1. Multi Set
A set whose elements may be repeated (do not have to be different) is called a multi set (double set).
Examples of multiple sets:
- A = {1, 1, 1, 2, 2, 3},
- B = {2, 2, 2},
- C = {2, 3, 4},
- D = {}.
Multi sets & Fuzzy sets
The multiplicity of an element in a multiset is the number of occurrences of that element in the multiset.
Examples of multiplicity:
M = { 1, 1, 1, 2, 2, 2, 3, 3, 1 }, the multiplicity of 1 is 4 and the multiplicity of 2 is 3, while the multiplicity of 3 is 2.
A set is a special example of a multiset, in which the multiplicity of each element is either 0 or 1. A set whose multiplicity of an element is 0 is the empty set.
Suppose P and Q are multisets, the operations that apply to the two multisets are as follows:
a.) P ∪ Q is a multiset whose element multiplicity is the same as the maximum multiplicity of the element in the sets P and Q.
Example:
- P = { a, a, a, c, d, d } and
- Q = { a, a, b, c, c }, then
- P ∪ Q = { a, a, a, b, c, c, d, d }
b.) P ∩ Q is a multiset whose element multiplicity is equal to the minimum multiplicity of that element in the sets P and Q.
Example:
- P = { a, a, a, c, d, d }, and
- Q = { a, a, b, c, c }, then
- P ∩ Q = { a, a, c }
c.) P -- Q is a multiset whose element multiplicity is equal to the element multiplicity in P minus its multiplicity in Q, this applies if the difference of the multiplicities is positive. If the difference is zero or negative then the element multiplicity is zero.
Example:
- P = { a, a, a, b, b, c, d, d, e }, and
- Q = { a, a, b, b, b, c, c, d, d, f }, then
- P -- Q = { a, e }
d.) P + Q, which is defined as the sum of two dual sets, is a multiset whose element multiplicity is equal to the sum of the element multiplicities in P and Q.
Example:
- P = { a, a, b, c, c }, and
- Q = { a, b, b, d }, then
- P + Q = { a, a, a, b, b, b, c, c, d }
2. Fuzzy set
For example, U is a universal set. A crisp set is a subset of U that distinguishes between members and non-members with a clear (definite) boundary.
Fuzzy set example:
A = {x | x ∈ Z dan x > 2} atau A = {3, 4, 5, ...}
Jelas bahwa 3 ∈ A dan 1∉ AIn a Fuzzy set, its members have certain membership values that are determined by the membership function. The membership function has a range between zero and one. The notation of the membership function is μA(x) = [0,1]
Example:
A = {5, 10, 20, 30, 40, 50, 60, 70, 80} , is a crisp set of ages in years.
The fuzzy sets “toddler”, “adult”, “young”, and “old” are subsets of A.
Table 1.2. Age Group Against Criteria in Fuzzy Set
| Elemen | Balita | Anak-anak | Muda | Dewasa | Tua |
|--------|--------|-----------|------|--------|-----|
| 5 | 0 | 1 | 1 | 0 | 0 |
| 10 | 0 | 1 | 1 | 0 | 0 |
| 20 | 0 | 0.2 | 0.8 | 0.8 | 0.1 |
| 30 | 0 | 0 | 0.5 | 1 | 0.2 |
| 40 | 0 | 0 | 0.2 | 1 | 0.4 |
| 50 | 0 | 0 | 0.1 | 1 | 0.6 |
| 60 | 0 | 0 | 0 | 1 | 0.8 |
| 70 | 0 | 0 | 0 | 1 | 1 |
| 80 | 0 | 0 | 0 | 1 | 1 |From the table above, note that:
- Toddler = { }
- Children = {5, 10, 20} with membership function μChildren = {1, 1, 0.2}
- Young = {5, 10, 20, 30, 40, 50} with membership function μYoung = {1, 1, 0.8, 0.5, 0.2, 0.1}
- Adult = {20, 30, 40, 50, 60, 70, 80} with membership function μAdult = {0.8, 1, 1, 1, 1, 1, 1, 1}
- Old = {20, 30, 40, 50, 60, 70, 80} with membership function μOld = {0.1, 0.2, 0.4, 0.6, 0.8, 1, 1}
There are several ways to express fuzzy sets, namely:
There are several ways (commonly used) to determine the membership function of a fuzzy set, including:
1. Sigmoid Function
Sigmoid function
2. Phi function
In this membership function, there is only one value of x that has a membership degree equal to 1, namely when x = c. The values around c have membership degrees that are still close to 1.
Phi function
3. Triangle Functions
Triangle function
4. Trapezoid Function
Trapezoid function
Some basic operations on Fuzzy sets are:
1. Complement:
Example :
μA(x) for Old = {0/5, 0/10, 0.1/20, 0.2/30, 0.4/40, 0.6/50, 0.8/60, 1/70, 1/80} μA(x)
= Not Old
= {1/5, 1/10, 0.9/20, 0.8/30, 0.6/40, 0.4/50, 0.2/60, 0/70, 0/80}
2. Union / Disjunction:
Example:
3. Slice (Intersection / Conjunction):
Example:
Understanding Fuzzy Systems
Specific Instructional Objectives
- Students are able to formulate problems containing facts with a certain degree of uncertainty into a Fuzzy System approach.
- Students are able to perform manual calculations and analysis to perform inference and defuzzification on Fuzzy Systems.
- Students are able to design software to solve problems using Fuzzy Systems.
1. Introduction
Humans tend to use language in terms of something that can be understood generally, rather than in the form of mathematical language that emphasizes accuracy. For example, we say:
"That thing is very heavy" rather than "That thing weighs 1500 kg."
Figure 7.1 - Fuzzy illustration
The representation of facts like the above uses ambiguous or fuzzy terms. For example, the word very heavy can have different meanings, how heavy?
Fuzzy system is a system that uses fuzzy sets to map an input into a certain output (black box). For example, if you know how good the service is at the restaurant, you can determine how much tip is appropriate to give to the waiter. Look at the picture below:
Figure 7.2 - Mapping input output with fuzzy
2. Linguistic Variables (Linguistic Variables)
Terms that represent vague facts as in the above examples are called linguistic variables. Table 7.1 shows other examples of linguistic variables and their typical possible values.
Table 7.1 - Examples of linguistic variables
In fuzzy expert systems, linguistic variables are used in fuzzy rules. Consider the example below:
R1 IF Low speed
THEN Make acceleration high
R2 IF Low air temperature
AND Sufficient pressure
THEN Make speed low
The range of possible values in a linguistic variable is called the universe of discourse. For example, "speed" in R1 can range from 0 to 200 km/h. Because "low speed" occupies a segment of the universe of discourse.
3. Fuzzy Set
Fuzzy sets are different from classical sets. Classical sets have clear boundaries (crisp sets), therefore membership of classical sets can be stated in only two ways, namely: being a member of the set or not. While in fuzzy sets, membership of an element in a set is further stated by its degree of membership. See Figure 7.3.
Figure 7.3 - Illustration of fuzzy and crisp sets
Figure 7.3 is a set of young people. Representation with crisp sets states that if someone is under 10 years old then he is a set of young people, otherwise he is classified as old. Conversely, by using fuzzy sets, the set of young people is determined by the degree of membership. For example, a person aged 2 years is a set of young people with a membership value of 0.95, or in other words very young, a person aged 10 years is a set of young people with a membership value of 0.5, while a person aged 18 years is a set of young people with a membership value of 0.2, or in other words less young.
The curve in Figure 7.3 can also be interpreted as a mapping from the input of a person's age to the output of a person's degree of youth. Specifically, this kind of curve is called a membership function. Another example that illustrates a fuzzy set is the fuzzy set of seasons in the northern hemisphere, as shown in Figure 7.4.
Figure 7.4 - Fuzzy set of seasons in the northern hemisphere
The astronomical definition of seasons is as shown in the left figure where there should be a drastic difference in each season, for example spring starts exactly in March and ends in June. However, each country in the northern hemisphere may experience the difference in seasons differently, for example spring may start in the second or third week of March and end in the first or second week of June, as shown in the right part of Figure 7.4. Other countries may experience different situations.
Figure 7.4 also shows that in the fuzzy set of seasons in the northern hemisphere there are several fuzzy subsets, namely: spring, summer, fall and winter. Also note that the second week of April (estimated in the figure) is a member of spring with a membership value of 0.8 and is a member of summer with a membership value of 0.2. Thus, an input can be considered to have partial membership in several membership functions. The implications of this will be discussed in the next sub-section.
Based on the examples above, the accurate definition of a fuzzy set is as follows:
Fuzzy Set: If X is a universe of discourse, an element of X is denoted as x. A fuzzy set A of X is characterized by a membership function μA(x) that associates each element x with a value of the degree of membership in A.
4. Membership Function
Membership Function (MF) is a curve that maps each point on the input (universe of discourse) to a membership value (or degree of membership) that has a value between 0 and 1 which is defined mathematically by the equation:
Each element x is mapped to a membership value by MF. This value is the degree of membership of x in the fuzzy set A.
Where the membership value of x is limited by:
As an example of the mapping of elements of x (in this case the height of a person) by MF to its membership values, is shown in Figure 7.5.
Figure 7.5 - Mapping of a person's height by MF
The membership value of the mapping result of element x by a membership function can have different values depending on the type of membership function. Commonly used membership functions are: triangle function, trapezoidal function, gaussian function, bell function and sigmoid function. The form of each membership function is as follows:
Triangle function. Triangular membership function is defined by 3 parameters a, b, c with the equation:
The triangle function with parameters: triangle(x;0.2,0.6,0.8) is shown in Figure 7.6.
Trapezoidal Function. The trapezoidal membership function is defined by 4 parameters a, b, c, d with the equation:
The triangle function with parameters: trapezoid(x;0.1,0.2,0.6,0.95) is shown in Figure 7.7.
Figure 7.6 - Triangle membership function
Figure 7.7 - Trapezoidal membership function.
Gaussian Function. The Gaussian membership function is defined by 2 parameters σ, and c with the equation:
The Gaussian function with parameters: gaussian(x;0.15,0.5) is shown in Figure 7.8.
Figure 7.8 - Gaussian membership function. σ= standard deviation, c=center
Bell Function. Bell-shaped membership function is defined by 3 parameters a, b and c with the equation:
The Bell function with parameters: bell(x;0.25,2.5,0.5) is shown in Figure 7.9. While the parameters a, b and c that specify the Bell function are shown in Figure 7.10.
Figure 7.9 - Bell membership function
Figure 7.10 - Location of parameters a,b and c in the Bell membership function.
Sigmoid Function. The Sigmoid membership function is defined by 2 parameters a and c with the equation:
If the value a > 0, then the sigmoid function will open to the right, whereas if a < 0 then the sigmoid function will open to the left. The Sigmoid function opens to the right with parameters: sigmoid(x;12,0.25) shown in Figure 7.11. Meanwhile, the Sigmoid function opens to the left with parameters: sigmoid(x;-12,0.75) shown in Figure 7.12.
Figure 7.11 - Sigmoid membership function opens to the right
Figure 7.12 - Sigmoid membership function opens to the left
5. Fuzzy Set Representation
As explained above, if we have a universe of discourse X with a fuzzy set A in it, then the set of elements of X expressed as {x1, x2, . . . , xn} can be mapped by the membership function μA(x) into their respective membership values with the value [0,1]. Suppose a fuzzy set is expressed as:
Where,
In a more compact way, the fuzzy set A can be represented as:
where the symbol "/" is a mapping from input xi to a membership value ai. For example:
HIGH = (0/125, 0.25/140, 0.7/150, 1/170, 1/190)
6. Fuzzy Set Operations
Operations on fuzzy sets are basically similar to operations on classical sets. For example, consider the fuzzy set operations in Figure 7.13, where the logical AND operation is replaced by min while the logical OR operation is replaced by max. It can be seen that the fuzzy set (which only has members 0 and 1) gives the same results as the operations on classical sets. Thus, these operations can be extended to fuzzy sets that have membership values [0,1].
Figure 7.13 - Fuzzy set operations
AND
The AND operation between two fuzzy sets A and B will produce an intersection between A and B on X which is defined as:
For example, consider the two fuzzy sets below:
TALL = (0/125, 0.2/140, 0.5/150, 0.8/170, 1/190) SHORT = (1/125, 0.8/140, 0.5/150
, 0.2/170, 0/190)
The result of the AND operation on the two fuzzy sets above:
μTINGGI ∧ μPENDEK(x) = (0/125, 0.2/140, 0.5/150, 0.2/170, 0/190)
OR
The OR operation between two fuzzy sets A and B will produce a union between A and B on X which is defined as:
Returning to the example above, the result of the OR operation on the TALL and SHORT fuzzy sets is:
μTINGGI ∨ μPENDEK(x) = (1/125, 0.8/140, 0.5/150, 0.8/170, 1/190)
NOT
The NOT operation on fuzzy set A will give the complement result of A, namely:
So the NOT operation for the fuzzy set HIGH in the previous example will produce the complement set of HIGH:
μNOT TINGGI(x) = (1/125, 0.8/140, 0.5/150, 0.2/170, 0/190)
Figure 7.14 - Fuzzy set operations with diagrams
The representation of fuzzy set operations in diagram form is shown in Figure 7.14.
Example
1. The fuzzy set about air temperature in Surabaya (with a universe of discourse between 20 and 40 degrees Celsius) is specified into three subsets, namely: low, medium and high. The membership function of each subset with its parameters is determined in the table below. Describe the fuzzy set!
