 Research
 Open Access
An extended data envelopment analysis for the decisionmaking
 XiaoLi Meng^{1, 2} and
 FuGui Shi^{1, 2}Email author
https://doi.org/10.1186/s1366001715020
© The Author(s) 2017
Received: 22 June 2017
Accepted: 6 September 2017
Published: 2 October 2017
Abstract
Based on the CCR model, we propose an extended data envelopment analysis to evaluate the efficiency of decision making units with historical input and output data. The contributions of the work are threefold. First, the input and output data of the evaluated decision making unit are variable over time, and time series method is used to analyze and predict the data. Second, there are many sample decision making units, which are divided into several ordered sample standards in terms of production strategy, and the constraint condition consists of one of the sample standards. Furthermore, the efficiency is illustrated by considering the efficiency relationship between the evaluated decision making unit and sample decision making units from constraint condition. Third, to reduce the computation complexity, we introduce an algorithm based on the binary search tree in the model to choose the sample standard that has similar behavior with the evaluated decision making unit. Finally, we provide two numerical examples to illustrate the proposed model.
Keywords
 data envelopment analysis
 sample standards
 time series analysis
 binary search tree
 decisionmaking
1 Introduction
In conventional data envelopment analysis (DEA) models, such as CCR model named after Charnes et al. [1] and BCC model proposed by Banker et al. [2], the inputs and outputs are assumed to be precise. In addition, the constraint condition consists of the evaluated decision making units (DMUs).
In practical studies, the input and output data of the evaluated DMUs are frequently variable over multiple time periods (time series data), and it is important to analyze the change of efficiency over time. For example, in the evaluation of travel agencies, transportation, ticket price, accommodation, and labor are always regarded as the inputs, whereas profits and satisfaction of tourists are the outputs. The inputs and outputs are affected by various influential factors, such as the tourism policy, investment of infrastructure, level of starred hotel, annual percapita income, and level of economic development. However, since the influential factors are variable over time, the inputs, outputs, and efficiencies of travel agencies are variable over time accordingly. Given the current upsurge in interest in DEA, it is surprising that the dynamic DEA attracts very little attention. The only methods we know of this area are Malmquist Productivity Index (MPI) and window analysis. MPI was originally proposed by Caves et al. [3] to estimate changes in the overall productivity growth of each DMU over a twoyear period by calculating the efficiency value. To deal with the productivity changes of DMUs over time, Färe et al. [4] constructed a DEAbased MPI by combining the efficiency measurement of Farrell [5] with the productivity measurement of Caves et al. Window analysis, proposed by Charnes et al. [6], is adopted to overcome the constraint of limited DMUs and is a benefit to detect the tendency of DMUs over long period with large inputs and outputs. Since then, some improved approaches on the DEAbased MPI or window analysis have been proposed [7–17]. However, both the DEAbased MPI and window analysis models suffer from one shortcoming: they neglect predicting efficiency of the evaluated DMU.
In many practical evaluation problems, efficiency of every evaluated DMU in a particular period may not be contrasted with the evaluated DMUs, but rather with sample standards determined by manufacturing parameters. The purpose of the contrast is not only to evaluate efficiency, but also to locate the standard with which the evaluated DMU has similar behavior. For instance, there are many grade standards for the evaluation of travel agencies. Travel agencies from the same region can be evaluated by the same standards separately, and those from different regions should not be evaluated by the same standards because of regional disparities. The standards should be formulated by the regional parameters. Taking outbound tourism as an example, it is an important part for travel agency business in developed regions, but it may not be contained in the travel agency business in some developing regions. Clearly, it is unreasonable that the outbound tourism is included in input measures to evaluate the travel agencies from different regions, and then grade standards in different regions should be formulated in terms of different manufacturing parameters. With these preparations, we then could use different standards to evaluate the level of travel agencies. However, in the existing DEA models, the constraint condition consists of the evaluated DMUs. Furthermore, we categorize DEA models into two types. The first type is the DEA models where the DMU under evaluation is included in the constraint condition [18–39]. The second type is the DEA models where the DMU under evaluation is not included in the constraint condition. For example, Andersen and Petersen [40] developed the superefficiency DEA model, which is identical to the BCC model, except that the DMU under evaluation is not included in the constraint condition. Superefficiency DEA model has been fully explored and applied [41–44].
Without such considerations, scholars will not be tempted to invest the effort in analyzing and predicting the development trend of the DMUs by contrasting with grade standards. In fact, managers can analyze and predict the development trend of input and output data based on historical data and then determine the level by contrasting with sample standards. Furthermore, to maximize profit and ensure proper resource allocation management, efforts can be made through improving influential factors. Therefore, it is a scenario that is worth considering in this case.
The rest of this paper is organized unfolded as follows. Section 2 introduces the CCR model and the time series method. In Section 3, an extended DEA model is proposed. In Section 4, the relationship between DEA efficiency and the production frontier is illustrated. In Section 5, the algorithm to determine sample standards is described. In Section 6, two numerical examples are given to illustrate the proposed model. At the end of the paper, some conclusions are drawn.
2 Preliminaries
2.1 CCR model
Definition 1
DEA efficient
If the optimal objective value of the evaluated DMU is equal to 1 and there is at least one optimal solution in which the optimal weight vectors of inputs and outputs are greater than 0, then the evaluated DMU is DEA efficient.
Definition 2
weak DEA efficient
If the optimal objective value of the evaluated DMU is equal to 1 and there is no optimal solution in which the optimal weight vectors of inputs and outputs are greater than 0, then the evaluated DMU is weak DEA efficient.
Definition 3
DEA inefficient
If the optimal objective value of the evaluated DMU is less than 1, then the evaluated DMU is DEA inefficient.
2.2 Time series method
3 An extended DEA model
Definition 4
DEA superefficient
An evaluated DMU is DEA superefficient if its optimal objective value is higher than 1 and there is at least one optimal solution in which the optimal weight vectors of inputs and outputs are greater than 0.
To determine the efficiency of the evaluated DMUs in the proposed model, the following theorems are given by considering the relationship between DEA efficiency and the optimal objective value.
Theorem 1
If the evaluated DMU is DEA superefficient by the kth standard, then the optimal objective value is greater than 1.
Theorem 2
The evaluated DMU is DEA efficient by all the combinations of sample DMUs in the kth standard if and only if there exists an optimal objective value that is equal to 1 and the optimal weight vectors of inputs and outputs are greater than 0.
Theorem 3
The evaluated DMU is weak DEA efficient by the kth standard if and only if the optimal objective value is equal to 1 and there does not exist any optimal solution in which the optimal weight vectors of inputs and outputs are greater than 0.
Theorem 4
The evaluated DMU is DEA inefficient by all the combinations of sample DMUs in the kth standard if and only if all optimal objective values are less than 1.
4 The relationship between DEA efficiency and the production frontier
In this section, we consider the case of two inputs and a single output to show the relationship between DEA efficiency and the production frontier. DEA efficiency is independent of the change of inputs and output by the same proportion, so we can change the inputs and output in the same proportion for each DMU until the output data of the evaluated DMUs and sample DMUs are equal. Next, the coordinate system is established with input 1 and input 2 as the x and y coordinate axes. For the DMU in the coordinate system, the closer it gets to the coordinate origin, the higher efficiency will be.
4.1 DEA efficiency and the production frontier in the conventional DEA models
4.2 DEA efficiency and the production frontier in the proposed model
Unlike conventional DEA models, in the proposed model, the constraint condition consists of one of the sample standards, and the production frontier is spanned by different combinations of sample DMUs from the constraint condition. To illustrate this, now we suppose that there are seven evaluated DMUs \(E _{3}\)\(E _{9}\) and that the kth standard is the constraint condition consisting of nine sample DMUs \(S _{4}\)\(S _{12}\).
The evaluated DMU \(E _{3}\) is closer to the coordinate origin than the most efficient production frontier, and then the efficiency of DMU \(E _{3}\) is higher than that of every sample DMU from the constraint condition. In such a case, the constraint condition consists of all sample DMUs of the kth standard, the optimal objective value is greater than 1, and the evaluated DMU \(E _{3}\) is DEA superefficient.
If the evaluated DMU is located between the most and least efficient production frontiers, then there is at least one optimal objective value equal to 1 for the evaluated DMU, such as DMU \(E _{4}\), \(E _{5}\), \(E _{6}\), \(E _{7}\), and \(E _{8}\). Clearly, in Figure 3, it is easy to see that DMU \(E _{4}\) is DEA superefficient by the least efficient production frontier \(S _{10}\)\(S _{12}\) and DEA inefficient by the most efficient production frontier \(S _{4}\)\(S _{7}\); then DMU \(E _{4}\) is DEA efficient (i.e., the optimal objective value of the evaluated DMU \(E _{4}\) is equal to 1, and the optimal weight vectors of inputs and outputs are greater than 0) by a combination of the kth standard. Similarly, there is an optimal objective value of DMU \(E _{5}\) equal to 1, and the optimal weight vectors of inputs and outputs are greater than 0. In the following, we consider the evaluated DMU \(E _{6}\), which is weak DEA efficient relative to the least efficient production frontier, and then there is an optimal objective value of the DMU \(E _{6}\) equal to 1. A similar analysis applies to the evaluated DMU \(E _{7}\); we can see that there is also an optimal objective value equal to 1. Finally, we take the evaluated DMU \(E _{8}\) into account, it can be expressed by a linear combination of DMU \(S _{10}\) and \(S _{11}\), and thus it is DEA efficient by the least efficient production frontier, and there is an optimal objective value equal to 1. The evaluated DMU \(E _{9}\) is located in the production possibility set of the least efficient production frontier but not located on the least efficient production frontier. Then DMU \(E _{9}\) is DEA inefficient, and the optimal objective value is less than 1. In fact, in the proposed model, the determined production frontier is spanned by the difference between the production possibility sets of the most and least efficient production frontiers.
5 Algorithm
 Step 0: :

Star with dividing the sample DMUs into m̄ ordered sample standards. Let \(t _{1}=1\), \(t _{2}=\bar{m}\).
 Step 1: :

Use the \(t _{1}\)th and \(t _{2}\)th standards to evaluate the evaluated DMU.
If the evaluated DMU is DEA efficient by the \(t _{1}\)th (or \(t _{2}\)th) standard, then
Stop  the evaluated DMU has similar behavior with the \(t _{1}\)th (or \(t _{2}\)th) standard;
If the evaluated DMU is DEA superefficient (or inefficient) by the \(t _{1}\)th and the \(t _{2}\)th standard, then
Stop  the evaluated DMU has not similar behavior with all the sample standards;
Else
Turn to Step 2.

 Step 2: :

If the evaluated DMU is DEA superefficient by the \(t _{1}\)th standard and inefficient by the \(t _{2}\)th standard, then

\(t _{3} \leftarrow[(t _{1}+t _{2})/2]\)

If the evaluated DMU is DEA inefficient by the \(t _{3}\)th standard, then

\(t _{2} \leftarrow t _{3}\); Turn to Step 2;


If the evaluated DMU is DEA superefficient by the \(t _{3}\)th standard, then

\(t _{1} \leftarrow t _{3}\); Turn to Step 2;


Else

Turn to Step 1.

If the evaluated DMU is DEA superefficient by the \(t _{1}\)th standard and weak efficient by the \(t _{2}\)th standard, then
Stop  the evaluated DMU has similar behavior with the \((t _{2}1)\)th standard.
If the evaluated DMU is DEA inefficient by the \(t _{1}\)th standard and superefficient by the \(t _{2}\)th standard, then
\(t _{3} \leftarrow[(t _{1}+t _{2})/2]\)

If the evaluated DMU is DEA superefficient by the \(t _{3}\)th standard, then

\(t _{2} \leftarrow t _{3}\); Turn to Step 2;


If the evaluated DMU is DEA inefficient by the \(t _{3}\)th standard, then

\(t _{1} \leftarrow t _{3}\); Turn to Step 2;


Else

Turn to Step 1.

If the evaluated DMU is DEA weak efficient by the \(t _{1}\)th standard and superefficient by the \(t _{2}\)th standard, then
Stop  the evaluated DMU has similar behavior with the \((t _{1}+1)\)th standard;

6 Illustrative examples
In this section, we present two numerical examples to illustrate the proposed model. For simplicity, “sample DMU” will be abbreviated to “SDMU”.
6.1 The first example
Production status in the last 23 years
Year  Input 1  Input 2  Output  Year  Input 1  Input 2  Output 

1  7.2  6.9  16.6  13  6.9  5.5  20.6 
2  7.8  5.1  20.1  14  7.9  5.8  20.7 
3  7.3  5.7  20.5  15  8.1  6.4  18.3 
4  6.5  6.7  20.0  16  6.8  6.2  19.1 
5  7.0  6.5  21.5  17  6.5  5.7  17.5 
6  7.2  5.5  16.6  18  6.9  5.6  15.0 
7  7.4  5.7  15.4  19  7.1  6.2  16.1 
8  7.1  6.0  17.7  20  7.0  6.5  20.1 
9  7.3  6.1  19.4  21  7.2  6.0  18.1 
10  7.5  6.3  16.4  22  7.2  5.9  17.4 
11  7.3  6.6  16.5  23  7.2  5.8  16.4 
12  6.4  6.1  20.1 
The sample standards
Standard 1  Standard 2  Standard 3  Standard 4  Standard 5  

\(\boldsymbol{\mathrm{SDMU}_{11}}\)  \(\boldsymbol{\mathrm{SDMU}_{12}}\)  \(\boldsymbol{\mathrm{SDMU}_{13}}\)  \(\boldsymbol{\mathrm{SDMU}_{21}}\)  \(\boldsymbol{\mathrm{SDMU}_{22}}\)  \(\boldsymbol{\mathrm{SDMU}_{23}}\)  \(\boldsymbol{\mathrm{SDMU}_{31}}\)  \(\boldsymbol{\mathrm{SDMU}_{32}}\)  \(\boldsymbol{\mathrm{SDMU}_{33}}\)  \(\boldsymbol{\mathrm{SDMU}_{41}}\)  \(\boldsymbol{\mathrm{SDMU}_{42}}\)  \(\boldsymbol{\mathrm{SDMU}_{43}}\)  \(\boldsymbol{\mathrm{SDMU}_{51}}\)  \(\boldsymbol{\mathrm{SDMU}_{52}}\)  \(\boldsymbol{\mathrm{SDMU}_{53}}\)  
Input1  1  3  2  1  3  2  1  3  2  1  3  2  1  3  2 
Input2  2  1  2  2  1  2  2  1  2  2  1  2  2  1  2 
Output  [0,1]  [0,2]  [0,3]  (1,2]  (2,3]  (3,4]  (2,3]  (3,4]  (4,5]  (3,4]  (4,5]  (5,6]  (4,5]  (5,6]  (6,7] 
The processes of evaluation
Step  Standard  Objective value  Result 

1  [k/2]=2  >1  efficient 
2  [(k + 2)/2]=3  >1  efficient 
3  [(k + 3)/2]=4  [0.9117,1.953]  weak efficient 
6.2 The second example
Strategic groups are always used in the strategic management of insurance companies, and groups companies have similar business models or similar combinations of strategies. An insurance company can ascertain major competitors, obtain the competitive situation, and then formulate production strategy by analyzing strategic groups [55].
6.2.1 Dividing the sample insurance companies into ordered strategic groups
Six strategic groups (from low to high)
Standard  Property Insurance Companies 

1  Bohai Property Insurance Company Limited; Chang an Property and liability Insurance Limited; Dubang Property & Casualty Insurance Company Limited; Nipponkoa Insurance Company (China) Limited; China Continent Insurance; Ancheng Property & Casualty Insurance Company Limited. 
2  Sinosafe Insurance Company Limited; Da Zhong Insurance Company Limited; Ming An Property & Casualty Insurance Company Limited; China Huanong Property & Casualty Insurance Company Limited; Liberty Insurance Company Limited. 
3  Huatai Insurance Company of China, Limited; Aioi Insurance Company Limited; American Chubb Group of Insurance; Bank of China Insurance Company Limited; Zurish Insurance, Beijing; Alltrust Insurance Company Limited. 
4  China Life Property & Casualty Insurance Company Limited; Tianan Property Insurance Company Limited; Dinghe Insurance; Sun Alliance Insurance Company; Generali China Insurance Company Limited; Sompo Japan Insurance (China) Company Limited. 
5  The Tokio Marine & Nichido Fire Insurance Company (China) Limited; Anxin Agricultural Insurance Company Limited; AIG General Insurance Company China Limited; Allianz Insurance Company Guangzhou Branch; Hyundai Insurance (China) Company Limited. 
6  Mitsui Sumitomo Insurance (China) Company Limited; Guoyuan Agricultural Insurance Company; China Export & Credit Insurance Corporation; Sunlight Mutual Insurance Company. 
6.2.2 Predicting production status of Samsung Fire & Marine Insurance (China) Company Ltd
The production status of Samsung F&M from 20082014
Year  TNE  FA  STEC  EIP  EP 

2008  62  1.67  2.28  65.78  55.68 
2009  75  2.25  2.95  92.22  81.66 
2010  91  2.43  9.2  87.94  68.42 
2011  138  9.16  13.75  125.03  233.71 
2012  192  10.01  19.88  134.23  102.67 
2013  226  17.52  24.8  142.87  247.14 
2014  325  19.27  28.1  195.56  217.11 
The predicted production status in 2014
2014  TNE  FA  STEC  EIP  EP 

Predicted  286  20.03  29.8  174.2  232.5 
6.2.3 Evaluating the production efficiency by the sample strategic groups
The evaluation processes and results
Step  Using predicted production status  Using actual production status  

Standard  Objective values  Strategic Groups  Standard  Objective values  Strategic Groups  
1  [6/2]=3  >1  No  [6/2]=3  >1  No 
2  [(6 + 2)/2]=4  >1  No  [(6 + 2)/2]=4  >1  No 
3  [(6 + 4)/2]=5  [0.62,2.69]  Yes  [(6 + 4)/2]=5  [0.73,2.21]  Yes 
7 Conclusions
In the conventional DEA model, the inputs and outputs are known exactly, and the constraint condition consists of the evaluated DMUs. However, in many real applications, the observed data of the evaluated DMUs are variable over time. The efficiency of every evaluated DMU in a particular period may not be contrasted with the evaluated DMUs, but with sample standards determined by production strategy. Moreover, the development trend of the evaluated DMU, which is an important index to the budgetary decisionmaking and management system, is often required to be predicted.
In this paper, we proposed an extended DEA model to evaluate the efficiency of DMUs with historical observed data of inputs and outputs. Firstly, based on the historical observed data, we introduced the time series method to analyze and predict the development trend of the evaluated DMUs. Secondly, in the proposed model, there are many sample DMUs, which are divided into several ordered sample standards in terms of manufacturing parameters, and the constraint condition consists of one of the sample standards. Finally, we employ the algorithm based on a binary search tree to determine the constraint condition in order to reduce the computation complexity. One of the most intriguing and appealing points mentioned is that the paper is suitable for the decisionmaking, whether the evaluated DMUs are hospitals, universities, branches of a bank, or whatever.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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