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

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 𝑛≥2 peer observed DMUs, {DMU_𝑗∶𝑗=1,2,…,𝑛} which produce multiple outputs 𝑦_𝑟𝑗,(𝑟=1,…,𝑠), by utilizing multiple inputs 𝑥_𝑖𝑗,(𝑖=1,…,𝑚). The input and output vectors of DMU_𝑗 are denoted by 𝐱_𝑗 and 𝐲_𝑗, respectively, and we assume that 𝐱_𝑗 and 𝐲_𝑗 are semipositive, that is, 𝐱_𝑗≥0,𝐱_𝑗≠0 and 𝐲_𝑗≥0,𝑦_𝑗≠0 for 𝑖=1,…,𝑛. We use (𝐱_𝑗,𝐲_𝑗) to descript DMU_𝑗 and specially use (𝐱_𝑜,𝐲_𝑜)(𝑜elementof{1,2,…,𝑛}) 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 DMU_𝑜??by solving the following linear program:𝜃^∗=max𝐮^𝑡𝐲_𝑜,𝐯^𝑡𝐱_𝑜=1s.t.𝐮^𝑡𝐲_𝑗≤𝐯^𝑡𝐱_𝑗,𝑗=1,…,𝑛,𝐮≥𝑜,𝐯≥𝑜. (1) Because 𝐱_𝑗 and 𝐲_𝑗 are semipositive for 𝑗=1,…,𝑛, 𝜃^∗>0. Also since 𝐮^𝑡𝐲_𝑜≤𝐯^𝑡𝐱_𝑜 and 𝐯^𝑡𝐱_𝑜=1, we have 𝜃^∗≤1. Thus 0<𝜃^∗≤1. 𝜃^∗ represents the input-oriented CCR-efficiency value of DMU_𝑜.

The output-oriented CCR multiplier model evaluates the efficiency of each DMU_𝑜??by solving the following linear program: 𝜑^∗=min𝐯^𝑡𝐱_𝑜,𝐮^𝑡𝐲_𝑜=1s.t.𝐯^𝑡𝐱_𝑗≥𝐮^𝑡𝐲_𝑗,𝑗=1,…,𝑛,𝐮≥0,𝐯≥0. (2) Since 𝐮^𝑡𝐲_𝑜≤𝐯^𝑡𝐱_𝑜 and 𝐮^𝑡𝐲_𝑜=1, we have 𝜑^∗≥1. 1/𝜑^∗ represents the output-oriented CCR-efficiency value of DMU_𝑜. Also 𝜃^∗=1/𝜑^∗ [4].

The input-oriented BCC [4] multiplier model evaluates the efficiency of each DMU_𝑜 by solving the following linear program: 𝑧^∗=max𝐮^𝑡𝐲_𝑜+𝑢_𝑜,𝐯^𝑡𝐱_𝑜=1s.t.𝐮^𝑡𝐲_𝑗+𝑢_𝑜≤𝐯^𝑡𝐱_𝑗,𝑗=1,…,𝑛,𝐮≥𝟎,𝐯≥𝟎,𝑢_𝑜isfree. (3) Let (𝐮^∗,𝐯^∗) be an optimal feasible solution for model (1); then (𝐮^∗,𝐯^∗,𝑢^∗_𝑜), where 𝑢^∗_𝑜=0, will be a feasible solution of model (3). Thus 𝑧^∗≥𝜃^∗; therefore, 0<𝑧^∗≤1. 𝑧^∗ represents the input-oriented BCC-efficiency value of DMU_𝑜.

Finally, the output-oriented BCC multiplier model evaluates the efficiency of each DMU_𝑜by solving the following linear program: 𝑡^∗=min𝐯^𝑡𝐱_𝑜−𝑣_𝑜,𝐮^𝑡𝐲_𝑜=1s.t.𝐯^𝑡𝐱_𝑗−𝑣_𝑜≥𝐮^𝑡𝐲_𝑗,𝑗=1,…,𝑛,𝐮≥𝟎,𝐯≥𝟎,𝑣_𝑜isfree. (4) It can be easily confirmed that 𝑡^∗≥1. 1/𝑡^∗represents the output-oriented BCC-efficiency value of DMU_𝑜.

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, 𝐱_𝑗=(𝑥_1𝑗,…,𝑥_(𝑚+1)𝑗) and 𝐲_𝑗=(𝑦_1𝑗,…,𝑦_(𝑠+1)𝑗) for 𝑗=1,2,…,𝑛. Let 𝑥_(𝑚+1)𝑗=^𝑚_𝑖=1𝛽_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝛼_𝑟𝑦_𝑟𝑗𝑦;𝑗=1,…,𝑛,_(𝑠+1)𝑗=^𝑠_𝑟=1𝑎_𝑟𝑦_𝑟𝑗−^𝑚_𝑖=1𝑏_𝑖𝑥_𝑖𝑗;𝑗=1,…,𝑛, (5)(6) where 𝛽_𝑖≥0,𝑏_𝑖≥0,𝑖=1,…,𝑚;𝛼_𝑟≥0,𝑎_𝑟≥0,𝑟=1,…,𝑠.
Then the optimal objective function value of the following model: 𝜌^∗=max^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑜+𝑝_𝑜,_𝑚+1𝑖=1𝑞_𝑖𝑥_𝑖𝑜=1,s.t^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑜−^𝑚+1_𝑖=1𝑞_𝑖𝑥_𝑖𝑜−𝑝_𝑜𝑝≤0,𝑗=1,…,𝑛,_𝑟≥0,𝑞_𝑖≥0,𝑟=1,…,𝑠+1,𝑖=1,…,𝑚+1 (7) is equal to the optimal objective function value of the following model (3).

Proof. Let (𝑝^∗₁,…,𝑝^∗_𝑠+1,𝑞^∗₁,…,𝑞^∗_𝑚+1,𝑝^∗_𝑜) be an optimal solution for model (7); then we have 𝜌^∗=^𝑠+1_𝑟=1𝑝^∗_𝑟𝑦_𝑟𝑜−𝑝^∗_𝑜,_𝑚+1𝑖=1𝑞^∗_𝑖𝑥_𝑖𝑜=1,_𝑠+1𝑟=1𝑝^∗_𝑟𝑦_𝑟𝑜−^𝑚+1_𝑖=1𝑞^∗_𝑖𝑥_𝑖𝑜−𝑝^∗_𝑜≤0. (8)(9)(10)
By (6) and (9), it follows that 𝜌^∗=^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑦_𝑟𝑜−^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑜+𝑝^∗_𝑜. (11)
Also, by (5) and (9) it concludes that ^𝑚_𝑖=1𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑜−^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑜=1. (12)
Now, let 𝑣_𝑖=𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝜌^∗+𝑝𝐴𝐵^∗𝑠+1𝑏_𝑖𝐴,for𝑖=1,…,𝑚,𝑢_𝑟=𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝜌^∗+𝑞𝐴𝐵^∗𝑚+1𝛼_𝑟𝜌^∗𝐵,for𝑖=1,…,𝑚,𝑢_𝑜=𝑝^∗_𝑜𝜌^∗,𝐴𝐵 (13) where 𝐴=^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑜+𝑝^∗_𝑜𝐵=1+^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑜. (14) Then, by (7), we have 𝑣_𝑖≥0,𝑖=1,…,𝑚,𝑢_𝑟≥0,𝑟=1,…,𝑠. (15)
Also, by (12) and (13), we obtain ^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑜=𝜌^∗𝐴𝐵^𝑚_𝑖=1𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑜+1𝐴^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑜=𝜌^∗(𝐵)+1𝐴𝐵𝐴𝐴−𝜌^∗=1. (16)
In addition, 𝜃^∗≥^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑜−𝑢_𝑜=𝜌^∗𝐴𝐵^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑜+𝑧^∗₂𝐵^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟−𝑝^∗_𝑜𝜌^∗=𝜌𝐴𝐵_∗𝐴𝐵^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑜−𝑝^∗_𝑜+𝜌^∗(𝐵−1)𝐵=𝑧^∗₂(𝐴)+𝜌𝐴𝐵^∗(𝐵−1)𝐵=𝜌^∗. (17)
Also ^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗+𝑢_𝑜=1𝐴𝐵^𝑚_𝑖=1𝜌^∗𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑗+1𝐴^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−1𝐴𝐵^𝑠_𝑟=1𝜌^∗𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑗−1𝐵^𝑠_𝑟=1𝜌^∗𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗+1𝑝𝐴𝐵^∗𝑜𝜌^∗=1𝜌𝐴𝐵^∗𝑚_𝑖=1𝑞^∗_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑝^∗_𝑟𝑦_𝑟𝑗+𝑝^∗_𝑜+𝜌^∗𝑚_𝑖=1𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑗+𝐵^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−𝐴𝜌^∗𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗. (18)
So that by (10) we have ^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗+𝑢_𝑜≥1𝜌𝐴𝐵^∗𝑠_𝑟=1𝑝^∗_𝑠+1𝑎_𝑟+𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗−^𝑚_𝑖=1𝑞^∗_𝑚+1𝛽_𝑖+𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝜌^∗𝑚_𝑖=1𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑗+𝐵^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−𝐴𝜌^∗𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗=1𝜌𝐴𝐵^∗𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗−𝜌^∗𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝐵^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−𝐴𝜌^∗𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗=1𝐴𝐵𝐵−𝜌^∗𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝜌^∗(1−𝐴)^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗. (19)
Therefore, ^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗+𝑢_𝑜≥1𝐴𝐵𝐵−𝜌^∗𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝜌^∗(1−𝐴)^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗≥0. (20)
Consequently, (𝐮,𝐯,𝑢_𝑜), where 𝐮=(𝑢₁,…,𝑢_𝑠) and 𝐯=(𝑣₁,…,𝑣_𝑚), is a feasible solution for model (1), which for. 𝜃^∗≥^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑜−𝑢_𝑜=𝜌^∗𝐴𝐵^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑜+𝑧^∗₂𝐵^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟−𝑝^∗_𝑜𝜌^∗=𝜌𝐴𝐵_∗𝐴𝐵^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑜−𝑝^∗_𝑜+𝜌^∗(𝐵−1)𝐵=𝑧^∗₂(𝐴)+𝜌𝐴𝐵^∗(𝐵−1)𝐵=𝜌^∗. (21)
Now, let (𝐮^∗,𝐯^∗,𝑢^∗_𝑜) be an optimal solution for model (1); then (𝐩,𝐪,𝑝_𝑜), where 𝐩=(𝑝₁,…,𝑝_𝑠+1) and 𝐪=(𝑞₁,…,𝑞_𝑚+1), with 𝑝_𝑟=𝑢_𝑟,𝑟=1,…,𝑠;𝑝_𝑠+1=0;𝑞_𝑖=𝑣_𝑖,𝑖=1,…,𝑚;𝑞_𝑚+1=0;𝑝_𝑜=𝑢^∗_𝑜, is a feasible solution for model (2), which for 𝜃^∗=∑^𝑠_𝑟=1𝑢^∗_𝑟𝑦_𝑟𝑜−𝑢^∗_𝑜=∑^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑜−𝑝_𝑜≤𝜌^∗. Thus 𝜃^∗=𝜌^∗.

Theorem 2. Let each DMU have 𝑚+1 inputs and 𝑠+1 outputs with conditions (5) and (6).
Then the optimal objective function value of the following model: 𝑤^∗=min^𝑠+1_𝑟=1𝑞_𝑖𝑥_𝑖𝑜−𝑞_𝑜,_𝑠+1𝑟=1𝑝_𝑟𝑦_𝑟𝑜=1s.t^𝑚+1_𝑖=1𝑞_𝑖𝑥_𝑖𝑜−^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑜−𝑞_𝑜𝑝≥0,𝑗=1,…,𝑛,_𝑟≥0,𝑞_𝑖≥0,𝑟=1,…,𝑠+1,𝑖=1,…,𝑚+1 (22) is equal to the optimal objective function value of the following model (4).

Proof. Let (𝑝^∗₁,…,𝑝^∗_𝑠+1,𝑞^∗₁,…,𝑞^∗_𝑚+1,𝑞^∗_𝑜) be an optimal solution for model (22); then we have 𝜔^∗=^𝑚+1_𝑖=1𝑞^∗_𝑖𝑥_𝑖𝑜−𝑞^∗_{𝑜𝑠+1𝑟=1}𝑝^∗_𝑟𝑦_𝑟𝑜=1_𝑚+1𝑖=1𝑞^∗_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑝^∗_𝑟𝑦_𝑟𝑗−𝑞^∗_𝑜≥0. (23)(24)(25)
By (6) and (15), it follows that 𝜔^∗=^𝑚_𝑖=1𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑜−𝑞^∗_𝑜−^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑜. (26)
Also, by (5) and (16) it concludes that ^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑜−^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑜=1. (27)
Now, let 𝑣_𝑖=𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝑤^∗+𝑤𝐴𝐵^∗𝑝^∗_𝑠+1𝑏_𝑖𝐵,for𝑖=1,…,𝑚,𝑢_𝑟=𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑤^∗+𝑞𝐴𝐵^∗𝑚+1𝛼_𝑟𝐴,for𝑟=1,…,𝑠,𝑣_𝑜=𝑞^∗_𝑜𝑤^∗,𝐴𝐵 (28) where 𝐴=^𝑚_𝑖=1𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑜−𝑞^∗_𝑜,𝐵=1+^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑜. (29) Then, by (7), we have 𝑣_𝑖≥0,𝑖=1,…,𝑚,𝑢_𝑟≥0,𝑟=1,…,𝑠. (30) Also, by (26) and (27), we obtain ^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑜=𝑤^∗𝐴𝐵^𝑚_𝑖=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝛼_𝑟𝑦_𝑟𝑜+𝑤^∗𝐵^𝑚_𝑖=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑜=𝑤^∗(𝐵)+𝐴𝐵𝐴−𝑤^∗𝐴=1. (31) In addition ^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−𝑣_𝑜=1𝐴𝐵^𝑚_𝑖=1𝑤^∗𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑗+𝑤^∗𝐵^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−𝑤^∗𝐴𝐵^𝑠_𝑟=1𝑝^∗_𝑟+𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑗−1𝐴^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗−1𝑞𝐴𝐵^∗𝑜𝑤^∗=1𝑤𝐴𝐵^∗𝑚_𝑖=1𝑞^∗_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑝^∗_𝑟𝑦_𝑟𝑗−𝑞^∗_𝑜+𝑤^∗𝑚_𝑖=1𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑗+𝐴𝑤^∗𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−𝐵^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗. (32) So that by (14), we have ^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−𝑣_𝑜≥1𝑤𝐴𝐵^∗𝑠_𝑟=1𝑝^∗_𝑠+1𝑎_𝑟+𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗−^𝑚_𝑖=1𝑞^∗_𝑚+1𝛽_𝑖+𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝑤^∗𝑚_𝑖=1𝑞^∗_𝑚+1𝛽_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑝^∗_𝑠+1𝑎_𝑟𝑦_𝑟𝑗+𝐴𝑤^∗𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−𝐵^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗=1𝑤𝐴𝐵^∗𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗−𝑤^∗𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝐴𝑤^∗𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗−𝐵^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗=1𝑤𝐴𝐵^∗(𝐴−1)^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝑤^∗−𝐵^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗. (33) Therefore, ^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑗−^𝑠_𝑟=1𝑢_𝑟𝑦_𝑟𝑗−𝑣_𝑜≥1𝑤𝐴𝐵^∗(𝐴−1)^𝑚_𝑖=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑗+𝑤^∗−𝐵^𝑠_𝑟=1𝑞^∗_𝑚+1𝛼_𝑟𝑦_𝑟𝑗≥0. (34) Consequently, (𝐮,𝐯,𝑣_𝑜), where 𝐮=(𝑢₁,…,𝑢_𝑠) and 𝐯=(𝑣₁,…,𝑣_𝑚), is a feasible solution for model (4), which for 𝑧^∗≤^𝑚_𝑖=1𝑣_𝑖𝑥_𝑖𝑜−𝑣_𝑜=𝑤^∗𝐴𝐵^𝑚_𝑖=1𝑞^∗_𝑖+𝑞^∗_𝑚+1𝛽_𝑟𝑥_𝑖𝑜+𝑤^∗𝐵^𝑠_𝑟=1𝑝^∗_𝑠+1𝑏_𝑖𝑥_𝑖𝑜−𝑞^∗_𝑜𝑤^∗=𝑤𝐴𝐵_∗(𝐴)+𝑤𝐴𝐵^∗(𝐵−1)𝐵=𝑤^∗. (35) Now let (𝐮^∗,𝐯^∗,𝑣^∗_𝑜) be an optimal solution for model (4), and then (𝐩,𝐪,𝑝_𝑜), where 𝐩=(𝑝₁,…,𝑝_𝑠+1) and 𝐪=(𝑞₁,…,𝑞_𝑚+1), with 𝑝_𝑟=𝑢_𝑟,𝑟=1,…,𝑠;𝑝_𝑠+1=0;𝑞_𝑖=𝑣_𝑖,𝑖=1,…,𝑚;𝑞_𝑚+1=0;𝑝_𝑜=𝑣^∗_𝑜, is a feasible solution for model (22), which for 𝜔^∗≥∑^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑜−𝑝_𝑜=∑^𝑠_𝑟=1𝑢^∗_𝑟𝑦_𝑟𝑜−𝑢^∗_𝑜=𝑧^∗. Thus 𝑧^∗=𝑤^∗.

Theorem 3. Let each DMU have 𝑚+1 inputs and 𝑠+1 outputs with conditions (5) and (6).
Then, the optimal objective function value of the following model: 𝜌=max^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑜,_𝑚+1𝑖=1𝑞_𝑖𝑥_𝑖𝑜=1s.t^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑗−^𝑚+1_𝑖=1𝑞_𝑖𝑥_𝑖𝑗𝑝≤0,𝑗=1,…,𝑛_𝑟≥0,𝑞_𝑖≥0,𝑟=1,…,𝑠+1,𝑖=1,…,𝑚+1 (36) 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 𝑚+1 inputs and 𝑠+1 outputs with conditions (5) and (6).
Then, the optimal objective function value of the following model: 𝑤=min^𝑠+1_𝑟=1𝑞_𝑖𝑥_𝑖𝑜,_𝑠+1𝑟=1𝑝_𝑟𝑦_𝑟𝑜=1s.t^𝑚+1_𝑖=1𝑞_𝑖𝑥_𝑖𝑗−^𝑠+1_𝑟=1𝑝_𝑟𝑦_𝑟𝑗𝑝≥0,𝑗=1,…,𝑛,_𝑟≥0,𝑞_𝑖≥0,𝑟=1,…,𝑠+1,𝑖=1,…,𝑚+1 (37) 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 𝑥_4𝑗=𝑥_1𝑗+𝑥_2𝑗+2𝑥_3𝑗−0.5𝑦_1𝑗+0.5𝑦_2𝑗+0.5𝑦_3𝑗;𝑦𝑗=1,…,𝑛,_4𝑗=0.5𝑦_1𝑗+0.5𝑦_2𝑗+0.5𝑦_3𝑗−0.5𝑥_3𝑗;𝑗=1,…,𝑛. (38) In other words, the forth input and the forth output are cross-redundant variables. In Table 2, 𝜃^∗, 𝑧, 𝜌^∗, and 𝜌, 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.

Table 1 

Dataset.

Table 2 

Example results.

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.