CN103377253A - Data envelopment analysis model based on space scanning method - Google Patents

Data envelopment analysis model based on space scanning method Download PDF

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CN103377253A
CN103377253A CN2012101304607A CN201210130460A CN103377253A CN 103377253 A CN103377253 A CN 103377253A CN 2012101304607 A CN2012101304607 A CN 2012101304607A CN 201210130460 A CN201210130460 A CN 201210130460A CN 103377253 A CN103377253 A CN 103377253A
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dea
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成刚
钱振华
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Abstract

A data envelopment analysis (DEA) model based on a space scanning method belongs to a technical efficiency analysis method in the field of management. Compared with a traditional DEA model, the DEA model provides rich and useful analyzing results and effectively solves the problem that efficiency value obtained by the traditional DEA model is not suitable for sorting of decision-making units. The method includes that (1) standardization conversion is conducted on analytical data to meet the requirement that a directional distance function DEA model analysis result unit is invariable; (2) in an Euclidean space, collection of a group of unit vectors is obtained evenly through the space scanning method; (3) by using each direction vector and standardization data set in the vector collection, the directional distance function DEA model is built, and an efficiency value and an average efficiency value of each model are calculated. The DEA model can be used for replacing the traditional DEA model and analyzing the technical efficiency of a decision-making unit in each field.

Description

A kind of DEA Model based on the spacescan method
Technical field
The present invention relates to a kind of DEA based on the spacescan method (DEA) model, overcome the problem that efficiency analysis result that traditional DEA model draws can not sort.Technical field of the present invention is the Analysis of technical efficiency in management field.
Background technology
DEA(Data Envelopment Analysis, DEA) is a kind of nonparametric technique method for analyzing efficiency based on being evaluated between object relatively.This analytical approach is the Charnes by the U.S., and Cooper and Rhodes proposed first in 1978.Because DEA is applied widely, particularly when the situation that the many inputs of analysis, fecund go out, has special advantage, thereby its range of application is expanded rapidly, contained at present the every field such as industry, agricultural, commerce, administration, education, health, physical culture, DEA develops into an important tool of merging mathematics, operational research, management, econometrics and computer science from initial a kind of analytical approach.
Traditional DEA model exists the efficiency analysis result who draws can not be used for the problem of ordering, reason is that traditional DEA model is upper definite a bit as the subpoint that is evaluated decision package (DMU) ahead of the curve, and make efficiency measurement as mark post to being evaluated DMU with this subpoint, the subpoint of different DMU is different, the efficiency value that calculates like this comparability each other is poor, thereby should not be used for ordering.In addition, traditional radially DEA model does not comprise slack variable to inefficient measurement, thereby makes the efficiency value that draws may over-evaluate the technical efficiency that is evaluated DMU.The SBM nonradial DEA model has taken into full account the slack variable of all inputs (outputs) to the measurement of efficiency value, distance is evaluated DMU subpoint farthest as reference on the forward position but it has adopted, this metering system, no matter be from gerentocratic angle, or from evaluated person's angle, all be to owe rational.
Summary of the invention
The object of the present invention is to provide a kind of technical efficiency evaluation method of sorting to being evaluated DMU of can be used for based on the DEA method, be not suitable for problem that DMU is sorted to solve efficiency analysis result that traditional DEA model draws, it is effective or non-effective to be that traditional DEA model can only be told DEA, does not possess the ability that DMU is sorted.
In order to achieve the above object, the present invention proposes the DEA model based on spacescan, the efficiency analysis result that it draws is evaluated DMU to compare with the forward position on all directions, its av eff value is integral body and the comprehensively reflection that is evaluated DMU and efficient forward position relation, and the av eff score of each DMU has comparability each other.The spacescan DEA model that the present invention proposes is a kind of new efficiency analysis technology based on the DEA method.
Step based on the DEA model of spacescan is: at first carry out standardized transformation to satisfy the as a result requirement of unit unchangeability of direction distance function DEA model analysis to analyzing data; Then in Euclidean space, obtain uniformly the set of one group of vector of unit length by the mode of spacescan; Utilize at last each direction vector and standardized data collection in the vector set, set up direction distance function DEA Model and calculate efficiency value and the av eff value of each model.
Embodiment
Embodiment based on the DEA model of spacescan comprises three steps: 1) data normalization; 2) set up the set of spacescan direction vector; 3) efficiency value of computer memory scanning DEA and average efficiency value.
1. data normalization.
The unit unchangeability refers to dimensionless (dimensionless) feature of efficiency measurement, and it is one of condition of DEA efficiency measurement method indispensability.The unit unchangeability of DEA refers to that the unit of efficiency value that its analysis draws and input, output data is irrelevant.If can be before setting up the DEA model, ultimate principle according to DEA, to dropping into and output data is carried out the nondimensionalization conversion, the efficiency measurement result of the DEA model of setting up with the data after the conversion after then must be that the unit that adopts with input, output data has nothing to do.DEA data normalization (data normalization) method:
Suppose to have n DMU, each DMU has the m kind to drop into and q kind output, is evaluated DMU 0Input and output be respectively x I0And y R0, any DMU jInput and output be respectively x IjAnd y Rj, DMU after the standardization jInput and output be respectively With
Figure 299259DEST_PATH_IMAGE002
The standardized transformation method is:
Figure 976228DEST_PATH_IMAGE003
Figure 414162DEST_PATH_IMAGE004
Figure 287309DEST_PATH_IMAGE005
。
In the said method, the use of absolute value makes above-mentioned standardized method can be applied to comprise the index of negative.DEA data normalization essence is to drop into and output data adopts respectively the input that is evaluated DMU0 and output numerical value as measuring unit, can be regarded as the change of input-output data unit.Therefore, every DEA efficiency measurement method with unit unchangeability (comprise radially with non-radially model), after adopting standardized data, the efficiency value that draws remains unchanged.That is to say DEA data normalization method kept with radially with the compatibility of non-radially model.
2. set up the set of spacescan direction vector.
Defective for the existence of DEA distance function, on the basis of DEA data normalization method and the definition of direction distance function model efficiency, this paper has proposed a kind of DEA model based on spacescan, it is in the geometric space of DEA model, make up the set of direction distance function model by setting one group of uniform direction vector in interval, because its algorithm implementation is similar to scanning called after spacescan DEA model in geometric space.
2.1 the definition of spacescan DEA model.
Vector of unit length refers to that length is 1 vector.In Euclidean space, the standardized vector of any non-vanishing vector u (normalized vector) is exactly the vector of unit length that is parallel to u:
Figure 494300DEST_PATH_IMAGE006
。
Because the efficiency measurement of direction distance function model is only relevant with the direction of direction vector, and irrelevant with the length of direction vector.In direction distance function model, when direction vector was replaced with its standardized vector, efficiency value remained unchanged, so has contained all direction vectors of direction distance function model on the collective entity of all vector of unit length.
With all the vector of unit length set expressions in the nonnegative quadrant of Euclidean space, set with take initial point as the center of circle, the nonnegative quadrant inside radius is that 1 hypersphere is corresponding.For example in two dimensional surface, gather Be expressed as take initial point as the center of circle radius quarter turn as 1, in three dimensions, set expression is for taking initial point as 1 1/8 sphere as center of circle radius.
In Euclidean space, can obtain uniformly by the mode of similar scanning the vector of unit length in the set.Evenly so-called, refer to that the analyzing spot that vector of unit length forms evenly distributes on hypersphere.Be designated as direction vector set ê by spacescan from the set of gathering the vector of unit length that obtains.ê
Figure 470663DEST_PATH_IMAGE008
, when sweep spacing (scanning interval) was infinitely small, ê infinitely approached.
[definition] will adopt the sets definition of the direction distance function model of the vector of unit length foundation among the set ê is spacescan model (directional vector scanning model, DVS), is designated as D.
The quantity of the direction vector that uses during the quantity of spacescan model element equals to scan, the quantity of sweep trace namely, how much it depends on the dimension of direction vector and the size of sweep spacing (the interval angle is designated as θ).The scanning dimension is n, and the sweep spacing angle is that the spacescan model of θ is designated as D n, its direction vector set is designated as ê n
Set ê nIn element (vector of unit length ê) be expressed as
ê n ＝ （ê 1, ê 2,…ê p…)
。
Each element in the spacescan model all can be used as an independently direction distance function model.Can be radially regarding element-specific in the spacescan model as with non-radially model.Direction distance function model when radially model is equivalent to each component value of direction vector and all equates, the SBM model then is equivalent to the direction distance function model of efficiency value minimum.
Can obtain effective forward position maximum efficiency value that conventional model is difficult to calculate by the spacescan model.In addition, the av eff value of spacescan model be evaluated DMU respectively from all directions when projection is done in the forward position, obtain the av eff value, with respect to the efficiency value that tradition calculates with single subpoint, preferably DMU is sorted.Because what the spacescan model adopted is the DEA standardized data, so the efficiency value that draws satisfies the requirement of unit unchangeability.
2.2 the implementation method of spacescan DEA model.
2.2.1 the algorithm of two-dimensional space scan model.
The algorithm of implementation space scan model is to scan under the definite condition of dimension and sweep spacing, finding the solution the process of all direction vectors among the direction vector collection ê.First from the simplest two-dimensional space scan model.
Quantitative relation between the angle ρ (representing with radian) between the quantity μ of sweep spacing θ, direction vector, an i direction vector (take horizontal ordinate as starting point) and the horizontal ordinate as:
。
Two-dimensional directional vector set ê 2Element ê with the coordinate figure (v of analyzing spot V 1, v 2) expression:
Figure 141761DEST_PATH_IMAGE012
Figure 169760DEST_PATH_IMAGE013
。
2.2.2 the algorithm of three dimensions scan model.
The three dimensions scan model is complicated than the two-dimensional space scan model, three-dimensional vector set ê 3Be distributed in the three dimensions as take initial point as the center of circle, radius is on 1 1/8 sphere.As the fixing coordinate figure v of three-dimensional vector 1The time (0≤v 1≤ 1), determines coordinate figure v 2And v 3Numerical value just change two-dimensional space scanning problem into.Work as v 1When increasing gradually with fixed intervals since 0, resulting by (v 1, v 2, v 3) the vector of unit length collection that consists of of coordinate figure is three-dimensional vector set ê 3Work as v 1=0 o'clock, by (v 2, v 3) the direction vector collection that consists of namely equals two-dimensional directional vector set ê 2
Three-dimensional vector set ê 3In direction vector evenly distribute at the analyzing spot that the Northern Hemisphere of 0-90 degree forms:
Scan line spacings equates that namely adjacent two latitude lines (sweep trace) equal sweep spacing θ with the angle that the centre of sphere forms.For example sweep spacing (θ) is 22.5 when spending, and forms north latitude 0 degree (equator), 22.5 degree, 45 degree, 67.5 degree and 5 scanning of 90 degree (arctic point) latitude line (point).
Adjacent two analyzing spots on the same sweep trace (latter two point except) also equal sweep spacing θ with the angle that the centre of sphere forms.
Can obtain the three-dimensional vector set by two-layer nested circulation ê 3, its algorithm is expressed as follows:
' the one-level circulation:
For i=0 to
Figure 289026DEST_PATH_IMAGE014
' cycle index is the maximum scan line number on single plane
' the secondary circulation:
For j = 0 to
Figure 644101DEST_PATH_IMAGE014
If
Figure 397162DEST_PATH_IMAGE016
Then
Figure 433251DEST_PATH_IMAGE017
Exit For ' withdraws from the secondary circulation
End If
If
Figure 59404DEST_PATH_IMAGE018
Then
Exit For ' withdraws from the secondary circulation
End If
Figure 118627DEST_PATH_IMAGE019
Figure 488429DEST_PATH_IMAGE020
Next j
Next I ‘。
Equal sweep spacing θ although above algorithm still can't satisfy the angle that any two adjacent analyzing spots on sphere and the centre of sphere form, the analyzing spot that this algorithm generates is approximate evenly to distribute.
2.2.3 the algorithm of n-dimensional space scan model.
N Wei Fangxiangxiangliangji ê nBe distributed in take initial point as the center of circle, radius be 1 1/2 nOn the n n-dimensional sphere n. ê nCan obtain by the nested circulation of n-1 layer, its algorithm is expressed as follows:
' the one-level circulation:
For i=0 to ' cycle index is the maximum scan line on each plane
Figure 130074DEST_PATH_IMAGE015
' the secondary circulation:
For j = 0 to
Figure 738910DEST_PATH_IMAGE014
If
Figure 646823DEST_PATH_IMAGE016
Then
Figure 595187DEST_PATH_IMAGE017
Exit For ' withdraws from the secondary circulation
End If
If
Figure 563143DEST_PATH_IMAGE018
Then
Exit For ' withdraws from the secondary circulation
End If
Figure 659275DEST_PATH_IMAGE021
……
' n-1 level circulation
For k = 0 to
Figure 557830DEST_PATH_IMAGE014
If Then
Figure 827454DEST_PATH_IMAGE023
Exit For ' withdraws from the circulation of n-1 level
End If
If
Figure 82986DEST_PATH_IMAGE024
Then
Exit For ' ' withdraws from the circulation of n-1 level
End If
Figure 598281DEST_PATH_IMAGE025
Next k
……
Next j
Next I ‘。
3. the efficiency value of computer memory scanning DEA and on average efficiency value.
Utilize the direction vector set that obtains in the step 2, each direction vector can be set up a direction distance function DEA model, calculates an efficiency value, then calculates its av eff value.
Direction distance function model is to the radially popularization of DEA model, and its linear programming equation is defined as follows (v and u represent respectively to drop into and the output direction vector, suppose constant returns to scale):
Figure 570010DEST_PATH_IMAGE027
Figure 375155DEST_PATH_IMAGE028
Figure 428562DEST_PATH_IMAGE029
Figure 206025DEST_PATH_IMAGE030
。
Adopt the measuring method of the direction distance function model efficiency value of DEA standardized data:
Figure 686685DEST_PATH_IMAGE031
Figure 244705DEST_PATH_IMAGE027
Figure 23174DEST_PATH_IMAGE032
Figure 983040DEST_PATH_IMAGE033
Figure 634601DEST_PATH_IMAGE030
Or
Figure 352022DEST_PATH_IMAGE034
Figure 747231DEST_PATH_IMAGE032
Figure 72481DEST_PATH_IMAGE030
。
Above-mentioned two formulas are of equal value.In above-mentioned definition, β v and β u represent to drop into the inefficiency degree with output.When the incidence vector v is got the input numerical value that is evaluated DMU, i.e. v=(1,1 ..., 1), output direction vector u got for 0 when vector, and direction distance function model is equivalent to the input orientation DEA model radially that adopts standardized data, efficiency value θ=1-β; When the incidence vector v is got 0 vector, output direction vector u gets the output numerical value that is evaluated DMU, i.e. u=(1,1 ... 1), direction distance function model is equivalent to the output directed radial DEA model that adopts standardized data, efficiency value θ=1/ (1+ β).The definition of above-mentioned direction distance function efficiency value has kept and the compatibility of model definition radially.
The av eff value of spacescan DEA model can adopt arithmetic mean or geometric mean.Arithmetic mean is expressed as
Figure 339514DEST_PATH_IMAGE036
；
Geometric mean is expressed as
Figure 476098DEST_PATH_IMAGE037
。
Wherein n represents the number of element (being direction vector) in the direction vector set, also the quantity of the direction distance function model of expression foundation.

Claims (4)

1. the DEA based on the spacescan method (DEA) model is characterized in that, the steps include: to carry out standardized transformation to analyzing data; In Euclidean space, obtain uniformly the set of one group of vector of unit length by the mode of spacescan; Utilize each direction vector and standardized data collection in the vector set, set up direction distance function DEA model and calculate efficiency value and the av eff value of each model.
2. the DEA model based on the spacescan method as claimed in claim 1, it is characterized in that, carry out standardization to analyzing data, so that the result of the direction distance function DEA model of setting up has possessed the unit unchangeability, the measuring unit that is analysis result and data is irrelevant, the data normalization method is: suppose to have n decision package (DMU), each DMU has the m kind to drop into and q kind output, is evaluated DMU 0Input and output be respectively x I0And y R0, any DMU jInput and output be respectively x IjAnd y Rj, DMU after the standardization jInput and output be respectively
Figure 2012101304607100001DEST_PATH_IMAGE002
With
Figure 2012101304607100001DEST_PATH_IMAGE004
, the standardized transformation method is (" || " expression takes absolute value):
Figure 2012101304607100001DEST_PATH_IMAGE008
Figure 2012101304607100001DEST_PATH_IMAGE010
。
3. the DEA model based on the spacescan method as claimed in claim 1 is characterized in that, in Euclidean space, obtains uniformly the set of one group of vector of unit length by the mode of spacescan; Evenly so-called, refer to that the analyzing spot that vector of unit length forms evenly distributes on the n n-dimensional sphere n.
4. the DEA model based on the spacescan method as claimed in claim 1 is characterized in that, has set up the set of a prescription to distance function DEA model, calculates efficiency value and the av eff value of each model; This not only provides more horn of plenty and useful analysis result, and efficiently solves efficiency value that traditional DEA model draws and be not suitable for problem that decision package is sorted.
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