CN113326998A - Index weight weighting method based on fuzzy hierarchy method and CRITIC method - Google Patents

Index weight weighting method based on fuzzy hierarchy method and CRITIC method Download PDF

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CN113326998A
CN113326998A CN202110240050.7A CN202110240050A CN113326998A CN 113326998 A CN113326998 A CN 113326998A CN 202110240050 A CN202110240050 A CN 202110240050A CN 113326998 A CN113326998 A CN 113326998A
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李珉
罗拥军
丁伟
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Jiangsu Anfang Electric Power Technology Co ltd
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Abstract

The index weight weighting method based on the fuzzy hierarchy method and the CRITIC method comprises the following steps: determining subjective weight based on a fuzzy hierarchy method, constructing a judgment matrix by utilizing triangular fuzzy numbers and calculating index weight; determining objective weight of the index by an entropy and Kendell correlation coefficient and adopting a CRITIC method; and determining the index combination weight based on the consistency of the subjective and objective weights, and obtaining an optimal combination weight model. The method utilizes the fuzzy consistency judgment matrix to compare the importance of each index on the basis of the fuzzy analytic hierarchy process and the CRITIC process, and solves the problems of consistency inspection and weight calculation of the judgment matrix of the analytic hierarchy process.

Description

Index weight weighting method based on fuzzy hierarchy method and CRITIC method
Technical Field
The invention relates to the field of warehouse management, in particular to an index weight weighting method based on a fuzzy hierarchy method and a CRITIC method.
Background
The storage, inspection and matching integrated indexes have certain ambiguity on the description of the problems, some indexes are difficult to be expressed deterministically by quantitative relations, and other indexes with quantitative numerical values have certain ambiguity on the evaluation. Fuzzy evaluation is an effective tool for evaluation by using fuzzy information.
The fuzzy evaluation first-choice needs to evaluate the membership degree of the index, namely, the standardization processing of the index. According to different index characteristics, the fuzzy evaluation has different membership functions. The membership function is classified according to the shape into a rectangular distribution function, a trapezoidal distribution function, an exponential distribution function, a K-th-order parabolic distribution function and the like; the index type is classified into a cost type membership function, a benefit type membership function, an interval type membership function, and the like. And selecting a rectangular distribution function and a trapezoidal distribution function as membership functions of fuzzy evaluation in consideration of index characteristics in an index system.
The analytic hierarchy process is a process of layering attributes and indexes of an evaluation object and synthesizing the indexes layer by layer to obtain a comprehensive evaluation index. In the hierarchical analysis, the weight of the index is determined by comparing the importance between the indexes, and the indexes are synthesized.
Disclosure of Invention
The invention provides an index weight weighting method based on a fuzzy hierarchy process and a CRITIC process, which is characterized in that on the basis of a fuzzy hierarchy process and a CRITIC process, a fuzzy consistency judgment matrix is used for comparing the importance of each index, and the problems of consistency check and weight calculation of the judgment matrix of the hierarchy process are solved.
The index weight weighting method based on the fuzzy hierarchy method and the CRITIC method comprises the following steps:
step 1, determining subjective weight based on a fuzzy hierarchy method, constructing a judgment matrix by utilizing triangular fuzzy numbers and calculating index weight;
step 2, determining objective weight of the index by an entropy and Kendell correlation coefficient and adopting a CRITIC method;
and 3, determining the index combination weight based on the consistency of the subjective and objective weights, and obtaining the optimal combination weight model.
Further, in step 1, firstly, a determination matrix is constructed by using a triangular fuzzy number, which is defined as follows:
let M be an element of E1，E1Representing the space formed by the whole fuzzy number, the membership function mu of Mm(x)， R→[0,1]Expressed as:
Figure BDA0002961774550000021
in the formula, l is less than or equal to M and less than or equal to u, l and u are respectively an upper boundary and a lower boundary supported by M, the M is called a triangular fuzzy number, the fuzzy number is written as M ═ l, M and u, and l-u | reflects the size of the fuzzy degree; using triangular fuzzy number as expert judgment matrix, i.e. fuzzy judgment matrix A ═ alphaij)n×nElement alpha in (1)ij＝(lij,mij,uij) Is one in mijThe different values have corresponding specific meanings in a closed interval of the median;
when the index set in the evaluation system is x ═ { x ═ x1,x2,K,xnAnd when the comparison is carried out, the important degree of pairwise comparison of indexes is obtained and is expressed as:
Figure RE-GDA0003187214450000031
then, index weight is calculated by utilizing a triangular fuzzy number correlation algorithm;
calculating the comprehensive synthetic triangular fuzzy number of each index:
Figure BDA0002961774550000032
calculating Si≥SkThe possibility of (2):
let V (S)i≥Sk) Is Si≥SkL, m, u represent the parameters of 3 triangular blur numbers, then:
Figure BDA0002961774550000033
in the formula, d is the highest point of the intersection of two triangular fuzzy membership functions, and S is calculatediProbability of greater than all other index composite values:
V(Si)＝V(Si≥S1,S2,K,Sn)＝minV(Si≥Sk)
i,k＝1,2,K,n,i≠k
and (3) calculating index weight:
let w' (α)i)＝V(Si) Then the weight vector can be expressed as W '═ W' (α)1),w′(α2),K,w′(αn))TThe normalized vector is normalized to obtain a normalized weight vector W ═ W (α ═ W1),w(α2),K,w(αn))T。
Further, in step 2, through the uncertainty of the entropy measurement information, for the ith evaluation index, the entropy is defined as:
Figure BDA0002961774550000034
Figure BDA0002961774550000035
in the formula: when f isikWhen 0, f is considered to beik lnfik＝0；
Entropy weight σ of i-th evaluation indexiIs defined as:
Figure BDA0002961774550000041
in the formula: sigma is more than or equal to 0iLess than or equal to 1 and
Figure BDA0002961774550000042
the smaller the entropy value of the ith evaluation index is, the larger the entropy weight of the index is, and the larger the difference of index values of different lines in the evaluation index is;
the degree of correlation of multi-column grade variables is measured by Kendel correlation coefficient:
vector ui＝(ui1,ui2,L,uiq) And uj＝(uj1,uj2,L,ujq) Having q elements, uiAnd ujThe corresponding k-th variable value in (1) is uikAnd ujk(k is more than or equal to 1 and less than or equal to q); falseLet XRiAnd XRjAre each uikAnd ujkAt uiAnd ujThe corresponding sorting value in the two vectors is changed into a variable (X)Ri, XRj) Set of collocation component variable pairs XRThen, the kendel correlation coefficient between the index i and the index j is defined as:
Figure BDA0002961774550000043
in the formula: a is a constant having a value of q (q-1)/2; n is a radical ofcAnd NdRespectively represent a set XRVariable pair element (X)Ri,XRj) The two variable sorting values are equal and unequal;
Figure BDA0002961774550000044
represents a vector uiThe number of the same variable value;
Figure BDA0002961774550000045
represents a vector ujThe number of the same variable value; kendell correlation coefficient gamma defining the ith index and all other indexesiComprises the following steps:
Figure BDA0002961774550000046
kendel correlation coefficient gamma of index iiThe larger the index is, the larger the rank correlation of the index with other indexes is indicated;
and (3) comprehensively integrating the entropy and the Kendell correlation coefficient, and determining the objective weight of the index by adopting a CRITIC method, wherein the objective weight of the ith index is expressed as:
Figure BDA0002961774550000051
further, in step 3, the subjective weight obtained by the fuzzy hierarchy method and the objective weight obtained by the CRITIC method are combined to generate a comprehensive weight through subjective and objective weighting.
Further, in step 3, it is assumed that there are v subjective weighting methods and m-v objective weighting methods in the m weighting methods, and the weight of the ith index under the s weighting method is γsi(ii) a According to the moment estimation theory, the expectation of the main weight and the expectation of the objective weight of the ith index are respectively S1(γi) And S2(γi)：
Figure BDA0002961774550000052
Figure BDA0002961774550000053
Then the objective and subjective weighting coefficients of the ith index are respectively tauiAnd
Figure BDA0002961774550000054
Figure BDA0002961774550000055
Figure BDA0002961774550000056
determining the comprehensive weight M of the ith index by taking the minimum sum of squared deviations between the integrated weight and the subjective and objective weights as a targetiThe consistency function of (1), i.e. the optimal combining weight model, is:
Figure BDA0002961774550000061
Figure BDA0002961774550000062
Mi≥0,i＝1,2,...,p
the invention achieves the following beneficial effects: the method is based on a fuzzy hierarchy method and a CRITIC method, and based on the fuzzy hierarchy method and the CRITIC method, the fuzzy consistency judgment matrix is used for comparing the importance of each index, so that the problems of consistency check and weight calculation of the judgment matrix of the hierarchical analysis method are solved.
Drawings
Fig. 1 is a flowchart of calculating the integrated weight according to the embodiment of the present invention.
FIG. 2 is a schematic diagram of a triangular fuzzy number in an embodiment of the present invention.
FIG. 3 is a schematic diagram of an intersection of two triangular fuzzy membership functions according to an embodiment of the present invention.
Detailed Description
The technical scheme of the invention is further explained in detail by combining the drawings in the specification.
The index weight weighting method based on the fuzzy hierarchy method and the CRITIC method comprises the following steps: step 1, determining subjective weight based on a fuzzy hierarchy method, constructing a judgment matrix by utilizing triangular fuzzy numbers and calculating index weight; step 2, determining objective weight of the index by an entropy and Kendell correlation coefficient and adopting a CRITIC method; and 3, determining the index combination weight based on the consistency of the subjective and objective weights, and obtaining an optimal combination weight model.
In step 1, the subjective weight determination based on the fuzzy hierarchy method specifically comprises the following steps:
the storage, inspection and matching integrated indexes have certain ambiguity on the description of the problems, some indexes are difficult to be expressed deterministically by quantitative relations, and other indexes with quantitative numerical values have certain ambiguity on the evaluation. Fuzzy evaluation is an effective tool for evaluation by using fuzzy information.
The fuzzy evaluation first-choice needs to evaluate the membership degree of the index, namely, the standardization processing of the index. According to different index characteristics, the fuzzy evaluation has different membership functions. The membership function is classified according to the shape into a rectangular distribution function, a trapezoidal distribution function, an exponential distribution function, a K-th-order parabolic distribution function and the like; the index type is classified into a cost type membership function, a benefit type membership function, an interval type membership function, and the like. And selecting a rectangular distribution function and a trapezoidal distribution function as membership functions of fuzzy evaluation in consideration of index characteristics in an index system.
The analytic hierarchy process is a process of layering attributes and indexes of an evaluation object and synthesizing the indexes layer by layer to obtain a comprehensive evaluation index. In the hierarchical analysis, the weight of the index is determined by comparing the importance between the indexes, and the indexes are synthesized.
The fuzzy analytic hierarchy process utilizes the fuzzy consistent judgment matrix to compare the importance of each index, improves the problems of consistency check and weight calculation of the analytic hierarchy process judgment matrix, and comprises the following calculation steps:
firstly, a judgment matrix is constructed by utilizing a triangular fuzzy number:
the triangular fuzzy number is a commonly used membership function form, which is defined as follows:
let M be an element of E1(E1Representing the space formed by the whole fuzzy number), the membership function mu of Mm(x)，R→[0,1]Can be expressed as:
Figure BDA0002961774550000071
in the formula: l is less than or equal to M and less than or equal to u, l and u are respectively an upper bound and a lower bound supported by M, the M is called a triangular fuzzy number, the fuzzy number can be written as M ═ l, M and u, and | l-u | reflects the size of the fuzzy degree, as shown in FIG. 2. The triangular fuzzy number is used to replace the traditional expert judgment matrix, namely the fuzzy judgment matrix A ═ alphaij)n×nElement alpha in (1)ij＝(lij,mij,uij) Is one in mijThe specific meanings of the interval representing the median are shown in the following table.
Figure BDA0002961774550000081
When the index set in the evaluation system is x ═ { x ═ x1,x2,K,xnAnd when the index is compared with the brightness, the importance degree of the index brightness contrast can be obtained, and the importance degree is represented as:
Figure RE-GDA0003187214450000082
calculating index weight by using a triangular fuzzy number correlation algorithm:
a. calculating the comprehensive synthetic triangular fuzzy number of each index:
Figure BDA0002961774550000083
b. calculating Si≥SkThe possibility of (2):
let V (S)i≥Sk) Is Si≥SkL, m, u represent the parameters of 3 triangular blur numbers, then:
Figure BDA0002961774550000084
in the formula: d is the highest point of the intersection of the two triangular fuzzy membership functions, as shown in FIG. 3. Find SiProbability of greater than all other index composite values:
V(Si)＝V(Si≥S1,S2,K,Sn)＝minV(Si≥Sk)
i,k＝1,2,K,n,i≠k
c. and (3) calculating index weight:
let w' (α)i)＝V(Si) Then the weight vector can be expressed as W '═ W' (α)1),w′(α2),K,w′(αn))TThe normalized vector is normalized to obtain a normalized weight vector W ═ W (α ═ W1),w(α2),K,w(αn))T。
Step 2, the objective weight determination based on the CRITIC method specifically comprises the following steps:
in order to reduce the too subjective influence of weight setting brought by a subjective weight weighting method, an objective weighting method can be introduced at the same time, and meanwhile, valuable information in index data can be effectively mined, so that the weighting process is more reasonable, and the weighting result is more accurate. The objective weight determination can therefore be performed using Critic's method.
The basic idea of the CRITIC method is to determine the objective weights of the indices based on the strength of contrast and the conflict between the evaluation indices. The contrast strength is expressed in the form of standard deviation, i.e. the size of the standard deviation indicates the size of the difference between the values of each scheme in the same index. The larger the standard deviation is, the larger the value difference between the schemes is. And the conflict between indexes is based on the correlation between indexes. If the two indexes have stronger positive correlation, the conflict between the two indexes is lower. Wherein, the contrast strength of the index is measured by different values of the index (namely, the difference of the index value); and the conflict between indexes is measured by the correlation of values of different indexes. The contrast strength of the evaluation indexes can be measured by using an entropy weight, and the conflict between the evaluation indexes can be measured by using a Kendall correlation Coefficient (Kendall Coefficient).
Entropy is commonly used in the uncertainty or disorder state of measurement information, and for the ith evaluation index, the entropy is defined as:
Figure BDA0002961774550000091
Figure BDA0002961774550000101
in the formula: when f isikWhen 0, f is considered to beiklnfik＝0。
Entropy weight σ of i-th evaluation indexiIs defined as:
Figure BDA0002961774550000102
in the formula: sigma is more than or equal to 0iLess than or equal to 1 and
Figure BDA0002961774550000103
from the above definition, it can be seen that the smaller the entropy value of the i-th evaluation index is, the larger the entropy weight of the index is, which indicates that the index values of different lines in the evaluation index are more different.
The Kendel correlation coefficient is a correlation coefficient used to measure the degree of correlation of multi-column rank variables. Vector ui＝(ui1,ui2,L,uiq) And uj＝(uj1,uj2,L,ujq) Having q elements, uiAnd ujThe corresponding k-th variable value in (1) is uikAnd ujk(k is more than or equal to 1 and less than or equal to q). Suppose XRiAnd XRjAre respectively uikAnd ujkAt uiAnd ujThe corresponding sorting value variable (X) in the two vectorsRi,XRj) Set of collocation component variable pairs XRThen, the kendell correlation coefficient between index i and index j can be defined as:
Figure BDA0002961774550000104
in the formula: a is a constant having a value of q (q-1)/2; n is a radical ofcAnd NdRespectively represent a set XRVariable pair element (X)Ri,XRj) The two variable sorting values are equal and unequal;
Figure BDA0002961774550000105
represents a vector uiThe number of the same variable value;
Figure BDA0002961774550000106
represents a vector ujHaving the same variable value. Thus, the Kendell correlation coefficient γ of the ith index with all other indexes is definediComprises the following steps:
Figure BDA0002961774550000111
kendel correlation coefficient gamma of index iiThe larger the index, the greater the rank correlation of the index with other indexes.
By integrating the entropy and Kendell correlation coefficient, and determining the objective weight of the index by using a CRITIC method, the objective weight of the ith index can be expressed as:
Figure BDA0002961774550000112
step 3, the index combination weight determination based on the consistency of the subjective and objective weights is specifically as follows:
based on the index system classified and screened in the previous section, a complete weight setting method and an evaluation method are required to be established for the selected evaluation system. For storage, examination and distribution integrated diagnosis and evaluation, the system index is large in size and has a hierarchical structure, and the index can be reasonably and effectively weighted by adopting comprehensive weight setting. Subjectively using a fuzzy analytic hierarchy process to fully utilize the guidance of expert experience on evaluation; the CRITIC method is objectively utilized to mine the information contained in the index data. The subjective and objective weighting is combined to generate the comprehensive weight. The process is shown in figure 1.
Suppose that v kinds of subjective weighting methods and m-v kinds of objective weighting methods exist in the m kinds of weighting methods, and the weight of the ith index under the s kind of weighting method is gammasi. The relative importance of subjective and objective weights is different for different evaluation indexes. Therefore, the expectation of the objective and the main weight of the ith index obtained according to the moment estimation theory is S1(γi) And S2(γi)：
Figure BDA0002961774550000113
Figure BDA0002961774550000121
Then the objective and subjective weighting coefficients of the ith index are respectively tauiAnd
Figure BDA0002961774550000126
Figure BDA0002961774550000122
Figure BDA0002961774550000123
determining the comprehensive weight M of the ith index by taking the minimum sum of squared deviations between the integrated weight and the subjective and objective weights as a targetiThe optimal combination weight model is as follows:
Figure BDA0002961774550000124
Figure BDA0002961774550000125
Mi≥0,i＝1,2,...,p
the above description is only a preferred embodiment of the present invention, and the scope of the present invention is not limited to the above embodiment, but equivalent modifications or changes made by those skilled in the art according to the present disclosure should be included in the scope of the present invention as set forth in the appended claims.

Claims (5)

1. The index weight weighting method based on the fuzzy hierarchy method and the CRITIC method is characterized in that: the method comprises the following steps:
step 1, determining subjective weight based on a fuzzy hierarchy method, constructing a judgment matrix by utilizing triangular fuzzy numbers and calculating index weight;
step 2, determining objective weight of the index by an entropy and Kendell correlation coefficient and adopting a CRITIC method;
and 3, determining the index combination weight based on the consistency of the subjective and objective weights, and obtaining an optimal combination weight model.
2. The method for weighting indexes based on fuzzy hierarchy method and CRITIC method as claimed in claim 1, wherein: in step 1, firstly, a judgment matrix is constructed by utilizing a triangular fuzzy number, wherein the triangular fuzzy number is defined as follows:
let M be an element of E1，E1Representing the space formed by the whole fuzzy number, the membership function mu of Mm(x)，R→[0,1]Expressed as:
Figure RE-FDA0003187214440000011
in the formula, l is less than or equal to M and less than or equal to u, l and u are respectively an upper boundary and a lower boundary supported by M, the M is called a triangular fuzzy number, the fuzzy number is written as M ═ l, M and u, and l-u | reflects the size of the fuzzy degree; using triangular fuzzy number as expert judgment matrix, i.e. fuzzy judgment matrix A ═ alphaij)n×nElement alpha in (1)ij＝(lij,mij,uij) Is one in mijThe different values have corresponding specific meanings in a closed interval of the median;
when the index set in the evaluation system is x ═ { x ═ x1,x2,K,xnAnd when the comparison is carried out, the importance degree of pairwise comparison of indexes is obtained and is expressed as:
Figure RE-FDA0003187214440000021
then, index weight is calculated by utilizing a triangular fuzzy number correlation algorithm;
calculating the comprehensive synthetic triangular fuzzy number of each index:
Figure RE-FDA0003187214440000022
calculating Si≥SkThe possibility of (2):
let V (S)i≥Sk) Is Si≥SkL, m, u represent the parameters of 3 triangular blur numbers, then:
Figure RE-FDA0003187214440000023
others
In the formula, d is the highest point of the intersection of two triangular fuzzy membership functions, and S is calculatediProbability of greater than all other index composite values:
V(Si)＝V(Si≥S1,S2,K,Sn)＝minV(Si≥Sk)
i,k＝1,2,K,n,i≠k
and (3) calculating index weight:
let w' (α)i)＝V(Si) Then the weight vector can be expressed as W '═ W' (α)1),w′(α2),K,w′(αn))TThe normalized vector is normalized to obtain a normalized weight vector W ═ W (α ═ W1),w(α2),K,w(αn))T。
3. The method for weighting indexes based on fuzzy hierarchy method and CRITIC method as claimed in claim 1, wherein: in step 2, through the uncertainty of the entropy measurement information, for the ith evaluation index, the entropy is defined as:
Figure FDA0002961774540000024
Figure FDA0002961774540000031
in the formula: when f isikWhen 0, f is considered to beiklnfik＝0；
Entropy weight σ of i-th evaluation indexiIs defined as:
Figure FDA0002961774540000032
in the formula: sigma is more than or equal to 0iLess than or equal to 1 and
Figure FDA0002961774540000033
the smaller the entropy value of the ith evaluation index is, the larger the entropy weight of the index is, which indicates that the index value difference of different lines in the evaluation index is larger;
the degree of correlation of multi-column grade variables is measured by Kendel correlation coefficient:
vector ui＝(ui1,ui2,L,uiq) And uj＝(uj1,uj2,L,ujq) Having q elements, uiAnd ujThe corresponding k-th variable value in (1) is uikAnd ujk(k is more than or equal to 1 and less than or equal to q); suppose XRiAnd XRjAre each uikAnd ujkAt uiAnd ujThe corresponding sorting value in the two vectors is changed into a variable (X)Ri,XRj) Set of collocation component variable pairs XRThen, the kendel correlation coefficient between the index i and the index j is defined as:
Figure FDA0002961774540000034
in the formula: a is a constant having a value of q (q-1)/2; n is a radical ofcAnd NdRespectively represent a set XRVariable pair element (X)Ri,XRj) The number of variable pairs with equal and unequal variable ranking values;
Figure FDA0002961774540000035
represents a vector uiThe number of the same variable value;
Figure FDA0002961774540000036
represents a vector ujThe number of the same variable value; kendell correlation coefficient gamma defining the ith index and all other indexesiComprises the following steps:
Figure FDA0002961774540000041
kendel correlation coefficient gamma of index iiThe larger the index is, the larger the rank correlation of the index with other indexes is indicated;
and (3) comprehensively integrating the entropy and the Kendell correlation coefficient, and determining the objective weight of the index by adopting a CRITIC method, wherein the objective weight of the ith index is expressed as:
Figure FDA0002961774540000042
4. the method for weighting indexes based on fuzzy hierarchy method and CRITIC method as claimed in claim 1, wherein: in step 3, the subjective weight obtained by the fuzzy hierarchy method and the objective weight obtained by the CRITIC method are combined to generate a comprehensive weight through subjective and objective weighting.
5. The method for weighting indexes based on fuzzy hierarchy method and CRITIC method as claimed in claim 1, wherein: suppose that v kinds of subjective weighting methods and m-v kinds of objective weighting methods exist in the m kinds of weighting methods, and the weight of the ith index under the s kind of weighting method is gammasi(ii) a According to the moment estimation theory, the expectation of the main weight and the expectation of the objective weight of the ith index are respectively S1(γi) And S2(γi)：
Figure FDA0002961774540000043
Figure FDA0002961774540000044
Then the objective and subjective weighting coefficients of the ith index are respectively tauiAnd
Figure FDA0002961774540000046
Figure FDA0002961774540000045
Figure FDA0002961774540000051
determining the comprehensive weight M of the ith index by taking the minimum sum of squared deviations between the integrated weight and the subjective and objective weights as a targetiThe consistency function of (1), i.e. the optimal combining weight model, is:
Figure FDA0002961774540000052
Figure FDA0002961774540000053
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CN110175789A (en) * 2019-05-31 2019-08-27 国网山东省电力公司经济技术研究院 Quan Fangfa is assigned based on fuzzy hierarchy and the power distribution network diagnostic assessment index weights of CRITIC method

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