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

Authors: Farhad Hosseinzadeh-Lotfi; Gholam-Reza Jahanshahloo; Mansour Mohammadpour
Journal: Journal of Applied Mathematics
 (2013)

---

## 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.


---

## Body

## 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 haven≥2 peer observed DMUs, {DMUj:j=1,2,…,n} which produce multiple outputs yrj,(r=1,…,s), by utilizing multiple inputs xij,(i=1,…,m). The input and output vectors of DMUj are denoted by xj and yj, respectively, and we assume that xj and yj are semipositive, that is, xj≥0,xj≠0 and yj≥0,yj≠0 for i=1,…,n. We use (xj,yj) to descript DMUj and specially use (xo,yo)(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 DMUo  by solving the following linear program:(1)θ*=maxutyo,=ii=ii===vtxo=1==s.t.utyj≤vtxj,j=1,…,n,==ii==iu≥o,v≥o.
Because xj and yj are semipositive for j=1,…,n, θ*>0. Also since utyo≤vtxo and vtxo=1, we have θ*≤1. Thus 0<θ*≤1. θ* represents the input-oriented CCR-efficiency value of DMUo.The output-oriented CCR multiplier model evaluates the efficiency of eachDMUo  by solving the following linear program:
(2)φ*=minvtxo,===ii==utyo=1===s.t.vtxj≥utyj,j=1,…,n,=iii==u≥0,v≥0.
Since utyo≤vtxo and utyo=1, we have φ*≥1. 1/φ* represents the output-oriented CCR-efficiency value of DMUo. Also θ*=1/φ* [4].The input-oriented BCC [4] multiplier model evaluates the efficiency of each DMUo by solving the following linear program:
(3)z*=maxutyo+uo,====i==vtxo=1=i=s.t.utyj+uo≤vtxj,j=1,…,n,=iii==u≥0,v≥0,uoisfree.
Let (u*,v*) be an optimal feasible solution for model (1); then (u*,v*,uo*), where uo*=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 DMUo.Finally, the output-oriented BCC multiplier model evaluates the efficiency of eachDMUoby solving the following linear program:
(4)t*=minvtxo-vo,===i===utyo=1=i=s.t.vtxj-vo≥utyj,j=1,…,n,==ii==u≥0,v≥0,voisfree.
It can be easily confirmed that t*≥1. 1/t*represents the output-oriented BCC-efficiency value of DMUo.


## 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,xj=(x1j,…,x(m+1)j) and yj=(y1j,…,y(s+1)j) for j=1,2,…,n. Let
(5)x(m+1)j=∑i=1mβixij-∑r=1sαryrj;j=1,…,n,(6)y(s+1)j=∑r=1saryrj-∑i=1mbixij;j=1,…,n,
where βi≥0,bi≥0,i=1,…,m;αr≥0,ar≥0,r=1,…,s.
Then the optimal objective function value of the following model:(7)ρ*=max∑r=1s+1pryro+po,∑i=1m+1qixio=1,s.t∑r=1s+1pryro-∑i=1m+1qixio-po≤0,j=1,…,n,pr≥0,qi≥0,r=1,…,s+1,i=1,…,m+1
is equal to the optimal objective function value of the following model (3).Proof.
Let(p1*,…,ps+1*,q1*,…,qm+1*,po*) be an optimal solution for model (7); then we have
(8)ρ*=∑r=1s+1pr*yro-po*,(9)∑i=1m+1qi*xio=1,(10)∑r=1s+1pr*yro-∑i=1m+1qi*xio-po*≤0.
By (6) and (9), it follows that
(11)ρ*=∑r=1s(pr*+ps+1*)yro-∑i=1mps+1*bixio+po*.
Also, by (5) and (9) it concludes that
(12)∑i=1m(qi*+qm+1*βi)xio-∑r=1sqm+1*αryro=1.
Now, let(13)v-i=(qi*+qm+1*βi)ρ*AB+ps+1*biA,fori=1,…,m,u-r=(pr*+ps+1*ar)ρ*AB+(qm+1*αr)ρ*B,fori=1,…,m,u-o=po*ρ*AB,
where
(14)A=∑r=1s(pr*+ps+1*ar)yro+po*B=1+∑r=1sqm+1*αryro.
Then, by (7), we have
(15)v-i≥0,i=1,…,m,u-r≥0,r=1,…,s.
Also, by (12) and (13), we obtain
(16)∑i=1mv-ixio=ρ*AB∑i=1m(qi*+qm+1*βi)xio+1A∑i=1m(ps+1*bi)xio=ρ*(B)AB+1A(A-ρ*)=1.
In addition,(17)θ*≥∑r=1su-ryro-u-o=ρ*AB∑r=1s(pr*+ps+1*ar)yro+z2*B∑r=1s(qm+1*αr)-po*ρ*AB=ρ*AB(∑r=1s(pr*+ps+1*ar)yro-po*)+ρ*(B-1)B=z2*AB(A)+ρ*(B-1)B=ρ*.
Also(18)∑i=1mv-ixij-∑r=1su-ryrj+u-o=1AB∑i=1mρ*(qi*+qm+1*βi)xij+1A∑i=1mps+1*bixij-1AB∑r=1sρ*(pr*+ps+1*ar)yrj-1B∑r=1sρ*(qm+1*αr)yrj+1ABpo*ρ*=1AB[ρ*(∑i=1mqi*xij-∑r=1spr*yrj+po*)+ρ*(∑i=1mqm+1*βixij-∑r=1sps+1*aryrj)+B∑i=1mps+1*bixij-Aρ*∑r=1s(qm+1*αr)yrj].
So that by (10) we have
(19)∑i=1mv-ixij-∑r=1su-ryrj+u-o≥1AB[ρ*(∑r=1s(ps+1*ar+qm+1*αr)yrj======i===-∑i=1m(qm+1*βi+ps+1*bi)xij)======+ρ*(∑i=1mqm+1*βixij-∑r=1sps+1*aryrj)======+B∑i=1mps+1*bixij-Aρ*∑r=1s(qm+1*αr)yrj]=1AB[ρ*∑r=1sqm+1*αryrj-ρ*∑i=1mps+1*bixij===i===+B∑i=1mps+1*bixij-Aρ*∑r=1s(qm+1*αr)yrj]=1AB[(B-ρ*)∑i=1mps+1*bixij+ρ*(1-A)∑r=1sqm+1*αryrj].
Therefore,(20)∑i=1mv-ixij-∑r=1su-ryrj+u-o≥1AB[(B-ρ*)∑i=1mps+1*bixij+ρ*(1-A)∑r=1sqm+1*αryrj]≥0.
Consequently,(u-,v-,u-o), where u-=(u-1,…,u-s) and v-=(v-1,…,v-m), is a feasible solution for model (1), which for.
(21)θ*≥∑r=1su-ryro-u-o=ρ*AB∑r=1s(pr*+ps+1*ar)yro+z2*B∑r=1s(qm+1*αr)-po*ρ*AB=ρ*AB(∑r=1s(pr*+ps+1*ar)yro-po*)+ρ*(B-1)B=z2*AB(A)+ρ*(B-1)B=ρ*.
Now, let(u*,v*,uo*) be an optimal solution for model (1); then (p-,q-,p-o), where p-=(p-1,…,p-s+1) and q-=(q-1,…,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=uo*, is a feasible solution for model (2), which for θ*=∑r=1sur*yro-uo*=∑r=1s+1p-ryro-p-o≤ρ*. Thus θ*=ρ*.Theorem 2.
Let each DMU havem+1 inputs and s+1 outputs with conditions (5) and (6).
Then the optimal objective function value of the following model:(22)w*=min∑r=1s+1qixio-qo,=uiumiiiin∑r=1s+1pryro=1==i=s.t∑i=1m+1qixio-∑r=1s+1pryro-qo≥0,j=1,…,n,pr≥0,qi≥0,r=1,…,s+1,i=1,…,m+1
is equal to the optimal objective function value of the following model (4).Proof.
Let(p1*,…,ps+1*,q1*,…,qm+1*,qo*) be an optimal solution for model (22); then we have
(23)ω*=∑i=1m+1qi*xio-qo*(24)∑r=1s+1pr*yro=1(25)∑i=1m+1qi*xij-∑r=1spr*yrj-qo*≥0.
By (6) and (15), it follows that
(26)ω*=∑i=1m(qi*+qm+1*βi)xio-qo*-∑r=1sqm+1*αryro.
Also, by (5) and (16) it concludes that
(27)∑r=1s(pr*+ps+1*ar)yro-∑i=1mps+1*bixio=1.
Now, let(28)v-i=(qi*+qm+1*βi)w*AB+w*ps+1*biB,fori=1,…,m,u-r=(pr*+ps+1*ar)w*AB+(qm+1*αr)A,forr=1,…,s,v-o=qo*w*AB,
where
(29)A=∑i=1m(qi*+qm+1*βi)xio-qo*,B=1+∑r=1sqm+1*αryro.
Then, by (7), we have
(30)v-i≥0,i=1,…,m,u-r≥0,r=1,…,s.
Also, by (26) and (27), we obtain
(31)∑r=1su-ryro=w*AB∑i=1m(pr*+ps+1*αr)yrokkkkkkkk+w*B∑i=1m(qm+1*αr)yro=w*(B)AB+(A-w*)A=1.
In addition
(32)∑i=1mv-ixij-∑r=1su-ryrj-v-o=1AB∑i=1mw*(qi*+qm+1*βi)xij+w*B∑i=1mps+1*bixij-w*AB∑r=1s(pr*+ps+1*ar)yrj-1A∑r=1s(qm+1*αr)yrj-1ABqo*w*=1AB[w*(∑i=1mqi*xij-∑r=1spr*yrj-qo*)llllllllllllllll+w*(∑i=1mqm+1*βixij-∑r=1sps+1*aryrj)llllllllllllllll+Aw*∑i=1mps+1*bixij-B∑r=1s(qm+1*αr)yrj].
So that by (14), we have
(33)∑i=1mv-ixij-∑r=1su-ryrj-v-o≥1AB[w*(∑r=1s(ps+1*ar+qm+1*αr)yrjjjjjjjjjjjjjjjjjjjij-∑i=1m(qm+1*βi+ps+1*bi)xij)llllllllllllllliiiii+w*(∑i=1mqm+1*βixij-∑r=1sps+1*aryrj)ggggggggggg+Aw*∑i=1mps+1*bixij-B∑r=1s(qm+1*αr)yrj]=1AB[w*∑r=1sqm+1*αryrj-w*∑i=1mps+1*bixijfffffffffffffffi+Aw*∑i=1mps+1*bixij-B∑r=1s(qm+1*αr)yrj]=1AB[w*(A-1)∑i=1mps+1*bixijhhhhhpppi+(w*-B)∑r=1sqm+1*αryrj].
Therefore,
(34)∑i=1mv-ixij-∑r=1su-ryrj-v-o≥1AB[w*(A-1)∑i=1mps+1*bixijlllllllllllllll+(w*-B)∑r=1sqm+1*αryrj]≥0.
Consequently, (u-,v-,v-o), where u-=(u-1,…,u-s) and v-=(v-1,…,v-m), is a feasible solution for model (4), which for
(35)z*≤∑i=1mv-ixio-v-o=w*AB∑i=1m(qi*+qm+1*βr)xio+w*B∑r=1s(ps+1*bi)xio-qo*w*AB=w*AB(A)+w*(B-1)B=w*.
Now let (u*,v*,vo*) be an optimal solution for model (4), and then (p-,q-,p-o), where p-=(p-1,…,p-s+1) and q-=(q-1,…,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=vo*, is a feasible solution for model (22), which for ω*≥∑r=1s+1p-ryro-p-o=∑r=1sur*yro-uo*=z*. Thus z*=w*.Theorem 3.
Let each DMU havem+1 inputs and s+1 outputs with conditions (5) and (6).
Then, the optimal objective function value of the following model:(36)ρ~=max∑r=1s+1pryro,yyyyyyyy∑i=1m+1qixio=1==iis.t∑r=1s+1pryrj-∑i=1m+1qixij≤0,j=1,…,npr≥0,qi≥0,r=1,…,s+1,i=1,…,m+1
is equal to the optimal objective function value of the following model (1).Proof.
This proof is similar to the proof of Theorem1.Theorem 4.
Let each DMU havem+1 inputs and s+1 outputs with conditions (5) and (6).
Then, the optimal objective function value of the following model:(37)w~=min∑r=1s+1qixio,=pppiimin∑r=1s+1pryro=1==iis.t∑i=1m+1qixij-∑r=1s+1pryrj≥0,j=1,…,n,pr≥0,qi≥0,r=1,…,s+1,i=1,…,m+1
is equal to the optimal objective function value of the following model (2).Proof.
This proof is similar to the proof of Theorem2.


## 4. Illustrative Example

In this section, we use the data recorded in Table1 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
(38)x4j=(x1j+x2j+2x3j)==-(0.5y1j+0.5y2j+0.5y3j);jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj=1,…,n,y4j=(0.5y1j+0.5y2j+0.5y3j)-0.5x3j;koooooooojjjjjjjjjjjjjjjjj=1,…,n.
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.
Inp 1
Inp 2
Inp 3
Inp 4
Out 1
Out 1
Out 3
Out 4
Unit 1
7
1
4
12.75
1
2.5
3
1.25
Unit 2
3
7
4
14.75
2.5
1
3
1.25
Unit 3
6
6
3
14.25
2.5
2
3
2.25
Unit 4
3
1
3
1.75
4
5.5
7
6.75
Unit 5
6
0.5
3
5.25
5
3.5
6
5.75
Unit 6
4
0.5
3
3.5
2
6
6
5.5
Unit 7
1.5
2.5
3
1.5
6
4
7
7
Unit 8
0.5
4
4
6.25
1.5
5
6
4.25
Unit 9
2.75
1.75
4
4
8
3
6
6.5
Unit 10
1
3
3
1
8
3
7
7.5
Unit 11
2
2
3
1.25
5.5
5
7
7.25
Unit 12
2.5
1.5
3
2
7
3
6
6.5
Unit 13
4.5
1.5
6
13
4
2
4
2
Unit 14
2
4
7
16.25
1.5
2
4
0.25
Unit 15
4
3
6
12.25
6.5
3.5
3.5
3.75
Unit 16
2
5
4
8.75
5
3.5
4
4.25
Unit 17
1.5
6
4
8.5
4.5
4.5
5
5
Unit 18
0.5
4
3
3
3.5
5.5
6
6
Unit 19
3.5
0.75
3
2.5
7.5
2.5
6
6.5
Unit 20
6
3.5
4
11
3.5
3.5
6
4.5Table 2
Example results.
θ
*
ρ
~
z
*
ρ
*
Unit 1
0.3461538
0.3461538
0.7500000
0.7500000
Unit 2
0.3214286
0.3214286
0.7500000
0.7500000
Unit 3
0.4285714
0.4285714
1.0000000
1.0000000
Unit 4
1.0000000
1.0000000
1.0000000
1.0000000
Unit 5
1.0000000
1.0000000
1.0000000
1.0000000
Unit 6
1.0000000
1.0000000
1.0000000
1.0000000
Unit 7
1.0000000
1.0000000
1.0000000
1.0000000
Unit 8
1.0000000
1.0000000
1.0000000
1.0000000
Unit 9
0.9973190
0.9973190
1.0000000
1.0000000
Unit 10
1.0000000
1.0000000
1.0000000
1.0000000
Unit 11
1.0000000
1.0000000
1.0000000
1.0000000
Unit 12
1.0000000
1.0000000
1.0000000
1.0000000
Unit 13
0.4444444
0.4444444
0.6666667
0.6666667
Unit 14
0.3809524
0.3809524
0.6666667
0.6666667
Unit 15
0.5607702
0.5607702
0.5714286
0.5594240
Unit 16
0.6052279
0.6052279
0.7500000
0.7500000
Unit 17
0.6847156
0.6847156
0.7500000
0.7500000
Unit 18
1.0000000
1.0000000
1.0000000
1.0000000
Unit 19
1.0000000
1.0000000
1.0000000
1.0000000
Unit 20
0.6428571
0.6428571
0.7500000
0.7500000


## 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.



---

*Source: 658635-2013-09-24.xml*