Ranking efficient DMUs using the Tchebycheff norm

Applied Mathematical Modelling 36 (2012) 46–56

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Ranking efficient DMUs using the Tchebycheff norm F. Rezai Balf a,⇑, H. Zhiani Rezai b, G.R. Jahanshahloo c, F. Hosseinzadeh Lotfi d a

Department of Mathematics, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran Department of Mathematics, Teacher Training University, Tehran, Iran d Department of Mathematics, Science and Research Branches, Islamic Azad University, Tehran, Iran b c

a r t i c l e

i n f o

Article history: Received 16 July 2008 Received in revised form 23 November 2010 Accepted 30 November 2010 Available online 23 February 2011 Keywords: Data envelopment analysis Ranking Efficiency

a b s t r a c t One problem that has been discussed frequently in data envelopment analysis (DEA) literature has been lack of discrimination in DEA applications, in particular when there are insufficient DMUs or the number of inputs and outputs is too high relative to the number of units. This is an additional reason for the growing interest in complete ranking techniques. In this paper a method for ranking extreme efficient decision making units (DMUs) is proposed. The method uses L1(or Tchebycheff) Norm, and it seems to have some superiority over other existing methods, because this method is able to remove the existing difficulties in some methods, such as Andersen and Petersen [2] (AP) that it is sometimes infeasible. The suggested model is always feasible. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Data envelopment analysis (DEA), a mathematical technique based on linear programming first introduced by Charnes et al. [1], is a way of determining the efficiency for a group of decision making units (DMUs) when measured over a set of multiple input and output. For a given set of input and output, DEA produces a single comprehensive measure of performance (efficiency score) for each DMU. This is done by constructing an empirically based best-practice or efficient frontier as a result of identifying a set of efficient DMUs (on the efficient frontier) and inefficient DMUs (not on the efficient frontier). Many methods have been proposed in order to rank best performers; such as Andersen and Petersen (AP) [2]. There are three problematic areas with this methodology. First, Andersen and Petersen refer to the DEA objective function value as a rank score for all units in the dual formulation of the super-efficiency model [2], despite the fact that each unit is evaluated according to different weights. Second, the super-efficient methodology can give ‘‘specialized’’ DMUs an excessively high ranking. To avoid this problem, Sueyoshi [3] introduced specific bounds on the weights in the super-efficient ranking model [3]. The third problem lies with an infeasibility issue, which if it occurs, means that the super-efficient technique cannot provide a complete ranking of all DMUs (see Thrall [4]). Zhu [5], Dula and Hichman [6] and Seiford and Zhu [7] prove under which conditions various super-efficient DEA models are infeasible. Mehrabian et al. [8] suggested a modification model in order to ensure feasibility. Also Hashimoto [9] developed a DEA super-efficient model with assurance region in order to rank the DMUs completely. Torgesen et al. [10] achieved a complete ranking of efficient DMUs by measuring their importance as a benchmark for inefficient DMUs.

⇑ Corresponding author. Tel./fax: +98 1232283538. E-mail address: [email protected] (F. Rezai Balf). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.11.077

F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

47

The remaining structure made in this article is as follow: In Section 2 we give a methodology of DEA and in Section 3 is presented an efficiency index base on infinity (Tchebycheff) norm. In Section 4 a model is introduced for ranking extreme efficient, a numerical example is given in Section 5, and finally in Section 6, conclusion is presented.

2. Methodology: DEA 2.1. DEA models Discussion of frontiers and efficiency measurement began formally with the seminal work of Farrell (1957), who provided definitions and a computational framework for both technical and allocative efficiency and inefficiency. Based Farrell’s (1957) work, the measurement of efficiency and the estimation of frontiers have developed explosively over the past several decades. The term DEA and the CCR model were first coined in Charnes et al. [1] and were followed by a phenomenal expansion of DEA in terms of its theory, methodology and application over the last decades. Consider n, DMUs each using m input to produce s output. Let xip(P0) represent input i of DMUp(p 2 {1, . . . , n}) and yrp(P0) represent output r of it. Its actual point of operation is given by (xp, yp) = (x1p, . . . , xmp, y1p, . . . , ysp) and the projected P P P P point of DMUp is given by ðX; YÞ ¼ ð nj¼1 kj x1j ; . . . ; nj¼1 kj xmj ; nj¼1 kj y1j ; . . . ; nj¼1 kj ysj Þ. The production possibility sets (PPS) Tc and Tv is defined as

( Tc ¼

ðX; YÞ : X P (

Tv ¼

ðX; YÞ : X P

n X

kj X j ;

Y6

n X

j¼1

j¼1

n X

n X

kj X j ;

Y6

j¼1

) kj Y j ;

kj Y j ;

j¼1

kj P 0; n X

j ¼ 1; . . . ; n ; )

kj ¼ 1;

kj P 0;

j ¼ 1; . . . ; n :

j¼1

Following some models are respectively presented like CCR (see Charnes et al. [1]) assess DMUs on Tc, some others, like BCC (see Banker et al. [11]) and ADD (see Charnes et al. [12]) on Tv in envelopment form:

Min h s:t:

n X

kj xij þ si ¼ hxip ;

i ¼ 1; . . . ; m;

kj yrj  sþr ¼ yrp ;

r ¼ 1; . . . ; s;

j¼1 n X j¼1

kj P 0;

j ¼ 1; . . . ; n;

si P 0;

i ¼ 1; . . . ; m;

sþr P 0;

r ¼ 1; . . . ; s:

ð1Þ

Min h s:t:

n X

kj xij þ si ¼ hxip ;

i ¼ 1; . . . ; m;

kj yrj  sþr ¼ yrp ;

r ¼ 1; . . . ; s;

j¼1 n X j¼1 n X

ð2Þ kj ¼ 1;

j¼1

kj P 0;

j ¼ 1; . . . ; n;

si P 0;

i ¼ 1; . . . ; m;

sþr P 0;

r ¼ 1; . . . ; s:

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F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

Max

m X

si þ

n X j¼1 n X j¼1 n X

sþr

r¼1

i¼1

s:t:

s X

kj xij þ si ¼ xip ;

i ¼ 1; . . . ; m;

kj yrj  sþr ¼ yrp ;

r ¼ 1; . . . ; s;

ð3Þ

kj ¼ 1;

j¼1

kj P 0; j ¼ 1; . . . ; n; si P 0; i ¼ 1; . . . ; m; sþr P 0; r ¼ 1; . . . ; s: DMUp is efficient-CCR and efficient-BCC if and only if in it’s evaluating by two models (1) and (2), respectively h ¼ 1; s ¼ 0ði ¼ 1; . . . ; mÞ and sþ ¼ 0ðr ¼ 1; . . . ; sÞ in optimal solution. Also it is efficient-ADD if and only if the value of i i objective function gains zero, in optimality. It is convenient that, we show DMUp is efficient-BCC if and only if it be efficient-ADD (for details see Cooper et al. [13]). 2.2. Review of ranking models In this subsection we are going to review AP and MAJ ranking models in data envelopment analysis. 2.2.1. AP model Anderson and Peterson [2], proposed the supper efficiency model. They omitted the efficient DMU from the PPS, Tc and ran CCR model for other units to rank them. Their proposed model is:

Min h s:t:

n X

kj xij þ si ¼ hxip ;

i ¼ 1; . . . ; m;

j¼1 j–0 n X

kj yrj  sþr ¼ yrp ;

r ¼ 1; . . . ; s ðAP modelÞ;

j¼1 j–p

kj P 0;

j ¼ 1; . . . ; n;

si

P 0;

i ¼ 1; . . . ; m;

sþr

P 0;

j – 0;

r ¼ 1; . . . ; s; ⁄

For efficient units h P 1 and for inefficient units 0 < h⁄ < 1. This model has two drawbacks: (1) AP model may be inefficient for special data in input oriented case. (2) This model is unstable for some DMUs which one of their data components is near to zero. 2.2.2. MAJ model To solve the important drawbacks of AP models, Mehrabian et al. [8], proposed another model for ranking efficient units. Their proposed model is:

Min 1 þ w s:t:

n X

kj xij 6 xip þ w;

i ¼ 1; . . . ; m ðMAJ modelÞ;

j¼1 j–p n X

kj yrj P yrp ;

r ¼ 1; . . . ; s

j¼1 j–0

kj P 0;

j ¼ 1; . . . ; n;

j–0

The necessary and sufficient conditions for feasibility of MAJ model is that in evaluating of DMUp, or yrp = 0, r = 1,. . .,s or there exists DMUj, j – p such that yrj – 0 [8].

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F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

3. An efficiency index base on infinity (Tchebycheff) norm Tavares and Antunes [14] were suggested a model for evaluate the efficiency, by Using L1-Norm. They have been used Tchebycheff norm in the objective function of ADD model. In other words, they have used following model for obtaining efficiency DMUs.

 !   n n X X   Min ðX p ; Y p Þ  kj xj ; kj yj    j¼1 j¼1 s:t:

n X

kj xij 6 xip ;

i ¼ 1; . . . ; m;

kj yrj P yrp ;

r ¼ 1; . . . ; s;

1

j¼1 n X

ð4Þ

j¼1 n X

kj ¼ 1;

j¼1

kj P 0;

j ¼ 1; . . . ; n

The objective function of model (4) minimizes the distance between DMUp(under evaluating unit) and its projected point on frontier efficiency. Also, they showed the model (4) can be expressed as follows:

Max U p s:t: U p þ

n X

kj xij 6 xip ;

i ¼ 1; . . . ; m;

j¼1

Up 

n X

kj yrj 6 yrp ;

r ¼ 1; . . . ; s;

ð5Þ

j¼1 n X

kj ¼ 1;

j¼1

kj P 0;

j ¼ 1; . . . ; n;

U p P 0; 1 where Up is efficiency value. DMUp is efficient if and only if Up = 0,[14]. With introducing efficiency index as hp ¼ 1þU , where p Up 2 [0, +1), DMUp is efficient in Tavares’s model (model 5) if and only if in optimal solution Up = 0 or hp = 1.

4. Proposed method   For ranking extreme efficient DMUP, first it will be removed from PPS (Tc) and then new PPS T 0c is defined as follow, this is shown in Fig. 1.

( T 0c

¼

ðX; YÞ : X P

X

kj X j ; Y 6

j–p

X

) kj Y j ; kj P 0;

j ¼ 1; . . . ; n

j–p

y

P

P

Tc′

x

O Fig. 1. Illustrate of T 0c .

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F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

It is specifying that if DMUpwith coordinates (Xp, Yp) be inside production possibility set (PPS), then we have:

9k P 0s:t:

n X

n X

kj X j 6 X p and

j¼1

kj Y j P Y p :

j¼1

But if the DMUp(under evaluation unit) lie outside PPS, then we have: n X

kj X j P X p and

j¼1

n X

! kj Y j 6 Y p ; kj  0; j ¼ 1; . . . ; n :

j¼1

In other words:

9i;

n X

kj xij > xip

or 9r;

j¼1

n X

kj yrj < yrp

j¼1

Now, suppose DMUp is outside of PPS. Hence, we want to obtain a point of PPS which dominated by DMUp. Thus, we have: P P 9k P 0s:t: nj¼1 kj X j P X p and nj¼1 kj Y j 6 Y p . So, the following model is suggested for ranking DMUp.

Min kP  Pk1 s:t:

n X

kj xij P xip ;

i ¼ 1; . . . ; m;

j¼1;j – p

ð6Þ

n X

kj yrj 6 yrp ;

r ¼ 1; . . . ; s;

j¼1;j – p

kj P 0 j ¼ 1; . . . ; n;

P  Pn n is a point in T 0c . Note that model (6) is where P = (Xp, Yp) is the point under evaluation and P ¼ j¼1;j – p kj xij ; j¼1;j – p kj yrj written based on domination between P and P. The objective function the model (6) is nonlinear. Following after manipulation model (6) is converted in a linear model, which can be easily solved. Let wðP; PÞ ¼ MaxjP  Pj. By definition of L1 norm, the objective of (6) reduces to MinwðP; PÞ ¼ Min MaxjP  Pj. We have

8( 91 ) ) (    <  = n n X X     @ A Min MaxjP  Pj ¼ Min Max kj xij  ; yrp  kj yrj  xip     :  ; j¼1;j – p j¼1;j – p i¼1;...;m r¼1;...;s 8 91 0 ! ! n n < X = X A ¼ Min@Max kj xij  xip ; yrp  kj yrj : j¼1;j – p ; j¼1;j – p 0

i¼1;...;m

r¼1;...;s

Let

8 < V P ¼ Max :

!

n X

kj xij  xip

j¼1;j – p

; yrp  i¼1;...;m

n X j¼1;j – p

9 =

! kj yrj r¼1;...;s

;

So,

Min V p n X

s:t: V p P

kj xij  xip ;

i ¼ 1; . . . ; m;

j¼1;j – p

V p P yrp 

n X

ð7Þ kj yrj ;

r ¼ 1; . . . ; s;

j¼1;j – p

kj P 0;

j ¼ 1; . . . ; n:

Note that the model (7) is independent of oriented (input-oriented and output-oriented), therefore, it is a superiority over other existence methods. Now, consider the set T 00c as follows (Fig. 2):

T 00c ¼ T 0c \ fðX; YÞjX P X p ; Y 6 Y p g; We claim that T 00c can be used instead of T 0c . In Theorem 1, we show that points in T 0c  T 00c are useless and may be abandoned. T 00c can be rewritten as follows:

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F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

Y

P

T c ′′

X

O Fig. 2. Illustrate of T 00c .

T 00c ¼

8 > > > > < > > > > :

n P

kj X j ;

j¼1;j – p n P

!

n P

kj Y j

:

j¼1;j – p

kj Y j 6 Y p ;

n P j¼1;j – p

kj P 0;

9 > > kj X j P X p ; > > =

j ¼ 1; . . . ; n;

j¼1;j – p

> > > > ;

Therefore, by using this T 00c , the above problem would be solved. b 2 T 0  T 00 then there exist P 2 T 00 such that wðP; PÞ 6 wðP; PÞ. b Theorem 1. If P c c c (

( Xb < X P Xb P X P P P or ðiiÞ: . Yb 6 Yb Yb > Yb P P P P ( n o X P ¼ X P > Xb  P and wðP; PÞ ¼ Max jX P  X P j; jY P  Y P j ¼ Max 0; jY P  Y b j 6 In the former case, define A as Y P ¼ Yb P P b MaxfjX P  Xb j; jY P  Y b jg ¼ wðP; PÞ.

P P  X P ¼ Xb b P Obviously wðP; PÞ. And in the other case define P as . Then:wðP; PÞ ¼ Max jX P  X P j;  PÞ 6 wðP; Y ¼ Y < Y P P P b jY P  Y P jg ¼ Max jX P  X P j; 0 6 MaxfjX P  Xb j; jY A  Y b jg ¼ wðP; PÞ. P A b h Again wðP; PÞ 6 wðP; PÞ. b 2 T 0  T 00 ;then ðiÞ: Proof. Let P c c

Theorem 2. In any optimal solution the model (7) , at least one of inputs (outputs) constraints is active. Proof. Consider dual the model (7) as follows:

Max s:t:

s X r¼1 s X r¼1 s X r¼1

m X

ur yrp 

v i xip

i¼1

ur þ

m X

v i ¼ 1;

ð8Þ

i¼1

ur yrj 

m X

v i xij 6 0;

j ¼ 1; . . . ; n;

j – p;

i¼1

ur P 0; v i P 0;

r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:

It is obvious that zero (zero vector in Rm+s) is not a solution for model (8). Therefore, suppose say ut > 0 (1 6 t 6 s), then using duality strong theorem, like constraint ofut in primal model is binding in optimality. The proof is complete. h Theorem 3. The projected point of DMUp in model (7) lies on the efficient frontier.   Proof. Suppose that k ; V p be optimal solution the obtained of model (7), then by Theorem 2 at Least one of m + s constraints is binding in optimality. If the binding constraint belong to inputs (outputs) constraints set, then we show the value optimal obtained of evaluating projected point by CCR model in input-oriented (output-oriented) is 1. Without lost problem generality, let us one of input constraints is binding in optimality, say, kth constraint. Therefore, we will have:

52

F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56 n X

V p ¼

kj xkj  xkp ;

j¼1;j – p

V p

n X

P

kj xij  xip ;

i ¼ 1; . . . ; m;

i–k

ð9Þ

j¼1;j – p n X

V p P yrp 

kj yrj ;

r ¼ 1; . . . ; s:

j¼1;j – p

Now, consider input oriented CCR model when projected point is evaluated as follows:

Min hp n X s:t: kj xij 6 hp ðxip þ V p Þ;

i ¼ 1; . . . ; m;

j¼1;j – p n X

ð10Þ kj yrj P yrp  V p ;

r ¼ 1; . . . ; s;

j¼1;j – p

kj P 0;

j ¼ 1; . . . ; n:

Suppose that ð k;  hP Þ is optimal solution of model (10), we will demonstrate that  hp ¼ 1 is optimal solution it. With contradict,  hp < 1, then n X

kj xij < xip þ V  ; p

i ¼ 1; . . . ; m;

kj y P y  V  ; rj rp p

r ¼ 1; . . . ; s:

j¼1;j – p n X

ð11Þ

j¼1;j – p

  Therefore, model (11) gives  k; V P is a feasible solution for problem (7), such that objective function value of model (7) is     equal V P . Therefore,  k; V P is also an optimal solution for model (7). It is contradict, because any constraint is not binding in inputs constraints. Consequently,  hp ¼ 1and projected point lie on the efficient frontier. h Theorem 4. Model (7) is always feasible and bounded. Proof. Constraints of model (7) can be written as

V p  ti ¼

n X

kj xij  xip

i ¼ 1; . . . ; m;

j¼1;j – p n X

V p  tþr ¼ yrp 

kj yij

r ¼ 1; . . . ; s;

j¼1;j – p

The slacks and Vp are always of opposite sign in the constraints. Hence, we can always obtain a solution to this problem in the following manner. Because all m + s constraints on the right hand of model (7) are positive, then set Vp equal to the maximum of these values and by using the slacks we attain the equalities required. In addition, obviously Vp is bounded below by zero. The proof is complete. The like model (7), we introduce a model as follow according to BCC model:

Min V p þ M

m X

di

i¼1 n X

s:t: V p þ di P

kj xij  xip ;

i ¼ 1; . . . ; m;

j¼1;j – p

V p P yrp 

n X

kj yrj ;

j¼1;j – p

V p 6 ytp ; n X kj ¼ 1; j¼1;j – p

kj P 0;

j ¼ 1; . . . ; n:

r ¼ 1; . . . ; s;

ð12Þ

53

F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

where M is a very large positive number and ytp = max{yrp}r=1,. . .,s. Note, there are some differences in BCC model for obtaining efficiency score (see Fig. 3.b) [15]. As, can be seen in Fig. 3.a DMUp is eliminated form PPS, Then side of foursquare is Vp. But in Fig. 3.b, the side of foursquare is not the real Vp value. We must include slacks di to the first group of constraints (Fig. 3.c). Since they only must use in especial cases, they appear with big M coefficients in objective. h

5. Example Consider 19 DMUs with two inputs and two outputs (Table 1). DMUs of 1, 2, 5, 9, 15 and 19 are CCR efficient. Results of ranking using Tchebycheff norm method are compared with AP and MAJ methods in Table 2. As shown in Table 2, DMU19 and DMU9 have highest and lowest rank in MAJ and Tchebycheff models, respectively. Meanwhile, both of them (DMU9 and DMU19) are infeasible in AP model.

y

O

x (a) The rank a DMU in Tv

y

x

O (b) Some difficulties in Tv

y

O

x (c) Particular case of the rank a DMU in Tv Fig. 3. The rank of a DMU in Tv in three cases.

54

F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56 Table 1 The value of inputs and outputs. DMUs

Input I

Input II

Output I

Output II

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

81 85 56.7 91 216 58 112.2 293.2 186.6 143.4 108.7 105.7 235 146.3 57 118.7 58 146 0

87.6 12.8 55.2 78.8 72 25.6 8.8 52 0 105.2 127 134.4 236.8 124 203 48.2 47.4 50.8 91.3

5191 3629 3302 3379 5368 1674 2350 6315 2865 7689 2165 3963 6643 4611 4869 3313 1853 4578 0

205 0 0 8 639 0 0 414 0 66 266 315 236 128 540 16 230 217 508

Table 2 The results of using different models for ranking of DMUs. DMUs AP

Value: Rank: Value: Rank: Value: Rank:

MAJ Tch. norm

1

2

5

9

15

19

115 4 105 5 11.94 5

174 1 109 3 22.72 2

130 3 110 2 22.70 3

Inf. – 104 6 10.07 6

133 2 106 4 18.68 4

Inf. – 128 1 60.90 1

Also, notice that, all DMUs are ranked very close to each other in MAJ and Tchebycheff models, while this is not in AP model. In model AP, DMU2 and DMU1 have first and last rank respectively. 6. Conclusion In this study a approach by using L1-Norm is suggested for ranking efficient DMUs. This model has the superiority relative to other methods. In this sense, it can be use in ranking both nonnegative data and value zero for some inputs of outputs. Therefore, it removes some infeasibility difficulty of the other models. The introduced model in this paper uses Tc, as it can be manipulated for Tv without any difficulty. It must be noted that some difficulties may arise using BCC model as it has shown in Fig. 3.b. This difficulty also will be removed by model (12), so that the values of slacks variables play important role in selection of ranking DMUs (Fig. 3c). Here we give a briefly of the processes model (4) to model (5). In general, DEA models assume that every input of a DMUp projected point should be verified in below inequalities: n X

kj xij 6 xip ;

i ¼ 1; . . . ; m;

kj yrj P yrp ;

r ¼ 1; . . . ; s;

j¼1 n X j¼1

By assumptions s i ¼ xip  jected point is given by

Pn

j¼1 kj xj ; i

¼ 1; . . . ; m; sþ r ¼

Pn

j¼1 kj xj

 yrp ; r ¼ 1; . . . ; s, the L1 distance between DMUp and its pro-

0   !     n n n X X X     @ kj xj ; kj yj  ¼ Max xip  kj xij  ðX p ; Y p Þ      j¼1 j¼1 j¼1 1

i¼1;...;m

    n X   ; yrp  kj yrj    j¼1

1

  A ¼ Max s ; . . . ; s ; sþ ; . . . ; sþ : 1 m 1 s r¼1;...;s

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F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

Assuming that each factor input of projection to be minimized and each factor input of projection to be maximized, then the following model is a Multiple Objective Linear Programming (MOLP) problem that reflects the efficiency of DMUp:

Max 

n X

n X

kj x1j ; . . . ; 

j¼1

s:t:

si

kj xmj ;

j¼1

¼ xip 

n X

n X

kj y1j ; . . . ;

j¼1

kj xj ;

i ¼ 1; . . . ; m;

kj xj  yrp ;

r ¼ 1; . . . ; s;

n X

!

kj ysj

j¼1

j¼1

sþr ¼

n X

ð13Þ

j¼1 n X

kj ¼ 1;

j¼1

si P 0;

i ¼ 1; . . . ; m;

sþr P 0;

r ¼ 1; . . . ; s;

kj P 0;

j ¼ 1; . . . ; n:

One of the processes generally used in MOLP to compute efficient solutions consists in minimizing a distance to a point in the objective function space that would optimize all the objective functions simultaneously and which is not feasible whenever the function are conflicting. The problem to be solved by using the L1metric is

(

Min Max

x1p  

n X

!

kj x1j ; . . . ; xmp  

j¼1 n X

si ¼ xip 

s:t:

n X

!

kj xmj ; y1p 

j¼1

kj xij ;

i ¼ 1; . . . ; m;

kj yrj  yrp ;

r ¼ 1; . . . ; s;

n X j¼1

kj y1j ; . . . ; ysp 

n X

)

kj ysj

j¼1

j¼1

sþr ¼

n X j¼1

n X

ð14Þ

kj ¼ 1;

j¼1

si P 0;

i ¼ 1; . . . ; m;

sþr P 0; r ¼ 1; . . . ; s; kj P 0; j ¼ 1; . . . ; n: The Min–Max objective function in (14) can be formulated as:

Min V s:t: V P si ;

i ¼ 1; . . . ; m

V P sþr ; r ¼ 1; . . . ; s; s n X kj ¼ 1; j¼1

si sþr

P 0;

i ¼ 1; . . . ; m;

P 0;

r ¼ 1; . . . ; s;

ð15Þ

kj P 0; j ¼ 1; . . . ; n: Vfree: By making V = Up the following formulation for (15) is obtained:

Max U p s:t: U p 6 si ;

i ¼ 1; . . . ; m;

U p 6 sþr ; r ¼ 1; . . . ; s; n X kj ¼ 1; j¼1

si P 0;

i ¼ 1; . . . ; m;

sþr P 0;

r ¼ 1; . . . ; s;

kj P 0;

j ¼ 1; . . . ; n;

U p free:

ð16Þ

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F. Rezai Balf et al. / Applied Mathematical Modelling 36 (2012) 46–56

The solution to (16) is always nonnegative. Therefore Up P 0. In finally model (16) is converted to model (17) as follows:

Max U p s:t: U p þ

n X

kj xij 6 xip ;

i ¼ 1; . . . ; m;

j¼1

Up 

n X

kj yrj 6 yrp ;

j¼1 n X

r ¼ 1; . . . ; s;

ð17Þ

kj ¼ 1;

j¼1

kj P 0;

j ¼ 1; . . . ; n;

U p P 0; References [1] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, Eur. J. Oper. Res. 2 (6) (1978) 429–444. [2] P. Andersen, N.C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Manage. Sci. 39 (1993) 1261–1264. [3] T. Sueyoshi, Data envelopment analysis non-parametric ranking test and index measurement: Slack-adjusted DEA and an application to Japanese agriculture cooperative, Omega Int. Manage. Sci. 27 (1999) 315–326. [4] R.M. Thrall, Duality, classification and slack in data envelopment analysis, The Ann. Oper. Res. 66 (1996) 109–138. [5] J. Zhu, Robustness of the efficient decision-making units in data envelopment analysis, Eur. J. Oper. Res. 90 (1996) 451–460. [6] J.H. Dula, B.L. Hickman, Effects of excluding the column being scored from the DEA envelopment LP technology matrix, J. Oper. Res. Soc. 48 (1997) 1001–1012. [7] L.M. Seiford, J. Zhu, Infeasibility of super-efficiency data envelopment analysis models, INFOR 37 (2) (1999) 174–187. [8] S. Mehrabian, M.R. Alirezaee, G.R. Jahanshahloo, A compelete efficiency ranking of decision making units in data envelopment analysis, Comput. Optimiz. Appl. 14 (1999) 261–266. [9] A. Hashimato, A ranked voting system using a DEA/AR exclusion model:A note, Eur. J. Oper. Res. 97 (1999) 600–604. [10] A.M. Torgersen, F.R. Forsund, S.A.C. Kittelsen, Slack-adjusted efficiency measures and ranking of efficient units, The J. Prod. Anal. 7 (1996) 379–398. [11] R.D. Banker, A. Charnes, W.W. Cooper, Some methods for estimating technical and scale inefficiencies in data envelopment analysis, Manage. Sci. 30 (9) (1984) 1078–1092. [12] A. Charnes, W.W. Cooper, B. Golany, L.M. Seiford, J. Stutz, Foundations of data envelopment analysis for Pareto–Koopmans efficient empirical production functions, J. Econ. 30 (1985) 91–107. [13] W.W. Cooper, S. Li, L.M. Seiford, K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, Reverences and DEA-solver Software, Kluwer Academic Publisher, Norwell, Mass, 1999. [14] G. Tavares, C.H. Antunes, A Tchebycheff DEA Model. Rutcor Research Report (2001). [15] S. Haag, P. Jaska, J. Semple, Sensitivity of efficiency classifications in the additive model of data envelopment analysis, J. Syst. Sci. 23 (1992) 789–798.

Further reading [16] A. Charnes, W.W. Cooper, S. Li, Using DEA to evaluate relative efficiencies in the economic performance of Chinese-key cities, Socio Econ. Plan. Sci. 23 (1989) 325–344. [17] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, N. Shoja, G. Tohidi, S. Razavian, Ranking by using L1 –norm in data envelopment analysis, Appl. Math. Comput. 153 (1) (2004) 215–224.