Applied Mathematics and Computation 163 (2005) 547–552

www.elsevier.com/locate/amc

Undesirable factors in efficiency measurement

A. Hadi Vencheh a,*, R. Kazemi Matin b,M. Tavassoli Kajani a

a Department of Mathematics, Azad University, P.O. Box 84815-119, Mobarekeh, Isfahan, Iranb Department of Mathematics, Azad University, P.O. Box 31485-313, Karaj, Iran

Abstract

Many production processes yield both desirable factors (inputs/outputs) and unde-

sirable ones. There are some models that evaluate efficiency level in the presence of

undesirable factors. The current models consider only undesirable outputs (inputs). In

this paper we propose a model for treating such factors in the framework of Data

Envelopment Analysis (DEA). The proposed model considers both of the undesirable

factors and we discuss efficiency measurement in the context of the model. A numerical

example is given.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Data envelopment analysis (DEA); Efficiency; Undesirable factors

1. Introduction

DEA was originally proposed by Charnes et al. [2] as a method for evalu-

ating the relative efficiency of Decision Making Units (DMUs) performing

essentially the same task. Each of the units uses multiple inputs to produce

multiple outputs. Classical DEA models rely on the assumption that inputs

have to be minimized and outputs have to be maximized. However, it wasmentioned already in [6] that the production process may also generate

undesirable outputs like smoke pollution or waste. Undesirable outputs may as

* Corresponding author.

E-mail address: ahadi@khuisf.ac.ir (A. Hadi Vencheh).

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.02.022

548 A. Hadi Vencheh et al. / Appl. Math. Comput. 163 (2005) 547–552

well appear in nonecological applications like health care (complications of

medical operations) and business (tax payment), cf. Smith [9]. A symmetriccase of inputs which should be maximized may also occur [1]. For example, the

aim of a recycling process is to use maximal quantity of the input waste.

F€are et al. [4] introduced a non-linear programming problem for efficiency

evaluation in the presence of undesirable outputs. Scheel [7] proposed some

radial measures which assume that any change of the output level will involve

both undesirable and desirable outputs. Jahanshahloo et al. [5] introduced a

model for inputs/outputs estimation when some factors are undesirable. Re-

cently Seiford and Zhu [8] proposed a DEA model, in the presence of unde-sirable outputs, to improve the performance via increasing the desirable

outputs and decreasing the undesirable outputs. The current models consider

the situation in which some outputs (inputs) are undesirable and all inputs

(outputs) are desirable. In this paper we propose a model for efficiency mea-

surement, our model considers the general case, that is, when some inputs and

outputs are undesirable simultaneously.

The paper is organized as follows: in Section 2 we introduce our model. In

Section 3 we incorporate undesirable factors in the model. Section 4 is devotedto a numerical example, in this section we illustrate our computational method

by a numerical example. Section 5 concludes.

2. The proposed model

We assume n DMUs, each of which consumes m inputs to produce soutputs. Let X 2 Rm�n

þ and Y 2 Rs�nþ be matrices containing the observed

input and output for n DMUs. We denote by xj (the jth column of X ) the

vector of inputs consumed by DMUj, and by xij the quantity of input iconsumed by DMUj. A similar notation is used for outputs. Now consider

the following model

ðPoÞ max u

s:t: ð1 � uÞxo PXk

ð1 þ uÞyo 6 Y k

ek ¼ 1

kP 0;

where o 2 f1; 2; . . . ; ng; refers to the unit under consideration and

e ¼ ð1; . . . ; 1Þ 2 Rn. The above model is non-radial [10]. ðPoÞ has a feasible

solution u ¼ 0; ko ¼ 1; kj ¼ 0ðj 6¼ oÞ. Hence the optimal u, denoted by u�, is

greater than 0. The dual problem of ðPoÞ is as follows:

A. Hadi Vencheh et al. / Appl. Math. Comput. 163 (2005) 547–552 549

ðDoÞ min ttxo � ltyo þ u0

s:t: ttxo þ ltyo ¼ 1

ttxj � ltyj þ u0 P 0 j ¼ 1; . . . ; n

l; t P 0; u0 free:

It is easy to see that problem ðDoÞ is equal to the following fractional pro-

gramming problem

ðFDoÞ min lo ¼ttxo � ltyo þ u0

ttxo þ ltyo

s:t:ttxj þ u0

ltyjP 1 j ¼ 1; . . . ; n

l; t P 0; u0 free:

We can write the objective function as follows:

lo ¼ttxoþu0

lt yo� 1

ttxolt yo

þ 1¼ fo � 1

go þ 1; ð1Þ

where

fo ¼ttxo þ u0

ltyo; go ¼

ttxoltyo

:

In fact foðgoÞ is the objective function in the fractional BCC (CCR) model.

From Eq. (1) we see that lo is always non-negative; for an efficient unit it is

zero.

By introducing slack vectors s� and sþ, model ðPoÞ is converted to the fol-

lowing model

ðQoÞ max u

s:t: Xk þ xou þ s� ¼ xoY k � you � sþ ¼ yoek ¼ 1

kP 0; s� P 0; sþ P 0:

Now suppose the output measures yrj; r ¼ 1; . . . ; s; be displaced by

wr; r ¼ 1; . . . ; s; and the input measures xij; i ¼ 1; . . . ;m; be displaced by

zi; i ¼ 1; . . . ;m: Let W be a matrix each column of which is w and Z be a matrix

each column of which is z. Then the linear programming problem for translated

data is given by

550 A. Hadi Vencheh et al. / Appl. Math. Comput. 163 (2005) 547–552

ðQoÞ max u

s:t: Xk þ �xou þ s� ¼ �xo

Y k � �you � sþ ¼ �yoek ¼ 1

k P 0; s� P 0; sþ P 0;

where Y ¼ Y þ W ;X ¼ X þ Z; �yo ¼ yo þ w and �xo ¼ xo þ z:

Theorem 1

(a) DMUo is efficient for ðQoÞ if and only if DMUo is efficient for ðQoÞ.(b) DMUo is inefficient for ðQoÞ if and only if DMUo is inefficient for ðQoÞ.

Proof. (a) When DMUo is efficient, we have u� ¼ 0: In this case the constraints

in problem ðQoÞ or ðQoÞ are the same as constraints in the Additive model

(ADD). Since Additive model is translation invariance [3] so ðQoÞ and ðQoÞ are

equivalent.

(b) Statement (b) is logically equivalent to statement (a). h

3. Incorporating undesirable factors in DEA

Now suppose some factors are undesirable so the data matrix can be rep-

resented as follows

Y�X

� �¼

Y D

Y U

�XD

�XU

2664

3775;

where Y DðXDÞ and Y U ðXU Þ represent the desirable and undesirable factors,

respectively. It is clear that we desire to increase the Y DðXU Þ and to decrease theY U ðXDÞ to improve the efficiency level. Following Seiford and Zhu [8] we

multiply each undesirable factor by ()1) and then find two proper translation

vectors v and w to convert negative data to positive. We get

Y�X

� �¼

Y D

YU

�XD

�XU

2664

3775;

where

�yUj ¼ �yUj þ v > 0; �xUj ¼ �xUj þ w > 0:

A. Hadi Vencheh et al. / Appl. Math. Comput. 163 (2005) 547–552 551

Employing the previous notations, model ðPoÞ becomes

Table

Input/o

Inpu

Out

Table

Efficien

DM

A

B

C

D

E

ðPoÞ max u

s:t: ð1 � uÞxDo PXDk

ð1 � uÞ�xUo PXUk

ð1 þ uÞyDo 6 Y Dk

ð1 þ uÞ�yUo 6 YUk

ek ¼ 1

kP 0:

Obviously, the above model expands desirable outputs and contracts unde-

sirable outputs. A similar discussion holds for the inputs.

4. An illustrative example

In this section we illustrative our method by an example.

Example. Consider Table 1.This example is taken from [5]. In this table we have five DMUs with two

inputs x1 and x2 and two outputs y1 and y2. Assume that the first input and the

second output are undesirable. Suppose v ¼ 15 and w ¼ 23. To measure the

efficiency level of each DMU we use problem ðPoÞ.Table 2 represents the efficiency level for each DMU.

1

utput levels

DMU A B C D E

t x1 5 10 15 20 7

x2 15 10 25 10 4

put y1 60 90 80 90 75

y2 7 11 6 5 9

2

cy levels

U u�

0.250

0.000

0.111

0.000

0.000

552 A. Hadi Vencheh et al. / Appl. Math. Comput. 163 (2005) 547–552

Hence B, D and E are efficient, whereas A and C are inefficient.

5. Conclusion

The current paper proposes a model for incorporating undesirable factors in

DEA. The proposed model evaluates the efficiency level of each DMU via

considering undesirable inputs and undesirable outputs simultaneously,

whereas other models lack this ability.

References

[1] K. Allen, DEA in the ecological––an overview, in: G. Westermann (Ed.), Data Envelopment

Analysis in the Service Sector, Wiesbaden, 1999, pp. 203–235.

[2] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, Eur.

J. Oper. Res. 2 (1978) 429–444.

[3] W.W. Cooper, L.M. Seiford, K. Tone, Data Envelopment Analysis : A Comprehensive Text

With Models, Applications, References and DEA-Solver Software, Kluwer Academic

Publisher, 1999.

[4] R. F€are, S. Grosskopf, C.A.K. Lovell, C. Pasurka, Multilateral productivity comparisons

when some outputs are undesirable : a nonparametric approach, Rev. Eco. Stat. 71 (1989) 90–

98.

[5] G.R. Jahanshahloo, A. Hadi Vencheh, A.A. Foroughi, R. Kazemi Matin, Inputs/outputs

estimation in DEA when some factors are undesirable, Appl. Math. Comput. (in press).

[6] T.C. Koopmans, Analysis of production as an efficient combination of activities, in: T.C.

Koopmans (Ed.), Activity Analysis of Production and Allocation, Cowles Commission, Wiley,

New York, 1951, pp. 33–97.

[7] H. Scheel, Undesirable outputs in efficiency valuations, Eur. J. Oper. Res. 132 (2001) 400–410.

[8] L.M. Seiford, J. Zhu, Modeling undesirable factors in efficiency evaluation, Eur. J. Oper. Res.

142 (2002) 16–20.

[9] P. Smith, Data envelopment analysis applied to financial statements, Omega: Int. J. Manage.

Sci. 18 (1990) 131–138.

[10] T. Joro, P. Korhonen, J. Wallenius, Structural comparison of data development analysis and

multiple objective linear programming, Manage. Sci. 44 (1998) 962–970.