Multi-criteria selection of alternatives using fuzzy inference rules. Part 1/3: Theory
Fuzzy Logic - Mathematical Foundations
Fuzzy set
Method Description
Let there be a set of elements U, and A - its fuzzy subset, the degree of belonging of its elements is determined by the value of X, taking the value from the interval [0; 1].
Thus, an object can be described by a set of criteria (parameters) X (1), X (2) ... X (n) belonging to the corresponding basic sets U (1), U (2) ... U (n). The set of criteria with the corresponding values ​​characterizes the acceptability S of the object relative to the task.
An example of such a statement:
d (i): “If X (1) = low and X (2) = good, then S = high”
for general case:
d (i): "If X (1) = A (1) and X (2) = A (2) and ... and X (p) = A (p), then S = B (i)”
')
Denote the intersection of the conditions X (1) = A (1), X (2) = A (2) ... X (n) = A (n) by X = A (i).
The intersection of fuzzy sets is the minimum of their membership functions:
Where
V = U (1) xU (2) x ... xU (p)
v = (u (1), u (2) ... u (p))
u (j) - the value of the element u (j) belonging to the fuzzy set A (i, j)
then the statement can be written in the form:
d (i): "If X = A (i), then S = B (i)"
Denote the set U or V by W, then A (i) is a fuzzy subset of W, and B (i) is a fuzzy subset of the unit interval I.
The implication of these fuzzy sets is expressed as
Where
H is a fuzzy subset on WxI
w belongs to W and i to I.
Similarly, the statements d (1), d (2) ... d (n) are transformed into sets H (1), H (2) ... H (n); their association is the set D.
For each pair (w, i) of WxI
Each alternative is described by a fuzzy subset C of W; its satisfactoryness is based on the computational inference rule:
G = C * D
where G is a fuzzy set of interval I. Then
Comparison of alternatives occurs with the help of point estimates.
For a fuzzy set A of I, we define an a-level set (a = [0,1]):
For each A (a), the average number of elements is calculated - M (A (a)):
Then the point value for the set A:
a_max is the value at which A has a maximum.
The best is the alternative with the highest point estimate.
In the second part of the article, the implementation of this method will be described; an example of his work is in the third.
Multi-criteria selection of alternatives using fuzzy inference rules. Implementation in Java. Part 2/3: Basic Algorithm
Multi-criteria selection of alternatives using fuzzy inference rules. Implementation in Java. Part 3/3: Example
Literature:
Borisov, Krumberg, Fedorov - “Decision Making Based on Fuzzy Models. Examples of use ”, 1990 (note: there are mistakes in the author’s calculations). Page 94-102
Fuzzy set
Method Description
Let there be a set of elements U, and A - its fuzzy subset, the degree of belonging of its elements is determined by the value of X, taking the value from the interval [0; 1].
Thus, an object can be described by a set of criteria (parameters) X (1), X (2) ... X (n) belonging to the corresponding basic sets U (1), U (2) ... U (n). The set of criteria with the corresponding values ​​characterizes the acceptability S of the object relative to the task.
An example of such a statement:
d (i): “If X (1) = low and X (2) = good, then S = high”
for general case:
d (i): "If X (1) = A (1) and X (2) = A (2) and ... and X (p) = A (p), then S = B (i)”
')
Denote the intersection of the conditions X (1) = A (1), X (2) = A (2) ... X (n) = A (n) by X = A (i).
The intersection of fuzzy sets is the minimum of their membership functions:
Where
V = U (1) xU (2) x ... xU (p)
v = (u (1), u (2) ... u (p))
u (j) - the value of the element u (j) belonging to the fuzzy set A (i, j)
then the statement can be written in the form:
d (i): "If X = A (i), then S = B (i)"
Denote the set U or V by W, then A (i) is a fuzzy subset of W, and B (i) is a fuzzy subset of the unit interval I.
The implication of these fuzzy sets is expressed as
Where
H is a fuzzy subset on WxI
w belongs to W and i to I.
Similarly, the statements d (1), d (2) ... d (n) are transformed into sets H (1), H (2) ... H (n); their association is the set D.
For each pair (w, i) of WxI
Each alternative is described by a fuzzy subset C of W; its satisfactoryness is based on the computational inference rule:
G = C * D
where G is a fuzzy set of interval I. Then
Comparison of alternatives occurs with the help of point estimates.
For a fuzzy set A of I, we define an a-level set (a = [0,1]):
For each A (a), the average number of elements is calculated - M (A (a)):
Then the point value for the set A:
a_max is the value at which A has a maximum.
The best is the alternative with the highest point estimate.
In the second part of the article, the implementation of this method will be described; an example of his work is in the third.
Multi-criteria selection of alternatives using fuzzy inference rules. Implementation in Java. Part 2/3: Basic Algorithm
Multi-criteria selection of alternatives using fuzzy inference rules. Implementation in Java. Part 3/3: Example
Literature:
Borisov, Krumberg, Fedorov - “Decision Making Based on Fuzzy Models. Examples of use ”, 1990 (note: there are mistakes in the author’s calculations). Page 94-102
Source: https://habr.com/ru/post/136160/
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