CN112700096A - Multi-attribute decision method based on triangular fuzzy number preference to scheme - Google Patents

Multi-attribute decision method based on triangular fuzzy number preference to scheme Download PDF

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CN112700096A
CN112700096A CN202011538450.8A CN202011538450A CN112700096A CN 112700096 A CN112700096 A CN 112700096A CN 202011538450 A CN202011538450 A CN 202011538450A CN 112700096 A CN112700096 A CN 112700096A
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decision
scheme
fuzzy number
attribute
similarity
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李�昊
刘欢
郑嘉成
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Harbin University of Science and Technology
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Harbin University of Science and Technology
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Abstract

The invention discloses a multi-attribute decision method with a preference to schemes based on triangular fuzzy numbers, which is characterized by comprising the following steps of: the method comprises the following steps: s1, converting the initial triangular fuzzy number decision matrix into a standard triangular fuzzy number decision matrix; s2, solving an attribute weight vector according to the nonlinear programming model established by the combination method; s3, constructing a weighted normalized triangular fuzzy number decision matrix; s4, calculating the positive and negative ideal points of each attribute according to the weighting matrix to form a triangular fuzzy number type positive and negative ideal decision scheme; s5, calculating the similarity between each alternative scheme and the triangular fuzzy number positive and negative ideal decision scheme; s6, calculating the total similarity of the similarity of each alternative scheme compared with the positive and negative ideal decision schemes in the scheme set; s7, the scheme sets are ranked and preferred according to the sequence of the overall similarity values from large to small, and the method can be used for the conditions that a decision maker has subjective preference on decision objects, such as talent assessment, project investment and the like.

Description

Multi-attribute decision method based on triangular fuzzy number preference to scheme
Technical Field
The invention relates to the field of multi-attribute decision problems, in particular to a multi-attribute decision problem widely existing in the technical fields of social economy and engineering.
Background
The multi-attribute decision is a class of decision problems widely existing in the technical fields of social economy and engineering, and in a decision process, the final judgment of a decision result is often influenced by preference information of a decision maker on a scheme object, so that an uncertain multi-attribute decision problem with known preference of a research scheme has important theoretical significance and practical value.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the multi-attribute decision problem that the attribute weight is unknown and subjective preference information and attribute values of a scheme are given in a triangular fuzzy number form, a triangular fuzzy number type multi-attribute decision method which is based on similarity and has preference on the scheme is provided.
To achieve the above object, fig. 1 shows a basic flow chart of the present invention, and the method includes the following steps:
s1, in order to unify the incommercity among different attribute values and eliminate the influence of different physical dimensions, converting the initial triangular fuzzy number type decision matrix into a standard triangular fuzzy number type decision matrix;
s2, solving an attribute weight vector according to the nonlinear programming model established by the combination method;
s3, constructing a weighted normalized triangular fuzzy number type decision matrix according to the attribute weight vector and the normalized triangular fuzzy number type decision matrix;
s4, calculating the positive and negative ideal points of each attribute by the weighted normalized triangular fuzzy number decision matrix to form a triangular fuzzy number positive and negative ideal decision scheme;
s5, calculating the similarity between each alternative scheme and the triangular fuzzy number positive and negative ideal decision scheme;
s6, calculating the total similarity of the similarity of each alternative scheme compared with the positive and negative ideal decision schemes in the scheme set;
and S7, performing quality judgment and screening sorting on the scheme sets according to the sequence of the overall similarity values from large to small.
The initial triangular fuzzy number type decision matrix in the step S1 is:
Figure RE-GDA0002968074020000011
wherein
Figure RE-GDA0002968074020000012
Figure RE-GDA0002968074020000013
Respectively represent
Figure RE-GDA0002968074020000014
A lower boundary and an upper boundary supported, and
Figure RE-GDA0002968074020000015
is composed of
Figure RE-GDA0002968074020000016
The median value of (a number representing the highest probability of taking a value in this interval);
the canonical triangular fuzzy number decision matrix is:
Figure RE-GDA0002968074020000017
the formula for carrying out the normalization processing is
Figure RE-GDA0002968074020000021
Or
Figure RE-GDA0002968074020000022
Wherein IjAnd (j ═ 1, 2) respectively represents a subscript set of common benefit and cost evaluation attributes, M ═ {1, 2, …, M }, and N ═ 1, 2, …, N }.
The nonlinear programming model obtained by using the combination method in the step S2 is:
Figure RE-GDA0002968074020000023
wherein d (W) is the total deviation of the attribute weight vector W selected such that the subjective preference information of the decision maker is different from the objective preference value (attribute value);
e (W) is the total deviation among all the alternative schemes, wherein lambda is more than or equal to 0 and less than or equal to 1, and represents the preference degree of a decision maker on the subjective and objective attribute weight determination method respectively;
Figure RE-GDA0002968074020000024
for decision maker to scheme XiIn the attribute ujObjective preference value and decision maker's opposite side scheme XiPreference information V ofiThe similarity between them;
s(xij，xkj) Solving the model by a Lagrange multiplier method for the similarity between the alternatives under the attribute j
Figure RE-GDA0002968074020000025
With respect to wjPartial derivative with mu and let it equal to 0 to obtain
Figure RE-GDA0002968074020000026
Figure RE-GDA0002968074020000027
The weight is thus obtained as:
Figure RE-GDA0002968074020000028
wherein j belongs to N, and the weight is normalized to obtain
Figure RE-GDA0002968074020000029
The formula for calculating the similarity of two canonical triangular fuzzy numbers is
Figure RE-GDA00029680740200000210
Max-type similarity, called two canonical triangular ambiguity numbers; or is
Figure RE-GDA0002968074020000031
The min-type similarity, referred to as the two canonical triangular ambiguity numbers, and can be seen,
Figure RE-GDA0002968074020000032
or
Figure RE-GDA0002968074020000033
The larger, the
Figure RE-GDA0002968074020000034
And
Figure RE-GDA0002968074020000035
the greater the similarity, the attribute weight vector obtained by solving the nonlinear programming model is: w ═ W1，w2，…，wn}。
The weighted normalized triangular fuzzy number type decision matrix in the step S3 is:
Figure RE-GDA0002968074020000036
the conceivable decision scheme in step S4 is:
Figure RE-GDA0002968074020000037
wherein the balance
Figure RE-GDA0002968074020000038
For positive ideal points, the negative ideal decision scheme is:
Figure RE-GDA0002968074020000039
wherein the content of the first and second substances,
Figure RE-GDA00029680740200000310
is a negative ideal point.
Similarity S (W) (X) between each alternative in the step S5 and the triangular fuzzy number type positive and negative ideal decision schemei，U+)、 S(W)(Xi，U-) And the similarity formula for calculating the two schemes is as follows:
Figure RE-GDA00029680740200000311
referred to as max-type similarity between schemes; or is
Figure RE-GDA00029680740200000312
Referred to as min-type similarity between schemes wherein
Figure RE-GDA00029680740200000313
The overall similarity in step S6 is RS (X)i) Where i ∈ M. The formula for calculating the overall similarity is
Figure RE-GDA00029680740200000314
Wherein i ∈ M.
Drawings
FIG. 1 is a schematic flow chart of the specific steps of the present invention;
Detailed Description
In the embodiment, for comparison, the method solves the problem of talent selection of an enterprise, and a decision maker needs to select talents with excellent talents to a leader on one hand; on the other hand, if the conditions are equivalent, the user wants to use the talents preferred by the user, and when a certain unit conducts examination selection on the cadres, 6 examination indexes (attributes) are firstly established: idedu thought1Degree of working state u2Work as wind u3Cultural level and knowledge structure u4Leader capacity u5Development ability u6(ii) a Then the people recommend and consult, each index is respectively scored, then statistical treatment is carried out, and 5 candidate persons X are determined from the scoresi(i ═ 1, 2, …, 5). Because the index values (attribute values) given by the masses to the same candidate are not completely the same, the attribute values of each candidate after statistical processing under each index (attribute) are given in the form of triangular fuzzy numbers, and the specific attribute values are shown in table 1.
TABLE 1 Attribute values for each candidate under various criteria
Figure RE-GDA0002968074020000041
The above method is used for processing and decision making:
step 1, normalizing the initial triangular fuzzy number type decision matrix to obtain a normalized decision matrix
Figure RE-GDA0002968074020000042
As shown in table 2.
Table 2 normalized decision information table x 10-1
Figure RE-GDA0002968074020000043
Step2, suppose the preference information of the decision maker for 5 candidates is V1＝[0.30,0.35,0.40]，V2＝[0.35,0.40,0.45], V3＝[0.35,0.40,0.50],V4＝[0.40,0.45,0.55],V5＝[0.40,0.50,0.60]，
The optimal attribute weight vector W is obtained from the above equation (7) (0.1638,0.1672,0.1673,0.1674,0.1691, 0.1652).
Step3 applying the attribute weight vector W to the canonical triangular fuzzy number decision matrix
Figure RE-GDA0002968074020000044
Constructing a weighted normalized triangular fuzzy number type decision matrix
Figure RE-GDA0002968074020000045
As shown in table 3.
Table 3 weighted normalized decision information table x 10-2
Figure RE-GDA0002968074020000046
Step 4, normalizing the triangular fuzzy number type decision matrix according to the weighting
Figure RE-GDA0002968074020000047
The triangular fuzzy number type positive ideal decision scheme and the negative ideal decision scheme formed by the positive ideal point sequence and the negative ideal point sequence are respectively as follows:
U+＝{[0.0701,0.0778,0.0853],[0.0727,0.0767,0.08],[0.0718,0.0771,0.0813],[0.0728,0.0774,0.0821], [0.0725,0.0766,0.0819],[0.074,0.0775,0.0807]}；
U-＝{[0.0623,0.0696,0.0768],[0.0671,0.0709,0.0758],[0.0671,0.0714,0.0755],[0.0673,0.0717,0.0771], [0.0693,0.0741,0.0785],[0.0662,0.0695,0.0733]}。
step 5 various alternatives XiTriangular fuzzy number type positive and negative ideal decision scheme U+、U-The similarity of (A) is respectively as follows:
S(W)(X1，U+)＝0.9745、S(W)(X1，U-)＝0.9478；
S(W)(X2，U+)＝0.9768、S(W)(X2，U-)＝0.9452；
S(W)(X3，U+)＝0.9628、S(W)(X3，U-)＝0.9596；
S(W)(X4，U+)＝0.9508、S(W)(X4，U-)＝0.9669；
S(W)(X5，U+)＝0.9674、S(W)(X5，U-)＝0.9551。
step 6: respective alternative XiAnd positive and negative ideal decision scheme U+、U-The overall similarity of the compared similarities in the scheme set is RS (X) respectively1)＝0.0056、RS(X2)＝0.0065、RS(X3)＝0.0006、RS(X4)＝-0.0033、RS(X5)＝0.0026。
Step 7: according to RS (X)i) Sequence pair scheme set (X) with large to small values of i ═ 1, 2, …, 5iAnd (1, 2, …, 5) performing quality judgment and screening sorting. The results are as follows:
X2＞X1＞X5＞X3＞X4
so the optimal candidate is X2。
The present invention is not limited to the above-described embodiments, which are merely preferred embodiments of the present invention, and the present invention is not limited thereto, and any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (7)

1. The multi-attribute decision method based on the triangular fuzzy number preference to the scheme is characterized in that: the method comprises the following steps:
s1: in order to unify the incommercity among different attribute values and eliminate the influence of different physical dimensions, the initial triangular fuzzy number type decision matrix is converted into a standard triangular fuzzy number type decision matrix;
s2: solving an attribute weight vector according to a nonlinear programming model established by a combination method;
s3: constructing a weighted normalized triangular fuzzy number type decision matrix according to the attribute weight vector and the normalized triangular fuzzy number type decision matrix;
s4: calculating positive and negative ideal points of each attribute by a weighted normalized triangular fuzzy number decision matrix to form a triangular fuzzy number positive and negative ideal decision scheme;
s5: solving the similarity between each alternative scheme and the triangular fuzzy number type positive and negative ideal decision schemes;
s6: calculating the overall similarity of the similarity of each alternative scheme compared with the positive and negative ideal decision schemes in the scheme set;
s7: and performing quality judgment and screening sorting on the scheme set according to the sequence of the overall similarity value from large to small.
2. The multi-attribute decision method based on triangulated fuzzy number pair scheme of claim 1, characterized by: the initial triangular fuzzy number type decision matrix in the step S1 is:
Figure FDA0002853873570000011
wherein
Figure FDA0002853873570000012
Figure FDA0002853873570000013
Respectively represent
Figure FDA00028538735700000113
A lower boundary and an upper boundary supported, and
Figure FDA0002853873570000014
is composed of
Figure FDA0002853873570000015
The median value of (a number representing the highest probability of taking a value in this interval); the canonical triangular fuzzy number decision matrix is:
Figure FDA0002853873570000016
Figure FDA0002853873570000017
the formula for carrying out the normalization processing is
Figure FDA0002853873570000018
Wherein IjAnd (j ═ 1, 2) respectively represents a subscript set of common benefit and cost evaluation attributes, M ═ {1, 2, …, M }, and N ═ 1, 2, …, N }.
3. The multi-attribute decision method based on triangulated fuzzy number pair scheme of claim 1, characterized by: the nonlinear programming model obtained by using the combination method in the step S2 is:
Figure FDA0002853873570000019
wherein D (W) is the total deviation of the attribute weight vector W which is selected to ensure that the subjective preference information of the decision maker is in accordance with the objective preference value (attribute value), E (W) is the total deviation among all the alternative schemes, 0 is more than or equal to lambda is less than or equal to 1, which represents the preference degree of the decision maker to the subjective and objective attribute weight determination method respectively,
Figure FDA00028538735700000110
for decision maker to scheme XiIn the attribute ujObjective preference value and decision maker's opposite side scheme XiPreference information V ofiSimilarity between them, s (x)ij，xkj) For the similarity between the alternatives under the attribute j, the formula for calculating the similarity of the two canonical triangular fuzzy numbers is
Figure FDA00028538735700000111
Max-type similarity, called two canonical triangular ambiguity numbers; or is
Figure FDA00028538735700000112
Called two canonical triangular fuzzy numbersThe min-type similarity of (A) and (B) is known,
Figure FDA0002853873570000021
or
Figure FDA0002853873570000022
The larger, the
Figure FDA0002853873570000023
And
Figure FDA0002853873570000024
the greater the similarity, the attribute weight vector obtained by solving the nonlinear programming model is: w ═ W1，w2，…，wn}。
4. The multi-attribute decision method based on triangulated fuzzy number pair scheme of claim 1, characterized by: the weighted normalized triangular fuzzy number type decision matrix in the step S3 is:
Figure FDA0002853873570000025
5. the multi-attribute decision method based on triangulated fuzzy number pair scheme of claim 1, characterized by: the conceivable decision scheme in step S4 is:
Figure FDA0002853873570000026
wherein the balance
Figure FDA0002853873570000027
For positive ideal points, the negative ideal decision scheme is:
Figure FDA0002853873570000028
wherein the balance
Figure FDA0002853873570000029
Is a negative ideal point.
6. The multi-attribute decision method based on triangulated fuzzy number pair scheme of claim 1, characterized by: similarity S (W) (X) between each alternative in the step S5 and the triangular fuzzy number type positive and negative ideal decision schemei，U+)、S(W)(Xi，U-) And the similarity formula for calculating the two schemes is as follows:
Figure FDA00028538735700000210
referred to as max-type similarity between schemes; or is
Figure FDA00028538735700000211
Referred to as min-type similarity between schemes wherein
Figure FDA00028538735700000212
7. The multi-attribute decision method based on triangulated fuzzy number pair scheme of claim 1, characterized by: the overall similarity in step S6 is RS (X)i) Wherein i belongs to M, and the formula for calculating the overall similarity is
Figure FDA00028538735700000213
Wherein i ∈ M.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115115142A (en) * 2022-08-24 2022-09-27 中国科学院地理科学与资源研究所 Ship emergency stop point planning method and device and electronic equipment

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115115142A (en) * 2022-08-24 2022-09-27 中国科学院地理科学与资源研究所 Ship emergency stop point planning method and device and electronic equipment

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