Liu Yuan College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao , Shandong, China

Transcription

1 doi: / Parameters and Relative Effective Frontier of Data Envelopment Analysis Method with the Instance of Resource Allocation Evaluation Liu Yuan College of Mining and Safety Engineering, Shandong of Science and, Qingdao , Shandong, China Yu Shuwei College of Economics and Management, Shandong of Science and, Qingdao , Shandong, China Abstract Data Envelopment Analysis method (DEA) is a non-parametric operational research method with strong obectivity. By analyzing the forming conditions and the calculation process of C2R model, the paper discusses the effect of parameters of DEA in the construction mechanism of the relative effective production frontier, and structures several DEA relative effective production Frontiers. Key words: Data Envelopment Analysis Method, Relative-Effective-Frontier, Resourse Allocation Evaluation 1. INTRODUCTION Data Envelopment Analysis (DEA) is an operational research method specially used to measure the relative efficiency of a certain number of multiple inputs and output organizations (Charnes, Cooper and Rhodes, 1978). This article analyzes characteristics of DEA by building the model for evaluation of the resource allocation in 1 engineering universities based on DEA (Liu and Yu, 2015). Suggesting that five of the thirteen universities is DEA result effective, Results of model indicate that it is possibly for some universities to take further development since they are in the state of increasing of the resource input s scale economy. The paper analyzes the process of DEA method and observes the weight coefficient as well as other parameters. According to these parameters, the Ineffective Production Frontier of Anhui of Science & are formed and located below the Relative efficient frontier hyper planes. 2. CONSTRUCTION OF EVALUATION MODEL According to DEA, the departments or units with the same goals and tasks, the same external environment, the same input and output contents are considered as a series of Decision Making Units (DMU). Through the calculations on multiple DMUs, the relative advantages and disadvantages of the input-output ratio of a certain DMU with respect of the others (Wang, 2001). In the study, C2R model of DEA was adopted to evaluate the advantages and disadvantages of operational efficiency of 1 universities. The evaluation process is described as below: Of all the evaluated DMUs consisting of several universities, a certain DMU would be ineffective relative to DEA if there is a specific weight coefficient which results in the sum of the proportional input-output ratios of the other DMUs is larger than the input-output ratio of this DMU (Wei, 2004). The comprehensive indicator evaluation was conducted in the study. For example, the comprehensive indicator for the input of the scientific research platform is: x ( n x ) / n (1) sci i1 Suppose the input and output indicators have weight vectors νt and μt, then the input-output ratio of the - th university will be: i T T ( k k i i k1 i1 h uy ) / ( vx ) ( u y ) / ( v x ), 1,2... n (2) Let us build a C2R model with the maximization of the input-output ratio of the 0 university among the universities (1 in total) participating in the evaluation as the goal: 27

2 max h 0 uy vx T 0 T 0 k k 1 i1 k0 uy k k k 1 s. t. 1, 1,2,,1 vx i i i1 u 0, v 0 The inequation is converted into an equation by adding the slack variable S+ or the surplus variable S or via other processing. Then we obtain: min n s.. t x s x 1 n y s y0 1 0, 1, 2, 1 0 uy vx i i0 withoutconstra int, s 0, s 0 In the resource allocation evaluation model of universities, the slack variables of each input and output indicator refer to the adusting variables to make the input or output indicator reach the relative optimum. In the multi-dimensional space of model, the optimal allocation conditions of the universities are presented T T by a series of points ( x, y ). A relatively effective production frontier formed by the effective points with the optimal allocation may be obtained during evaluation. On the relatively effective production frontier of the model, λ makes all the effective points connected with one another and the slack variables make the effective frontier form an envelope along each dimension. As for the other universities without the optimal allocation, θ represents the distance from a DMU to the effective frontier or the envelope.. EVALUATION PROCESS AND RESULTS The study selected the data reflecting the development status of the universities and established an indicator system by constituting comprehensive indicators with direct indicators(liu and Yu, 2015). The comprehensive indicators include three input items, i.e. disciplinary resources, teacher resources and scientific research platform; and three output items, i.e. research output, personnel training and social services(wei, 1989). The value of each comprehensive indicator was obtained by averaging several direct indicators. The data was collected from Educational Statistics Yearbook 2014 and 2015 and websites of the universities. The details about the indicators are given in Table 1. Grade I Indicator Table 1. Indicator System for the Evaluation Model for Resource Allocation of the Universities Grade II Indicator Grade II Indicator (Comprehensive Indicator) Grade III Indicator Code Scientific research platform Input indicators Disciplinary resources Teacher resources Xsci Xsb Xtec Academicians from CAS and CAE Postdoctoral Research Station State Key Disciplines National Experimental Teaching Center Undergraduate Programs Number of doctor stations Number of master stations Total number of staff Number of professors Number of associate professors Grade III Indicator Code x1 x2 x x4 x5 x6 x7 x8 x9 x10 () (4) 274

3 Output indicators Student output Research output Social services Ystd Ysio Ysoc Doctorial tutors Number of teachers with doctoral degree National awards for teachers Scale of in-school students Scale of undergraduates Scale of postgraduates Scale of doctoral candidates Total scale of graduate students Scale of unior college students Scale of foreign students State-level scientific research proects (completed) Provincial scientific research proects (completed) Publishing academic works Publishing academic papers Textbook compiling Patent for invention Scientific research funds (year) Total amount of scientific research funds State-level science and technology awards Provincial science and technology awards Important international cooperation of universities Member units of school board Important school-enterprise cooperation Important institution-locality cooperation x11 x12 x1 y1 y2 y y4 y5 y6 y7 y8 y9 y10 y11 y12 y1 y14 y15 y16 y17 y18 y19 y20 y21 With direct indicators of China of Mining and as the base, those of the other universities were compared with the former first and then averaged to form the comprehensive indicators. In this way, six comprehensive indicators of each university were obtained. Through reasonable modification on some of such indicators, we have the data about the modified comprehensive indictors of the universities. Table 2. Modified Comprehensive Indicators of Universities Input Indicator Output Indicator Xsb Xtec Xsci Ysoc Ysio Ystd China of Mining and (Xuzhou) China of Mining and (Beiing) Xiangtan Chongqing China of Geosciences Liaoning Technical Shandong of Science and Xi'an of Science and Henan Polytechnic Anhui of Science and North China of Science and Hunan of Science and Inner Mongolia of Based on C2R Model, the evaluation results of engineering universities are as shown in Table. 275

4 Table. Evaluation Results of Resource Allocation Conditions of the Universities Based on C2R Model DMU Σλ Scale Efficiency θ Technical Efficiency DEA result China of Mining and No (Xuzhou) Ascending 0.52 Ineffective Ineffective China of Mining and No (Beiing) Ascending 0.82 Ineffective Ineffective Xiangtan No. 1.2 Descending 0.91 Ineffective Ineffective Chongqing No.4 1 Economical 1 Effective Effective China of Geosciences No.5 1 Economical 1 Effective Effective Liaoning Technical No.6 1 Economical 1 Effective Effective Shandong of Science and No Ascending 0.66 Ineffective Ineffective Xi'an of Science and No.8 1 Economical 1 Effective Effective Henan Polytechnic No Ascending 0.60 Ineffective Ineffective Anhui of Science and No Ascending 0.44 Ineffective Ineffective North China of Science and No Ascending 0.85 Ineffective Ineffective Hunan of Science and No.12 1 Economical 1 Effective Effective Inner Mongolia of No Ascending 0.5 Ineffective Ineffective In the evaluation results, the technical efficiency indicator θ and the scale efficiency indicator λ reflect the resource allocation of the evaluated DMU. The technical efficiency evaluated by DEA refers to keeping the output unchanged without reducing any input item, or adding an output item with the input unchanged. The more approximate the value of θ is to 1, the higher of the technical efficiency of the appropriate DMU will be. The value of DMU in term of technical efficiency is 1. The scale efficiency parameter of DEA Model is λ. When the input items of a certain DMU cause changes of the output items in proportion, the scale of this DMU is effective. According to the evaluation results, five of the thirteen universities have reasonable resource allocation (DEA result effective), and the remaining involve uneconomical use of educational resources. Most of the universities are in the state of increasing of the resource input s scale economy, indicating it is possible to obtain the increase of talents and effective output in terms of scientific research by expanding the input scale of educational resources. 4. PARAMETER SIGNIFICANCE OF THE EVALUATION MOSEL The evaluation process and parameter significance of DEA will be analyzed with Anhui of Science and as an example. Within the range of 1 DMU constituted by the involved universities, the values of comprehensive indicators for Anhui of Science and are given in Table 4. Table 4. Values of Comprehensive Indicators for Anhui of Science and Name xsb xtec xsci ysoc ysio ystd Anhui of Science and According to the values of comprehensive indicators for Anhui of Science and, the expression for evaluating its resource allocation by means of DEA was established. The weighted value formed by the input indicators and weight coefficient of the university is: x i (5) And, the weighted value formed by the output indicators and weight coefficient of the university is: y (6) To evaluate the reasonableness of the resource allocation of Anhui of Science and based on DEA, ust refers to the planning and calculation process by using the combination of input and output of other universities to substitute that of Anhui of Science and, within the range of 1 276

5 DMU. The purpose of such process is to find a set of weight coefficient (ν, μ) for input and output, so that θ is minimized. From the evaluation results of the resource allocation of Anhui of Science and by means of C2R model, it can be seen that the presence of the input and output weight coefficient and the proportion coefficient allows that the output indicators of Anhui of Science and may be substituted by a certain proportion of output indicators of Hunan of Science and, Liaoning Technical and Chongqing. Besides, the weighted value is less in the total input. Therefore, DEA is ineffective to Anhui of Science and and its resource allocation may be further optimized. The data about input and output indicators of the three universities are listed in Table 5. Table 5. Data List about Input and Output Comprehensive Indicators of Three Universities Used as Reference in the Evaluation on Anhui of Science and by Means of DEA Name xsb xtec xsci ysoc ysio ystd Chongqing Liaoning Technical Hunan of Science and Through calculations with respect of the model, a series of parameters were obtained, including the DEA efficiency, the proportionality coefficient for the optimization process of involved reference universities and Anhui of Science and, along with the input and output weight coefficients for the resource optimization allocation of Anhui of Science and. According to the input and output combination of DMU of the three universities, substituting the evaluated DMU, the expectation value for the resource optimization allocation of Anhui of Science and may be calculated. The proportions of input and output combinations of the three reference universities obtained by planning and calculation are given below: Table 6. Proportionality Coefficients of Three Reference Universities Involved in the Optimization Process of Anhui of Science and Name Chongqing Liaoning Technical Hunan of Science and λ Through the planning and calculation process, the parameters used to evaluate and optimize the resource allocation of Anhui of Science and were obtained, including the DEA efficiency θ and the input indicator value, as shown below: Table 7. Efficiency Parameters and Input and Output Weight Coefficient for Evaluation on Anhui of Science and by Means of C2R Model DEA Efficiency xsb xtec xsci ysoc ysio ystd θ ν1 ν2 ν μ1 μ2 μ The plus and minus signs are used in the weight coefficients mainly to balance the equation. In the output indicator parameters for Anhui of Science and, obtained by planning and calculation based on its C2R model, the weight coefficients of social service and research outputs were both 0. The optimization values for the indicators with the weight coefficient 0 are still depending on the proportion coefficient and the values of reference DMUs. The expressions of the input indicators for the three reference universities substituting that of Anhui of Science and are: x x x x x x x x x x (7) 277

6 After the values of the input indicators and parameters are taken into the expressions, the input weight value of Chongqing is obtained as: ( ) (8) λ ν x ν x ν x Through calculation, the input weight value of Chongqing is Similarly, the input weight value of Liaoning Technical is , and that of Hunan of Science and is The sum of the input weight values of the three reference universities is , which is namely the value of θ. The input weight value of Anhui of Science and is: (9) The resulting value is 1. The DEA efficiency value θ based on C2R Model is namely , meaning that the sum of the input weight values of the three reference universities as the DMUs is θ times of Anhui of Science and as a DMU. The value of θ indicates the ineffectiveness of the resource allocation. In a two-dimensional graph, θ is the slope of the production frontier function corresponding to the DMU. The weight value of the output indicator of the three reference universities is: y y y y The resulting value is The output weight value of the evaluated DMU, namely Anhui of Science and is: (10) y (11) The two values equal to each other, so the planning conditions hold. According to the planning and calculation results, the expectation value for the input and output indicators of the evaluated DMU may be obtained. The comprehensive indicators of Anhui of Science and and the optimization targets obtained by DEA are listed as below. The expectation values for the resource optimization allocation of Anhui of Science and obtained from C2R Model are given in Table 8. Table 8. Data about Comprehensive Indictor of Anhui of Science and and Expectation Values for Optimization Obtained from C2R Model DEA Parameters xsci xsb xtec ystd ysio ysoc Original data Slack movement Radial movement Expectation value Wherein, the expectation value is the target for the resource optimization allocation of Anhui of Science and obtained from the evaluation based on C2R Model. Compared to the original data, the expectation values of the three input indicators are less. However, the expectation values of the other two indicators except the output indicator of personnel training both exceed the original data, which indicates that the three input resources of Anhui of Science and have the problem of wasting of resources. Besides, the output indicators of Anhui of Science and have the possibility of growth if the input resources are more economical. The data about input and output indicators as well as expectation values for optimization of Anhui of Science and are presented in a radar chart as Figure 1. Figure 1 shows the comparison between the data about comprehensive indicators of Anhui of Science and and the expectation value for optimization obtained from C2R Model. In the hexagon, the upper three groups of data refer to input indicators, and the lower three groups of data refer to output indicators. The dotted line stands for the expectation value of the indicators and the full line stands for the original data. Obviously, the expectation values of the three input indicators are less than the original data, while those of the other two output indicators have been increased except that the personnel training output comprehensive indicator equals to the original data. 278

7 Figure 1. Expectation Values for Optimization of Anhui of Science and 5. RELATIVELY EFFECTIVE FRONTIER OF THE EVALUATION MODEL The evaluation results based on DEA may be restored to the multi-dimensional space, where the dimension is determined by the total number of input and output items. If there is a total of three input and output items, the evaluation results may be restored to Relatively Effective Frontier Envelope Plane in a three-dimensional space. When there are more input and output items, the Relatively Effective Frontier Envelope Plane will be a higherdimension image. More than three input and output items were used in DEA during the study of engineering universities. In the evaluation results of Anhui of Science and, there are four input and output indicators with non-zero weight. To present the Relatively Effective Frontier Envelope Plane obtained from the evaluation based on DEA, the evaluation results of Anhui of Science and by using C 2 R model were plotted in a three-dimensional graph. By removing the indicators, a three-dimensional proection of Relatively Effective Frontier Envelope Plane was obtained. That was not a real Relatively Effective Frontier Envelope Plane, but it could provide the visual impression of the evaluation process based on DEA. After a variable is removed, the function of Relatively Effective Frontier Envelope Plane of Anhui of Science and by using C 2 R Model will be: y x x x x x x x x x The coordinates for the points corresponding to the optimization results of the three reference universities as DMUs with DEA effective and the evaluated Anhui of Science and are respectively: Table 9. Proected Coordinates in Three-dimensional Coordinate System for Effective DMU with DEA DMU x1/x x2/x y/x Chongqing Liaoning Technical Hunan of Science and Optimization points of Anhui of Science and The Ineffective Envelope Plane function for Anhui of Science and as the DMU is: (12) y 2 x2 1 x 1 x x x x x x x 2 1 (1) 279

8 The coordinates corresponding to Anhui of Science and as the DMU is: Table 10. Proected Coordinates in Three-dimensional Coordinate System for Anhui of Science and as the DMU Evaluated DMU x1/x x2/x y/x Anhui of Science and The Relatively Effective Frontier Envelope Plane of Anhui of Science and obtained from the evaluation by using C 2 R Model is as shown below. Figure 2. Relatively Effective Frontier Envelope Plane of Anhui of Science and Obtained from the Evaluation by Using C2R Model Figure. Comparison between Relatively Ineffective Production Frontier and Relatively Effective Frontier Envelope Plane of Anhui of Science and The graph is generated by inputting the data into the software Origin, and it shows the proection of Relatively Effective Frontier Envelope Plane in the three-dimensional coordinate system for Anhui of Science and as the DMU and the other three reference universities based on DEA. To form a three-dimensional graph, the original function was changed to some degree, so the changes of the four variables were presented in a three-dimensional graph. The mathematical meanings of the three coordinate axes are illustrated in the graph below. In the graph, Points A, B and C respectively stand for the locations of Chongqing, Hunan of Science and and Liaoning Technical and Chongqing as DMUs in the three-dimensional coordinate system. The three points are the DMUs with DEA effective, located on the Relatively Effective Frontier Envelope Plane. X' refers to the location which the resource optimization allocation of Anhui of Science and can reach, and is located on the same plane. X is the location of Anhui of Science and as DMU, located below the envelope plane, and the resource allocation is ineffective. The Relatively Ineffective Production Frontier and Relatively Effective Frontier Envelope Plane of Anhui of Science and were further compared, as shown in the figure below. The two planes look parallel to each other due to the angle of composition, but actually, they are not. The lower plane refers to the production frontier of Anhui of Science and represented by the evaluated DMU. According to the udgment by C 2 R Model, a more optimized relatively effective production frontier based on the operation conditions of the other three universities than Relatively Effective Frontier Envelope Plane of Anhui of Science and. Anhui of Science and should adust its operating mode into the resource allocation state corresponding to the plane. In most cases, the multi-dimensional relatively efficient frontier is hard to form since there are many input and output indicators of the evaluated obect. Therefore, it is a simpler and effective method to construct proects in a two-dimensional space. The proection of relatively effective frontier of C 2 R Model is in y = x. Therefore, the effective DMUs with DEA will be presented as a series of points falling on the function y = x in the two-dimensional coordinate system, at a specific weight. As for the ineffective DMUs with DEA, will fall 280

9 below the function of y = x at a specific weight. Each DMU has a specific weight coefficient, so the proection generated during the evaluation of different DMUs is specific as well. The proection of the evaluation results of Anhui of Science and by C 2 R Model in a two-dimensional coordinate system is as follows: Figure 4. Proection of the Evaluation Results of Anhui of Science and by C2R Model According to the weight coefficients ν T and μ T obtained from the evaluation on Anhui of Science and by using C 2 R Model, the proection of the input and output combination of the four universities as well as the relatively efficient frontier envelope in the two-dimensional coordinate system. In the process of Anhui of Science and using C 2 R Model, the indicators of the other three universities were used as the reference. Therefore, there are a total of 8 points. The meaning of each point and line is described as follows: Point A (2.74, 2.74) is the proection for the input and output combination of Chongqing. The proportion coefficient of this participating in the evaluation of Anhui of Science and is , =0.0997, so A' is determined as (0.0997,0.0997). Similarly, B and B' as well as C and C' respectively stands for the information about Hunan of Science and and Liaoning Technical participating in the evaluation and optimization of resource allocation of Anhui of Science and. The six points are all on the relatively effective frontier envelope plane represented by y=x of C 2 R Model. Point X (1, 0.441) and X' (0.441, 0.441) indicate the evaluation and optimization conditions of Anhui of Science and is also the value of θ. The non-optimized points of Anhui of Science and are on the line of y=θx. The output weight value of Point X obtained from the optimization of Anhui of Science and by using C2R Model remained unchanged. The input weight value decreased from 1 to 0.441, falling on the relatively effective frontier envelope plane of y=x. 6. CONCLUSION This article built DEA model to evaluate the resource allocation of 1 typical universities, udged effective and ineffective decision making unit among them. By analyzing the operation process and parameters of the model, this article draws the the relatively ineffective production frontier and the relatively effective frontier of dea model in d space. The relatively effective production frontier, which is a Connection between several effective decision making units, represents the effective running state of obects evaluated by the model. The model found ineffective descion making unit located in the Relatively Ineffective Production Frontier which is below the Relatively effective Production Frontier. The high dimensional graphics formed by DEA model can be proected in the 2 dimensional space. In the 2 dimensional space, the comparsion between the effective and ineffective running state is enhanced, however other information is ignored. 281

10 REFERENCES Charnes A, Cooper W.W, Rhodes E (1978) Measuring the Efficiency of Decision Making units, European Journal of Operational Research, 2(6), pp Liu Yuan and Yu Shuwei (2015) Research on Resource-Allocation of Coal Mining Education Using Data Envelopment Analysis, Journal of Shandong of Science and, 2015(04), pp Wang Jianhong (2001) Empirical Research on Performance Evaluation of Colleges and Universities, China Higher Education, Beiing. Wei Quanling (2004) Data Envelopment Analysis, Science Press, Beiing. Wei Quanling (1989) DEM Method for Evaluation of Relative Effectiveness, China Renmin Press, Beiing. 282