Abstract
Purpose
The purpose of this paper is to build a novel data envelopment analysis (DEA) model to evaluate the efficiencies of decision making units (DMUs).
Design/methodology/approach
Using the Choquet integrals as aggregating tool, the authors give a novel DEA model to evaluate the efficiencies of DMUs.
Findings
It extends DEA model to evaluate the DMU with interactive variables (inputs or outputs), the classical DEA model is a special form. At last, the authors use the numerical examples to illustrate the performance of the proposed model.
Practical implications
The proposed DEA model can be used to evaluate the efficiency of the DMUs with multiple interactive inputs and outputs.
Originality/value
This paper introduce a new DEA model to evaluate the DMU with interactive variables (inputs or outputs), the classical DEA model is a special form.
Keywords
Citation
Ji, A., Liu, H., Qiu, H.-j. and Lin, H. (2015), "Data envelopment analysis with interactive variables", Management Decision, Vol. 53 No. 10, pp. 2390-2406. https://doi.org/10.1108/MD-11-2014-0631
Publisher
:Emerald Group Publishing Limited
Copyright © 2015, Authors. Published by Emerald Group Publishing Limited. This work is published under the Creative Commons Attribution (CC BY 3.0) Licence. Anyone may reproduce, distribute, translate and create derivative works of the article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licenses/by/3.0/legalcode .
1. Introduction
Data envelopment analysis (DEA) is a common methods for evaluating the performance of organizational units. DEA is a non-parametric optimizing mathematical method that was first introduced by Charnes and Cooper Rhodes (1978), then was developed by Banker et al. (1984). According to DEA, it combines and transforms multiple inputs and outputs into a single efficiency index, efficiency will be calculated relatively among a homogeneous collection of decision-making units (DMUs) based on some equal inputs and outputs. This approach first establishes an “efficient frontier” formed by a set of DMUs that exhibit best practices and then assigns the efficiency level to other non-frontier units according to their distances to the efficient frontier. The basic idea has since generated a wide range of variations in measuring efficiency. Nowadays, various DEA efficiency models are available for different types of measuring requirement, such as the constant returns to scale model (Charnes and Cooper Rhodes, 1978), the variable-returns-to-scale (Banker et al., 1984) model, the additive model (Charnes et al., 1985) the slacks-based measures and the free disposal hull model (Bardhan et al., 1996), etc. It also has been applied to various industrial and non-industrial contexts, such as banking (Cooper et al., 2008; Kao and Liu, 2009), education (Ray and Jeon, 2008), hospital (Ancarani et al., 2009), and so on.
In all the classical DEA models, the combination form of the multiple inputs (or outputs) is a linear weighted sum, it needs an assumption that there is no interaction among the contributions from individual information sources so that the joint contribution is just the simple sum of contributions from individual information sources. But in practice, the variables (inputs or outputs) are usually strongly correlated, there are interactions among variables. In general, the inherent interaction cannot be ignored in the efficiency evaluation. For example, in the efficiency evaluation of Greek commercial banks (Varias and Sofianopoulou, 2012), the inputs (such as fixed assets and total assets) are closely related, if the weighted sum of multiple inputs (or outputs) is used in the DEA model, then the evaluation results may have deviations. The selection of variables (inputs or output) has significant impacts on DEA efficiency estimates (Nataraja and Johnson, 2011), Dyson et al. (2001) showed that the omission of a highly correlated variable or adding of uncorrelated variable can have a significant impact on the efficiency estimates of some production units.
The collinearity of variables is a special interaction, the collinearity of variables can affect the evaluation result of DEA (Jukti, 1995; Fengxia Dong and Mitchell, 2015).
Some researchers have proposed several models to solve these problems. Adler and Golany (2001, 2002) suggested using the principal component analysis (PCA) to produce uncorrelated linear combinations of original inputs and outputs, and construct a PCA-based DEA model. Adler and Yazhemsky (2010) concluded that PCA-DEA outperforms classical DEA by comparing their discrimination performance in a simulation exercise. Independent component analysis (ICA) is an another information aggregation tool to resolve the problem of variables correlation, ICA is a novel statistical technique used to extract independent variables from observed multivariate statistical data where no relevant data mixture mechanisms are available (Hyvarinen and Oja, 2000). Using ICA to extract independent variables (inputs or outputs) in DEA efficiency measurement (Kao et al., 2011) can partly overcome the effect of variables correlation.
Although the above two DEA models can avoid efficiency misjudgment for the DMUs with interactive variables (inputs or outputs), but they divide efficiency measurement into two stages, first use the variables (inputs, outputs) of the DMUs to generate independent variables (inputs, outputs) and then construct DEA model for the generated independent variables (inputs, outputs), which lead to high computation complexities. Especially, in the two DEA models, the variables (inputs, outputs) of DEA model are not practical variables of the DMUs, but are the variables produced by the original variables (in PCA-based DEA model, they are uncorrelated combinations of original variables; in ICA-based DEA model, they are the basis vector of original variables), the results of efficiency evaluations have fewer guiding for production activities.
The weighted sum is the Lebesgue-like integrals on discrete spaces, in which the global contribution toward the objective is regarded as the sum of contributions from each attribute. It needs a basic supposition that there is no interaction among attributes. None of the classical aggregation tools are related to interactions among attributes. Choquet integral (Choquet, 1954) is a generalization of the Lebesgue integral, when the fuzzy measure is additive, the Choquet integrals coincide with Lebesgue integrals. Choquet integral with respect to efficiency measures (also called fuzzy measures or non-additive measures) is a non-linear integral (Wang et al., 2010), it can provide the attribute aggregation tool which considers interactions among attributes.
In this paper, we shall discuss the efficiency evaluation model with interaction variables (inputs or outputs), in which Choquet integrals are used to aggregate the multiple inputs and outputs into a single efficiency index. In Section 2, we present some preliminaries. In Section 3, we propose the DEA model with interaction variables. In Section 4, we apply the proposed DEA model to the efficiency evaluation.
2. Background
In this section, some preliminaries for DEA and Choquet integral are presented.
2.1. The DEA model and cross-efficiency evaluation
Charnes, Cooper, and Rhodes introduced the method of DEA to address the problem of efficiency measurement for DMUs with multiple inputs and multiple outputs. They have applied in the efficiency evaluation for the non-market agencies like schools, hospitals, which produce identifiable and measurable outputs from measurable inputs but generally lack market prices of outputs (and often of some inputs as well). Suppose that there are n DMUs, each producing m outputs from s inputs. DMU 0, the DMU to be evaluated. DMU k use the input bundle x k=(x 1k , x 2k ,…, x sk ) to produce the output bundle yk=(y 1k , y 2k ,…, y mk ). The weighted sums are used to aggregate of inputs and outputs, the measure of efficiency of any DMU is obtained as the maximum of ratio of total weighted outputs to the total weighted inputs subject to the condition that the similar ratios for every DMU be less than or equal to unity. Its mathematical model is as follows:
Subject to:
This is a linear fractional functional programming problem, by the C2 transformation (Charnes and Cooper, 1962), the linear fractional functional programming problem (1) can be transformed to the following equivalent linear programming:
Subject to:
The above efficiency evaluation models are called CCR models which are treated in input-oriented forms. The efficient DMUs in CCR model are called CCR efficient.
The dual form of the programming (2) is:
Subject to:
Banker, Charnes and Cooper (BCC) model (Banker et al., 1984) adds an additional constant variable, w 0, in order to permit variable returns-to-scale:
Subject to:
The dual form of the programming (4) is:
Subject to:
It can be seen from DEA models (2) (or (4)) that the essence of CCR (BCC) models is that the DMU evaluated tries to find out its weight vector to maximizing its weighted output with the constraints that its weighted input is fixed as unity and the weighted output is not larger than the weighted input for all DMUs. In other words, each DMU seeks its favorite weight vector to its own advantage.
To overcome the inability of DEA in discriminating between DEA efficient units, cross-efficiency evaluation (Sexton et al., 1986; Doyle and Green, 1994, 1995) has been suggested in the DEA literature. In the cross-efficiency evaluation, each DMU determines a set of input and output weights individually, leading to n sets of weights for n DMUs. The n sets of weights are then used to assess the efficiencies of the n DMUs, resulting in n efficiency values for every DMU. The n efficiency values for each DMU are finally averaged as an overall efficiency value of the DMU. It is believed that the cross-efficiency evaluation can guarantee a unique ordering for the DMUs. Due to its good discrimination power, the cross-efficiency evaluation has been used in a variety of applications, including preference voting and project ranking (Green et al., 1996), industrial robot selection (Baker and Talluri, 1997). The cross-efficiency evaluation has also been deeply studied theoretically (Ramón and Ruiz, 2010; Cook and Zhu, 2014).
In the first stage, the optimal weights of inputs and outputs are calculated for each DMU using the classical DEA formulation (CCR model or BBC model). Given the results of the first stage, the weights used by the DMU can be utilized for calculating the peer rated efficiency for each of the other DMUs. The peer evaluation score, ϑ pk , indicates the efficiency score for DMU k using the weights obtained by DMU p :
The cross-efficiency scores can be summarized in a cross-efficiency matrix. Note that 0ϑ pk 1 and the elements in the diagonal, ϑ pp , represent the standard DEA efficiency score. Then the overall efficiency value of the DMU k is calculated by:
2.2. Efficiency measures and the Choquet integral
The concept of efficiency measure was initiated in the 1950s (Choquet, 1954) and has been well developed since 1970s (Denneberg, 1994; Wang and Klir, 2008; Wang et al., 2010).
Considering the non-linear relationships, particularly interactions among attributes, the non-linear integrals can be used as data aggregation tools. Studies of non-linear integrals can be found in literature, such as (Wang and Klir, 2008; Wang et al., 2010; Denneberg, 1994). The weighted sum is also referred to as the Lebesgue-like integral on a finite space (Wang et al., 2010) The Choquet integral (Choquet, 1954; Wang et al., 2010), one of non-linear integrals, is more appropriate for information fusion and data mining applications because it contains very important information regarding the interactions among attributes in the database. It can be utilized to aggregate the values of feature attributes according to the values of the set function μ. Thus, the Choquet integral with respect to an efficiency measure is chosen as the non-linear data aggregation tool.
Let finite set X={x 1, x 2, …, x n } be the attributes in a multi-dimensional data set. Several important types of efficiency measure are defined as the follows (Denneberg, 1994; Wang and Klir, 2008; Wang et al., 2010).
Definition 1. An efficiency measure μ defined on X is a set function μ: P(X)→[0,+∞) satisfying μ()=0 (where P(X) denotes the power set of X); μ is called a monotone measure if it satisfies μ()=0 and μ(E)μ(F) for E⊂F, where E, F are any sets in P(X).
The fuzzy measure (Sugeno, 1974) is a monotone efficiency measure on X.
Definition 2. An efficiency measure is said to be regular if μ(X)=1.
Definition 3. A signed efficiency measure μ defined on X is a set function μ: P(X)→(–∞, +∞) satisfying μ()=0.
For efficiency measure μ, A∈P(X), B∈P(X), there are three cases:
Case 1: If μ(A∪B) > μ(A)+μ(B), implying that A and B have a multiplicative effect.
Case 2: If μ(A∪B)=μ(A)+μ(B), implying that A and B have an additive effect.
Case 3: If μ(A∪B) < μ(A)+μ(B), implying that A and B have a substitutive effect.
The efficiency measure is often used with the non-linear integral for aggregating information evaluation by considering the influence of the substitutive and multiplication effect among all attributes (see the example 1 below, the worker x 1 and x 2 have a multiplicative effect; the worker x 2 and x 3 have a substitutive effect).
Definition 4. Let (X, P(X), μ) be an efficiency measure space, f: X→(−∞, +∞) be a measurable real-valued function, then the Choquet integral of f(x)with respect to the efficiency measure μ is defined as follows:
Where (c) indicates the type of integral being the Choquet integral and F α ={x∣f(x)α, x∈X}, α∈[0, ∞), (Equation 15) are the Riemann integral.When f are non-negative functions, (Equation 16)
The Choquet integral is a non-linear integral, it is generalization of the Lebesgue integral and coincide with the Lebesgue integral for additive efficiency measure.
When X={x 1, x 2, …, x n } is a finite set, the values of f, i.e., f(x 1), f(x 2), …, f(x n ), can be sorted in a non-decreasing order so that (Equation 17), where (Equation 18) is a certain permutation of {x 1, x 2, …, x n }, then the Choquet integral is obtained by:
Where (Equation 20) and (Equation 21).
We can easily verify the formula (6) is equivalent to:
where A n+1=.
Example 1 Let {x 1, x 2, x 3} be the set of three workers, function f: X→[0, ∞) be the numbers of hours of their work in a certain day with:
Individual and joint efficiency measure μ: P(X)→[0,+∞) are as following:
(C)∫fdμ=3×17+1×13+2×5=74
Because of the interactions among attributes, it is not equal to the weighted sum: 6×5+3×6+4×7=76.
3. DEA with interactive variables
According to the actual production activities, the efficiency measures are positive and monotone, therefore, in the following, let us suppose that the efficiency measures are always positive and monotone. For a collection of DMUs, the attribute set of inputs is X={x 1, x 2,…, x s }, the attribute set of outputs is Y={y 1, y 2, …, y m }. Taking into account the interaction among the multiple inputs (outputs), the efficiency measure μ({x i }) represent the efficiency of the input index x i , the efficiency measure μ(A) (A is not a single point set) represent the joint efficiency of attribute set A⊂X; and it is similar to the efficiency measure ν on P(Y). Let f k (x i) denote the numerical information that kth DMU obtained from input attribute x i, g k (y j ) denote the numerical information that kth DMU obtained from output attribute y j . The kth DMU use the input bundle f k (x)=(f k (x1), f k (x2),…, f k (x s )) to produce the output bundle g k (y)=(g k (y1), g k (y2),…, g k (y m )). The proposed measure of the efficiency of any DMU is obtained as the maximum of the ratio of the aggregate outputs to aggregate inputs, where the aggregate inputs (or outputs) are calculated by the Choquet integral. The efficiency measure μ and ν can be determined by the following optimization problem ((CH-CCR)I Model):
Subject to:
The efficiency ratio ranges from zero to one, the k-DMU is considered relatively efficient if it receives a score of one. Thus, each unit will choose weights so as to maximize self-efficiency, given the constraints.
The result of the DEA is the determination of the hyper-planes that define an envelope surface. DMUs that lie on the surface determine the envelope and are deemed efficient, whilst those that do not are deemed inefficient.
We utilize the following transformations:
The set functions λ: X→[0,+∞), ω: Y→[0,+∞) are all also non-negative monotone efficiency measures.
By using the transformations (9), the (CH-CCR)I model (8) can be changed into the following equivalent CH-CCR model:
Subject to:
When the efficiency measures λ on X, ω on Y are all additive, the CH-CCR model (10) is coincide with the classical CCR model, therefore CH-CCR model is a generalization of the classical CCR.
The CH-CCR model (10) are in essence a linear programming. For the Choquet integral (c)∫f k dλ (or (c)∫g k dω), once the integrand f k (or g k ) is given, the calculation of its Choquet integral only involves the value of λ(or ω) at the sets in a chain from the universal set X (or Y) to the empty set. In order to express the (c)∫f k dλ (or (c)∫g k dω) in an explicit linear form of the unknown parameters that are the value of λ(or ω), we introduce an alternate calculation formula for the Choquet integral as follows (Wang et al., 2010):
Where (Equation 30) if j is expressed in terms of binary digits j s j s–1…j 2 j 1 for every j=1, 2, …, 2s−1 and:
Similarly,
where (Equation 33) if j is expressed in terms of binary digits j s j s−1…j 2 j 1 for every j=1, 2,…, 2m−1 and:
Therefore, through the above transformations, the CH-CCR model (10) can transform a linear programming with decision variables λj (j=1, 2,…, 2s−1) and ω j (j=1, 2,…, 2m−1):
Subject to:
For the simple efficiency evaluation problem (less DMUs, less inputs, less outputs), we can solve the optimization problem (10) directly; for a reasonably large efficiency evaluation problem (large DMUs or large inputs or large outputs), we can use the genetic algorithm (Wang et al., 2010) to solve the optimization problem (12).
Definition 5. The performance of DMU 0 is efficient iff there exist the optimal solution μ on P(X), ν on P(Y) in (CH−CCR)I Model, such that:
The Definition 5 is equivalent to the following definition.
Definition 5 ' . The performance of DMU 0 is CH-CCR efficient if there exist the optimal solution λ on P(X), ω on P(Y) in (10), such that: h 0=(c)=∫g 0 dω=1,(c)∫f 0 dλ=1.
In order to obtain the overall efficiency value for every DMU, the optimal efficiency measures of inputs and outputs are calculated for each DMU using the CH-CCR model. Given the results of the first stage, the efficiency measures used by the DMU p are denoted by λ p on P(X), ω p on P(Y), they can be utilized for calculating the peer rated efficiency for each of the other DMUs. The peer evaluation score, ϑ pk , indicates the efficiency score for DMU k using the efficiency measure obtained by DMU p :
The cross-efficiency scores can be summarized in the cross-efficiency matrix. Note that 0ϑ pk 1 and the elements in the diagonal, ϑ pp , represent the standard DEA efficiency score. Then the overall efficiency value of the DMU k is calculated by:
4. Illustration with numerial examples
In this section, we provide three numerical examples to show the performance of the proposed models. One simple example is given to illustrate the proposed models is consistent with the classical DEA when the interaction (correlations) between the input (output) variables are very low, and then we provide two numerical examples that have frequently appeared in the related literature to show the performance of the proposed models is different from the classical DEA for highly correlated input (or output) variables.
Example 1: in this example, we are to evaluate the community health center (DMU) of Hebei Province in China, evaluating input indexes are confirmed as the public expenditure (10,000 Yuan) X 1, the number of medical staffs X 2 and the fixed assets (10,000 Yuan) X 3; the output index is considered as the numbers of medical service (thousands) (including inpatient service and childhood immunization) Y 1, the numbers of management of chronic diseases (thousands) Y 2. The data are shown in Table I.
The correlation matrix of the inputs is:
The correlation matrix of the outputs is:
Applying the CCR and CH-CCR models given before to the data in Table I, we obtain the results shown in Table II.
As shown in Table II, the efficiency evaluation of CCR and CH-CCR models are same (although the efficiency value are different), and the rank of the overall efficiency is also the same. The reason may be that the correlations between the input (output) variables are very low, so we may consider the input (output) variables are mutually independent and there little interactions between variables (inputs and outputs).
Example 2: the data set is taken from a previous study (Guo, Jia and Qiu) for comparing the efficiency of eight retail enterprises of Chengdu city in China, evaluating input indexes are confirmed as production flexibility (day) X 1, order advance (day) X 2, and guaranteeing cost or the returned merchandise cost (10,000 Yuan) X 3; the output index is considered as complete execution rate of orders (per cent) Y 1, satisfaction rate of customer (per cent) Y2, and intelligence capital rate (per cent) Y 3. DMU1 (j=1, 2, …, 8) means each retail enterprise in Chengdu city. Data are shown in Table III.
The correlation matrix of the inputs is:
The correlation matrix of the outputs is:
Applying the CCR and CH-CCR models given before to the data in Table III, we obtain the results shown in Tables IV-VI.
From the correlation matrix of inputs and outputs, there are interaction between some variables (inputs or outputs). As shown in Tables IV and V, the efficiency evaluation of CCR and CH-CCR models are slightly different, both the CCR and CH-CCR models define DMU 1, DMU 3 and DMU 8 as efficient, the DMU 5 is efficient in CCR model, but inefficient in CH-CCR model, the differences are probably caused by the interaction of variables (inputs or outputs). On the other hand, it can be seen from Table V that the input x 2, x 3 and the output y 3 have no contribution to the efficiency evaluation independently, but their joint contribution with other indexes cannot be ignored. And from Table VI, it can be seen that because the interaction of variables are little and the efficiency differences among the DMUs are not big enough, so in the cross-evaluation, although the overall efficiency values are different, but the rank of the overall efficiency is the same.
Example 3: a real data set taken from a previous study by Sherman and Gold (1985) for comparing the efficiency of 14 bank branches is given in Table VII. The comparison is based on three input and four output as follows:
Input 1: rent (thousands of dollars).
Input 2: full time equivalent personnel.
Input 3: supplies (thousands of dollars).
Output 1: loan applications, new pass-book loans, life insurance sales.
Output 2: new accounts, closed accounts.
Output 3: travelers checks sold, bonds sold, bonds redeemed.
Output 4: deposits, withdrawals, checks sold, treasury checks issued, B per cent checks, loan payments, pass-book loan payments, life insurance payments, mortgage payments.
The correlation matrix of the inputs is:
The correlation matrix of the outputs is:
Applying the CCR and CH-CCR models given before to the data in Table VII, we obtain the results shown in Table VI.
The results in Table VIII show that the efficiency evaluation of CCR and CH-CCR models are different, the DMU3 is efficient in CCR model, but inefficient in CH-CCR model. And in the overall efficiency evaluation, because the interaction of variables are big enough, the ranks of the DMU3, DMU8 and DMU5, DMU7, DMU13 are different.
From the above numerial examples, we conclude that the interaction between variables (inputs or outputs) can affect the efficiency evaluation. When there are no any interaction between the variables (inputs and outputs), the proposed models is consistent with the classical DEA, and the efficiency evaluation of the proposed models is different from the classical DEA for highly correlated input (or output) variables.
The numerial examples also show that the proposed model is a generalization of the classical CCR.
5. Conclusion
In this paper, a novel DEA model is proposed for evaluating the efficiencies of DMUs, and the Choquet integrals are used to aggregate the multiple inputs and outputs into a single efficiency index. This model extends DEA model to evaluate the DMU with interactive variables (inputs or outputs), the classical DEA model is a special form. Besides, the proposed model is used to analysis some numerical examples and the results show good performance. In the future, the other DEA model with interactive variables would be studied.
Corresponding author
Dr Hui Liu can be contacted at: jab@hbu.edu.cn
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Further reading
Guo, J. , Jia, L. and Qiu, L. (2006), “Research on supply chain performance evaluation based on DEA/AHP model”, Proceedings of the 2006 IEEE Asia-Pacific Conference on Services Computing (APSCC’06), Guangzhou, 12-15 December, 0-7695-2751-5/06.
Acknowledgements
© Aibing Ji, Hui Liu, Hong-jie Qiu and Haobo Lin. Published by Emerald Group Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 3.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial & non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/3.0/legalcode
National Social Science Fund (14BJY010), science and technology research and development programs of Hebei Province (12276104D-3) and Research Foundation of College of Medicine in Hebei University (2012A3003).