A Modified DDF-Based Super-Efficiency Modelhandling Negative Data

In conventional DEA models, inputs and outputs are assumed to be non-negative while negative data may occur in some DEA application such as the performance analysis of socially responsible and mutual funds; and the macroeconomic performance where “rate of growth of GDP per capita” can be either negative or positive. To handle the negative data and provide a measure of efficiency for all units, many researches have been studied. In this paper, the radial super-efficiency model based on Directional Distance Function (DDF) is modified to provide a complete ranking order of the DMUs (including efficient and inefficient ones). The proposed model shows more reliability on differentiating efficient DMUs from inefficient ones via a new super-efficiency measure. The properties of proposed model include feasibility, monotonicity and unit invariance. Moreover, the model can produce positive outputs when data are non-negative. Apart from numerical examples, an empirical study in bank sector demonstrates the superiority of the proposed model.


Introduction
Data Envelopment Analysis (DEA) is a powerful tool in the context of production management for performance measurement. The purpose of DEA is to measure the relative efficiency of a set of decision making units (DMUs) where multiple inputs convert into multiple outputs (Charneset al. (1978)). Conventional DEA models assume non-negative values for inputs and outputs. However, there are many applications in which one or more inputs and/or outputs are necessarily negative such as the performance analysis of socially responsible and mutual funds (Basso & Funari (2014)), and the macroeconomic performance where "rate of growth of GDP per capita" can be either negative or positive (Lovell(1995)).In DEA literature, there have been various approaches for dealing with unrestricted in sign variables. Pastor (1996) approached negative data using a translation invariance classification, for the first time. That is, in light of the translation in variance property in basic DEA models such as the additive model, the original negative data can be equivalently converted to positive data by adding a constant number. However, many DEA models such as CCR may not have this property to be applied as a treatment of negative data (Ali & Seiford, 1990). A number of significant contributions have been developed in the DEA literature to address the occurrence of negative data (e.g., Kerstens Lin and Chen (2017) proposed a novel DDFbased VRS radial super-efficiency DEA model which is feasible and is able to handle negative data. They claimed that their proposed model can provide a measure of efficiency for all DMUs in the presence of negative data. This paper highlights some cases that their model is not responding for ranking of DMUs for example when DMUs consume the same inputs. Apart from Hadi-Vencheh and Esmaeilzadeh (2013), and Lin and Chen (2017), super-efficiency models with negative data have received no attention in the literature. The contribution of this paper is fivefold: 1 Note that zero data can make the super-efficiency model under CCR infeasible (Lee and Zhu (2012)) 1.
A modified DDF based super-efficiency model interacting with negative data is proposed.

2.
The proposed model is always feasible and conveys good properties such as unit invariance, monotonicity, and providing positive outputs when data are nonnegative.

3.
The proposed model can provide a ranking order for all DMUs via a new superefficiency measure and produce improved targets for inefficient units.

4.
By using different changing rates for inputs and outputs in the proposed model, DMU reaches the frontier with maximum potential in inputs and outputs.

5.
This study shows that in distinguishing DMUs to efficient and inefficient ones, proposed model shows higher reliability than the other super-efficiency model compared in this study.
The rest of the paper is outlined as follows. Section 2 briefly presents the concept of DDF,DDF-based super-efficiency model and the model proposed by Lin and Chen (2017). In Section 3, a modified DDF-based super-efficiency model handling negative data is introduced. In section 4, the proposed model is applied to a numerical example. The penultimate section is devoted to an illustration application and finally Section 6 concludes this study.
The reference bundle ( , ) can be chosen in an arbitrary way and this makes the DDF varies with reference to the evaluated DMU. The VRS DEA formulation for model (1) is as follows: ∑ ≥ + =1 , r = 1, 2, … , s, , ≥ 0, j = 1, 2, … , n.. Model (2) combines the features of both an input-and output-oriented models in which each input and output of the unit under assessment are decreased and increased respectively, at the same time by the same portion β.The factor * as the optimal value of βin model (2) is the Nerlove-Luenberger (N-L) measure of technical inefficiency for the evaluated DMU. By implication, its efficiency equals 1 − * (Ray (2008)).

Super-efficiency model based on DDF
The idea behind the super-efficiency method is that a DMU under analysis is excluded from the reference set so that the efficientDMUs can receive scores greater than or equal to the unity while the score for the inefficient DMUs do not change. In so doing, the super-efficiency version of model (1) Ray (2008) defined the super-efficiency score of the evaluated DMUo equals1 − * , where * is the optimum value of model (3). The smaller the value of * , more efficient the DMUo is. For any efficient DMUo, 1 − * is no less than 1.
The direction vector ( , ) should be non-negative and non-zero, and can be chosen in anarbitrary way (Chen et al. (2013), Ray (2008)). I fall input and output data are nonnegative, the standard DDF for the DMUo is adopted by choosing ( , ) as ( , ) (Chambers et al. (1998)) and the N-L super-efficiency model (NLS model) is obtained. The NLS model is very often feasible for non-negative data, but it fails in two cases (Ray (2008)

Proposed model by Lin and Chen(2017)
Lin and Chen (2017) showed that in the presence of negative data, both the NLS and LCS models might be infeasible. This is because their related direction vectors might be negative, which could result in the DMUo to be further away from the super-efficiency frontier and thus lead to infeasibility. Accordingly, they choose a new direction vector which is always non-negative and non-zero, independent of inputs and outputs being nonnegative or not. Their proposed model is as follows: To exemplify the Lin and Chen's proposed model (4), let us consider the numerical example presented in Table1 where there are eight DMUs with one positive input( ), and two free in sign-valued outputs( 1 2 ).
The results of applying model (4) to the DMUs in Table 1 are presented in Table 2. The optimal values of 1 − * besides the optimal slack values (s * ; 1 * , 2 * )are shown in columns two-five. The input and outputs projections( * ; 1 * , 2 * ) are represented in the columns six-eight. Projection points are computed by inserting the optimal value in the right-hand side of the input and output inequalities in model (4).  Table 2 reports that * = * = * = * = * = 0, * = −0.1364, * = −0.1200 and * = −0.2657. DMUs A, G and H are Pareto-efficient, while DMUs B, C, D, E and F are inefficient due to the optimal slack-values. Table 1 shows that all the DMUs are on the frontier in their input components meaning that input level is efficient; but due to illogical results for DMUs A, G and H the input projections are not on the efficient frontier, as represented in the sixth column of Table 2. This is because + > 0, = 1, 2, … , for each ∈ {1, 2, … , } and model (4) uses a unified changing rate for both inputs and outputs. Thus, when DMUs consume the same inputs, our expectation is * = 0 and * = 1 for all DMUs whether efficient or inefficient. This demonstrates that the optimal values of * and the projection points for DMUs A, G and H are illogical results. Consequently, using the 1 − * as the super-efficiency measure, model (4) is unable to provide a complete ranking order for all DMUs. Note that this expectation is not true, when DMUs produce the same outputs; because in this case − = 0, = 1, 2, … , for each ∈ {1, 2, … , }and the output constraints in model (4) is disappeared due to convexity constraint.

Proposed super-efficiency model
The single input and both outputs cannot be moved at the same rate to the frontier due to the fact that input level is already efficient, as shown in Table 1. The proposed model by Lin and Chen (2017) is unable to provide a complete ranking order for all the DMUs when DMUs consume the same inputs. Different rates and should be used for inputs and outputs, respectively. To this end, the proposed model is as follows: In order to the evaluated DMU reach to the super-efficiency frontier, following conditions are required. If DMUo is efficient, inputs should be increased and outputs should be contracted, which means ≤ 0and ≤ 0. And if DMUo is inefficient, inputs should be contracted and outputs should be increased, which means ≥ 0and ≥ 0. These conditions are incorporated by enforcing . ≥ 0 into the model (5). Due to the constraint of . ≥ 0, model (5) is a non-linear programming problem. This non-linear programming problem can be transformed to a linear programming problem using the following procedure. Two binary variables, w and z are introduced and the non-linear constraint . ≥ 0is transformed into a set of linear constraints as follows: where M is a sufficiently large number. Obviously, w=1 and z=0 signify ≥ 0and ≥ 0, respectively; and w=0 and z=1 signify ≤ 0 and ≤ 0, respectively. By substitution of this set of the linear constraints for . ≥ 0, model (5) becomes a mixed integer linear programming problem. When the evaluated DMU moves simultaneously to the frontier through direction of and , the non-zero slacks may be survived. To verify the existence of non-zero slack(s), a non-Archimedean infinitesimal with sum of slacks is incorporated into the objective function of model (5) to reflect the optimal slack calculation process in the standard DEA model. (5) is always feasible and the following inequalities are hold: a) 0 ≤ * < 1and0 ≤ * for ( , ) ∈ ;and also b) −1 ≤ * < 0and−1 ≤ * < 0 for( , ) .

Proposition 1.Model
Proof. The proof is given in Appendix A. Corollary 1.Let * and * be the optimal solutions of model (5), then 1− * 1+ * ≥ 0. According to corollary 1, the measure of super-efficiency can be defined as following: If * > 1, then DMUo is an extreme efficient unit.
(b) If * = 1andthe optimal value of the slacks generated by model (5) are zero, then DMUo is a non-extreme efficient unit.
(c) If * = 1and the optimal value of the slacks produce by model (5) are non-zero, then DMUo is a weak efficient unit.
From model (5)  Further examination of the proposed method is made by applying DMUs in Table 1. Table 3 reports the results when proposed model is applied to the numerical example in Table1. The optimal solutions of the proposed model i.e. the optimal values of * and * are shown in the second and third columns of Table 3, respectively; and the superefficiency measure ( * )is presented in the fourth column. The columns five-seven of Table 3 show the projection point for a DMU under evaluation. 2 Note that Whenβ y * = −1, * diverges to infinity, hence the super-efficiency measure is assumed to be infinity. The results show that DMUs A, G and H are efficient; since their supper-efficiency measures are greater than one. However, DMUs B, C, D, E and F are inefficient, since their supper-efficiency measures are less than one. Using different changing rates for input and outputs, proposed model provides β x * = 0 for all DMUs, unlike the model (4). Column five shows thatx * = 1for all the DMUs, and this logical outcome was expected. The proposed model provided ranking order for all the DMUs, shown in column eight: ≻ ≻ ≻ ≻ ≻ ≻ ≻ .

Numerical example
In this section, data set of "the notional effluent processing system" from Sharp et al. (2007) is used to show the applicability and merits of the proposed model.
The data set is presented in Table 4. There are 13 DMUs, with two inputs {x1, x2} and three outputs {y1, y2, y3}:one positive input (cost), one non-positive input (effluent), one positive output (saleable output), and two non-positive outputs (methane and CO2).  (5) on data sets used in Table  4. The optimal values of * , * and the super-efficiency measure * are represented in the columns four-sixin Table 5, respectively. DMUs C, G, H, K and M are efficient, since their super-efficiency measures are greater than 1. Other DMUs are inefficient, since their supper-efficiency measures are less than 1. The ranking order for DMU M is superior to other DMUs, as shown in the seventh column in Table 5. As it is shown in the second and the sixth columns in Table 5, both models are feasible for all DMUs and they can differentiate the performance of both efficient and inefficient DMUs for used data set. The ranking orders of both models are close; however their super-efficiency measures are different. This is due to the fact that, in proposed model i.e., model (5) different rates, and for inputs and outputs respectively are used, while the same rates are used for both inputs and outputs in the model (4). The superefficiency measure provided by Model (4) for DMU I is 1.0000 however, the measure of * as the measure of super-efficiency yielded by model (5) is 0.9912. This shows that model (5) is more responsive than model (4) and it can differentiate the DMUs more discretely. Table 6 shows the target input-output values of inefficient DMUs, determined by the model (5). The proposed model demonstrates that in each inefficient DMU, the inputs and the outputs should be reduced and expanded, respectively, in order to tend to the super-efficiency frontier. Hence, the proposed model can provide improved target inputs and outputs for all the inefficient DMUs.  (2017) calculated the improved targets for inefficient DMUs. By comparison of their results and the results obtained using proposed model, shown in Table 6, it can be concluded that for some of the DMUs the targets obtained using model (4) is more improved (in some components) than the one obtained using model (5). In other DMUs the proposed model provided more improved targets (in some components) than the model (4). These variations are due to having different directions in their movements to reach the super-efficiency frontier.

An empirical application
In this section the proposed model is illustrated by applying it to a real world data of the 61 banks in the GCC 3 countries. In this evaluation, the input variables are total assets, capital and deposits. The output variables are loans and equity in each branch. Note that the last output could take both positive and negative values among the banks. For full definitions of variables see Emrouznejad and Anouze (2010). Table 7 below shows the descriptive statistics of the variables. The outcomes after applying assumed data set in Model (4) and in model (5) are reported in Table 8. From the second, third and the sixth columns in Table 8, VE model is infeasible for DMUs 1, 9 and 46. Both models (4) and (5) (5) is more precise and responsive than model (4) in discriminating the DMUs. From Table  8,all the super-efficiency scores yielded by model (5) for inefficient(efficient) DMUs are less than or equal (bigger than or equal)to those generated by model (4) as shown in Figure 1.Thus, super-efficiency scores vary from 0.9194 to 1.2310 under the Lin and Chen's model, whereas they vary from 0.4433 to 1.3004 under our proposed model. As can be seen,in general, the super-efficiency scores obtained from model (4) is around 1.0000 for inefficient DMUs, whereas these scores yielded from model (5) have bigger changing ranges for inefficient ones. From Table 8 Table 9 shows the target input-output values of inefficient DMUs, determined by both models.  The proposed model demonstrates that in each inefficient DMU, the inputs and the outputs should be reduced and expanded, respectively, in order to tend to the superefficiency frontier.
From the theoretical analyses it is concluded that, the same as Lin and Chen's model, the proposed model can deal with the data set with free in sign values and can provide improved targets for inefficient DMUs.

Conclusion
Conventional DEA models are introduced to evaluate DMUs with non-negative data, while in practice there are important DMUs with negative data and they need to be evaluated. Recently, Lin and Chen (2017) proposed a novel DDF-based VRS radial super-efficiency DEA model which is feasible and is able to handle negative data. They claimed that their proposed model can provide a measure of efficiency for all DMUs. In this study, it is shown that although their proposed model can overcome the common infeasibility problem, the model has failing in some cases. It is unable to provide a complete ranking order and logical results in such a case that all DMUs consume the same inputs. This is because(i) in this model a unified changing rate for both inputs and outputs is used and (ii) the input improvement direction is strictly positive. In this study, a modified radial DDF-based super-efficiency model is proposed to provide a complete ranking order for all the DMUs via a new super-efficiency measure.
Following conclusion is made after redefining a i and b r .
Proposition 3.If inputs (outputs) of the DMUo are reduced (increased), the optimal value of model (5) does not increase for a i and b r defined in (12) and (13).
Proof. If specified input reduction and output expansion happens, the direction vector is (x io − ∆x io + a i , y ro + ∆y ro − b r ). The following statement is made by having the definitions of (12) and (13). x io − ∆x io + a i > 0, i = 1, 2, . . . ,m, and y ro + ∆y ro − b r > 0, r = 1, 2, . . . , s. Consequently the corresponding model (5) Assume the optimal solution of model (14) as (λ ′ , ′ , ′ ). A similar derivation made in (10) is applied for input constraint of the model (14) Thus, ′ < 1. A similar derivation made in (11) is applied for output constraint of the model (14) using (13) as following.