Modifying the convexity condition in Data Envelopment Analysis (DEA) Modificación de la condición de convexidad en el Análisis Envolvente de Datos (AED)

Conventional Data Envelopment Analysis (DEA) models are based on a production possibility set (PPS) that satisfies various postulates. Extension or modification of these axioms leads to different DEA models. In this paper, our focus concentrates on the convexity axiom, leaving the other axioms unmodified. Modifying or extending the convexity condition can lead to a different PPS. This adaptation is followed by a two-step procedure to evaluate the efficiency of a unit based on the resulting PPS. The proposed frontier is located between two standard, well-known DEA frontiers. The model presented can differentiate between units more finely than the standard variable return to scale (VRS) model. In order to illustrate the strengths of the proposed model, a real data set describing Iranian banks was employed. The results show that this alternative model outperforms the standard VRS model and increases the discrimination power of (VRS) models.


INTRODUCTION
In standard microeconomic theory, the concept of a production function forms the basis for a description of input-output relationships in a firm. The production function shows the maximum amount of outputs that can be achieved by combining various quantities of inputs. Thus initially the problem is the construction of an empirical production function or frontier based on the observed data. Data envelopment analysis (DEA) has been recognized as an effective nonparametric mathematical optimization technique for measuring the relative efficiency of a group of comparable decision-making units (DMUs) with multiple inputs and outputs (Charnes et al., 1994), DEA has been used in many contexts including education systems, health care units, agricultural productions, military logistics and many other applications (Charnes et al., 1994;Alder et al., 2006;Emrouznejad et al., 2008;Lozano et al., 2013;Asmild et al., 2013). Conventional DEA models assume real-valued and non-negative inputs and outputs vectors. Besides, each DMU is expressed by the notation ( , ) xy. The first component is regarded to input and the second can specify outputs. The set of feasible activities is called the production possibility set (PPS) and is denoted by P and satisfied in the axioms of Envelopment, free disposability, constant return to scale (CRS) or unbounded ray and semi positive linear combination of activities to construct P. Regarding to these axioms, the ray from the origin through the highest point is the CRS efficient frontier. As pointed out correctly in Podinovski, 2004 CRS models require full proportionality assumptions between all inputs and outputs. The pioneering CCR 1 model proposed by Charnes et al., 1978 satisfied in the above mentioned postulates. Various extension of axioms has been proposed in DEA literature. Among the modified variation, BCC 2 model was presented by Banker, 1984. The BCC model has its production frontiers spanned by the convex hull of the existing DMUs. In other words, the axiom of CRS (Constant Return to Scale) was extended and modified. The frontier has piecewise linear and concave characteristics. Also satisfies in variable return to scale (VRS) properties. Other extension and modification of CRS property can be found in (Seiford et al., 1990;Petersen., 1990;Bogetoft, 1996;Bogetoft et al., 2000;Färe et al., 1985) Theoretically, these classic models evaluate proportional efficiency by maximizing ratio of the weighted sum of its outputs to the weighted sum of its inputs, subject to the condition that this ratio does not exceed one for any DMU. Another variation of DEA models is Additive model. This model has the same production possibility set as CCR and BCC models and their variants but treats the input excesses and output shortfalls directly in the objective function. A slack-based measure of efficiency (SBM) is another version of DEA models. This measure makes its efficiency evaluation as effected in objective, invariant to the measures used for different inputs and outputs (Cooper et al., 2007). Free disposal Hull (FDH) model assumes a nonconvex possibility set and firstly presented by Deprins et al., 2006. This model ignores the convexity axioms and then has extended by Tulkens, 2006 As it can be seen the modification and extensions of axioms leads to different models. As another example, modifying free disposability leads to notation of weak disposability which was demonstrated by Kuosmanen, 2005 and extended by various authors. Among them, refer to (Podinovski et al., 2011;Färe et al., 2004) . For the notation of congestion can refer to Cherchye et al., 2001 As it can be seen different setting of axioms lead to different models. One of the most modified axioms was convexity. When the attention restricts to convexity axioms, the most applied BCC model is imagined. In this model, the convex hull of existing units' parallel to variable return to scale (VRS) characteristic evaluates the efficiency. Reserving the other axioms, that is to say, Envelopment, free disposability and VRS, a dilemma can raise. If the construction of convex hull is modified, how the efficiency frontier changes. What's more, to what extent, the efficiency score does change? This article tackles with this dilemma. Looking over the existing studies, this paper focuses on DEA axioms while the convexity axioms is modified or relaxed. In other words, the postulates of Envelopment, free disposability also variable return to scale are preserved. On the other hand, the convexity condition has been modified and extended. This extension is imposed as an interval with upper and lower bound. Equipped with the relaxed axioms, a two-step procedure is presented to evaluate the efficiency. The first stage surveys the minimum quantity of boundaries which might be used in the second stage. A linear model is proposed to evaluate the efficiency score in the second stage. As the relaxed axioms claim the frontier of proposed method can be inserted between two classic DEA model, i.e. CCR and BCC model. The results show that this model can estimate the efficiency of DMUs more accurate than BCC model and operates as well as this model. The real example of 37 Iranian bank branches supports the idea behind this modified axiom. The rest of this paper is organized as follows: In the following section, the axioms of standard DEA models are presented. A proposed two-phase model with modified convexity condition is presented in the third section. An empirical example highlights the model strengths in section 4. Conclusion will end the paper.

PRELIMINARIES
In DEA each observed DMU is characterized by a pair of non-negative input and output vectors The classic (Charnes et al., 1978) DEA model assumes that the underlying production possibility set (PPS) denoted by {( , ) can produce } ms T x y x R y R    and satisfies the following axioms: Free disposability: ( , ) x y T  , ( , ) According to the minimum extrapolation principle (Banker et al., 1984), the DEA production possibility set (PPS) is the intersection of all sets ms SR   that satisfy the maintained axioms. Under the maintained assumptions (1) -(4), the minimum extrapolation PPS can be explicitly stated as: If the study turns to efficiency estimations of DMUs and attention has restricted to classic Farrell input efficiency measurement defined as: programming problem of this evaluation is called CCR (Charnes et al., 1978) input-oriented and has the following format: This model is a constant return to scale (CRS) program and it assumes that the status of all input/output variables are known prior to solving the model. The efficiency ratio o  ranges between zero and one, with o DMU being considered relatively efficient if it receives a score of one. From a managerial perspective, this model delivers assessments and targets with an output maximization orientation. In addition to these basic postulates, technology T also be assumed to satisfy some of the following returns to scale axioms: Under the minimal assumptions of free disposability (2) and convexity (4), the minimum extrapolation PPS is referred to as the variable return to scale (VRS) technology, formally stated as: Regarding to this technology, the mathematical programming problem foe estimating the efficiency of DMUs has the following format: The above model is an input oriented BCC (Banker et al., 1984) model. A similar model can be formulated to present output orientation VRS technology. If in addition to axioms (2) and (4) the assumption of NIRS (5) is imposed, the resulting minimum extrapolation PPS is the NIRS technology: On the other hand, if instead of (5) one assumes NDRS (6), then the minimum extrapolation PPS is the NDRS technology: As it can be seen relaxation of axiom (3) leads to models of variable and non-increasing decreasing) returns to scale (Seiford et al., 1990) In addition to these four classical DEA technology, many variations of axioms (2) -(4) have been presented in the literature. Relaxation of axiom (2) leads to models of weak disposability (Kuosmanen, 2005) and congestion (Cherchye et al., 2001). Relaxation of (4) leads to free disposable hull (Deprins et al., 2006) and free replicable hull models (Tulkens, 2006). The above technologies and models are common models of relaxation axioms. In following section relaxation of convexity axiom will be discussed from another perspective.

METHODOOLOGY
Suppose that we have , .
x y T x x and y y 

5) Minimal extrapolation: For each
T satisfying in axioms 1-4, we have TT   An Algebraic representation of the PPS for technology DEA CRS T , which satisfying the axioms 1-5, is given as Relaxation the convexity condition, the variable return to scale (VRS) technology was presented by Banker et al., 1984 and formally stated as: . In the spirit of relaxing convexity axiom, the goal comes to specification of  . Toward this end, a two-phase procedure is established.

First step:
Looking for the minimum quantity for variable  falls behind the scope of proposed first stage.
It is worth to note that the proposed technology has the minimum extrapolation interpretation under adapted set of axioms. To measure efficiency improvement, a modified input efficiency measure is needed. Our attention has been restricted to Farrell input efficiency measure defined as: The unit under evaluation is denoted by ( , ) oo xy. Applying this measure directly to RVRS T can yield the modified input efficiency scores relative to the adapted reference technology. Also it can be computed by solving the following model and solver software.
Symbol  denotes a non-Archimedean infinitesimal and parameter  plays the role of abatement factor on inputs, as it does on model (2). One important feature of model (3) Model (4) (3). One important conclusion is that intensity variable is same in both models. This unchanged feature makes the proposed frontier acts between two known frontiers.

Theorem2:
The efficiency score of model (4) is not worse than that of BCC model (model (2)). That is ) θ (θ NEW BCC  .

Proof:
The last constraint of model (4)   be a feasible solution of BCC model (model (2)), it is also a feasible solution of model (4). Comparing both models' feasible space, it is concluded that the space of BCC model (model (2)) is the subset of space of model (4). Hence, the optimal value of BCC model (model (2)) is not better than the optimal value of model (4). That is Theorem 3: In model (4), at least there exit one efficient unit.

Proof:
Consider the dual format of BCC model (model (2) The relation (7) admits that there must be at least one constraint for which the optimal weight ) , (   v u leads to equality between the left and right hand side. Now, consider the dual format of proposed model (4). The dual form is stated as follows: Which is impossible. Therefore, the first constraint and the objective function of the dual model (8)   and this completes the proof. Briefly, a two-stage approach can simplify the procedure of finding the minimum quantity of proposed lower and upper bound. Our next objective is to characterize PPS that satisfies the minimum extrapolation principle subject to the properties 1-4. To illustrate, consider a simple numerical example in a single-input single-output case. Table1 shows the five DMUs. This example aptly illustrates that a sequential application of the axioms can generate a monotonic and convex hull of PPS. Figure 1   T are verified by this simple example. By applying envelopment, free disposability, constant return to scale and convexity parallel to minimum extrapolation the black linear frontier represents the DEA CRS frontier. The frontier was demonstrated as CCR frontier. Applying envelopment, free disposability and convexity again imposing variable return to scale parallel to minimum extrapolation. The piece-wise linear frontier depicts DEA VRS frontier. This frontier shows BCC frontier. For the purpose of comparison, the adapted axioms are implemented on this data set. Regarding to axiomatic foundation (envelopment, free disposability and variable return to scale) with relaxed convexity along with minimum extrapolation, the proposed frontier has the following format. Figure 2 represents the frontiers on the sample data set.  Figure 2 shows the estimated frontier of model (4) have a character which comes closer to DEA CRS (CCR model) rather than DEA VRS (BCC model). What's more, it is worth to note that the proposed frontier has some similar behavior as DEAVRS (BCC model). This behavior can be characterized as the effect of constraint (4-3). Since this constraint allows the intensity variable  alters in described optimal minimum bound. With reference to Figure 2, unit D2 has lied on three frontiers. Interestingly, the proposed frontier in unit D2 is tangent to the frontier between BCC and CCR frontier. This property is consistent with Theorem (3). One of the important feature of proposed model (model (4)) can be driven with regarding to Figure 2. The proposed frontier is upper or tangent to the DEA VRS frontier (BCC model). Therefore, the efficiency measured by model (4) is not worse than that measured by BCC model (model (2)). Clearly, this property supports Theorem2. The example illustrates the need to modify the input efficiency measure in the case of relaxation convexity. The radial efficiency scores obtained with our proposed model come close to those of DEA VRS (BCC) models, but there are some notable differences, particularly with DMUs 1, 3 and 4. DMU#1 is an efficient unit in DEA VRS model while its radial efficiency is 0.91 in proposed method. The benchmarks obtained by our proposed model are very close to the DEA VRS model. However, there exist notable differences between the computed targets in our proposed model with those obtained by DEA VRS formulation. Solving the DEA VRS model (model (2)), we obtain the radial efficiency of 0.53 and reference point (2.67, 3.01) with intensity variables 1 = 0.33 2 = 0.67 for inefficient unit D5. On the hand, as the Figure 2 shows the radial efficiency for this inefficient unit with regard to relaxed convexity technology is 0.46 and reference point (2.29, 2.27). This example demonstrates that proposed technology, which are justified by the adapted axiomatic analysis, can lead to efficiency scores and performance targets not greater that the known DEA VRS method. The application of the next section demonstrates that the proposed formulations can yield substantially different results.

AN EMPIRICAL EXAMPLE
This section illustrates the proposed model in assessing 37 Iranian bank branches. Four factors are selected as inputs: personnel (Staff) privilege ) (x 1 , benefit payment ) (x 2 and delayed demands ) (x 3 , and one factor recorded as output: interest (y1). For comparison, three alternative models were computed: CCR model (model (1)), BCC model (model (2)) and our proposed model (model (4)). The obtained radial input efficiency scores are presented in Table 3. Implementing our proposed model (4)   The last column of Table 3 shows the results of proposed method. The results of our proposed model come close to BCC model, but there are some notable differences, particularly units 30, 26, 11, 5, 4 and 3 are efficient in BCC model but in the proposed model are inefficient. Only six out of 37 units exactly give the same efficiency score as BCC model (2) does. In general, the efficiency scores of our proposed model (4) is always smaller than those of BCC model (2) and larger than those of CCR model (model (1)). This suggests that the relaxed convexity axioms can enhance the discriminatory power of the model. The application also demonstrates that the BCC model leads to overestimated efficiency assessments. Finally, we would like to emphasize on statistical analysis. The last three rows of Table 3 depict average, variance and standard deviation respectively. The average of the proposed method (model (4)) is smaller than BBC model and larger than CCR model. That is to say, this average is inserted between two known models' average. This claims that the proposed model is able to identify the efficiency distribution. The variance of the proposed model (4) is same as the BCC model. Also, both of them are smaller than CCR variance quantity. What's more, the standard deviation of proposed model (4) is extremely small. This claims that the efficiency scores obtained by proposed model (4) tend to be close to the average. In practice, dispersion of efficiency scores in proposed model (4) is extremely lower than the CCR model. In other words, a low standard deviation indicates that the efficiency scores are spread out over a tighter range. Also, the results acknowledge that our proposed model (4) increase discrimination power of DEA VRS models.

CONCLUSION
Based on the widespread application of Data envelopment analysis (DEA) in performance estimation, it is worth to provide a suitable model to improve the efficiency. Each of the standard DEA models is constructed on specific postulates. Considering some axiomatic foundations, a production possibility set (PPS) is defined. In this paper we have presented an axiomatic foundation for a DEA model by relaxing the axiom of convexity. After modifying the notions of convexity, a two-step approach has been identified to yield a convex and monotone frontier. The proposed frontier has inserted between constant return to scale (CRS) DEA model and variable return to scale (VRS) DEA models. Although the efficiency score of