the club?”. The answer depends on how much he pays, and it will be: “full membership”, “half membership” or “quarter a membership”, etc...

Did you distinguish between the two questions? In the first one, we chose a person randomly and studied the probability that he is a member of the club. Whereas we care about the degree of membership for each one of the members, in the second one.

This is simply the difference between randomness and fuzziness, but we will give another example to explain it more clearly, and let this example be similar to those we usually use in probability theory.

Suppose we have a box with four red balls and three blue ones. If we pull a random ball out of the box, then the probability that we get a red ball equals 4 over 7. This is something we know very well.

However, do all the red balls have the same degree of redness?

From the fuzzy logic point of view, we forget about the random pull trial, and think about determining the degree of redness for each one of the four balls. In other words, we will set a degree of membership for each ball in the set of “Red balls”. As for the blue balls in the box, they will get a zero degree of membership in the set of “Red balls”.

For instance, in the following image, we give approximate values for the degrees of membership for several balls of different degrees of redness in the set of “Red balls”.

In short, fuzziness has nothing to do with random trials, rather it cares about making a logical decision about the degree of membership of something in some group, while probability and randomness study the probable results of a certain random experiment.

One may now ask: What is the difference between their applications in real life?

Consider a foodstuff factory, which we want to study its production process mathematically. Actually, there is a lot of randomness here. For instance, we don’t know for sure how many customers will come every day, how many boxes or pieces they will demand or when the machines in the factory will break down! However, probability theory helps us to get some expectations for all that, in order to produce the appropriate amount such that the demands of the customers are met as much as possible without losing goods because of expiration.

On the other hand, fuzziness helps us deal with the vague demands more flexibly. One of the demands may be: “about one hundred boxes” or “almost one thousand pieces”. Such unclear demands can be represented and studied using the tools of fuzzy logic without having to specify an exact amount. (See the previous article to know more about representing such fuzzy numbers)

Furthermore, fuzziness may help in such factory to represent the filled amount of some product in each piece. For instance, we sometimes see the written filled amount on some product: “200g ±2%”. This is due to that some machines cannot measure the filled amount accurately. Although this difference may not affect the customers a lot, it may cause a big loss or gain for the factory. Therefore, it would be very beneficial if we were able to represent this fuzzy difference mathematically. After that, other mathematical tools, such as mathematical optimization and linear programming, help us to use these fuzzy numbers to get the maximum gain or minimum loss.

At this point, we reach the end of our journey with fuzzy logic in this series, which we hope it clarified the concept interestingly. Your questions and notes are welcome at any time. Follow us for more articles about new and exciting subjects.

Marwa Tuffaha

marwat.math@gmail.com

Reference: Zadeh, L.A., 1965. Fuzzy sets. Information and control, 8(3), pp.338-353.