CN104123467A - GRA-TOPSIS model evaluation method based on expert preferences - Google Patents

Abstract

The invention discloses a GRA-TOPSIS model evaluation method based on expert preferences. The GRA-TOPSIS model evaluation method based on expert preferences is characterized by comprising the first step of calculating a grey relational coefficient and the second step of evaluating a TOPSIS model on the basis of the weighted grey relational coefficient. By means of the GRA-TOPSIS model evaluation method based on expert preferences, the non-linear relationship between evaluation indexes can be reflected effectively, computation is simplified, objective factors affecting the weight can be reflected accurately, and therefore the evaluation precision of the TOPSIS model is improved.

Description

A kind of evaluation method of the GRA-TOPSIS model based on expert's preference

Technical field

The present invention relates to data evaluation method field, be specially a kind of evaluation method of the GRA-TOPSIS model based on expert's preference.

Background technology

Multiple attribute decision making (MADM) is the mathematical method that a kind of use solves limited scheme multiobjectives decision.Multiple attribute decision making (MADM) comprises structure decision matrix, data pre-service, and the steps such as option screening, are intended to by a series of mathematical operation and data processing, for the decision-making of this scheme provides foundation.TOPSIS method approaches the sort method of ideal solution, in order to solve Multiple Attribute Decision Problems.Because the numerous areas such as economy, social science and engineering exist the problem of multiple attribute decision making (MADM) in a large number, therefore this decision-making technique is just widely used.The basic step of TOPSIS method is: first, obtain original matrix; Then by vectorial specification handles, obtain the decision matrix after standard; Then this decision matrix is weighted; And then calculating plus-minus ideal solutions; Calculate on this basis the distance of ideal solution and negative ideal solution; And calculate comprehensive evaluation index; Finally according to quality, sort.

Although TOPSIS evaluation model is widely used, the same with other evaluation models, it also has some limitations, and comprising:

(1) the nonlinearities change problem of real data.Traditional TOPSIS evaluation model presents linear changing relation in order to make between attribute and the effectiveness of attribute, adopt Euclidean distance to calculate the difference between evaluation of programme and ideal solution, the result finally drawing is the solution of a rigidity, and the effectiveness of actual achievement data is not linear change, therefore can produce larger error.

(2) accuracy problem of weight.Traditional method of weighting is to carry out direct weighting based on raw data and existing mathematical formulae or decision matrix, Information Entropy for example, such mode makes the weight of evaluation index depart from actual value because having ignored other objective factors of weighing factor, thereby causes the applicability of TOPSIS evaluation model and accuracy to reduce.

(3) operability and the property simplified problem.Trend from research, in the improvement of TOPSIS evaluation model, the probing into of weight calculation more and more focused on integrated with method of obtaining of data, but seriously ignored the operability in data handling procedure and the property simplified, and increased the weight of as the fussy degree of achievement data in weighting work for the treatment of.

Summary of the invention

The present invention is in order to overcome above-mentioned the deficiencies in the prior art, a kind of evaluation method of the GRA-TOPSIS model based on expert's preference has been proposed, the effective nonlinear relationship between reflected appraisal index, and simplify calculated amount, and can accurately reflect the objective factor of weighing factor, thereby improve the evaluation precision of TOPSIS evaluation model.

For solving the weak point of existing issue, the present invention adopts following technical scheme to be:

The feature of the evaluation method of a kind of GRA-TOPSIS model based on expert's preference of the present invention is to carry out as follows:

Step 1, calculating grey incidence coefficient ζ _i(k):

1.1, ordered series of numbers set X={x is compared in definition _i,k| i=1,2 ..., m; K=1,2 ..., n}; M represents the sum of evaluation object, and n represents the sum of evaluation index; x _i,kk the evaluation index that represents i evaluation object;

1.2, utilize formula (1) to described evaluation index x _i,kcarry out standardization and obtain achievement data x _i,k':

1.3, utilize formula (2) to obtain the utility function η based on expert's preference _i,k:

${\mathrm{\η}}_{i,k}=\left\{\begin{array}{cc}{x_{i,k}}^{\′}& {x_{i,k}}^{\′}>x^{\exp }\\ {x_{i,k}}^{\′}{({x_{i,k}}^{\′}-x^{\exp }+1)}^{\mathrm{\δ}}& {x_{i,k}}^{\′}\≤x^{\exp }\end{array}\right.---\left(2\right)$

In formula (2), δ represents described achievement data x _i,k' preference extent index; δ value is to be greater than 1 integer; x ^expexpression is to described achievement data x _i,k' expectation value; x ^exp∈ [0,1], η _i,k∈ [0,1];

1.4, utilize the represented analytical hierarchy process of formula (3) to described utility function η _i,kbe weighted, obtain weighting matrix t _i,k:

t _i,k＝(ω _k×η _i,k) _m×n (3)

In formula (3), ω _krepresent described achievement data x _i,k' weight; ω _k∈ [0,1];

1.5, definition reference sequence set t ₀={ t _{0, k}| k=1,2 ..., n}, t _{0, k}be illustrated in described weighting matrix t _i,kin the maximal value of k row element;

1.6, utilize formula (4) to obtain the grey incidence coefficient ζ of i evaluation object _i(k):

${\mathrm{\ζ}}_i\left(k\right)=\frac{\underset{i}{\min }\underset{k}{\min }|t_{0,k}-t_{i,k}|+\mathrm{\ρ}\underset{i}{\max }\underset{k}{\max }|t_{0,k}-t_{i,k}|}{|t_{0,k}-t_{i,k}|+\mathrm{\ρ}\underset{i}{\max }\underset{k}{\max }|t_{0,k}-t_{i,k}|}---\left(4\right)$

In formula (4), ρ is resolution ratio, ρ ∈ [0,1];

The evaluation of step 2, the TOPSIS model based on grey association coefficient:

2.1, utilize formula (5) and formula (6) to obtain respectively the grey incidence coefficient ζ of described i evaluation object _i(k) positive ideal solution with negative ideal solution

${\mathrm{\ζ}}_0^{+}=\left\{(\underset{1\≤i\≤m}{\max }{\mathrm{\ζ}}_i\left(k\right),k=\mathrm{1,2},...,n)\right\}---\left(5\right)$

${\mathrm{\ζ}}_0^{-}=\left\{(\underset{1\≤i\≤m}{\min }{\mathrm{\ζ}}_i\left(k\right),k=\mathrm{1,2},...,n)\right\}---\left(6\right)$

2.2, utilize formula (7) and formula (8) to obtain respectively the grey incidence coefficient ζ of described i evaluation object _i(k) with described positive ideal solution between distance and the grey incidence coefficient ζ of described i evaluation object _i(k) with described negative ideal solution between distance

$d_i^{+}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{+}\left(k\right){]}^2},(1\≤i\≤m)---\left(7\right)$

$d_i^{-}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{-}\left(k\right){]}^2},(1\≤i\≤m)---\left(8\right)$

2.3, utilize (9) to obtain the approach degree of i evaluation object

$C_i^{*}=\frac{d_i^{+}}{d_i^{-}+d_i^{+}}---\left(9\right)$

With described approach degree evaluation result as described GRA-TOPSIS model based on expert's preference.

Compared with prior art, beneficial effect of the present invention is embodied in:

1, the present invention is by obtaining the grey incidence coefficient based on expert's preference, make the linear relationship of evaluation result between more can embodiment of evaluation index, made up the too defect of rigidity of traditional TOPSIS evaluation model solution, make evaluation result can more reflect objective reality situation, and grey incidence coefficient and existing TOPSIS evaluation model are carried out to effective combination, utilize TOPSIS evaluation model can effectively solve the feature of quantizating index inconsistence problems, by objective factor, be comprehensively an approach degree parameter, thereby improve precision and the accuracy of model evaluation.

2, the present invention composes power by analytical hierarchy process to attribute utility value, overcome prior art and made the processing procedure loaded down with trivial details problem that becomes because of integrated all kinds of tax power methods, analytical hierarchy process is workable, not only simplify the complexity of existing method of weighting, and improved achievement data precision after weighting is processed.

3, the present invention, by structure utility function, has taken into full account the selectivity preference degree of expert to achievement data property value, thereby can more accurately reflect the utility value of achievement data, has overcome the too defect of rigidity of the solution that obtains in prior art.

4, the present invention can be used for engineering, and enterprise operation and management, in the numerous areas such as macro economic analysis, has wide range of applications.For example: the evaluation of engineering project, the evaluation of the passenger of airline satisfaction, service quality, the evaluation of innovative Enterprise Innovation Capability, the evaluation of the Finance Environment in city, etc., for Project Benefit evaluation, enterprise management decision-making, macroeconomy operating analysis decision-making provide quantitative foundation.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the inventive method;

Fig. 2 is utility function figure of the present invention.

Embodiment

In the present embodiment, as shown in Figure 1, a kind of evaluation method of the GRA-TOPSIS model based on expert's preference is to carry out as follows:

Step 1, calculating grey incidence coefficient ζ _i(k):

1.1, ordered series of numbers set X={x is compared in definition _i,k| i=1,2 ..., m; K=1,2 ..., n}; M represents the sum of evaluation object, and n represents the sum of evaluation index; x _i,kk the evaluation index that represents i evaluation object; Compare optional empirical data, production data or the decision data etc. selected of ordered series of numbers;

1.2, utilize formula (1) to evaluation index x _i,kcarry out standardization and obtain achievement data x _i,k':

, may there is no dimension in dissimilar achievement data; The data of different dimensions can not directly be used for comparative evaluation.Therefore, need to take different normalization methods for dissimilar index, eliminate the impact of the different dimensions of different indexs, be convenient to direct comparison and evaluation between each index value.The standardization of achievement data is mainly that achievement data standard is turned to the number being under the jurisdiction of on interval, interval [0,1].

Desired value more the more excellent index of high praise object benefit type index; The more excellent index of the lower evaluation object of desired value is cost type index.The standardized method of article benefit type of the present invention and cost type index.

1.3, utilize formula (2) to obtain the utility function η based on expert's preference _i,k:

${\mathrm{\η}}_{i,k}=\left\{\begin{array}{cc}{x_{i,k}}^{\′}& {x_{i,k}}^{\′}>x^{\exp }\\ {x_{i,k}}^{\′}{({x_{i,k}}^{\′}-x^{\exp }+1)}^{\mathrm{\δ}}& {x_{i,k}}^{\′}\≤x^{\exp }\end{array}\right.---\left(2\right)$

In formula (2), δ represents achievement data x _i,k' preference extent index; δ value is to be greater than 1 integer; x ^expexpression is to achievement data x _i,k' expectation value; x ^exp∈ [0,1]; Utility function η _i,kthe attribute utility value obtaining is estimated in expression to expert's preference; η _i,k∈ [0,1];

As shown in Figure 2, the fluctuation range of utility function is expected x ^expcontrol.Work as x _i,k> x ^exptime, the preference of utility function value is the variation tendency of even linear increment; Work as x _i,k≤ x ^exptime, the preference degree of utility function value is subject to the impact of parameter δ.δ value is less, and the earthquake intensity of utility function is larger; Otherwise δ value is larger, the earthquake intensity of utility function is less.

1.4, utilize the represented analytical hierarchy process of formula (3) to utility function η _i,kbe weighted, obtain weighting matrix t _i,k:

t _i,k＝(ω _k×η _i,k) _m×n (3)

In formula (3), ω _krepresent achievement data x _i,k' weight; ω _k∈ [0,1];

1.5, definition reference sequence set t ₀={ t _{0, k}| k=1,2 ..., n}, t _{0, k}be illustrated in weighting matrix t _i,kin the maximal value of k row element;

1.6, utilize formula (4) to obtain the grey incidence coefficient ζ of i evaluation object _i(k):

${\mathrm{\ζ}}_i\left(k\right)=\frac{\underset{i}{\min }\underset{k}{\min }|t_{0,k}-t_{i,k}|+\mathrm{\ρ}\underset{i}{\max }\underset{k}{\max }|t_{0,k}-t_{i,k}|}{|t_{0,k}-t_{i,k}|+\mathrm{\ρ}\underset{i}{\max }\underset{k}{\max }|t_{0,k}-t_{i,k}|}---\left(4\right)$

In formula (4), ρ is resolution ratio, ρ ∈ [0,1];

The evaluation of step 2, the TOPSIS model based on grey association coefficient:

2.1, utilize formula (5) and formula (6) to obtain respectively the grey incidence coefficient ζ of i evaluation object _i(k) positive ideal solution with negative ideal solution

${\mathrm{\ζ}}_0^{+}=\left\{(\underset{1\≤i\≤m}{\max }{\mathrm{\ζ}}_i\left(k\right),k=\mathrm{1,2},...,n)\right\}---\left(5\right)$

${\mathrm{\ζ}}_0^{-}=\left\{(\underset{1\≤i\≤m}{\min }{\mathrm{\ζ}}_i\left(k\right),k=\mathrm{1,2},...,n)\right\}---\left(6\right)$

2.2, utilize formula (7) and formula (8) to obtain respectively the grey incidence coefficient ζ of i evaluation object _i(k) with positive ideal solution between distance and the grey incidence coefficient ζ of i evaluation object _i(k) with negative ideal solution between distance

$d_i^{+}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{+}\left(k\right){]}^2},(1\≤i\≤m)---\left(7\right)$

$d_i^{-}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{-}\left(k\right){]}^2},(1\≤i\≤m)---\left(8\right)$

2.3, utilize (9) to obtain the approach degree of i evaluation object

$C_i^{*}=\frac{d_i^{+}}{d_i^{-}+d_i^{+}}---\left(9\right)$

With approach degree evaluation result as the GRA-TOPSIS model based on expert's preference.

Below by citing an actual example to be further explained explanation:

Take Anhui Province's electronic information, equipment is manufactured and the innovative enterprise of i=12 family of three industries of new material is research sample, evaluates the innovation ability of 12Jia enterprise by the present invention.

Set k=18 evaluation index x _i,kbe respectively: entrepreneur's creativity consciousness level, employee's creativity consciousness level, R & D funds, R & D personnel, R & D science and technology mechanism, raise outside research funding, external scientific research cooperative Input Level, innovative information acquisition capacity, patented technology output level, industry standard formulation, new product (service) income accounts for the ratio of sales revenue, new product (service) income increase rate, overall labour productivity, staffs training level, intellectual property right management level, R & D personnel satisfaction, business growth index, Green develop index.

Step 1, calculating grey incidence coefficient ζ _i(k):

1.1, by table 1 and table 2, determine relatively ordered series of numbers set X:

Table 1 index x _{i, 1}-x _{i, 8}compare ordered series of numbers

Table 2 index x _{i, 9}-x _{i, 18}compare ordered series of numbers

1.2, utilize formula (1) to evaluation index x _i,kcarry out standardization and obtain x _i,k':

1.3 calculate the utility value of evaluation index

Utilize formula (2) to calculate 12 innovative enterprises at the utility value of 18 indexs, as shown in Table 3 and Table 4, wherein, expert selects the expectation x of preference to the structure of acquisition ^expbe set to respectively 0.7 and 2 with parameter δ.

Table 3 index x _{i, 1}-x _{i, 8}utility value matrix

Table 4 index x _{i, 9}-x _{i, 18}utility value matrix

1.4, utilize the represented analytical hierarchy process of formula (3) to utility function η _i,kbe weighted, thus the weights omega of 18 evaluation indexes of acquisition _k:

ω _k＝(0.1893，0.1519，0.1333，0.1034，0.0767，0.0556，0.0541，0.0737，0.1465，0.1171，0.1348，0.0972，0.0894，0.0499，0.0512，0.0554，0.0559，0.0399)

1.5, determine reference sequence t ₀:

t ₀＝(0.1120,0.0899,0.0789,0.0612,0.0454,0.0329,0.0320,0.0436,0.0867,0.0693,0.0884,0.0575,0.0529,0.0295,0.0303,0.0328,0.0331,0.0236)

1.6, utilize formula (4) to obtain the grey incidence coefficient ζ of i evaluation object _i, then calculate the grey incidence coefficient of weighting (k).Its result is as shown in table 5 and table 6:

Table 5 index x _i1-x _i8grey incidence coefficient

Table 6 index x _i9-x _i18grey incidence coefficient

The evaluation of step 2, the TOPSIS model based on grey association coefficient:

Utilize formula (5) and formula (6) to obtain respectively the grey incidence coefficient ζ of i evaluation object _i(k) positive ideal solution with negative ideal solution and calculate respectively the distance of positive ideal solution with the distance to negative ideal solution

$\begin{array}{c}{d}_i^{+}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{+}\left(k\right){]}^2}=(\mathrm{1.8399,1.3153,1.8898,1.8539,1.6895,1.8319,1.4326},\\ \mathrm{1.6119,1.8761,1.6777,1.5822,1.8177})\end{array}$

$\begin{array}{c}{d}_i^{-}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{-}\left(k\right){]}^2}=(\mathrm{0.4540,1.3582,0.2826,0.4236,0.9038,0.5728,1.2227},\\ \mathrm{0.9495,0.2673,0.8175,1.1903,0.6119})\end{array}$

On this basis, by formula (9), calculated the relative approach degree of 12 innovative Enterprise Innovation Capability indexs:

$\begin{array}{c}{C}_i^{*}=(\mathrm{0.8021,0.4920,0.8699,0.8140,0.6515,0.7618,0.5395,0.6293,0.8753,0.6724},\\ \mathrm{0.5707,0.7482})\end{array}$

be worth littlely, represent that corresponding evaluation object is more excellent, represent that herein innovation ability is stronger.Press value ascending (innovation ability of corresponding evaluation object by by force to weak) is followed successively by this Numerical examples explanation, by building reasonable, perfect index rating system, uses the weighting GRA-TOPSIS model based on expert's preference, can carry out scientific and effective evaluation to the innovation ability of innovative enterprise.

Claims (1)

1. an evaluation method for the GRA-TOPSIS model based on expert's preference, is characterized in that carrying out as follows:

Step 1, calculating grey incidence coefficient ζ _i(k):

1.1, ordered series of numbers set X={x is compared in definition _i,k| i=1,2 ..., m; K=1,2 ..., n}; M represents the sum of evaluation object, and n represents the sum of evaluation index; x _i,kk the evaluation index that represents i evaluation object;

1.2, utilize formula (1) to described evaluation index x _i,kcarry out standardization and obtain achievement data x _i,k':

1.3, utilize formula (2) to obtain the utility function η based on expert's preference _i,k:

${\mathrm{\η}}_{i,k}=\left\{\begin{array}{cc}{x_{i,k}}^{\′}& {x_{i,k}}^{\′}>x^{\exp }\\ {x_{i,k}}^{\′}{({x_{i,k}}^{\′}-x^{\exp }+1)}^{\mathrm{\δ}}& {x_{i,k}}^{\′}\≤x^{\exp }\end{array}\right.---\left(2\right)$

In formula (2), δ represents described achievement data x _i,k' preference extent index; δ value is to be greater than 1 integer; x ^expexpression is to described achievement data x _i,k' expectation value; x ^exp∈ [0,1], η _i,k∈ [0,1];

1.4, utilize the represented analytical hierarchy process of formula (3) to described utility function η _i,kbe weighted, obtain weighting matrix t _i,k:

t _i,k＝(ω _k×η _i,k) _m×n (3)

In formula (3), ω _krepresent described achievement data x _i,k' weight; ω _k∈ [0,1];

1.5, definition reference sequence set t ₀={ t _{0, k}| k=1,2 ..., n}, t _{0, k}be illustrated in described weighting matrix t _i,kin the maximal value of k row element;

1.6, utilize formula (4) to obtain the grey incidence coefficient ζ of i evaluation object _i(k):

${\mathrm{\ζ}}_i\left(k\right)=\frac{\underset{i}{\min }\underset{k}{\min }|t_{0,k}-t_{i,k}|+\mathrm{\ρ}\underset{i}{\max }\underset{k}{\max }|t_{0,k}-t_{i,k}|}{|t_{0,k}-t_{i,k}|+\mathrm{\ρ}\underset{i}{\max }\underset{k}{\max }|t_{0,k}-t_{i,k}|}---\left(4\right)$

In formula (4), ρ is resolution ratio, ρ ∈ [0,1];

The evaluation of step 2, the TOPSIS model based on grey association coefficient:

2.1, utilize formula (5) and formula (6) to obtain respectively the grey incidence coefficient ζ of described i evaluation object _i(k) positive ideal solution with negative ideal solution

${\mathrm{\ζ}}_0^{+}=\left\{(\underset{1\≤i\≤m}{\max }{\mathrm{\ζ}}_i\left(k\right),k=\mathrm{1,2},...,n)\right\}---\left(5\right)$

${\mathrm{\ζ}}_0^{-}=\left\{(\underset{1\≤i\≤m}{\min }{\mathrm{\ζ}}_i\left(k\right),k=\mathrm{1,2},...,n)\right\}---\left(6\right)$

2.2, utilize formula (7) and formula (8) to obtain respectively the grey incidence coefficient ζ of described i evaluation object _i(k) with described positive ideal solution between distance and the grey incidence coefficient ζ of described i evaluation object _i(k) with described negative ideal solution between distance

$d_i^{+}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{+}\left(k\right){]}^2},(1\≤i\≤m)---\left(7\right)$

$d_i^{-}=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\Σ}}}[{\mathrm{\ζ}}_i\left(k\right)-{\mathrm{\ζ}}_i^{-}\left(k\right){]}^2},(1\≤i\≤m)---\left(8\right)$

2.3, utilize (9) to obtain the approach degree of i evaluation object

$C_i^{*}=\frac{d_i^{+}}{d_i^{-}+d_i^{+}}---\left(9\right)$

With described approach degree evaluation result as described GRA-TOPSIS model based on expert's preference.