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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 658635, 7 pages

http://dx.doi.org/10.1155/2013/658635

## An Extension of Cross Redundancy of Interval Scale Outputs and Inputs in DEA

^{1}Department of Mathematics, Science and Research Branch, Islamic Azad University, Hesarak, Poonak, Tehran, Iran^{2}Faculty of Mathematical Science and Computer Engineering, University for Teacher Education, 599 Taleghani Avenue, Tehran 15618, Iran

Received 19 November 2012; Accepted 22 May 2013

Academic Editor: Hadi Nasseri

Copyright © 2013 Farhad Hosseinzadeh-Lotfi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is well known that data envelopment analysis (DEA) models are sensitive to selection of input and output variables. As the number of variables increases, the ability to discriminate between the decision making units (DMUs) decreases. Thus, to preserve the discriminatory power of a DEA model, the number of inputs and outputs should be kept at a reasonable level. There are many cases in which an interval scale output in the sample is derived from the subtraction of nonnegative linear combination of ratio scale outputs and nonnegative linear combination of ratio scale inputs. There are also cases in which an interval scale input is derived from the subtraction of nonnegative linear combination of ratio scale inputs and nonnegative linear combination of ratio scale outputs. Lee and Choi (2010) called such interval scale output and input a cross redundancy. They proved that the addition or deletion of a cross-redundant output variable does not affect the efficiency estimates yielded by the CCR or BCC models. In this paper, we present an extension of cross redundancy of interval scale outputs and inputs in DEA models. We prove that the addition or deletion of a cross-redundant output and input variable does not affect the efficiency estimates yielded by the CCR or BCC models.

#### 1. Introduction

In many DEA applications, such as income, an interval scale output in the sample is derived from the subtraction of nonnegative linear combination of ratio scale outputs and nonnegative linear combination of ratio scale inputs. There are also many cases, like cost, in which an interval scale input is derived from the subtraction of nonnegative linear combination of ratio scale inputs and nonnegative linear combination of ratio scale outputs, although the effect of such dependencies on DEA is not clear. Lee and Choi [1] called such interval scale output and input a *cross redundancy*. They proved that the addition or deletion of a cross-redundant output variable does not affect the efficiency estimates yielded by the CCR or BCC models. Francisco J. López [2] generalized the contributions of Lee and Choi by introducing specific definitions and conducting some additional analysis on the impact of the presence of other types of linear dependencies among the inputs and outputs of a DEA model. In this paper, we deal with cross-redundant output and input variables simultaneously in DEA models. We prove that the addition or deletion of a cross-redundant output and input variable does not affect the efficiency estimates yielded by the CCR or BCC models. The paper is organized as follows. In Section 2, we introduce preliminaries of DEA. In Section 3, we present our main results. In Section 4, we will illustrate that the addition or deletion of cross-redundant output variable and input variable does not affect the efficiency estimates yielded by the CCR or BCC models. Conclusions are summarized in Section 5.

#### 2. Preliminaries

Suppose that we have peer observed DMUs, which produce multiple outputs , by utilizing multiple inputs . The input and output vectors of are denoted by and , respectively, and we assume that and are semipositive, that is, and for . We use to descript and specially use as the DMU under evaluation. Throughout this paper, vectors will be denoted by bold letters.

The input-oriented CCR [3] multiplier model evaluates the efficiency of each ??by solving the following linear program: Because and are semipositive for , . Also since and , we have . Thus . represents the input-oriented CCR-efficiency value of .

The output-oriented CCR multiplier model evaluates the efficiency of each ??by solving the following linear program: Since and , we have . represents the output-oriented CCR-efficiency value of . Also [4].

The input-oriented BCC [4] multiplier model evaluates the efficiency of each by solving the following linear program: Let be an optimal feasible solution for model (1); then , where , will be a feasible solution of model (3). Thus ; therefore, . represents the input-oriented BCC-efficiency value of .

Finally, the output-oriented BCC multiplier model evaluates the efficiency of each by solving the following linear program: It can be easily confirmed that . represents the output-oriented BCC-efficiency value of .

#### 3. Main Results

In this section, we prove that the addition or deletion of a cross-redundant output variable and/or input variable does not affect the efficiency estimates yielded by the BCC multiplier model in input- and output-oriented versions. Similarly, it can be proved that the addition or deletion of cross-redundant variable does not affect efficiency estimates yielded by the CCR multiplier model in input- and output-oriented versions.

Theorem 1. * Let each DMU have m + 1 inputs and s + 1 outputs, that is, and for . Let
**
where **Then the optimal objective function value of the following model:
**
is equal to the optimal objective function value of the following model (3).*

*Proof. *Let be an optimal solution for model (7); then we have

By (6) and (9), it follows that

Also, by (5) and (9) it concludes that

Now, let
where
Then, by (7), we have

Also, by (12) and (13), we obtain

In addition,

Also

So that by (10) we have

Therefore,

Consequently, , where and , is a feasible solution for model (1), which for.

Now, let be an optimal solution for model (1); then , where and , with , is a feasible solution for model (2), which for . Thus .

Theorem 2. * Let each DMU have inputs and outputs with conditions (5) and (6).**Then the optimal objective function value of the following model:
**
is equal to the optimal objective function value of the following model (4).*

*Proof. *Let be an optimal solution for model (22); then we have

By (6) and (15), it follows that

Also, by (5) and (16) it concludes that

Now, let
where
Then, by (7), we have
Also, by (26) and (27), we obtain
In addition
So that by (14), we have
Therefore,
Consequently, , where and , is a feasible solution for model (4), which for
Now let be an optimal solution for model (4), and then , where and , with , is a feasible solution for model (22), which for . Thus .

Theorem 3. *Let each DMU have inputs and outputs with conditions (5) and (6).**Then, the optimal objective function value of the following model:
**
is equal to the optimal objective function value of the following model (1).*

*Proof. *This proof is similar to the proof of Theorem 1.

Theorem 4. * Let each DMU have inputs and outputs with conditions (5) and (6).**Then, the optimal objective function value of the following model:
**
is equal to the optimal objective function value of the following model (2).*

*Proof. *This proof is similar to the proof of Theorem 2.

#### 4. Illustrative Example

In this section, we use the data recorded in Table 1 to illustrate that the addition or deletion of a cross-redundant output variable and input variable does not affect the efficiency estimates yielded by the CCR or BCC models. These correspond to 20 DMUs, whose efficiency is assessed using four inputs and four outputs where In other words, the forth input and the forth output are cross-redundant variables. In Table 2, , , , and , respectively, record the efficiency measure provided by model (1), model (3), model (7), and model (36). It is evident from Table 2 that the addition or deletion of cross-redundant output variable and/or input variable does not affect the efficiency estimates yielded by the input-oriented CCR or BCC multiplier models.

#### 5. Conclusions

In this paper, we have studied the effect of the cross redundancy between interval scale input and output variables on the efficiency estimates yielded by the CCR multiplier model in input- and output-oriented versions and the BCC multiplier model in input- and output-oriented versions. We proved that the addition or deletion of a cross-redundant output variable and input variable does not affect the efficiency estimates yielded by the input-oriented BCC multiplier model and the output-oriented BCC multiplier model. Similarly, it can be proved that the addition or deletion of cross-redundant variable does not affect efficiency estimates yielded by the CCR multiplier model in input- and output-oriented versions.

#### References

- K. Lee and K. Choi, “Cross redundancy and sensitivity in DEA models,”
*Journal of Productivity Analysis*, vol. 34, no. 2, pp. 151–165, 2010. View at Publisher · View at Google Scholar · View at Scopus - F. J. López, “Generalizing cross redundancy in data envelopment analysis,”
*European Journal of Operational Research*, vol. 214, no. 3, pp. 716–721, 2011. - A. Charnes, W. W. Cooper, and E. Rhodes, “Measuring the efficiency of decision making units,”
*European Journal of Operational Research*, vol. 2, no. 6, pp. 429–444, 1978. - R. D. Banker, A. Charnes, and W. W. Cooper, “Some models for estimating technical and scale inefficiencies in data envelopment analysis,”
*Management Science*, vol. 30, no. 9, pp. 1078–1092, 1984. View at Scopus