CN115511193A - Multi-attribute decision method based on interval spherical fuzzy set fuzzy entropy - Google Patents

Multi-attribute decision method based on interval spherical fuzzy set fuzzy entropy Download PDF

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CN115511193A
CN115511193A CN202211219876.6A CN202211219876A CN115511193A CN 115511193 A CN115511193 A CN 115511193A CN 202211219876 A CN202211219876 A CN 202211219876A CN 115511193 A CN115511193 A CN 115511193A
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fuzzy
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刘欢
肖小
吕强
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Harbin University of Science and Technology
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Abstract

The invention discloses a multi-attribute decision method based on a new fuzzy entropy of an interval value spherical fuzzy set, which is characterized by comprising the following steps of: the method comprises the following steps: s1, carrying out standardization processing on an interval spherical fuzzy number decision matrix to form a standard interval spherical fuzzy number evaluation matrix; s2, solving attribute weight through fuzzy entropy of interval spherical fuzzy number; s3, determining a positive ideal solution and a negative ideal solution through an interval spherical fuzzy number scoring function; s4, solving the group benefit value and the individual regret value of each alternative scheme through a distance function, an attribute weight and positive and negative ideal solutions; and S5, determining the sequence of the alternative schemes by using a VIKOR method through the group benefit value and the individual regret value to obtain an optimal scheme or a compromise scheme, wherein the method can solve the evaluation value into the objective weight of the interval value spherical fuzzy number so that the decision result is more reasonable. The method can be applied to various multi-attribute decision scenes, such as project investment, building site selection, talent selection and the like.

Description

Multi-attribute decision method based on interval spherical fuzzy set fuzzy entropy
Technical Field
The invention provides a fuzzy entropy and multi-attribute decision method based on an interval value spherical fuzzy set, particularly relates to a multi-attribute decision problem widely existing in the technical fields of social economy and engineering, and belongs to the field of multi-attribute decision.
Background
The interval value spherical fuzzy set is one of the more advanced tools for processing fuzzy information at present, compared with the traditional fuzzy set, the interval value spherical fuzzy set contains more uncertain information, and after the interval value spherical fuzzy set is combined with multi-attribute decision, the attribute degree of a decision maker can be more widely represented. One important problem faced in solving the multi-attribute decision-making problem is the determination of attribute weights. In the existing interval value spherical fuzzy multi-attribute decision, the attribute weight is a subjective weight directly given by a decision maker, so that the decision maker has serious subjectivity and cannot objectively and completely reflect the importance of each attribute. The invention provides a fuzzy entropy of an interval value spherical fuzzy set, and objective weights of all attributes are calculated according to the fuzzy entropy of an evaluation value, and finally the obtained objective weights are applied to a VIKOR method to sequence and optimize alternative schemes.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problem of interval value spherical fuzzy multi-attribute decision-making with unknown attribute weight, in order to consider the influence of objective attribute weight on a decision-making result, a method for calculating fuzzy entropy of an interval value spherical fuzzy set is provided, and the objective weight of the attribute can be obtained based on the fuzzy entropy. Through a distance measurement function of interval value spherical fuzzy numbers, the alternative schemes can be ranked and preferred by combining the VIKOR method and the attribute weight.
To achieve the above object, fig. 1 shows a basic flowchart of the present invention, and the method includes the following steps:
s1, carrying out standardization processing on an interval spherical fuzzy number shape decision matrix to form a standard interval spherical fuzzy number evaluation matrix;
s2, determining a fuzzy entropy of the interval spherical fuzzy number, and solving attribute weight through the fuzzy entropy;
s3, obtaining the score and the accuracy of the evaluation value through an interval spherical fuzzy number score function and an accuracy function so as to determine a positive ideal solution and a negative ideal solution;
s4, determining the distance between the evaluation value and the positive and negative ideal solutions through a distance function, and solving the group benefit value and the individual regret value of each alternative scheme by utilizing the attribute weight;
s5, sorting the alternative schemes by using a VIKOR method through the group interest value and the individual regret value to obtain an optimal scheme or a compromise scheme;
the spherical fuzzy numerical decision matrix of the initial interval value in the step S1 is as follows: o = (h) ij ) m×n Total m alternatives a = { a = { (a) 1 ,A 2 ,…,A m H, n attributes C = { C = } 1 ,C 2 ,…,C n Therein of
Figure RE-GDA0003946232040000011
μ ij 、υ ij And pi ij Respectively represent h ij Degree of membership, degree of non-membership and degree of hesitation,
Figure RE-GDA0003946232040000012
and
Figure RE-GDA0003946232040000013
respectively represent the upper and lower bounds of the membership degree interval,
Figure RE-GDA0003946232040000014
and
Figure RE-GDA0003946232040000015
respectively representing the upper and lower boundaries of the non-membership degree and the upper and lower boundaries of the hesitation degree;
the evaluation matrix of the spherical fuzzy number is as follows:
Figure RE-GDA0003946232040000016
if C i For the benefit type attribute, then
Figure RE-GDA0003946232040000017
If C i For a cost-type attribute, then
Figure RE-GDA0003946232040000021
In the step S2, an evaluation value h of a jth decision attribute of an ith scheme in the sphere fuzzy number evaluation matrix of the canonical interval is evaluated ij The fuzzy entropy of (a) is:
Figure RE-GDA0003946232040000022
wherein the content of the first and second substances,
Figure RE-GDA0003946232040000023
satisfy as the interval spherical fuzzy number h ij The following 4 constraints of fuzzy entropy:
(1)
Figure RE-GDA0003946232040000024
is a distinct set;
(2)
Figure RE-GDA0003946232040000025
and is provided with
Figure RE-GDA0003946232040000026
(3)
Figure RE-GDA0003946232040000027
(4)
Figure RE-GDA0003946232040000028
And is
Figure RE-GDA0003946232040000029
Or
Figure RE-GDA00039462320400000210
If the attribute C j In all the alternatives, the larger the sum of the fuzzy entropy compared with other attributes is, the less information can be provided by the attribute, and the smaller the contribution of the scheme preference is, the less weight should be given, so that the interval value spherical fuzzy entropy is given the following weight:
Figure RE-GDA00039462320400000211
thus, a weight vector ω of the decision matrix can be obtained (m) {ω 1 ,ω 2 ,…,ω m }。
Sphere fuzzy number alpha of interval value in S3 j ＝{[a j ,b j ],[c j ,d j ],[e j ,f j ]The score function of } is:
Figure RE-GDA00039462320400000212
the exact function is:
Figure RE-GDA00039462320400000213
α > β when S (α) > S (β); h (α) > H (β) when S (α) = S (β) then α > β; α ≈ β if S (α) = S (β) and H (α) = H (β).
The positive and negative ideal solutions of each scheme are determined by the formulas (3) and (4).
The positive ideal solution is:
Figure RE-GDA00039462320400000214
the negative ideal solution is:
Figure RE-GDA00039462320400000215
the spherical fuzzy number alpha of the interval value in the step S4 1 ＝{[a 1 ,b 1 ],[c 1 ,d 1 ],[e 1 ,f 1 ]And α 2 ＝{[a 2 ,b 2 ],[c 2 ,d 2 ],[e 2 ,f 2 ]The distance function between } is:
Figure RE-GDA00039462320400000216
according to the distance formula, the group benefit value S of each scheme can be obtained i And individual regret value R i :
Figure RE-GDA0003946232040000031
Figure RE-GDA0003946232040000032
In the formula: s i Indicating the group benefit of the alternatives, S i The smaller the population benefit is; r i Unfortunately for the individual, R i The smaller the size, the smaller the subject suffers; omega j As a weight of each attribute.
The sequence of the VIKOR method of each scheme in the step S5 is as follows:
the benefit ratio Q is obtained through the group benefit value and the individual regret value i The calculation formula is as follows:
Figure RE-GDA0003946232040000033
in the formula:
Figure RE-GDA0003946232040000034
the maximum group benefit for each scheme;
Figure RE-GDA0003946232040000035
minimum group benefit for each protocol;
Figure RE-GDA0003946232040000036
minimal individual regret for each protocol;
Figure RE-GDA0003946232040000037
is the largest individual regret for each protocol;
Figure RE-GDA0003946232040000038
is the decision preference of the decision maker.
Respectively put each scheme according to S i ，R i ，Q i The sorting is carried out from small to large, and the scheme is superior at the top.
If Q i Optimal solution Y in the ranking 1 With sub-optimal solution Y 2 Satisfying both the acceptable dominance criterion and the acceptable stability criterion, then Y 1 Is an optimal scheme; if acceptable stability criteria are not met, then Y 1 And Y 2 Are all trade-off schemes; if the acceptable stability criteria are met but the acceptable dominance criteria are not met, then all solutions that do not meet the acceptable dominance criteria are overall optimal.
Acceptable dominance criteria are: q (Y) 2 )+Q(Y 1 ) DQ wherein DQ = 1/(n-1);
the acceptable stability criteria are: y is 1 Is S i Rank or R i The top ranked objects in the ranking.
Drawings
FIG. 1 is a schematic flow chart of the specific steps of the present invention;
Detailed Description
Example for comparison, the method of the present invention is used to solve investment policy choices of a certain multinational company, and 4 existing investment schemes need evaluation choices. These 4 investment plans correspond to 4 possible investment sites, respectively: pakistan A 1 Iran A. Var. 2 Cacique A 3 Bangladesh A 4 . The comfort zone C needs to be evaluated from 4 evaluation criteria (attributes) 1 Government regulation C 2 Benefits of people C 3 Market competition C 4 The investment scenario described above is evaluated. All attributes are assumed to be profitable. The decision maker gives the related opinions in the form of interval value spherical fuzzy numbers, and the specific attribute values are shown in table 1.
TABLE 1 sphere fuzzy decision matrix for interval values
Figure RE-GDA0003946232040000039
The above method is used for processing and decision making:
step 1, normalizing the interval value spherical fuzzy number type decision matrix to obtain a normalized decision matrix
Figure RE-GDA0003946232040000041
As shown in table 2.
TABLE 2 normalized decision information Table
Figure RE-GDA0003946232040000042
Step2, solving the fuzzy entropy of the interval value spherical fuzzy number in the normalized decision matrix according to the formula (2) to obtain a fuzzy entropy matrix Em ×n As shown in table 3.
TABLE 3 fuzzy entropy matrix
Figure RE-GDA0003946232040000043
The decision matrix weight vector is obtained according to formula (2) and table 3 as follows: ω = {0.2567,0.1991,0.3231,0.2212}
And Step 3, obtaining a decision information score matrix according to the score formula (3) as shown in the table 4.
Figure RE-GDA0003946232040000044
Obtaining a positive ideal decision scheme and a negative ideal decision scheme according to the decision information scoring matrix and the precise function respectively as follows:
Figure RE-GDA0003946232040000045
Figure RE-GDA0003946232040000046
and Step 4, calculating the group benefit and the individual regret of each scheme according to the formulas (7), (8) and (9) and the positive and negative ideal solutions.
The population benefit for each alternative was: s = {0.7450,0.7801,0.2212,0.9384}
Individuals with each alternative regret to be: r = {0.3231,0.2476,0.2212,0.1526}
Step 6 the individual protocols were ranked by the VIKOR method.
Let the decision preference be
Figure RE-GDA0003946232040000047
The benefit ratios for each scheme are: q = {0.8652,0.6682,0.2012,0.5};
the scenario resulting from the benefit ratio Q is ranked as: a. The 3 >A 4 >A 2 >A 4 According to the decision rule of VIKOR, A 3 Is also first in the group benefit ranking, meeting acceptable stability criteria; from Q 3 -Q 4 ＝0.2988<0.3333, it can be concluded that the results do not meet acceptable dominance criteria, therefore, A 3 And A 4 All are acceptable ideal schemes.

Claims (6)

1. The multi-attribute decision method based on the new fuzzy entropy of the interval value spherical fuzzy set is characterized by comprising the following steps: the method comprises the following steps:
s1, carrying out standardization processing on an interval spherical fuzzy number shape decision matrix to form a standard interval spherical fuzzy number evaluation matrix;
s2, determining a fuzzy entropy of the interval spherical fuzzy number, and solving attribute weight through the fuzzy entropy;
s3, obtaining the score and the accuracy of the evaluation value through the interval spherical fuzzy number score function and the accuracy function so as to determine a positive ideal solution and a negative ideal solution;
s4, determining the distance between the evaluation value and the positive and negative ideal solutions through a distance function, and solving the group benefit value and the individual regret value of each alternative scheme by utilizing the attribute weight;
and S5, sequencing the alternatives by utilizing a VIKOR method through the group interest value and the individual regret value to obtain an optimal scheme or a compromise scheme.
2. The multi-attribute decision method based on the new fuzzy entropy of the interval value spherical fuzzy set as claimed in claim 1, characterized in that: the spherical fuzzy number type decision matrix of the initial interval value in the step S1 is as follows: o = (h) ij ) m×n Total m alternatives a = { a = { (a) 1 ,A 2 ,…,A m N attributes C = { C = } 1 ,C 2 ,…,C n Therein of
Figure FDA0003876799620000011
μ ij 、v ij And pi ij Respectively represent h ij Degree of membership, degree of non-membership and degree of hesitation,
Figure FDA0003876799620000012
and
Figure FDA0003876799620000013
respectively represent the upper and lower bounds of the membership degree interval,
Figure FDA0003876799620000014
and
Figure FDA0003876799620000015
respectively representing the upper and lower bounds of the non-membership degree and the upper and lower bounds of the hesitation degree; the evaluation matrix of the spherical fuzzy number in the standard interval is as follows:
Figure FDA0003876799620000016
if C i Is a benefit type attribute, then h ij ＝
Figure FDA0003876799620000017
If C i For a cost-type attribute, then
Figure FDA0003876799620000018
3. The multi-attribute decision method based on the new fuzzy entropy of the interval value spherical fuzzy set according to claim 1, characterized in that: in the step S2, an evaluation value h of a jth decision attribute of an ith scheme in the sphere fuzzy number evaluation matrix of the canonical interval is determined ij The fuzzy entropy of (a) is:
Figure FDA0003876799620000019
then, the weight of the jth attribute can be expressed as:
Figure FDA00038767996200000110
thus, a weight vector ω of the decision matrix can be obtained (m) {ω 1 ,ω 2 ,…,ω m }。
4. The multi-attribute decision method based on the new fuzzy entropy of the interval value spherical fuzzy set according to claim 1, characterized in that: the interval value spherical fuzzy number alpha in the step S3 j ＝{[a j ,b j ],[c j ,d j ],[e j ,f j ]The score function of is:
Figure FDA00038767996200000111
the exact function is:
Figure FDA00038767996200000112
the positive and negative ideal solutions for each solution are determined by equation (3), with higher values of the scoring function being preferred. The higher the value of the exact function is when the score functions are equal, the better.
The positive ideal solution is
Figure FDA00038767996200000113
The negative ideal solution is
Figure FDA00038767996200000114
5. The multi-attribute decision method based on the new fuzzy entropy of the interval value spherical fuzzy set as claimed in claim 1, characterized in that: the spherical fuzzy number alpha of the interval value in the step S4 1 ＝{[a 1 ,b 1 ],[c 1 ,d 1 ],[e 1 ,f 1 ]And α 2 ＝{[a 2 ,b 2 ],[c 2 ,d 2 ],[e 2 ,f 2 ]The distance function between } is:
Figure FDA00038767996200000115
according to the distance formula, the group benefit value S of each scheme can be obtained i And individual regret value R i :
Figure FDA0003876799620000021
Figure FDA0003876799620000022
In the formula: s i Indicating the population benefit of the alternative, S i The smaller the population benefit is; r i Unfortunately for the individual, R i The smaller the size, the smaller the subject regret; omega j As a weight of each attribute.
6. The multi-attribute decision method based on the new fuzzy entropy of the interval value spherical fuzzy set as claimed in claim 1, characterized in that: the sequence of the VIKOR method of each scheme in the step S5 is as follows:
the benefit ratio Q is derived from the group benefit value and the individual regret value i The calculation formula is as follows:
Figure FDA0003876799620000023
in the formula:
Figure FDA0003876799620000024
is the maximum population benefit for each protocol;
Figure FDA0003876799620000025
minimum group benefit for each protocol;
Figure FDA0003876799620000026
is the smallest individual for each protocol;
Figure FDA0003876799620000027
is the largest individual regret for each protocol;
Figure FDA0003876799620000028
is the decision preference of the decision maker.
Respectively put each scheme according to S i ，R i ，Q i Sorting from small to large, the scheme is superior in the front, if Q i Optimal solution Y in the ranking 1 With sub-optimal solution Y 2 Satisfying both the acceptable dominance criterion and the acceptable stability criterion, then Y 1 Is an optimal scheme; if acceptable stability criteria are not met, then Y 1 And Y 2 Are all trade-off schemes; if the acceptable stability criteria are met but the acceptable dominance criteria are not, then all solutions that do not meet the acceptable dominance criteria are the best overallThe advantages are good. Acceptable dominance criteria are: q (Y) 2 )+Q(Y 1 ) DQ wherein DQ = 1/(n-1); acceptable stability criteria are: y is 1 Is S i Rank or R i The top ranked objects in the ranking.
CN202211219876.6A 2022-10-04 2022-10-04 Multi-attribute decision method based on interval spherical fuzzy set fuzzy entropy Pending CN115511193A (en)

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