Data envelopment analysis cross-efficiency model in fuzzy environments

Abstract

The fuzzy data envelopment analysis (FDEA) method lacks sufficient discrimination power to rank efficient decision making units (DMUs) with fuzzy data; moreover, it evaluates DMUs using only self-evaluation. This paper develops the fuzzy cross efficiency (FCE) DEA model that combines self-evaluation with peer-evaluation to eliminate the weaknesses of traditional FDEA. This method solves the efficiency evaluation problem in fuzzy environments from a new perspective. An effective method is provided to solve this model, and the ranking results of DMUs at different α-levels are obtained. Finally, an example is given to illustrate the proposed method in greater detail.

Keywords

Data envelopment analysis (DEA)fuzzy cross efficiency (FCE)ranking

1 Introduction

Based on the concept of management efficiency, Charnes et al. [1] proposed data envelopment analysis (DEA) as a new decision-making method to evaluate the relative efficiency of decision-making units (DMUs) with inputs and outputs. The original model, namely, the Charnes-Cooper-Rhodes (CCR) model, has received extensive attention and has been researched by many scholars. The Banker-Chames- Cooper (BCC) model was proposed by Banker et al. [2] to evaluate the relative efficiency of DMUs under variable returns to scale assumption. The CCR and BCC models are classified as traditional DEA methods, but they both have their disadvantages. In general, these disadvantages can be attributed to insufficient discrimination power in ranking DMUs efficiently. Moreover, the efficiency of each DMU is usually exaggerated by self-evaluation, which can lead to distorted results.

To solve these problems, the cross efficiency model was proposed by Sexton et al. [3]. This model is a powerful assessment tool that combines self-evaluation with peer-evaluation. In this method, the efficiency of each DMU is calculated based on self-determined weights and those based on other DMUs. However, the original cross efficiency model has several flaws; specifically, the solution to this model may not be unique. To solve this problem, benevolent and aggressive formations were proposed by Doyle and Green [4] as secondary goals to optimize the input and output weights. The benevolent formation regards other DMUs as partners and aims to maximize their efficiency while maintaining its own optimal efficiency. The aggressive formation regards other DMUs as competitors and maximizes its own efficiency while minimizing the efficiency of others. Game theory was introduced by Liang et al. into the cross efficiency model to find the best DMU [5]. The neutral DEA model was proposed by Wang et al. [6] to optimize the input and output weights for each DMU from its own point of view without being aggressive or benevolent to the other DMUs. Yang et al. [7] suggested that the results of benevolent and aggressive formations compose an interval cross efficiency matrix. By using this matrix, the rankings of all DMUs can be obtained. DEA cross evaluation was developed by Contreras [8] to optimize the rank position of DMUs. A cross efficiency aggregation method was developed by Yang et al. [9] based on an evidential-reasoning approach to reflect a decision maker’s preferences. Dotoli et al. [10] presented a new cross efficiency method to evaluate different elements under certainty by integrating the cross efficiency technique and the fuzzy logic framework. This method was then applied to evaluate the performance of healthcare systems in Italy.

In recent decades, cross efficiency has been used as a performance measurement tool in many fields [11 –13]. However, these models measure DMU efficiency based solely on crisp data and are essentially ineffective for dealing with imprecise data. In the real world, inputs and outputs of DMUs are always imprecise and sometimes may even be linguistic variables such as “good”, “tall”, etc. Therefore, it is of great significance to study DEA in fuzzy environments. The fuzzy data envelopment analysis (FDEA) model was developed by Guo and Tanaka [14] to transform fuzzy constraints into crisp constraints. An alternative FDEA approach was developed by Zerafat et al. [15] to retain model fuzziness by maximizing the membership functions of inputs and outputs. The slack-based measurement was introduced in FDEA by Chen et al. [16] to estimate management achievements in banking. Momeni et al. [17] proposed a new fuzzy network slacks-based DEA model to evaluate the performance of supply chains with internal linking activities in fuzzy environments. An integrated DEA enhanced Russell measure model in fuzzy environments was developed by Majid et al. [18] to solve the sustainable supplier selection problem.

According to Hatami-Marbini et al. [19], the methods proposed for dealing with fuzzy data in DEA can be divided into four categories according to their approach: the tolerance approach, the α-level based approach, the fuzzy ranking approach, and the possibility approach. These methods have their own advantages and disadvantages, but no one has an absolute advantage over another. The disadvantages faced by traditional DEA must also be addressed by FDEA. As one of the main approaches for addressing the drawbacks of traditional DEA, the cross efficiency method with imprecise data is very rare. In fact, research conducted by Dotoli et al. [10] is the only research we could find on evaluating cross efficiency in fuzzy environments. However, Dotoli’s method is complicated, requiring a lot of computations, and multiple defuzzifications may lead to information loss.

This paper proposes a new fuzzy cross efficiency (FCE) DEA model to rank all DMUs by combining self-evaluation with peer-evaluation in fuzzy environments. Not only is this method simple to apply, but it also retains as much uncertain information as possible for the decision maker. The remainder of this paper is organized as follows. In Section 2, we review DEA, FDEA, and cross efficiency models. Section 3 proposes the FCE model and demonstrates how it is solved, and Section 4 gives a numerical example for illustration. Finally, conclusions and the direction of future research are given in Section 5.

2 Background

2.1 DEA and FDEA

DEA is a non-parametric decision-making method that measures the efficiency of DMUs with multiple inputs and outputs. Assume that there are n DMUs and that each DMU has m inputs and s outputs. The input and output values of DMU_j (j = 1, …, n) are represented by x_ij (i = 1, …, m) and y_rj (r = 1, …, s), respectively. The CCR efficiency of DMU_k is given by the following model.

$\begin{matrix} Max θ_{kk} = \sum_{r = 1}^{s} u_{rk} y_{rk} \\ s . t . \sum_{i = 1}^{m} v_{ik} x_{ik} = 1; \\ \sum_{r = 1}^{s} u_{rk} y_{rj} - \sum_{i = 1}^{m} v_{ik} x_{ij} \leq 0; j = 1, \dots, n; \\ u_{rk}, v_{ik} \geq 0; r = 1, \dots, s; i = 1, \dots, m; \end{matrix}$ (1) where v_ik and u_rk are the weights associated with the ith input and rth output of DMU_k, respectively. If the value of $θ_{kk}^{*}$ is 1, DMU_k is efficient; otherwise, it is inefficient. $θ_{kk}^{*}$ is the optimal solution toEquation (1).

When the inputs and outputs data in Equation (1) are fuzzy numbers, the model is transformed into the fuzzy CCR model.

$\begin{matrix} Max θ_{kk} = \sum_{r = 1}^{s} u_{rk} {\tilde{y}}_{rk} \\ s . t . \sum_{i = 1}^{m} v_{ik} {\tilde{x}}_{ik} = 1; \\ \sum_{r = 1}^{s} u_{rk} {\tilde{y}}_{rj} - \sum_{i = 1}^{m} v_{ik} {\tilde{x}}_{ij} \leq 0; j = 1, \dots, n; \\ u_{rk}, v_{ik} \geq 0; r = 1, \dots, s; i = 1, \dots, m; \end{matrix}$ (2) where “∼” indicates the fuzziness.

2.2 Cross efficiency model

The cross efficiency model introduces a secondary goal to optimize the input and output weights based on the results of the CCR model. In the cross efficiency method, the results of the CCR model are viewed as the self-evaluation results; however, the efficiency should be evaluated from the perspectives of other DMUs, i.e., peer-evaluation. The aggressive and benevolent formulations developed by Doyle and Green [4] are well-known cross efficiency models. Specifically, $\begin{array}{l} Min \sum_{r = 1}^{s} u_{r k} (\sum_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{n} y_{r j}) \\ s .t . \sum_{i = 1}^{m} v_{i k} \sum_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{n} x_{i j} = 1; \\ \sum_{r = 1}^{s} u_{r k} y_{r k} - θ_{k k}^{*} \sum_{i = 1}^{m} v_{i k} x_{i k} = 0; \\ \sum_{r = 1}^{s} u_{r k} y_{r j} - \sum_{i = 1}^{m} v_{i k} x_{i j} \leq 0; j = 1, \dots, n; j \neq k; \\ u_{r k}, v_{i k} \geq 0; r = 1, \dots, s; i = 1, \dots, m; \end{array}$ (3)

Equation (3) is the aggressive formulation of cross efficiency; if the minimized objective function is converted to a maximum, Equation (3) becomes the following benevolent formulation: $\begin{array}{l} Max \sum_{r = 1}^{s} u_{r k} (\sum_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{n} y_{r j}) \\ s .t . \sum_{i = 1}^{m} v_{i k} \sum_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{n} x_{i j} = 1; \\ \sum_{r = 1}^{s} u_{r k} y_{r k} - θ_{k k}^{*} \sum_{i = 1}^{m} v_{i k} x_{i k} = 0; \\ \sum_{r = 1}^{s} u_{r k} y_{r j} - \sum_{i = 1}^{m} v_{i k} x_{i j} \leq 0; j = 1, \dots, n; j \neq k; \\ u_{r k}, v_{i k} \geq 0; r = 1, \dots, s; i = 1, \dots, m; \end{array}$ (4)

If the efficiency of DMU_k equals $θ_{kk}^{*}$ , Equation (3) minimizes the cross efficiencies of the other DMUs, while Equation (4) maximizes them. In this case, the optimal input and output weights determined by Equations (3) and (4) are used to calculate the aggressive and benevolent cross efficiency of other DMUs.

When the inputs and outputs data are fuzzy numbers, Equations (3) and (4) can be converted to the FCE models, i.e., x_ij (i = 1, …, m), y_rj (r = 1, …, s), and $θ_{kk}^{*}$ are replaced with ${\tilde{x}}_{ij} (i = 1, \dots, m)$ , ${\tilde{y}}_{rj} (r = 1, \dots, s)$ , and ${\tilde{θ}}_{kk}^{*}$ , respectively, similar to Equations (1) and (2). However, neither the aggressive nor benevolent formulations are a perfect remedy. First, in reality, the relationships among DMUs are phenomenally complex and cannot simply be attributed to cooperation or competition. Second, the results of different formulations may be different, and no adequate theoretical evidence exists to support which formulation should be chosen. Third, according to available information, the technology used to evaluate FCE is immature.

3 Fuzzy cross efficiency model

3.1 Approach to fuzzy data

According to Hatami-Marbini et al. [19], there are many methods for dealing with fuzzy data. In this paper, the α-level-based approach is chosen to turn a fuzzy number into an interval number by using α-cuts. This approach, proposed by Meada et al. [20], has caused serious concern and has been widely used in fuzzy environments. The general procedure of the α-level-based approach is as follows.

Assume that $\tilde{M}$ is an arbitrary fuzzy number, the function A represents its membership function, and $S (\tilde{M})$ represents its support set. For a given α-level (0 ≤ α ≤ 1), the α-cut set of $\tilde{M}$ is represented by [M^L, M^U] _α. The values of M^L and M^U are determined by $M^{L} = Min {m \in S (\tilde{M}) | A (m) \geq α},$ (5) $M^{U} = Max {m \in S (\tilde{M}) | A (m) \geq α} .$ (6)

In this way, $\tilde{M}$ is transformed into an interval number, and a model with fuzzy data can be solved using the solution of the interval data. The α-level is determined by the needs of the decision-makers. This method can retain sufficient information for decision makers.

3.2 Solution to the fuzzy CCR model

For a given α-level, the input and output data of DMUs are represented by [X^L, X^U] _α and [Y^L, Y^U] _α, respectively, and Equations (7) and (8), proposed by Dimitris [21], are used to solve the DEA model with this data. In these models, the upper and lower bounds of CCR efficiency of DMU_k are given by

$\begin{matrix} Max θ_{kk}^{U} = \sum_{r = 1}^{s} u_{rk} y_{rk}^{U} \\ s . t . \sum_{i = 1}^{m} v_{ik} x_{ik}^{L} = 1; \\ \sum_{r = 1}^{s} u_{rk} y_{rk}^{U} - \sum_{i = 1}^{m} v_{ik} x_{ik}^{L} \leq 0; \\ \sum_{r = 1}^{s} u_{rk} y_{rj}^{L} - \sum_{i = 1}^{m} v_{ik} x_{ij}^{U} \leq 0; j = 1, \dots, n; j \neq k; \\ u_{rk}, v_{ik} \geq 0 r = 1, \dots, s; i = 1, \dots, m; \end{matrix}$ (7) and $\begin{matrix} Max θ_{kk}^{L} = \sum_{r = 1}^{s} u_{rk} y_{rk}^{L} \\ s . t . \sum_{i = 1}^{m} v_{ik} x_{ik}^{U} = 1; \\ \sum_{r = 1}^{s} u_{rk} y_{rk}^{L} - \sum_{i = 1}^{m} v_{ik} x_{ik}^{U} \leq 0; \\ \sum_{r = 1}^{s} u_{rk} y_{rj}^{U} - \sum_{i = 1}^{m} v_{ik} x_{ij}^{L} \leq 0; j = 1, \dots, n; j \neq k; \\ u_{rk}, v_{ik} \geq 0; r = 1, \dots, s; i = 1, \dots, m; \end{matrix}$ (8)

The CCR efficiency interval $[θ_{kk}^{L *}, θ_{kk}^{U *}]_{α}$ is a result of Equations (7) and (8). If $θ_{kk}^{L *} = θ_{kk}^{U *} = 1$ , the DMU_k is fuzzy efficient. However, there may have many fuzzy efficient DMUs among all DMUs; unfortunately, these models do not have enough power to distinguish them. Futhermore, These models have some of the same flaws as traditional DEA. For example, only self-evaluation is taken into account. Therefore, a secondary goal is required to solve these problems.

3.3 Solution to the fuzzy cross efficiency model

Clearly, if the inputs and outputs of DMU_k are $x_{ik}^{L} (i = 1, 2, \dots, m)$ , and $y_{ik}^{U} (i = 1, 2, \dots, s)$ , respectively, and if other DMUs’ inputs are maximal with minimal outputs, the optimal efficiency $θ_{kk}^{U *}$ of DMU_k can be obtained. Conversely, if the other DMUs are considered collectively, the minimum efficiency of the entire collection may be obtained under the same circumstances. Similarly, if the inputs and outputs of DMU_k are $x_{ik}^{U} (i = 1, 2, \dots, m)$ and $y_{ik}^{L} (i = 1, 2, \dots, s)$ , respectively, and if the other DMUs’ inputs are minimal with maximal outputs, the optimal overall efficiency of other DMUs can be obtained. At this time, the optimal efficiency of DMU_k is $θ_{kk}^{L *}$ . Combining the advantages of the benevolent and aggressive formations of cross efficiency, the FCE model can be obtained as follows: $\begin{matrix} Max \sum_{r = 1}^{s} u_{rk} (\sum_{j = 1 j \neq k}^{n} y_{rj}^{U}) \\ s . t . \sum_{i = 1}^{m} v_{ik} \sum_{j = 1 j \neq k}^{n} x_{ij}^{L} = 1; \\ \sum_{r = 1}^{s} u_{rk} y_{rk}^{L} - θ_{kk}^{L *} \sum_{i = 1}^{m} v_{ik} x_{ik}^{U} = 0; \\ \sum_{r = 1}^{s} u_{rk} y_{rj}^{U} - \sum_{i = 1}^{m} v_{ik} x_{ij}^{L} \leq 0; j = 1, \dots, n; j \neq k; \\ u_{rk}, v_{ik} \geq 0; r = 1, \dots, s; i = 1, \dots, m; \end{matrix}$ (9) and $\begin{array}{l} Min \sum_{r = 1}^{s} u_{r k} (\sum_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{n} y_{r j}^{L}) \\ s .t . \sum_{i = 1}^{m} v_{i k} \sum_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{n} x_{i j}^{U} = 1; \\ \sum_{r = 1}^{s} u_{r k} y_{r k}^{U} - θ_{k k}^{U *} \sum_{i = 1}^{m} v_{i k} x_{i k}^{L} = 0; \\ \sum_{r = 1}^{s} u_{r k} y_{r j}^{L} - \sum_{i = 1}^{m} v_{i k} x_{i j}^{U} \leq 0; j = 1, \dots, n; j \neq k; \\ u_{r k}, v_{i k} \geq 0; r = 1, \dots, s; i = 1, \dots, m; \end{array}$ (10)

The purpose of Equation (9) is to find the optimal weights that ensure that the overall cross efficiency of other DMUs is maximal, while simultaneously satisfying the efficiency of DMU_k is $θ_{kk}^{L *}$ . These weights are then used to determine the upper bounds of the FCE value of DMU_j (j = 1, 2, …, n ; j ≠ k) from the perspective of DMU_k. On the other hand, Equation (10) minimizes the overall FCE value of other DMUs to calculate the lower bounds of DMU_j, while simultaneously satisfying the efficiency of DMU_k is $θ_{kk}^{U *}$ . In Equations (9) and (10), the second constraints are used to ensure that the efficiency of DMU_k is equal to its optimal CCR efficiency. Clearly, all weights must be greater than zero.

For DMU_k, the values of the optimal efficiency $θ_{kj}^{*}$ of DMU_j (j = 1, 2, …, n ; j ≠ k) are given by. $θ_{kj}^{*} = \frac{\sum_{r = 1}^{s} u_{rk}^{*} y_{rj}^{U}}{\sum_{i = 1}^{m} v_{ik}^{*} x_{ij}^{L}} k, j = 1, \dots, n; k \neq j;$ (11) where $v_{ik}^{*}$ and $u_{rk}^{*}$ are the optimal weights in Equation (9). The values of the worst efficiency $θ_{kj}^{* *}$ of DMU_j (j = 1, 2, …, n ; j ≠ k) are given by $θ_{kj}^{* *} = \frac{\sum_{r = 1}^{s} u_{rk}^{* *} y_{rj}^{L}}{\sum_{i = 1}^{m} v_{ik}^{* *} x_{ij}^{U}} k, j = 1, \dots, n; k \neq j;$ (12) where $v_{ik}^{* *}$ and $u_{rk}^{* *}$ are the optimal weights in Equation (10). In particular, $θ_{kj}^{*}$ and $θ_{kj}^{* *}$ are not proper upper and lower bounds of DMU_j, respectively, and the interval composed of $θ_{kj}^{*}$ and $θ_{kj}^{* *}$ is only an estimated range of the efficiency of DMU_j from the perspective of DMU_k.

Finally, the FCE upper bound of DMU_j is given by ${\bar{θ}}_{j}^{U} = \frac{1}{n} (θ_{jj}^{U *} + \sum_{k = 1 k \neq j}^{n} θ_{kj}^{*}) j = 1, \dots, n;$ (13)

Here, the upper bounds of DMU_j’s FCE is considered as an average of self-evaluation’s optimal efficiency and peer-evaluation’s optimal efficiency when DMU_j’s inputs are minimal and its outputs are maximal. Similarly, the lower bound of DMU_j is given by ${\bar{θ}}_{j}^{L} = \frac{1}{n} (θ_{jj}^{L *} + \sum_{k = 1 k \neq j}^{n} θ_{kj}^{* *}) j = 1, \dots, n;$ (14)

The lower bounds of DMU_j’s FCE is an average of self-evaluation’s optimal efficiency and peer-evaluation’s worst efficiency when DMU_j’s inputs are maximal and its outputs are minimal. Therefore, the FCE of DMU_j is given by $[{\bar{θ}}_{j}^{L}, {\bar{θ}}_{j}^{U}]_{α}$ .

3.4 Ranking DMUs according to the fuzzy cross efficiency interval

Recall that for a given α-level, the FCE of DMU_j is $[{\bar{θ}}_{j}^{L}, {\bar{θ}}_{j}^{U}]_{α}$ . A minimax regret-based approach (MRA) for comparing and ranking interval efficiency was proposed by Wang [22]. This approach is summarized as follows.

For DMU_k (k = 1, …, n), assume $b = max_{j \neq k} {{\bar{θ}}_{j}^{U}}$ ; then the decision-maker suffers losses and regret if ${\bar{θ}}_{k}^{L} < b$ . The maximum efficiency losses is given by $r_{k}^{U} = max_{j \neq k} {{\bar{θ}}_{j}^{U}} - {\bar{θ}}_{k}^{L}$ . Otherwise, the decision maker suffers no loss and r_k = 0. Therefore, the efficiency losses of DMU_k is given by $r_{k}^{U} = max [max_{j \neq k} {{\bar{θ}}_{j}^{U}} - {\bar{θ}}_{k}^{L}, 0]$ (15)

Clearly, the decision-maker will make a decision based on the principle of $min_{k} {r_{k}^{U}}$ .

Based on the above analysis, the general procedure of MRA is as follows.

Step 1: Select the DMU with the lowest r^U value by calculating r^U for each efficiency interval. Suppose DMU_i1 (1 ≤ i1 ≤ n) is selected.

Step 2: Eliminate DMU_i1 from consideration, recalculate r^U in the remaining DMUs, and reselect the DMU with the lowest r^U. Suppose the most desirable DMU is DMU_i2 (1 ≤ i2 ≤ n, i2 ≠ i1).

Step 3: Repeat the above steps until only one DMU_in remains. As a result, for a given α-level, the ranks of the DMUs are obtained and are represented by DMU_i1 ≻ DMU_i2 ≻ ⋯ ≻ DMU_in.

4 An illustrative example

In this section, a numerical example is provided to show the function of the FCE model and to illustrate how the model is solved.

To simplify calculations, the input and output data of the DMUs are assumed to be triangular fuzzy numbers, [l, m, u], which are generated in an arbitrary way. These values are listed in Table 1.

To better show the calculation process, we assume α= 0.75. Using Equations (7–10), we obtain the results shown in Tables 2–4.

The bold numbers in Tables 2 and 3 are the efficiency values obtained through self-evaluation. In Tables 2 and 3, all of the efficiency values obtained by peer-evaluation are lower than the upper bound of the self-evaluation efficiency interval, which is a reasonable phenomenon. Note from Table 4 that the ranking based on the FDEA model can hardly be distinguished for all of the DMUs. For example, the value of the fuzzy CCR efficiency of DMU2 is equal to that of DMU7; thus, we cannot identify which is more effective. This problem is solved by the ranking of the FCE. To be clear, for the FDEA model, this problem is even more serious than for the DEA model with crisp data. This is because that the constraints become weaker in the fuzzy CCR model. Furthermore, there are some differences between the two ranking results, which are caused by the differences between the self-evaluation result and peer-evaluation result.

In a fuzzy environment, different decision makers have different attitudes; thus, the efficiencies and rankings at different α-levels should be calculated separately for the DMUs. These results are shown in Table 5.

According to the longitudinal comparison in Table 6, the ranking is stable at different α-levels. For this kind of stability, the ranking achieved by FCE is noticeably better than that of FDEA. However, for the ranking of FCE, it also has some differences between different α-levels. For example, if α ≤ 0.25, DMU8 ≻ DMU6; otherwise, the order is reversed. This implies that different α-levels change results, even though the change may be limited. Moreover, as the α-level increases, the efficiency uncertainty decreases. The horizontal comparison shown in Table 6 proves that the FCE has more distinguishing power. For example, when α ≥ 0.75, DMU2 ≻ DMU7 for FCE, while DMU2=DMU7 for FDEA. Furthermore, the ranking of DMU7 is very different for FDEA and FCE. This is because the inputs and outputs, which favor DMU7, are heavily weighted by FDEA. Thus, we obtain a more reasonable result using FCE.

5 Conclusion

In this paper, we developed a new FCE model for dealing with imprecise data in fuzzy environments. The significance of turning a fuzzy number into an interval number using the α-level-based approach is that decision-makers can make decisions based on their own preferences and real-world situations. The FCE model was used to calculate the efficiency interval of DMUs, which successfully overcomes the flaws of traditional FDEA. The MRA was used to order all DMUs according to their efficiency interval. Finally, an illustrative example was given to show the advantages, potential, solution, and significance of the FCE model by comparing it with the FDEA model.

Inevitably, there are some limitations of the proposed model. First, the α-level-based approach requires a certain level of effort and experience from the decision maker. Second, the interval of FCE is not sufficiently precise. We plan to address these problems in our future research.

Footnotes

Acknowledgments

This research is supported by the National Natural Science Foundation of China under the Grant No. 71371053, the Research Fund for the Doctoral Program of Higher Education of China under the Grant No. 20123514110012, Humanities and Social Science Foundation of the Ministry of Education under the Grant No. 14YJC630056, and Natural Science Foundation of Fujian Province, China under the Grant No. 2014J01264.

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