Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 1619798, 13 pages
https://doi.org/10.1155/2017/1619798

Research Article

A Revised Model of the Neutral DEA Model and Its Extension

Peng Liu, Li-Fang Wang, and Jian Chang

School of Management, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Li-Fang Wang; lifang@nwpu.edu.cn

Received 11 February 2017; Accepted 30 March 2017; Published 13 June 2017

Academic Editor: Vladimir Turetsky

Copyright © 2017 Peng Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The neutral data envelopment analysis (DEA) model is an alternative way to determine the weights in DEA cross-efficiency evaluation, while avoiding the difficulty in making a choice between the aggressive and benevolent formulations. However, the weights determined by the neutral model merely make the efficiency of part output bigger than other sets of weights. The neutral model is not able to make the efficiency of each output of the DMU biggest among the favorable weights. This neutral model is not purely “neutral” and not most favorable to the DMU. We proposed a revised model for the neutral model. Based on the idea that the DMU should choose a set of weights to maximize its own efficiency, this paper proposes a new cross-efficiency model. The weights determined by the two models are neutral, neither aggressive nor benevolent.

1. Introduction

Data envelopment analysis proposed by Charnes et al. [1] is a nonparametric method to identify the production frontier and to measure the efficiency of homogenous decision-making units (DMUs) with multiple inputs and multiple outputs. DEA traditional model (CCR and BCC) measures the relative efficiency of DMU relative to its own subpoint on the production frontier. As long as the DMU is on the production frontier, it is the DEA efficient point and the CCR value is equal to 1. Since there is often more than one DMU on the production frontier, more than one DMU is often evaluated as DEA efficient and cannot be discriminated any further. Lacking of the discrimination power becomes the major drawback of the traditional DEA model.

To solve this problem, some scholars have extended the traditional DEA model and proposed some new approaches to improve the discrimination power of traditional DEA model. One such method is DEA super efficiency model [2] that will lead to the efficiency value of efficient DMUs to be larger than 1, which will further distinguish the efficient DMUs. Another method is DEA common-weight evaluation [3]. Unlike the traditional DEA model in which each DMU uses its own favorable weights to measure the efficiency, the DEA common-weight evaluation adopts the common set of weights to measure the efficiency of all DMUs. There are still other methods, such as the multivariate statistics ranking technique [4], the benchmark ranking method [5], the DEA multicriteria decision-making method [6], and the context-dependent DEA technique [7].

DEA cross-efficiency evaluation as an extension to DEA is a commonly used method to improve the discriminative power of the traditional DEA model. The cross-efficiency evaluation utilizes self-evaluation and peer-evaluation to evaluate the efficiency of each DMU. The concept of cross-efficiency evaluation was first proposed by Sexton et al. [8] and was later developed by Doyle and Green [9, 10]. In the cross-efficiency evaluation, each DMU determines a set of weights among all the favorable weights based on some ways. That will lead to sets of weights. sets of weights will lead to efficiency values for each DMU. The final average efficiency value for each DMU will be computed by assembling the efficiency values. The rank of DMUs is determined by the final average efficiency value of DMU. It is believed that the overall efficiency value calculated by cross-efficiency evaluation can guarantee a unique ordering for the DMUs and can be used with few DMUs (e.g., four or five) to produce a unique ranking ordering [10].

Because of the powerful discrimination among DMUs, the cross-efficiency evaluation has been widely applied in efficiency evaluation of nursing homes [8], project ranking and preference voting [11], efficiency evaluation of countries in the Olympic Games (Wu, Liang, and Chen 2009), fixed cost and resource allocation [12], portfolio selection in the Korean stock market [13], supplier selection under uncertainty [14], and so on. However, the nonunique solution to the traditional DEA model (CCR) limits the application of DEA ross-efficiency. How to solve this problem is the focus of the research of DEA cross-efficiency. Using the secondary goal models is the main method to solve this question. Up to now, there are many secondary goal models proposed by many scholars such as aggressive and benevolent models proposed by Doyle and Green [9], the neutral model by Wang (2010), the weight-balanced model by Wu et al. [15], the maximin model by Lim [16], and the satisfaction degree model by Wu et al. [17]. The aggressive and benevolent models proposed by Doyle and Green [9] are two classical ways among the many models. The aggressive (benevolent) model minimizes (maximizes) the average efficiency of other DMUs. The neutral DEA cross-efficiency model suggested by Ying-Ming Wang (2010) offers the new idea to solve this question. Wang considered that the particular DMU should choose a set of weights which is most favorable to itself not considering whether it is favorable or not to other DMUs. The neutral model maximinimizes the efficiency of each output.

To avoid the difficulty in making a choice between the aggressive and benevolent formulations in DEA cross-efficiency evaluation, Wang and Chin [18] put forward a neutral DEA model for the cross-efficiency evaluation, which maximinimizes the relative efficiency of each output. The neutral DEA model was proposed on the idea that “when a DMU is given an opportunity to decide a set of input and output weights, what the DMU is concerned most about is whether the weights can be as favorable as possible to itself.” However, the weights determined by the neutral DEA model can just make the efficiency of part outputs higher. That leads to the neutral DEA model proposed by Wang and Chin [18] being not very consistent with its modeling idea. To address this issue, we propose a revised model. We further propose a new model based on the idea that according to self-interest principle the DMU will select a set of input and output weights among the favorable weights to maximize its own efficiency.

The rest of the paper is organized as follows. Section 2 briefly introduces the cross-efficiency evaluation and its aggressive and benevolent formulations. Section 3 reviews the neutral DEA mode suggested by Wang and Chin [18] and proposes a revised model. Section 4 proposes a new model based on the self-interest principle. Section 5 makes a comparison among these above models through the numerical examples. Section 6 offers the conclusions.

2. The Cross-Efficiency Evaluation

Suppose there are DMUs to be evaluated with inputs and outputs. Denote by and the input and output values of , based on the input and output ratio, whose efficiencies are defined as

where and are, respectively, the input and output weights.

Consider a DMU, say, , , whose DEA CCR-efficiency value can be computed by the following CCR model [1]:

When the value is equal to 1, it means that is DEA efficient. The input and output weights determined by the model are favorable to . Because the weights determined by CCR model are favorable to the , the CCR value has been viewed as the self-evaluation.

By using Charnes and Cooper transformation (Charnes and Cooper, 1962), model (2) can be equivalently transformed into the linear program (LP) below for solution:

Let and be the optimal solution to the above model. Then, is referred to as the CCR-efficiency value of , which is the self-evaluated efficiency of As such, is referred to as a cross-efficiency value of according to the favorable weights of and reflects the peer-evaluation of to .

Model (3) is solved times, each time for each DMU. As a result, there will be sets of input and output weights available for and each DMU will have cross-efficiency values that are peer-evaluation and one CCR-efficiency value that is self-evaluation, which form a cross-efficiency matrix, as shown in Table 1, where are the CCR-efficiencies of the ; that is, . The average efficiency of DMU is measured by the one CCR-efficiency value and cross-efficiency values; namely,

Based on their average cross-efficiencies, the DMUs can usually be fully ranked.

Table 1: Cross-efficiency matrix for DMUs [18].

DMU
Target DMU
Average cross-efficiency

1
2

1

2

It is noticed that model (3) may have multiple optimal solutions. This nonuniqueness of input and output weights would damage the use of cross-efficiency evaluation. To solve this problem, one solution suggested by Sexton et al. [8] is to introduce a secondary goal to gain a unique set of the input and output weights among the multiple optimal solutions determined by the CCR-efficiency model (3). The most commonly used secondary goal functions were suggested by Doyle and Green [9] and are shown below:

Model (5) is commonly called the aggressive formulation for cross-efficiency evaluation which aims to minimize the cross-efficiencies of the other DMUs, whereas model (6) is known as the benevolent cross-efficiency evaluation which maximizes the cross-efficiencies of the other DMUs. Since the two models obtain the input and output weights in two different ways, there is obviously no guarantee that they can lead to the same efficiency ranking or conclusion for the DMUs.

3. The Neutral DEA Model

To avoid the difficulty in making a choice between the aggressive and benevolent formulations, Wang and Chin [18] put forward a neutral DEA model based on the idea that when a DMU is given an opportunity to decide a set of input and output weights, what the DMU is concerned most about is whether the weights can be as favorable as possible to itself. It should not care too much about how to be aggressive or benevolent to the other DMUs. The neutral model is shown below:

where is the efficiency of the th output of . Obviously, the goal of the above model has nothing to do with the other DMUs. It determines the input and output weights just from the viewpoint of itself. The DM has therefore not to make any difficult yet subjective choice between the aggressive and benevolent formulations. Through Charnes and Cooper transformation, model (7) can be converted into the following LP for solution:

where (), , and are decision variables. Like all DEA models for cross-efficiency evaluation, model (8) needs to be solved times, each time for one different DMU. Accordingly, there will be sets of input and output weights available for cross-efficiency evaluation.

Since is fixed, the efficiencies of all the outputs determined by this neutral model are not biggest among the favorable weights. For example, if we want to measure the efficiency of the DMUs which consume two inputs to produce two outputs, we supposed that one of the DMUs such as is the DEA efficiency; that is to say is equal to 1. If there are two sets of weights that are favorable to , the first set of weights leads to the efficiency of and being 1/2 and 1/2 that is computed through the formulation according to model (7) and second set of weights leads to the efficiency of and being 1/3 and 2/3. Because of the first set of weights is 1/2 and of the second set of weights is 1/3, so the DMUA will choose the first set of weights to calculate the cross-efficiency of DMUs but the first set of weights determined by model (7) just ensures that the efficiency of is bigger (1/2 > 1/3, 1/2 < 2/3). Obviously model (7) just can ensure that the efficiencies of all the outputs are close to each other. Model (7) proposed by Wang and Chin [18] is not fully followed by its modeling idea that when a DMU is given an opportunity to decide a set of input and output weights, what the DMU is concerned most about is whether the weights can be as favorable as possible to itself.

Based on the above analysis, we suggest a revised model. The revised model is shown below:

Obviously, model (9) is more consistent with the modeling idea proposed by Wang and Chin.

4. The New DEA Model

According to the weights determined by the second goal to be aggressive or benevolent to the other DMUs, Doyle and Green propose the aggressive and benevolent DEA cross-efficiency model. However, according to the self-interest principle in economics, the DMU tends to choose a set of input and output weights to maximize its own efficiency among the favorable weights. Based upon this point of view, we construct the following DEA model for cross-efficiency evaluation. The model is shown below:

The above model standardized the input and output weights determined by the CCR model. It also means that the DMU chooses a set of weights to maximize its efficiency among the favorable weights where both of the sum of the inputs weights and sum of the outputs weights are all equal to 1. Model (10) recovers the CCR value at the same base that both of the sums of the inputs weights and outputs weights are equal to 1 and then compares the original and absolute values.

5. Numerical Examples

In this section, we provide two numerical examples to illustrate the practical applications of the proposed DEA model in the cross-efficiency evaluation and make a comparison among the mentioned DEA cross-efficiency evaluation models.

Example 1 (efficiency evaluation of seven departments in a university (Wong and Beasley, 1990)). Seven academic departments (DMUs) in a university are evaluated in the case of three inputs and three outputs given below and Table 2 shows their data:

 : number of academic staff : academic staff salaries in thousands of pounds : support staff salaries in thousands of pounds : number of undergraduate students : number of graduate students : number of research papers

Table 2: Data for seven academic departments and their CCR values [18].

Department (DMU)
Inputs
Outputs
CCR-efficiency

1
12
400
20
65
35
17
1

2
19
750
70
139
41
40
1

3
42
1500
70
225
68
75
1

4
15
600
100
90
12
17
0.8197

5
45
2000
250
253
145
130
1

6
19
730
50
132
45
45
1

7
41
2350
600
305
159
97
1

Since the CCR-efficiencies evaluate six of the seven academic departments as efficient and cannot discriminate among them any further, cross-efficiencies are computed. Tables 3 and 4 show the aggressive and benevolent cross-efficiencies results, which are, respectively, obtained through model (5) and model (6). It is very clearly shown that the two different formulations result in the different efficiency rankings for the seven academic departments. In particular, the aggressive cross-efficiencies in Table 4 evaluate DMU₁ as the most efficient academic department, whereas DMU1 is in the third place evaluated by the benevolent cross-efficiencies in Table 5. The two different efficiency rankings to some extent damage the practical application of cross-efficiency evaluation which claims to produce a unique ordering for DMUs.

Table 3: Aggressive cross-efficiency matrix of the seven academic departments in a university [18].

Department (DMU)
Target DMU
Average
Ranking

1
2
3
4
5
6
7

1
1.00
0.85
0.93
0.69
1.00
0.93
0.75
0.8788
1

2
0.33
1.00
0.62
1.00
0.70
0.84
0.56
0.7219
4

3
0.56
0.85
1.00
0.74
0.56
1.00
0.42
0.7301
3

4
0.07
0.76
0.28
0.82
0.24
0.44
0.21
0.4018
7

5
0.33
0.66
0.31
0.76
1.00
0.48
0.83
0.6259
5

6
0.51
1.00
0.82
0.95
0.79
1.00
0.61
0.8126
2

7
0.15
0.60
0.16
0.99
0.98
0.28
1.00
0.5966
6

Table 4: Benevolent cross-efficiency matrix of the seven academic departments in a university [18].

Department (DMU)
Target DMU
Average
Ranking

1
2
3
4
5
6
7

1
1.00
0.92
1.00
0.69
1.00
1.00
1.00
0.9442
3

2
0.98
1.00
0.85
1.00
0.85
0.98
0.98
0.9486
2

3
0.76
0.77
1.00
0.73
0.67
0.77
0.77
0.7827
6

4
0.64
0.70
0.45
0.82
0.41
0.64
0.64
0.6160
7

5
0.94
0.90
0.50
0.77
1.00
0.94
0.94
0.8534
5

6
1.00
1.00
1.00
0.95
0.91
1.00
1.00
0.9801
1

7
1.00
1.00
0.29
1.00
1.00
1.00
1.00
0.8992
4

Table 5: Neutral cross-efficiency matrix of the seven academic departments in a university [18].

Department (DMU)
Target DMU
Average
Ranking

1
2
3
4
5
6
7

1
1.0000
0.92
1.00
0.69
1.00
1.00
0.94
0.9362
2

2
0.93
1.00
0.62
1.00
0.85
0.95
0.97
0.9026
3

3
0.76
0.77
1.00
0.73
0.67
0.75
0.75
0.7763
6

4
0.61
0.70
0.27
0.82
0.41
0.54
0.60
0.5649
7

5
0.84
0.89
0.34
0.76
1.00
0.99
0.96
0.8272
5

6
0.96
1.00
0.83
0.95
0.91
1.00
1.00
0.9493
1

7
0.82
1.00
0.17
1.00
1.00
1.00
1.00
0.8552
4

To avoid the difficulty in making a subjective choice between the two different formulations for cross-efficiency evaluation, Wang and Chin [18] proposed a neutral DEA model (7) to evaluate the seven academic departments. The results are shown in Table 5. Since these efficiencies are neither aggressive nor benevolent, they are more neutral. The average cross-efficiencies in Table 5 clearly show that DMU₁ is in the second place.

We propose a revised model (9) to reevaluate the cross-efficiencies of the seven academic departments. The results are shown in Table 6. The results show that the DMU₆ is the most efficient academic department. Based on the self-interest principle, we suggest model (10). The cross-efficiencies in Table 7 calculated by model (10) also evaluate the DMU₆ as the most efficient academic department. However, the ranks of the DMU₁ and DMU2 and DMU3 and DMU5 are different by the two different formulations.

Table 6: Cross-efficiency matrix of the seven academic departments in a university by model (9).

Department (DMU)
Target DMU
Average
Ranking

1
2
3
4
5
6
7

1
1.0000
0.8330
0.9657
0.6878
0.9032
0.8492
1.0000
0.8913
2

2
0.5814
1.0000
0.8567
1.0000
0.8013
1.0000
0.6826
0.8460
3

3
0.5217
0.8331
1.0000
0.7351
0.7287
0.8397
0.5378
0.7423
5

4
0.1898
0.7991
0.4665
0.8197
0.3636
0.7548
0.2514
0.5207
7

5
0.7337
0.6893
0.4846
0.7646
1.0000
0.6911
0.9589
0.7603
4

6
0.6834
0.9981
0.9998
0.9506
0.9256
1.0000
0.7593
0.9024
1

7
0.5695
0.6773
0.2904
0.9986
0.7252
0.6395
1.0000
0.7001
6

Table 7: Cross-efficiency matrix of the seven academic departments in a university by the new DEA model (10).

Department
(DMU)
Target DMU
Average
Ranking

1
2
3
4
5
6
7

1
1.0000
0.8241
0.9685
0.6877
0.7070
0.9103
0.7136
0.8302
3

2
0.6445
1.0000
0.6286
1.0000
0.7890
0.8629
0.9958
0.8458
2

3
0.6889
0.7942
1.0000
0.7352
0.6606
0.9508
0.7641
0.7991
4

4
0.2508
0.7719
0.2022
0.8197
0.4106
0.4632
0.7400
0.5226
7

5
0.6254
0.7663
0.5668
0.7651
1.0000
0.5303
0.9025
0.7366
5

6
0.7914
0.9855
0.9218
0.9509
0.8751
1.0000
0.9915
0.9309
1

7
0.3702
0.8015
0.2105
1.0000
0.8570
0.3210
1.0000
0.6515
6

Using the FPI to measure the false positiveness of DMUs, the results of the FPI value are shown in Table 8, where the FPI was computed by (Baker and Talluri, 1997)

It is seen that none of the six DEA efficient academic departments is highly false positive. However, the non-DEA efficient academic department DMU₄ shows high false positiveness. This means that there is a significant difference between the self-evaluation and peer-evaluation of DMU₄. Usually, a lower value of the FPI is more preferred. The most efficient DMU is undoubtedly the one with the lowest FPI value, as highlighted in Table 8.

Table 8: False positiveness of the seven academic departments in a university.

Department (DMU)
FPI (%) for cross-efficiency evaluation

Aggressive
Benevolent
Neutral
Revised
New

1
13.79
5.91
6.81
12.2
20.45

2
38.52
5.41
10.79
18.2
18.23

3
36.96
27.76
28.81
34.72
25.14

4
104.01
33.07
45.11
57.42
56.85

5
59.76
17.18
20.88
31.53
35.76

6
23.06
2.03
5.34
10.82
7.42

7
67.62
11.21
16.94
42.84
53.49

Example 2 (efficiency evaluation of 14 international passenger airlines (Tofallis, 1997)). Fourteen major passenger airlines are evaluated in terms of three inputs and two outputs that are defined below:

 : aircraft capacity in ton-kilometres : operating cost : nonflight assets such as reservation system, facilities, and current assets : passenger kilometres : nonpassenger revenue

Table 9 shows the input and output data of the 14 passenger airlines in the year 1990 along with their CCR-efficiencies, which evaluate seven out of the 14 passenger airlines as DEA efficient and cannot distinguish them any further.

Table 9: Data for 14 passenger airlines and CCR values [18].

Airline (DMU)
Inputs
Outputs
CCR-efficiency

1
5273
3239
2003
26,677
697
0.8684

2
5895
4225
4557
3081
539
0.3379

3
24,099
9560
6267
124,055
1266
0.9475

4
13,565
7499
3213
64,734
1563
0.9581

5
5183
1880
783
23,604
513
1.0000

6
19,080
8032
3272
95,011
572
0.9766

7
4603
3457
2360
22,112
969
1.0000

8
12,097
6779
6474
52,363
2001
0.8588

9
6587
3341
3581
26,504
1297
0.9477

10
5654
1878
1916
19,277
972
1.0000

11
12,599
8098
3310
41,925
3398
1.0000

12
5728
2481
2254
27,754
982
1.0000

13
4715
1792
2485
31,332
543
1.0000

14
22,793
9874
4145
122,528
1404
1.0000

Tables 10 and 11 show, respectively, the aggressive and benevolent cross-efficiencies of the 14 passenger airlines. It is found that the two different formulations for cross-efficiency evaluation lead to two different efficiency rankings again. According to the aggressive cross- efficiencies in Table 10, the DMU₅ is the most efficient passenger airline. However, the benevolent cross-efficiencies in Table 11 evaluate DMU₁₁ as the most efficient passenger airline. The inconsistent results in efficiency ranking obviously cause confusion for the DM. This issue causes the choice difficulty for the DM.

Table 10: Aggressive cross-efficiency matrix of 14 passenger airlines [18].

Airline (DMU)
Target DMU
Average cross-efficiency
Rank

1
2
3
4
5
6
7
8
9
10
11
12
13
14

1
0.8684
0.4501
0.6225
0.8684
0.4418
0.4726
0.7679
0.7881
0.7031
0.4158
0.3390
0.7043
0.4711
0.4726
0.5990
12

2
0.1719
0.3379
0.0472
0.1719
0.0224
0.0247
0.2770
0.2724
0.2808
0.2465
0.1152
0.2789
0.0417
0.0247
0.1652
14

3
0.8826
0.1942
0.9475
0.8826
0.6567
0.6898
0.6468
0.6833
0.6225
0.2559
0.1968
0.6261
0.7422
0.6898
0.6226
11

4
0.9581
0.4259
0.7034
0.9581
0.6683
0.6973
0.7629
0.7850
0.6991
0.4027
0.4739
0.7016
0.4937
0.6973
0.6734
7

5
0.9653
0.3658
1.0000
0.9653
1.0000
1.0000
0.7011
0.7359
0.7778
0.5272
0.6382
0.7820
0.7181
1.0000
0.7983
1

6
0.8818
0.1108
0.9563
0.8818
0.9632
0.9766
0.5745
0.6084
0.5099
0.1376
0.1703
0.5141
0.6766
0.9766
0.6385
9

7
0.9211
0.7781
0.4773
0.9211
0.3108
0.3382
1.0000
1.0000
0.8395
0.5416
0.4000
0.8383
0.3658
0.3382
0.6478
8

8
0.7813
0.6114
0.5162
0.7813
0.2683
0.2924
0.8415
0.8588
0.8208
0.5703
0.3011
0.8194
0.4418
0.2924
0.5855
13

9
0.7855
0.7278
0.5076
0.7855
0.2455
0.2677
0.8881
0.9072
0.9477
0.7501
0.3528
0.9452
0.4537
0.2677
0.6309
10

10
0.7821
0.6354
0.6520
0.7821
0.3338
0.3564
0.7650
0.7944
1.0000
1.0000
0.4942
1.0000
0.5871
0.3564
0.6813
6

11
1.0000
1.0000
0.4287
1.0000
0.4202
0.4418
1.0000
1.0000
1.0000
0.8107
1.0000
1.0000
0.2961
0.4418
0.7742
2

12
0.9462
0.6336
0.7500
0.9462
0.4085
0.4395
0.9082
0.9395
0.9998
0.7647
0.4244
1.0000
0.6398
0.4395
0.7314
5

13
1.0000
0.4257
1.0000
1.0000
0.4183
0.4555
0.9511
1.0000
1.0000
0.5855
0.2129
1.0000
1.0000
0.4555
0.7503
3

14
1.0000
0.2277
1.0000
1.0000
0.9806
1.0000
0.6919
0.7275
0.6478
0.2747
0.3300
0.6521
0.7097
1.0000
0.7316
4

Table 11: Benevolent cross-efficiency matrix of 14 passenger airlines [18].

Airline (DMU)
Target DMU
Average cross-efficiency
Rank

1
2
3
4
5
6
7
8
9
10
11
12
13
14

1
0.8684
0.4501
0.6225
0.8684
0.8492
0.4726
0.8108
0.7881
0.7031
0.7512
0.8684
0.7713
0.8684
0.8684
0.7543
12

2
0.1719
0.3379
0.0472
0.1719
0.1735
0.0247
0.2479
0.2724
0.2808
0.2058
0.1719
0.2025
0.1719
0.1719
0.1894
14

3
0.8826
0.1942
0.9475
0.8826
0.8844
0.6898
0.7232
0.6833
0.6225
0.7846
0.8826
0.8072
0.8826
0.8826
0.7678
9

4
0.9581
0.4259
0.7034
0.9581
0.9413
0.6973
0.8228
0.7850
0.6991
0.8113
0.9581
0.8341
0.9581
0.9581
0.8222
6

5
0.9653
0.3658
1.0000
0.9653
1.0000
1.0000
0.7704
0.7359
0.7778
1.0000
0.9653
1.0000
0.9653
0.9653
0.8912
3

6
0.8818
0.1108
0.9563
0.8818
0.8780
0.9766
0.6615
0.6084
0.5099
0.7176
0.8818
0.7478
0.8818
0.8818
0.7554
11

7
0.9211
0.7781
0.4773
0.9211
0.8795
0.3382
1.0000
1.0000
0.8395
0.7808
0.9211
0.8012
0.9211
0.9211
0.8214
7

8
0.7813
0.6114
0.5162
0.7813
0.7703
0.2924
0.8458
0.8588
0.8208
0.7532
0.7813
0.7631
0.7813
0.7813
0.7242
13

9
0.7855
0.7278
0.5076
0.7855
0.7889
0.2677
0.8782
0.9072
0.9477
0.8375
0.7855
0.8369
0.7855
0.7855
0.7590
10

10
0.7821
0.6354
0.6520
0.7821
0.8250
0.3564
0.7780
0.7944
1.0000
1.0000
0.7821
0.9719
0.7821
0.7821
0.7803
8

11
1.0000
1.0000
0.4287
1.0000
1.0000
0.4418
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9193
1

12
0.9462
0.6336
0.7500
0.9462
0.9602
0.4395
0.9362
0.9395
0.9998
1.0000
0.9462
1.0000
0.9462
0.9462
0.8850
4

13
1.0000
0.4257
1.0000
1.0000
1.0000
0.4555
1.0000
1.0000
1.0000
0.9843
1.0000
1.0000
1.0000
1.0000
0.9190
2

14
1.0000
0.2277
1.0000
1.0000
1.0000
1.0000
0.7795
0.7275
0.6478
0.8569
1.0000
0.8838
1.0000
1.0000
0.8659
5

In comparison with the aggressive and benevolent cross-efficiencies in Tables 10 and 11, the neutral cross-efficiencies are presented in Table 12, which are obtained by the neutral DEA model (8). Neutral cross-efficiencies in Table 12 evaluate DMU₁₁ as the most efficient airline.

Table 12: Neutral cross-efficiency matrix of 14 passenger airlines [18].

Airline (DMU)
Target DMU
Average cross-efficiency
Rank

1
2
3
4
5
6
7
8
9
10
11
12
13
14

1
0.8684
0.4501
0.6225
0.8684
0.4907
0.4726
0.7744
0.7881
0.7031
0.6603
0.8639
0.7533
0.7031
0.8492
0.7049
11

2
0.1719
0.3379
0.0472
0.1719
0.1027
0.0247
0.2716
0.2724
0.2808
0.1641
0.1723
0.2054
0.2808
0.1735
0.1912
14

3
0.8826
0.1942
0.9475
0.8826
0.4856
0.6898
0.6606
0.6833
0.6225
0.7894
0.8830
0.7871
0.6225
0.8844
0.7154
10

4
0.9581
0.4259
0.7034
0.9581
0.7093
0.6973
0.7704
0.7850
0.6991
0.7109
0.9542
0.8137
0.6991
0.9413
0.7733
7

5
0.9653
0.3658
1.0000
0.9653
1.0000
1.0000
0.7091
0.7359
0.7778
1.0000
0.9730
1.0000
0.7778
1.0000
0.8764
2

6
0.8818
0.1108
0.9563
0.8818
0.6162
0.9766
0.5895
0.6084
0.5099
0.7125
0.8809
0.7209
0.5099
0.8780
0.7024
12

7
0.9211
0.7781
0.4773
0.9211
0.4738
0.3382
1.0000
1.0000
0.8395
0.6340
0.9110
0.7829
0.8395
0.8795
0.7711
8

8
0.7813
0.6114
0.5162
0.7813
0.3741
0.2924
0.8434
0.8588
0.8208
0.6645
0.7787
0.7542
0.8208
0.7703
0.6906
13

9
0.7855
0.7278
0.5076
0.7855
0.4036
0.2677
0.8865
0.9072
0.9477
0.7504
0.7863
0.8374
0.9477
0.7889
0.7378
9

10
0.7821
0.6354
0.6520
0.7821
0.5587
0.3564
0.7632
0.7944
1.0000
1.0000
0.7918
0.9969
1.0000
0.8250
0.7813
6

11
1.0000
1.0000
0.4287
1.0000
1.0000
0.4418
0.9901
1.0000
1.0000
0.7970
1.0000
1.0000
1.0000
1.0000
0.9041
1

12
0.9462
0.6336
0.7500
0.9462
0.5418
0.4395
0.9115
0.9395
0.9998
0.9399
0.9494
1.0000
0.9998
0.9602
0.8541
4

13
1.0000
0.4257
1.0000
1.0000
0.3800
0.4555
0.9652
1.0000
1.0000
1.0000
1.0000
0.9860
1.0000
1.0000
0.8723
3

14
1.0000
0.2277
1.0000
1.0000
0.7507
1.0000
0.7058
0.7275
0.6478
0.8286
1.0000
0.8598
0.6478
1.0000
0.8140
5

The cross-efficiencies in Table 13 computed by the revised model (9) show that the DMU₁₁ is the most efficient airline. Table 14 obtained by solving the new DEA cross-efficiency model (10) evaluates the DMU₁₃ as the most efficient airline.

Table 13: Cross-efficiency matrix of 14 passenger airlines by model (9).

Airline (DMU)
Target DMU
Average cross-efficiency
Rank

1
2
3
4
5
6
7
8
9
10
11
12
13
14

1
0.8684
0.4501
0.6225
0.8684
0.7705
0.4726
0.7962
0.7881
0.7031
0.7473
0.8307
0.7513
0.8322
0.8534
0.7396
11

2
0.1719
0.3379
0.0472
0.1719
0.1988
0.0247
0.2630
0.2724
0.2808
0.2059
0.1805
0.2075
0.1818
0.1719
0.1940
14

3
0.8826
0.1942
0.9475
0.8826
0.8064
0.6898
0.6972
0.6833
0.6225
0.7796
0.8674
0.7810
0.8628
0.8820
0.7556
10

4
0.9581
0.4259
0.7034
0.9581
0.8375
0.6973
0.7985
0.7850
0.6991
0.8063
0.9154
0.8089
0.9145
0.9464
0.8039
6

5
0.9653
0.3658
1.0000
0.9653
1.0000
1.0000
0.7476
0.7359
0.7778
0.9956
1.0000
0.9906
0.9879
0.9882
0.8943
3

6
0.8818
0.1108
0.9563
0.8818
0.7528
0.9766
0.6273
0.6084
0.5099
0.71156
0.8485
0.7124
0.8431
0.8792
0.7357
12

7
0.9211
0.7781
0.4773
0.9211
0.79858
0.3382
1.0000
1.0000
0.8395
0.7775
0.8599
0.7843
0.8674
0.8896
0.8038
7

8
0.7813
0.6114
0.5162
0.7813
0.7552
0.2924
0.8532
0.8588
0.8208
0.7510
0.7686
0.7562
0.7717
0.7701
0.7206
13

9
0.7855
0.7278
0.5076
0.7855
0.8250
0.2677
0.8954
0.9072
0.9477
0.8361
0.7999
0.8408
0.8012
0.7836
0.7651
9

10
0.7821
0.6354
0.6520
0.7821
0.9564
0.3564
0.78670
0.7944
1.0000
1.0000
0.8558
0.9971
0.8481
0.8077
0.8039
8

11
1.0000
1.0000
0.4287
1.0000
1.0000
0.4418
1.0000
1.0000
1.0000
0.9970
1.0000
0.9990
1.0000
1.0000
0.9190
1

12
0.9462
0.6336
0.7500
0.9462
0.9894
0.4395
0.9370
0.9395
0.9998
0.9974
0.9691
1.0000
0.9664
0.9519
0.8904
4

13
1.0000
0.4257
1.0000
1.0000
0.9850
0.4555
1.0000
1.0000
1.0000
0.9809
1.0000
0.9869
1.0000
0.9940
0.9163
2

14
1.0000
0.2277
1.0000
1.0000
0.8880
1.0000
0.7459
0.7275
0.6478
0.8507
0.9734
0.8512
0.9674
1.0000
0.8485
5

Table 14: Cross-efficiency matrix of 14 passenger airlines by the new DEA model (10).

Airline (DMU)
Target DMU
Average cross-efficiency
Rank

1
2
3
4
5
6
7
8
9
10
11
12
13
14

1
0.8684
0.4501
0.6225
0.8684
0.5914
0.4726
0.7707
0.7881
0.7031
0.6231
0.6892
0.7392
0.6820
0.7414
0.6864
12

2
0.1719
0.3379
0.0472
0.1719
0.0813
0.0247
0.2756
0.2724
0.2808
0.1893
0.1606
0.2245
0.0825
0.0814
0.1716
14

3
0.8826
0.1942
0.9475
0.8826
0.7563
0.6898
0.6520
0.6833
0.6225
0.6755
0.6784
0.7374
0.7973
0.8851
0.7203
9

4
0.9581
0.4259
0.7034
0.9581
0.7371
0.6973
0.7663
0.7850
0.6991
0.6543
0.8022
0.7785
0.7126
0.8440
0.7516
6

5
0.9653
0.3658
1.0000
0.9653
1.0000
1.0000
0.7053
0.7359
0.7778
0.8835
0.8927
0.9258
0.7414
0.9161
0.8482
2

6
0.8818
0.1108
0.9563
0.8818
0.8500
0.9766
0.5798
0.6084
0.5099
0.5807
0.6884
0.6537
0.7710
0.9260
0.7125
10

7
0.9211
0.7781
0.4773
0.9211
0.5061
0.3382
1.0000
1.0000
0.8395
0.6404
0.7235
0.7995
0.6584
0.6759
0.7342
8

8
0.7813
0.6114
0.5162
0.7813
0.4637
0.2924
0.8429
0.8588
0.8208
0.6668
0.6136
0.7731
0.6269
0.6081
0.6612
13

9
0.7855
0.7278
0.5076
0.7855
0.4706
0.2677
0.8890
0.9072
0.9477
0.7732
0.6492
0.8675
0.6022
0.5814
0.6973
11

10
0.7821
0.6354
0.6520
0.7821
0.6102
0.3564
0.7668
0.7944
1.0000
1.0000
0.7373
1.0000
0.5660
0.6085
0.7351
7

11
1.0000
1.0000
0.4287
1.0000
0.6707
0.4418
0.9990
1.0000
1.0000
0.8257
1.0000
1.0000
0.5189
0.6476
0.8237
3

12
0.9462
0.6336
0.7500
0.9462
0.6643
0.4395
0.9111
0.9395
0.9998
0.9193
0.7953
1.0000
0.7498
0.7772
0.8194
4

13
1.0000
0.4257
1.0000
1.0000
0.6654
0.4555
0.9566
1.0000
1.0000
0.9372
0.7171
1.0000
1.0000
0.9166
0.8624
1

14
1.0000
0.2277
1.0000
1.0000
0.9288
1.0000
0.6973
0.7275
0.6478
0.7045
0.8146
0.79241
0.8334
1.0000
0.8124
5

Table 15 shows the values of FPI of the 14 passenger airlines, from which it is seen that none of the seven DEA efficient passenger airlines is highly false positive, but a non-DEA efficient passenger airline DMU₂ reveals the highest false positiveness value. This implies that the self-evaluation of DMU₂ is significantly different from its peer-evaluation. The most efficient passenger airline is the DMU with the lowest FPI value, as highlighted in Table 15.

Table 15: False positiveness of the 14 passenger airlines.

Airline (DMU)
FPI (%) for cross-efficiency evaluation

Aggressive
Benevolent
Neutral
Revised
New

1
44.98
15.12
26.10
17.41
26.52

2
104.52
78.39
111.58
74.18
96.91

3
104.52
23.40
40.38
25.40
31.54

4
42.28
16.53
26.30
19.18
27.47

5
25.26
12.21
17.49
11.82
17.90

6
52.96
29.28
53.16
32.74
37.07

7
54.36
21.74
35.55
24.41
36.20

8
46.67
18.59
29.76
19.18
29.89

9
50.23
24.86
35.18
23.87
35.91

10
46.77
28.16
32.27
24.39
36.04

11
29.16
8.78
14.54
8.81
21.40

12
36.72
13.00
20.40
12.31
22.04

13
33.28
8.82
20.90
9.13
15.96

14
36.69
15.48
29.70
17.86
23.09

6. Conclusions

Cross-efficiency evaluation is an important and popular tool to evaluate and rank the efficiency of the DMUs. The classical DEA cross-efficiency evaluation formulations are the aggressive and benevolent models proposed by the Doyle and Green. However, the two models often cause different ranking of the DMUs. This issue causes a choice difficulty for the DM between the two different formulations. To solve this issue, Wang and Chin [18] propose a neutral model. Through the deep analysis of this neutral model, we suggest a revised model. Based on the self-interest principle, we further propose a new model. The weights are determined by the new model purely from its own point of view without the need to be aggressive or benevolent to the other DMUs. The model is more logical and more in line with code of the conduct of rational DM.

Because the models proposed in this paper are the nonlinear programming models, it is a little difficult to compute relative to linear programming models. This paper determines the importance level of the outputs by the different weights of output computed by the neutral model. There may be some other feasible ways such as the prices of the outputs.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work was supported by Science and Technology Projects of State Grid of China (Grant no. 52110415000Q).

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