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A new model is introduced in the process of evaluating efficiency value of decision making units (DMUs) through data envelopment analysis (DEA) method. Two virtual DMUs called ideal point DMU and anti-ideal point DMU are combined to form a comprehensive model based on the DEA method. The ideal point DMU is taking self-assessment system according to efficiency concept. The anti-ideal point DMU is taking other-assessment system according to fairness concept. The two distinctive ideal point models are introduced to the DEA method and combined through using variance ration. From the new model, a reasonable result can be obtained. Numerical examples are provided to illustrate the new constructed model and certify the rationality of the constructed model through relevant analysis with the traditional DEA model.

Data envelopment analysis (DEA) is an effective nonparametric statistical method for processing evaluation problems of multiple inputs and outputs. The first DEA model, CCR model, was created in 1978, which has been widely used over these years. People use the DEA method that has been innovated and improved based on its original model to evaluate decision making units (DMU). The formal point is using self-assessment system and getting the best principle which benefits itself to get the efficiency. It is found that many problems occurred in the process of researching the CCR model [

However, the cross efficiency is still not perfect. Scholars have focused on two aspects: one is the selection of competition and cooperation; the other is the aggregation of cross efficiency matrix. Paths for the former one are as follows: (a) Take the original logical relationship system as reference such as benevolent type, aggressive type, and neutral type. (b) Build a new reference system, such as new ideal point. The latter one is aggregated with multiattribute evaluation methods. In practice, different methods should be taken into consideration.

Wu et al. (2009) proposed a neutral cross efficiency model to analyze the relation between decision making units from a neutral perspective. In their neutral model, the efficiency was obtained by maximizing the ratio of the input indexes between one DMU and the combination of other DMUs [

In China, scholars also perform the DEA cross efficiency recently. Wu and Liang (2006) analyzed the defects when using the final average cross efficiency to evaluate DMUs. They improved the cross efficiency method with the final cross efficiency weight coefficient by using the cooperative games theory and the coalition games theory [

After summarizing the study above, this paper uses ideal points and multiattribute evaluations, combining ideal and anti-ideal points as an external reference system in the other-assessment stage to play a supporting role for the efficiency evaluation of all DMUs. The ideal point is based on efficiency and the anti-ideal point is based on fairness. The self-assessment is called upper level system and the model with two ideal points which is the core of the other-assessment is called second-level system. Crucially, the efficiency is still taken as the core of the combination of the DEA and two ideal point models. To achieve the sense of the other-assessment, efficiency and fairness should be focused on the second-level evaluations simultaneously, and we need to aggregate the second-level reference units based on positive ideal and negative anti-ideal points.

The main purpose of this paper is introducing a model to evaluate the DEA efficiency of DMUs. In the new model we tend to suggest the ideal and anti-ideal points. We will propose two point view models, combine them into a new synthesis model, and then calculate the examples by using the new synthesis model of efficiency cross. The rest of this paper is organized as follows. Section

DEA is an efficient nonparametric evaluation method for processing evaluation problems of multiple inputs and outputs. After the construction of the first DEA model, CCR model, the DEA method is verified to be an effective efficiency evaluation method.

Assume that there are

CCR model, as a basic model, can be used to get an optimal target value, but the nonunique optimal target weight will lead to the nonunique model efficiency value. To cover this shortage, there are aggressive and benevolent types of cross efficiency models. The model can be expressed as follows:

The benevolent cross efficiency model is to achieve the maximum objective function; that is,

The aggressive cross efficiency model is to achieve the minimum objective function; that is,

Aggressive and benevolent efficiency models have the same constraints (model (

DEA cross efficiency is a way to evaluate self and others. It introduces an external reference system in order to prevent DMUs from excessively paying attention to its own advantages and avoiding disadvantages. The commonly used external reference system is the ideal point method. Chinglai and Kwangsun first put forward the idea of Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), which is the basis of the ideal point model. The center part of the ideal point model is that the best choice from all units is the one which is closer to the ideal point and further from the anti-ideal point [

There are many scholars studying the DEA cross efficiency together with the ideal and anti-ideal points; in addition, they get different method to combine the cross efficiency model and ideal and anti-ideal points models. One of the combination approaches is firstly using cross efficiency model to get the calculation results and then combining the results with ideal and anti-ideal points models, which is just taking the result of ideal and anti-ideal points models, respectively, put forward by Wang and Chin [

When performing cross efficiency analysis for different decision making units based on the principle of ideal and anti-ideal points, we invent another new DMU to revise its input and output with some characters, and with the different principle we get model (

Therein,

Therein,

Through analyzing similarities and differences of aggressive and benevolent cross efficiency models and according to model (

According to efficiency concept, the efficiency of ideal point can be defined as

According to this model, we can get an evaluation matrix:

According to fairness concept, the efficiency of anti-ideal point can be defined as

We can get another evaluation matrix:

The above-mentioned

The ideal point model discussed in this text pursues the maximum output under the minimum investment, which is achieved by improving the aggressive type based on the core of efficiency. The anti-ideal point model is created for companies to avoid pursuing excessive outputs by considering the improved benevolent type on a fair and equitable basis and for restraining them with the maximum input index and the minimum output index. A conventional assessment method is to separate them apart. In this paper, we combine them together, considering both efficiency and fairness to achieve an efficient assessment which does not violate the efficient assessment orientation of the DEA method.

There are many ways to aggregate date. Calculation of the result of the positive ideal and negative anti-ideal points and the general efficient calculation are mentioned in this article. In the general calculation process, there are two methods: weighted arithmetic mean and arithmetic–geometric mean. The notion of weighted mean refers to the fact that the observed value can have a strong impact on the evaluation result; the larger the value is, the deeper the impact will be. And it possesses the complementary or typical values of a set of numbers by using the product of their values. Geometric mean is a tape of mean or average, which indicated the equilibrium tendency of the whole calculation. With a minimized index value, the mean value of the system may equal zero. Features of these two kinds of mean are consistent with the start point and features of positive ideal and negative anti-ideal point. Functionality and equilibrium are the two features of a system operation. By combining these two different ways of calculations with the model calculation, an assessment closer to reality can be easily accepted by people. In summary, an overall efficiency value calculation model can be obtained:

According to the above-mentioned model, we can obtain the optimal efficiency evaluation vector:

The evaluation recognition can be enhanced by using the ration of each variance and the sum of the variances which determines the weight, so we have

From analyzing the problems of the traditional CCR model and researching into the ideal point model, we can achieve the assumption of the new improved model by combining cross efficiency value under the condition of ideal point with that of anti-ideal point.

We now illustrate the applications of the proposed DEA models and ideal and anti-ideal points model using two numerical examples. One is a simple DEA efficiency-rating example, and the other is a complicated performance rating case with China’s metal manufacturing industry.

Data for seven DMUs with three inputs and three outputs.

DMU | | | | | | |
---|---|---|---|---|---|---|

| 12 | 400 | 20 | 60 | 35 | 17 |

| 19 | 750 | 70 | 139 | 41 | 40 |

| 42 | 1500 | 70 | 225 | 68 | 75 |

| 15 | 600 | 100 | 90 | 12 | 17 |

| 45 | 2000 | 250 | 253 | 145 | 130 |

| 18 | 730 | 50 | 132 | 45 | 45 |

| 41 | 2350 | 600 | 305 | 159 | 97 |

As can be seen from the rating results of Table _{1} through DMU_{3} and DMU_{5} through DMU_{7} as DEA efficient units, which means they perform equally well. However, it cannot discriminate among them any further. In order to rank the six DEA efficient units, now, we use the proposed DEA models to reevaluate these seven DMUs. We calculate the ranking of the respective models under the ideal and anti-ideal points and the comprehensive ranking. The resulting efficiency rating and the ranking are presented in Table

Efficiency rating and the proposed model values for the seven DMUs.

DMU | CCR | Benevolent | Aggressive | Ideal and anti-ideal points | ||||
---|---|---|---|---|---|---|---|---|

Efficiency | Rank | Efficiency | Rank | Efficiency | Rank | Efficiency | Rank | |

| 1 | 1 | 0.9123 | 3 | 0.7457 | 3 | 0.7473 | 4 |

| 1 | 1 | 0.9182 | 2 | 0.7834 | 2 | 0.8399 | 2 |

| 1 | 1 | 0.7793 | 6 | 0.6983 | 5 | 0.7210 | 5 |

| 0.8197 | 7 | 0.5778 | 7 | 0.5212 | 7 | 0.5769 | 7 |

| 1 | 1 | 0.8895 | 5 | 0.6918 | 6 | 0.7157 | 6 |

| 1 | 1 | 1 | 1 | 0.8661 | 1 | 0.9095 | 1 |

| 1 | 1 | 0.8901 | 4 | 0.7360 | 4 | 0.7666 | 3 |

It is clear from Table _{1} through DMU_{3} and DMU_{5} through DMU_{7} to be not completely the same. In addition, we take the ration of each variance and the sum of the variances getting the comprehensive efficiency. Based on the approach, the following ranking order can be obtained.

DMU_{6} _{2} _{7} _{1} _{3} _{5} _{4}, where the symbol “_{5} _{3}, however, the DEA model based on anti-ideal point assesses that DMU_{3} _{5}. The proposed model is considered both efficient (ideal point) and fair (anti-ideal point); the final ranking is DMU_{3} _{5}. All of the three DEA models, which are benevolent DEA model, aggressive DEA model, and ideal and anti-ideal points DEA model, agree that DMU_{6} is the best DMU and DMU_{4} is the worst DMU.

Through Table

DEA efficiency correlation test.

Spearman’s rho | Benevolent | Aggressive | Ideal & anti-ideal points |
---|---|---|---|

Benevolent | |||

Correlation | 1.000 | .964 | .929 |

Sig. (bilateral) | .000 | .003 | |

Aggressive | |||

Correlation | .964 | 1.000 | .964 |

Sig. (bilateral) | .000 | ||

Ideal & anti-ideal points | |||

Correlation | .929 | .964 | 1.000 |

Sig. (bilateral) | .003 | .000 |

Data for twenty-nine DMUs with three inputs and three outputs.

DMU | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

| 1.0000 | 0.8615 | 0.3778 | 0.6999 | 0.5401 | 0.1636 | 0.3994 | 0.5530 | 0.1115 |

| 0.1325 | 0.9033 | 0.4627 | 0.1934 | 0.4929 | 0.3918 | 0.4147 | 0.1805 | 0.1232 |

| 0.5260 | 0.8275 | 0.4582 | 1.0000 | 0.3989 | 0.1566 | 0.6249 | 0.1963 | 0.1138 |

| 0.1199 | 0.7958 | 0.6509 | 0.1992 | 0.3455 | 0.2025 | 0.3143 | 0.1000 | 0.1147 |

| 0.1653 | 0.5995 | 0.5093 | 0.2311 | 0.3936 | 0.1955 | 0.3140 | 0.1558 | 0.1240 |

| 0.1118 | 0.2704 | 0.1000 | 0.1348 | 0.4850 | 0.2541 | 0.2797 | 0.2383 | 0.2412 |

| 0.1574 | 0.4337 | 0.2593 | 0.1287 | 0.2352 | 0.3100 | 1.0000 | 1.0000 | 0.1832 |

| 0.1103 | 0.5008 | 0.3008 | 0.1525 | 0.5701 | 0.2599 | 0.6326 | 0.1614 | 0.1656 |

| 0.1185 | 0.4528 | 0.6124 | 0.2038 | 0.4761 | 0.2053 | 0.3702 | 0.1292 | 0.1595 |

| 0.1023 | 0.3989 | 0.1882 | 0.1408 | 0.6753 | 0.2155 | 0.6769 | 0.2640 | 0.1675 |

| 0.1256 | 0.4072 | 0.3392 | 0.1645 | 0.4029 | 0.2114 | 0.4979 | 0.1870 | 0.1523 |

| 0.1276 | 0.5038 | 0.7720 | 0.2113 | 0.4390 | 0.1738 | 0.2313 | 0.1099 | 0.1267 |

| 0.1022 | 0.2820 | 0.3069 | 0.1507 | 0.7561 | 0.1564 | 0.4748 | 0.2319 | 0.2255 |

| 0.1065 | 0.4393 | 0.4164 | 0.1575 | 0.6612 | 0.2986 | 0.3056 | 0.1082 | 0.1515 |

| 0.1269 | 0.6168 | 0.2408 | 0.1440 | 0.4388 | 0.6869 | 0.2748 | 0.3282 | 0.1115 |

| 0.1533 | 0.7862 | 0.4799 | 0.2251 | 0.4298 | 0.3047 | 0.4377 | 0.1163 | 0.1175 |

| 0.1179 | 0.5188 | 0.2765 | 0.1484 | 0.4556 | 0.2488 | 0.5382 | 0.3218 | 0.1702 |

| 0.1179 | 0.3204 | 0.3163 | 0.1583 | 0.4300 | 0.1782 | 0.6388 | 0.2062 | 0.1769 |

| 0.1415 | 0.3499 | 0.1166 | 0.1000 | 0.2643 | 0.2928 | 0.3847 | 0.3925 | 0.1831 |

| 0.1257 | 0.5399 | 0.5380 | 0.1764 | 0.4225 | 0.5014 | 0.2477 | 0.2126 | 0.1306 |

| 0.1281 | 0.3000 | 0.2011 | 0.1273 | 0.3879 | 0.1333 | 0.4839 | 0.4246 | 0.1924 |

| 0.1151 | 0.4305 | 0.2607 | 0.1475 | 0.4632 | 1.0000 | 0.2472 | 0.1998 | 0.2056 |

| 0.1066 | 0.3780 | 0.3058 | 0.1502 | 0.6645 | 0.3584 | 0.2691 | 0.1998 | 0.1363 |

| 0.1547 | 0.8324 | 0.6759 | 0.2691 | 0.3882 | 0.1245 | 0.2970 | 0.5145 | 0.1226 |

| 0.1291 | 0.4833 | 0.5727 | 0.1820 | 0.4311 | 0.3336 | 0.2041 | 0.1403 | 0.1357 |

| 0.1244 | 0.6089 | 0.3394 | 0.1611 | 0.4079 | 0.2191 | 0.4133 | 0.2062 | 0.1262 |

| 0.1022 | 0.3801 | 0.2214 | 0.1448 | 0.7779 | 0.2486 | 0.5813 | 0.2191 | 0.1490 |

| 0.1100 | 0.8030 | 0.6977 | 0.2066 | 0.7037 | 0.2658 | 0.6461 | 0.1741 | 0.1161 |

| 0.1835 | 0.7399 | 0.4916 | 0.2834 | 0.1000 | 0.3468 | 0.5062 | 0.7585 | 0.1381 |

The traditional CCR model evaluates 7 of 29 DMUs to be DEA efficient and cannot distinguish them further, so many DMUs are being rated as DEA efficient. The original intention of ranking the DMUs cannot be realized. The DEA models with benevolent and aggressive are chosen to reevaluate the performance of the 29 DMUs, and we also still use the ideal and anti-ideal point model to get the efficiency of the 29 DMUs. The efficiency of the 29 DMUs with four models is shown in Table _{13} has the best overall performance, which is followed by DMU_{7}, DMU_{10}, and DMU_{27}, while DMU_{1} has the worst performance followed by DMU_{3}, DMU_{28}, and DMU_{25}. The rankings of the 29 DMUs with four models are shown in Table

Efficiency rating and rank for the twenty-nine DMUs.

DMU | CCR | Benevolent | Aggressive | Ideal and anti-ideal points | ||||
---|---|---|---|---|---|---|---|---|

Efficiency | Rank | Efficiency | Rank | Efficiency | Rank | Efficiency | Rank | |

| 0.4015 | 26 | 0.1366 | 29 | 0.1212 | 29 | 0.1257 | 29 |

| 0.6486 | 21 | 0.5016 | 21 | 0.4643 | 21 | 0.4544 | 21 |

| 0.3635 | 28 | 0.1746 | 28 | 0.1559 | 28 | 0.1612 | 28 |

| 0.5169 | 24 | 0.3889 | 24 | 0.3630 | 23 | 0.3552 | 24 |

| 0.3912 | 27 | 0.3550 | 26 | 0.3190 | 26 | 0.3312 | 25 |

| 1.0000 | 1 | 0.7808 | 8 | 0.7115 | 6 | 0.7611 | 6 |

| 1.0000 | 1 | 0.9451 | 3 | 0.8517 | 3 | 0.8892 | 2 |

| 0.9120 | 10 | 0.7910 | 7 | 0.7109 | 7 | 0.7130 | 9 |

| 0.6623 | 20 | 0.5507 | 19 | 0.5015 | 19 | 0.5168 | 19 |

| 1.0000 | 1 | 0.9529 | 2 | 0.8543 | 2 | 0.8689 | 3 |

| 0.7085 | 18 | 0.6209 | 18 | 0.5512 | 18 | 0.5711 | 17 |

| 0.4663 | 25 | 0.3973 | 23 | 0.3629 | 24 | 0.3750 | 23 |

| 1.0000 | 1 | 0.9579 | 1 | 0.8709 | 1 | 0.9112 | 1 |

| 0.8048 | 16 | 0.6644 | 14 | 0.5961 | 14 | 0.6131 | 15 |

| 0.9029 | 11 | 0.6236 | 17 | 0.5578 | 17 | 0.5681 | 18 |

| 0.5227 | 23 | 0.4210 | 22 | 0.3791 | 22 | 0.3796 | 22 |

| 0.8377 | 14 | 0.7082 | 10 | 0.6465 | 10 | 0.6576 | 10 |

| 0.9678 | 8 | 0.7951 | 6 | 0.7030 | 8 | 0.7288 | 7 |

| 1.0000 | 1 | 0.6626 | 15 | 0.5928 | 15 | 0.6239 | 14 |

| 0.6769 | 19 | 0.5414 | 20 | 0.4825 | 20 | 0.5030 | 20 |

| 0.9523 | 9 | 0.7460 | 9 | 0.6748 | 9 | 0.7144 | 8 |

| 1.0000 | 1 | 0.8908 | 4 | 0.7706 | 5 | 0.8167 | 5 |

| 0.8716 | 12 | 0.7025 | 11 | 0.6244 | 12 | 0.6532 | 11 |

| 0.8221 | 15 | 0.6811 | 13 | 0.6130 | 13 | 0.6451 | 12 |

| 0.5776 | 22 | 0.3606 | 25 | 0.3421 | 25 | 0.3294 | 26 |

| 0.7363 | 17 | 0.6421 | 16 | 0.5786 | 16 | 0.6037 | 16 |

| 1.0000 | 1 | 0.8566 | 5 | 0.7845 | 4 | 0.8173 | 4 |

| 0.3012 | 29 | 0.2484 | 27 | 0.2274 | 27 | 0.2333 | 27 |

| 0.8457 | 13 | 0.7025 | 11 | 0.6338 | 11 | 0.6441 | 13 |

DEA efficiency correlation test.

Spearmen’s rho | CCR | Benevolent | Aggressive | Ideal and anti-ideal point |
---|---|---|---|---|

CCR | ||||

Correlation | 1.000 | .936 | .940 | .940 |

Sig. (bilateral) | .000 | .000 | .000 | |

Benevolent | ||||

Correlation | .936 | 1.000 | .997 | .994 |

Sig. (bilateral) | .000 | .000 | .000 | |

Aggressive | ||||

Correlation | .940 | .997 | 1.000 | .995 |

Sig. (bilateral) | .000 | .000 | .000 | |

Ideal and anti-ideal point | ||||

Correlation | .940 | .994 | .995 | 1.000 |

Sig. (bilateral) | .000 | .000 | .000 |

According to Table

Combining the benevolent and aggressive efficiency models of DEA cross efficiency with the ideal point method, based on two different aspects of efficiency and fairness, we improve the benevolent cross efficiency model and aggressive cross efficiency model, respectively. By combining these two models, we receive a general efficiency value. The improved method is more scientific and reasonable for evaluating the efficiency value of decision making units. It ensures a balanced development by promoting decision making units from both efficiency and fairness levels instead of blindly developing good projects. The validity and scientific nature of improved models and the improvement of cross efficiency evaluations can also be testified by verifying calculation examples and their relevant results.

There are no conflicts of interest regarding the publication of this manuscript.

This paper is supported by Liaoning Education Department fund item “regional innovation efficiency evaluation and promotion strategy of Liaoning province” (serial no. W2014026), Liaoning Social Planning item “prediction and driving factors of carbon emission in Liaoning province” (serial no. L15BJY035), Shenyang Municipal Science and Technology Bureau item “prediction and driving factors of carbon emission in Shenyang” (serial no. F16-233-5-08), and Liaoning Provincial Financial Research fund item “analysis and countermeasures of dynamic impact on Liaoning based on the comprehensive model of ‘camp changed to increase’” (serial no. 16C003).