CN115099624A - Multi-attribute decision-making system based on intuition fuzzy entropy and interval fuzzy entropy - Google Patents

Multi-attribute decision-making system based on intuition fuzzy entropy and interval fuzzy entropy Download PDF

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CN115099624A
CN115099624A CN202210736580.5A CN202210736580A CN115099624A CN 115099624 A CN115099624 A CN 115099624A CN 202210736580 A CN202210736580 A CN 202210736580A CN 115099624 A CN115099624 A CN 115099624A
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许晓曾
李梦
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Chongqing Technology and Business University
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Abstract

The invention discloses a multi-attribute decision making system based on intuitionistic fuzzy entropy and interval fuzzy entropy, which comprises a decision information acquisition module, a fuzzy entropy calculation module and a decision result generation module; the decision information acquisition module is used for acquiring a plurality of decision information matrixes of the target and transmitting the decision information matrixes to the fuzzy entropy calculation module; the fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient, and transmits the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient to the decision result generation module; and the decision result generation module calculates the comprehensive evaluation result of each decision information matrix according to the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient, and obtains the optimal decision according to the comprehensive evaluation result of the decision information matrix. The hesitation information entropy expected by mathematics provided by the patent is very effective in practical application, and the difference between the entropy and the entropy added with the traditional fuzzy entropy and the standard deviation is very small.

Description

Multi-attribute decision-making system based on intuition fuzzy entropy and interval fuzzy entropy
Technical Field
The invention relates to the field of multi-attribute decision making, in particular to a multi-attribute decision making system based on intuitionistic fuzzy entropy and interval fuzzy entropy.
Background
The concept of interval intuitive fuzzy sets was first proposed by ataanassov. The method is characterized in that a certain object is difficult to be depicted by a certain precise value in consideration of the characteristics of complexity and uncertainty of the object and the limitation of the cognitive level of people, and interval values can more flexibly process the complex objects, so that the concept of interval intuitive fuzzy sets is provided. Since then, the interval intuitive fuzzy set is widely applied to the fields of scheme decision, image processing, machine learning, logic planning and the like. The key problems of the interval intuitive fuzzy set applied in scheme decision are mainly divided into 2 categories:
the first category of problems with attribute weight determination: the solution is divided into 3 types, one is about the case that the attribute weight of the solution is known, the other is about the case that the attribute weight is partially known, and the last is about the preference problem of the solution in the case that the attribute weight is completely unknown. Aiming at the interval fuzzy multi-attribute decision problem with completely unknown attribute weight, an optimal scheme is obtained mainly by establishing a linear programming model and a fuzzy entropy weight method.
In the second category of sorting problem of the number of intervals, Pavel takes the median point of the number of intervals as a sorting basis, and although the calculation amount is small, the median point cannot sufficiently reflect the characteristics of the number of intervals, and much information is lost. The integrated method of the Xuezui for interval intuition and fuzziness is researched, a score function and an accurate function are defined, and the method is applied to the field of decision making. However, the scoring function does not take the hesitation into account, and therefore inaccurate or even misjudgment of the ordering may occur when the patterns are ordered. The bamboo Wen starts from the geometric meaning of interval intuitive fuzzy numbers, proposes a scoring function containing parameters, but the value of the parameters is generally fixed to 0.5, resulting in that the value of the scoring function only depends on the upper interval of membership and non-membership, thereby losing the information of the lower interval. Aiming at the problem that the scoring function described above cannot correctly sort some interval numbers, the invention provides a new scoring function by comprehensively considering the influence of the membership degree of an interval intuitive fuzzy number, the absolute difference value of non-membership degree, useful information and abstaining right information on decision making, and can solve the limitation of the previous scoring function. However, there is still a problem of sequencing failure for some number of intervals.
Thus, the characterization and quantification of ambiguity is an important issue in many system models and designs affecting uncertainty metrics.
Disclosure of Invention
The invention aims to provide a multi-attribute decision system based on intuition fuzzy entropy and interval fuzzy entropy, which comprises a decision information acquisition module, a fuzzy entropy calculation module and a decision result generation module;
the decision information acquisition module is used for acquiring a plurality of decision information matrixes of targets (such as optimal sequencing of ecological agricultural areas and the like) and transmitting the decision information matrixes to the fuzzy entropy calculation module;
the fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient, and transmits the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient to the decision result generation module;
and the decision result generation module calculates the comprehensive evaluation result of each decision information matrix according to the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient, and obtains the optimal decision according to the comprehensive evaluation result of the decision information matrix.
Preferably, the decision information matrix is denoted as R i (u jk ) (ii) a i represents the number of decision makers, j represents the evaluation number, and k represents the benefit number;
the fuzzy entropy calculation module is used for determining a decision information matrix R i (u jk ) The step of processing to obtain the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient comprises the following steps:
1) calculating the expectation and standard deviation of the hesitation degree of the decision maker and the expectation and variance of the hesitation degree of the benefit attribute of each decision information matrix;
wherein the desired vector is noted as
Figure BDA0003715711530000021
The vector of standard deviations is denoted S k (π i (X))＝{S(π 1 (X)),S(π 2 (X)) }, k ═ 1, 2; k-1 represents the expert hesitation degree, and k-2 represents the benefit attribute hesitation degree;
2) respectively inputting the respective expectation and standard deviation of the hesitation degree and the benefit attribute weight of the decision maker into a fuzzy entropy calculation model, and calculating a fuzzy entropy;
the fuzzy entropy calculation model is stored in a decision maker hesitation degree weight coefficient and benefit attribute weight coefficient calculation module;
3) establishing a decision maker hesitation degree weight r ij The formula is calculated, namely:
Figure BDA0003715711530000022
wherein (E) 1 ) ij Representing fuzzy entropy corresponding to the hesitation degree of the decision maker;
4) according to the data of the formula (1), calculating a hesitation degree weight coefficient matrix of each decision maker, namely:
Figure BDA0003715711530000023
in the formula, r 4n Representing a hesitation degree weight coefficient of the decision maker;
5) establishing a benefit attribute weight coefficient w ij The calculation formula, namely:
Figure BDA0003715711530000031
wherein i represents a method, and k represents the number of benefit attributes; (E) 2 ) ij Representing fuzzy entropy corresponding to the benefit attribute;
6) according to the data of the formula (2), calculating to obtain a hesitation degree weight coefficient matrix of each benefit attribute, namely:
Figure BDA0003715711530000032
in the formula, w 4k And a hesitation degree weight coefficient representing the benefit attribute.
Preferably, the fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels, which are respectively as follows:
Figure BDA0003715711530000033
Figure BDA0003715711530000034
Figure BDA0003715711530000035
Figure BDA0003715711530000036
in the formula,
Figure BDA0003715711530000037
respectively outputting 4 intuitive fuzzy entropy calculation submodels; hesitation degree pi of fuzzy set A at point x A (x)＝1-u A (x)-ν A (x)；μ A (x):X→[0,1],ν A (x):X→[0,1]Respectively representing the membership degree and the non-membership degree of the fuzzy set A; e f (A) A distance representing a degree of membership and a degree of non-membership; s (pi) A (X)) is a hesitation degree pi A (x) Standard deviation of (2).
Preferably, the hesitation degree intuitive fuzzy entropy and the benefit attribute intuitive fuzzy entropy of the decision maker are as follows:
Figure BDA0003715711530000038
preferably, when the fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result of the decision information matrix is as follows:
Figure BDA0003715711530000039
in the formula of lambda i Is a weight vector; r i (u jk ) Is decision information.
Preferably, the fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels, which are respectively as follows:
Figure BDA0003715711530000041
Figure BDA0003715711530000042
Figure BDA0003715711530000043
Figure BDA0003715711530000044
in the formula,
Figure BDA0003715711530000045
respectively calculating the output of the submodels for the 4 intervals of fuzzy entropy;
Figure BDA0003715711530000046
distances representing degrees of membership and non-degrees of membership; degree of hesitation of interval
Figure BDA0003715711530000047
Superscript U, L represents an upper triangular matrix and a lower triangular matrix, respectively;
Figure BDA0003715711530000048
Figure BDA0003715711530000049
is degree of hesitation of interval
Figure BDA00037157115300000410
Degree of hesitation of interval
Figure BDA00037157115300000411
Standard deviation of (2).
Preferably, the intuitional fuzzy entropy of the decision maker hesitation degree interval and the intuitional fuzzy entropy of the benefit attribute are as follows:
Figure BDA00037157115300000412
preferably, when the interval fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result of the decision information matrix is as follows:
Figure BDA00037157115300000413
in the formula, λ i Is a weight vector;
Figure BDA00037157115300000414
is decision information.
Preferably, the decision information is obtained by expert experience.
Preferably, the system also comprises a database for storing data of the decision information acquisition module, the fuzzy entropy calculation module and the decision result generation module.
The technical effect of the invention is needless to say, the patent establishes several general frameworks about the entropy of the Intuitive Fuzzy Sets (IFSs) and the interval intuitive fuzzy sets (IVIFSs), the frameworks mainly comprise two parts of fuzzy entropy and probability information entropy, and theoretical proofs are given.
This patent provides new decision-making system to current decision-making system's limitation, has solved some problems that exist on the multi-objective selection, if: the subjectivity is too high, and sometimes a user only selects the target according to the impression of the target, and a set of objective evaluation system is not established; the weight of the evaluation factors is difficult to determine, and the importance degree of each evaluation factor is different when the target is selected, so that the user has a correct measurement rule for the weight of each index;
the invention considers the random uncertainty of different experts in scoring the same target, provides an improved multi-attribute decision-making system based on intuition fuzzy entropy and interval fuzzy entropy, can correctly and reasonably calculate the weight of each index attribute, reduces the influence of the uncertainty on the decision-making result to a certain extent, and provides a more accurate scheme for multi-target selection.
Experimental results show that the hesitation information entropy expected by mathematics provided by the patent is very effective in practical application, and the difference between the entropy and the entropy added with the traditional fuzzy entropy and the standard deviation is very small.
Drawings
Fig. 1 is an information acquisition function for uncertainty of information.
Detailed Description
The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.
Example 1:
referring to fig. 1, a multi-attribute decision system based on intuitive fuzzy entropy and interval fuzzy entropy includes a decision information obtaining module, a fuzzy entropy calculating module, and a decision result generating module;
the system is used for carrying out optimal sequencing on the ecological agriculture area.
The decision information acquisition module is used for acquiring a plurality of decision information matrixes of the target and transmitting the decision information matrixes to the fuzzy entropy calculation module; the target includes a number of ecological agricultural zones to be selected.
The fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient, and transmits the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient to the decision result generation module;
and the decision result generation module calculates the comprehensive evaluation result of each decision information matrix according to the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient, and obtains the optimal decision according to the comprehensive evaluation result of the decision information matrix.
The decision information matrix is marked as R i (u jk ) (ii) a i represents the number of decision makers, j represents the evaluation number, and k represents the benefit number;
the fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient, and the method comprises the following steps:
1) calculating the expectation and standard deviation of the hesitation degree of the decision maker and the expectation and variance of the hesitation degree of the benefit attribute of each decision information matrix;
wherein the desired vector is noted as
Figure BDA0003715711530000051
The vector of standard deviations is denoted S k (π i (X))＝{S(π 1 (X)),S(π 2 (X)),...,S(π n (X)) }, k ═ 1, 2; n is the dimension of the decision information matrix; k-1 represents the hesitation degree of the expert, and k-2 represents the hesitation degree of the benefit attribute;
2) inputting respective expectation and standard deviation of the hesitation degree and benefit attribute weight of the decision maker into a fuzzy entropy calculation model, and calculating fuzzy entropy
Figure BDA0003715711530000061
The fuzzy entropy calculation model is stored in a decision maker hesitation degree weight coefficient and benefit attribute weight coefficient calculation module;
3) establishing a decision maker hesitation degree weight r ij The calculation formula, namely:
Figure BDA0003715711530000062
wherein (E) 1 ) ij Representing fuzzy entropy corresponding to the hesitation degree of the decision maker;
4) according to the data of the formula (1), calculating to obtain a hesitation degree weight coefficient matrix of each decision maker, namely:
Figure BDA0003715711530000063
in the formula, r 4n Representing a hesitation degree weight coefficient of the decision maker;
5) establishing a benefit attribute weight coefficient w ij The calculation formula, namely:
Figure BDA0003715711530000064
wherein i represents a method, and k represents the number of benefit attributes; (E) 2 ) ij Representing fuzzy entropy corresponding to the benefit attribute;
6) according to the data of the formula (2), calculating to obtain a hesitation degree weight coefficient matrix of each benefit attribute, namely:
Figure BDA0003715711530000065
in the formula, w 4k And a hesitation degree weight coefficient representing the benefit attribute.
The fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels which are respectively as follows:
Figure BDA0003715711530000066
Figure BDA0003715711530000067
Figure BDA0003715711530000068
Figure BDA0003715711530000069
in the formula,
Figure BDA00037157115300000610
respectively outputting 4 intuitive fuzzy entropy calculation submodels; hesitation degree pi of fuzzy set A at point x A (x)＝1-u A (x)-ν A (x)；μ A (x):X→[0,1],ν A (x):X→[0,1]Respectively representing the membership degree and the non-membership degree of the fuzzy set A; e f (A) A distance representing a degree of membership and a degree of non-membership; s (pi) A (X)) is a hesitation degree pi A (x) Standard deviation of (2).
The intuitive fuzzy entropy of the hesitation degree and the intuitive fuzzy entropy of the benefit attribute of the decision maker are as follows:
Figure BDA0003715711530000071
when the fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result of the decision information matrix is as follows:
Figure BDA0003715711530000072
in the formula, λ i Is a weight vector; r is i (u jk ) Is decision information.
The decision information is obtained by an expert experience method.
The system also comprises a database used for storing data of the decision information acquisition module, the fuzzy entropy calculation module and the decision result generation module.
Example 2:
a multi-attribute decision system based on intuition fuzzy entropy and interval fuzzy entropy comprises a decision information acquisition module, a fuzzy entropy calculation module and a decision result generation module;
the decision information acquisition module is used for acquiring a plurality of decision information matrixes of targets (such as optimal sequencing of ecological agricultural areas) and transmitting the decision information matrixes to the fuzzy entropy calculation module;
the fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient, and transmits the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient to the decision result generation module;
and the decision result generation module calculates the comprehensive evaluation result of each decision information matrix according to the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient, and obtains the optimal decision according to the comprehensive evaluation result of the decision information matrix.
The step that the interval fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient comprises the following steps:
1) calculating the expectation and standard deviation of the hesitation degree of the decision maker and the expectation and variance of the hesitation degree of the benefit attribute of each decision information matrix;
wherein the desired vector is noted as
Figure BDA0003715711530000073
The vector of standard deviations is denoted S k (π i (X))＝{S(π 1 (X)),S(π 2 (X)),...,S(π n (X)) }, k ═ 1, 2; n is the dimension of the decision information matrix; k-1 represents the expert hesitation degree, and k-2 represents the benefit attribute hesitation degree;
2) inputting respective expectation and standard deviation of the hesitation degree and benefit attribute weight of the decision maker into a fuzzy entropy calculation model, and calculating fuzzy entropy
Figure BDA0003715711530000074
The interval fuzzy entropy calculation model is stored in a decision maker hesitation degree weight coefficient and benefit attribute weight coefficient calculation module;
3) establishing a decision maker hesitation degree weight r ij The calculation formula, namely:
Figure BDA0003715711530000081
wherein i represents the decision information of the decision maker, and j represents the number of the decision maker;
4) according to the data of the formula (1), calculating a hesitation degree weight coefficient matrix of each decision maker, namely:
Figure BDA0003715711530000082
5) establishing a benefit attribute weight coefficient w ij The calculation formula, namely:
Figure BDA0003715711530000083
wherein i represents a method, and k represents the number of benefit attributes;
6) according to the data of the formula (2), calculating to obtain a hesitation degree weight coefficient matrix of each benefit attribute, namely:
Figure BDA0003715711530000084
the fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels which are respectively as follows:
Figure BDA0003715711530000085
Figure BDA0003715711530000086
Figure BDA0003715711530000087
Figure BDA0003715711530000088
in the formula,
Figure BDA0003715711530000089
respectively calculating the output of the submodels for the 4 intervals of fuzzy entropy;
Figure BDA00037157115300000810
distances representing degrees of membership and non-degrees of membership; degree of hesitation of interval
Figure BDA00037157115300000811
Superscript U, L represents upper triangular matrix and lower triangular matrix, respectively;
Figure BDA00037157115300000812
Figure BDA00037157115300000813
is degree of hesitation of interval
Figure BDA00037157115300000814
Degree of hesitation of interval
Figure BDA0003715711530000091
Standard deviation of (2).
The intuitive fuzzy entropy of the hesitation degree and the intuitive fuzzy entropy of the benefit attribute of the decision maker are as follows:
Figure BDA0003715711530000092
when the interval fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result of the decision information matrix is as follows:
Figure BDA0003715711530000093
in the formula, λ i Is a weight vector;
Figure BDA0003715711530000094
is decision information.
The decision information is obtained by an expert experience method.
The system also comprises a database used for storing data of the decision information acquisition module, the fuzzy entropy calculation module and the decision result generation module.
Example 3:
a multi-attribute decision system based on intuition fuzzy entropy and interval fuzzy entropy comprises a decision information acquisition module, a fuzzy entropy calculation module and a decision result generation module;
the decision information acquisition module is used for acquiring a plurality of decision information matrixes of targets (such as optimal sequencing of ecological agricultural areas and the like) and transmitting the decision information matrixes to the fuzzy entropy calculation module;
the fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient, and transmits the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient to the decision result generation module;
and the decision result generation module calculates the comprehensive evaluation result of each decision information matrix according to the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient, and obtains the optimal decision according to the comprehensive evaluation result of the decision information matrix.
The decision information matrix is marked as R i (u jk ) (ii) a i represents the number of decision makers, j represents the evaluation number, and k represents the benefit number;
the fuzzy entropy calculation module is used for determining a decision information matrix R i (u jk ) The step of processing to obtain the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient comprises the following steps:
1) calculating the expectation and standard deviation of the hesitation degree of the decision maker and the expectation and variance of the hesitation degree of the benefit attribute of each decision information matrix;
wherein the desired vector is noted as
Figure BDA0003715711530000095
The vector of standard deviations is denoted S k (π i (X))＝{S(π 1 (X)),S(π 2 (X)) }, k ═ 1, 2; k-1 represents the hesitation degree of the expert, and k-2 represents the hesitation degree of the benefit attribute;
2) respectively inputting the respective expectation and standard deviation of the hesitation degree and the benefit attribute weight of the decision maker into a fuzzy entropy calculation model, and calculating a fuzzy entropy;
the fuzzy entropy calculation model is stored in a decision maker hesitation degree weight coefficient and benefit attribute weight coefficient calculation module;
3) establishing a decision maker hesitation degree weight r ij Formula for calculationNamely:
Figure BDA0003715711530000101
wherein (E) 1 ) ij Representing fuzzy entropy corresponding to the hesitation degree of the decision maker;
4) according to the data of the formula (1), calculating to obtain a hesitation degree weight coefficient matrix of each decision maker, namely:
Figure BDA0003715711530000102
in the formula, r 4n Representing a hesitation degree weight coefficient of the decision maker;
5) establishing a benefit attribute weight coefficient w ij The calculation formula, namely:
Figure BDA0003715711530000103
wherein i represents a method, and k represents the number of benefit attributes; (E) 2 ) ij Representing fuzzy entropy corresponding to the benefit attribute;
6) according to the data of the formula (2), calculating to obtain a hesitation degree weight coefficient matrix of each benefit attribute, namely:
Figure BDA0003715711530000104
in the formula, w 4k And a hesitation degree weight coefficient representing the benefit attribute.
The fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels which are respectively as follows:
Figure BDA0003715711530000105
Figure BDA0003715711530000106
Figure BDA0003715711530000107
Figure BDA0003715711530000108
in the formula,
Figure BDA0003715711530000109
respectively outputting 4 intuitive fuzzy entropy calculation submodels; hesitation degree pi of fuzzy set A at point x A (x)＝1-u A (x)-ν A (x)；μ A (x):X→[0,1],ν A (x):X→[0,1]Respectively representing the membership degree and the non-membership degree of the fuzzy set A; e f (A) A distance representing a degree of membership and a degree of non-membership; s (Pi) A (X)) is a hesitation degree pi A (x) Standard deviation of (2).
The intuitive fuzzy entropy of the hesitation degree and the intuitive fuzzy entropy of the benefit attribute of the decision maker are as follows:
Figure BDA0003715711530000111
when the fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result of the decision information matrix is as follows:
Figure BDA0003715711530000112
in the formula, λ i Is a weight vector; r i (u jk ) Is decision information.
The fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels which are respectively as follows:
Figure BDA0003715711530000113
Figure BDA0003715711530000114
Figure BDA0003715711530000115
Figure BDA0003715711530000116
in the formula,
Figure BDA0003715711530000117
respectively calculating the output of the submodels for the 4 intervals of fuzzy entropy;
Figure BDA0003715711530000118
a distance representing a degree of membership and a degree of non-membership; degree of hesitation of interval
Figure BDA0003715711530000119
Superscript U, L represents an upper triangular matrix and a lower triangular matrix, respectively;
Figure BDA00037157115300001110
Figure BDA00037157115300001111
is degree of interval hesitation
Figure BDA00037157115300001112
Degree of hesitation of interval
Figure BDA00037157115300001113
Standard deviation of (2).
The intuitionistic fuzzy entropy of the decision maker hesitation degree interval and the intuitionistic fuzzy entropy of the benefit attribute are as follows:
Figure BDA00037157115300001114
when the interval fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result of the decision information matrix is as follows:
Figure BDA00037157115300001115
in the formula, λ i Is a weight vector;
Figure BDA00037157115300001116
is decision information.
The decision information is obtained by expert experience.
The system also comprises a database used for storing data of the decision information acquisition module, the fuzzy entropy calculation module and the decision result generation module.
Example 4:
a multi-attribute decision-making system based on intuition fuzzy entropy and interval fuzzy entropy is disclosed in an embodiment 1, wherein the basic theory is as follows:
1) background of the theory
Definition 1.A is an intuitive fuzzy set A-IFS, any element X ∈ X in A having the form: a ═ tone<x，u A (x)，v A (x))|x∈X}
Wherein: mu.s A (x)：X→[0，1]，ν A (x)：X→[0，1]And 0 is less than or equal to mu A (x)+v A (x)≤1，
Figure BDA0003715711530000121
Which represent the degree of membership and the degree of non-membership of the fuzzy set a, respectively.
Another parameter in the intuitive blur set A (x)＝1-u A (x)-v A (x) X ∈ X, which indicates the hesitation of a at point X, and, obviously,
Figure BDA0003715711530000122
0≤π A (x) Less than or equal to 1, when pi A (x) When 0, the intuitive blur set degenerates into a blur set.
For any two intuitive fuzzy sets A in X 1 And A 2 The following definitions are provided:
1)
Figure BDA0003715711530000123
to pair
Figure BDA0003715711530000124
If and only if
Figure BDA0003715711530000125
And is
Figure BDA0003715711530000126
2)A 1 ＝A 2 If and only if
Figure BDA0003715711530000127
And is provided with
Figure BDA0003715711530000128
3) Complement set
Figure BDA0003715711530000129
4) This three-dimensional array
Figure BDA00037157115300001210
Referred to as ataassov intuitive fuzzy values.
As an important metric for the Intuitive Fuzzy Set (IFS), A 1 ，A 2 ∈IFS，X＝{x i 1,2, …, n, Szmidt defines Hamming distances as follows:
Figure BDA00037157115300001211
definition 2 for a definition on the number set X
Figure BDA00037157115300001212
Figure BDA00037157115300001213
It has the following form:
Figure BDA00037157115300001214
therein are provided with
Figure BDA00037157115300001215
And is aligned with
Figure BDA00037157115300001216
Its interval hesitation degree is expressed as
Figure BDA00037157115300001217
And is provided with
Figure BDA00037157115300001218
Figure BDA00037157115300001219
For convenience, this is expressed as follows:
Figure BDA00037157115300001220
Figure BDA00037157115300001221
and, to
Figure BDA00037157115300001222
Similarly, for any two intervals belonging to X, the fuzzy set is intuitive
Figure BDA00037157115300001223
Namely, it is
Figure BDA00037157115300001224
This example is defined as follows:
1)
Figure BDA00037157115300001225
if and only if
Figure BDA00037157115300001226
And for
Figure BDA00037157115300001227
Is provided with
Figure BDA00037157115300001228
2)
Figure BDA0003715711530000131
If and only if
Figure BDA0003715711530000132
And is
Figure BDA0003715711530000133
3)
Figure BDA0003715711530000134
Complement of
Figure BDA0003715711530000135
4) Three-dimensional array
Figure BDA0003715711530000136
Called an ataassov interval intuitive fuzzy value.
For X ═ X i |i＝1，2，…，n}，
Figure BDA0003715711530000137
And
Figure BDA0003715711530000138
defined as:
Figure BDA0003715711530000139
the embodiment redefines the axiom of A-IFSs:
definition 3. a real function E: ifs (x) → [0, 1], e (a) is referred to as an entropy on ifs (x) if it satisfies the following properties:
(F1) e (a) ═ 0 if and only if a is a distinct set;
(E2) e (A) is 1 and only
Figure BDA00037157115300001310
All have u A (x i )＝v A (x i )＝0；
(E3) E (A). ltoreq.E (B) if A has less ambiguity than B, that is:
if it is
Figure BDA00037157115300001311
Then pair
Figure BDA00037157115300001312
Has u B (x i )≤v B (x i )，π A (x)≤π B (x) Or if
Figure BDA00037157115300001313
Then pair
Figure BDA00037157115300001314
Has u B (x i )≥v B (x i )，π A (x)≤π B (x)；
(E4)E(A)＝E(A C ).
Similarly, the axioms of A-IVIFSs are redefined:
definition 4. a real function
Figure BDA00037157115300001315
IVIFS(X)→[0，1]，
Figure BDA00037157115300001316
Referred to as an entropy on IVIFS (X) and satisfies the following properties:
Figure BDA00037157115300001317
if and only if
Figure BDA00037157115300001318
Is a distinct set;
Figure BDA00037157115300001319
if and only if pairs
Figure BDA00037157115300001320
Are all provided with
Figure BDA00037157115300001321
Figure BDA00037157115300001322
Figure BDA00037157115300001323
If it is not
Figure BDA00037157115300001324
Ratio of degree of blur of
Figure BDA00037157115300001325
Small, that is to say if
Figure BDA00037157115300001326
Then pair
Figure BDA00037157115300001327
Is provided with
Figure BDA00037157115300001328
And is provided with
Figure BDA00037157115300001329
Figure BDA00037157115300001330
If it is
Figure BDA00037157115300001331
Then pair
Figure BDA00037157115300001332
Is provided with
Figure BDA00037157115300001333
Figure BDA00037157115300001334
And is
Figure BDA00037157115300001335
Figure BDA00037157115300001336
2) Research foundation
To pair
Figure BDA0003715711530000141
Documents [4,10,11 ]]Some entropy models are proposed to measure the intuitive fuzzy sets under the condition of satisfying the definitions 3 and 4, and have different membership degrees and non-membership degrees especially under the condition of the same hesitation degree. Szmid [10,12 ]]Two non-probabilistic intuitive fuzzy entropies are proposed:
Figure BDA0003715711530000142
Figure BDA0003715711530000143
fuzzy entropy based on distance and hesitation degree including membership degree and non-membership degree: this entropy is applied to any element A of the fuzzy set i E IFSs (i ═ 1,2, …, n), which is defined as follows:
Figure BDA0003715711530000144
equation (5) defines one element x in the fuzzy set i For any set A in the fuzzy set i ε IFS (X), which is expressed as follows:
Figure BDA0003715711530000145
corresponding fuzzy set of interval
Figure BDA0003715711530000146
Liu [9 ]]The definition of equation (4) is extended:
Figure BDA0003715711530000147
but the entropy (7) is within
Figure BDA0003715711530000148
The A-IFSs are degraded without satisfying definition (3). Thus, entropy is proposed
Figure BDA0003715711530000149
Figure BDA00037157115300001410
Then, a new interval intuitive fuzzy entropy is provided, and an interval intuitive fuzzy set
Figure BDA00037157115300001411
Is provided with
Figure BDA00037157115300001412
Then for the collection
Figure BDA00037157115300001413
The interval intuitive fuzzy entropy of (a) is expressed as follows:
Figure BDA0003715711530000151
3) several general frameworks for the measurement of intuitive fuzzy sets and Interval intuitive fuzzy sets
A new entropy model is provided for the intuitive fuzzy entropy and the interval intuitive fuzzy entropy, and the new entropy model comprises the combination of membership degree, the distance between the membership degree and the non-membership degree and the hesitation degree. However, it does not reflect the probabilistic numerical characteristic of the hesitation degree and the information characteristic of the hesitation degree. For example, one metric may be used in multi-attribute decision making (MADM), with the score of each expert having a little randomness when many experts evaluate alternatives. His hesitation can be seen as a random variable. How does this randomness be expressed? This embodiment may be reflected in the mathematical expectation and variance of hesitations. As can be seen from the information entropy model, the larger the hesitation degree is, the more information is reflected by the information entropy, so that it can be reflected by the information entropy. Therefore, the study of the uncertainty (PI, II) caused by non-ambiguity is important. In general, a robust system should contain all three (FI, PI, and II) types of uncertainty.
3.1) intuitive fuzzy entropy measure
This embodiment proposes several new fuzzy set metric frameworks that contain the three above-mentioned uncertainty metrics for humans, the model is as follows:
Figure BDA0003715711530000152
Figure BDA0003715711530000153
Figure BDA0003715711530000154
where A ∈ IFSs, E in the formula (11-13) f (A) Represents a fuzzy entropy measure which is mainly measured by the distance between membership and non-membership. Such as: e f (A) Can be represented by the following model:
Figure BDA0003715711530000155
is defined as:
Figure BDA0003715711530000161
the model is as follows:
Figure BDA0003715711530000162
the entropy is:
Figure BDA0003715711530000163
e in the formula (11-13) pi (A) The entropy is represented including the probability uncertainty (PI) and the information uncertainty (II) of the hesitation. Pi A (x i )，x i epsilon.A denotes that A is in x i Degree of hesitation of, pi A (x i ) Can be regarded as a sample value of A and it corresponds to entropy E pi (A) Can be influenced by pi A (x i ) Mathematical expectation of
Figure BDA0003715711530000164
And standard deviation S (pi) A (X)) are measured, and their calculation formula is as follows:
Figure BDA0003715711530000165
Figure BDA0003715711530000166
in the above formula
Figure BDA0003715711530000167
And S (pi) A (X)) measures the probability uncertainty in a.
In combination with the definition of Shannon's information entropy:
Figure BDA0003715711530000168
p i by using
Figure BDA0003715711530000169
Or S (pi) A (X)) substituted, by definition, when
Figure BDA00037157115300001610
Or S (pi) A (X)) should also increase monotonically, and the maximum value of entropy, max E (a) ═ 1, so this embodiment redefines the entropy function, ensuring that E (a) is a monotonically increasing function.
Figure BDA00037157115300001611
The information entropy function is defined as:
f(x)＝x(1-ln x) (21)
it is specified that 0 ln 0 is 0, and its function image is shown in fig. 1.
Based on the above analysis, the present embodiment proposes several entropy models that measure the intuitive blur set. To pair
Figure BDA00037157115300001616
As defined below:
Figure BDA00037157115300001612
Figure BDA00037157115300001613
Figure BDA00037157115300001614
Figure BDA00037157115300001615
in the formula E f (A) Can use E fzl (A)，E fldl (A)，E fzj (A)，E fpp (A) And (4) performing isentropic expression.
Theorem 1. pair
Figure BDA0003715711530000171
Function defined by equations (22-25)
Figure BDA0003715711530000172
Is an entropy model of the intuitive fuzzy set.
And (3) proving that: the equations (22-25) are similar, and this example demonstrates (22). If it is not
Figure BDA0003715711530000173
Is that of the intuitive fuzzy set, it should satisfy definition 3. Herein this example takes E f (A) Is an entropy of the fuzzy set, so it satisfies properties (E1) - (E4), as demonstrated primarily in the following example:
Figure BDA0003715711530000174
(E1) the method comprises the following steps If A is a distinct set, u A (x i )＝1，v A (x i )＝1，π A (x i ) Is equal to 0, and therefore has
Figure BDA0003715711530000175
S(π A (X)) ═ 0. Into equation (22) with E f (A)＝0，
Figure BDA0003715711530000176
I.e. with E pi (A) 0. On the other hand, if E fpi (A) 0, only when E f (A) 0 and E pi (A) Equal to 0, therefore
Figure BDA0003715711530000177
S(π A (X)) -0, so a is a distinct set.
(E2) The method comprises the following steps If u is A (x i )＝v A (x i ) 0, then pi A (x i ) 1, then E f (A)＝1，
Figure BDA0003715711530000178
Thus E pi (A) Now suppose that
Figure BDA0003715711530000179
From the formula (22), easily obtained
Figure BDA00037157115300001710
Here E f (A) As 1, the present embodiment only needs to satisfy
Figure BDA00037157115300001711
Since each of f (x) and x (1-ln) is a monotonically increasing function, it is considered that
Figure BDA00037157115300001712
I.e. pi A (x i ) 1, so u A (x i )＝v A (x i )＝0，
Figure BDA00037157115300001713
(E3) The method comprises the following steps To pair
Figure BDA00037157115300001714
If it is
Figure BDA00037157115300001715
And u is B (x i )≤v B (x i )，π A (x i )≤π B (x i )，
Figure BDA00037157115300001716
Then u is A (x i )≤u B (x i )≤v B (x i )≤v A (x i )，
Figure BDA00037157115300001717
S 2 (π B (X))-S 2 (π A (X))＝E(π B (X) 2 )-(E(π B (X))) 2 -(E(π A (X) 2 )-(E(π A (X))) 2 )
＝(E(π B (X) 2 )-E(π A (X) 2 ))-((E(π B (X))) 2 -(E(π A (X))) 2 )
To pair
Figure BDA00037157115300001718
π B (x i )≥π A (x i ) Easy to obtain E (π) B (X) 2 )≥E(π A (X) 2 ) And (E (π) B (X))) 2 ≥(E(π A (X))) 2 Then S is 2 (π B (X))≥S 2 (π A (X)), namely S (π) B (X))≥S(π A (X)) is
Figure BDA00037157115300001719
From the formula (22), a
Figure BDA0003715711530000181
E fpi (B)≥E fpi (A) In that respect Similarly, if
Figure BDA0003715711530000182
When is to
Figure BDA0003715711530000183
Has u B (x i )≥v B (x i )，π A (x i )≤π B (x i ) Having E of fpi (B)≥E fpi (A)。
(E4) Based on symmetry, E is easily obtained fpi (A)＝E fpi (A C )。
3.2) general framework of Interval intuitive fuzzy sets (IVIFSs) metrics
From the general frameworks (11) - (13) of the intuitive fuzzy sets, similarly, the general framework of the corresponding interval intuitive fuzzy sets (iviffs) metric is given herein.
Figure BDA0003715711530000184
Figure BDA0003715711530000185
Figure BDA0003715711530000186
Here, the
Figure BDA0003715711530000187
Function(s)
Figure BDA0003715711530000188
Is an interval intuitive fuzzy entropy that defines the distance between membership and non-membership. For example, it may be the entropy model proposed by and of lie:
Figure BDA0003715711530000189
entropy:
Figure BDA00037157115300001810
or:
Figure BDA00037157115300001811
based on the above analysis, the present embodiment proposes an entropy model of several measurement interval intuitive fuzzy sets,
Figure BDA00037157115300001812
entropy is defined as follows:
Figure BDA00037157115300001813
Figure BDA0003715711530000191
Figure BDA0003715711530000192
Figure BDA0003715711530000193
of formula (32-35)
Figure BDA0003715711530000194
Respectively can use
Figure BDA0003715711530000195
And (4) showing.
Theorem 2. in pairs
Figure BDA0003715711530000196
Formula (II)(32-35) the real function defined
Figure BDA0003715711530000197
Figure BDA0003715711530000198
Is interval intuitive fuzzy entropy.
And (3) proving that: the demonstration of the entropy models (32-35) is similar, and the present example demonstrates the model (32) simply.
If it is used
Figure BDA0003715711530000199
Is the entropy of IVIFSs, it should satisfy definition 4, where
Figure BDA00037157115300001910
Expressing the entropy of the fuzzy set of intuitive intervals, thus satisfying the property
Figure BDA00037157115300001911
This example mainly demonstrates that:
Figure BDA00037157115300001912
Figure BDA00037157115300001913
if it is used
Figure BDA00037157115300001914
Is a distinct set, then there are
Figure BDA00037157115300001915
Figure BDA00037157115300001916
And is
Figure BDA00037157115300001917
Thus is provided with
Figure BDA00037157115300001918
And
Figure BDA00037157115300001919
Figure BDA00037157115300001920
as can be derived from the equation (32),
Figure BDA00037157115300001921
namely that
Figure BDA00037157115300001922
And is
Figure BDA00037157115300001923
Therefore, it is not only easy to use
Figure BDA00037157115300001924
On the other hand, only when
Figure BDA00037157115300001925
When the temperature of the water is higher than the set temperature,
Figure BDA00037157115300001926
then
Figure BDA00037157115300001927
Namely that
Figure BDA00037157115300001928
Is a distinct set.
Figure BDA00037157115300001929
If it is not
Figure BDA00037157115300001930
Then
Figure BDA00037157115300001931
Is easy to obtain
Figure BDA00037157115300001932
Namely, it is
Figure BDA00037157115300001933
In turn, assume that
Figure BDA00037157115300001934
From equation (32), it follows:
Figure BDA00037157115300001935
since the function f (x) x (1-lnx) monotonically increases, it is possible to reduce the number of times that the function f (x) monotonically increases
Figure BDA00037157115300001936
Namely, it is
Figure BDA00037157115300001937
Thus, pair
Figure BDA0003715711530000201
Figure BDA0003715711530000202
To pair
Figure BDA0003715711530000203
If it is
Figure BDA0003715711530000204
And is
Figure BDA0003715711530000205
Figure BDA0003715711530000206
And
Figure BDA0003715711530000207
then the present embodiment can obtain the inequality u A (x i )≤u B (x i )≤v B (x i )≤v A (x i ) And are and
Figure BDA0003715711530000208
Figure BDA0003715711530000209
due to the fact that
Figure BDA00037157115300002010
Can obtain the product
Figure BDA00037157115300002011
And is
Figure BDA00037157115300002012
Then
Figure BDA00037157115300002013
Namely that
Figure BDA00037157115300002014
Figure BDA00037157115300002015
In a similar manner to that described above,
Figure BDA00037157115300002016
is composed of formula (32)
Figure BDA00037157115300002017
Figure BDA00037157115300002018
Figure BDA00037157115300002019
Figure BDA00037157115300002020
Similarly, if
Figure BDA00037157115300002021
And is
Figure BDA00037157115300002022
And
Figure BDA00037157115300002023
can also obtain
Figure BDA00037157115300002024
(E4) From the symmetry, it is clear
Figure BDA00037157115300002025
Example 5:
the verification experiment of the multi-attribute decision system based on the intuitive fuzzy entropy and the interval fuzzy entropy described in the embodiments 1 and 4 comprises the following contents:
the entropy model framework proposed by the patent can be used for measurement of decision maker weights and expert weights for multi-attribute decision making. Obviously, if the larger the amount of information of a sample, the smaller its entropy, the larger the entropy weight of that attribute. To compare the effectiveness of the models, the present example made the following experiments:
example 1 Hubei province can be roughly divided into 7 agroecological regions according to the difference of environment and natural resources, and respectively uses alpha j And (j ═ 1,2, …, 7). The present embodiment considers the preferred ordering of the several ecoagricultural regions based on the statistics of the ecoagricultural regions. One composed of three experts R i (i-1, 2,3) given a weight vector λ -0.5, 0.2,0.3 according to their rank T . The attributes considered in the evaluation are for eachRegion alpha j Ecological, economic and social benefits C k (k ═ 1,2,3), assuming that the importance of these several benefits is completely unknown. The personal decision matrix for the expert's personal opinion of agroecological regional attributes is as follows:
Figure BDA0003715711530000211
Figure BDA0003715711530000212
Figure BDA0003715711530000213
wherein,
Figure BDA0003715711530000214
next, this embodiment measures the hesitation weight coefficient and the attribute weight r of the expert respectively using the entropy models (22-25) proposed in this embodiment i ,w k (i, k ═ 1,2,3), and finally, a preferred ranking is made on the ecoagricultural areas.
In the first step, the decision maker weight is composed of two aspects, one is the expert importance, which is known; the other is randomness of evaluation of a decision maker, for example, if a certain decision maker is likely to score higher on the whole, or the hesitation degree of each attribute is large, the scoring quality of the decision maker is not high, and the weight of evaluation of the decision maker is smaller. First, the expectation and standard deviation of the hesitations of three decision makers are calculated:
Figure BDA0003715711530000221
substituting into model (22-25), calculating corresponding entropy as E (R)
Figure BDA0003715711530000222
In the second step, the larger the entropy, the higher the ambiguity, and the smaller the weight. So the decision maker weight r i Calculated according to the following formula
Figure BDA0003715711530000223
Substituting the data to respectively obtain the random influence weight of each decision maker on each model
Figure BDA0003715711530000224
Third, similarly, the benefit attribute weight may be calculated as follows:
Figure BDA0003715711530000225
fourthly, calculating the comprehensive score of each model of each region by the following formula
Figure BDA0003715711530000226
Figure BDA0003715711530000227
Finally, the higher the composite score, the better the composite benefit in the region. From matrix Z fpi (u j ) The data can show that the comprehensive benefit ordering of several entropies proposed by the embodiment is consistent. The ecological environment of each region is ordered as follows:
α 1 ＞α 4 ＞α 2 ＞α 5 ＞α 3 ＞α 7 ＞α 6
in order to check the validity of the measure of the frame entropy proposed in this embodiment, a comparative analysis was performed on the four models proposed in this embodiment and the model results proposed in the prior art. The main results are shown in table 1.
The model presented in table 1 and the literature [5] model demonstrated that λ ═ (0.5,0.2,0.3)
Figure BDA0003715711530000231
As can be seen from the data in Table 1, the algorithm proposed in this embodiment has an additional weight coefficient caused by the randomness of the hesitation of the decision maker, so that all the composite scores are smaller, but their preferred ranks are the same. For the four proposed model entropies, their results are very similar, namely the probability information entropy part E of hesitation pi (A) The contribution of the contribution is much larger, the distance E between degree of membership and degree of non-membership f (A) The influence is not great; the influence of the expected information of the hesitation degree on the weight coefficient is large, and the influence of the standard deviation is small. Therefore, in practical application, if the data is better, the entropy model can be selected
Figure BDA0003715711530000232
If the requirement on the precision is high, the calculation can be selected
Figure BDA0003715711530000233
The weights are calculated.
Example 5:
the verification experiments of the multi-attribute decision making system based on the intuitionistic fuzzy entropy and the interval fuzzy entropy described in the embodiments 2 and 4 comprise the following contents:
by using
Figure BDA0003715711530000234
Figure BDA0003715711530000235
An interval fuzzy set matrix representing the decision maker,
Figure BDA0003715711530000241
Figure BDA0003715711530000242
Figure BDA0003715711530000243
calculating decision maker and attribute weights r using a proposed entropy model (32-35) i ,w k (i, k ═ 1,2,3), and the agroecological regions are preferably ranked by the composite attribute score.
First, calculate the hesitation and standard deviation of the decision maker
Figure BDA0003715711530000244
S L (π i (X))＝(0.0587 0.0660 0.0849),S U (π i (X))＝(0.0921 0.1003 0.0957)
Substituting the models (32-35) to obtain the corresponding entropy of the models as
Figure BDA0003715711530000245
Step 2, the larger the entropy is, the smaller the ambiguity is, and the hesitation degree weight r of the decision maker i The following process is performed to make the following processes,
Figure BDA0003715711530000246
Figure BDA0003715711530000247
step 3, similarly, the attribute weight coefficient w i Can obtain
Figure BDA0003715711530000251
Step 4, calculating the comprehensive attribute value of each ecological agricultural area by using the following formula
Figure BDA0003715711530000252
Figure BDA0003715711530000253
Step 5, finally, according to the comprehensive score
Figure BDA0003715711530000254
The larger the model is, the more optimal the ecological agricultural area is, and from the conclusion of the empirical analysis, the comprehensive sequencing conclusion of the proposed models is consistent, and all the conclusions are alpha 1 ＞α 4 ＞α 2 ＞α 5 ＞α 3 ＞α 7 ＞α 6 。
In order to check the validity of the model, the proposed model was analyzed in comparison with the models mentioned in the prior art. Their analytical data are shown in Table 2.
Table 2 interval intuition fuzzy set different measurement method experimental result λ ═ (0.5,0.2,0.3)
Figure BDA0003715711530000255
As can be seen from Table 2, the algorithm proposed in this embodiment has an additional weight coefficient caused by the randomness of the hesitations of the decision maker, so that all the composite scores are smaller, but their preferred ranks are consistent. Taking the example as a research object, the comprehensive scores of the four proposed model entropies are very similar, namely the probability information entropy part of the hesitation degree
Figure BDA0003715711530000256
The occupied effect is much larger, and the distance between the membership degree and the non-membership degree
Figure BDA0003715711530000257
The influence on the comprehensive score is small; the influence of the expected information of the hesitation degree on the weight coefficient is large, and the influence of the standard deviation is small. Therefore, in practical application, if the data is better, the entropy model can be selected
Figure BDA0003715711530000258
If the requirement on the precision is high, the calculation can be selected
Figure BDA0003715711530000259
The weights are calculated.
And (4) conclusion:
in this embodiment, three measurement frames and four entropy models are provided for the intuitive fuzzy set and the interval intuitive fuzzy set, and theoretical proofs are made. The advantages of these models are:
(1) the metric framework includes the traditional degree of membership and the distance difference part E of the degree of membership f (A) And
Figure BDA0003715711530000261
and E f (A) And
Figure BDA0003715711530000262
can be embodied by directly applying related research results.
(2) Example analysis shows that the probability information entropy E of the hesitation degree provided by the embodiment pi (A) And
Figure BDA0003715711530000263
the method plays an absolute role in multi-attribute decision making; and the proposed entropy model E 4 (π i (A))，
Figure BDA0003715711530000264
Simple and effective.
(3) The proposed metric framework applies to both Intuitive Fuzzy Sets (IFS) and interval fuzzy sets (IVIFS).

Claims (10)

1.A multi-attribute decision-making system based on intuition fuzzy entropy and interval fuzzy entropy is characterized in that: the fuzzy entropy calculation system comprises the decision information acquisition module, a fuzzy entropy calculation module and a decision result generation module.
The decision information acquisition module is used for acquiring a plurality of decision information matrixes of the target and transmitting the decision information matrixes to the fuzzy entropy calculation module.
The fuzzy entropy calculation module processes the decision information matrix to obtain a decision maker hesitation degree weight coefficient and a benefit attribute hesitation degree weight coefficient, and transmits the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient to the decision result generation module;
and the decision result generation module calculates the comprehensive evaluation result of each decision information matrix according to the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient, and obtains the optimal decision according to the comprehensive evaluation result of the decision information matrix.
2. The multi-attribute decision system based on intuitive fuzzy entropy and interval fuzzy entropy as claimed in claim 1, wherein the decision information matrix is denoted as R i (u jk ) (ii) a i represents the number of decision makers, j represents the evaluation number, and k represents the benefit number;
the fuzzy entropy calculation module is used for determining a decision information matrix R i (u jk ) The step of processing to obtain the decision maker hesitation degree weight coefficient and the benefit attribute hesitation degree weight coefficient comprises the following steps:
1) calculating the expectation and standard deviation of the hesitation degree of the decision maker and the expectation and variance of the hesitation degree of the benefit attribute of each decision information matrix;
wherein the desired vector is noted as
Figure FDA0003715711520000011
The vector of standard deviations is denoted S k (π i (X))＝{S(π 1 (X)),S(π 2 (X)) }, k ═ 1, 2; k-1 represents the hesitation degree of the expert, and k-2 represents the hesitation degree of the benefit attribute;
2) and (4) inputting the respective expectation and standard deviation of the hesitation degree and the benefit attribute weight of the decision maker into a fuzzy entropy calculation model, and calculating the fuzzy entropy.
The fuzzy entropy calculation model is stored in a decision maker hesitation degree weight coefficient and benefit attribute weight coefficient calculation module;
3) establishing a decision maker hesitation degree weight r ij The calculation formula, namely:
Figure FDA0003715711520000012
wherein (E) 1 ) ij Representing fuzzy entropy corresponding to the hesitation degree of the decision maker;
4) according to the data of the formula (1), calculating a hesitation degree weight coefficient matrix r (R) of each decision maker, namely:
Figure FDA0003715711520000013
in the formula, r 4n Representing a hesitation degree weight coefficient of the decision maker;
5) establishing a benefit attribute weight coefficient w ij The calculation formula, namely:
Figure FDA0003715711520000021
wherein i represents a method, and k represents the number of benefit attributes; (E) 2 ) ij Representing fuzzy entropy corresponding to the benefit attribute;
6) according to the data of the formula (2), calculating to obtain a hesitation degree weight coefficient matrix w (C) of each benefit attribute, namely:
Figure FDA0003715711520000022
in the formula, w 4k And a hesitation degree weight coefficient representing the benefit attribute.
3. The multi-attribute decision making system based on intuitive fuzzy entropy and interval fuzzy entropy of claim 2, wherein the fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels, which are respectively as follows:
Figure FDA0003715711520000023
Figure FDA0003715711520000024
Figure FDA0003715711520000025
Figure FDA0003715711520000026
in the formula,
Figure FDA0003715711520000027
respectively outputting 4 intuitive fuzzy entropy calculation submodels; hesitation degree pi of fuzzy set A at point x A (x)＝1-u A (x)-v A (x)；μ A (x):X→[0,1],v A (x):X→[0,1]Respectively representing the membership degree and the non-membership degree of the fuzzy set A; e f (A) A distance representing a degree of membership and a degree of non-membership; s (pi) A (X)) is a hesitation degree pi A (x) Standard deviation of (d).
4. The multi-attribute decision system based on intuitionistic fuzzy entropy and interval fuzzy entropy of claim 3, wherein the degree of hesitation intuitionistic fuzzy entropy and the benefit attribute intuitionistic fuzzy entropy of the decision maker are as follows:
Figure FDA0003715711520000028
5. the multi-attribute decision-making system based on intuitive fuzzy entropy and interval fuzzy entropy of claim 4, wherein when the fuzzy entropy calculation model comprises 4 intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result Z of the decision information matrix fpi (u j ) As follows:
Figure FDA0003715711520000031
in the formula of lambda i Is a weight vector; r i (u jk ) Is decision information.
6. The multi-attribute decision making system based on intuitive fuzzy entropy and interval fuzzy entropy of claim 2, wherein the interval fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels, which are respectively as follows:
Figure FDA0003715711520000032
Figure FDA0003715711520000033
Figure FDA0003715711520000034
Figure FDA0003715711520000035
in the formula,
Figure FDA0003715711520000036
respectively calculating the output of the submodels for the 4 intervals of fuzzy entropy;
Figure FDA0003715711520000037
a distance representing a degree of membership and a degree of non-membership; degree of hesitation of interval
Figure FDA0003715711520000038
Superscript U, L represents an upper triangular matrix and a lower triangular matrix, respectively;
Figure FDA0003715711520000039
Figure FDA00037157115200000310
is degree of hesitation of interval
Figure FDA00037157115200000311
Degree of hesitation of interval
Figure FDA00037157115200000312
Standard deviation of (d).
7. The multi-attribute decision system based on intuitionistic fuzzy entropy and interval fuzzy entropy of claim 6, wherein the degree of hesitation of the decision maker intuitionistic fuzzy entropy and the benefit attribute intuitionistic fuzzy entropy are as follows:
Figure FDA00037157115200000313
8. the multi-attribute decision-making system based on intuitive fuzzy entropy and interval fuzzy entropy of claim 7, wherein when the interval fuzzy entropy calculation model comprises 4 interval intuitive fuzzy entropy calculation submodels, the comprehensive evaluation result of the decision information matrix
Figure FDA00037157115200000314
As follows:
Figure FDA00037157115200000315
in the formula, λ i Is a weight vector;
Figure FDA00037157115200000316
is decision information.
9. The multi-attribute decision making system based on intuitive fuzzy entropy and interval fuzzy entropy of claim 1, wherein: the decision information is obtained by expert experience.
10. The multi-attribute decision making system based on intuitive fuzzy entropy and interval fuzzy entropy of claim 1, wherein: the system also comprises a database used for storing data of the decision information acquisition module, the fuzzy entropy calculation module and the decision result generation module.
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